Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-07T18:37:41.075Z Has data issue: false hasContentIssue false
  • English
  • Français

The BMAP/GI/1 queue with server set-up times and server vacations

Part of: Special processes

Published online by Cambridge University Press:  01 July 2016

Josep M. Ferrandiz
Josep M. Ferrandiz*
Affiliation:
Hewlett-Packard Laboratories
*
∗Postal address: Hewlett-Packard Laboratories, Filton Road, Stoke Gifford, Bristol, BS12 6QZ, UK.
Get access Rights & Permissions [Opens in a new window]

Abstract

Using Palm-martingale calculus, we derive the workload characteristic function and queue length moment generating function for the BMAP/GI/1 queue with server vacations. In the queueing system under study, the server may start a vacation at the completion of a service or at the arrival of a customer finding an empty system. In the latter case we will talk of a server set-up time. The distribution of a set-up time or of a vacation period after a departure leaving a non-empty system behind is conditionally independent of the queue length and workload. Furthermore, the distribution of the server set-up times may be different from the distribution of vacations at service completion times. The results are particularized to the M/GI/1 queue and to the BMAP/GI/1 queue (without vacations).

Keywords

PALM-MARTINGALE CALCULUSQUEUES WITH VACATIONSBATCH MARKOVIAN ARRIVAL PROCESSESSERVER SET-UP

MSC classification

Primary: 60K25: Queueing theory
Type
Research Article
Information
Advances in Applied Probability , Volume 25 , Issue 1 , March 1993 , pp. 235 - 254
Copyright
Copyright © Applied Probability Trust 1993 

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] Baccelli, F. and Brémaud, P. (1987) Palm Probabilities and Stationary Queues. Lecture Notes in Statistics 41, Springer-Verlag, Heidelberg.CrossRefGoogle Scholar
[2] Boxma, O. J. (1989) Workloads and waiting times in single-server systems with multiple customer classes. QUESTA 5, 185–214.Google Scholar
[3] Brémaud, P. (1981) Point Processes and Queues: Martingale Dynamics. Springer-Verlag, New York.CrossRefGoogle Scholar
[4] Brémaud, P. (1989) Characteristics of queueing systems observed at events and the connections between stochastic intensity and Palm probability. QUESTA 5, 99–112.Google Scholar
[5] Doshi, B. T. (1986) Queueing systems with vacations—A survey. QUESTA 1, 29–66.Google Scholar
[6] Heffes, H. and Lucantoni, D. M. (1986) A Markov modulated characterization of packetized voice and data traffic and related statistical multiplexer performance. IEEE J. Sel. Areas Comm. 4, 856–868.CrossRefGoogle Scholar
[7] Keilson, J. and Servi, L. D. (1989) Blocking probability for M/G/1 vacation systems with occupancy level dependent schedules. Operat. Res. 37, 134–140.CrossRefGoogle Scholar
[8] Lucantoni, D. M. (1991) New results on the single server queue with a batch markovian arrival process. Stoch. Models 7, 1–46.CrossRefGoogle Scholar
[9] Lucantoni, D. M. Meier-Hellstern, K. S. and Neuts, M. F. (1990) A single server-queue with server vacations and a class of non-renewal arrival processes. Adv. Appl. Prob. 22, 676–705.CrossRefGoogle Scholar
[10] Neuts, M. F. (1979) A versatile Markovian point process. J. Appl. Prob. 16, 764–769.CrossRefGoogle Scholar
[11] Neuts, M. F. (1981) Matrix-Geometric Solutions in Stochastic Models. Johns Hopkins University Press, Baltimore.Google Scholar
[12] Neuts, M. F. (1989) Structured Stochastic Matrices of M/G/1 Type and their Applications, Volume 4 of Probability : Pure and Applied. Marcel Dekker, New York.Google Scholar
[13] Ramaswami, V. (1980) The N/G/1 queue and its detailed analysis. Adv. Appl. Prob. 12, 222–261.CrossRefGoogle Scholar