Abstract
We present a framework for representing a queue at arrival epochs as a Harris recurrent Markov chain (HRMC). The input to the queue is a marked point process governed by a HRMC and the queue dynamics are formulated by a general recursion. Such inputs include the cases of i.i.d, regenerative, Markov modulated, Markov renewal and the output from some queues as well. Since a HRMC is regenerative, the queue inherits the regenerative structure. As examples, we consider split & match, tandem, G/G/c and more general skip forward networks. In the case of i.i.d. input, we show the existence of regeneration points for a Jackson type open network having general service and interarrivai time distributions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
S. Asmussen,Applied Probability and Queues (John Wiley & Sons, 1987).
S. Asmussen and H. Thorisson, A Markov chain approach to periodic queues, J. Appl. Prob. 24 (1987) 215–225.
K.B. Athreya and P. Ney, A new approach to the limit theory of recurrent Markov chains, Tran. Amer. Math. Soc. 245 (1978) 493–501.
Burman and Smith, An asymptotic analysis of queueing systems with Markov modulated arrivals, Opns. Res., 34 1 (1986).
F. Charlot, M. Ghidouche and M. Hamami, Irre'ductibilite et re'currence au sens de Harris des 'temps d'attente des files GI/GI/q, Z. War. ver. Geb. 43 (1978) 187–203.
P. Franken, D. Konig, U. Arndt and V. Schmidt,Queues and Point Processes (Akademie-Verlag, Berlin, 1981).
P.W. Glynn and D.L. Igelhart, Simulation output analysis for general state space Markov chains, Appl. Prob.-Computer Science, The Interface I (1981).
P.W. Glynn and D.L. Igelhart, Estimation of steady state central moments by the regenerative method, O.R. Letters 5, 6 (1986) 271–276.
D.P. Heyman and S.Jr. Stidham, The relation between customer and time averages in queues, Opns. Res., 28 (1980) 983–994.
J. Kiefer and J. Wolfowitz, On the theory of queues with many servers, Trans. Amer. Math. Soc., 78 (1955) 1–18.
R.M. Loynes, The stability of a queue with non-independent interarrivai and service times, Proc. Camb. Philos. Soc., 58 (1962) 497–520.
M. Miyazawa, Time and customer processes in queues with stationary input, J. Appl. Prob., 14 (1977) 349–357.
M. Miyazawa, A formal approach to queueing processes in the steady state and their applications, J. Appl. Prob., 16 (1979) 332–346.
M. Miyazawa, The derivation of invariance relations in complex queueing systems with stationary inputs, Adv. Appl. Prob., 15 (1983) 874–885.
M.F. Neuts,Matrix-Geometric Solutions in Stochastic Models (Johns Hopkins University Press, Baltimore, London, 1981).
E. Nummelin, A splitting technique for Harris recurrent Markov chains, Z. War. ver. Geb., 43 (1978) 309–318.
E. Nummelin, A conservation property for general GI/G/1 queues with application to tandem queues, Adv. Appl. Prob., 11 (1979) 660–672.
E. Nummelin, Regeneration in tandem queues, Adv. Appl. Prob., 13 (1981) 221–230.
E. Nummelin,General Irreducible Markov Chains and Non-Negative Operators (Cambridge University Press, Cambridge, 1984).
G.S. Shedler,Regeneration and Networks of Queues (Springer Verlag, 1987).
K. Sigman, Regeneration in tandem queues with multi-server stations. To appear in J. Appl. Prob. (June 1988).
R. Wolff, Conditions for finite ladder height and delay moments, Opns. Res., 32, 4 (1984) 909–916.
R. Wolff, Sample-path derivations of the excess, age and spread distributions, To appear in J. Appl. Prob.
Author information
Authors and Affiliations
Additional information
A revised version of the author's winning paper of the 1986 George E. Nicholson Prize (awarded by the Operations Research Society of America).
Rights and permissions
About this article
Cite this article
Sigman, K. Queues as Harris recurrent Markov chains. Queueing Syst 3, 179–198 (1988). https://doi.org/10.1007/BF01189048
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01189048