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An analytical solution for the discrete time single server system with semi-Markovian arrivals

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In this paper we derive an analytical solution for the stationary distribution of the number of customers and the idle time in a single server system with semi-Markovian arrival processes in discrete time domain (SM/G/1). This kind of arrival process enables us to take autocorrelations into account, with various applications for the modeling of communication and manufacturing systems. It will be shown that the distribution of the customer number can be represented as a linear combination of geometric distributions. Thus a simple calculation of higher moments of the customer number is possible.

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  1. [1]

    E. Arjas, On a fundamental identity in the theory of semi-Markov processes, Adv. Appl. Prob. 4 (1972) 258–270.

  2. [2]

    E. Çinlar, Queues with semi-Markovian arrivals, J. Appl. Prob. 4 (1967) 365–379.

  3. [3]

    M.L. Chaudhry and M. Agarwal, Exact and approximate numerical solutions of steady-state distributions arising in the queueGI/G/1, Queueing Syst. 10 (1992) 105–152.

  4. [4]

    M.L. Chaudhry, U.C. Gupta and M. Agarwal, Exact and approximate numerical solutions to steady-state single-server queues:M/G/1 — a unified approach, Queueing Syst. 10 (1992) 351–380.

  5. [5]

    M.L. Chaudhry, Alternative numerical solutions of stationary queueing-time distributions in discrete-time queues: GI/G/1, J. Oper. Res. Soc. 44 (1993) 1035–1051.

  6. [6]

    D.R. Cox and V. Isham,Point Processes (Chapman and Hall, London, 1980).

  7. [7]

    W. Ding and P. Decker, Waiting time distribution of a discreteSSMP/G/1 queue and its applications in ATM systems,Proc. Semin. Int. Teletraffic Congress, New Jersey (1990) Paper 9.4.

  8. [8]

    W. Feller,An Introduction to Probability Theory and Its Applications, Vol. 1 and 2 (Wiley, New York, 1971).

  9. [9]

    B. Gopinath and J.A. Morrison, Discrete-time single server queues with correlated input, AT & T Bell Lab. Techn. J. 56 (1977) 1743–1768.

  10. [10]

    W.K. Grassmann and J.L. Jain, Numerical solutions of the waiting time distribution and idle time distribution of the arithmeticGI/G/1 queue, Oper. Res. 37 (1989) 141–150.

  11. [11]

    G. Ha\linger, A polynomial factorization approach to the discrete timeGI/G/1/(N) queue size distribution, Perf. Eval. to appear.

  12. [12]

    G. Ha\linger, Analysis of the discrete time single server queue with semi-Markovian input: Waiting time and busy period, Internal Report, Tech. Univ. Darmstadt, Dept. of Computer Science (1992).

  13. [13]

    G. Ha\linger and E.S. Rieger, Analysis of open queueing networks with discrete time renewal processes,Proc. MMB Conf., IFB 286 (Springer, Heidelberg, 1991) pp. 15–29 (in German).

  14. [14]

    H. Heffes and M. Lucantoni, A Markov modulated characterization of packetized voice and data traffic and related statistical multiplexer performance, IEEE J. Sel. Areas Commun. SAC-4 (1986) 856–867.

  15. [15]

    P. Henrici,Applied and Computational Complex Analysis, Vol. 1 (Wiley, New York, 1974).

  16. [16]

    C. Herrmann, Analysis of the discrete-timeSMP/D/1/s finite buffer queue with applications in ATM,IEEE INFOCOM'93, San Francisco (1993) pp. 160–167.

  17. [17]

    L. Kleinrock,Queueing Systems, Vol. 1 and 2 (Wiley, New York, 1975/76).

  18. [18]

    D.E. Knuth,Concrete Mathematics (Addison-Wesley, Reading, MA, 1989).

  19. [19]

    P.J. Kuehn, Approximate analysis of general queueing networks by decomposition, IEEE Trans. Commun. COM-27 (1979) 113–126.

  20. [20]

    S.-Q. Li, A general solution technique for discrete queueing analysis of multimedia traffic on ATM, IEEE Trans. Commun. COM-39 (1991) 1115–1132.

  21. [21]

    D.M. Lucantoni, K.S. Meier-Hellstern and M.F. Neuts, A single-server queue with server vacations and a class of non-renewal arrival processes, Adv. Appl. Prob. 22 (1990) 676–705.

  22. [22]

    J. Ponstein, Theory and numerical solution of a discrete queueing problem, Statistica Neerlandica 20 (1974) 139–152.

  23. [23]

    N.U. Prabhu and L.C. Tang, Markov-modulated single server queueing systems, JAP Special Volume (1994), to appear.

  24. [24]

    W.H. Press, B.P. Flannery, S.A. Teukolsky and W.T. Vetterling,Numerical Recipes in C (Cambridge University Press, Cambridge, 1989).

  25. [25]

    E.S. Rieger, An analytical solution to the discrete time single server queue with semi-Markovian arrivals, Ph.D. thesis (in German), Techn. Univ. Darmstadt (1992).

  26. [26]

    B. Sengupta, The semi-Markovian queue: theory and applications, Commun. Statist. Stoch. Models 6 (1990) 383–413.

  27. [27]

    J.H. de Smit, The single server semi-Markov queue, Stoch. Proc. Appl. 22 (1986) 37–50.

  28. [28]

    L. Takács, A storage process with semi-Markov input, Adv. Appl. Prob. 7 (1975) 830–844.

  29. [29]

    H.C. Tijms,Stochastic Modeling and Analysis (Wiley, New York, 1986).

  30. [30]

    R.C. Tucker, Accurate method for analysis of a packet-speech multiplexer with limited delay, IEEE Trans. Commun. COM-36 (1988) 479–483.

  31. [31]

    R.W. Wolff,Stochastic Modeling and the Theory of Queues (Prentice-Hall, Englewood Cliffs, 1989).

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Rieger, E.S., Haßlinger, G. An analytical solution for the discrete time single server system with semi-Markovian arrivals. Queueing Syst 18, 69–105 (1994).

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  • Discrete time analysis
  • semi-Markovian arrivals
  • stationary queue size and idle time distribution
  • roots
  • polynomial factorization