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Muto, Kiyoshi; Miyazaki, Haruo; Seki, Yoichi; Kimura, Yoshihiro; Shibata, Yukio

Lattice path counting and \(M/M/c\) queueing systems. (English) Zbl 0836.60099

Queueing Syst. 19, No. 1-2, 193-214 (1995).
The transient solution of \(M/M/c\) queue in a closed form is obtained for the probability of exactly \(i\) arrivals and \(j\) departures within a time interval of length \(t\). The method is based on the well-known combinatorial considerations: the authors count the paths from the origin to \((i,j)\) that have exactly \(r(d)\) \(x\)-steps whose depth from the line \(y = x\) is \(d = 0, \dots, c - 1\). Some numerical examples are considered.
Reviewer: J.Morozov (Petrozavodsk)

Cited in 6 Documents

MSC:

60K25 Queueing theory (aspects of probability theory)
90B22 Queues and service in operations research

Keywords:

transient solution; queue size; lattice path counting; depth of part; numerical examples
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References:

[1] O.J. Boxma, The joint arrival and departure process for theM/M/1 queue, Stat. Neerlandica 38 (1984) 199-208. · Zbl 0546.60092 · doi:10.1111/j.1467-9574.1984.tb01110.x
[2] W. Böhm and S.G. Mohanty, The transient solution ofM/M/1 queues under(M, N)-policy. A combinatorial approach, J. Stat. Plann. Inference 34 (1993) 23-33. · Zbl 0777.05004 · doi:10.1016/0378-3758(93)90031-Z
[3] D.G. Champernowne, An elementary method of the solution of the queueing problem with a single server and constant parameter, J. Roy. Stat. Soc. Ser. B18 (1956) 125-128. · Zbl 0073.13003
[4] D. Gross and C.M. Harris,Fundamentals of Queueing Theory (Wiley, New York, 1985). · Zbl 0658.60122
[5] J. Medhi,Stochastic Models in Queueing Theory (Academic Press, New York, 1991). · Zbl 0743.60100
[6] S.G. Mohanty,Lattice Path Counting and Applications (Academic Press, New York, 1979). · Zbl 0455.60013
[7] S.G. Mohanty and W. Panny, A discrete-time analogue of theM/M/1 queue and the transient solution: A geometric approach, Sankhy?, Ser. A, 52 (1990) 364-370. · Zbl 0732.60102
[8] P.R. Parthasarathy, A transient solution to anM/M/1 queue: a simple approach, Adv. Appl. Prob. (1987) 997-998. · Zbl 0632.60098
[9] P.R. Parthasarathy and M. Sharafali, Transient solution to the many server Poisson queue: a simple approach, J. Appl. Prob. 26 (1989) 584-594. · Zbl 0693.60076 · doi:10.2307/3214415
[10] C.D. Pegden and M. Rosenshine, Some new results for theM/M/1 queue, Manag. Sci. 28 (1982) 821-828. · Zbl 0484.60075 · doi:10.1287/mnsc.28.7.821
[11] T.L. Saaty, Time dependent solution of many server Poisson queue, Oper. Res. 8 (1960) 755-772. · Zbl 0105.11702 · doi:10.1287/opre.8.6.755
[12] K. Sen and J.L. Jain, Combinatorial approach to Markovian queueing models, J. Stat. Plann. Inference 34 (1993) 269-279. · Zbl 0766.60117 · doi:10.1016/0378-3758(93)90011-T
[13] K. Sen, J.L. Jain and J.M. Gupta, Lattice path approach to transient solution ofM/M/1 with (0,k) control policy, J. Stat. Plann. Inference 34 (1993) 259-267. · Zbl 0765.60099 · doi:10.1016/0378-3758(93)90010-4
[14] O.P. Sharma,Markovian Queues (Ellis Horwood, Chichester, 1990).
[15] L. Takács, A generalization of the ballot problem and its application in the theory of queues, J. Amer. Stat. Ass. 57 (1962) 327-337. · Zbl 0109.36702 · doi:10.2307/2281642
[16] L. Takács, A single server queues with recurrent input and exponentially distributed service times, Oper. Res. 10 (1962) 395-399. · Zbl 0285.60089 · doi:10.1287/opre.10.3.395
[17] L. Takács, A combinatorial method in the theory of queues, SIAM J. Appl. Math. 10 (1962) 691-694. · Zbl 0118.13503 · doi:10.1137/0110053
[18] D. Towsley, An application of the reflection principle to the transient analysis of theM/M/1 queue, Naval. Res. Logist. 34 (1987) 451-456. · Zbl 0665.60103 · doi:10.1002/1520-6750(198706)34:3<451::AID-NAV3220340310>3.0.CO;2-8
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