Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-26T17:21:31.697Z Has data issue: false hasContentIssue false
  • English
  • Français

On the subexponential properties in stationary single-server queues: a Palm-martingale approach

Part of: Mathematical economics Special processes Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Naoto Miyoshi
Naoto Miyoshi*
Affiliation:
Tokyo Institute of Technology
*
∗ Postal address: Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, Tokyo, 152-8552, Japan. Email address: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper studies the subexponential properties of the stationary workload, actual waiting time and sojourn time distributions in work-conserving single-server queues when the equilibrium residual service time distribution is subexponential. This kind of problem has been previously investigated in various queueing and insurance risk settings. For example, it has been shown that, when the queue has a Markovian arrival stream (MAS) input governed by a finite-state Markov chain, it has such subexponential properties. However, though MASs can approximate any stationary marked point process, it is known that the corresponding subexponential results fail in the general stationary framework. In this paper, we consider the model with a general stationary input and show the subexponential properties under some additional assumptions. Our assumptions are so general that the MAS governed by a finite-state Markov chain inherently possesses them. The approach used here is the Palm-martingale calculus, that is, the connection between the notion of Palm probability and that of stochastic intensity. The proof is essentially an extension of the M/GI/1 case to cover ‘Poisson-like’ arrival processes such as Markovian ones, where the stochastic intensity is admitted.

Keywords

Stationary work-conserving single-server queuesubexponential distributionPalm-martingale calculusstochastic intensity kernel

MSC classification

Primary: 60K25: Queueing theory
Secondary: 60G55: Point processes 60G70: Extreme value theory; extremal processes 91B30: Risk theory, insurance
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 36 , Issue 3 , September 2004 , pp. 872 - 892
Copyright
Copyright © Applied Probability Trust 2004 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Ash, R. B. (1972). Real Analysis and Probability (Prob. Math. Statist. 11). Academic Press, New York.Google Scholar
[2] Asmussen, S. and Koole, G. (1993). Marked point processes as limits of Markovian arrival streams. J. Appl. Prob. 30, 365–372.Google Scholar
[3] Asmussen, S., Henriksen, L. F. and Klüppelberg, C. (1994). Large claims approximations for risk processes in a Markovian environment. Stoch. Process. Appl. 54, 29–43.Google Scholar
[4] Asmussen, S., Schmidli, H. and Schmidt, V. (1999). Tail probabilities for non-standard risk and queueing processes with subexponential jumps. Adv. Appl. Prob. 31, 422–447.Google Scholar
[5] Baccelli, F. and Brémaud, P. (2003). Elements of Queueing Theory: Palm-Martingale Calculus and Stochastic Recurrences, 2nd edn. Springer, Berlin.Google Scholar
[6] Beneš, V. E. (1957). On queues with Poisson arrivals. Ann. Math. Statist. 28, 670–677.Google Scholar
[7] Billingsley, P. (1995). Probability and Measure, 3rd edn. John Wiley, New York.Google Scholar
[8] Brémaud, P., (1981). Point Processes and Queues. Martingale Dynamics. Springer, New York.Google Scholar
[9] Brémaud, P., (1989). Characteristics of queueing systems observed at events and the connection between stochastic intensity and Palm probability. Queueing Systems Theory Appl. 5, 99–112.Google Scholar
[10] Cooper, R. B. and Niu, S.-C. (1986). Beneš's formula for M/G/1-FIFO ‘explained’ by preemptive-resume LIFO. J. Appl. Prob. 23, 550–554.Google Scholar
[11] Embrechts, P. and Veraverbeke, N. (1982). Estimates for the probability of ruin with special emphasis on the possibility of large claims. Insurance Math. Econom. 1, 55–72.Google Scholar
[12] Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events for Insurance and Finance. Springer, Berlin.Google Scholar
[13] Jelenković, P. R. and Lazar, A. A. (1998). Subexponential asymptotics of a Markov-modulated random walk with queueing applications. J. Appl. Prob. 35, 325–347.Google Scholar
[14] Loynes, R. M. (1962). The stability of queues with non-independent inter-arrival and service times. Proc. Camb. Phil. Soc. 58, 497–520.Google Scholar
[15] Miyoshi, N. (2001). On the stationary workload distribution of work-conserving single-server queues: a general formula via stochastic intensity. J. Appl. Prob. 38, 793–798.Google Scholar
[16] Pakes, A. G. (1975). On the tails of waiting-time distributions. J. Appl. Prob. 12, 555–564.Google Scholar
[17] Papangelou, F. (1974). The conditional intensity of general point processes and an application to line processes. Z. Wahrscheinlichkeitsth. 28, 207–226.Google Scholar
[18] Rolski, T., Schmidli, H., Schmidt, V. and Teugels, J. (1999). Stochastic Processes for Insurance and Finance. John Wiley, Chichester.Google Scholar
[19] Sigman, K. (1999). Appendix: a primer on heavy-tailed distributions. Queueing Systems Theory Appl. 33, 261–275.Google Scholar
[20] Sigman, K. (ed.) (1999). Special issue on queues with heavy-tailed distributions (Queueing Systems Theory Appl. 33).Google Scholar
[21] Takine, T. (2001). Subexponential asymptotics of the waiting time distribution in a single-server queue with multiple Markovian arrival streams. Stoch. Models 17, 429–448.Google Scholar