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Martingale Methods for Analysing Single-Server Queues

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Abstract

In this paper we presents a martingale method for analysing queues of M/G/1 type, which have been generalised so that the system passes through a series of phases on which the service behaviour may differ. The analysis uses the process embedded at departures to create a martingale, which makes possible the calculation of the probability generating function of the stationary occupancy distribution. Specific examples are given, for instance, a model of an unreliable queueing system, and an example of a queue-length-threshold overload-control system.

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References

  1. J. Abate and W. Whitt, Numerical inversion of probability generating functions, Oper. Res. Lett. 12 (1992) 245-251.

    Google Scholar 

  2. F. Baccelli, Exponential martingales and Wald's formula for two-queue networks, J. Appl. Probab. 23 (1986) 812-819.

    Google Scholar 

  3. F. Baccelli and A.M. Makowski, Direct martingale arguments for stability: The M/GI/1 case, Systems Control Lett. 6 (1985) 181-186.

    Google Scholar 

  4. F. Baccelli and A.M. Makowski, Dynamic, transient and stationary behaviour of the M/GI/1 queue via martingales, Ann. Probab. 17(4) (1989) 1691-1699.

    Google Scholar 

  5. F. Baccelli and A.M.Makowski, Martingale relations for the M/GI/1 queue with Markov modulated Poisson input, Stochastic Process. Appl. 38 (1991) 99-133.

    Google Scholar 

  6. J.W. Cohen, The Single Server Queue (North-Holland, Amsterdam, 1969).

    Google Scholar 

  7. R.B. Cooper, Introduction to Queueing Theory (Macmillan, New York, 1972).

    Google Scholar 

  8. J.N. Daigle, Queue length distributions from probability generating functions via discrete Fourier transforms, Oper. Res. Lett. 8 (August 1989) 229-236.

    Google Scholar 

  9. J.N. Daigle and M. Roughan, Queue-length distributions for multi-priority queueing systems, in: IEEE INFOCOM'99, 1999.

  10. J. Dshalalow, Multi-channel queueing systems with infinite waiting rooms and stochastic control, J. Appl. Probab. 26 (1989) 345-362.

    Google Scholar 

  11. G.H. Golub and C.F. van Loan, Matrix Computations (North Oxford Academic, 1983).

  12. W.-B. Gong, A. Yan and C.G. Cassandras, The M/G/1 queue with queue-length dependent arrival rate, Commun. Statist. Stochastic Models 8(4) (1992) 733-741.

    Google Scholar 

  13. O.C. Ibe and K.S. Trivedi, Two queues with alternating service and server breakdown, Queueing Systems 7 (1990) 253-268.

    Google Scholar 

  14. M. Kijima and N. Makimoto, Computation of the quasi-stationary distributions in M(n)/GI/1/K and GI/M(n)/1/K queues, Queueing Systems, 11 (1992) 255-272.

    Google Scholar 

  15. M. Kijima and N. Makimoto, A unifed approach to GI/M(n)/1/K and M(n)/G/1/K queues via finite quasi-birth-death processes, Commun. Statist. Stochastic Models 8(2) (1992) 269-288.

    Google Scholar 

  16. L. Kleinrock, Queueing Systems, Vol. I: Theory (Wiley, New York 1975).

    Google Scholar 

  17. R.O. LaMaire, M/G/1/N vacation model with varying e-limited service discipline, Queueing Systems 11 (1992) 357-375.

    Google Scholar 

  18. S.-Q. Li, Overload control in a finite message storage buffer, IEEE Trans. Commun. 37(12) (1989) 1330-1338.

    Google Scholar 

  19. J.D.C. Little, A proof of the queueing formula: L = λw, Oper. Res. 9(3) (1961) 383-387.

    Google Scholar 

  20. N.A.Marlow and M. Tortorella, Some general characteristics of two-state reliability models for maintained systems, J. Appl. Probab. 32 (1995) 805-820.

    Google Scholar 

  21. J.A. Morrison, Two-server queue with one server idle below a threshold, Queueing Systems 7 (1990) 325-336.

    Google Scholar 

  22. T. Nakagawa and S. Osaki, Markov renewal processes with some non-regeneration points and their applications to reliability theory, Microelectronics Reliability 15 (1976) 633-636.

    Google Scholar 

  23. M.F. Neuts, A queueing model for a storage buffer in which the arrival rate is controlled by a switch with random delay, Performance Evaluation 5 (1985) 243-256.

    Google Scholar 

  24. M.F. Neuts, Structured Stochastic Matrices of M/G/1 Type and Their Applications (Marcel Dekker, New York, 1989).

    Google Scholar 

  25. J. Neveu, Discrete-Time Martingales (North-Holland, Amsterdam, 1975).

    Google Scholar 

  26. J.H. Park, The analysis of the MX/G/1 queue by a martingale method, Master thesis, Korea Advanced Institute of Science and Technology (1990).

  27. C.E.M. Pearce and M. Roughan, Forward delay times in multi-phase discrete-time renewal processes, Asia Pacific J. Oper. Res. 14(1) (1997) 1-10.

    Google Scholar 

  28. R. Pyke, Markov renewal processes: Definitions and preliminary properties, Ann. Math. Statist. 32 (1961) 1231-1242.

    Google Scholar 

  29. R. Pyke, Markov renewal processes with finitely many states, Ann. Math. Statist. 32 (1961) 1242-1259.

    Google Scholar 

  30. M. Roughan, An analysis of a modified M/G/1 queue using a martingale technique, J. Appl. Probab. 33 (March 1996) 224-238.

    Google Scholar 

  31. M. Roughan and C.E.M. Pearce, Analysis of a hysteretic overload control, in: ITC-16, Edinburgh, 1999, pp. 293-302.

  32. M. Roughan and C.E.M. Pearce, A martingale analysis of hysteretic overload control, Adv. in Performance Anal. 3(1) (2000) 1-30.

    Google Scholar 

  33. M.P. Rumsewicz and D.E. Smith, A comparison of SS7 congestion control options during mass call-in situations, IEEE/ACM Trans. Networks 3(1) (1995) 1-9.

    Google Scholar 

  34. J.A. Schormans, J.M. Pitts and E.M. Scharf, A priority queue with superimposed geometric batch arrivals, Comm. Statist. Stochastic Models 9(1) (1993) 105-122.

    Google Scholar 

  35. L. Takács, Introduction to the Theory of Queues (Oxford Univ. Press, Oxford, 1962).

    Google Scholar 

  36. H. Takagi, Time dependent process of M/G/1 vacation models with exhaustive service, J. Appl. Probab. 29 (1992) 418-429.

    Google Scholar 

  37. R.W. Wolff, Poisson arrivals see time averages, Oper. Res. 30 (1982) 223-231.

    Google Scholar 

  38. G.F. Yeo, Single server queue with modified service mechanisms, J. Australian Math. Soc. 2 (1962) 499-507.

    Google Scholar 

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Authors and Affiliations

  1. AT&T Labs – Research, 180 Park Av, Florham Park, NJ, 07932, USA

    M. Roughan

  2. Department of Applied Mathematics, University of Adelaide, Adelaide, 5005, Australia

    C.E.M. Pearce

Authors
  1. M. Roughan
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  2. C.E.M. Pearce
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Roughan, M., Pearce, C. Martingale Methods for Analysing Single-Server Queues. Queueing Systems 41, 205–239 (2002). https://doi.org/10.1023/A:1015851021001

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