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(1975)
Analysis of queueing system
L. Zadeh (1999)
Fuzzy sets as a basis for a theory of possibilityFuzzy Sets and Systems, 100
(2005)
Fuzzy probability new approach and applications, SpringerVerlag
S. Karlin, P. Morse (1958)
Queues, Inventories, And Maintenance
(1958)
Queues, inventories and maintenance. New York: Wiley (Operation Research Society of America
(1999)
State-dependent Queues
(1985)
Fundamental of queueing theory
John White, J. Schmidt, G. Bennett (1975)
Analysis of queueing systems
(1988)
The general solutions of the truncated non markovian queues
R. Stanford (1982)
The set of limiting distributions for a Markov chain with fuzzy transition probabilitiesFuzzy Sets and Systems, 7
Fuzzy Inf. Eng. (2011) 4: 379-384 DOI 10.1007/s12543-011-0092-7 ORIGINAL ARTICLE Truncated Erlangian Queueing System with Fuzzy Arrival Rate, Balking and Reneging A. Pourdarvish· M. Shokry Received: 26 January 2011/ Revised: 29 July 2011/ Accepted: 20 October 2011/ © Springer-Verlag Berlin Heidelberg and Fuzzy Information and Engineering Branch of the Operations Research Society of China Abstract In this paper, we derive the performance measures of the truncated Erlan- gian service queueing system with state-dependent rate, balking and reneging with fuzzy arrival rate λ so for the (M/E /1/N(α,β)), we have obtained P the steady n r n,s state probabilities in the system with the unit in the service being at stage s (s=1, 2, ··· , r); k is the boundary state for the number of customer more than the one that the rate of service increases. We treat this queueing system for general values of r, k and N and derive the Fuzzy eﬀective measures for the operation of the system at any time. Keywords Fuzzy numbers · Arrival rate · Service rate · Fuzzy variables · Perfor- mance measures 1. Introduction This paper considers the queueing system M/E /1/N with state-dependent service rate, balking and reneging concepts. The Erlang distribution, denoted by E is a special case of the gamma distribution and is named after A.K. Erlang who pioneered queueing system theory for its application to congestion in telephone networks. The non-truncated queue: M/E /1 was solved by Morse [2] at r = 2 and white et al [4], who obtained the solution in the series expansion. Al Seedy [1] gave an analytical solution of the queue: M/E /1/N with balking only. This work has been followed by Kotb [3] who studied the analytical solution of the state-dependent Erlangian queue: M/E /1/N with balking by using a very useful lemma. In this paper, we treat the analytical solution of the queue: M/E /1/N(α,β) for ﬁnite capacity considering by using a recurrence relations. We obtain P , the probabilities that there are n units in n,s A. Pourdarvish () Department of Statistics, Faculty of Mathematical Science, University of Mazandaran, 47416-95447, Babolsar, Iran email: a.pourdarvish@umz.ac.ir M. Shokry Department of Statistics, University of Mazandaran, 47416-95447, Babolsar, Iran 380 A. Pourdarvish · M. Shokry (2011) the system and the unit in service occupies stage s (1 ≤ s ≤ r in terms of P ). The probability of an empty system P is also obtained. The discipline considered is ﬁrst in ﬁrst out (FIFO). 2. The Problem Analysis Consider the single-channel service time Erlangian queue having r-service stages each with rate μ , with the state-dependent and reneging in the form rμ , n = 1, ⎪ (1) μ = ⎪ rμ + (n− 1)α, 2 ≤ n ≤ k,μ <μ , n (1) (1) (2) rμ + (n− 1)α, k+ 1 ≤ n ≤ N. (2) This means that the units are served with two diﬀerent rates rμ or rμ depending 1 2 on the number of units in the system whether 1 ≤ n ≤ k or k+ 1 ≤ n ≤ N, respectively. Also, consider an exponential inter arrival pattern with rate λ . Assume (1−β)to be the probability that a unit balks (does not enter the queue), where: β =P (a unit joins the queue), 0 ≤ β ≤ 1, 1 ≤ n ≤ N; for n = 0,β = 1, it is clear that: λ, n = 0, λ = n ⎪ βλ, 1 ≤ n ≤ N. Assume the probabilities : P = P (n units in the system and the unit in service being in stage s), n,s where: 1 ≤ n ≤ N, 1 ≤ s ≤ r , P =probability of an empty system, i.e., the dally probability. The steady-state diﬀerence equations are: n = 0 → λP − rμ P = 0. (1) 0 (1) 1,1 (rμ +βλ)P − rμ P = 0, 1 ≤ s ≤ r− 1, (1) 1,s (1) 1,s+1 n = 1 → (2) (rμ +βλ)P −λP − (rμ +α)P = 0, s = r. (1) 1,r 0 (1) 2,1 2 ≤ n ≤ k− 1, (i)1 ≤ s ≤ r− 1, (ii) s = r, (i)(rμ + (n− 1)α+βλ)P −βλP − (rμ + (n− 1)α)P = 0, ⎨ (1) n,s n−1,s (1) n,s+1 (3) (ii)(rμ + (n− 1)α+βλ)P −βλP − (rμ + nα)P = 0. (1) n,r n−1,r (1) n+1,1 n = k, (i)1 ≤ s ≤ r− 1, (ii) s = r, (i)(rμ + (k− 1)α+βλ)P −βλP − (rμ + (k− 1)α)P = 0, ⎨ (1) k,s k−1,s (1) k,s+1 (4) (ii)(rμ + (k− 1)α+βλ)P −βλP − (rμ + kα)P = 0. (1) k,r k−1,r (2) k+1,1 1 ≤ n ≤ N − 1, (i)1 ≤ s ≤ r− 1, (ii) s = r, (i)(rμ + (n− 1)α+βλ)P −βλP − (rμ + (n− 1)α)P = 0, ⎨ (2) n,s n−1,s (2) n,s+1 (5) (ii)(rμ + (n− 1)α+βλ)P −βλP − (rμ + nα)P = 0. (2) n,r n−1,r (2) n+1,1 Fuzzy Inf. Eng. (2011) 4: 379-384 381 n = N, (i)1 ≤ s ≤ r− 1, (ii) s = r, (i)(rμ + (N − 1)α)P −βλP − (rμ + (N − 1)α)P = 0, (2) n,s n−1,s (2) n,s+1 (6) (ii)(rμ + (N − 1)α)P −βλP = 0. (2) n,r n−1,r Summing (2) over s and using (1), gives: βλ n = 2 → P = P . (7) 2,1 1,s (rμ +α) (1) s=1 Summing (3) over s, using (7) and adding the results obtaining for 2 ≤ n ≤ k− 1, leads to: βλ 3 ≤ n ≤ k → P = P . (8) n,1 n−1,s rμ + (n− 1)α (1) s=1 Similarly, summing (4) over s and using (8) at n = k, yields βλ n = k+ 1 → P = P . (9) k+1,1 k,s (rμ + kα) (2) s=1 Summing (5) over s and using (9): βλ k+ 2 ≤ n ≤ N → P = P . (10) n,1 n−1,s (rμ + (n− 1)α) (2) s=1 From equation, one can easily show that P = φ P . Making use of Equation (2), 1,1 1 0 yields s−1 1 ≤ s ≤ r → P = φ (1+βφ ) P . (11) 1,s 1 1 0 Upon using the ﬁrst equation of (3) and (8) we get the recurrence relation. 2 ≤ n ≤ k, ⎧ ⎫ r s−1 ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ s−1 P = βφ (1+βφ ) P − ( ) P . (12) n,s n n ⎪ n−1,i n−1,i⎪ ⎪ ⎪ ⎩ ⎭ 1+βφ i=1 i=1 Also, from the ﬁrst equation of (5) and (10), we obtain: k+ 1 ≤ n ≤ N − 1, ⎧ ⎫ r s−1 ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ s−1 P = βφ (1+βφ ) P − ( ) P . (13) ⎪ ⎪ n,s n n n−1,i n−1,i ⎪ ⎪ ⎩ ⎭ 1+βφ i=1 i=1 Finally, using Equation (6) and Equation (10) at n = N, gives: 1 ≤ s ≤ r → P = βφ P , (14) N,s N N−1,i i=s where: ⎪ λ ⎪ , 1 ≤ n ≤ k, ⎪ rμ + (n− 1)α (1) φ = n ⎪ , k+ 1 ≤ n ≤ N. rμ + (n− 1)α (2) 382 A. Pourdarvish · M. Shokry (2011) Equations (11)-(14) are the required recurrence relations giving all probabilities in terms of P which its self may now be determined by using the normalizing condition: N r P + P = 1. (15) 0 n,s n=1 s=1 Hence all the probabilities are completely known in terms of the queue parameters. 3. Example The following example illustrates the theoretical results. In the system: M/E /1/N with state-dependent, balking and reneging, let N = 4, K = 2, r = 3 (i.e., the queue M/E /1/4(α,β)), in Equations (11)-(15). The results are P = a P , P = a P , P = a P , 1,1 1 0 1,2 2 0 1,3 3 0 P = b P , P = b P , P = b P , 2,1 1 0 2,2 2 0 2,3 3 0 P = c P , P = c P , P = c P , 3,1 1 0 3,2 2 0 3,3 3 0 P = d P , P = d P , P = d P , 4,1 1 0 4,2 2 0 4,3 3 0 where: a = φ , a = φ (1+βφ ) , a = φ(1+βφ ) , 1 1 2 1 1 3 1 b = βφ (a + a + a ), 1 2 1 2 3 b = βφ (1+βφ ) (a + a + a )− ( )a , 2 2 2 1 2 3 1 1+βφ 1 1 b = βφ (1+βφ ) (a + a + a )− ( )a − ( ) a , 3 2 2 1 2 3 1 2 1+βφ 1+βφ 2 2 c = βφ (b + b + b ), 1 3 1 2 3 c = βφ (1+βφ ) b + b + b − ( )b , 2 3 3 1 2 3 1 1+βφ 1 1 c = βφ (1+βφ ) b + b + b − ( )b − ( ) b , 3 3 3 1 2 3 1 2 1+βφ 1+βφ 3 3 d = βφ (c + c + c ), d = βφ (c + c ), d = βφ c , 1 4 1 2 3 2 4 2 3 3 4 3 λ λ φ = ,φ = , 1 2 3μ 3μ +α (1) (1) λ λ φ = ,φ = . 3 4 3μ + 2α 3μ + 3α (2) (2) From the normalizing condition: 3 3 3 3 P + P + P + P + P = 1, 0 1,s 2,s 3,s 4,s s=1 s=1 s=1 s=1 we have: −1 P = {1+ a + a + a + b + b + b + c + c + c + d + d + d } . 0 1 2 3 1 2 3 1 2 3 1 2 3 Therefore, the expected numbers in the system and in the queue are, respectively: 4 3 L = nP = ((a + a + a )+ 2(b + b + b ) n,s 1 2 3 1 2 3 n=1 s=1 Fuzzy Inf. Eng. (2011) 4: 379-384 383 +3(c + c + c )+ 4(d + d + d ))P , (16) 1 2 3 1 2 3 0 4 3 L = (n− 1)P = ((b + b + b )+ 2(c + c + c ) q n,s 1 2 3 1 2 3 n=1 s=1 +3(d + d + d ))P . 1 2 3 0 Also the expected waiting time in the system and the queue are obtained as follows: L 1 W = , W = , λ = (L− L )μ, μ = (μ +μ ), q q (1) (2) ´ ´ λ λ whereλ is the mean rate of units actually entering the system. 4. Special Cases Case1 : In the above described model selecting {α = 0,β = 1, N →∞}, we shall reach to an M/E /1 model. Case2 : Selecting {α = 0,μ = μ = μ, K = N}, the model M/E /1/N(β) will be (1) (2) r approached which would be the Erlangian system with ﬁnite capacity and Balking phenomena. 5. System withλ In this section, we shall derive the eﬀective measures of the system when the state dependent arrival rate is a fuzzy number. But μ ,μ are two crisp service rates, (1) (2) which are predetermined. It may be noted that this derivation shall be done for the ﬁx number of Erlangian phases, i.e., r = 3 here because in fuzziﬁed form we have to consider the related function to be monotone. Now it remains to consider monotony of a, b, c, d, p as the function of λ. This may be easily done by diﬀerentiating i i i i 0 respect toλ, see that they are all increasing functions ofλ. So on the basis ofα− cuts we shall have: ⎧ ⎫ 4 3 ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ L[α] = nP | A . ⎪ ⎪ n,s ⎪ ⎪ ⎩ ⎭ n=1 s=1 Using the Relation (16), we now have: ⎧ ⎫ 3 3 3 3 ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ L[α] = ( a + 2 b + 3 c + 4 d )P | A , ⎪ i i i i 0 ⎪ ⎪ ⎪ ⎩ ⎭ i=1 i=1 i=1 1 where the condition A is: ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ ˜ ˜ A = (P , P ) ∈ P[α], P + P = 1, a ∈ a ˜ [α],··· , d ∈ d [α] . ⎪ ⎪ 0 n,s 0 n,s i i i i ⎪ ⎪ ⎩ ⎭ n,s Now we shall obtain the lower and upper α− cuts by substituting another suﬃx l or u in the appropriate parameters; so L[α] = L , L , l,α u,α L = (a + a + a + 2(b + b + b )+ 3(c + c + c ) l,α 1,l 2,l 3,l 1,l 2,l 3,l 1,l 2,l 3,l 384 A. Pourdarvish · M. Shokry (2011) +4(d + d + d ))P , 1,l 2,l 3,l 0,l L = (a + a + a + 2(b + b + b )+ 3(c + c + c ) u,α 1,u 2,u 3,u 1,u 2,u 3,u 1,u 2,u 3,u +4(d + d + d ))P . 1,u 2,u 3,u 0,u And we have: 4 3 L [α] = (n− 1)P | A , q n,s n=1 s=1 L [α] = {((b + b + b )+ 2(c + c + c )+ 3(d + d + d ))P | A}, q 1 2 3 1 2 3 1 2 3 0 L [α] = L , L , q q,(l,α) q,(u,α) L = ((b + b + b )+ 2(c + c + c )+ 3(d + d + d ))P , q,(l,α) 1,l 2,l 3,l 1,l 2,l 3,l 1,l 2,l 3,l 0,l L = ((b + b + b )+ 2(c + c + c )+ 3(d + d + d ))P . q,(u,α) 1,u 2,u 3,u 1,u 2,u 3,u 1,u 2,u 3,u 0,u 6. Conclusion In this paper, we considered an (M/E /1/N(α,β)), queueing system with balking, reneging and fuzzy parameter. We developed the equations of the steady state prob- abilities and derived some performance measures of the system. One example was presented to demonstrate how the fuzzy parameters of the model inﬂuence the behav- ior of the system. Acknowledgements The authors would like to thank two referees for their comments which led to im- provements of this paper. References 1. Alseedy R O (1988) The general solutions of the truncated non markovian queues: Ph.D. Thesis, Faculty of Science, Tanta university 2. Morse P M (1958) Queues, inventories and maintenance. New York: Wiley (Operation Research Society of America) 3. Kotb K A M (1999) State-dependent Queues. Ph.D. Thesis, Faculty of Science, Tanta University 4. White J A, Schmidt J W, Bennett G K (1975) Analysis of queueing system. New York: Academic Press 5. Buckley J J, Feuring T, Hayashi Y (2005) Fuzzy probability new approach and applications, Springer- Verlag Berlin Heidelberg 6. Zadeh L A (1978) Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets and Systems 1: 3-28 7. Stanford R E (1982) The set of limiting distributions for a markov chain with fuzzy transition proba- bilities. Fuzzy Sets and Systems 7: 71-78 8. Gross D, Harris C(1985) Fundamental of queueing theory. John Wiley and Sons. University of Michigan
Fuzzy Information and Engineering – Taylor & Francis
Published: Dec 1, 2011
Keywords: Fuzzy numbers; Arrival rate; Service rate; Fuzzy variables; Performance measures
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