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Lattice path counting andM/M/c queueing systems

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We apply the lattice path counting method to the analysis of the transientM/M/c queueing system. A closed-form solution is obtained for the probability of exactlyi arrivals andj departures within a time interval of lengtht in anM/M/c queueing system that is empty at the initial time. The derivation of the probability is based on the counting of paths from the origin to(i,j) on thexy-plane, that have exactly rd x-steps whose depth from the liney=x isd (d=0,1,...,c−1). The closed-form solution has an expression useful for numerical calculation.

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  1. [1]

    O.J. Boxma, The joint arrival and departure process for theM/M/1 queue, Stat. Neerlandica 38 (1984) 199–208.

  2. [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.

  3. [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.

  4. [4]

    D. Gross and C.M. Harris,Fundamentals of Queueing Theory (Wiley, New York, 1985).

  5. [5]

    J. Medhi,Stochastic Models in Queueing Theory (Academic Press, New York, 1991).

  6. [6]

    S.G. Mohanty,Lattice Path Counting and Applications (Academic Press, New York, 1979).

  7. [7]

    S.G. Mohanty and W. Panny, A discrete-time analogue of theM/M/1 queue and the transient solution: A geometric approach, Sankhyā, Ser. A, 52 (1990) 364–370.

  8. [8]

    P.R. Parthasarathy, A transient solution to anM/M/1 queue: a simple approach, Adv. Appl. Prob. (1987) 997–998.

  9. [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.

  10. [10]

    C.D. Pegden and M. Rosenshine, Some new results for theM/M/1 queue, Manag. Sci. 28 (1982) 821–828.

  11. [11]

    T.L. Saaty, Time dependent solution of many server Poisson queue, Oper. Res. 8 (1960) 755–772.

  12. [12]

    K. Sen and J.L. Jain, Combinatorial approach to Markovian queueing models, J. Stat. Plann. Inference 34 (1993) 269–279.

  13. [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.

  14. [14]

    O.P. Sharma,Markovian Queues (Ellis Horwood, Chichester, 1990).

  15. [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.

  16. [16]

    L. Takács, A single server queues with recurrent input and exponentially distributed service times, Oper. Res. 10 (1962) 395–399.

  17. [17]

    L. Takács, A combinatorial method in the theory of queues, SIAM J. Appl. Math. 10 (1962) 691–694.

  18. [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.

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Muto, K., Miyazaki, H., Seki, Y. et al. Lattice path counting andM/M/c queueing systems. Queueing Syst 19, 193–214 (1995). https://doi.org/10.1007/BF01148946

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  • M/M/c queue
  • transient solution
  • lattice path counting
  • depth of path