Analysis of the N-policy GI/M/1/K Queueing Systems with Working Breakdowns and Repairs

Analysis of the N-policy GI/M/1/K Queueing Systems with Working Breakdowns and Repairs Abstract In this paper, we present an algorithmic approach to the analysis of the finite-capacity GI/M/1 queue with working breakdowns under N-policy. When there are no customers in the system, the server is turned off. If the number of customers in the system reaches threshold N, then the server is turned on and working. The recently introduced working breakdown involves serving newly arrived customers at a lower service rate in cases where the server breaks down. Service times during busy and breakdown periods are exponentially distributed. When a breakdown occurs, the failed server is not repaired until there are no customers remaining in the system. In this type of queueing system, we compute the steady-state probabilities at arbitrary and pre-arrival epochs using the supplementary variable method. We propose an algorithm for computing the steady-state probabilities at pre-arrival epochs and develop system performance measures. Finally, numerical analysis is used to evaluate the effects of various system parameters on system performance measures. 1. INTRODUCTION The widely discussed N-policy, introduced by Yadin and Naor [1], is a queueing system with control policies where the server is turned off when the system is emptied, and the server is turned on if the queue length reaches or exceeds threshold N. Wang and Ke [2] applied the supplementary variable technique and recursive method to investigate the N-policy M/G/1 queues with finite capacity and infinite capacity. Using the same technique, Ke and Wang [3] addressed the N-policy G/M/1 queue with finite capacity. Jain et al. [4] elucidated the reliability characteristics of a machine repair problem with reneging under N-policy. A review on the optimal control of queues under N-policy can be found in Tadj and Choudhury [5]. Wang et al. [6] discussed an N-policy queue in a fuzzy environment using fuzzy set theory, where the arrival and service rates are fuzzy numbers. Vijaya Laxmi and Suchitra [7] considered the N-policy GI/M(n)/1 queue with state dependent service rates, working vacations and Bernoulli schedule vacation interruptions, where the server continues working a slower rate during vacation periods. Haridass and Arumuganathan [8] considered the Mx/G/1 retrial queueing system with modified vacations under N-policy, where the server takes no more than M vacations when the number of customers in orbit falls below threshold N. Wang et al. [9] recently analyzed the equilibrium strategic behavior of customers and social optimization in an M/M/1 retrial queue with constant retrial rate under N-policy. All of the above papers assumed that the server is reliable; however, in many real-life situations, servers are subject to unpredictable breakdowns. Servers cease providing service during breakdown periods; i.e. service is interrupted. The N-policy queues with server breakdowns have been discussed by several researchers, including Wang and Ke [10], Pearn et al. [11], Choudhury et al. [12], Choudhury and Tadj [13] and Singh et al. [14]. Details pertaining to queueing models under N-policy can also be found in the survey by Jayachitra and Albert [15]. Yang and Wu [16] applied the matrix–geometric method to study the N-policy M/M/1 queue with an unreliable server and working vacations. Based on the principle of maximum entropy, Jain et al. [17] approximated stationary probability distributions of the system size for the N-policy Mx/G/1 queue with server breakdowns and delayed phase repair under a Bernoulli vacation schedule. Most recently, Wang et al. [18] analyzed the discrete-time Geo/G/1 queue with server breakdowns under random N-policy, where threshold N is a random number that varies in different cycles. In a pioneering study, Kalidass and Kasturi [19] analyzed the M/M/1 queue with working breakdowns, in which the server lowers the service rate in the event of a breakdown during a busy period. Kim and Lee [20] derived the system size and sojourn time distributions for the M/G/1 queue with disasters and working breakdowns. Liu and Song [21] conducted analysis on a batch arrival M/M/1 queue with working breakdowns. Liou [22] applied the matrix–geometric method to compute the steady-state probabilities for an M/M/1 queue with working breakdowns, in which customers may balk and renege. Yen et al. [23] used the Laplace transform technique to study the reliability-based measures of a warm standby repairable system with working breakdowns. In recent years, numerous researches have analyzed working breakdown queues, the details of which may be found in Chen et al. [24, 25], Jiang and Liu [26] and Li and Zhang [27]. It is important to investigate queueing systems with a finite buffer due to the fact that in many situations, the waiting space is limited. Ke [28] considered a GI/M/1 queue with finite capacity and N-policy, in which the server takes multiple vacations when the system empties. The operating characteristics of the N-policy GI/M/1 queue with finite capacity and startup times were investigated by Ke [29]. The proposed queueing model of this paper is different from Ke [28, 29]. The differences are that (1) our paper does not address server vacations and startup times; and (2) Ke [28, 29] were not attentive to server breakdowns. A number of studies have examined finite-capacity GI/M/1 queues [30–36]; however, little research has been dedicated to finite-capacity GI/M/1 queueing models with server breakdowns. To the best of our knowledge, this is the first study to apply steady-state analysis to a finite-capacity GI/M/1 queue with working breakdowns under N-policy. We adopted the supplementary variable technique, which is a simple and straightforward approach to compute the stationary probabilities at arbitrary epochs [37, 38]. The remainder of the paper is organized as follows. In the next section, we present a description of the model and define the notation. In Section 3, we formulate the mathematical model of the system using the supplementary variable technique. Section 4 outlines an algorithm for the computation of steady-state probabilities at pre-arrival epochs. In Section 5, we deduce the steady-state probabilities at arbitrary epochs. In Section 6, we develop various system performance measures and perform sensitivity analysis. Conclusions are drawn in Section 7. 2. MODEL AND NOTATION In this study, we consider an N-policy GI/M/1/K queue with working breakdowns. The system comprises a finite waiting space K and an unreliable server. The inter-arrival times of customers are independent and identically distributed (i.i.d.) random variables with a general distribution function A(x), probability density function a(x), mean inter-arrival time a1=1/λ, Laplace–Stieltjes transform (LST) A*(θ)=∫0∞e−θxdA(x) and A*(j)(θ) is the jth-order (j≥1) derivative of A*(θ) with respect to θ. Arriving customers are served on a first-come first-served (FCFS) basis, and the server can serve only one customer at a time. The server may break down during busy periods, and breakdown occurs according to a Poisson process with parameter α. When a breakdown occurs, the server continues working at a reduced service rate. The service times during the normal busy and breakdown periods follow exponential distributions with mean rates of μb and μd(<μb), respectively. During a breakdown period, the failed server is repaired only after the system becomes empty. The repair times of the failed server are exponentially distributed with mean rate β. Moreover, it is assumed that various stochastic processes involved in the queueing system are mutually independent. 2.1. Practical justification of the model To illustrate the potential application of the proposed queueing model, we consider a facility in which a single product is fabricated using a single production machine. Raw materials arriving at the facility are processed by the production machine. When the quantity of raw materials achieves the upper bound of the storage capacity, newly arrived raw materials are not allowed to enter the facility. For the purpose of saving energy, the production machine is turned off when there are no raw materials in the facility. If there are N or more raw materials in the system, then the production machine is turned on. During the processing of raw materials, the production machine may break down at any time; however, it may still be able to operate, albeit at a lower processing rate. To avoid production interruptions, the failed machine is repaired until there are no raw materials to process. This production system could be appropriately modeled using the proposed queueing system. 3. ANALYSIS OF THE MODEL We first establish the differential equations in a steady state by treating the remaining inter-arrival time as a supplementary variable. Let N(t) be the number of customers in the system and X(t) be remaining inter-arrival time for the next arrival at time t. The server state at time t is described as Y(t)={0,iftheserveristurnedoff.1,iftheserveristurnedonandinnormalworkingstate,2,iftheserverisinbreakdownstate. We define P0,n(x,t)dx=Pr{N(t)=n,Y(t)=0,x<X(t)≤x+dx},n=0,1,…,N−1,x≥0 P1,n(x,t)dx=Pr{N(t)=n,Y(t)=1,x<X(t)≤x+dx},n=1,2,…,K,x≥0 P2,n(x,t)dx=Pr{N(t)=n,Y(t)=2,x<X(t)≤x+dx},n=0,1,…,K,x≥0 P0,n(t)=∫0∞P0,n(x,t)dx,n=0,1,…,N−1, P1,n(t)=∫0∞P1,n(x,t)dx,n=1,2,…,K, P2,n(t)=∫0∞P2,n(x,t)dx,n=0,1,…,K, In steady-state, the steady-state probabilities can be written as follows: P0,n=limt→∞P0,n(t),n=0,1,…,N−1, P1,n=limt→∞P1,n(t),n=1,2,…,K, P2,n=limt→∞P2,n(t),n=0,1,…,K. Using the method of supplementary variable, the steady-state differential-difference equations governing the system can be written as follows: −ddxP0,0(x)=μbP1,1(x)+βP2,0(x), (1) −ddxP0,n(x)=a(x)P0,n−1(0),1≤n≤N−1, (2) −ddxP1,1(x)=−(μb+α)P1,1(x)+μbP1,2(x), (3) −ddxP1,n(x)=−(μb+α)P1,n(x)+μbP1,n+1(x)+a(x)P1,n−1(0),2≤n≤N−1, (4) −ddxP1,N(x)=−(μb+α)P1,N(x)+a(x)P0,N−1(0)+a(x)P1,N−1(0)+μbP1,N+1(x), (5) −ddxP1,n(x)=−(μb+α)P1,n(x)+a(x)P1,n−1(0)+μbP1,n+1(x),N+1≤n≤K−1, (6) −ddxP1,K(x)=−(μb+α)P1,K(x)+a(x)(P1,K−1(0)+P1,K(0)), (7) −ddxP2,0(x)=−βP2,0(x)+μdP2,1(x), (8) −ddxP2,n(x)=−μdP2,n(x)+μdP2,n+1(x)+αP1,n(x)+a(x)P2,n−1(0),1≤n≤K−1, (9) −ddxP2,K(x)=−μdP2,K(x)+αP1,K(x)+a(x)(P2,K−1(0)+P2,K(0)). (10) Let us introduce the following LSTs of a(x), P0,n(x), P1,n(x) and P2,n(x): A*(θ)=∫0∞e−θxa(x)dx,P0,n*(θ)=∫0∞e−θxP0,n(x)dx,P1,n*(θ)=∫0∞e−θxP1,n(x)dx,P2,n*(θ)=∫0∞e−θxP2,n(x)dx,θ≥0, so that P0,n=P0,n*(0)=∫0∞P0,n(x)dx,n=0,1,…,N−1. P1,n=P1,n*(0)=∫0∞P1,n(x)dx,n=1,2,…,K. P2,n=P2,n*(0)=∫0∞P2,n(x)dx,n=0,1,…,K. Multiplying Equations (1–10) by e−θx on both sides and integrating over x, we obtain −θP0,0*(θ)=μbP1,1*(θ)+βP2,0*(θ)−P0,0(0), (11) −θP0,n*(θ)=A*(θ)P0,n−1(0)−P0,n(0),1≤n≤N−1, (12) (μb+α−θ)P1,1*(θ)=μbP1,2*(θ)−P1,1(0), (13) (μb+α−θ)P1,n*(θ)=A*(θ)P1,n−1(0)+μbP1,n+1*(θ)−P1,n(0),2≤n≤N−1, (14) (μb+α−θ)P1,N*(θ)=A*(θ)P0,N−1(0)+A*(θ)P1,N−1(0)+μbP1,N+1*(θ)−P1,N(0), (15) (μb+α−θ)P1,n*(θ)=A*(θ)P1,n−1(0)+μbP1,n+1*(θ)−P1,n(0),N+1≤n≤K−1, (16) (μb+α−θ)P1,K*(θ)=A*(θ)(P1,K−1(0)+P1,K(0))−P1,K(0), (17) (β−θ)P2,0*(θ)=μdP2,1*(θ)−P2,0(0), (18) (μd−θ)P2,n*(θ)=μdP2,n+1*(θ)+αP1,n*(θ)+A*(θ)P2,n−1(0)−P2,n(0),1≤n≤K−1, (19) (μd−θ)P2,K*(θ)=αP1,K*(θ)+A*(θ)(P2,K−1(0)+P2,K(0))−P2,K(0). (20) Substituting θ=0 into Equation (12) gives P0,0(0)=P0,1(0)=⋯=P0,N−1(0). (21) Adding Equations (11–20), it yields ∑n=0N−1P0,n*(θ)+∑n=1KP1,n*(θ)+∑n=0KP2,n*(θ)=1−A*(θ)θ(∑n=0N−1P0,n(0)+∑n=1KP1,n(0)+∑n=0KP2,n(0)). (22) On taking limit as θ→0 in Equation (22), we obtain ∑n=0N−1P0,n*(0)+∑n=1KP1,n*(0)+∑n=0KP2,n*(0)=1λ(∑n=0N−1P0,n(0)+∑n=1KP1,n(0)+∑n=0KP2,n(0)). (23) Here, the normalization condition implies ∑n=0N−1P0,n*(0)+∑n=1KP1,n*(0)+∑n=0KP2,n*(0)=1. (24) Using Equation (24), Equation (23) can be written as ∑n=0N−1P0,n(0)+∑n=1KP1,n(0)+∑n=0KP2,n(0)=λ. (25) Remark 1 Equation (25) shows that the expected number of entries into the system per unit time is equal to the mean arrival rate λ. Substituting θ=μb+α into Equation (17), we have P1,K−1(0)=(1−A*(μb+α))P1,K(0)A*(μb+α). (26) Similarly, substituting θ=μb+α into Equations (14–16), we obtain P1,n−1(0)=P1,n(0)−δn,NA*(μb+α)P0,N−1(0)−μbP1,n+1*(μb+α)A*(μb+α),2≤n≤K−1, (27) where δn,N denotes the Kronecker delta function. From Equation (17), P1,K*(θ) can be written as follows: P1,K*(θ)={−A*(1)(θ)(P1,K−1(0)+P1,K(0)),ifθ=μb+α,A*(θ)(P1,K−1(0)+P1,K(0))−P1,K(0)(μb+α−θ),ifθ≠μb+α. (28) For 2≤n≤K−1, from Equations (14–16), we have P1,n*(θ)={−(A*(1)(θ)P1,n−1(0)+δn,NA*(1)(θ)P0,N−1(0)+μbP1,n+1*(1)(θ)),ifθ=μb+α,A*(θ)P1,n−1(0)+δn,NA*(θ)P0,N−1(0)+μbP1,n+1*(θ)−P1,n(0)(μb+α−θ),ifθ≠μb+α. (29) Substituting θ=μd into Equations (19) and (20) yields P2,n−1(0)=P2,n(0)−μdP2,n+1*(μd)−αP1,n*(μd)A*(μd),1≤n≤K−1, (30) P2,K−1(0)=(1−A*(μd))P2,K(0)−αP1,K*(μd)A*(μd). (31) Using Equation (20), it is easy to show P2,K*(θ)={P2,K*(θ)=−(αP1,K*(1)(θ)+A*(1)(θ)(P2,K−1(0)+P2,K(0))),ifθ=μd,αP1,K*(θ)+A*(θ)(P2,K−1(0)+P2,K(0))−P2,K(0)(μd−θ),ifθ≠μd. (32) From Equation (19), for 1≤n≤K−1, we obtain P2,n*(θ)={−(μdP2,n+1*(1)(θ)+αP1,n*(1)(θ)+A*(1)(θ)P2,n−1(0)),ifθ=μd,μdP2,n+1*(θ)+αP1,n*(θ)+A*(θ)P2,n−1(0)−P2,n(0)(μd−θ),ifθ≠μd. (33) Our objective is to determine the steady-state probabilities P0,n*(0), P1,n*(0) and P2,n*(0), which are computed with help of P1,n(0) and P2,n(0). Finding P1,n(0) and P2,n(0) requires that we derive the jth-order derivative of P1,n*(θ) and P2,n*(θ) with respect to θ. By differentiating Equations (14–17) j times with respect to θ and then setting θ=μb+α, we obtain P1,n*(j)(μb+α)=−A*(j+1)(μb+α)P1,n−1(0)+δn,NA*(j+1)(μb+α)P0,N−1(0)+μbP1,n+1*(j+1)(μb+α)j+1,j≥1,2≤n≤K−1, (34) P1,K*(j)(μb+α)=−A*(j+1)(μb+α)(P1,K−1(0)+P1,K(0))j+1,j≥1, (35) where A*(j)(θ)=djdθjA*(θ). Similarity, for θ≠μb+α, differentiating Equations (14–17) yields P1,n*(j)(θ)=A*(j)(θ)P1,n−1(0)+δn,NA*(j)(θ)P0,N−1(0)+μbP1,n+1*(j)(θ)+jP1,n*(j−1)(θ)(μb+α−θ),j≥1,2≤n≤K−1, (36) P1,K*(j)(θ)=A*(j)(θ)(P1,K−1(0)+P1,K(0))+jP1,K*(j−1)(θ)(μb+α−θ),j≥1, (37) where P1,n*(0)(θ)=P1,n*(θ). In sequel, we differentiate Equations (19) and (20) j times with respect to θ. Then, the jth-order derivatives of P2,n*(θ) at θ=μd are given by P2,n*(j)(μd)=−μdP2,n+1*(j+1)(μd)+αP1,n*(j+1)(μd)+A*(j+1)(μd)P2,n−1(0)j+1,j≥1,1≤n≤K−1, (38) P2,K*(j)(μd)=−αP1,K*(j+1)(μd)+A*(j+1)(μd)(P2,K−1(0)+P2,K(0))j+1,j≥1, (39) For θ≠μd, differentiating Equations (19) and (20) j times with respect to θ gives P2,n*(j)(θ)=μdP2,n+1*(j)(θ)+αP1,n*(j)(θ)+A*(j)(θ)P2,n−1(0)+jP2,n*(j−1)(θ)(μd−θ),j≥1,1≤n≤K−1, (40) P2,K*(j)(θ)=αP1,K*(j)(θ)+A*(j)(θ)(P2,K−1(0)+P2,K(0))+jP2,K*(j−1)(θ)(μd−θ),j≥1, (41) where P2,n*(0)(θ)=P2,n*(θ). 4. STEADY-STATE PROBABILITY DISTRIBUTION AT PRE-ARRIVAL EPOCHS Let Pi,n−(i=0,1,2) be the steady-state probability that an arrival finds n customers in the system when the server is in state i. In accordance with the arguments proposed by Goswami et al. [32] and Banik et al. [39], we have the relationship between Pi,n−(i=0,1,2) and Pi,n(0)(i=0,1,2) as follows: P0,n−=1λP0,n(0),0≤n≤N−1, (42) P1,n−=1λP1,n(0),1≤n≤K, (43) P2,n−=1λP2,n(0),0≤n≤K. (44) Before determining the pre-arrival epoch probabilities Pi,n−(i=0,1,2), it is necessary to evaluate P0,n(0)(0≤n≤N−1), P1,n(0)(1≤n≤K) and P2,n(0)(0≤n≤K). However, it is difficult to obtain the analytical forms of P0,n(0), P1,n(0) and P2,n(0). We overcome this difficulty by using the results in Section 3 to formulate an algorithm for numerically computing the pre-arrival epoch probabilities. The step-by-step procedure of the algorithm is described as follows: Step 1: Express P1,n(0) and P1,n*(θ) in terms of P0,N−1(0) and P1,K(0) as follows: P1,n(0)={ξnP1,K(0),ifN≤n≤K,ξnP1,K(0)+ϕnP0,N−1(0),if1≤n≤N−1, P1,n*(θ)={Ψn(θ)P1,K(0),ifN≤n≤K,ϒn(θ)P0,N−1(0)+Ψn(θ)P1,K(0),if2≤n≤N−1. Step 1.1: Calculate ξn using Equations (26) and (27) as follows: ξK=1,ξK−1=1−A*(μb+α)A*(μb+α),ξn−1=ξn−μbΨn+1(μb+α)A*(μb+α),1≤n≤K−1. Step 1.2: Calculate ϕn using Equation (27) as follows: ϕN−1=−1,ϕN−2=ϕN−1A*(μb+α),ϕn−1=ϕn−μbϒn+1(μb+α)A*(μb+α),1≤n≤N−2. Step 1.3: Calculate ϒn(θ) and Ψn(θ) using Equations (28) and (29) as follows:  if n=Kthen   if θ=μb+αthen     ΨK(θ)=−A*(1)(θ)(ξK−1+ξK)   else     ΨK(θ)=A*(θ)(ξK−1+ξK)−ξK(μb+α−θ)   end if  else if N≤n≤K−1then   if θ=μb+αthen     Ψn(θ)=−(A*(1)(θ)ξn−1+μbΨn+1(1)(θ))   else     Ψn(θ)=A*(θ)ξn−1+μbΨn+1(θ)−ξn(μb+α−θ)   end if  else if n=N−1then   if θ=μb+αthen     ϒN−1(θ)=−(A*(1)(θ)ϕN−2)     ΨN−1(θ)=−(A*(1)(θ)ξN−2+μbΨN(1)(θ))   else     ϒN−1(θ)=A*(θ)ϕN−2−ϕN−1(μb+α−θ)     ΨN−1(θ)=A*(θ)ξN−2+μbΨN(θ)−ξN−1(μb+α−θ)   end if  else if 2≤n≤N−2then   if θ=μb+αthen     ϒn(θ)=−(A*(1)(θ)ϕn−1+μbϒn+1(1)(θ))     Ψn(θ)=−(A*(1)(θ)ξn−1+μbΨn+1(1)(θ))   else     ϒn(θ)=A*(θ)ϕn−1+μbϒn+1(θ)−ϕn(μb+α−θ)     Ψn(θ)=A*(θ)ξn−1+μbΨn+1(θ)−ξn(μb+α−θ)  end if  if n=Kthen   if θ=μb+αthen     ΨK(θ)=−A*(1)(θ)(ξK−1+ξK)   else     ΨK(θ)=A*(θ)(ξK−1+ξK)−ξK(μb+α−θ)   end if  else if N≤n≤K−1then   if θ=μb+αthen     Ψn(θ)=−(A*(1)(θ)ξn−1+μbΨn+1(1)(θ))   else     Ψn(θ)=A*(θ)ξn−1+μbΨn+1(θ)−ξn(μb+α−θ)   end if  else if n=N−1then   if θ=μb+αthen     ϒN−1(θ)=−(A*(1)(θ)ϕN−2)     ΨN−1(θ)=−(A*(1)(θ)ξN−2+μbΨN(1)(θ))   else     ϒN−1(θ)=A*(θ)ϕN−2−ϕN−1(μb+α−θ)     ΨN−1(θ)=A*(θ)ξN−2+μbΨN(θ)−ξN−1(μb+α−θ)   end if  else if 2≤n≤N−2then   if θ=μb+αthen     ϒn(θ)=−(A*(1)(θ)ϕn−1+μbϒn+1(1)(θ))     Ψn(θ)=−(A*(1)(θ)ξn−1+μbΨn+1(1)(θ))   else     ϒn(θ)=A*(θ)ϕn−1+μbϒn+1(θ)−ϕn(μb+α−θ)     Ψn(θ)=A*(θ)ξn−1+μbΨn+1(θ)−ξn(μb+α−θ)  end if  if n=Kthen   if θ=μb+αthen     ΨK(θ)=−A*(1)(θ)(ξK−1+ξK)   else     ΨK(θ)=A*(θ)(ξK−1+ξK)−ξK(μb+α−θ)   end if  else if N≤n≤K−1then   if θ=μb+αthen     Ψn(θ)=−(A*(1)(θ)ξn−1+μbΨn+1(1)(θ))   else     Ψn(θ)=A*(θ)ξn−1+μbΨn+1(θ)−ξn(μb+α−θ)   end if  else if n=N−1then   if θ=μb+αthen     ϒN−1(θ)=−(A*(1)(θ)ϕN−2)     ΨN−1(θ)=−(A*(1)(θ)ξN−2+μbΨN(1)(θ))   else     ϒN−1(θ)=A*(θ)ϕN−2−ϕN−1(μb+α−θ)     ΨN−1(θ)=A*(θ)ξN−2+μbΨN(θ)−ξN−1(μb+α−θ)   end if  else if 2≤n≤N−2then   if θ=μb+αthen     ϒn(θ)=−(A*(1)(θ)ϕn−1+μbϒn+1(1)(θ))     Ψn(θ)=−(A*(1)(θ)ξn−1+μbΨn+1(1)(θ))   else     ϒn(θ)=A*(θ)ϕn−1+μbϒn+1(θ)−ϕn(μb+α−θ)     Ψn(θ)=A*(θ)ξn−1+μbΨn+1(θ)−ξn(μb+α−θ)  end if  if n=Kthen   if θ=μb+αthen     ΨK(θ)=−A*(1)(θ)(ξK−1+ξK)   else     ΨK(θ)=A*(θ)(ξK−1+ξK)−ξK(μb+α−θ)   end if  else if N≤n≤K−1then   if θ=μb+αthen     Ψn(θ)=−(A*(1)(θ)ξn−1+μbΨn+1(1)(θ))   else     Ψn(θ)=A*(θ)ξn−1+μbΨn+1(θ)−ξn(μb+α−θ)   end if  else if n=N−1then   if θ=μb+αthen     ϒN−1(θ)=−(A*(1)(θ)ϕN−2)     ΨN−1(θ)=−(A*(1)(θ)ξN−2+μbΨN(1)(θ))   else     ϒN−1(θ)=A*(θ)ϕN−2−ϕN−1(μb+α−θ)     ΨN−1(θ)=A*(θ)ξN−2+μbΨN(θ)−ξN−1(μb+α−θ)   end if  else if 2≤n≤N−2then   if θ=μb+αthen     ϒn(θ)=−(A*(1)(θ)ϕn−1+μbϒn+1(1)(θ))     Ψn(θ)=−(A*(1)(θ)ξn−1+μbΨn+1(1)(θ))   else     ϒn(θ)=A*(θ)ϕn−1+μbϒn+1(θ)−ϕn(μb+α−θ)     Ψn(θ)=A*(θ)ξn−1+μbΨn+1(θ)−ξn(μb+α−θ)  end if Step 1.4: Calculate ϒn(j)(θ) and Ψn(j)(θ) using Equations (34–37) as follows:  if n=Kthen   if θ=μb+αthen     ΨK(j)(θ)=−A*(j+1)(θ)(ξK−1+ξK)j+1   else     ΨK*(j)(θ)=A*(j)(θ)(ξK−1+ξK)+jΨK*(j−1)(θ)(μb+α−θ)   end if  else if N≤n≤K−1then   if θ=μb+αthen     Ψn(j)(θ)=−A*(j+1)(θ)ξn−1+μbΨn+1(j+1)(θ)j+1   else     Ψn(j)(θ)=A*(j)(θ)ξn−1+μbΨn+1(j)(θ)+jΨn(j−1)(θ)(μb+α−θ)   end if  else if n=N−1then   if θ=μb+αthen     ϒN−1(j)(θ)=−A*(j+1)(θ)ϕN−2j+1     ΨN−1(j)(θ)=−A*(j+1)(θ)ξN−2+μbΨN(j+1)(θ)j+1   else     ϒN−1(j)(θ)=A*(j)(θ)ϕN−2+jϒN−1(j−1)(θ)(μb+α−θ)     ΨN−1(j)(θ)=A*(j)(θ)ξN−2+μbΨN(j)(θ)+jΨN−1(j−1)(θ)(μb+α−θ)   end if  else if 2≤n≤N−2then   if θ=μb+αthen     ϒn(j)(θ)=−A*(j+1)(θ)ϕn−1+μbϒn+1(j+1)(θ)j+1     Ψn(j)(θ)=−A*(j+1)(θ)ξn−1+μbΨn+1(j+1)(θ)j+1   else     ϒn(j)(θ)=A*(j)(θ)ϕn−1+μbϒn+1(j)(θ)+jϒn(j−1)(θ)(μb+α−θ)     Ψn(j)(θ)=A*(j)(θ)ξn−1+μbΨn+1(j)(θ)+jΨn(j−1)(θ)(μb+α−θ)   end if  end if  if n=Kthen   if θ=μb+αthen     ΨK(j)(θ)=−A*(j+1)(θ)(ξK−1+ξK)j+1   else     ΨK*(j)(θ)=A*(j)(θ)(ξK−1+ξK)+jΨK*(j−1)(θ)(μb+α−θ)   end if  else if N≤n≤K−1then   if θ=μb+αthen     Ψn(j)(θ)=−A*(j+1)(θ)ξn−1+μbΨn+1(j+1)(θ)j+1   else     Ψn(j)(θ)=A*(j)(θ)ξn−1+μbΨn+1(j)(θ)+jΨn(j−1)(θ)(μb+α−θ)   end if  else if n=N−1then   if θ=μb+αthen     ϒN−1(j)(θ)=−A*(j+1)(θ)ϕN−2j+1     ΨN−1(j)(θ)=−A*(j+1)(θ)ξN−2+μbΨN(j+1)(θ)j+1   else     ϒN−1(j)(θ)=A*(j)(θ)ϕN−2+jϒN−1(j−1)(θ)(μb+α−θ)     ΨN−1(j)(θ)=A*(j)(θ)ξN−2+μbΨN(j)(θ)+jΨN−1(j−1)(θ)(μb+α−θ)   end if  else if 2≤n≤N−2then   if θ=μb+αthen     ϒn(j)(θ)=−A*(j+1)(θ)ϕn−1+μbϒn+1(j+1)(θ)j+1     Ψn(j)(θ)=−A*(j+1)(θ)ξn−1+μbΨn+1(j+1)(θ)j+1   else     ϒn(j)(θ)=A*(j)(θ)ϕn−1+μbϒn+1(j)(θ)+jϒn(j−1)(θ)(μb+α−θ)     Ψn(j)(θ)=A*(j)(θ)ξn−1+μbΨn+1(j)(θ)+jΨn(j−1)(θ)(μb+α−θ)   end if  end if View Large  if n=Kthen   if θ=μb+αthen     ΨK(j)(θ)=−A*(j+1)(θ)(ξK−1+ξK)j+1   else     ΨK*(j)(θ)=A*(j)(θ)(ξK−1+ξK)+jΨK*(j−1)(θ)(μb+α−θ)   end if  else if N≤n≤K−1then   if θ=μb+αthen     Ψn(j)(θ)=−A*(j+1)(θ)ξn−1+μbΨn+1(j+1)(θ)j+1   else     Ψn(j)(θ)=A*(j)(θ)ξn−1+μbΨn+1(j)(θ)+jΨn(j−1)(θ)(μb+α−θ)   end if  else if n=N−1then   if θ=μb+αthen     ϒN−1(j)(θ)=−A*(j+1)(θ)ϕN−2j+1     ΨN−1(j)(θ)=−A*(j+1)(θ)ξN−2+μbΨN(j+1)(θ)j+1   else     ϒN−1(j)(θ)=A*(j)(θ)ϕN−2+jϒN−1(j−1)(θ)(μb+α−θ)     ΨN−1(j)(θ)=A*(j)(θ)ξN−2+μbΨN(j)(θ)+jΨN−1(j−1)(θ)(μb+α−θ)   end if  else if 2≤n≤N−2then   if θ=μb+αthen     ϒn(j)(θ)=−A*(j+1)(θ)ϕn−1+μbϒn+1(j+1)(θ)j+1     Ψn(j)(θ)=−A*(j+1)(θ)ξn−1+μbΨn+1(j+1)(θ)j+1   else     ϒn(j)(θ)=A*(j)(θ)ϕn−1+μbϒn+1(j)(θ)+jϒn(j−1)(θ)(μb+α−θ)     Ψn(j)(θ)=A*(j)(θ)ξn−1+μbΨn+1(j)(θ)+jΨn(j−1)(θ)(μb+α−θ)   end if  end if  if n=Kthen   if θ=μb+αthen     ΨK(j)(θ)=−A*(j+1)(θ)(ξK−1+ξK)j+1   else     ΨK*(j)(θ)=A*(j)(θ)(ξK−1+ξK)+jΨK*(j−1)(θ)(μb+α−θ)   end if  else if N≤n≤K−1then   if θ=μb+αthen     Ψn(j)(θ)=−A*(j+1)(θ)ξn−1+μbΨn+1(j+1)(θ)j+1   else     Ψn(j)(θ)=A*(j)(θ)ξn−1+μbΨn+1(j)(θ)+jΨn(j−1)(θ)(μb+α−θ)   end if  else if n=N−1then   if θ=μb+αthen     ϒN−1(j)(θ)=−A*(j+1)(θ)ϕN−2j+1     ΨN−1(j)(θ)=−A*(j+1)(θ)ξN−2+μbΨN(j+1)(θ)j+1   else     ϒN−1(j)(θ)=A*(j)(θ)ϕN−2+jϒN−1(j−1)(θ)(μb+α−θ)     ΨN−1(j)(θ)=A*(j)(θ)ξN−2+μbΨN(j)(θ)+jΨN−1(j−1)(θ)(μb+α−θ)   end if  else if 2≤n≤N−2then   if θ=μb+αthen     ϒn(j)(θ)=−A*(j+1)(θ)ϕn−1+μbϒn+1(j+1)(θ)j+1     Ψn(j)(θ)=−A*(j+1)(θ)ξn−1+μbΨn+1(j+1)(θ)j+1   else     ϒn(j)(θ)=A*(j)(θ)ϕn−1+μbϒn+1(j)(θ)+jϒn(j−1)(θ)(μb+α−θ)     Ψn(j)(θ)=A*(j)(θ)ξn−1+μbΨn+1(j)(θ)+jΨn(j−1)(θ)(μb+α−θ)   end if  end if View Large Step 2: Express P2,n(0) and P2,n*(θ) in terms of P0,N−1(0), P1,K(0) and P2,K(0) as follows: P2,n(0)={ωnP1,K(0)+ψnP2,K(0),ifN≤n≤K,τnP0,N−1(0)+ωnP1,K(0)+ψnP2,K(0),if0≤n≤N−1, P2,n*(θ)={Ωn(θ)P1,K(0)+Φn(θ)P2,K(0),ifN≤n≤K,Ζn(θ)P0,N−1(0)+Ωn(θ)P1,K(0)+Φn(θ)P2,K(0),if1≤n≤N−1. Step 2.1: Calculate ωn and ψn using Equations (30) and (31) as follows: ψK=1,ωK=0,ψK−1=(1−A*(μd))ψKA*(μd),ωK−1=(1−A*(μd))ωK−αΨK(μd)A*(μd) ωn−1=ωn−μdΩn+1(μd)−αΨn(μd)A*(μd),1≤n≤K−1, ψn−1=ψn−μdΦn+1(μd)A*(μd),1≤n≤K−1. Step 2.2: Calculate τn using Equation (31) as follows: τN−1=0,τN−2=τN−1−αϒN−1(μd)A*(μd), τn−1=τn−μdΖn+1(μd)−αϒn(μd)A*(μd),1≤n≤N−2. Step 2.3: Calculate Ζn(θ), Ωn(θ) and Φn(θ) using Equations (32) and (33) as follows:  if n=Kthen   if θ=μdthen     ΩK(θ)=−(αΨK(1)(θ)+A*(1)(θ)ωK−1)     ΦK(θ)=−(A*(1)(θ)(ψK−1+ψK))   else     ΩK(θ)=αΨK(θ)+A*(θ)ωK−1(μd−θ)     ΦK(θ)=A*(θ)(ψK−1+ψK)−ψK(μd−θ)   end if  else if N≤n≤K−1then   if θ=μdthen     Ωn(θ)=−(μdΩn+1(1)(θ)+αΨn(1)(θ)+A*(1)(θ)ωn−1)     Φn(θ)=−(μdΦn+1(1)(θ)+A*(1)(θ)ψn−1)   else     Ωn(θ)=μdΩn+1(θ)+αΨn(θ)+A*(θ)ωn−1−ωn(μd−θ)     Φn(θ)=μdΦn+1(θ)+A*(θ)ψn−1−ψn(μd−θ)  else if n=N−1then   if θ=μdthen     ΖN−1(θ)=−(αϒN−1(1)(θ)+A*(1)(θ)τN−2)     ΩN−1(θ)=−(μdΩN(1)(θ)+αΨN−1(1)(θ)+A*(1)(θ)ωN−2)     ΦN−1(θ)=−(μdΦN(1)(θ)+A*(1)(θ)ψN−2)   else     ΖN−1(θ)=αϒN−1(θ)+A*(θ)τN−2−τN−1(μd−θ)     ΩN−1(θ)=μdΩN(θ)+αΨN−1(θ)+A*(θ)ωN−2−ωN−1(μd−θ)     ΦN−1(θ)=μdΦN(θ)+A*(θ)ψN−2−ψN−1(μd−θ)   end if  else if 1≤n≤N−2then   if θ=μdthen     Ζn(θ)=−(μdΖn+1(1)(θ)+αϒn(1)(θ)+A*(1)(θ)τn−1)     Ωn(θ)=−(μdΩn+1(1)(θ)+αΨn(1)(θ)+A*(1)(θ)ωn−1)     Φn(θ)=−(μdΦn+1(1)(θ)+A*(1)(θ)ψn−1)   else     Ζn(θ)=μdΖn+1(θ)+αϒn(θ)+A*(θ)τn−1−τn(μd−θ)     Ωn(θ)=μdΩn+1(θ)+αΨn(θ)+A*(θ)ωn−1−ωn(μd−θ)     Φn(θ)=μdΦn+1(θ)+A*(θ)ψn−1−ψn(μd−θ)   end if  end if  if n=Kthen   if θ=μdthen     ΩK(θ)=−(αΨK(1)(θ)+A*(1)(θ)ωK−1)     ΦK(θ)=−(A*(1)(θ)(ψK−1+ψK))   else     ΩK(θ)=αΨK(θ)+A*(θ)ωK−1(μd−θ)     ΦK(θ)=A*(θ)(ψK−1+ψK)−ψK(μd−θ)   end if  else if N≤n≤K−1then   if θ=μdthen     Ωn(θ)=−(μdΩn+1(1)(θ)+αΨn(1)(θ)+A*(1)(θ)ωn−1)     Φn(θ)=−(μdΦn+1(1)(θ)+A*(1)(θ)ψn−1)   else     Ωn(θ)=μdΩn+1(θ)+αΨn(θ)+A*(θ)ωn−1−ωn(μd−θ)     Φn(θ)=μdΦn+1(θ)+A*(θ)ψn−1−ψn(μd−θ)  else if n=N−1then   if θ=μdthen     ΖN−1(θ)=−(αϒN−1(1)(θ)+A*(1)(θ)τN−2)     ΩN−1(θ)=−(μdΩN(1)(θ)+αΨN−1(1)(θ)+A*(1)(θ)ωN−2)     ΦN−1(θ)=−(μdΦN(1)(θ)+A*(1)(θ)ψN−2)   else     ΖN−1(θ)=αϒN−1(θ)+A*(θ)τN−2−τN−1(μd−θ)     ΩN−1(θ)=μdΩN(θ)+αΨN−1(θ)+A*(θ)ωN−2−ωN−1(μd−θ)     ΦN−1(θ)=μdΦN(θ)+A*(θ)ψN−2−ψN−1(μd−θ)   end if  else if 1≤n≤N−2then   if θ=μdthen     Ζn(θ)=−(μdΖn+1(1)(θ)+αϒn(1)(θ)+A*(1)(θ)τn−1)     Ωn(θ)=−(μdΩn+1(1)(θ)+αΨn(1)(θ)+A*(1)(θ)ωn−1)     Φn(θ)=−(μdΦn+1(1)(θ)+A*(1)(θ)ψn−1)   else     Ζn(θ)=μdΖn+1(θ)+αϒn(θ)+A*(θ)τn−1−τn(μd−θ)     Ωn(θ)=μdΩn+1(θ)+αΨn(θ)+A*(θ)ωn−1−ωn(μd−θ)     Φn(θ)=μdΦn+1(θ)+A*(θ)ψn−1−ψn(μd−θ)   end if  end if View Large  if n=Kthen   if θ=μdthen     ΩK(θ)=−(αΨK(1)(θ)+A*(1)(θ)ωK−1)     ΦK(θ)=−(A*(1)(θ)(ψK−1+ψK))   else     ΩK(θ)=αΨK(θ)+A*(θ)ωK−1(μd−θ)     ΦK(θ)=A*(θ)(ψK−1+ψK)−ψK(μd−θ)   end if  else if N≤n≤K−1then   if θ=μdthen     Ωn(θ)=−(μdΩn+1(1)(θ)+αΨn(1)(θ)+A*(1)(θ)ωn−1)     Φn(θ)=−(μdΦn+1(1)(θ)+A*(1)(θ)ψn−1)   else     Ωn(θ)=μdΩn+1(θ)+αΨn(θ)+A*(θ)ωn−1−ωn(μd−θ)     Φn(θ)=μdΦn+1(θ)+A*(θ)ψn−1−ψn(μd−θ)  else if n=N−1then   if θ=μdthen     ΖN−1(θ)=−(αϒN−1(1)(θ)+A*(1)(θ)τN−2)     ΩN−1(θ)=−(μdΩN(1)(θ)+αΨN−1(1)(θ)+A*(1)(θ)ωN−2)     ΦN−1(θ)=−(μdΦN(1)(θ)+A*(1)(θ)ψN−2)   else     ΖN−1(θ)=αϒN−1(θ)+A*(θ)τN−2−τN−1(μd−θ)     ΩN−1(θ)=μdΩN(θ)+αΨN−1(θ)+A*(θ)ωN−2−ωN−1(μd−θ)     ΦN−1(θ)=μdΦN(θ)+A*(θ)ψN−2−ψN−1(μd−θ)   end if  else if 1≤n≤N−2then   if θ=μdthen     Ζn(θ)=−(μdΖn+1(1)(θ)+αϒn(1)(θ)+A*(1)(θ)τn−1)     Ωn(θ)=−(μdΩn+1(1)(θ)+αΨn(1)(θ)+A*(1)(θ)ωn−1)     Φn(θ)=−(μdΦn+1(1)(θ)+A*(1)(θ)ψn−1)   else     Ζn(θ)=μdΖn+1(θ)+αϒn(θ)+A*(θ)τn−1−τn(μd−θ)     Ωn(θ)=μdΩn+1(θ)+αΨn(θ)+A*(θ)ωn−1−ωn(μd−θ)     Φn(θ)=μdΦn+1(θ)+A*(θ)ψn−1−ψn(μd−θ)   end if  end if  if n=Kthen   if θ=μdthen     ΩK(θ)=−(αΨK(1)(θ)+A*(1)(θ)ωK−1)     ΦK(θ)=−(A*(1)(θ)(ψK−1+ψK))   else     ΩK(θ)=αΨK(θ)+A*(θ)ωK−1(μd−θ)     ΦK(θ)=A*(θ)(ψK−1+ψK)−ψK(μd−θ)   end if  else if N≤n≤K−1then   if θ=μdthen     Ωn(θ)=−(μdΩn+1(1)(θ)+αΨn(1)(θ)+A*(1)(θ)ωn−1)     Φn(θ)=−(μdΦn+1(1)(θ)+A*(1)(θ)ψn−1)   else     Ωn(θ)=μdΩn+1(θ)+αΨn(θ)+A*(θ)ωn−1−ωn(μd−θ)     Φn(θ)=μdΦn+1(θ)+A*(θ)ψn−1−ψn(μd−θ)  else if n=N−1then   if θ=μdthen     ΖN−1(θ)=−(αϒN−1(1)(θ)+A*(1)(θ)τN−2)     ΩN−1(θ)=−(μdΩN(1)(θ)+αΨN−1(1)(θ)+A*(1)(θ)ωN−2)     ΦN−1(θ)=−(μdΦN(1)(θ)+A*(1)(θ)ψN−2)   else     ΖN−1(θ)=αϒN−1(θ)+A*(θ)τN−2−τN−1(μd−θ)     ΩN−1(θ)=μdΩN(θ)+αΨN−1(θ)+A*(θ)ωN−2−ωN−1(μd−θ)     ΦN−1(θ)=μdΦN(θ)+A*(θ)ψN−2−ψN−1(μd−θ)   end if  else if 1≤n≤N−2then   if θ=μdthen     Ζn(θ)=−(μdΖn+1(1)(θ)+αϒn(1)(θ)+A*(1)(θ)τn−1)     Ωn(θ)=−(μdΩn+1(1)(θ)+αΨn(1)(θ)+A*(1)(θ)ωn−1)     Φn(θ)=−(μdΦn+1(1)(θ)+A*(1)(θ)ψn−1)   else     Ζn(θ)=μdΖn+1(θ)+αϒn(θ)+A*(θ)τn−1−τn(μd−θ)     Ωn(θ)=μdΩn+1(θ)+αΨn(θ)+A*(θ)ωn−1−ωn(μd−θ)     Φn(θ)=μdΦn+1(θ)+A*(θ)ψn−1−ψn(μd−θ)   end if  end if View Large Step 2.4: Calculate Ζn(j)(θ), Ωn(j)(θ) and Φn(j)(θ) using Equations (38–41) as follows:  if n=Kthen   if θ=μdthen     ΩK(j)(θ)=−αΨK(j+1)(θ)+A*(j+1)(θ)ωK−1j+1     ΦK(j)(θ)=−A*(j+1)(θ)(ψK−1+ψK)j+1   else     ΩK(j)(θ)=αΨK(j)(θ)+A*(j)(θ)ωK−1+jΩK(j−1)(θ)(μd−θ)     ΦK(j)(θ)=A*(j)(θ)(ψK−1+ψK)+jΦK(j−1)(θ)(μd−θ)   end if  else if N≤n≤K−1then   if θ=μdthen     Ωn(j)(θ)=−μdΩn+1(j+1)(θ)+αΨn(j+1)(θ)+A*(j+1)(θ)ωn−1j+1,     Φn(j)(θ)=−μdΦn+1(j+1)(θ)+A*(j+1)(θ)ψn−1j+1,   else     Ωn(j)(θ)=μdΩn+1(j)(θ)+αΨn(j)(θ)+A*(j)(θ)ωn−1+jΩn(j−1)(θ)(μd−θ),     Φn(j)(θ)=μdΦn+1(j)(θ)+A*(j)(θ)ψn−1+jΦn(j−1)(θ)(μd−θ),   end if  else if n=N−1then   if θ=μdthen     ΖN−1(j)(θ)=−αϒN−1(j+1)(θ)+A*(j+1)(θ)τN−2j+1     ΩN−1(j)(θ)=−μdΩN(j+1)(θ)+αΨN−1(j+1)(θ)+A*(j+1)(θ)ωN−2j+1     ΦN−1(j)(θ)=−μdΦN(j+1)(θ)+A*(j+1)(θ)ψN−2j+1   else     ΖN−1(j)(θ)=αϒN−1(j)(θ)+A*(j)(θ)τN−2+jΖN−1(j−1)(θ)(μd−θ)     ΩN−1(j)(θ)=μdΩN(j)(θ)+αΨN−1(j)(θ)+A*(j)(θ)ωN−2+jΩN−1(j−1)(θ)(μd−θ)     ΦN−1(j)(θ)=μdΦN(j)(θ)+A*(j)(θ)ψN−2+jΦN−1(j−1)(θ)(μd−θ)   end if  else if 1≤n≤N−2then   if θ=μdthen     Ζn(j)(θ)=−μdΖn+1(j+1)(θ)+αϒn(j+1)(θ)+A*(j+1)(θ)τn−1j+1     Ωn(j)(θ)=−μdΩn+1(j+1)(θ)+αΨn(j+1)(θ)+A*(j+1)(θ)ωn−1j+1     Φn(j)(θ)=−μdΦn+1(j+1)(θ)+A*(j+1)(θ)ψn−1j+1   else     Ζn(j)(θ)=μdΖn+1(j)(θ)+αϒn(j)(θ)+A*(j)(θ)τn−1+jΖn(j−1)(θ)(μd−θ)     Ωn(j)(θ)=μdΩn+1(j)(θ)+αΨn(j)(θ)+A*(j)(θ)ωn−1+jΩn(j−1)(θ)(μd−θ)     Φn(j)(θ)=μdΦn+1(j)(θ)+A*(j)(θ)ψn−1+jΦn(j−1)(θ)(μd−θ)   end if  end if  if n=Kthen   if θ=μdthen     ΩK(j)(θ)=−αΨK(j+1)(θ)+A*(j+1)(θ)ωK−1j+1     ΦK(j)(θ)=−A*(j+1)(θ)(ψK−1+ψK)j+1   else     ΩK(j)(θ)=αΨK(j)(θ)+A*(j)(θ)ωK−1+jΩK(j−1)(θ)(μd−θ)     ΦK(j)(θ)=A*(j)(θ)(ψK−1+ψK)+jΦK(j−1)(θ)(μd−θ)   end if  else if N≤n≤K−1then   if θ=μdthen     Ωn(j)(θ)=−μdΩn+1(j+1)(θ)+αΨn(j+1)(θ)+A*(j+1)(θ)ωn−1j+1,     Φn(j)(θ)=−μdΦn+1(j+1)(θ)+A*(j+1)(θ)ψn−1j+1,   else     Ωn(j)(θ)=μdΩn+1(j)(θ)+αΨn(j)(θ)+A*(j)(θ)ωn−1+jΩn(j−1)(θ)(μd−θ),     Φn(j)(θ)=μdΦn+1(j)(θ)+A*(j)(θ)ψn−1+jΦn(j−1)(θ)(μd−θ),   end if  else if n=N−1then   if θ=μdthen     ΖN−1(j)(θ)=−αϒN−1(j+1)(θ)+A*(j+1)(θ)τN−2j+1     ΩN−1(j)(θ)=−μdΩN(j+1)(θ)+αΨN−1(j+1)(θ)+A*(j+1)(θ)ωN−2j+1     ΦN−1(j)(θ)=−μdΦN(j+1)(θ)+A*(j+1)(θ)ψN−2j+1   else     ΖN−1(j)(θ)=αϒN−1(j)(θ)+A*(j)(θ)τN−2+jΖN−1(j−1)(θ)(μd−θ)     ΩN−1(j)(θ)=μdΩN(j)(θ)+αΨN−1(j)(θ)+A*(j)(θ)ωN−2+jΩN−1(j−1)(θ)(μd−θ)     ΦN−1(j)(θ)=μdΦN(j)(θ)+A*(j)(θ)ψN−2+jΦN−1(j−1)(θ)(μd−θ)   end if  else if 1≤n≤N−2then   if θ=μdthen     Ζn(j)(θ)=−μdΖn+1(j+1)(θ)+αϒn(j+1)(θ)+A*(j+1)(θ)τn−1j+1     Ωn(j)(θ)=−μdΩn+1(j+1)(θ)+αΨn(j+1)(θ)+A*(j+1)(θ)ωn−1j+1     Φn(j)(θ)=−μdΦn+1(j+1)(θ)+A*(j+1)(θ)ψn−1j+1   else     Ζn(j)(θ)=μdΖn+1(j)(θ)+αϒn(j)(θ)+A*(j)(θ)τn−1+jΖn(j−1)(θ)(μd−θ)     Ωn(j)(θ)=μdΩn+1(j)(θ)+αΨn(j)(θ)+A*(j)(θ)ωn−1+jΩn(j−1)(θ)(μd−θ)     Φn(j)(θ)=μdΦn+1(j)(θ)+A*(j)(θ)ψn−1+jΦn(j−1)(θ)(μd−θ)   end if  end if View Large  if n=Kthen   if θ=μdthen     ΩK(j)(θ)=−αΨK(j+1)(θ)+A*(j+1)(θ)ωK−1j+1     ΦK(j)(θ)=−A*(j+1)(θ)(ψK−1+ψK)j+1   else     ΩK(j)(θ)=αΨK(j)(θ)+A*(j)(θ)ωK−1+jΩK(j−1)(θ)(μd−θ)     ΦK(j)(θ)=A*(j)(θ)(ψK−1+ψK)+jΦK(j−1)(θ)(μd−θ)   end if  else if N≤n≤K−1then   if θ=μdthen     Ωn(j)(θ)=−μdΩn+1(j+1)(θ)+αΨn(j+1)(θ)+A*(j+1)(θ)ωn−1j+1,     Φn(j)(θ)=−μdΦn+1(j+1)(θ)+A*(j+1)(θ)ψn−1j+1,   else     Ωn(j)(θ)=μdΩn+1(j)(θ)+αΨn(j)(θ)+A*(j)(θ)ωn−1+jΩn(j−1)(θ)(μd−θ),     Φn(j)(θ)=μdΦn+1(j)(θ)+A*(j)(θ)ψn−1+jΦn(j−1)(θ)(μd−θ),   end if  else if n=N−1then   if θ=μdthen     ΖN−1(j)(θ)=−αϒN−1(j+1)(θ)+A*(j+1)(θ)τN−2j+1     ΩN−1(j)(θ)=−μdΩN(j+1)(θ)+αΨN−1(j+1)(θ)+A*(j+1)(θ)ωN−2j+1     ΦN−1(j)(θ)=−μdΦN(j+1)(θ)+A*(j+1)(θ)ψN−2j+1   else     ΖN−1(j)(θ)=αϒN−1(j)(θ)+A*(j)(θ)τN−2+jΖN−1(j−1)(θ)(μd−θ)     ΩN−1(j)(θ)=μdΩN(j)(θ)+αΨN−1(j)(θ)+A*(j)(θ)ωN−2+jΩN−1(j−1)(θ)(μd−θ)     ΦN−1(j)(θ)=μdΦN(j)(θ)+A*(j)(θ)ψN−2+jΦN−1(j−1)(θ)(μd−θ)   end if  else if 1≤n≤N−2then   if θ=μdthen     Ζn(j)(θ)=−μdΖn+1(j+1)(θ)+αϒn(j+1)(θ)+A*(j+1)(θ)τn−1j+1     Ωn(j)(θ)=−μdΩn+1(j+1)(θ)+αΨn(j+1)(θ)+A*(j+1)(θ)ωn−1j+1     Φn(j)(θ)=−μdΦn+1(j+1)(θ)+A*(j+1)(θ)ψn−1j+1   else     Ζn(j)(θ)=μdΖn+1(j)(θ)+αϒn(j)(θ)+A*(j)(θ)τn−1+jΖn(j−1)(θ)(μd−θ)     Ωn(j)(θ)=μdΩn+1(j)(θ)+αΨn(j)(θ)+A*(j)(θ)ωn−1+jΩn(j−1)(θ)(μd−θ)     Φn(j)(θ)=μdΦn+1(j)(θ)+A*(j)(θ)ψn−1+jΦn(j−1)(θ)(μd−θ)   end if  end if  if n=Kthen   if θ=μdthen     ΩK(j)(θ)=−αΨK(j+1)(θ)+A*(j+1)(θ)ωK−1j+1     ΦK(j)(θ)=−A*(j+1)(θ)(ψK−1+ψK)j+1   else     ΩK(j)(θ)=αΨK(j)(θ)+A*(j)(θ)ωK−1+jΩK(j−1)(θ)(μd−θ)     ΦK(j)(θ)=A*(j)(θ)(ψK−1+ψK)+jΦK(j−1)(θ)(μd−θ)   end if  else if N≤n≤K−1then   if θ=μdthen     Ωn(j)(θ)=−μdΩn+1(j+1)(θ)+αΨn(j+1)(θ)+A*(j+1)(θ)ωn−1j+1,     Φn(j)(θ)=−μdΦn+1(j+1)(θ)+A*(j+1)(θ)ψn−1j+1,   else     Ωn(j)(θ)=μdΩn+1(j)(θ)+αΨn(j)(θ)+A*(j)(θ)ωn−1+jΩn(j−1)(θ)(μd−θ),     Φn(j)(θ)=μdΦn+1(j)(θ)+A*(j)(θ)ψn−1+jΦn(j−1)(θ)(μd−θ),   end if  else if n=N−1then   if θ=μdthen     ΖN−1(j)(θ)=−αϒN−1(j+1)(θ)+A*(j+1)(θ)τN−2j+1     ΩN−1(j)(θ)=−μdΩN(j+1)(θ)+αΨN−1(j+1)(θ)+A*(j+1)(θ)ωN−2j+1     ΦN−1(j)(θ)=−μdΦN(j+1)(θ)+A*(j+1)(θ)ψN−2j+1   else     ΖN−1(j)(θ)=αϒN−1(j)(θ)+A*(j)(θ)τN−2+jΖN−1(j−1)(θ)(μd−θ)     ΩN−1(j)(θ)=μdΩN(j)(θ)+αΨN−1(j)(θ)+A*(j)(θ)ωN−2+jΩN−1(j−1)(θ)(μd−θ)     ΦN−1(j)(θ)=μdΦN(j)(θ)+A*(j)(θ)ψN−2+jΦN−1(j−1)(θ)(μd−θ)   end if  else if 1≤n≤N−2then   if θ=μdthen     Ζn(j)(θ)=−μdΖn+1(j+1)(θ)+αϒn(j+1)(θ)+A*(j+1)(θ)τn−1j+1     Ωn(j)(θ)=−μdΩn+1(j+1)(θ)+αΨn(j+1)(θ)+A*(j+1)(θ)ωn−1j+1     Φn(j)(θ)=−μdΦn+1(j+1)(θ)+A*(j+1)(θ)ψn−1j+1   else     Ζn(j)(θ)=μdΖn+1(j)(θ)+αϒn(j)(θ)+A*(j)(θ)τn−1+jΖn(j−1)(θ)(μd−θ)     Ωn(j)(θ)=μdΩn+1(j)(θ)+αΨn(j)(θ)+A*(j)(θ)ωn−1+jΩn(j−1)(θ)(μd−θ)     Φn(j)(θ)=μdΦn+1(j)(θ)+A*(j)(θ)ψn−1+jΦn(j−1)(θ)(μd−θ)   end if  end if View Large Step 3: Substituting θ=μb+α into Equation (13), then P0,N−1(0) can be expressed in terms of P1,K(0) as follows: P0,N−1(0)=ΔP1,K(0), where Δ=ξ1−μbΨ2(μb+α)μbϒ2(μb+α)−ϕ1. Step 4: Substituting θ=β into Equation (18), P2,K(0) can be expressed in terms of P1,K(0) as follows: P2,K(0)=ΘP1,K(0), where Θ=(τ0−μdΖ1(β))Δ−(μdΩ1(β)−ω0)μdΦ1(β)−ψ0. Step 5: Determine the only unknown quantity P1,K(0). From Equation (25), we know that P1,K(0)=λNΔ+∑n=1N−1(ξn+ϕnΔ)+∑n=NKξn+∑n=0N−1(τnΔ+ωn+ψnΘ)+∑n=NK(ωn+ψnΘ). Step 6: Compute the probabilities P0,n(0), P1,n(0) and P2,n(0) as follows: P0,n(0)=ΔP1,K(0),0≤n≤N−1, P1,n(0)={ξnP1,K(0),ifN≤n≤K,(ξn+ϕnΔ)P1,K(0),if1≤n≤N−1, P2,n(0)={(ωn+ψnΘ)P1,K(0),ifN≤n≤K,(τnΔ+ωn+ψnΘ)P1,K(0),if0≤n≤N−1. Step 7: Compute the pre-arrival epoch probabilities P0,n−, P1,n− and P2,n− using Equations (42–44). 5. STEADY-STATE PROBABILITY DISTRIBUTION AT ARBITRARY EPOCHS To obtain the steady-state probabilities at arbitrary epochs, we begin by differentiating Equations (12) with respect to θ and then setting θ=0. It follows P0,n=−A*(1)(0)P0,n−1(0)=a1P0,n−1(0)=P0,n−1−,1≤n≤N−1. (45) From Equations (21) and (45), one can easily get P0,1=P0,2=⋯=P0,N−1. (46) Next, we develop relations between pre-arrival and arbitrary epoch probabilities. Substituting θ=0 into Equations (13–20) and in conjunction with Equations (42–44), we get P1,1=μbμb+αP1,2−λμb+αP1,1−, (47) P1,n=1μb+αμbP1,n+1+λμb+α(P1,n−1−−P1,n−),2≤n≤N−1, (48) P1,N=1μb+αμbP1,N+1+λμb+α(P0,N−1−+P1,N−1−−P1,N−), (49) P1,n=1μb+αμbP1,n+1+λμb+α(P1,n−1−−P1,n−),N+1≤n≤K−1, (50) P1,K=λμb+αP1,K−1−, (51) P2,0=μdβP2,1−λβP2,0−, (52) P2,n=P2,n+1+αμdP1,n+λμd(P2,n−1−−P2,n−),1≤n≤K−1, (53) P2,K=αμdP1,K+λμdP2,K−1−. (54) Once the pre-arrival epoch probabilities are known, the steady-state probabilities at arbitrary epochs can be obtained using Equations (46–54). Remark 2 When α=0, we can obtain the results of the N-policy GI/M/1/K queue with a reliable server (see Ke and Wang [3]). 6. SYSTEM PERFORMANCE MEASURES AND SENSITIVITY ANALYSIS We develop a number of system performance measures based on steady-state probabilities to evaluate the proposed N-policy GI/M/1/K queue with working breakdowns. These measures can be used to improve the effectiveness of the system. Next, we present numerical results to explore the sensitivity of system performance measures. 6.1. System performance measures Let us define the system performance measures of the queueing model under study as follows: LS≡ the expected number of customers in the system; PL≡ the blocking probability of the system; WS≡ the expected waiting time in the system, PI≡ the probability that the server is turned off, PN≡ the probability that the server is in normal busy period, PD≡ the probability that the server is in breakdown period. LS≡ the expected number of customers in the system; PL≡ the blocking probability of the system; WS≡ the expected waiting time in the system, PI≡ the probability that the server is turned off, PN≡ the probability that the server is in normal busy period, PD≡ the probability that the server is in breakdown period. View Large LS≡ the expected number of customers in the system; PL≡ the blocking probability of the system; WS≡ the expected waiting time in the system, PI≡ the probability that the server is turned off, PN≡ the probability that the server is in normal busy period, PD≡ the probability that the server is in breakdown period. LS≡ the expected number of customers in the system; PL≡ the blocking probability of the system; WS≡ the expected waiting time in the system, PI≡ the probability that the server is turned off, PN≡ the probability that the server is in normal busy period, PD≡ the probability that the server is in breakdown period. View Large The expressions for LS, PL, WS, PI, PN and PD are given by LS=∑n=1N−1n⋅P0,n+∑n=1Kn⋅(P1,n+P2,n)=N(N−1)2P0,1+∑n=1Kn⋅(P1,n+P2,n). (55) PL=P1,K−+P2,K−, (56) WS=LSλeff, (57) PI=∑n=0N−1P0,n, (58) PN=∑n=1KP1,n, (59) PD=∑n=0KP2,n, (60) where λeff=λ(1−PL) represents the effective arrival rate. 6.2. Sensitivity analysis In the following, we present numerical examples illustrating the effects of various system parameters on some performance measures, including the expected number of customers in the system (LS), the blocking probability of the system (PL) and the expected waiting time in the system (WS). Three inter-arrival time distributions are considered: exponential distribution with a*(s)=λ/(λ+s), three-stage Erlang distribution with a*(s)=(3λ/(3λ+s))3 and deterministic distribution with a*(s)=e−s/λ. We fix K=12, N=3,6,9 and consider the following five cases: Case 1: μb=5.0, μd=2.0, α=0.1, β=4.0, and select various values of λ. Case 2: λ=2.0, μd=2.0, α=0.1, β=4.0, and select various values of μb. Case 3: λ=2.0, μb=5.0, α=0.1, β=4.0, and select various values of μd. Case 4: λ=2.0, μb=5.0, μd=2.0, β=4.0, and select various values of α. Case 5: λ=2.0, μb=5.0, μd=2.0, α=0.1, and select various values of β. Tables 1–5 summarize the numerical results for the above five cases, illustrating the effects induced by changes in various system parameters under three inter-arrival time distributions. It can be seen in Table 1 that LS, WS and PL increase with an increase in the mean arrival rate λ under all three inter-arrival time distributions. Intuitively, we know that an increase in the mean arrival rate results in a more congested system. Tables 2 and 3 show that LS, WS and PL decrease with an increase in the mean service rates μb and μd under all three inter-arrival time distributions. In Table 4, one can see that the larger the mean breakdown rate α is, the larger LS, WS and PL are under all three inter-arrival time distributions. Thus, larger values of α could have a negative effect on system performance measures. Table 5 shows that LS, WS and PL tend to decrease when the mean repair rate β increases, except for N=9. In Tables 1–5, we observe that LS, WS, and PL tend to increase with an increase in threshold value N under all three inter-arrival time distributions. One way to enhance customer satisfaction would be to reduce the waiting time of customers. Increasing the values of μb and μd and decreasing the value of threshold N would be expected to reduce waiting time. The above sensitivity analysis could be highly beneficial for performance evaluations in actual implementations. Table 1. Performance measures for various values of λ under three different inter-arrival time distributions (Case 1). λ N=3 N=6 N=9 LS WS PL LS WS PL LS WS PL N-policy M/M/1/12 queue with working breakdowns 2.0 3.7108 1.9244 0.0359 5.0466 2.6453 0.0461 6.1066 3.2358 0.0564 2.5 7.4607 3.6001 0.1711 8.0104 3.9073 0.1800 8.3980 4.1289 0.1864 3.0 9.7451 4.7841 0.3210 9.8562 4.8545 0.3232 9.9366 4.9055 0.3248 3.5 10.6073 5.2716 0.4251 10.6275 5.2854 0.4255 10.6428 5.2959 0.4258 4.0 10.9863 5.4826 0.4990 10.9903 5.4854 0.4991 10.9935 5.4877 0.4992 N-policy E3/M/1/12 queue with working breakdowns 2.0 3.9893 2.0514 0.0277 5.3458 2.7727 0.0360 6.3542 3.3214 0.0434 2.5 8.9416 4.4016 0.1874 9.1962 4.5487 0.1913 9.3661 4.6468 0.1938 3.0 10.6166 5.2919 0.3313 10.6354 5.3039 0.3316 10.6494 5.3126 0.3318 3.5 11.1056 5.5500 0.4283 11.1073 5.5511 0.4283 11.1086 5.5520 0.4283 4.0 11.3326 5.6658 0.5000 11.3328 5.6659 0.5000 11.3330 5.6660 0.5000 N-policy D/M/1/12 queue with working breakdowns 2.0 4.2456 2.1731 0.0231 5.5749 2.8737 0.0300 6.5268 3.3853 0.0360 2.5 9.8235 4.8824 0.1952 9.9222 4.9402 0.1966 9.9880 4.9785 0.1975 3.0 10.9912 5.4928 0.3330 10.9942 5.4947 0.3330 10.9965 5.4961 0.3331 3.5 11.3328 5.6662 0.4285 11.3330 5.6663 0.4285 11.3331 5.6663 0.4285 4.0 11.5000 5.7500 0.5000 11.5000 5.7500 0.5000 11.5000 5.7500 0.5000 λ N=3 N=6 N=9 LS WS PL LS WS PL LS WS PL N-policy M/M/1/12 queue with working breakdowns 2.0 3.7108 1.9244 0.0359 5.0466 2.6453 0.0461 6.1066 3.2358 0.0564 2.5 7.4607 3.6001 0.1711 8.0104 3.9073 0.1800 8.3980 4.1289 0.1864 3.0 9.7451 4.7841 0.3210 9.8562 4.8545 0.3232 9.9366 4.9055 0.3248 3.5 10.6073 5.2716 0.4251 10.6275 5.2854 0.4255 10.6428 5.2959 0.4258 4.0 10.9863 5.4826 0.4990 10.9903 5.4854 0.4991 10.9935 5.4877 0.4992 N-policy E3/M/1/12 queue with working breakdowns 2.0 3.9893 2.0514 0.0277 5.3458 2.7727 0.0360 6.3542 3.3214 0.0434 2.5 8.9416 4.4016 0.1874 9.1962 4.5487 0.1913 9.3661 4.6468 0.1938 3.0 10.6166 5.2919 0.3313 10.6354 5.3039 0.3316 10.6494 5.3126 0.3318 3.5 11.1056 5.5500 0.4283 11.1073 5.5511 0.4283 11.1086 5.5520 0.4283 4.0 11.3326 5.6658 0.5000 11.3328 5.6659 0.5000 11.3330 5.6660 0.5000 N-policy D/M/1/12 queue with working breakdowns 2.0 4.2456 2.1731 0.0231 5.5749 2.8737 0.0300 6.5268 3.3853 0.0360 2.5 9.8235 4.8824 0.1952 9.9222 4.9402 0.1966 9.9880 4.9785 0.1975 3.0 10.9912 5.4928 0.3330 10.9942 5.4947 0.3330 10.9965 5.4961 0.3331 3.5 11.3328 5.6662 0.4285 11.3330 5.6663 0.4285 11.3331 5.6663 0.4285 4.0 11.5000 5.7500 0.5000 11.5000 5.7500 0.5000 11.5000 5.7500 0.5000 Table 1. Performance measures for various values of λ under three different inter-arrival time distributions (Case 1). λ N=3 N=6 N=9 LS WS PL LS WS PL LS WS PL N-policy M/M/1/12 queue with working breakdowns 2.0 3.7108 1.9244 0.0359 5.0466 2.6453 0.0461 6.1066 3.2358 0.0564 2.5 7.4607 3.6001 0.1711 8.0104 3.9073 0.1800 8.3980 4.1289 0.1864 3.0 9.7451 4.7841 0.3210 9.8562 4.8545 0.3232 9.9366 4.9055 0.3248 3.5 10.6073 5.2716 0.4251 10.6275 5.2854 0.4255 10.6428 5.2959 0.4258 4.0 10.9863 5.4826 0.4990 10.9903 5.4854 0.4991 10.9935 5.4877 0.4992 N-policy E3/M/1/12 queue with working breakdowns 2.0 3.9893 2.0514 0.0277 5.3458 2.7727 0.0360 6.3542 3.3214 0.0434 2.5 8.9416 4.4016 0.1874 9.1962 4.5487 0.1913 9.3661 4.6468 0.1938 3.0 10.6166 5.2919 0.3313 10.6354 5.3039 0.3316 10.6494 5.3126 0.3318 3.5 11.1056 5.5500 0.4283 11.1073 5.5511 0.4283 11.1086 5.5520 0.4283 4.0 11.3326 5.6658 0.5000 11.3328 5.6659 0.5000 11.3330 5.6660 0.5000 N-policy D/M/1/12 queue with working breakdowns 2.0 4.2456 2.1731 0.0231 5.5749 2.8737 0.0300 6.5268 3.3853 0.0360 2.5 9.8235 4.8824 0.1952 9.9222 4.9402 0.1966 9.9880 4.9785 0.1975 3.0 10.9912 5.4928 0.3330 10.9942 5.4947 0.3330 10.9965 5.4961 0.3331 3.5 11.3328 5.6662 0.4285 11.3330 5.6663 0.4285 11.3331 5.6663 0.4285 4.0 11.5000 5.7500 0.5000 11.5000 5.7500 0.5000 11.5000 5.7500 0.5000 λ N=3 N=6 N=9 LS WS PL LS WS PL LS WS PL N-policy M/M/1/12 queue with working breakdowns 2.0 3.7108 1.9244 0.0359 5.0466 2.6453 0.0461 6.1066 3.2358 0.0564 2.5 7.4607 3.6001 0.1711 8.0104 3.9073 0.1800 8.3980 4.1289 0.1864 3.0 9.7451 4.7841 0.3210 9.8562 4.8545 0.3232 9.9366 4.9055 0.3248 3.5 10.6073 5.2716 0.4251 10.6275 5.2854 0.4255 10.6428 5.2959 0.4258 4.0 10.9863 5.4826 0.4990 10.9903 5.4854 0.4991 10.9935 5.4877 0.4992 N-policy E3/M/1/12 queue with working breakdowns 2.0 3.9893 2.0514 0.0277 5.3458 2.7727 0.0360 6.3542 3.3214 0.0434 2.5 8.9416 4.4016 0.1874 9.1962 4.5487 0.1913 9.3661 4.6468 0.1938 3.0 10.6166 5.2919 0.3313 10.6354 5.3039 0.3316 10.6494 5.3126 0.3318 3.5 11.1056 5.5500 0.4283 11.1073 5.5511 0.4283 11.1086 5.5520 0.4283 4.0 11.3326 5.6658 0.5000 11.3328 5.6659 0.5000 11.3330 5.6660 0.5000 N-policy D/M/1/12 queue with working breakdowns 2.0 4.2456 2.1731 0.0231 5.5749 2.8737 0.0300 6.5268 3.3853 0.0360 2.5 9.8235 4.8824 0.1952 9.9222 4.9402 0.1966 9.9880 4.9785 0.1975 3.0 10.9912 5.4928 0.3330 10.9942 5.4947 0.3330 10.9965 5.4961 0.3331 3.5 11.3328 5.6662 0.4285 11.3330 5.6663 0.4285 11.3331 5.6663 0.4285 4.0 11.5000 5.7500 0.5000 11.5000 5.7500 0.5000 11.5000 5.7500 0.5000 Table 2. Performance measures for various values of μb under three different inter-arrival time distributions (Case 2). μb N=3 N=6 N=9 LS WS PL LS WS PL LS WS PL N-policy M/M/1/12 queue with working breakdowns 4.5 3.9489 2.0545 0.0389 5.2316 2.7516 0.0494 6.2436 3.3216 0.0602 5.0 3.7108 1.9244 0.0359 5.0466 2.6453 0.0461 6.1066 3.2358 0.0564 5.5 3.5116 1.8163 0.0333 4.8888 2.5552 0.0434 5.9882 3.1625 0.0532 6.0 3.3417 1.7245 0.0311 4.7517 2.4774 0.0410 5.8844 3.0986 0.0505 6.5 3.1948 1.6454 0.0292 4.6312 2.4093 0.0389 5.7923 3.0423 0.0480 N-policy E3/M/1/12 queue with working breakdowns 4.5 4.2058 2.1673 0.0297 5.5060 2.8618 0.0380 6.4722 3.3908 0.0456 5.0 3.9893 2.0514 0.0277 5.3458 2.7727 0.0360 6.3542 3.3214 0.0434 5.5 3.8044 1.9528 0.0259 5.2068 2.6956 0.0342 6.2517 3.2612 0.0415 6.0 3.6438 1.8674 0.0244 5.0839 2.6277 0.0326 6.1607 3.2080 0.0398 6.5 3.5023 1.7924 0.0231 4.9740 2.5670 0.0312 6.0790 3.1604 0.0382 N-policy D/M/1/12 queue with working breakdowns 4.5 4.4445 2.2783 0.0246 5.7168 2.9512 0.0314 6.6291 3.4437 0.0375 5.0 4.2456 2.1731 0.0231 5.5749 2.8737 0.0300 6.5268 3.3853 0.0360 5.5 4.0728 2.0818 0.0218 5.4498 2.8054 0.0287 6.4361 3.3334 0.0346 6.0 3.9200 2.0014 0.0207 5.3377 2.7444 0.0275 6.3001 3.2596 0.0336 6.5 3.7832 1.9295 0.0196 5.2360 2.6892 0.0265 6.0449 3.1244 0.0326 μb N=3 N=6 N=9 LS WS PL LS WS PL LS WS PL N-policy M/M/1/12 queue with working breakdowns 4.5 3.9489 2.0545 0.0389 5.2316 2.7516 0.0494 6.2436 3.3216 0.0602 5.0 3.7108 1.9244 0.0359 5.0466 2.6453 0.0461 6.1066 3.2358 0.0564 5.5 3.5116 1.8163 0.0333 4.8888 2.5552 0.0434 5.9882 3.1625 0.0532 6.0 3.3417 1.7245 0.0311 4.7517 2.4774 0.0410 5.8844 3.0986 0.0505 6.5 3.1948 1.6454 0.0292 4.6312 2.4093 0.0389 5.7923 3.0423 0.0480 N-policy E3/M/1/12 queue with working breakdowns 4.5 4.2058 2.1673 0.0297 5.5060 2.8618 0.0380 6.4722 3.3908 0.0456 5.0 3.9893 2.0514 0.0277 5.3458 2.7727 0.0360 6.3542 3.3214 0.0434 5.5 3.8044 1.9528 0.0259 5.2068 2.6956 0.0342 6.2517 3.2612 0.0415 6.0 3.6438 1.8674 0.0244 5.0839 2.6277 0.0326 6.1607 3.2080 0.0398 6.5 3.5023 1.7924 0.0231 4.9740 2.5670 0.0312 6.0790 3.1604 0.0382 N-policy D/M/1/12 queue with working breakdowns 4.5 4.4445 2.2783 0.0246 5.7168 2.9512 0.0314 6.6291 3.4437 0.0375 5.0 4.2456 2.1731 0.0231 5.5749 2.8737 0.0300 6.5268 3.3853 0.0360 5.5 4.0728 2.0818 0.0218 5.4498 2.8054 0.0287 6.4361 3.3334 0.0346 6.0 3.9200 2.0014 0.0207 5.3377 2.7444 0.0275 6.3001 3.2596 0.0336 6.5 3.7832 1.9295 0.0196 5.2360 2.6892 0.0265 6.0449 3.1244 0.0326 Table 2. Performance measures for various values of μb under three different inter-arrival time distributions (Case 2). μb N=3 N=6 N=9 LS WS PL LS WS PL LS WS PL N-policy M/M/1/12 queue with working breakdowns 4.5 3.9489 2.0545 0.0389 5.2316 2.7516 0.0494 6.2436 3.3216 0.0602 5.0 3.7108 1.9244 0.0359 5.0466 2.6453 0.0461 6.1066 3.2358 0.0564 5.5 3.5116 1.8163 0.0333 4.8888 2.5552 0.0434 5.9882 3.1625 0.0532 6.0 3.3417 1.7245 0.0311 4.7517 2.4774 0.0410 5.8844 3.0986 0.0505 6.5 3.1948 1.6454 0.0292 4.6312 2.4093 0.0389 5.7923 3.0423 0.0480 N-policy E3/M/1/12 queue with working breakdowns 4.5 4.2058 2.1673 0.0297 5.5060 2.8618 0.0380 6.4722 3.3908 0.0456 5.0 3.9893 2.0514 0.0277 5.3458 2.7727 0.0360 6.3542 3.3214 0.0434 5.5 3.8044 1.9528 0.0259 5.2068 2.6956 0.0342 6.2517 3.2612 0.0415 6.0 3.6438 1.8674 0.0244 5.0839 2.6277 0.0326 6.1607 3.2080 0.0398 6.5 3.5023 1.7924 0.0231 4.9740 2.5670 0.0312 6.0790 3.1604 0.0382 N-policy D/M/1/12 queue with working breakdowns 4.5 4.4445 2.2783 0.0246 5.7168 2.9512 0.0314 6.6291 3.4437 0.0375 5.0 4.2456 2.1731 0.0231 5.5749 2.8737 0.0300 6.5268 3.3853 0.0360 5.5 4.0728 2.0818 0.0218 5.4498 2.8054 0.0287 6.4361 3.3334 0.0346 6.0 3.9200 2.0014 0.0207 5.3377 2.7444 0.0275 6.3001 3.2596 0.0336 6.5 3.7832 1.9295 0.0196 5.2360 2.6892 0.0265 6.0449 3.1244 0.0326 μb N=3 N=6 N=9 LS WS PL LS WS PL LS WS PL N-policy M/M/1/12 queue with working breakdowns 4.5 3.9489 2.0545 0.0389 5.2316 2.7516 0.0494 6.2436 3.3216 0.0602 5.0 3.7108 1.9244 0.0359 5.0466 2.6453 0.0461 6.1066 3.2358 0.0564 5.5 3.5116 1.8163 0.0333 4.8888 2.5552 0.0434 5.9882 3.1625 0.0532 6.0 3.3417 1.7245 0.0311 4.7517 2.4774 0.0410 5.8844 3.0986 0.0505 6.5 3.1948 1.6454 0.0292 4.6312 2.4093 0.0389 5.7923 3.0423 0.0480 N-policy E3/M/1/12 queue with working breakdowns 4.5 4.2058 2.1673 0.0297 5.5060 2.8618 0.0380 6.4722 3.3908 0.0456 5.0 3.9893 2.0514 0.0277 5.3458 2.7727 0.0360 6.3542 3.3214 0.0434 5.5 3.8044 1.9528 0.0259 5.2068 2.6956 0.0342 6.2517 3.2612 0.0415 6.0 3.6438 1.8674 0.0244 5.0839 2.6277 0.0326 6.1607 3.2080 0.0398 6.5 3.5023 1.7924 0.0231 4.9740 2.5670 0.0312 6.0790 3.1604 0.0382 N-policy D/M/1/12 queue with working breakdowns 4.5 4.4445 2.2783 0.0246 5.7168 2.9512 0.0314 6.6291 3.4437 0.0375 5.0 4.2456 2.1731 0.0231 5.5749 2.8737 0.0300 6.5268 3.3853 0.0360 5.5 4.0728 2.0818 0.0218 5.4498 2.8054 0.0287 6.4361 3.3334 0.0346 6.0 3.9200 2.0014 0.0207 5.3377 2.7444 0.0275 6.3001 3.2596 0.0336 6.5 3.7832 1.9295 0.0196 5.2360 2.6892 0.0265 6.0449 3.1244 0.0326 Table 3. Performance measures for various values of μd under three different inter-arrival time distributions (Case 3). μd N=3 N=6 N=9 LS WS PL LS WS PL LS WS PL N-policy M/M/1/12 queue with working breakdowns 1.0 10.9747 10.9444 0.4986 10.9819 10.9552 0.4988 10.9873 10.9625 0.4989 1.5 8.2923 5.3272 0.2217 8.6951 5.6400 0.2292 8.9664 5.8510 0.2338 2.0 3.7108 1.9244 0.0359 5.0466 2.6453 0.0461 6.1066 3.2358 0.0564 2.5 2.1947 1.1022 0.0044 3.6783 1.8534 0.0077 5.0314 2.5520 0.0142 3.0 1.8434 0.9224 0.0007 3.3246 1.6654 0.0019 4.7448 2.3870 0.0061 N-policy E3/M/1/12 queue with working breakdowns 1.0 11.3320 11.3305 0.4999 11.3324 11.3310 0.4999 11.3326 11.3313 0.4999 1.5 9.7290 6.4114 0.2413 9.8601 6.5177 0.2436 9.9440 6.5844 0.2449 2.0 3.9893 2.0514 0.0277 5.3458 2.7727 0.0360 6.3542 3.3214 0.0434 2.5 1.9990 1.0006 0.0011 3.5206 1.7644 0.0023 4.9169 2.4710 0.0051 3.0 1.7179 0.8590 0.0001 3.2167 1.6087 0.0002 4.6628 2.3341 0.0012 N-policy D/M/1/12 queue with working breakdowns 1.0 11.4999 11.4998 0.5000 11.4999 11.4999 0.5000 11.5000 11.4999 0.5000 1.5 10.4179 6.9228 0.2476 10.4540 6.9524 0.2482 10.4777 6.9713 0.2485 2.0 4.2456 2.1731 0.0231 5.5749 2.8737 0.0300 6.5268 3.3853 0.0360 2.5 1.8920 0.9463 0.0003 3.4030 1.7028 0.0007 4.8116 2.4109 0.0021 3.0 1.6769 0.8385 0.0000 3.1629 1.5815 0.0000 4.6095 2.3054 0.0003 μd N=3 N=6 N=9 LS WS PL LS WS PL LS WS PL N-policy M/M/1/12 queue with working breakdowns 1.0 10.9747 10.9444 0.4986 10.9819 10.9552 0.4988 10.9873 10.9625 0.4989 1.5 8.2923 5.3272 0.2217 8.6951 5.6400 0.2292 8.9664 5.8510 0.2338 2.0 3.7108 1.9244 0.0359 5.0466 2.6453 0.0461 6.1066 3.2358 0.0564 2.5 2.1947 1.1022 0.0044 3.6783 1.8534 0.0077 5.0314 2.5520 0.0142 3.0 1.8434 0.9224 0.0007 3.3246 1.6654 0.0019 4.7448 2.3870 0.0061 N-policy E3/M/1/12 queue with working breakdowns 1.0 11.3320 11.3305 0.4999 11.3324 11.3310 0.4999 11.3326 11.3313 0.4999 1.5 9.7290 6.4114 0.2413 9.8601 6.5177 0.2436 9.9440 6.5844 0.2449 2.0 3.9893 2.0514 0.0277 5.3458 2.7727 0.0360 6.3542 3.3214 0.0434 2.5 1.9990 1.0006 0.0011 3.5206 1.7644 0.0023 4.9169 2.4710 0.0051 3.0 1.7179 0.8590 0.0001 3.2167 1.6087 0.0002 4.6628 2.3341 0.0012 N-policy D/M/1/12 queue with working breakdowns 1.0 11.4999 11.4998 0.5000 11.4999 11.4999 0.5000 11.5000 11.4999 0.5000 1.5 10.4179 6.9228 0.2476 10.4540 6.9524 0.2482 10.4777 6.9713 0.2485 2.0 4.2456 2.1731 0.0231 5.5749 2.8737 0.0300 6.5268 3.3853 0.0360 2.5 1.8920 0.9463 0.0003 3.4030 1.7028 0.0007 4.8116 2.4109 0.0021 3.0 1.6769 0.8385 0.0000 3.1629 1.5815 0.0000 4.6095 2.3054 0.0003 Table 3. Performance measures for various values of μd under three different inter-arrival time distributions (Case 3). μd N=3 N=6 N=9 LS WS PL LS WS PL LS WS PL N-policy M/M/1/12 queue with working breakdowns 1.0 10.9747 10.9444 0.4986 10.9819 10.9552 0.4988 10.9873 10.9625 0.4989 1.5 8.2923 5.3272 0.2217 8.6951 5.6400 0.2292 8.9664 5.8510 0.2338 2.0 3.7108 1.9244 0.0359 5.0466 2.6453 0.0461 6.1066 3.2358 0.0564 2.5 2.1947 1.1022 0.0044 3.6783 1.8534 0.0077 5.0314 2.5520 0.0142 3.0 1.8434 0.9224 0.0007 3.3246 1.6654 0.0019 4.7448 2.3870 0.0061 N-policy E3/M/1/12 queue with working breakdowns 1.0 11.3320 11.3305 0.4999 11.3324 11.3310 0.4999 11.3326 11.3313 0.4999 1.5 9.7290 6.4114 0.2413 9.8601 6.5177 0.2436 9.9440 6.5844 0.2449 2.0 3.9893 2.0514 0.0277 5.3458 2.7727 0.0360 6.3542 3.3214 0.0434 2.5 1.9990 1.0006 0.0011 3.5206 1.7644 0.0023 4.9169 2.4710 0.0051 3.0 1.7179 0.8590 0.0001 3.2167 1.6087 0.0002 4.6628 2.3341 0.0012 N-policy D/M/1/12 queue with working breakdowns 1.0 11.4999 11.4998 0.5000 11.4999 11.4999 0.5000 11.5000 11.4999 0.5000 1.5 10.4179 6.9228 0.2476 10.4540 6.9524 0.2482 10.4777 6.9713 0.2485 2.0 4.2456 2.1731 0.0231 5.5749 2.8737 0.0300 6.5268 3.3853 0.0360 2.5 1.8920 0.9463 0.0003 3.4030 1.7028 0.0007 4.8116 2.4109 0.0021 3.0 1.6769 0.8385 0.0000 3.1629 1.5815 0.0000 4.6095 2.3054 0.0003 μd N=3 N=6 N=9 LS WS PL LS WS PL LS WS PL N-policy M/M/1/12 queue with working breakdowns 1.0 10.9747 10.9444 0.4986 10.9819 10.9552 0.4988 10.9873 10.9625 0.4989 1.5 8.2923 5.3272 0.2217 8.6951 5.6400 0.2292 8.9664 5.8510 0.2338 2.0 3.7108 1.9244 0.0359 5.0466 2.6453 0.0461 6.1066 3.2358 0.0564 2.5 2.1947 1.1022 0.0044 3.6783 1.8534 0.0077 5.0314 2.5520 0.0142 3.0 1.8434 0.9224 0.0007 3.3246 1.6654 0.0019 4.7448 2.3870 0.0061 N-policy E3/M/1/12 queue with working breakdowns 1.0 11.3320 11.3305 0.4999 11.3324 11.3310 0.4999 11.3326 11.3313 0.4999 1.5 9.7290 6.4114 0.2413 9.8601 6.5177 0.2436 9.9440 6.5844 0.2449 2.0 3.9893 2.0514 0.0277 5.3458 2.7727 0.0360 6.3542 3.3214 0.0434 2.5 1.9990 1.0006 0.0011 3.5206 1.7644 0.0023 4.9169 2.4710 0.0051 3.0 1.7179 0.8590 0.0001 3.2167 1.6087 0.0002 4.6628 2.3341 0.0012 N-policy D/M/1/12 queue with working breakdowns 1.0 11.4999 11.4998 0.5000 11.4999 11.4999 0.5000 11.5000 11.4999 0.5000 1.5 10.4179 6.9228 0.2476 10.4540 6.9524 0.2482 10.4777 6.9713 0.2485 2.0 4.2456 2.1731 0.0231 5.5749 2.8737 0.0300 6.5268 3.3853 0.0360 2.5 1.8920 0.9463 0.0003 3.4030 1.7028 0.0007 4.8116 2.4109 0.0021 3.0 1.6769 0.8385 0.0000 3.1629 1.5815 0.0000 4.6095 2.3054 0.0003 Table 4. Performance measures for various values of α under three different inter-arrival time distributions (Case 4). α N=3 N=6 N=9 LS WS PL LS WS PL LS WS PL N-policy M/M/1/12 queue with working breakdowns 0.05 2.9714 1.5206 0.0229 4.4243 2.2828 0.0310 5.6524 2.9431 0.0397 0.1 3.7108 1.9244 0.0359 5.0466 2.6453 0.0461 6.1066 3.2358 0.0564 0.2 4.5188 2.3783 0.0500 5.6657 3.0175 0.0612 6.5381 3.5236 0.0722 0.3 4.9514 2.6269 0.0576 5.9753 3.2081 0.0687 6.7498 3.6683 0.0800 0.4 5.2209 2.7837 0.0622 6.1621 3.3246 0.0732 6.8779 3.7570 0.0847 N-policy E3/M/1/12 queue with working breakdowns 0.05 3.1823 1.6207 0.0182 4.6832 2.4021 0.0252 5.8798 3.0355 0.0315 0.1 3.9893 2.0514 0.0277 5.3458 2.7727 0.0360 6.3542 3.3214 0.0434 0.2 4.8141 2.5003 0.0373 5.9514 3.1187 0.0459 6.7673 3.5761 0.0538 0.3 5.2334 2.7319 0.0422 6.2372 3.2845 0.0505 6.9597 3.6965 0.0586 0.4 5.4871 2.8732 0.0451 6.4049 3.3825 0.0532 7.0742 3.7688 0.0615 N-policy D/M/1/12 queue with working breakdowns 0.05 3.4044 1.7294 0.0157 4.9050 2.5070 0.0218 6.0551 3.1117 0.0270 0.1 4.2456 2.1731 0.0231 5.5749 2.8737 0.0300 6.5268 3.3853 0.0360 0.2 5.0554 2.6066 0.0303 6.1479 3.1922 0.0370 6.9078 3.6101 0.0433 0.3 5.4490 2.8196 0.0337 6.4070 3.3377 0.0402 7.0763 3.7107 0.0465 0.4 5.6817 2.9462 0.0358 6.5563 3.4220 0.0420 7.2052 3.7867 0.0486 α N=3 N=6 N=9 LS WS PL LS WS PL LS WS PL N-policy M/M/1/12 queue with working breakdowns 0.05 2.9714 1.5206 0.0229 4.4243 2.2828 0.0310 5.6524 2.9431 0.0397 0.1 3.7108 1.9244 0.0359 5.0466 2.6453 0.0461 6.1066 3.2358 0.0564 0.2 4.5188 2.3783 0.0500 5.6657 3.0175 0.0612 6.5381 3.5236 0.0722 0.3 4.9514 2.6269 0.0576 5.9753 3.2081 0.0687 6.7498 3.6683 0.0800 0.4 5.2209 2.7837 0.0622 6.1621 3.3246 0.0732 6.8779 3.7570 0.0847 N-policy E3/M/1/12 queue with working breakdowns 0.05 3.1823 1.6207 0.0182 4.6832 2.4021 0.0252 5.8798 3.0355 0.0315 0.1 3.9893 2.0514 0.0277 5.3458 2.7727 0.0360 6.3542 3.3214 0.0434 0.2 4.8141 2.5003 0.0373 5.9514 3.1187 0.0459 6.7673 3.5761 0.0538 0.3 5.2334 2.7319 0.0422 6.2372 3.2845 0.0505 6.9597 3.6965 0.0586 0.4 5.4871 2.8732 0.0451 6.4049 3.3825 0.0532 7.0742 3.7688 0.0615 N-policy D/M/1/12 queue with working breakdowns 0.05 3.4044 1.7294 0.0157 4.9050 2.5070 0.0218 6.0551 3.1117 0.0270 0.1 4.2456 2.1731 0.0231 5.5749 2.8737 0.0300 6.5268 3.3853 0.0360 0.2 5.0554 2.6066 0.0303 6.1479 3.1922 0.0370 6.9078 3.6101 0.0433 0.3 5.4490 2.8196 0.0337 6.4070 3.3377 0.0402 7.0763 3.7107 0.0465 0.4 5.6817 2.9462 0.0358 6.5563 3.4220 0.0420 7.2052 3.7867 0.0486 Table 4. Performance measures for various values of α under three different inter-arrival time distributions (Case 4). α N=3 N=6 N=9 LS WS PL LS WS PL LS WS PL N-policy M/M/1/12 queue with working breakdowns 0.05 2.9714 1.5206 0.0229 4.4243 2.2828 0.0310 5.6524 2.9431 0.0397 0.1 3.7108 1.9244 0.0359 5.0466 2.6453 0.0461 6.1066 3.2358 0.0564 0.2 4.5188 2.3783 0.0500 5.6657 3.0175 0.0612 6.5381 3.5236 0.0722 0.3 4.9514 2.6269 0.0576 5.9753 3.2081 0.0687 6.7498 3.6683 0.0800 0.4 5.2209 2.7837 0.0622 6.1621 3.3246 0.0732 6.8779 3.7570 0.0847 N-policy E3/M/1/12 queue with working breakdowns 0.05 3.1823 1.6207 0.0182 4.6832 2.4021 0.0252 5.8798 3.0355 0.0315 0.1 3.9893 2.0514 0.0277 5.3458 2.7727 0.0360 6.3542 3.3214 0.0434 0.2 4.8141 2.5003 0.0373 5.9514 3.1187 0.0459 6.7673 3.5761 0.0538 0.3 5.2334 2.7319 0.0422 6.2372 3.2845 0.0505 6.9597 3.6965 0.0586 0.4 5.4871 2.8732 0.0451 6.4049 3.3825 0.0532 7.0742 3.7688 0.0615 N-policy D/M/1/12 queue with working breakdowns 0.05 3.4044 1.7294 0.0157 4.9050 2.5070 0.0218 6.0551 3.1117 0.0270 0.1 4.2456 2.1731 0.0231 5.5749 2.8737 0.0300 6.5268 3.3853 0.0360 0.2 5.0554 2.6066 0.0303 6.1479 3.1922 0.0370 6.9078 3.6101 0.0433 0.3 5.4490 2.8196 0.0337 6.4070 3.3377 0.0402 7.0763 3.7107 0.0465 0.4 5.6817 2.9462 0.0358 6.5563 3.4220 0.0420 7.2052 3.7867 0.0486 α N=3 N=6 N=9 LS WS PL LS WS PL LS WS PL N-policy M/M/1/12 queue with working breakdowns 0.05 2.9714 1.5206 0.0229 4.4243 2.2828 0.0310 5.6524 2.9431 0.0397 0.1 3.7108 1.9244 0.0359 5.0466 2.6453 0.0461 6.1066 3.2358 0.0564 0.2 4.5188 2.3783 0.0500 5.6657 3.0175 0.0612 6.5381 3.5236 0.0722 0.3 4.9514 2.6269 0.0576 5.9753 3.2081 0.0687 6.7498 3.6683 0.0800 0.4 5.2209 2.7837 0.0622 6.1621 3.3246 0.0732 6.8779 3.7570 0.0847 N-policy E3/M/1/12 queue with working breakdowns 0.05 3.1823 1.6207 0.0182 4.6832 2.4021 0.0252 5.8798 3.0355 0.0315 0.1 3.9893 2.0514 0.0277 5.3458 2.7727 0.0360 6.3542 3.3214 0.0434 0.2 4.8141 2.5003 0.0373 5.9514 3.1187 0.0459 6.7673 3.5761 0.0538 0.3 5.2334 2.7319 0.0422 6.2372 3.2845 0.0505 6.9597 3.6965 0.0586 0.4 5.4871 2.8732 0.0451 6.4049 3.3825 0.0532 7.0742 3.7688 0.0615 N-policy D/M/1/12 queue with working breakdowns 0.05 3.4044 1.7294 0.0157 4.9050 2.5070 0.0218 6.0551 3.1117 0.0270 0.1 4.2456 2.1731 0.0231 5.5749 2.8737 0.0300 6.5268 3.3853 0.0360 0.2 5.0554 2.6066 0.0303 6.1479 3.1922 0.0370 6.9078 3.6101 0.0433 0.3 5.4490 2.8196 0.0337 6.4070 3.3377 0.0402 7.0763 3.7107 0.0465 0.4 5.6817 2.9462 0.0358 6.5563 3.4220 0.0420 7.2052 3.7867 0.0486 Table 5. Performance measures for various values of β under three different inter-arrival time distributions (Case 5). β N=3 N=6 N=9 LS WS PL LS WS PL LS WS PL N-policy M/M/1/12 queue with working breakdowns 2.5 3.8054 1.9770 0.0376 5.0802 2.6659 0.0472 6.1033 3.2363 0.0570 3.0 3.7659 1.9550 0.0369 5.0663 2.6574 0.0468 6.1046 3.2361 0.0568 3.5 3.7347 1.9377 0.0363 5.0552 2.6506 0.0464 6.1057 3.2359 0.0566 4.0 3.7108 1.9244 0.0359 5.0466 2.6453 0.0461 6.1066 3.2358 0.0564 4.5 3.6922 1.9141 0.0355 5.0409 2.6418 0.0459 6.1060 3.2359 0.0565 N-policy E3/M/1/12 queue with working breakdowns 2.5 4.1029 2.1128 0.0290 5.3791 2.7921 0.0367 6.3450 3.3178 0.0438 3.0 4.0557 2.0873 0.0285 5.3655 2.7842 0.0364 6.3487 3.3192 0.0437 3.5 4.0183 2.0670 0.0280 5.3544 2.7777 0.0362 6.3518 3.3205 0.0435 4.0 3.9893 2.0514 0.0277 5.3458 2.7727 0.0360 6.3542 3.3214 0.0434 4.5 3.9662 2.0389 0.0274 5.3391 2.7688 0.0358 6.3556 3.3219 0.0434 N-policy D/M/1/12 queue with working breakdowns 2.5 4.3695 2.2391 0.0243 5.6052 2.8909 0.0305 6.5078 3.3761 0.0362 3.0 4.3172 2.2112 0.0238 5.5924 2.8836 0.0303 6.5194 3.3819 0.0361 3.5 4.2770 2.1897 0.0234 5.5826 2.8780 0.0301 6.5226 3.3833 0.0361 4.0 4.2456 2.1731 0.0231 5.5749 2.8737 0.0300 6.5268 3.3853 0.0360 4.5 4.2205 2.1597 0.0229 5.5689 2.8702 0.0299 6.5302 3.3869 0.0360 β N=3 N=6 N=9 LS WS PL LS WS PL LS WS PL N-policy M/M/1/12 queue with working breakdowns 2.5 3.8054 1.9770 0.0376 5.0802 2.6659 0.0472 6.1033 3.2363 0.0570 3.0 3.7659 1.9550 0.0369 5.0663 2.6574 0.0468 6.1046 3.2361 0.0568 3.5 3.7347 1.9377 0.0363 5.0552 2.6506 0.0464 6.1057 3.2359 0.0566 4.0 3.7108 1.9244 0.0359 5.0466 2.6453 0.0461 6.1066 3.2358 0.0564 4.5 3.6922 1.9141 0.0355 5.0409 2.6418 0.0459 6.1060 3.2359 0.0565 N-policy E3/M/1/12 queue with working breakdowns 2.5 4.1029 2.1128 0.0290 5.3791 2.7921 0.0367 6.3450 3.3178 0.0438 3.0 4.0557 2.0873 0.0285 5.3655 2.7842 0.0364 6.3487 3.3192 0.0437 3.5 4.0183 2.0670 0.0280 5.3544 2.7777 0.0362 6.3518 3.3205 0.0435 4.0 3.9893 2.0514 0.0277 5.3458 2.7727 0.0360 6.3542 3.3214 0.0434 4.5 3.9662 2.0389 0.0274 5.3391 2.7688 0.0358 6.3556 3.3219 0.0434 N-policy D/M/1/12 queue with working breakdowns 2.5 4.3695 2.2391 0.0243 5.6052 2.8909 0.0305 6.5078 3.3761 0.0362 3.0 4.3172 2.2112 0.0238 5.5924 2.8836 0.0303 6.5194 3.3819 0.0361 3.5 4.2770 2.1897 0.0234 5.5826 2.8780 0.0301 6.5226 3.3833 0.0361 4.0 4.2456 2.1731 0.0231 5.5749 2.8737 0.0300 6.5268 3.3853 0.0360 4.5 4.2205 2.1597 0.0229 5.5689 2.8702 0.0299 6.5302 3.3869 0.0360 Table 5. Performance measures for various values of β under three different inter-arrival time distributions (Case 5). β N=3 N=6 N=9 LS WS PL LS WS PL LS WS PL N-policy M/M/1/12 queue with working breakdowns 2.5 3.8054 1.9770 0.0376 5.0802 2.6659 0.0472 6.1033 3.2363 0.0570 3.0 3.7659 1.9550 0.0369 5.0663 2.6574 0.0468 6.1046 3.2361 0.0568 3.5 3.7347 1.9377 0.0363 5.0552 2.6506 0.0464 6.1057 3.2359 0.0566 4.0 3.7108 1.9244 0.0359 5.0466 2.6453 0.0461 6.1066 3.2358 0.0564 4.5 3.6922 1.9141 0.0355 5.0409 2.6418 0.0459 6.1060 3.2359 0.0565 N-policy E3/M/1/12 queue with working breakdowns 2.5 4.1029 2.1128 0.0290 5.3791 2.7921 0.0367 6.3450 3.3178 0.0438 3.0 4.0557 2.0873 0.0285 5.3655 2.7842 0.0364 6.3487 3.3192 0.0437 3.5 4.0183 2.0670 0.0280 5.3544 2.7777 0.0362 6.3518 3.3205 0.0435 4.0 3.9893 2.0514 0.0277 5.3458 2.7727 0.0360 6.3542 3.3214 0.0434 4.5 3.9662 2.0389 0.0274 5.3391 2.7688 0.0358 6.3556 3.3219 0.0434 N-policy D/M/1/12 queue with working breakdowns 2.5 4.3695 2.2391 0.0243 5.6052 2.8909 0.0305 6.5078 3.3761 0.0362 3.0 4.3172 2.2112 0.0238 5.5924 2.8836 0.0303 6.5194 3.3819 0.0361 3.5 4.2770 2.1897 0.0234 5.5826 2.8780 0.0301 6.5226 3.3833 0.0361 4.0 4.2456 2.1731 0.0231 5.5749 2.8737 0.0300 6.5268 3.3853 0.0360 4.5 4.2205 2.1597 0.0229 5.5689 2.8702 0.0299 6.5302 3.3869 0.0360 β N=3 N=6 N=9 LS WS PL LS WS PL LS WS PL N-policy M/M/1/12 queue with working breakdowns 2.5 3.8054 1.9770 0.0376 5.0802 2.6659 0.0472 6.1033 3.2363 0.0570 3.0 3.7659 1.9550 0.0369 5.0663 2.6574 0.0468 6.1046 3.2361 0.0568 3.5 3.7347 1.9377 0.0363 5.0552 2.6506 0.0464 6.1057 3.2359 0.0566 4.0 3.7108 1.9244 0.0359 5.0466 2.6453 0.0461 6.1066 3.2358 0.0564 4.5 3.6922 1.9141 0.0355 5.0409 2.6418 0.0459 6.1060 3.2359 0.0565 N-policy E3/M/1/12 queue with working breakdowns 2.5 4.1029 2.1128 0.0290 5.3791 2.7921 0.0367 6.3450 3.3178 0.0438 3.0 4.0557 2.0873 0.0285 5.3655 2.7842 0.0364 6.3487 3.3192 0.0437 3.5 4.0183 2.0670 0.0280 5.3544 2.7777 0.0362 6.3518 3.3205 0.0435 4.0 3.9893 2.0514 0.0277 5.3458 2.7727 0.0360 6.3542 3.3214 0.0434 4.5 3.9662 2.0389 0.0274 5.3391 2.7688 0.0358 6.3556 3.3219 0.0434 N-policy D/M/1/12 queue with working breakdowns 2.5 4.3695 2.2391 0.0243 5.6052 2.8909 0.0305 6.5078 3.3761 0.0362 3.0 4.3172 2.2112 0.0238 5.5924 2.8836 0.0303 6.5194 3.3819 0.0361 3.5 4.2770 2.1897 0.0234 5.5826 2.8780 0.0301 6.5226 3.3833 0.0361 4.0 4.2456 2.1731 0.0231 5.5749 2.8737 0.0300 6.5268 3.3853 0.0360 4.5 4.2205 2.1597 0.0229 5.5689 2.8702 0.0299 6.5302 3.3869 0.0360 7. CONCLUSIONS In this paper, we used the supplementary variable method to analyze the N-policy GI/M/1/K queue with working breakdowns. We proposed an algorithm for computing the steady-state probabilities at pre-arrival epochs. The steady-state probabilities at arbitrary epochs can be derived using the relationship between pre-arrival and arbitrary epoch probabilities. We also developed system performance measures. Numerical examples were performed to enable sensitivity analysis of the system performance measures. Our findings in this study could be useful in computer, telecommunication, production and manufacturing systems. In future work, it would be interesting to extend the proposed queueing model to include single (multiple) vacation(s) and impatient customers. REFERENCES 1 Yadin , M. and Naor , P. ( 1963 ) Queueing systems with a removable service station . Oper. Res. Q. , 14 , 393 – 405 . Google Scholar CrossRef Search ADS 2 Wang , K.-H. and Ke , J.-C. ( 2000 ) A recursive method to the optimal control of an M/G/1 queueing system with finite capacity and infinite capacity . Appl. Math. Model. , 24 , 899 – 914 . Google Scholar CrossRef Search ADS 3 Ke , J.-C. and Wang , K.-H. ( 2002 ) A recursive method for the N policy G/M/1 queueing system with finite capacity . Eur. J. Oper. Res. , 142 , 577 – 594 . Google Scholar CrossRef Search ADS 4 Jain , M. , Kulshrestha , R. and Maheshwari , S. ( 2004 ) N-policy for a machine repair system with spares and reneging . Appl. Math. Model. , 28 , 513 – 531 . Google Scholar CrossRef Search ADS 5 Tadj , L. and Choudhury , G. ( 2005 ) Optimal design and control of queues . Top , 13 , 359 – 412 . Google Scholar CrossRef Search ADS 6 Wang , T.-Y. , Yang , D.-Y. and Li , M.-J. ( 2010 ) Fuzzy analysis for the N-policy queues with infinite capacity . Int. J. Inf. Manag. Sci , 21 , 41 – 56 . 7 Vijaya Laxmi , P. and Suchitra , V. ( 2014 ) Finite buffer GI/M(n)/1 queue with Bernoulli-schedule vacation interruption under N-Policy . Int. Sch. Res. Notices , 2014 , 392317 . http://dx.doi.org/10.1155/2014/392317 . Google Scholar PubMed 8 Haridass , M. and Arumuganathan , R. ( 2015 ) Analysis of a single server batch arrival retrial queueing system with modified vacations and N-policy . Rairo-Oper. Res. , 49 , 279 – 296 . Google Scholar CrossRef Search ADS 9 Wang , J. , Zhang , X. and Huang , P. ( 2017 ) Strategic behavior and social optimization in a constant retrial queue with the N-policy . Eur. J. Oper. Res. , 256 , 841 – 849 . Google Scholar CrossRef Search ADS 10 Wang , K.-H. and Ke , J.-C. ( 2002 ) Control policies of an M/G/1 queueing system with a removable and non‐reliable server . Int. Trans. Oper. Res. , 9 , 195 – 212 . Google Scholar CrossRef Search ADS 11 Pearn , W.L. , Ke , J.-C. and Chang , Y.-C. ( 2004 ) Sensitivity analysis of the optimal management policy for a queuing system with a removable and non-reliable server . Comput. Ind. Eng. , 46 , 87 – 99 . Google Scholar CrossRef Search ADS 12 Choudhury , G. , Ke , J.-C. and Tadj , L. ( 2009 ) The N-policy for an unreliable server with delaying repair and two phases of service . J. Comput. Appl. Math. , 231 , 349 – 364 . Google Scholar CrossRef Search ADS 13 Choudhury , G. and Tadj , L. ( 2011 ) The optimal control of an Mx/G/1 unreliable server queue with two phases of service and Bernoulli vacation schedule . Math. Comput. Model. , 54 , 673 – 688 . Google Scholar CrossRef Search ADS 14 Singh , C.J. , Jain , M. and Kumar , B. ( 2013 ) Analysis of queue with two phases of service and m phases of repair for server breakdown under N-policy . Int. J. Serv. Oper. Manag. , 16 , 373 – 406 . 15 Jayachitra , P. and Albert , A.J. ( 2014 ) Recent developments in queueing models under N-policy: a short survey . Int. J. Math. Arch , 5 , 227 – 233 . 16 Yang , D.-Y. and Wu , C.-H. ( 2015 ) Cost-minimization analysis of a working vacation queue with N-policy and server breakdowns . Comput. Ind. Eng. , 82 , 151 – 158 . Google Scholar CrossRef Search ADS 17 Jain , M. , Sharma , G.C. and Rani , V. ( 2016 ) Maximum entropy approach for an unreliable MX/G/1 queueing system with Bernoulli vacation, restricted admission and delayed phase repair under N-policy . Int. J. Serv. Oper. Manag , 23 , 407 – 442 . 18 Wang , T.-Y. , Liu , T.-H. and Chang , F.-M. ( 2017 ) Analysis of a random N-policy Geo/G/1 queue with the server subject to repairable breakdowns . J. Ind. Prod. Eng. , 34 , 19 – 29 . 19 Kalidass , K. and Kasturi , R. ( 2012 ) A queue with working breakdowns . Comput. Ind. Eng. , 63 , 779 – 783 . Google Scholar CrossRef Search ADS 20 Kim , B.K. and Lee , D.H. ( 2014 ) The M/G/1 queue with disasters and working breakdowns . Appl. Math. Model. , 38 , 1788 – 1798 . Google Scholar CrossRef Search ADS 21 Liu , Z. and Song , Y. ( 2014 ) The MX/M/1 queue with working breakdown . Rairo-Oper. Res. , 48 , 399 – 413 . Google Scholar CrossRef Search ADS 22 Liou , C.-D. ( 2015 ) Markovian queue optimisation analysis with an unreliable server subject to working breakdowns and impatient customers . Int. J. Syst. Sci. , 46 , 2165 – 2182 . Google Scholar CrossRef Search ADS 23 Yen , T.-C. , Chen , W.-L. and Chen , J.-Y. ( 2016 ) Reliability and sensitivity analysis of the controllable repair system with warm standbys and working breakdown . Comput. Ind. Eng. , 97 , 84 – 92 . Google Scholar CrossRef Search ADS 24 Chen , J.-Y. , Wang , K.-H. , Sheu , S.-P. and Chou , W.-K. ( 2014 ) Maximum entropy analysis to the N Policy M/G/1 queue with working breakdowns . J. Test. Eval. , 43 , 1211 – 1220 . 25 Chen , J.-Y. , Yen , T.-C. and Wang , K.-H. ( 2016 ) Cost optimization of a single-server queue with working breakdowns under the N policy . J. Test. Eval. , 44 , 2059 – 2067 . 26 Jiang , T. and Liu , L. ( 2017 ) The GI/M/1 queue in a multi-phase service environment with disasters and working breakdowns . Int. J. Comput. Math. , 94 , 707 – 726 . Google Scholar CrossRef Search ADS 27 Li , T. and Zhang , L. ( 2017 ) An M/G/1 retrial G-queue with general retrial times and working breakdowns . Math. Comput. Appl. , 22 , 15 . 28 Ke , J.-C. ( 2003 ) The analysis of a general input queue with N policy and exponential vacations . Queueing Syst. , 45 , 135 – 160 . Google Scholar CrossRef Search ADS 29 Ke , J.-C. ( 2003 ) The operating characteristic analysis on a general input queue with N policy and a startup time . Math. Method Oper. Res. , 57 , 235 – 254 . Google Scholar CrossRef Search ADS 30 Chao , X. and Rahman , A. ( 2006 ) Analysis and computational algorithm for queues with state-dependent vacations I: G/M(n)/1/K . J. Syst. Sci. Complex. , 19 , 36 – 53 . Google Scholar CrossRef Search ADS 31 Zhang , M. and Hou , Z. ( 2011 ) Steady state analysis of the GI/M/1/N queue with a variant of multiple working vacations . Comput. Ind. Eng. , 61 , 1296 – 1301 . Google Scholar CrossRef Search ADS 32 Goswami , V. , Laxmi , P.V. and Jyothsna , K. ( 2013 ) Analysis of GI/M(n)/1/N queue with state-dependent multiple working vacations . Opsearch , 50 , 106 – 124 . Google Scholar CrossRef Search ADS 33 Laxmi , P.V. and Suchitra , V. ( 2014 ) A recursive algorithm for state dependent GI/M/1/N queue with Bernoulli-schedule vacation . J. Math. Model. Algorithms Oper. Res. , 13 , 283 – 299 . Google Scholar CrossRef Search ADS 34 Goswami , V. ( 2015 ) Study of customers’ impatience in a GI/M/1/N queue with working vacations . Int. J. Manage. Sci. Eng. Manag , 10 , 144 – 154 . 35 Kempa , W.M. ( 2017 ) A comprehensive study on the queue-size distribution in a finite-buffer system with a general independent input flow . Perform. Eval. , 108 , 1 – 15 . Google Scholar CrossRef Search ADS 36 Laxmi , P.V. and Indira , S. ( 2017 ) Ant colony optimisation for impatient customer queue under N-policy and Bernoulli schedule vacation interruption . Inte. J. Math. Oper. Res. , 10 , 167 – 189 . Google Scholar CrossRef Search ADS 37 Niu , Z. and Takahashi , Y. ( 1999 ) A finite‐capacity queue with exhaustive vacation/close‐down/setup times and Markovian arrival processes . Queueing Syst , 31 , 1 – 23 . Google Scholar CrossRef Search ADS 38 Gupta , U.C. and Sikdar , K. ( 2004 ) A finite capacity bulk service queue with single vacation and Markovian arrival process . Int. J. Stoch. Anal. , 2004 , 337 – 357 . Google Scholar CrossRef Search ADS 39 Banik , A.D. , Gupta , U.C. and Pathak , S.S. ( 2007 ) On the GI/M/1/N queue with multiple working vacations—analytic analysis and computation . Appl. Math. Model. , 31 , 1701 – 1710 . Google Scholar CrossRef Search ADS Author notes Handling editor: Antonio Fernandez Anta © The British Computer Society 2018. All rights reserved. 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Analysis of the N-policy GI/M/1/K Queueing Systems with Working Breakdowns and Repairs

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© The British Computer Society 2018. All rights reserved. For permissions, please email: journals.permissions@oup.com
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0010-4620
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Abstract

Abstract In this paper, we present an algorithmic approach to the analysis of the finite-capacity GI/M/1 queue with working breakdowns under N-policy. When there are no customers in the system, the server is turned off. If the number of customers in the system reaches threshold N, then the server is turned on and working. The recently introduced working breakdown involves serving newly arrived customers at a lower service rate in cases where the server breaks down. Service times during busy and breakdown periods are exponentially distributed. When a breakdown occurs, the failed server is not repaired until there are no customers remaining in the system. In this type of queueing system, we compute the steady-state probabilities at arbitrary and pre-arrival epochs using the supplementary variable method. We propose an algorithm for computing the steady-state probabilities at pre-arrival epochs and develop system performance measures. Finally, numerical analysis is used to evaluate the effects of various system parameters on system performance measures. 1. INTRODUCTION The widely discussed N-policy, introduced by Yadin and Naor [1], is a queueing system with control policies where the server is turned off when the system is emptied, and the server is turned on if the queue length reaches or exceeds threshold N. Wang and Ke [2] applied the supplementary variable technique and recursive method to investigate the N-policy M/G/1 queues with finite capacity and infinite capacity. Using the same technique, Ke and Wang [3] addressed the N-policy G/M/1 queue with finite capacity. Jain et al. [4] elucidated the reliability characteristics of a machine repair problem with reneging under N-policy. A review on the optimal control of queues under N-policy can be found in Tadj and Choudhury [5]. Wang et al. [6] discussed an N-policy queue in a fuzzy environment using fuzzy set theory, where the arrival and service rates are fuzzy numbers. Vijaya Laxmi and Suchitra [7] considered the N-policy GI/M(n)/1 queue with state dependent service rates, working vacations and Bernoulli schedule vacation interruptions, where the server continues working a slower rate during vacation periods. Haridass and Arumuganathan [8] considered the Mx/G/1 retrial queueing system with modified vacations under N-policy, where the server takes no more than M vacations when the number of customers in orbit falls below threshold N. Wang et al. [9] recently analyzed the equilibrium strategic behavior of customers and social optimization in an M/M/1 retrial queue with constant retrial rate under N-policy. All of the above papers assumed that the server is reliable; however, in many real-life situations, servers are subject to unpredictable breakdowns. Servers cease providing service during breakdown periods; i.e. service is interrupted. The N-policy queues with server breakdowns have been discussed by several researchers, including Wang and Ke [10], Pearn et al. [11], Choudhury et al. [12], Choudhury and Tadj [13] and Singh et al. [14]. Details pertaining to queueing models under N-policy can also be found in the survey by Jayachitra and Albert [15]. Yang and Wu [16] applied the matrix–geometric method to study the N-policy M/M/1 queue with an unreliable server and working vacations. Based on the principle of maximum entropy, Jain et al. [17] approximated stationary probability distributions of the system size for the N-policy Mx/G/1 queue with server breakdowns and delayed phase repair under a Bernoulli vacation schedule. Most recently, Wang et al. [18] analyzed the discrete-time Geo/G/1 queue with server breakdowns under random N-policy, where threshold N is a random number that varies in different cycles. In a pioneering study, Kalidass and Kasturi [19] analyzed the M/M/1 queue with working breakdowns, in which the server lowers the service rate in the event of a breakdown during a busy period. Kim and Lee [20] derived the system size and sojourn time distributions for the M/G/1 queue with disasters and working breakdowns. Liu and Song [21] conducted analysis on a batch arrival M/M/1 queue with working breakdowns. Liou [22] applied the matrix–geometric method to compute the steady-state probabilities for an M/M/1 queue with working breakdowns, in which customers may balk and renege. Yen et al. [23] used the Laplace transform technique to study the reliability-based measures of a warm standby repairable system with working breakdowns. In recent years, numerous researches have analyzed working breakdown queues, the details of which may be found in Chen et al. [24, 25], Jiang and Liu [26] and Li and Zhang [27]. It is important to investigate queueing systems with a finite buffer due to the fact that in many situations, the waiting space is limited. Ke [28] considered a GI/M/1 queue with finite capacity and N-policy, in which the server takes multiple vacations when the system empties. The operating characteristics of the N-policy GI/M/1 queue with finite capacity and startup times were investigated by Ke [29]. The proposed queueing model of this paper is different from Ke [28, 29]. The differences are that (1) our paper does not address server vacations and startup times; and (2) Ke [28, 29] were not attentive to server breakdowns. A number of studies have examined finite-capacity GI/M/1 queues [30–36]; however, little research has been dedicated to finite-capacity GI/M/1 queueing models with server breakdowns. To the best of our knowledge, this is the first study to apply steady-state analysis to a finite-capacity GI/M/1 queue with working breakdowns under N-policy. We adopted the supplementary variable technique, which is a simple and straightforward approach to compute the stationary probabilities at arbitrary epochs [37, 38]. The remainder of the paper is organized as follows. In the next section, we present a description of the model and define the notation. In Section 3, we formulate the mathematical model of the system using the supplementary variable technique. Section 4 outlines an algorithm for the computation of steady-state probabilities at pre-arrival epochs. In Section 5, we deduce the steady-state probabilities at arbitrary epochs. In Section 6, we develop various system performance measures and perform sensitivity analysis. Conclusions are drawn in Section 7. 2. MODEL AND NOTATION In this study, we consider an N-policy GI/M/1/K queue with working breakdowns. The system comprises a finite waiting space K and an unreliable server. The inter-arrival times of customers are independent and identically distributed (i.i.d.) random variables with a general distribution function A(x), probability density function a(x), mean inter-arrival time a1=1/λ, Laplace–Stieltjes transform (LST) A*(θ)=∫0∞e−θxdA(x) and A*(j)(θ) is the jth-order (j≥1) derivative of A*(θ) with respect to θ. Arriving customers are served on a first-come first-served (FCFS) basis, and the server can serve only one customer at a time. The server may break down during busy periods, and breakdown occurs according to a Poisson process with parameter α. When a breakdown occurs, the server continues working at a reduced service rate. The service times during the normal busy and breakdown periods follow exponential distributions with mean rates of μb and μd(<μb), respectively. During a breakdown period, the failed server is repaired only after the system becomes empty. The repair times of the failed server are exponentially distributed with mean rate β. Moreover, it is assumed that various stochastic processes involved in the queueing system are mutually independent. 2.1. Practical justification of the model To illustrate the potential application of the proposed queueing model, we consider a facility in which a single product is fabricated using a single production machine. Raw materials arriving at the facility are processed by the production machine. When the quantity of raw materials achieves the upper bound of the storage capacity, newly arrived raw materials are not allowed to enter the facility. For the purpose of saving energy, the production machine is turned off when there are no raw materials in the facility. If there are N or more raw materials in the system, then the production machine is turned on. During the processing of raw materials, the production machine may break down at any time; however, it may still be able to operate, albeit at a lower processing rate. To avoid production interruptions, the failed machine is repaired until there are no raw materials to process. This production system could be appropriately modeled using the proposed queueing system. 3. ANALYSIS OF THE MODEL We first establish the differential equations in a steady state by treating the remaining inter-arrival time as a supplementary variable. Let N(t) be the number of customers in the system and X(t) be remaining inter-arrival time for the next arrival at time t. The server state at time t is described as Y(t)={0,iftheserveristurnedoff.1,iftheserveristurnedonandinnormalworkingstate,2,iftheserverisinbreakdownstate. We define P0,n(x,t)dx=Pr{N(t)=n,Y(t)=0,x<X(t)≤x+dx},n=0,1,…,N−1,x≥0 P1,n(x,t)dx=Pr{N(t)=n,Y(t)=1,x<X(t)≤x+dx},n=1,2,…,K,x≥0 P2,n(x,t)dx=Pr{N(t)=n,Y(t)=2,x<X(t)≤x+dx},n=0,1,…,K,x≥0 P0,n(t)=∫0∞P0,n(x,t)dx,n=0,1,…,N−1, P1,n(t)=∫0∞P1,n(x,t)dx,n=1,2,…,K, P2,n(t)=∫0∞P2,n(x,t)dx,n=0,1,…,K, In steady-state, the steady-state probabilities can be written as follows: P0,n=limt→∞P0,n(t),n=0,1,…,N−1, P1,n=limt→∞P1,n(t),n=1,2,…,K, P2,n=limt→∞P2,n(t),n=0,1,…,K. Using the method of supplementary variable, the steady-state differential-difference equations governing the system can be written as follows: −ddxP0,0(x)=μbP1,1(x)+βP2,0(x), (1) −ddxP0,n(x)=a(x)P0,n−1(0),1≤n≤N−1, (2) −ddxP1,1(x)=−(μb+α)P1,1(x)+μbP1,2(x), (3) −ddxP1,n(x)=−(μb+α)P1,n(x)+μbP1,n+1(x)+a(x)P1,n−1(0),2≤n≤N−1, (4) −ddxP1,N(x)=−(μb+α)P1,N(x)+a(x)P0,N−1(0)+a(x)P1,N−1(0)+μbP1,N+1(x), (5) −ddxP1,n(x)=−(μb+α)P1,n(x)+a(x)P1,n−1(0)+μbP1,n+1(x),N+1≤n≤K−1, (6) −ddxP1,K(x)=−(μb+α)P1,K(x)+a(x)(P1,K−1(0)+P1,K(0)), (7) −ddxP2,0(x)=−βP2,0(x)+μdP2,1(x), (8) −ddxP2,n(x)=−μdP2,n(x)+μdP2,n+1(x)+αP1,n(x)+a(x)P2,n−1(0),1≤n≤K−1, (9) −ddxP2,K(x)=−μdP2,K(x)+αP1,K(x)+a(x)(P2,K−1(0)+P2,K(0)). (10) Let us introduce the following LSTs of a(x), P0,n(x), P1,n(x) and P2,n(x): A*(θ)=∫0∞e−θxa(x)dx,P0,n*(θ)=∫0∞e−θxP0,n(x)dx,P1,n*(θ)=∫0∞e−θxP1,n(x)dx,P2,n*(θ)=∫0∞e−θxP2,n(x)dx,θ≥0, so that P0,n=P0,n*(0)=∫0∞P0,n(x)dx,n=0,1,…,N−1. P1,n=P1,n*(0)=∫0∞P1,n(x)dx,n=1,2,…,K. P2,n=P2,n*(0)=∫0∞P2,n(x)dx,n=0,1,…,K. Multiplying Equations (1–10) by e−θx on both sides and integrating over x, we obtain −θP0,0*(θ)=μbP1,1*(θ)+βP2,0*(θ)−P0,0(0), (11) −θP0,n*(θ)=A*(θ)P0,n−1(0)−P0,n(0),1≤n≤N−1, (12) (μb+α−θ)P1,1*(θ)=μbP1,2*(θ)−P1,1(0), (13) (μb+α−θ)P1,n*(θ)=A*(θ)P1,n−1(0)+μbP1,n+1*(θ)−P1,n(0),2≤n≤N−1, (14) (μb+α−θ)P1,N*(θ)=A*(θ)P0,N−1(0)+A*(θ)P1,N−1(0)+μbP1,N+1*(θ)−P1,N(0), (15) (μb+α−θ)P1,n*(θ)=A*(θ)P1,n−1(0)+μbP1,n+1*(θ)−P1,n(0),N+1≤n≤K−1, (16) (μb+α−θ)P1,K*(θ)=A*(θ)(P1,K−1(0)+P1,K(0))−P1,K(0), (17) (β−θ)P2,0*(θ)=μdP2,1*(θ)−P2,0(0), (18) (μd−θ)P2,n*(θ)=μdP2,n+1*(θ)+αP1,n*(θ)+A*(θ)P2,n−1(0)−P2,n(0),1≤n≤K−1, (19) (μd−θ)P2,K*(θ)=αP1,K*(θ)+A*(θ)(P2,K−1(0)+P2,K(0))−P2,K(0). (20) Substituting θ=0 into Equation (12) gives P0,0(0)=P0,1(0)=⋯=P0,N−1(0). (21) Adding Equations (11–20), it yields ∑n=0N−1P0,n*(θ)+∑n=1KP1,n*(θ)+∑n=0KP2,n*(θ)=1−A*(θ)θ(∑n=0N−1P0,n(0)+∑n=1KP1,n(0)+∑n=0KP2,n(0)). (22) On taking limit as θ→0 in Equation (22), we obtain ∑n=0N−1P0,n*(0)+∑n=1KP1,n*(0)+∑n=0KP2,n*(0)=1λ(∑n=0N−1P0,n(0)+∑n=1KP1,n(0)+∑n=0KP2,n(0)). (23) Here, the normalization condition implies ∑n=0N−1P0,n*(0)+∑n=1KP1,n*(0)+∑n=0KP2,n*(0)=1. (24) Using Equation (24), Equation (23) can be written as ∑n=0N−1P0,n(0)+∑n=1KP1,n(0)+∑n=0KP2,n(0)=λ. (25) Remark 1 Equation (25) shows that the expected number of entries into the system per unit time is equal to the mean arrival rate λ. Substituting θ=μb+α into Equation (17), we have P1,K−1(0)=(1−A*(μb+α))P1,K(0)A*(μb+α). (26) Similarly, substituting θ=μb+α into Equations (14–16), we obtain P1,n−1(0)=P1,n(0)−δn,NA*(μb+α)P0,N−1(0)−μbP1,n+1*(μb+α)A*(μb+α),2≤n≤K−1, (27) where δn,N denotes the Kronecker delta function. From Equation (17), P1,K*(θ) can be written as follows: P1,K*(θ)={−A*(1)(θ)(P1,K−1(0)+P1,K(0)),ifθ=μb+α,A*(θ)(P1,K−1(0)+P1,K(0))−P1,K(0)(μb+α−θ),ifθ≠μb+α. (28) For 2≤n≤K−1, from Equations (14–16), we have P1,n*(θ)={−(A*(1)(θ)P1,n−1(0)+δn,NA*(1)(θ)P0,N−1(0)+μbP1,n+1*(1)(θ)),ifθ=μb+α,A*(θ)P1,n−1(0)+δn,NA*(θ)P0,N−1(0)+μbP1,n+1*(θ)−P1,n(0)(μb+α−θ),ifθ≠μb+α. (29) Substituting θ=μd into Equations (19) and (20) yields P2,n−1(0)=P2,n(0)−μdP2,n+1*(μd)−αP1,n*(μd)A*(μd),1≤n≤K−1, (30) P2,K−1(0)=(1−A*(μd))P2,K(0)−αP1,K*(μd)A*(μd). (31) Using Equation (20), it is easy to show P2,K*(θ)={P2,K*(θ)=−(αP1,K*(1)(θ)+A*(1)(θ)(P2,K−1(0)+P2,K(0))),ifθ=μd,αP1,K*(θ)+A*(θ)(P2,K−1(0)+P2,K(0))−P2,K(0)(μd−θ),ifθ≠μd. (32) From Equation (19), for 1≤n≤K−1, we obtain P2,n*(θ)={−(μdP2,n+1*(1)(θ)+αP1,n*(1)(θ)+A*(1)(θ)P2,n−1(0)),ifθ=μd,μdP2,n+1*(θ)+αP1,n*(θ)+A*(θ)P2,n−1(0)−P2,n(0)(μd−θ),ifθ≠μd. (33) Our objective is to determine the steady-state probabilities P0,n*(0), P1,n*(0) and P2,n*(0), which are computed with help of P1,n(0) and P2,n(0). Finding P1,n(0) and P2,n(0) requires that we derive the jth-order derivative of P1,n*(θ) and P2,n*(θ) with respect to θ. By differentiating Equations (14–17) j times with respect to θ and then setting θ=μb+α, we obtain P1,n*(j)(μb+α)=−A*(j+1)(μb+α)P1,n−1(0)+δn,NA*(j+1)(μb+α)P0,N−1(0)+μbP1,n+1*(j+1)(μb+α)j+1,j≥1,2≤n≤K−1, (34) P1,K*(j)(μb+α)=−A*(j+1)(μb+α)(P1,K−1(0)+P1,K(0))j+1,j≥1, (35) where A*(j)(θ)=djdθjA*(θ). Similarity, for θ≠μb+α, differentiating Equations (14–17) yields P1,n*(j)(θ)=A*(j)(θ)P1,n−1(0)+δn,NA*(j)(θ)P0,N−1(0)+μbP1,n+1*(j)(θ)+jP1,n*(j−1)(θ)(μb+α−θ),j≥1,2≤n≤K−1, (36) P1,K*(j)(θ)=A*(j)(θ)(P1,K−1(0)+P1,K(0))+jP1,K*(j−1)(θ)(μb+α−θ),j≥1, (37) where P1,n*(0)(θ)=P1,n*(θ). In sequel, we differentiate Equations (19) and (20) j times with respect to θ. Then, the jth-order derivatives of P2,n*(θ) at θ=μd are given by P2,n*(j)(μd)=−μdP2,n+1*(j+1)(μd)+αP1,n*(j+1)(μd)+A*(j+1)(μd)P2,n−1(0)j+1,j≥1,1≤n≤K−1, (38) P2,K*(j)(μd)=−αP1,K*(j+1)(μd)+A*(j+1)(μd)(P2,K−1(0)+P2,K(0))j+1,j≥1, (39) For θ≠μd, differentiating Equations (19) and (20) j times with respect to θ gives P2,n*(j)(θ)=μdP2,n+1*(j)(θ)+αP1,n*(j)(θ)+A*(j)(θ)P2,n−1(0)+jP2,n*(j−1)(θ)(μd−θ),j≥1,1≤n≤K−1, (40) P2,K*(j)(θ)=αP1,K*(j)(θ)+A*(j)(θ)(P2,K−1(0)+P2,K(0))+jP2,K*(j−1)(θ)(μd−θ),j≥1, (41) where P2,n*(0)(θ)=P2,n*(θ). 4. STEADY-STATE PROBABILITY DISTRIBUTION AT PRE-ARRIVAL EPOCHS Let Pi,n−(i=0,1,2) be the steady-state probability that an arrival finds n customers in the system when the server is in state i. In accordance with the arguments proposed by Goswami et al. [32] and Banik et al. [39], we have the relationship between Pi,n−(i=0,1,2) and Pi,n(0)(i=0,1,2) as follows: P0,n−=1λP0,n(0),0≤n≤N−1, (42) P1,n−=1λP1,n(0),1≤n≤K, (43) P2,n−=1λP2,n(0),0≤n≤K. (44) Before determining the pre-arrival epoch probabilities Pi,n−(i=0,1,2), it is necessary to evaluate P0,n(0)(0≤n≤N−1), P1,n(0)(1≤n≤K) and P2,n(0)(0≤n≤K). However, it is difficult to obtain the analytical forms of P0,n(0), P1,n(0) and P2,n(0). We overcome this difficulty by using the results in Section 3 to formulate an algorithm for numerically computing the pre-arrival epoch probabilities. The step-by-step procedure of the algorithm is described as follows: Step 1: Express P1,n(0) and P1,n*(θ) in terms of P0,N−1(0) and P1,K(0) as follows: P1,n(0)={ξnP1,K(0),ifN≤n≤K,ξnP1,K(0)+ϕnP0,N−1(0),if1≤n≤N−1, P1,n*(θ)={Ψn(θ)P1,K(0),ifN≤n≤K,ϒn(θ)P0,N−1(0)+Ψn(θ)P1,K(0),if2≤n≤N−1. Step 1.1: Calculate ξn using Equations (26) and (27) as follows: ξK=1,ξK−1=1−A*(μb+α)A*(μb+α),ξn−1=ξn−μbΨn+1(μb+α)A*(μb+α),1≤n≤K−1. Step 1.2: Calculate ϕn using Equation (27) as follows: ϕN−1=−1,ϕN−2=ϕN−1A*(μb+α),ϕn−1=ϕn−μbϒn+1(μb+α)A*(μb+α),1≤n≤N−2. Step 1.3: Calculate ϒn(θ) and Ψn(θ) using Equations (28) and (29) as follows:  if n=Kthen   if θ=μb+αthen     ΨK(θ)=−A*(1)(θ)(ξK−1+ξK)   else     ΨK(θ)=A*(θ)(ξK−1+ξK)−ξK(μb+α−θ)   end if  else if N≤n≤K−1then   if θ=μb+αthen     Ψn(θ)=−(A*(1)(θ)ξn−1+μbΨn+1(1)(θ))   else     Ψn(θ)=A*(θ)ξn−1+μbΨn+1(θ)−ξn(μb+α−θ)   end if  else if n=N−1then   if θ=μb+αthen     ϒN−1(θ)=−(A*(1)(θ)ϕN−2)     ΨN−1(θ)=−(A*(1)(θ)ξN−2+μbΨN(1)(θ))   else     ϒN−1(θ)=A*(θ)ϕN−2−ϕN−1(μb+α−θ)     ΨN−1(θ)=A*(θ)ξN−2+μbΨN(θ)−ξN−1(μb+α−θ)   end if  else if 2≤n≤N−2then   if θ=μb+αthen     ϒn(θ)=−(A*(1)(θ)ϕn−1+μbϒn+1(1)(θ))     Ψn(θ)=−(A*(1)(θ)ξn−1+μbΨn+1(1)(θ))   else     ϒn(θ)=A*(θ)ϕn−1+μbϒn+1(θ)−ϕn(μb+α−θ)     Ψn(θ)=A*(θ)ξn−1+μbΨn+1(θ)−ξn(μb+α−θ)  end if  if n=Kthen   if θ=μb+αthen     ΨK(θ)=−A*(1)(θ)(ξK−1+ξK)   else     ΨK(θ)=A*(θ)(ξK−1+ξK)−ξK(μb+α−θ)   end if  else if N≤n≤K−1then   if θ=μb+αthen     Ψn(θ)=−(A*(1)(θ)ξn−1+μbΨn+1(1)(θ))   else     Ψn(θ)=A*(θ)ξn−1+μbΨn+1(θ)−ξn(μb+α−θ)   end if  else if n=N−1then   if θ=μb+αthen     ϒN−1(θ)=−(A*(1)(θ)ϕN−2)     ΨN−1(θ)=−(A*(1)(θ)ξN−2+μbΨN(1)(θ))   else     ϒN−1(θ)=A*(θ)ϕN−2−ϕN−1(μb+α−θ)     ΨN−1(θ)=A*(θ)ξN−2+μbΨN(θ)−ξN−1(μb+α−θ)   end if  else if 2≤n≤N−2then   if θ=μb+αthen     ϒn(θ)=−(A*(1)(θ)ϕn−1+μbϒn+1(1)(θ))     Ψn(θ)=−(A*(1)(θ)ξn−1+μbΨn+1(1)(θ))   else     ϒn(θ)=A*(θ)ϕn−1+μbϒn+1(θ)−ϕn(μb+α−θ)     Ψn(θ)=A*(θ)ξn−1+μbΨn+1(θ)−ξn(μb+α−θ)  end if  if n=Kthen   if θ=μb+αthen     ΨK(θ)=−A*(1)(θ)(ξK−1+ξK)   else     ΨK(θ)=A*(θ)(ξK−1+ξK)−ξK(μb+α−θ)   end if  else if N≤n≤K−1then   if θ=μb+αthen     Ψn(θ)=−(A*(1)(θ)ξn−1+μbΨn+1(1)(θ))   else     Ψn(θ)=A*(θ)ξn−1+μbΨn+1(θ)−ξn(μb+α−θ)   end if  else if n=N−1then   if θ=μb+αthen     ϒN−1(θ)=−(A*(1)(θ)ϕN−2)     ΨN−1(θ)=−(A*(1)(θ)ξN−2+μbΨN(1)(θ))   else     ϒN−1(θ)=A*(θ)ϕN−2−ϕN−1(μb+α−θ)     ΨN−1(θ)=A*(θ)ξN−2+μbΨN(θ)−ξN−1(μb+α−θ)   end if  else if 2≤n≤N−2then   if θ=μb+αthen     ϒn(θ)=−(A*(1)(θ)ϕn−1+μbϒn+1(1)(θ))     Ψn(θ)=−(A*(1)(θ)ξn−1+μbΨn+1(1)(θ))   else     ϒn(θ)=A*(θ)ϕn−1+μbϒn+1(θ)−ϕn(μb+α−θ)     Ψn(θ)=A*(θ)ξn−1+μbΨn+1(θ)−ξn(μb+α−θ)  end if  if n=Kthen   if θ=μb+αthen     ΨK(θ)=−A*(1)(θ)(ξK−1+ξK)   else     ΨK(θ)=A*(θ)(ξK−1+ξK)−ξK(μb+α−θ)   end if  else if N≤n≤K−1then   if θ=μb+αthen     Ψn(θ)=−(A*(1)(θ)ξn−1+μbΨn+1(1)(θ))   else     Ψn(θ)=A*(θ)ξn−1+μbΨn+1(θ)−ξn(μb+α−θ)   end if  else if n=N−1then   if θ=μb+αthen     ϒN−1(θ)=−(A*(1)(θ)ϕN−2)     ΨN−1(θ)=−(A*(1)(θ)ξN−2+μbΨN(1)(θ))   else     ϒN−1(θ)=A*(θ)ϕN−2−ϕN−1(μb+α−θ)     ΨN−1(θ)=A*(θ)ξN−2+μbΨN(θ)−ξN−1(μb+α−θ)   end if  else if 2≤n≤N−2then   if θ=μb+αthen     ϒn(θ)=−(A*(1)(θ)ϕn−1+μbϒn+1(1)(θ))     Ψn(θ)=−(A*(1)(θ)ξn−1+μbΨn+1(1)(θ))   else     ϒn(θ)=A*(θ)ϕn−1+μbϒn+1(θ)−ϕn(μb+α−θ)     Ψn(θ)=A*(θ)ξn−1+μbΨn+1(θ)−ξn(μb+α−θ)  end if Step 1.4: Calculate ϒn(j)(θ) and Ψn(j)(θ) using Equations (34–37) as follows:  if n=Kthen   if θ=μb+αthen     ΨK(j)(θ)=−A*(j+1)(θ)(ξK−1+ξK)j+1   else     ΨK*(j)(θ)=A*(j)(θ)(ξK−1+ξK)+jΨK*(j−1)(θ)(μb+α−θ)   end if  else if N≤n≤K−1then   if θ=μb+αthen     Ψn(j)(θ)=−A*(j+1)(θ)ξn−1+μbΨn+1(j+1)(θ)j+1   else     Ψn(j)(θ)=A*(j)(θ)ξn−1+μbΨn+1(j)(θ)+jΨn(j−1)(θ)(μb+α−θ)   end if  else if n=N−1then   if θ=μb+αthen     ϒN−1(j)(θ)=−A*(j+1)(θ)ϕN−2j+1     ΨN−1(j)(θ)=−A*(j+1)(θ)ξN−2+μbΨN(j+1)(θ)j+1   else     ϒN−1(j)(θ)=A*(j)(θ)ϕN−2+jϒN−1(j−1)(θ)(μb+α−θ)     ΨN−1(j)(θ)=A*(j)(θ)ξN−2+μbΨN(j)(θ)+jΨN−1(j−1)(θ)(μb+α−θ)   end if  else if 2≤n≤N−2then   if θ=μb+αthen     ϒn(j)(θ)=−A*(j+1)(θ)ϕn−1+μbϒn+1(j+1)(θ)j+1     Ψn(j)(θ)=−A*(j+1)(θ)ξn−1+μbΨn+1(j+1)(θ)j+1   else     ϒn(j)(θ)=A*(j)(θ)ϕn−1+μbϒn+1(j)(θ)+jϒn(j−1)(θ)(μb+α−θ)     Ψn(j)(θ)=A*(j)(θ)ξn−1+μbΨn+1(j)(θ)+jΨn(j−1)(θ)(μb+α−θ)   end if  end if  if n=Kthen   if θ=μb+αthen     ΨK(j)(θ)=−A*(j+1)(θ)(ξK−1+ξK)j+1   else     ΨK*(j)(θ)=A*(j)(θ)(ξK−1+ξK)+jΨK*(j−1)(θ)(μb+α−θ)   end if  else if N≤n≤K−1then   if θ=μb+αthen     Ψn(j)(θ)=−A*(j+1)(θ)ξn−1+μbΨn+1(j+1)(θ)j+1   else     Ψn(j)(θ)=A*(j)(θ)ξn−1+μbΨn+1(j)(θ)+jΨn(j−1)(θ)(μb+α−θ)   end if  else if n=N−1then   if θ=μb+αthen     ϒN−1(j)(θ)=−A*(j+1)(θ)ϕN−2j+1     ΨN−1(j)(θ)=−A*(j+1)(θ)ξN−2+μbΨN(j+1)(θ)j+1   else     ϒN−1(j)(θ)=A*(j)(θ)ϕN−2+jϒN−1(j−1)(θ)(μb+α−θ)     ΨN−1(j)(θ)=A*(j)(θ)ξN−2+μbΨN(j)(θ)+jΨN−1(j−1)(θ)(μb+α−θ)   end if  else if 2≤n≤N−2then   if θ=μb+αthen     ϒn(j)(θ)=−A*(j+1)(θ)ϕn−1+μbϒn+1(j+1)(θ)j+1     Ψn(j)(θ)=−A*(j+1)(θ)ξn−1+μbΨn+1(j+1)(θ)j+1   else     ϒn(j)(θ)=A*(j)(θ)ϕn−1+μbϒn+1(j)(θ)+jϒn(j−1)(θ)(μb+α−θ)     Ψn(j)(θ)=A*(j)(θ)ξn−1+μbΨn+1(j)(θ)+jΨn(j−1)(θ)(μb+α−θ)   end if  end if View Large  if n=Kthen   if θ=μb+αthen     ΨK(j)(θ)=−A*(j+1)(θ)(ξK−1+ξK)j+1   else     ΨK*(j)(θ)=A*(j)(θ)(ξK−1+ξK)+jΨK*(j−1)(θ)(μb+α−θ)   end if  else if N≤n≤K−1then   if θ=μb+αthen     Ψn(j)(θ)=−A*(j+1)(θ)ξn−1+μbΨn+1(j+1)(θ)j+1   else     Ψn(j)(θ)=A*(j)(θ)ξn−1+μbΨn+1(j)(θ)+jΨn(j−1)(θ)(μb+α−θ)   end if  else if n=N−1then   if θ=μb+αthen     ϒN−1(j)(θ)=−A*(j+1)(θ)ϕN−2j+1     ΨN−1(j)(θ)=−A*(j+1)(θ)ξN−2+μbΨN(j+1)(θ)j+1   else     ϒN−1(j)(θ)=A*(j)(θ)ϕN−2+jϒN−1(j−1)(θ)(μb+α−θ)     ΨN−1(j)(θ)=A*(j)(θ)ξN−2+μbΨN(j)(θ)+jΨN−1(j−1)(θ)(μb+α−θ)   end if  else if 2≤n≤N−2then   if θ=μb+αthen     ϒn(j)(θ)=−A*(j+1)(θ)ϕn−1+μbϒn+1(j+1)(θ)j+1     Ψn(j)(θ)=−A*(j+1)(θ)ξn−1+μbΨn+1(j+1)(θ)j+1   else     ϒn(j)(θ)=A*(j)(θ)ϕn−1+μbϒn+1(j)(θ)+jϒn(j−1)(θ)(μb+α−θ)     Ψn(j)(θ)=A*(j)(θ)ξn−1+μbΨn+1(j)(θ)+jΨn(j−1)(θ)(μb+α−θ)   end if  end if  if n=Kthen   if θ=μb+αthen     ΨK(j)(θ)=−A*(j+1)(θ)(ξK−1+ξK)j+1   else     ΨK*(j)(θ)=A*(j)(θ)(ξK−1+ξK)+jΨK*(j−1)(θ)(μb+α−θ)   end if  else if N≤n≤K−1then   if θ=μb+αthen     Ψn(j)(θ)=−A*(j+1)(θ)ξn−1+μbΨn+1(j+1)(θ)j+1   else     Ψn(j)(θ)=A*(j)(θ)ξn−1+μbΨn+1(j)(θ)+jΨn(j−1)(θ)(μb+α−θ)   end if  else if n=N−1then   if θ=μb+αthen     ϒN−1(j)(θ)=−A*(j+1)(θ)ϕN−2j+1     ΨN−1(j)(θ)=−A*(j+1)(θ)ξN−2+μbΨN(j+1)(θ)j+1   else     ϒN−1(j)(θ)=A*(j)(θ)ϕN−2+jϒN−1(j−1)(θ)(μb+α−θ)     ΨN−1(j)(θ)=A*(j)(θ)ξN−2+μbΨN(j)(θ)+jΨN−1(j−1)(θ)(μb+α−θ)   end if  else if 2≤n≤N−2then   if θ=μb+αthen     ϒn(j)(θ)=−A*(j+1)(θ)ϕn−1+μbϒn+1(j+1)(θ)j+1     Ψn(j)(θ)=−A*(j+1)(θ)ξn−1+μbΨn+1(j+1)(θ)j+1   else     ϒn(j)(θ)=A*(j)(θ)ϕn−1+μbϒn+1(j)(θ)+jϒn(j−1)(θ)(μb+α−θ)     Ψn(j)(θ)=A*(j)(θ)ξn−1+μbΨn+1(j)(θ)+jΨn(j−1)(θ)(μb+α−θ)   end if  end if View Large Step 2: Express P2,n(0) and P2,n*(θ) in terms of P0,N−1(0), P1,K(0) and P2,K(0) as follows: P2,n(0)={ωnP1,K(0)+ψnP2,K(0),ifN≤n≤K,τnP0,N−1(0)+ωnP1,K(0)+ψnP2,K(0),if0≤n≤N−1, P2,n*(θ)={Ωn(θ)P1,K(0)+Φn(θ)P2,K(0),ifN≤n≤K,Ζn(θ)P0,N−1(0)+Ωn(θ)P1,K(0)+Φn(θ)P2,K(0),if1≤n≤N−1. Step 2.1: Calculate ωn and ψn using Equations (30) and (31) as follows: ψK=1,ωK=0,ψK−1=(1−A*(μd))ψKA*(μd),ωK−1=(1−A*(μd))ωK−αΨK(μd)A*(μd) ωn−1=ωn−μdΩn+1(μd)−αΨn(μd)A*(μd),1≤n≤K−1, ψn−1=ψn−μdΦn+1(μd)A*(μd),1≤n≤K−1. Step 2.2: Calculate τn using Equation (31) as follows: τN−1=0,τN−2=τN−1−αϒN−1(μd)A*(μd), τn−1=τn−μdΖn+1(μd)−αϒn(μd)A*(μd),1≤n≤N−2. Step 2.3: Calculate Ζn(θ), Ωn(θ) and Φn(θ) using Equations (32) and (33) as follows:  if n=Kthen   if θ=μdthen     ΩK(θ)=−(αΨK(1)(θ)+A*(1)(θ)ωK−1)     ΦK(θ)=−(A*(1)(θ)(ψK−1+ψK))   else     ΩK(θ)=αΨK(θ)+A*(θ)ωK−1(μd−θ)     ΦK(θ)=A*(θ)(ψK−1+ψK)−ψK(μd−θ)   end if  else if N≤n≤K−1then   if θ=μdthen     Ωn(θ)=−(μdΩn+1(1)(θ)+αΨn(1)(θ)+A*(1)(θ)ωn−1)     Φn(θ)=−(μdΦn+1(1)(θ)+A*(1)(θ)ψn−1)   else     Ωn(θ)=μdΩn+1(θ)+αΨn(θ)+A*(θ)ωn−1−ωn(μd−θ)     Φn(θ)=μdΦn+1(θ)+A*(θ)ψn−1−ψn(μd−θ)  else if n=N−1then   if θ=μdthen     ΖN−1(θ)=−(αϒN−1(1)(θ)+A*(1)(θ)τN−2)     ΩN−1(θ)=−(μdΩN(1)(θ)+αΨN−1(1)(θ)+A*(1)(θ)ωN−2)     ΦN−1(θ)=−(μdΦN(1)(θ)+A*(1)(θ)ψN−2)   else     ΖN−1(θ)=αϒN−1(θ)+A*(θ)τN−2−τN−1(μd−θ)     ΩN−1(θ)=μdΩN(θ)+αΨN−1(θ)+A*(θ)ωN−2−ωN−1(μd−θ)     ΦN−1(θ)=μdΦN(θ)+A*(θ)ψN−2−ψN−1(μd−θ)   end if  else if 1≤n≤N−2then   if θ=μdthen     Ζn(θ)=−(μdΖn+1(1)(θ)+αϒn(1)(θ)+A*(1)(θ)τn−1)     Ωn(θ)=−(μdΩn+1(1)(θ)+αΨn(1)(θ)+A*(1)(θ)ωn−1)     Φn(θ)=−(μdΦn+1(1)(θ)+A*(1)(θ)ψn−1)   else     Ζn(θ)=μdΖn+1(θ)+αϒn(θ)+A*(θ)τn−1−τn(μd−θ)     Ωn(θ)=μdΩn+1(θ)+αΨn(θ)+A*(θ)ωn−1−ωn(μd−θ)     Φn(θ)=μdΦn+1(θ)+A*(θ)ψn−1−ψn(μd−θ)   end if  end if  if n=Kthen   if θ=μdthen     ΩK(θ)=−(αΨK(1)(θ)+A*(1)(θ)ωK−1)     ΦK(θ)=−(A*(1)(θ)(ψK−1+ψK))   else     ΩK(θ)=αΨK(θ)+A*(θ)ωK−1(μd−θ)     ΦK(θ)=A*(θ)(ψK−1+ψK)−ψK(μd−θ)   end if  else if N≤n≤K−1then   if θ=μdthen     Ωn(θ)=−(μdΩn+1(1)(θ)+αΨn(1)(θ)+A*(1)(θ)ωn−1)     Φn(θ)=−(μdΦn+1(1)(θ)+A*(1)(θ)ψn−1)   else     Ωn(θ)=μdΩn+1(θ)+αΨn(θ)+A*(θ)ωn−1−ωn(μd−θ)     Φn(θ)=μdΦn+1(θ)+A*(θ)ψn−1−ψn(μd−θ)  else if n=N−1then   if θ=μdthen     ΖN−1(θ)=−(αϒN−1(1)(θ)+A*(1)(θ)τN−2)     ΩN−1(θ)=−(μdΩN(1)(θ)+αΨN−1(1)(θ)+A*(1)(θ)ωN−2)     ΦN−1(θ)=−(μdΦN(1)(θ)+A*(1)(θ)ψN−2)   else     ΖN−1(θ)=αϒN−1(θ)+A*(θ)τN−2−τN−1(μd−θ)     ΩN−1(θ)=μdΩN(θ)+αΨN−1(θ)+A*(θ)ωN−2−ωN−1(μd−θ)     ΦN−1(θ)=μdΦN(θ)+A*(θ)ψN−2−ψN−1(μd−θ)   end if  else if 1≤n≤N−2then   if θ=μdthen     Ζn(θ)=−(μdΖn+1(1)(θ)+αϒn(1)(θ)+A*(1)(θ)τn−1)     Ωn(θ)=−(μdΩn+1(1)(θ)+αΨn(1)(θ)+A*(1)(θ)ωn−1)     Φn(θ)=−(μdΦn+1(1)(θ)+A*(1)(θ)ψn−1)   else     Ζn(θ)=μdΖn+1(θ)+αϒn(θ)+A*(θ)τn−1−τn(μd−θ)     Ωn(θ)=μdΩn+1(θ)+αΨn(θ)+A*(θ)ωn−1−ωn(μd−θ)     Φn(θ)=μdΦn+1(θ)+A*(θ)ψn−1−ψn(μd−θ)   end if  end if View Large  if n=Kthen   if θ=μdthen     ΩK(θ)=−(αΨK(1)(θ)+A*(1)(θ)ωK−1)     ΦK(θ)=−(A*(1)(θ)(ψK−1+ψK))   else     ΩK(θ)=αΨK(θ)+A*(θ)ωK−1(μd−θ)     ΦK(θ)=A*(θ)(ψK−1+ψK)−ψK(μd−θ)   end if  else if N≤n≤K−1then   if θ=μdthen     Ωn(θ)=−(μdΩn+1(1)(θ)+αΨn(1)(θ)+A*(1)(θ)ωn−1)     Φn(θ)=−(μdΦn+1(1)(θ)+A*(1)(θ)ψn−1)   else     Ωn(θ)=μdΩn+1(θ)+αΨn(θ)+A*(θ)ωn−1−ωn(μd−θ)     Φn(θ)=μdΦn+1(θ)+A*(θ)ψn−1−ψn(μd−θ)  else if n=N−1then   if θ=μdthen     ΖN−1(θ)=−(αϒN−1(1)(θ)+A*(1)(θ)τN−2)     ΩN−1(θ)=−(μdΩN(1)(θ)+αΨN−1(1)(θ)+A*(1)(θ)ωN−2)     ΦN−1(θ)=−(μdΦN(1)(θ)+A*(1)(θ)ψN−2)   else     ΖN−1(θ)=αϒN−1(θ)+A*(θ)τN−2−τN−1(μd−θ)     ΩN−1(θ)=μdΩN(θ)+αΨN−1(θ)+A*(θ)ωN−2−ωN−1(μd−θ)     ΦN−1(θ)=μdΦN(θ)+A*(θ)ψN−2−ψN−1(μd−θ)   end if  else if 1≤n≤N−2then   if θ=μdthen     Ζn(θ)=−(μdΖn+1(1)(θ)+αϒn(1)(θ)+A*(1)(θ)τn−1)     Ωn(θ)=−(μdΩn+1(1)(θ)+αΨn(1)(θ)+A*(1)(θ)ωn−1)     Φn(θ)=−(μdΦn+1(1)(θ)+A*(1)(θ)ψn−1)   else     Ζn(θ)=μdΖn+1(θ)+αϒn(θ)+A*(θ)τn−1−τn(μd−θ)     Ωn(θ)=μdΩn+1(θ)+αΨn(θ)+A*(θ)ωn−1−ωn(μd−θ)     Φn(θ)=μdΦn+1(θ)+A*(θ)ψn−1−ψn(μd−θ)   end if  end if  if n=Kthen   if θ=μdthen     ΩK(θ)=−(αΨK(1)(θ)+A*(1)(θ)ωK−1)     ΦK(θ)=−(A*(1)(θ)(ψK−1+ψK))   else     ΩK(θ)=αΨK(θ)+A*(θ)ωK−1(μd−θ)     ΦK(θ)=A*(θ)(ψK−1+ψK)−ψK(μd−θ)   end if  else if N≤n≤K−1then   if θ=μdthen     Ωn(θ)=−(μdΩn+1(1)(θ)+αΨn(1)(θ)+A*(1)(θ)ωn−1)     Φn(θ)=−(μdΦn+1(1)(θ)+A*(1)(θ)ψn−1)   else     Ωn(θ)=μdΩn+1(θ)+αΨn(θ)+A*(θ)ωn−1−ωn(μd−θ)     Φn(θ)=μdΦn+1(θ)+A*(θ)ψn−1−ψn(μd−θ)  else if n=N−1then   if θ=μdthen     ΖN−1(θ)=−(αϒN−1(1)(θ)+A*(1)(θ)τN−2)     ΩN−1(θ)=−(μdΩN(1)(θ)+αΨN−1(1)(θ)+A*(1)(θ)ωN−2)     ΦN−1(θ)=−(μdΦN(1)(θ)+A*(1)(θ)ψN−2)   else     ΖN−1(θ)=αϒN−1(θ)+A*(θ)τN−2−τN−1(μd−θ)     ΩN−1(θ)=μdΩN(θ)+αΨN−1(θ)+A*(θ)ωN−2−ωN−1(μd−θ)     ΦN−1(θ)=μdΦN(θ)+A*(θ)ψN−2−ψN−1(μd−θ)   end if  else if 1≤n≤N−2then   if θ=μdthen     Ζn(θ)=−(μdΖn+1(1)(θ)+αϒn(1)(θ)+A*(1)(θ)τn−1)     Ωn(θ)=−(μdΩn+1(1)(θ)+αΨn(1)(θ)+A*(1)(θ)ωn−1)     Φn(θ)=−(μdΦn+1(1)(θ)+A*(1)(θ)ψn−1)   else     Ζn(θ)=μdΖn+1(θ)+αϒn(θ)+A*(θ)τn−1−τn(μd−θ)     Ωn(θ)=μdΩn+1(θ)+αΨn(θ)+A*(θ)ωn−1−ωn(μd−θ)     Φn(θ)=μdΦn+1(θ)+A*(θ)ψn−1−ψn(μd−θ)   end if  end if View Large Step 2.4: Calculate Ζn(j)(θ), Ωn(j)(θ) and Φn(j)(θ) using Equations (38–41) as follows:  if n=Kthen   if θ=μdthen     ΩK(j)(θ)=−αΨK(j+1)(θ)+A*(j+1)(θ)ωK−1j+1     ΦK(j)(θ)=−A*(j+1)(θ)(ψK−1+ψK)j+1   else     ΩK(j)(θ)=αΨK(j)(θ)+A*(j)(θ)ωK−1+jΩK(j−1)(θ)(μd−θ)     ΦK(j)(θ)=A*(j)(θ)(ψK−1+ψK)+jΦK(j−1)(θ)(μd−θ)   end if  else if N≤n≤K−1then   if θ=μdthen     Ωn(j)(θ)=−μdΩn+1(j+1)(θ)+αΨn(j+1)(θ)+A*(j+1)(θ)ωn−1j+1,     Φn(j)(θ)=−μdΦn+1(j+1)(θ)+A*(j+1)(θ)ψn−1j+1,   else     Ωn(j)(θ)=μdΩn+1(j)(θ)+αΨn(j)(θ)+A*(j)(θ)ωn−1+jΩn(j−1)(θ)(μd−θ),     Φn(j)(θ)=μdΦn+1(j)(θ)+A*(j)(θ)ψn−1+jΦn(j−1)(θ)(μd−θ),   end if  else if n=N−1then   if θ=μdthen     ΖN−1(j)(θ)=−αϒN−1(j+1)(θ)+A*(j+1)(θ)τN−2j+1     ΩN−1(j)(θ)=−μdΩN(j+1)(θ)+αΨN−1(j+1)(θ)+A*(j+1)(θ)ωN−2j+1     ΦN−1(j)(θ)=−μdΦN(j+1)(θ)+A*(j+1)(θ)ψN−2j+1   else     ΖN−1(j)(θ)=αϒN−1(j)(θ)+A*(j)(θ)τN−2+jΖN−1(j−1)(θ)(μd−θ)     ΩN−1(j)(θ)=μdΩN(j)(θ)+αΨN−1(j)(θ)+A*(j)(θ)ωN−2+jΩN−1(j−1)(θ)(μd−θ)     ΦN−1(j)(θ)=μdΦN(j)(θ)+A*(j)(θ)ψN−2+jΦN−1(j−1)(θ)(μd−θ)   end if  else if 1≤n≤N−2then   if θ=μdthen     Ζn(j)(θ)=−μdΖn+1(j+1)(θ)+αϒn(j+1)(θ)+A*(j+1)(θ)τn−1j+1     Ωn(j)(θ)=−μdΩn+1(j+1)(θ)+αΨn(j+1)(θ)+A*(j+1)(θ)ωn−1j+1     Φn(j)(θ)=−μdΦn+1(j+1)(θ)+A*(j+1)(θ)ψn−1j+1   else     Ζn(j)(θ)=μdΖn+1(j)(θ)+αϒn(j)(θ)+A*(j)(θ)τn−1+jΖn(j−1)(θ)(μd−θ)     Ωn(j)(θ)=μdΩn+1(j)(θ)+αΨn(j)(θ)+A*(j)(θ)ωn−1+jΩn(j−1)(θ)(μd−θ)     Φn(j)(θ)=μdΦn+1(j)(θ)+A*(j)(θ)ψn−1+jΦn(j−1)(θ)(μd−θ)   end if  end if  if n=Kthen   if θ=μdthen     ΩK(j)(θ)=−αΨK(j+1)(θ)+A*(j+1)(θ)ωK−1j+1     ΦK(j)(θ)=−A*(j+1)(θ)(ψK−1+ψK)j+1   else     ΩK(j)(θ)=αΨK(j)(θ)+A*(j)(θ)ωK−1+jΩK(j−1)(θ)(μd−θ)     ΦK(j)(θ)=A*(j)(θ)(ψK−1+ψK)+jΦK(j−1)(θ)(μd−θ)   end if  else if N≤n≤K−1then   if θ=μdthen     Ωn(j)(θ)=−μdΩn+1(j+1)(θ)+αΨn(j+1)(θ)+A*(j+1)(θ)ωn−1j+1,     Φn(j)(θ)=−μdΦn+1(j+1)(θ)+A*(j+1)(θ)ψn−1j+1,   else     Ωn(j)(θ)=μdΩn+1(j)(θ)+αΨn(j)(θ)+A*(j)(θ)ωn−1+jΩn(j−1)(θ)(μd−θ),     Φn(j)(θ)=μdΦn+1(j)(θ)+A*(j)(θ)ψn−1+jΦn(j−1)(θ)(μd−θ),   end if  else if n=N−1then   if θ=μdthen     ΖN−1(j)(θ)=−αϒN−1(j+1)(θ)+A*(j+1)(θ)τN−2j+1     ΩN−1(j)(θ)=−μdΩN(j+1)(θ)+αΨN−1(j+1)(θ)+A*(j+1)(θ)ωN−2j+1     ΦN−1(j)(θ)=−μdΦN(j+1)(θ)+A*(j+1)(θ)ψN−2j+1   else     ΖN−1(j)(θ)=αϒN−1(j)(θ)+A*(j)(θ)τN−2+jΖN−1(j−1)(θ)(μd−θ)     ΩN−1(j)(θ)=μdΩN(j)(θ)+αΨN−1(j)(θ)+A*(j)(θ)ωN−2+jΩN−1(j−1)(θ)(μd−θ)     ΦN−1(j)(θ)=μdΦN(j)(θ)+A*(j)(θ)ψN−2+jΦN−1(j−1)(θ)(μd−θ)   end if  else if 1≤n≤N−2then   if θ=μdthen     Ζn(j)(θ)=−μdΖn+1(j+1)(θ)+αϒn(j+1)(θ)+A*(j+1)(θ)τn−1j+1     Ωn(j)(θ)=−μdΩn+1(j+1)(θ)+αΨn(j+1)(θ)+A*(j+1)(θ)ωn−1j+1     Φn(j)(θ)=−μdΦn+1(j+1)(θ)+A*(j+1)(θ)ψn−1j+1   else     Ζn(j)(θ)=μdΖn+1(j)(θ)+αϒn(j)(θ)+A*(j)(θ)τn−1+jΖn(j−1)(θ)(μd−θ)     Ωn(j)(θ)=μdΩn+1(j)(θ)+αΨn(j)(θ)+A*(j)(θ)ωn−1+jΩn(j−1)(θ)(μd−θ)     Φn(j)(θ)=μdΦn+1(j)(θ)+A*(j)(θ)ψn−1+jΦn(j−1)(θ)(μd−θ)   end if  end if View Large  if n=Kthen   if θ=μdthen     ΩK(j)(θ)=−αΨK(j+1)(θ)+A*(j+1)(θ)ωK−1j+1     ΦK(j)(θ)=−A*(j+1)(θ)(ψK−1+ψK)j+1   else     ΩK(j)(θ)=αΨK(j)(θ)+A*(j)(θ)ωK−1+jΩK(j−1)(θ)(μd−θ)     ΦK(j)(θ)=A*(j)(θ)(ψK−1+ψK)+jΦK(j−1)(θ)(μd−θ)   end if  else if N≤n≤K−1then   if θ=μdthen     Ωn(j)(θ)=−μdΩn+1(j+1)(θ)+αΨn(j+1)(θ)+A*(j+1)(θ)ωn−1j+1,     Φn(j)(θ)=−μdΦn+1(j+1)(θ)+A*(j+1)(θ)ψn−1j+1,   else     Ωn(j)(θ)=μdΩn+1(j)(θ)+αΨn(j)(θ)+A*(j)(θ)ωn−1+jΩn(j−1)(θ)(μd−θ),     Φn(j)(θ)=μdΦn+1(j)(θ)+A*(j)(θ)ψn−1+jΦn(j−1)(θ)(μd−θ),   end if  else if n=N−1then   if θ=μdthen     ΖN−1(j)(θ)=−αϒN−1(j+1)(θ)+A*(j+1)(θ)τN−2j+1     ΩN−1(j)(θ)=−μdΩN(j+1)(θ)+αΨN−1(j+1)(θ)+A*(j+1)(θ)ωN−2j+1     ΦN−1(j)(θ)=−μdΦN(j+1)(θ)+A*(j+1)(θ)ψN−2j+1   else     ΖN−1(j)(θ)=αϒN−1(j)(θ)+A*(j)(θ)τN−2+jΖN−1(j−1)(θ)(μd−θ)     ΩN−1(j)(θ)=μdΩN(j)(θ)+αΨN−1(j)(θ)+A*(j)(θ)ωN−2+jΩN−1(j−1)(θ)(μd−θ)     ΦN−1(j)(θ)=μdΦN(j)(θ)+A*(j)(θ)ψN−2+jΦN−1(j−1)(θ)(μd−θ)   end if  else if 1≤n≤N−2then   if θ=μdthen     Ζn(j)(θ)=−μdΖn+1(j+1)(θ)+αϒn(j+1)(θ)+A*(j+1)(θ)τn−1j+1     Ωn(j)(θ)=−μdΩn+1(j+1)(θ)+αΨn(j+1)(θ)+A*(j+1)(θ)ωn−1j+1     Φn(j)(θ)=−μdΦn+1(j+1)(θ)+A*(j+1)(θ)ψn−1j+1   else     Ζn(j)(θ)=μdΖn+1(j)(θ)+αϒn(j)(θ)+A*(j)(θ)τn−1+jΖn(j−1)(θ)(μd−θ)     Ωn(j)(θ)=μdΩn+1(j)(θ)+αΨn(j)(θ)+A*(j)(θ)ωn−1+jΩn(j−1)(θ)(μd−θ)     Φn(j)(θ)=μdΦn+1(j)(θ)+A*(j)(θ)ψn−1+jΦn(j−1)(θ)(μd−θ)   end if  end if  if n=Kthen   if θ=μdthen     ΩK(j)(θ)=−αΨK(j+1)(θ)+A*(j+1)(θ)ωK−1j+1     ΦK(j)(θ)=−A*(j+1)(θ)(ψK−1+ψK)j+1   else     ΩK(j)(θ)=αΨK(j)(θ)+A*(j)(θ)ωK−1+jΩK(j−1)(θ)(μd−θ)     ΦK(j)(θ)=A*(j)(θ)(ψK−1+ψK)+jΦK(j−1)(θ)(μd−θ)   end if  else if N≤n≤K−1then   if θ=μdthen     Ωn(j)(θ)=−μdΩn+1(j+1)(θ)+αΨn(j+1)(θ)+A*(j+1)(θ)ωn−1j+1,     Φn(j)(θ)=−μdΦn+1(j+1)(θ)+A*(j+1)(θ)ψn−1j+1,   else     Ωn(j)(θ)=μdΩn+1(j)(θ)+αΨn(j)(θ)+A*(j)(θ)ωn−1+jΩn(j−1)(θ)(μd−θ),     Φn(j)(θ)=μdΦn+1(j)(θ)+A*(j)(θ)ψn−1+jΦn(j−1)(θ)(μd−θ),   end if  else if n=N−1then   if θ=μdthen     ΖN−1(j)(θ)=−αϒN−1(j+1)(θ)+A*(j+1)(θ)τN−2j+1     ΩN−1(j)(θ)=−μdΩN(j+1)(θ)+αΨN−1(j+1)(θ)+A*(j+1)(θ)ωN−2j+1     ΦN−1(j)(θ)=−μdΦN(j+1)(θ)+A*(j+1)(θ)ψN−2j+1   else     ΖN−1(j)(θ)=αϒN−1(j)(θ)+A*(j)(θ)τN−2+jΖN−1(j−1)(θ)(μd−θ)     ΩN−1(j)(θ)=μdΩN(j)(θ)+αΨN−1(j)(θ)+A*(j)(θ)ωN−2+jΩN−1(j−1)(θ)(μd−θ)     ΦN−1(j)(θ)=μdΦN(j)(θ)+A*(j)(θ)ψN−2+jΦN−1(j−1)(θ)(μd−θ)   end if  else if 1≤n≤N−2then   if θ=μdthen     Ζn(j)(θ)=−μdΖn+1(j+1)(θ)+αϒn(j+1)(θ)+A*(j+1)(θ)τn−1j+1     Ωn(j)(θ)=−μdΩn+1(j+1)(θ)+αΨn(j+1)(θ)+A*(j+1)(θ)ωn−1j+1     Φn(j)(θ)=−μdΦn+1(j+1)(θ)+A*(j+1)(θ)ψn−1j+1   else     Ζn(j)(θ)=μdΖn+1(j)(θ)+αϒn(j)(θ)+A*(j)(θ)τn−1+jΖn(j−1)(θ)(μd−θ)     Ωn(j)(θ)=μdΩn+1(j)(θ)+αΨn(j)(θ)+A*(j)(θ)ωn−1+jΩn(j−1)(θ)(μd−θ)     Φn(j)(θ)=μdΦn+1(j)(θ)+A*(j)(θ)ψn−1+jΦn(j−1)(θ)(μd−θ)   end if  end if View Large Step 3: Substituting θ=μb+α into Equation (13), then P0,N−1(0) can be expressed in terms of P1,K(0) as follows: P0,N−1(0)=ΔP1,K(0), where Δ=ξ1−μbΨ2(μb+α)μbϒ2(μb+α)−ϕ1. Step 4: Substituting θ=β into Equation (18), P2,K(0) can be expressed in terms of P1,K(0) as follows: P2,K(0)=ΘP1,K(0), where Θ=(τ0−μdΖ1(β))Δ−(μdΩ1(β)−ω0)μdΦ1(β)−ψ0. Step 5: Determine the only unknown quantity P1,K(0). From Equation (25), we know that P1,K(0)=λNΔ+∑n=1N−1(ξn+ϕnΔ)+∑n=NKξn+∑n=0N−1(τnΔ+ωn+ψnΘ)+∑n=NK(ωn+ψnΘ). Step 6: Compute the probabilities P0,n(0), P1,n(0) and P2,n(0) as follows: P0,n(0)=ΔP1,K(0),0≤n≤N−1, P1,n(0)={ξnP1,K(0),ifN≤n≤K,(ξn+ϕnΔ)P1,K(0),if1≤n≤N−1, P2,n(0)={(ωn+ψnΘ)P1,K(0),ifN≤n≤K,(τnΔ+ωn+ψnΘ)P1,K(0),if0≤n≤N−1. Step 7: Compute the pre-arrival epoch probabilities P0,n−, P1,n− and P2,n− using Equations (42–44). 5. STEADY-STATE PROBABILITY DISTRIBUTION AT ARBITRARY EPOCHS To obtain the steady-state probabilities at arbitrary epochs, we begin by differentiating Equations (12) with respect to θ and then setting θ=0. It follows P0,n=−A*(1)(0)P0,n−1(0)=a1P0,n−1(0)=P0,n−1−,1≤n≤N−1. (45) From Equations (21) and (45), one can easily get P0,1=P0,2=⋯=P0,N−1. (46) Next, we develop relations between pre-arrival and arbitrary epoch probabilities. Substituting θ=0 into Equations (13–20) and in conjunction with Equations (42–44), we get P1,1=μbμb+αP1,2−λμb+αP1,1−, (47) P1,n=1μb+αμbP1,n+1+λμb+α(P1,n−1−−P1,n−),2≤n≤N−1, (48) P1,N=1μb+αμbP1,N+1+λμb+α(P0,N−1−+P1,N−1−−P1,N−), (49) P1,n=1μb+αμbP1,n+1+λμb+α(P1,n−1−−P1,n−),N+1≤n≤K−1, (50) P1,K=λμb+αP1,K−1−, (51) P2,0=μdβP2,1−λβP2,0−, (52) P2,n=P2,n+1+αμdP1,n+λμd(P2,n−1−−P2,n−),1≤n≤K−1, (53) P2,K=αμdP1,K+λμdP2,K−1−. (54) Once the pre-arrival epoch probabilities are known, the steady-state probabilities at arbitrary epochs can be obtained using Equations (46–54). Remark 2 When α=0, we can obtain the results of the N-policy GI/M/1/K queue with a reliable server (see Ke and Wang [3]). 6. SYSTEM PERFORMANCE MEASURES AND SENSITIVITY ANALYSIS We develop a number of system performance measures based on steady-state probabilities to evaluate the proposed N-policy GI/M/1/K queue with working breakdowns. These measures can be used to improve the effectiveness of the system. Next, we present numerical results to explore the sensitivity of system performance measures. 6.1. System performance measures Let us define the system performance measures of the queueing model under study as follows: LS≡ the expected number of customers in the system; PL≡ the blocking probability of the system; WS≡ the expected waiting time in the system, PI≡ the probability that the server is turned off, PN≡ the probability that the server is in normal busy period, PD≡ the probability that the server is in breakdown period. LS≡ the expected number of customers in the system; PL≡ the blocking probability of the system; WS≡ the expected waiting time in the system, PI≡ the probability that the server is turned off, PN≡ the probability that the server is in normal busy period, PD≡ the probability that the server is in breakdown period. View Large LS≡ the expected number of customers in the system; PL≡ the blocking probability of the system; WS≡ the expected waiting time in the system, PI≡ the probability that the server is turned off, PN≡ the probability that the server is in normal busy period, PD≡ the probability that the server is in breakdown period. LS≡ the expected number of customers in the system; PL≡ the blocking probability of the system; WS≡ the expected waiting time in the system, PI≡ the probability that the server is turned off, PN≡ the probability that the server is in normal busy period, PD≡ the probability that the server is in breakdown period. View Large The expressions for LS, PL, WS, PI, PN and PD are given by LS=∑n=1N−1n⋅P0,n+∑n=1Kn⋅(P1,n+P2,n)=N(N−1)2P0,1+∑n=1Kn⋅(P1,n+P2,n). (55) PL=P1,K−+P2,K−, (56) WS=LSλeff, (57) PI=∑n=0N−1P0,n, (58) PN=∑n=1KP1,n, (59) PD=∑n=0KP2,n, (60) where λeff=λ(1−PL) represents the effective arrival rate. 6.2. Sensitivity analysis In the following, we present numerical examples illustrating the effects of various system parameters on some performance measures, including the expected number of customers in the system (LS), the blocking probability of the system (PL) and the expected waiting time in the system (WS). Three inter-arrival time distributions are considered: exponential distribution with a*(s)=λ/(λ+s), three-stage Erlang distribution with a*(s)=(3λ/(3λ+s))3 and deterministic distribution with a*(s)=e−s/λ. We fix K=12, N=3,6,9 and consider the following five cases: Case 1: μb=5.0, μd=2.0, α=0.1, β=4.0, and select various values of λ. Case 2: λ=2.0, μd=2.0, α=0.1, β=4.0, and select various values of μb. Case 3: λ=2.0, μb=5.0, α=0.1, β=4.0, and select various values of μd. Case 4: λ=2.0, μb=5.0, μd=2.0, β=4.0, and select various values of α. Case 5: λ=2.0, μb=5.0, μd=2.0, α=0.1, and select various values of β. Tables 1–5 summarize the numerical results for the above five cases, illustrating the effects induced by changes in various system parameters under three inter-arrival time distributions. It can be seen in Table 1 that LS, WS and PL increase with an increase in the mean arrival rate λ under all three inter-arrival time distributions. Intuitively, we know that an increase in the mean arrival rate results in a more congested system. Tables 2 and 3 show that LS, WS and PL decrease with an increase in the mean service rates μb and μd under all three inter-arrival time distributions. In Table 4, one can see that the larger the mean breakdown rate α is, the larger LS, WS and PL are under all three inter-arrival time distributions. Thus, larger values of α could have a negative effect on system performance measures. Table 5 shows that LS, WS and PL tend to decrease when the mean repair rate β increases, except for N=9. In Tables 1–5, we observe that LS, WS, and PL tend to increase with an increase in threshold value N under all three inter-arrival time distributions. One way to enhance customer satisfaction would be to reduce the waiting time of customers. Increasing the values of μb and μd and decreasing the value of threshold N would be expected to reduce waiting time. The above sensitivity analysis could be highly beneficial for performance evaluations in actual implementations. Table 1. Performance measures for various values of λ under three different inter-arrival time distributions (Case 1). λ N=3 N=6 N=9 LS WS PL LS WS PL LS WS PL N-policy M/M/1/12 queue with working breakdowns 2.0 3.7108 1.9244 0.0359 5.0466 2.6453 0.0461 6.1066 3.2358 0.0564 2.5 7.4607 3.6001 0.1711 8.0104 3.9073 0.1800 8.3980 4.1289 0.1864 3.0 9.7451 4.7841 0.3210 9.8562 4.8545 0.3232 9.9366 4.9055 0.3248 3.5 10.6073 5.2716 0.4251 10.6275 5.2854 0.4255 10.6428 5.2959 0.4258 4.0 10.9863 5.4826 0.4990 10.9903 5.4854 0.4991 10.9935 5.4877 0.4992 N-policy E3/M/1/12 queue with working breakdowns 2.0 3.9893 2.0514 0.0277 5.3458 2.7727 0.0360 6.3542 3.3214 0.0434 2.5 8.9416 4.4016 0.1874 9.1962 4.5487 0.1913 9.3661 4.6468 0.1938 3.0 10.6166 5.2919 0.3313 10.6354 5.3039 0.3316 10.6494 5.3126 0.3318 3.5 11.1056 5.5500 0.4283 11.1073 5.5511 0.4283 11.1086 5.5520 0.4283 4.0 11.3326 5.6658 0.5000 11.3328 5.6659 0.5000 11.3330 5.6660 0.5000 N-policy D/M/1/12 queue with working breakdowns 2.0 4.2456 2.1731 0.0231 5.5749 2.8737 0.0300 6.5268 3.3853 0.0360 2.5 9.8235 4.8824 0.1952 9.9222 4.9402 0.1966 9.9880 4.9785 0.1975 3.0 10.9912 5.4928 0.3330 10.9942 5.4947 0.3330 10.9965 5.4961 0.3331 3.5 11.3328 5.6662 0.4285 11.3330 5.6663 0.4285 11.3331 5.6663 0.4285 4.0 11.5000 5.7500 0.5000 11.5000 5.7500 0.5000 11.5000 5.7500 0.5000 λ N=3 N=6 N=9 LS WS PL LS WS PL LS WS PL N-policy M/M/1/12 queue with working breakdowns 2.0 3.7108 1.9244 0.0359 5.0466 2.6453 0.0461 6.1066 3.2358 0.0564 2.5 7.4607 3.6001 0.1711 8.0104 3.9073 0.1800 8.3980 4.1289 0.1864 3.0 9.7451 4.7841 0.3210 9.8562 4.8545 0.3232 9.9366 4.9055 0.3248 3.5 10.6073 5.2716 0.4251 10.6275 5.2854 0.4255 10.6428 5.2959 0.4258 4.0 10.9863 5.4826 0.4990 10.9903 5.4854 0.4991 10.9935 5.4877 0.4992 N-policy E3/M/1/12 queue with working breakdowns 2.0 3.9893 2.0514 0.0277 5.3458 2.7727 0.0360 6.3542 3.3214 0.0434 2.5 8.9416 4.4016 0.1874 9.1962 4.5487 0.1913 9.3661 4.6468 0.1938 3.0 10.6166 5.2919 0.3313 10.6354 5.3039 0.3316 10.6494 5.3126 0.3318 3.5 11.1056 5.5500 0.4283 11.1073 5.5511 0.4283 11.1086 5.5520 0.4283 4.0 11.3326 5.6658 0.5000 11.3328 5.6659 0.5000 11.3330 5.6660 0.5000 N-policy D/M/1/12 queue with working breakdowns 2.0 4.2456 2.1731 0.0231 5.5749 2.8737 0.0300 6.5268 3.3853 0.0360 2.5 9.8235 4.8824 0.1952 9.9222 4.9402 0.1966 9.9880 4.9785 0.1975 3.0 10.9912 5.4928 0.3330 10.9942 5.4947 0.3330 10.9965 5.4961 0.3331 3.5 11.3328 5.6662 0.4285 11.3330 5.6663 0.4285 11.3331 5.6663 0.4285 4.0 11.5000 5.7500 0.5000 11.5000 5.7500 0.5000 11.5000 5.7500 0.5000 Table 1. Performance measures for various values of λ under three different inter-arrival time distributions (Case 1). λ N=3 N=6 N=9 LS WS PL LS WS PL LS WS PL N-policy M/M/1/12 queue with working breakdowns 2.0 3.7108 1.9244 0.0359 5.0466 2.6453 0.0461 6.1066 3.2358 0.0564 2.5 7.4607 3.6001 0.1711 8.0104 3.9073 0.1800 8.3980 4.1289 0.1864 3.0 9.7451 4.7841 0.3210 9.8562 4.8545 0.3232 9.9366 4.9055 0.3248 3.5 10.6073 5.2716 0.4251 10.6275 5.2854 0.4255 10.6428 5.2959 0.4258 4.0 10.9863 5.4826 0.4990 10.9903 5.4854 0.4991 10.9935 5.4877 0.4992 N-policy E3/M/1/12 queue with working breakdowns 2.0 3.9893 2.0514 0.0277 5.3458 2.7727 0.0360 6.3542 3.3214 0.0434 2.5 8.9416 4.4016 0.1874 9.1962 4.5487 0.1913 9.3661 4.6468 0.1938 3.0 10.6166 5.2919 0.3313 10.6354 5.3039 0.3316 10.6494 5.3126 0.3318 3.5 11.1056 5.5500 0.4283 11.1073 5.5511 0.4283 11.1086 5.5520 0.4283 4.0 11.3326 5.6658 0.5000 11.3328 5.6659 0.5000 11.3330 5.6660 0.5000 N-policy D/M/1/12 queue with working breakdowns 2.0 4.2456 2.1731 0.0231 5.5749 2.8737 0.0300 6.5268 3.3853 0.0360 2.5 9.8235 4.8824 0.1952 9.9222 4.9402 0.1966 9.9880 4.9785 0.1975 3.0 10.9912 5.4928 0.3330 10.9942 5.4947 0.3330 10.9965 5.4961 0.3331 3.5 11.3328 5.6662 0.4285 11.3330 5.6663 0.4285 11.3331 5.6663 0.4285 4.0 11.5000 5.7500 0.5000 11.5000 5.7500 0.5000 11.5000 5.7500 0.5000 λ N=3 N=6 N=9 LS WS PL LS WS PL LS WS PL N-policy M/M/1/12 queue with working breakdowns 2.0 3.7108 1.9244 0.0359 5.0466 2.6453 0.0461 6.1066 3.2358 0.0564 2.5 7.4607 3.6001 0.1711 8.0104 3.9073 0.1800 8.3980 4.1289 0.1864 3.0 9.7451 4.7841 0.3210 9.8562 4.8545 0.3232 9.9366 4.9055 0.3248 3.5 10.6073 5.2716 0.4251 10.6275 5.2854 0.4255 10.6428 5.2959 0.4258 4.0 10.9863 5.4826 0.4990 10.9903 5.4854 0.4991 10.9935 5.4877 0.4992 N-policy E3/M/1/12 queue with working breakdowns 2.0 3.9893 2.0514 0.0277 5.3458 2.7727 0.0360 6.3542 3.3214 0.0434 2.5 8.9416 4.4016 0.1874 9.1962 4.5487 0.1913 9.3661 4.6468 0.1938 3.0 10.6166 5.2919 0.3313 10.6354 5.3039 0.3316 10.6494 5.3126 0.3318 3.5 11.1056 5.5500 0.4283 11.1073 5.5511 0.4283 11.1086 5.5520 0.4283 4.0 11.3326 5.6658 0.5000 11.3328 5.6659 0.5000 11.3330 5.6660 0.5000 N-policy D/M/1/12 queue with working breakdowns 2.0 4.2456 2.1731 0.0231 5.5749 2.8737 0.0300 6.5268 3.3853 0.0360 2.5 9.8235 4.8824 0.1952 9.9222 4.9402 0.1966 9.9880 4.9785 0.1975 3.0 10.9912 5.4928 0.3330 10.9942 5.4947 0.3330 10.9965 5.4961 0.3331 3.5 11.3328 5.6662 0.4285 11.3330 5.6663 0.4285 11.3331 5.6663 0.4285 4.0 11.5000 5.7500 0.5000 11.5000 5.7500 0.5000 11.5000 5.7500 0.5000 Table 2. Performance measures for various values of μb under three different inter-arrival time distributions (Case 2). μb N=3 N=6 N=9 LS WS PL LS WS PL LS WS PL N-policy M/M/1/12 queue with working breakdowns 4.5 3.9489 2.0545 0.0389 5.2316 2.7516 0.0494 6.2436 3.3216 0.0602 5.0 3.7108 1.9244 0.0359 5.0466 2.6453 0.0461 6.1066 3.2358 0.0564 5.5 3.5116 1.8163 0.0333 4.8888 2.5552 0.0434 5.9882 3.1625 0.0532 6.0 3.3417 1.7245 0.0311 4.7517 2.4774 0.0410 5.8844 3.0986 0.0505 6.5 3.1948 1.6454 0.0292 4.6312 2.4093 0.0389 5.7923 3.0423 0.0480 N-policy E3/M/1/12 queue with working breakdowns 4.5 4.2058 2.1673 0.0297 5.5060 2.8618 0.0380 6.4722 3.3908 0.0456 5.0 3.9893 2.0514 0.0277 5.3458 2.7727 0.0360 6.3542 3.3214 0.0434 5.5 3.8044 1.9528 0.0259 5.2068 2.6956 0.0342 6.2517 3.2612 0.0415 6.0 3.6438 1.8674 0.0244 5.0839 2.6277 0.0326 6.1607 3.2080 0.0398 6.5 3.5023 1.7924 0.0231 4.9740 2.5670 0.0312 6.0790 3.1604 0.0382 N-policy D/M/1/12 queue with working breakdowns 4.5 4.4445 2.2783 0.0246 5.7168 2.9512 0.0314 6.6291 3.4437 0.0375 5.0 4.2456 2.1731 0.0231 5.5749 2.8737 0.0300 6.5268 3.3853 0.0360 5.5 4.0728 2.0818 0.0218 5.4498 2.8054 0.0287 6.4361 3.3334 0.0346 6.0 3.9200 2.0014 0.0207 5.3377 2.7444 0.0275 6.3001 3.2596 0.0336 6.5 3.7832 1.9295 0.0196 5.2360 2.6892 0.0265 6.0449 3.1244 0.0326 μb N=3 N=6 N=9 LS WS PL LS WS PL LS WS PL N-policy M/M/1/12 queue with working breakdowns 4.5 3.9489 2.0545 0.0389 5.2316 2.7516 0.0494 6.2436 3.3216 0.0602 5.0 3.7108 1.9244 0.0359 5.0466 2.6453 0.0461 6.1066 3.2358 0.0564 5.5 3.5116 1.8163 0.0333 4.8888 2.5552 0.0434 5.9882 3.1625 0.0532 6.0 3.3417 1.7245 0.0311 4.7517 2.4774 0.0410 5.8844 3.0986 0.0505 6.5 3.1948 1.6454 0.0292 4.6312 2.4093 0.0389 5.7923 3.0423 0.0480 N-policy E3/M/1/12 queue with working breakdowns 4.5 4.2058 2.1673 0.0297 5.5060 2.8618 0.0380 6.4722 3.3908 0.0456 5.0 3.9893 2.0514 0.0277 5.3458 2.7727 0.0360 6.3542 3.3214 0.0434 5.5 3.8044 1.9528 0.0259 5.2068 2.6956 0.0342 6.2517 3.2612 0.0415 6.0 3.6438 1.8674 0.0244 5.0839 2.6277 0.0326 6.1607 3.2080 0.0398 6.5 3.5023 1.7924 0.0231 4.9740 2.5670 0.0312 6.0790 3.1604 0.0382 N-policy D/M/1/12 queue with working breakdowns 4.5 4.4445 2.2783 0.0246 5.7168 2.9512 0.0314 6.6291 3.4437 0.0375 5.0 4.2456 2.1731 0.0231 5.5749 2.8737 0.0300 6.5268 3.3853 0.0360 5.5 4.0728 2.0818 0.0218 5.4498 2.8054 0.0287 6.4361 3.3334 0.0346 6.0 3.9200 2.0014 0.0207 5.3377 2.7444 0.0275 6.3001 3.2596 0.0336 6.5 3.7832 1.9295 0.0196 5.2360 2.6892 0.0265 6.0449 3.1244 0.0326 Table 2. Performance measures for various values of μb under three different inter-arrival time distributions (Case 2). μb N=3 N=6 N=9 LS WS PL LS WS PL LS WS PL N-policy M/M/1/12 queue with working breakdowns 4.5 3.9489 2.0545 0.0389 5.2316 2.7516 0.0494 6.2436 3.3216 0.0602 5.0 3.7108 1.9244 0.0359 5.0466 2.6453 0.0461 6.1066 3.2358 0.0564 5.5 3.5116 1.8163 0.0333 4.8888 2.5552 0.0434 5.9882 3.1625 0.0532 6.0 3.3417 1.7245 0.0311 4.7517 2.4774 0.0410 5.8844 3.0986 0.0505 6.5 3.1948 1.6454 0.0292 4.6312 2.4093 0.0389 5.7923 3.0423 0.0480 N-policy E3/M/1/12 queue with working breakdowns 4.5 4.2058 2.1673 0.0297 5.5060 2.8618 0.0380 6.4722 3.3908 0.0456 5.0 3.9893 2.0514 0.0277 5.3458 2.7727 0.0360 6.3542 3.3214 0.0434 5.5 3.8044 1.9528 0.0259 5.2068 2.6956 0.0342 6.2517 3.2612 0.0415 6.0 3.6438 1.8674 0.0244 5.0839 2.6277 0.0326 6.1607 3.2080 0.0398 6.5 3.5023 1.7924 0.0231 4.9740 2.5670 0.0312 6.0790 3.1604 0.0382 N-policy D/M/1/12 queue with working breakdowns 4.5 4.4445 2.2783 0.0246 5.7168 2.9512 0.0314 6.6291 3.4437 0.0375 5.0 4.2456 2.1731 0.0231 5.5749 2.8737 0.0300 6.5268 3.3853 0.0360 5.5 4.0728 2.0818 0.0218 5.4498 2.8054 0.0287 6.4361 3.3334 0.0346 6.0 3.9200 2.0014 0.0207 5.3377 2.7444 0.0275 6.3001 3.2596 0.0336 6.5 3.7832 1.9295 0.0196 5.2360 2.6892 0.0265 6.0449 3.1244 0.0326 μb N=3 N=6 N=9 LS WS PL LS WS PL LS WS PL N-policy M/M/1/12 queue with working breakdowns 4.5 3.9489 2.0545 0.0389 5.2316 2.7516 0.0494 6.2436 3.3216 0.0602 5.0 3.7108 1.9244 0.0359 5.0466 2.6453 0.0461 6.1066 3.2358 0.0564 5.5 3.5116 1.8163 0.0333 4.8888 2.5552 0.0434 5.9882 3.1625 0.0532 6.0 3.3417 1.7245 0.0311 4.7517 2.4774 0.0410 5.8844 3.0986 0.0505 6.5 3.1948 1.6454 0.0292 4.6312 2.4093 0.0389 5.7923 3.0423 0.0480 N-policy E3/M/1/12 queue with working breakdowns 4.5 4.2058 2.1673 0.0297 5.5060 2.8618 0.0380 6.4722 3.3908 0.0456 5.0 3.9893 2.0514 0.0277 5.3458 2.7727 0.0360 6.3542 3.3214 0.0434 5.5 3.8044 1.9528 0.0259 5.2068 2.6956 0.0342 6.2517 3.2612 0.0415 6.0 3.6438 1.8674 0.0244 5.0839 2.6277 0.0326 6.1607 3.2080 0.0398 6.5 3.5023 1.7924 0.0231 4.9740 2.5670 0.0312 6.0790 3.1604 0.0382 N-policy D/M/1/12 queue with working breakdowns 4.5 4.4445 2.2783 0.0246 5.7168 2.9512 0.0314 6.6291 3.4437 0.0375 5.0 4.2456 2.1731 0.0231 5.5749 2.8737 0.0300 6.5268 3.3853 0.0360 5.5 4.0728 2.0818 0.0218 5.4498 2.8054 0.0287 6.4361 3.3334 0.0346 6.0 3.9200 2.0014 0.0207 5.3377 2.7444 0.0275 6.3001 3.2596 0.0336 6.5 3.7832 1.9295 0.0196 5.2360 2.6892 0.0265 6.0449 3.1244 0.0326 Table 3. Performance measures for various values of μd under three different inter-arrival time distributions (Case 3). μd N=3 N=6 N=9 LS WS PL LS WS PL LS WS PL N-policy M/M/1/12 queue with working breakdowns 1.0 10.9747 10.9444 0.4986 10.9819 10.9552 0.4988 10.9873 10.9625 0.4989 1.5 8.2923 5.3272 0.2217 8.6951 5.6400 0.2292 8.9664 5.8510 0.2338 2.0 3.7108 1.9244 0.0359 5.0466 2.6453 0.0461 6.1066 3.2358 0.0564 2.5 2.1947 1.1022 0.0044 3.6783 1.8534 0.0077 5.0314 2.5520 0.0142 3.0 1.8434 0.9224 0.0007 3.3246 1.6654 0.0019 4.7448 2.3870 0.0061 N-policy E3/M/1/12 queue with working breakdowns 1.0 11.3320 11.3305 0.4999 11.3324 11.3310 0.4999 11.3326 11.3313 0.4999 1.5 9.7290 6.4114 0.2413 9.8601 6.5177 0.2436 9.9440 6.5844 0.2449 2.0 3.9893 2.0514 0.0277 5.3458 2.7727 0.0360 6.3542 3.3214 0.0434 2.5 1.9990 1.0006 0.0011 3.5206 1.7644 0.0023 4.9169 2.4710 0.0051 3.0 1.7179 0.8590 0.0001 3.2167 1.6087 0.0002 4.6628 2.3341 0.0012 N-policy D/M/1/12 queue with working breakdowns 1.0 11.4999 11.4998 0.5000 11.4999 11.4999 0.5000 11.5000 11.4999 0.5000 1.5 10.4179 6.9228 0.2476 10.4540 6.9524 0.2482 10.4777 6.9713 0.2485 2.0 4.2456 2.1731 0.0231 5.5749 2.8737 0.0300 6.5268 3.3853 0.0360 2.5 1.8920 0.9463 0.0003 3.4030 1.7028 0.0007 4.8116 2.4109 0.0021 3.0 1.6769 0.8385 0.0000 3.1629 1.5815 0.0000 4.6095 2.3054 0.0003 μd N=3 N=6 N=9 LS WS PL LS WS PL LS WS PL N-policy M/M/1/12 queue with working breakdowns 1.0 10.9747 10.9444 0.4986 10.9819 10.9552 0.4988 10.9873 10.9625 0.4989 1.5 8.2923 5.3272 0.2217 8.6951 5.6400 0.2292 8.9664 5.8510 0.2338 2.0 3.7108 1.9244 0.0359 5.0466 2.6453 0.0461 6.1066 3.2358 0.0564 2.5 2.1947 1.1022 0.0044 3.6783 1.8534 0.0077 5.0314 2.5520 0.0142 3.0 1.8434 0.9224 0.0007 3.3246 1.6654 0.0019 4.7448 2.3870 0.0061 N-policy E3/M/1/12 queue with working breakdowns 1.0 11.3320 11.3305 0.4999 11.3324 11.3310 0.4999 11.3326 11.3313 0.4999 1.5 9.7290 6.4114 0.2413 9.8601 6.5177 0.2436 9.9440 6.5844 0.2449 2.0 3.9893 2.0514 0.0277 5.3458 2.7727 0.0360 6.3542 3.3214 0.0434 2.5 1.9990 1.0006 0.0011 3.5206 1.7644 0.0023 4.9169 2.4710 0.0051 3.0 1.7179 0.8590 0.0001 3.2167 1.6087 0.0002 4.6628 2.3341 0.0012 N-policy D/M/1/12 queue with working breakdowns 1.0 11.4999 11.4998 0.5000 11.4999 11.4999 0.5000 11.5000 11.4999 0.5000 1.5 10.4179 6.9228 0.2476 10.4540 6.9524 0.2482 10.4777 6.9713 0.2485 2.0 4.2456 2.1731 0.0231 5.5749 2.8737 0.0300 6.5268 3.3853 0.0360 2.5 1.8920 0.9463 0.0003 3.4030 1.7028 0.0007 4.8116 2.4109 0.0021 3.0 1.6769 0.8385 0.0000 3.1629 1.5815 0.0000 4.6095 2.3054 0.0003 Table 3. Performance measures for various values of μd under three different inter-arrival time distributions (Case 3). μd N=3 N=6 N=9 LS WS PL LS WS PL LS WS PL N-policy M/M/1/12 queue with working breakdowns 1.0 10.9747 10.9444 0.4986 10.9819 10.9552 0.4988 10.9873 10.9625 0.4989 1.5 8.2923 5.3272 0.2217 8.6951 5.6400 0.2292 8.9664 5.8510 0.2338 2.0 3.7108 1.9244 0.0359 5.0466 2.6453 0.0461 6.1066 3.2358 0.0564 2.5 2.1947 1.1022 0.0044 3.6783 1.8534 0.0077 5.0314 2.5520 0.0142 3.0 1.8434 0.9224 0.0007 3.3246 1.6654 0.0019 4.7448 2.3870 0.0061 N-policy E3/M/1/12 queue with working breakdowns 1.0 11.3320 11.3305 0.4999 11.3324 11.3310 0.4999 11.3326 11.3313 0.4999 1.5 9.7290 6.4114 0.2413 9.8601 6.5177 0.2436 9.9440 6.5844 0.2449 2.0 3.9893 2.0514 0.0277 5.3458 2.7727 0.0360 6.3542 3.3214 0.0434 2.5 1.9990 1.0006 0.0011 3.5206 1.7644 0.0023 4.9169 2.4710 0.0051 3.0 1.7179 0.8590 0.0001 3.2167 1.6087 0.0002 4.6628 2.3341 0.0012 N-policy D/M/1/12 queue with working breakdowns 1.0 11.4999 11.4998 0.5000 11.4999 11.4999 0.5000 11.5000 11.4999 0.5000 1.5 10.4179 6.9228 0.2476 10.4540 6.9524 0.2482 10.4777 6.9713 0.2485 2.0 4.2456 2.1731 0.0231 5.5749 2.8737 0.0300 6.5268 3.3853 0.0360 2.5 1.8920 0.9463 0.0003 3.4030 1.7028 0.0007 4.8116 2.4109 0.0021 3.0 1.6769 0.8385 0.0000 3.1629 1.5815 0.0000 4.6095 2.3054 0.0003 μd N=3 N=6 N=9 LS WS PL LS WS PL LS WS PL N-policy M/M/1/12 queue with working breakdowns 1.0 10.9747 10.9444 0.4986 10.9819 10.9552 0.4988 10.9873 10.9625 0.4989 1.5 8.2923 5.3272 0.2217 8.6951 5.6400 0.2292 8.9664 5.8510 0.2338 2.0 3.7108 1.9244 0.0359 5.0466 2.6453 0.0461 6.1066 3.2358 0.0564 2.5 2.1947 1.1022 0.0044 3.6783 1.8534 0.0077 5.0314 2.5520 0.0142 3.0 1.8434 0.9224 0.0007 3.3246 1.6654 0.0019 4.7448 2.3870 0.0061 N-policy E3/M/1/12 queue with working breakdowns 1.0 11.3320 11.3305 0.4999 11.3324 11.3310 0.4999 11.3326 11.3313 0.4999 1.5 9.7290 6.4114 0.2413 9.8601 6.5177 0.2436 9.9440 6.5844 0.2449 2.0 3.9893 2.0514 0.0277 5.3458 2.7727 0.0360 6.3542 3.3214 0.0434 2.5 1.9990 1.0006 0.0011 3.5206 1.7644 0.0023 4.9169 2.4710 0.0051 3.0 1.7179 0.8590 0.0001 3.2167 1.6087 0.0002 4.6628 2.3341 0.0012 N-policy D/M/1/12 queue with working breakdowns 1.0 11.4999 11.4998 0.5000 11.4999 11.4999 0.5000 11.5000 11.4999 0.5000 1.5 10.4179 6.9228 0.2476 10.4540 6.9524 0.2482 10.4777 6.9713 0.2485 2.0 4.2456 2.1731 0.0231 5.5749 2.8737 0.0300 6.5268 3.3853 0.0360 2.5 1.8920 0.9463 0.0003 3.4030 1.7028 0.0007 4.8116 2.4109 0.0021 3.0 1.6769 0.8385 0.0000 3.1629 1.5815 0.0000 4.6095 2.3054 0.0003 Table 4. Performance measures for various values of α under three different inter-arrival time distributions (Case 4). α N=3 N=6 N=9 LS WS PL LS WS PL LS WS PL N-policy M/M/1/12 queue with working breakdowns 0.05 2.9714 1.5206 0.0229 4.4243 2.2828 0.0310 5.6524 2.9431 0.0397 0.1 3.7108 1.9244 0.0359 5.0466 2.6453 0.0461 6.1066 3.2358 0.0564 0.2 4.5188 2.3783 0.0500 5.6657 3.0175 0.0612 6.5381 3.5236 0.0722 0.3 4.9514 2.6269 0.0576 5.9753 3.2081 0.0687 6.7498 3.6683 0.0800 0.4 5.2209 2.7837 0.0622 6.1621 3.3246 0.0732 6.8779 3.7570 0.0847 N-policy E3/M/1/12 queue with working breakdowns 0.05 3.1823 1.6207 0.0182 4.6832 2.4021 0.0252 5.8798 3.0355 0.0315 0.1 3.9893 2.0514 0.0277 5.3458 2.7727 0.0360 6.3542 3.3214 0.0434 0.2 4.8141 2.5003 0.0373 5.9514 3.1187 0.0459 6.7673 3.5761 0.0538 0.3 5.2334 2.7319 0.0422 6.2372 3.2845 0.0505 6.9597 3.6965 0.0586 0.4 5.4871 2.8732 0.0451 6.4049 3.3825 0.0532 7.0742 3.7688 0.0615 N-policy D/M/1/12 queue with working breakdowns 0.05 3.4044 1.7294 0.0157 4.9050 2.5070 0.0218 6.0551 3.1117 0.0270 0.1 4.2456 2.1731 0.0231 5.5749 2.8737 0.0300 6.5268 3.3853 0.0360 0.2 5.0554 2.6066 0.0303 6.1479 3.1922 0.0370 6.9078 3.6101 0.0433 0.3 5.4490 2.8196 0.0337 6.4070 3.3377 0.0402 7.0763 3.7107 0.0465 0.4 5.6817 2.9462 0.0358 6.5563 3.4220 0.0420 7.2052 3.7867 0.0486 α N=3 N=6 N=9 LS WS PL LS WS PL LS WS PL N-policy M/M/1/12 queue with working breakdowns 0.05 2.9714 1.5206 0.0229 4.4243 2.2828 0.0310 5.6524 2.9431 0.0397 0.1 3.7108 1.9244 0.0359 5.0466 2.6453 0.0461 6.1066 3.2358 0.0564 0.2 4.5188 2.3783 0.0500 5.6657 3.0175 0.0612 6.5381 3.5236 0.0722 0.3 4.9514 2.6269 0.0576 5.9753 3.2081 0.0687 6.7498 3.6683 0.0800 0.4 5.2209 2.7837 0.0622 6.1621 3.3246 0.0732 6.8779 3.7570 0.0847 N-policy E3/M/1/12 queue with working breakdowns 0.05 3.1823 1.6207 0.0182 4.6832 2.4021 0.0252 5.8798 3.0355 0.0315 0.1 3.9893 2.0514 0.0277 5.3458 2.7727 0.0360 6.3542 3.3214 0.0434 0.2 4.8141 2.5003 0.0373 5.9514 3.1187 0.0459 6.7673 3.5761 0.0538 0.3 5.2334 2.7319 0.0422 6.2372 3.2845 0.0505 6.9597 3.6965 0.0586 0.4 5.4871 2.8732 0.0451 6.4049 3.3825 0.0532 7.0742 3.7688 0.0615 N-policy D/M/1/12 queue with working breakdowns 0.05 3.4044 1.7294 0.0157 4.9050 2.5070 0.0218 6.0551 3.1117 0.0270 0.1 4.2456 2.1731 0.0231 5.5749 2.8737 0.0300 6.5268 3.3853 0.0360 0.2 5.0554 2.6066 0.0303 6.1479 3.1922 0.0370 6.9078 3.6101 0.0433 0.3 5.4490 2.8196 0.0337 6.4070 3.3377 0.0402 7.0763 3.7107 0.0465 0.4 5.6817 2.9462 0.0358 6.5563 3.4220 0.0420 7.2052 3.7867 0.0486 Table 4. Performance measures for various values of α under three different inter-arrival time distributions (Case 4). α N=3 N=6 N=9 LS WS PL LS WS PL LS WS PL N-policy M/M/1/12 queue with working breakdowns 0.05 2.9714 1.5206 0.0229 4.4243 2.2828 0.0310 5.6524 2.9431 0.0397 0.1 3.7108 1.9244 0.0359 5.0466 2.6453 0.0461 6.1066 3.2358 0.0564 0.2 4.5188 2.3783 0.0500 5.6657 3.0175 0.0612 6.5381 3.5236 0.0722 0.3 4.9514 2.6269 0.0576 5.9753 3.2081 0.0687 6.7498 3.6683 0.0800 0.4 5.2209 2.7837 0.0622 6.1621 3.3246 0.0732 6.8779 3.7570 0.0847 N-policy E3/M/1/12 queue with working breakdowns 0.05 3.1823 1.6207 0.0182 4.6832 2.4021 0.0252 5.8798 3.0355 0.0315 0.1 3.9893 2.0514 0.0277 5.3458 2.7727 0.0360 6.3542 3.3214 0.0434 0.2 4.8141 2.5003 0.0373 5.9514 3.1187 0.0459 6.7673 3.5761 0.0538 0.3 5.2334 2.7319 0.0422 6.2372 3.2845 0.0505 6.9597 3.6965 0.0586 0.4 5.4871 2.8732 0.0451 6.4049 3.3825 0.0532 7.0742 3.7688 0.0615 N-policy D/M/1/12 queue with working breakdowns 0.05 3.4044 1.7294 0.0157 4.9050 2.5070 0.0218 6.0551 3.1117 0.0270 0.1 4.2456 2.1731 0.0231 5.5749 2.8737 0.0300 6.5268 3.3853 0.0360 0.2 5.0554 2.6066 0.0303 6.1479 3.1922 0.0370 6.9078 3.6101 0.0433 0.3 5.4490 2.8196 0.0337 6.4070 3.3377 0.0402 7.0763 3.7107 0.0465 0.4 5.6817 2.9462 0.0358 6.5563 3.4220 0.0420 7.2052 3.7867 0.0486 α N=3 N=6 N=9 LS WS PL LS WS PL LS WS PL N-policy M/M/1/12 queue with working breakdowns 0.05 2.9714 1.5206 0.0229 4.4243 2.2828 0.0310 5.6524 2.9431 0.0397 0.1 3.7108 1.9244 0.0359 5.0466 2.6453 0.0461 6.1066 3.2358 0.0564 0.2 4.5188 2.3783 0.0500 5.6657 3.0175 0.0612 6.5381 3.5236 0.0722 0.3 4.9514 2.6269 0.0576 5.9753 3.2081 0.0687 6.7498 3.6683 0.0800 0.4 5.2209 2.7837 0.0622 6.1621 3.3246 0.0732 6.8779 3.7570 0.0847 N-policy E3/M/1/12 queue with working breakdowns 0.05 3.1823 1.6207 0.0182 4.6832 2.4021 0.0252 5.8798 3.0355 0.0315 0.1 3.9893 2.0514 0.0277 5.3458 2.7727 0.0360 6.3542 3.3214 0.0434 0.2 4.8141 2.5003 0.0373 5.9514 3.1187 0.0459 6.7673 3.5761 0.0538 0.3 5.2334 2.7319 0.0422 6.2372 3.2845 0.0505 6.9597 3.6965 0.0586 0.4 5.4871 2.8732 0.0451 6.4049 3.3825 0.0532 7.0742 3.7688 0.0615 N-policy D/M/1/12 queue with working breakdowns 0.05 3.4044 1.7294 0.0157 4.9050 2.5070 0.0218 6.0551 3.1117 0.0270 0.1 4.2456 2.1731 0.0231 5.5749 2.8737 0.0300 6.5268 3.3853 0.0360 0.2 5.0554 2.6066 0.0303 6.1479 3.1922 0.0370 6.9078 3.6101 0.0433 0.3 5.4490 2.8196 0.0337 6.4070 3.3377 0.0402 7.0763 3.7107 0.0465 0.4 5.6817 2.9462 0.0358 6.5563 3.4220 0.0420 7.2052 3.7867 0.0486 Table 5. Performance measures for various values of β under three different inter-arrival time distributions (Case 5). β N=3 N=6 N=9 LS WS PL LS WS PL LS WS PL N-policy M/M/1/12 queue with working breakdowns 2.5 3.8054 1.9770 0.0376 5.0802 2.6659 0.0472 6.1033 3.2363 0.0570 3.0 3.7659 1.9550 0.0369 5.0663 2.6574 0.0468 6.1046 3.2361 0.0568 3.5 3.7347 1.9377 0.0363 5.0552 2.6506 0.0464 6.1057 3.2359 0.0566 4.0 3.7108 1.9244 0.0359 5.0466 2.6453 0.0461 6.1066 3.2358 0.0564 4.5 3.6922 1.9141 0.0355 5.0409 2.6418 0.0459 6.1060 3.2359 0.0565 N-policy E3/M/1/12 queue with working breakdowns 2.5 4.1029 2.1128 0.0290 5.3791 2.7921 0.0367 6.3450 3.3178 0.0438 3.0 4.0557 2.0873 0.0285 5.3655 2.7842 0.0364 6.3487 3.3192 0.0437 3.5 4.0183 2.0670 0.0280 5.3544 2.7777 0.0362 6.3518 3.3205 0.0435 4.0 3.9893 2.0514 0.0277 5.3458 2.7727 0.0360 6.3542 3.3214 0.0434 4.5 3.9662 2.0389 0.0274 5.3391 2.7688 0.0358 6.3556 3.3219 0.0434 N-policy D/M/1/12 queue with working breakdowns 2.5 4.3695 2.2391 0.0243 5.6052 2.8909 0.0305 6.5078 3.3761 0.0362 3.0 4.3172 2.2112 0.0238 5.5924 2.8836 0.0303 6.5194 3.3819 0.0361 3.5 4.2770 2.1897 0.0234 5.5826 2.8780 0.0301 6.5226 3.3833 0.0361 4.0 4.2456 2.1731 0.0231 5.5749 2.8737 0.0300 6.5268 3.3853 0.0360 4.5 4.2205 2.1597 0.0229 5.5689 2.8702 0.0299 6.5302 3.3869 0.0360 β N=3 N=6 N=9 LS WS PL LS WS PL LS WS PL N-policy M/M/1/12 queue with working breakdowns 2.5 3.8054 1.9770 0.0376 5.0802 2.6659 0.0472 6.1033 3.2363 0.0570 3.0 3.7659 1.9550 0.0369 5.0663 2.6574 0.0468 6.1046 3.2361 0.0568 3.5 3.7347 1.9377 0.0363 5.0552 2.6506 0.0464 6.1057 3.2359 0.0566 4.0 3.7108 1.9244 0.0359 5.0466 2.6453 0.0461 6.1066 3.2358 0.0564 4.5 3.6922 1.9141 0.0355 5.0409 2.6418 0.0459 6.1060 3.2359 0.0565 N-policy E3/M/1/12 queue with working breakdowns 2.5 4.1029 2.1128 0.0290 5.3791 2.7921 0.0367 6.3450 3.3178 0.0438 3.0 4.0557 2.0873 0.0285 5.3655 2.7842 0.0364 6.3487 3.3192 0.0437 3.5 4.0183 2.0670 0.0280 5.3544 2.7777 0.0362 6.3518 3.3205 0.0435 4.0 3.9893 2.0514 0.0277 5.3458 2.7727 0.0360 6.3542 3.3214 0.0434 4.5 3.9662 2.0389 0.0274 5.3391 2.7688 0.0358 6.3556 3.3219 0.0434 N-policy D/M/1/12 queue with working breakdowns 2.5 4.3695 2.2391 0.0243 5.6052 2.8909 0.0305 6.5078 3.3761 0.0362 3.0 4.3172 2.2112 0.0238 5.5924 2.8836 0.0303 6.5194 3.3819 0.0361 3.5 4.2770 2.1897 0.0234 5.5826 2.8780 0.0301 6.5226 3.3833 0.0361 4.0 4.2456 2.1731 0.0231 5.5749 2.8737 0.0300 6.5268 3.3853 0.0360 4.5 4.2205 2.1597 0.0229 5.5689 2.8702 0.0299 6.5302 3.3869 0.0360 Table 5. Performance measures for various values of β under three different inter-arrival time distributions (Case 5). β N=3 N=6 N=9 LS WS PL LS WS PL LS WS PL N-policy M/M/1/12 queue with working breakdowns 2.5 3.8054 1.9770 0.0376 5.0802 2.6659 0.0472 6.1033 3.2363 0.0570 3.0 3.7659 1.9550 0.0369 5.0663 2.6574 0.0468 6.1046 3.2361 0.0568 3.5 3.7347 1.9377 0.0363 5.0552 2.6506 0.0464 6.1057 3.2359 0.0566 4.0 3.7108 1.9244 0.0359 5.0466 2.6453 0.0461 6.1066 3.2358 0.0564 4.5 3.6922 1.9141 0.0355 5.0409 2.6418 0.0459 6.1060 3.2359 0.0565 N-policy E3/M/1/12 queue with working breakdowns 2.5 4.1029 2.1128 0.0290 5.3791 2.7921 0.0367 6.3450 3.3178 0.0438 3.0 4.0557 2.0873 0.0285 5.3655 2.7842 0.0364 6.3487 3.3192 0.0437 3.5 4.0183 2.0670 0.0280 5.3544 2.7777 0.0362 6.3518 3.3205 0.0435 4.0 3.9893 2.0514 0.0277 5.3458 2.7727 0.0360 6.3542 3.3214 0.0434 4.5 3.9662 2.0389 0.0274 5.3391 2.7688 0.0358 6.3556 3.3219 0.0434 N-policy D/M/1/12 queue with working breakdowns 2.5 4.3695 2.2391 0.0243 5.6052 2.8909 0.0305 6.5078 3.3761 0.0362 3.0 4.3172 2.2112 0.0238 5.5924 2.8836 0.0303 6.5194 3.3819 0.0361 3.5 4.2770 2.1897 0.0234 5.5826 2.8780 0.0301 6.5226 3.3833 0.0361 4.0 4.2456 2.1731 0.0231 5.5749 2.8737 0.0300 6.5268 3.3853 0.0360 4.5 4.2205 2.1597 0.0229 5.5689 2.8702 0.0299 6.5302 3.3869 0.0360 β N=3 N=6 N=9 LS WS PL LS WS PL LS WS PL N-policy M/M/1/12 queue with working breakdowns 2.5 3.8054 1.9770 0.0376 5.0802 2.6659 0.0472 6.1033 3.2363 0.0570 3.0 3.7659 1.9550 0.0369 5.0663 2.6574 0.0468 6.1046 3.2361 0.0568 3.5 3.7347 1.9377 0.0363 5.0552 2.6506 0.0464 6.1057 3.2359 0.0566 4.0 3.7108 1.9244 0.0359 5.0466 2.6453 0.0461 6.1066 3.2358 0.0564 4.5 3.6922 1.9141 0.0355 5.0409 2.6418 0.0459 6.1060 3.2359 0.0565 N-policy E3/M/1/12 queue with working breakdowns 2.5 4.1029 2.1128 0.0290 5.3791 2.7921 0.0367 6.3450 3.3178 0.0438 3.0 4.0557 2.0873 0.0285 5.3655 2.7842 0.0364 6.3487 3.3192 0.0437 3.5 4.0183 2.0670 0.0280 5.3544 2.7777 0.0362 6.3518 3.3205 0.0435 4.0 3.9893 2.0514 0.0277 5.3458 2.7727 0.0360 6.3542 3.3214 0.0434 4.5 3.9662 2.0389 0.0274 5.3391 2.7688 0.0358 6.3556 3.3219 0.0434 N-policy D/M/1/12 queue with working breakdowns 2.5 4.3695 2.2391 0.0243 5.6052 2.8909 0.0305 6.5078 3.3761 0.0362 3.0 4.3172 2.2112 0.0238 5.5924 2.8836 0.0303 6.5194 3.3819 0.0361 3.5 4.2770 2.1897 0.0234 5.5826 2.8780 0.0301 6.5226 3.3833 0.0361 4.0 4.2456 2.1731 0.0231 5.5749 2.8737 0.0300 6.5268 3.3853 0.0360 4.5 4.2205 2.1597 0.0229 5.5689 2.8702 0.0299 6.5302 3.3869 0.0360 7. CONCLUSIONS In this paper, we used the supplementary variable method to analyze the N-policy GI/M/1/K queue with working breakdowns. We proposed an algorithm for computing the steady-state probabilities at pre-arrival epochs. The steady-state probabilities at arbitrary epochs can be derived using the relationship between pre-arrival and arbitrary epoch probabilities. We also developed system performance measures. Numerical examples were performed to enable sensitivity analysis of the system performance measures. Our findings in this study could be useful in computer, telecommunication, production and manufacturing systems. In future work, it would be interesting to extend the proposed queueing model to include single (multiple) vacation(s) and impatient customers. REFERENCES 1 Yadin , M. and Naor , P. ( 1963 ) Queueing systems with a removable service station . Oper. Res. Q. , 14 , 393 – 405 . Google Scholar CrossRef Search ADS 2 Wang , K.-H. and Ke , J.-C. ( 2000 ) A recursive method to the optimal control of an M/G/1 queueing system with finite capacity and infinite capacity . Appl. Math. Model. , 24 , 899 – 914 . Google Scholar CrossRef Search ADS 3 Ke , J.-C. and Wang , K.-H. ( 2002 ) A recursive method for the N policy G/M/1 queueing system with finite capacity . Eur. J. Oper. Res. , 142 , 577 – 594 . Google Scholar CrossRef Search ADS 4 Jain , M. , Kulshrestha , R. and Maheshwari , S. ( 2004 ) N-policy for a machine repair system with spares and reneging . Appl. Math. Model. , 28 , 513 – 531 . Google Scholar CrossRef Search ADS 5 Tadj , L. and Choudhury , G. ( 2005 ) Optimal design and control of queues . Top , 13 , 359 – 412 . Google Scholar CrossRef Search ADS 6 Wang , T.-Y. , Yang , D.-Y. and Li , M.-J. ( 2010 ) Fuzzy analysis for the N-policy queues with infinite capacity . Int. J. Inf. Manag. Sci , 21 , 41 – 56 . 7 Vijaya Laxmi , P. and Suchitra , V. ( 2014 ) Finite buffer GI/M(n)/1 queue with Bernoulli-schedule vacation interruption under N-Policy . Int. Sch. Res. Notices , 2014 , 392317 . http://dx.doi.org/10.1155/2014/392317 . Google Scholar PubMed 8 Haridass , M. and Arumuganathan , R. ( 2015 ) Analysis of a single server batch arrival retrial queueing system with modified vacations and N-policy . Rairo-Oper. Res. , 49 , 279 – 296 . Google Scholar CrossRef Search ADS 9 Wang , J. , Zhang , X. and Huang , P. ( 2017 ) Strategic behavior and social optimization in a constant retrial queue with the N-policy . Eur. J. Oper. Res. , 256 , 841 – 849 . Google Scholar CrossRef Search ADS 10 Wang , K.-H. and Ke , J.-C. ( 2002 ) Control policies of an M/G/1 queueing system with a removable and non‐reliable server . Int. Trans. Oper. Res. , 9 , 195 – 212 . Google Scholar CrossRef Search ADS 11 Pearn , W.L. , Ke , J.-C. and Chang , Y.-C. ( 2004 ) Sensitivity analysis of the optimal management policy for a queuing system with a removable and non-reliable server . Comput. Ind. Eng. , 46 , 87 – 99 . Google Scholar CrossRef Search ADS 12 Choudhury , G. , Ke , J.-C. and Tadj , L. ( 2009 ) The N-policy for an unreliable server with delaying repair and two phases of service . J. Comput. Appl. Math. , 231 , 349 – 364 . Google Scholar CrossRef Search ADS 13 Choudhury , G. and Tadj , L. ( 2011 ) The optimal control of an Mx/G/1 unreliable server queue with two phases of service and Bernoulli vacation schedule . Math. Comput. Model. , 54 , 673 – 688 . Google Scholar CrossRef Search ADS 14 Singh , C.J. , Jain , M. and Kumar , B. ( 2013 ) Analysis of queue with two phases of service and m phases of repair for server breakdown under N-policy . Int. J. Serv. Oper. Manag. , 16 , 373 – 406 . 15 Jayachitra , P. and Albert , A.J. ( 2014 ) Recent developments in queueing models under N-policy: a short survey . Int. J. Math. Arch , 5 , 227 – 233 . 16 Yang , D.-Y. and Wu , C.-H. ( 2015 ) Cost-minimization analysis of a working vacation queue with N-policy and server breakdowns . Comput. Ind. Eng. , 82 , 151 – 158 . Google Scholar CrossRef Search ADS 17 Jain , M. , Sharma , G.C. and Rani , V. ( 2016 ) Maximum entropy approach for an unreliable MX/G/1 queueing system with Bernoulli vacation, restricted admission and delayed phase repair under N-policy . Int. J. Serv. Oper. Manag , 23 , 407 – 442 . 18 Wang , T.-Y. , Liu , T.-H. and Chang , F.-M. ( 2017 ) Analysis of a random N-policy Geo/G/1 queue with the server subject to repairable breakdowns . J. Ind. Prod. Eng. , 34 , 19 – 29 . 19 Kalidass , K. and Kasturi , R. ( 2012 ) A queue with working breakdowns . Comput. Ind. Eng. , 63 , 779 – 783 . Google Scholar CrossRef Search ADS 20 Kim , B.K. and Lee , D.H. ( 2014 ) The M/G/1 queue with disasters and working breakdowns . Appl. Math. Model. , 38 , 1788 – 1798 . Google Scholar CrossRef Search ADS 21 Liu , Z. and Song , Y. ( 2014 ) The MX/M/1 queue with working breakdown . Rairo-Oper. Res. , 48 , 399 – 413 . Google Scholar CrossRef Search ADS 22 Liou , C.-D. ( 2015 ) Markovian queue optimisation analysis with an unreliable server subject to working breakdowns and impatient customers . Int. J. Syst. Sci. , 46 , 2165 – 2182 . Google Scholar CrossRef Search ADS 23 Yen , T.-C. , Chen , W.-L. and Chen , J.-Y. ( 2016 ) Reliability and sensitivity analysis of the controllable repair system with warm standbys and working breakdown . Comput. Ind. Eng. , 97 , 84 – 92 . Google Scholar CrossRef Search ADS 24 Chen , J.-Y. , Wang , K.-H. , Sheu , S.-P. and Chou , W.-K. ( 2014 ) Maximum entropy analysis to the N Policy M/G/1 queue with working breakdowns . J. Test. Eval. , 43 , 1211 – 1220 . 25 Chen , J.-Y. , Yen , T.-C. and Wang , K.-H. ( 2016 ) Cost optimization of a single-server queue with working breakdowns under the N policy . J. Test. Eval. , 44 , 2059 – 2067 . 26 Jiang , T. and Liu , L. ( 2017 ) The GI/M/1 queue in a multi-phase service environment with disasters and working breakdowns . Int. J. Comput. Math. , 94 , 707 – 726 . Google Scholar CrossRef Search ADS 27 Li , T. and Zhang , L. ( 2017 ) An M/G/1 retrial G-queue with general retrial times and working breakdowns . Math. Comput. Appl. , 22 , 15 . 28 Ke , J.-C. ( 2003 ) The analysis of a general input queue with N policy and exponential vacations . Queueing Syst. , 45 , 135 – 160 . Google Scholar CrossRef Search ADS 29 Ke , J.-C. ( 2003 ) The operating characteristic analysis on a general input queue with N policy and a startup time . Math. Method Oper. Res. , 57 , 235 – 254 . Google Scholar CrossRef Search ADS 30 Chao , X. and Rahman , A. ( 2006 ) Analysis and computational algorithm for queues with state-dependent vacations I: G/M(n)/1/K . J. Syst. Sci. Complex. , 19 , 36 – 53 . Google Scholar CrossRef Search ADS 31 Zhang , M. and Hou , Z. ( 2011 ) Steady state analysis of the GI/M/1/N queue with a variant of multiple working vacations . Comput. Ind. Eng. , 61 , 1296 – 1301 . Google Scholar CrossRef Search ADS 32 Goswami , V. , Laxmi , P.V. and Jyothsna , K. ( 2013 ) Analysis of GI/M(n)/1/N queue with state-dependent multiple working vacations . Opsearch , 50 , 106 – 124 . Google Scholar CrossRef Search ADS 33 Laxmi , P.V. and Suchitra , V. ( 2014 ) A recursive algorithm for state dependent GI/M/1/N queue with Bernoulli-schedule vacation . J. Math. Model. Algorithms Oper. Res. , 13 , 283 – 299 . Google Scholar CrossRef Search ADS 34 Goswami , V. ( 2015 ) Study of customers’ impatience in a GI/M/1/N queue with working vacations . Int. J. Manage. Sci. Eng. Manag , 10 , 144 – 154 . 35 Kempa , W.M. ( 2017 ) A comprehensive study on the queue-size distribution in a finite-buffer system with a general independent input flow . Perform. Eval. , 108 , 1 – 15 . Google Scholar CrossRef Search ADS 36 Laxmi , P.V. and Indira , S. ( 2017 ) Ant colony optimisation for impatient customer queue under N-policy and Bernoulli schedule vacation interruption . Inte. J. Math. Oper. Res. , 10 , 167 – 189 . Google Scholar CrossRef Search ADS 37 Niu , Z. and Takahashi , Y. ( 1999 ) A finite‐capacity queue with exhaustive vacation/close‐down/setup times and Markovian arrival processes . Queueing Syst , 31 , 1 – 23 . Google Scholar CrossRef Search ADS 38 Gupta , U.C. and Sikdar , K. ( 2004 ) A finite capacity bulk service queue with single vacation and Markovian arrival process . Int. J. Stoch. Anal. , 2004 , 337 – 357 . Google Scholar CrossRef Search ADS 39 Banik , A.D. , Gupta , U.C. and Pathak , S.S. ( 2007 ) On the GI/M/1/N queue with multiple working vacations—analytic analysis and computation . Appl. Math. Model. , 31 , 1701 – 1710 . Google Scholar CrossRef Search ADS Author notes Handling editor: Antonio Fernandez Anta © The British Computer Society 2018. All rights reserved. For permissions, please email: journals.permissions@oup.com This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/about_us/legal/notices)

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The Computer JournalOxford University Press

Published: May 26, 2018

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