Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-26T06:11:43.075Z Has data issue: false hasContentIssue false
  • English
  • Français

Approximation of periodic queues

Published online by Cambridge University Press:  01 July 2016

Tomasz Rolski
Tomasz Rolski*
Affiliation:
University of Wrocław
*
∗Postal address: Mathematical Institute, University of Wroclaw, pl. Grunwaldzki 2/4, 50–384 Wroclaw, Poland.
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we demonstrate how some characteristics of queues with the periodic Poisson arrivals can be approximated by the respective characteristics in queues with Markov modulated input. These Markov modulated queues were recently studied by Regterschot and de Smit (1984). The approximation theorems are given in terms of the weak convergence of some characteristics and their uniform integrability. The approximations are applicable for the following characteristics: mean workload, mean workload at the time of day, mean delay, mean queue size.

Keywords

MARKOV MODULATED QUEUEWORKLOADWORKLOAD AT THE TIME OF DAYUNIFORM INTEGRABILITY
Type
Research Article
Information
Advances in Applied Probability , Volume 19 , Issue 3 , September 1987 , pp. 691 - 707
Copyright
Copyright © Applied Probability Trust 1987 

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

Borovkov, A. A. (1972) Stochastic Processes in Queueing Theory (in Russian). Nauka, Moskva.Google Scholar
Borovkov, A. A. (1980) Asymptotic Methods in Queueing Theory (in Russian). Nauka, Moskva.Google Scholar
Brumelle, S. L. (1971) On the relation between customer and time averages in queues. J. Appl. Prob. 8, 508–520.Google Scholar
Cohen, J. W. (1969) The Single Server Queue. North-Holland, Amsterdam.Google Scholar
Franken, P., König, D., Arndt, U. and Schmidt, V. (1981) Queues and Point Processes. Akademie-Verlag, Berlin.Google Scholar
Grandell, J. (1977) Point processes and random measures. Adv. Appl. Prob. 9, 502–526.Google Scholar
Harrison, J. M. and Lemoine, A. J. (1977) Limit theorems for periodic queues. J. Appl. Prob. 14, 566–576.CrossRefGoogle Scholar
Heyman, D. (1982) On Ross&s conjecture about queues with non-stationary Poisson arrivals. J. Appl. Prob. 19, 245–249.CrossRefGoogle Scholar
Kallenberg, O. (1983) Random Measures. Academic Press. London.Google Scholar
Kamae, T. and Krengel, U. (1978) Stochastic partial ordering. Ann. Prob. 6, 1044–1049.Google Scholar
Lemoine, A. J. (1981) On queues with periodic Poisson input. J. Appl. Prob. 18, 889–900.CrossRefGoogle Scholar
Regterschot, G. J. K. and De Smit, J. H. A. (1984) The queue M/G/1 with Markov modulated arrivals and services. Math. Operat. Res. Google Scholar
Rolski, T. (1981a) Stationary Random Processes Associated with Point Processes. Lecture Notes in Statistics 5. Springer-Verlag, New York.Google Scholar
Rolski, T. (1981b) Queues with non-stationary input stream: Ross&s conjecture. Adv. Appl. Prob. 13, 603–618.Google Scholar
Stoyan, D. (1983) Comparison Methods for Queues and Other Stochastic Models (ed. Daley, D.). Wiley, Chichester.Google Scholar
Szczotka, W. (1986) The joint distribution of the waiting time and the queue size for a single server queue. Dissertationes Math. CCXL VIII.Google Scholar
Whitt, W. (1980) Some useful functional limit theorems. Math. Operat. Res. 5, 67–85.Google Scholar