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Age Process, Workload Process, Sojourn Times, and Waiting Times in a Discrete Time SM[K]/PH[K]/1/FCFS Queue

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Abstract

In this paper, we study a discrete time queueing system with multiple types of customers and a first-come-first-served (FCFS) service discipline. Customers arrive according to a semi-Markov arrival process and the service times of individual customers have PH-distributions. A GI/M/1 type Markov chain for a generalized age process of batches of customers is introduced. The steady state distribution of the GI/M/1 type Markov chain is found explicitly and, consequently, the steady state distributions of the age of the batch in service, the total workload in the system, waiting times, and sojourn times of different batches and different types of customers are obtained. We show that the generalized age process and a generalized total workload process have the same steady state distribution. We prove that the waiting times and sojourn times have PH-distributions and find matrix representations of those PH-distributions. When the arrival process is a Markov arrival process with marked transitions, we construct a QBD process for the age process and the total workload process. The steady state distributions of the waiting times and the sojourn times, both at the batch level and the customer level, are obtained from the steady state distribution of the QBD process. A number of numerical examples are presented to gain insight into the waiting processes of different types of customers.

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References

  1. S.A. Alfa, Discrete time queues and matrix-analytic methods, TOP 10(2) (2002) 147–210.

    Google Scholar 

  2. S. Asmussen, Phase-type representation in random walk and queueing problems, Annals of Probability 20(2) (1992) 772–789.

    Google Scholar 

  3. S. Asmussen and C. O’Cinneide, Representation for matrix-geometric and matrix-exponential steady-state distributions with applications to many-server queues, Stochastic Models 14 (1998) 369–387.

    Google Scholar 

  4. S. Asmussen and G. Koole, Marked point processes as limits of Markovian arrival streams, J. Appl. Probab. 30 (1993) 365–372.

    Google Scholar 

  5. D. Bini and B. Meini, On cyclic reduction applied to a class of Toeplitz-like matrices arising in queueing problems, in: Computations with Markov Chains, eds. W.J. Stewart (Kluwer Academic Publisher, 1996) pp. 21–38.

  6. E. Cinlar, Queues with semi-Markov arrivals, J. Appl. Prob. 4 (1967) 365–379.

    Google Scholar 

  7. J.W. Cohen, The Single Server Queue (North-Holland Amsterdam, 1982).

  8. D.V. Cortizo, J. Garcia, C. Blondia and B. Van Houdt, FIFO by sets ALOHA (FS-ALOHA): A collision resolution algorithm for the contention channel in wireless ATM systems, Performance Evaluation 36–37 (1999) 401–427.

    Google Scholar 

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

    Article  Google Scholar 

  10. G. Fayolle, V.A. Malyshev and M.V. Menshikov, Topics in the Constructive Theory of Countable Markov Chains (Cambridge University Press, 1995).

  11. H.R. Gail, S.L. Hantler and B.A. Taylor, Solutions of the basic matrix equation for M/G/1 and G/M/1 type Markov chains, Stochastic Models 10 (1994) 1–43.

    Google Scholar 

  12. H.R. Gail, S.L. Hantler and B.A. Taylor, Non-skip-free M/G/1 and G/M/1 type Markov chains, Adv. Appl. Probab. 29 (1997) 733–758.

    Google Scholar 

  13. F.R. Gantmacher, The Theory of Matrices (Chelsea, New York, 1959).

    Google Scholar 

  14. W.K. Grassmann and J.L. Jain, Numerical solutions of waiting time distribution and idle time distribution of the arithmetic GI/G/1 queue, Operations Research 37 (1989) 141–150.

    Google Scholar 

  15. Qi-Ming HE, Queues with marked customers, Adv. Appl. Prob. 28 (1996) 567–587.

    Google Scholar 

  16. Qi-Ming HE, Quasi-birth-and-death Markov processes with a tree structure and the MMAP[ K]/ PH[K]/N/LCFS non-preemptive queue, European Journal of Operational Research 120(3) (2000) 641–656.

    Article  Google Scholar 

  17. Qi-Ming HE, The versatility of MMAP[K] and the MMAP[K]/ G[K]/1 queue, Queueing Systems 38(4) (2001) 397–418.

    Article  Google Scholar 

  18. Qi-Ming HE, Workload process, waiting times, and sojourn times in a discrete time MMAP[K]/ SM[K]/1/FCFS queue, Stochastic Models 20(4) (2004) 415–437.

    Article  Google Scholar 

  19. Qi-Ming HE, Age process, sojourn times, waiting times, and queue lengths in a continuous time SM[K]/PH[K]/1/FCFS queue (submitted for publication) (2003).

  20. Qi-Ming HE and A.S. Alfa, The MMAP[K]/PH[K]/1 queue with a last-come-first-served preemptive service discipline, Queueing Systems 28 (1998) 269–291.

    Google Scholar 

  21. Qi-Ming HE and M.F. Neuts, Markov chains with marked transitions, Stochastic Processes and their Applications 74(1) (1998) 37–52.

    Article  Google Scholar 

  22. G. Latouche and V. Ramaswami, A logarithmic reduction algorithm for quasi-birth-and-death process, Journal of Applied Probability 30 (1993) 650–674.

    Google Scholar 

  23. G. Latouche and V. Ramaswami, Introduction to Matrix Analytic Methods in Stochastic Modelling (ASA & SIAM, Philadelphia, USA, 1999).

    Google Scholar 

  24. R.M. Loynes, The stability of a queue with non-independent interarrival and service times, Proc. Cambridge Philos. Soc. 58 (1962) 497–520.

    Google Scholar 

  25. M.F. Neuts, Generalizations of the Pollaczek-Khinchin integral method in the theory of queues, Adv. Appl. Prob. 18 (1986) 952–990.

    Google Scholar 

  26. M.F. Neuts, Matrix-Geometric Solutions in Stochastic Models: An algorithmic Approach (The Johns Hopkins University Press, Baltimore, 1981).

    Google Scholar 

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

    Google Scholar 

  28. B. Sengupta, Markov processes whose steady state distribution is matrix-exponential with an application to the GI/PH/1 queue, Adv. Appl. Prob. 21 (1989) 159–180.

    Google Scholar 

  29. B. Sengupta, Phase-type representations for matrix-geometric solutions, Stochastic Models 6 (1990) 163–167.

    Google Scholar 

  30. B. Sengupta, The semi-Markovian queue: Theory and applications, Stochastic Models 6 (1990) 383–413.

    Google Scholar 

  31. T. Takine, Queue length distribution in a FIFO single-server queue with multiple arrival streams having different service time distributions, Queueing System 39 (2001) 349–375.

    Article  Google Scholar 

  32. T. Takine, A recent progress in algorithmic analysis of FIFO queues with Markovian arrival streams, J. Korean Math. Soc. 38(4) (2001) 807–842.

    Google Scholar 

  33. T. Takine and T. Hasegawa, The workload in a MAP/G/1 queue with state-dependent services: Its applications to a queue with preemptive resume priority, Stochastic Models 10(1) (1994) 183–204.

    Google Scholar 

  34. B. Van Houdt and C. Blondia, Stability and performance of stack algorithms for random access communication modeled as a tree structured QBD Markov chain, Stochastic Models 17 (2001) 247–270.

    Article  Google Scholar 

  35. B. Van Houdt and C. Blondia, The delay distribution of a type k customer in a FCFS MMAP[K]/PH[K]/1 queue, Journal of Applied Probability 39(1) (2002) 213–222.

    Article  Google Scholar 

  36. B. Van Houdt and C. Blondia, The waiting time distribution of a type k customer in a FCFS MMAP[K]/PH[K]/2 queue (manuscript), (2002).

  37. B. Van Houdt and C. Blondia, The waiting time distribution of a type k customer in a discrete time MMAP[K]/PH[K]/c (c = 1, 2) queue using QBDs, Stochastic models 20 (2004) 55–69.

    Article  Google Scholar 

  38. Y.Q. Zhao, W. Li and W.J. Braun, Censoring, factorization, and spectral analysis for transition matrices with block-repeating entries, Technical report (No. 355), Laboratory for Research in Statistics and Probability, Carleton University and University of Ottawa, (2001).

  39. T. Yang and M. Chaudhry, On the steady-state queue size distributions of discrete-time GI/G/1 queue, Adv. Appl. Probab. 28 (1996) 1177–1200.

    Google Scholar 

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Authors and Affiliations

  1. Department of Industrial Engineering, Dalhousie University, Halifax, Nova Scotia, Canada, B3J 2X4

    Qi-Ming He

  2. School of Economics and Management, Tsinghua University, Beijing, P.R. China

    Qi-Ming He

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Correspondence to Qi-Ming He.

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AMS subject classification: 60K25, 60J10

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He, QM. Age Process, Workload Process, Sojourn Times, and Waiting Times in a Discrete Time SM[K]/PH[K]/1/FCFS Queue. Queueing Syst 49, 363–403 (2005). https://doi.org/10.1007/s11134-005-6972-y

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  • DOI: https://doi.org/10.1007/s11134-005-6972-y

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