TY - JOUR
AU1 - Yang,, Dong-Yuh
AU2 - Cho,, Yi-Chun
AB - 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