Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-23T10:38:24.494Z Has data issue: false hasContentIssue false
  • English
  • Français

The transient behaviour of a single server queue with batch arrivals

Published online by Cambridge University Press:  09 April 2009

P. J. Brockwell
P. J. Brockwell
Affiliation:
University of Melbourne.
Rights & Permissions [Opens in a new window]

Summary

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A queue at which arrivals occur randomly in batches of fixed size r and for which the service times are independent negative exponential variates is considered. Expressions are obtained for the moments of the transient waiting time distribution and the distribution of the number of customers in the system just before the nth batch arrives. The distribution of the number of customers served in a busy period is also determined.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 3 , Issue 2 , May 1963 , pp. 241 - 248
Copyright
Copyright © Australian Mathematical Society 1963

References

[1]Ledermann, W. and Reuter, G. E. H., Spectral Theory for the Differential Equations of Simple Birth and Death Processes, Phil. Trans. Roy. Soc. London, (A), 246, 326–369 (1954).Google Scholar
[2]Bailey, N. T. J., A Continuous Time Treatment of a Simple Queue using Generating Functions, J.R.S.S. (B), 16, 288–291 (1954).Google Scholar
[3]Luchak, G., The Solution of the Single Channel Queueing Equation Characterized by a Time-dependent Poisson-distributed Arrival Rate and a General Class of Holding Times, Operations Research, 4, 711–732 (1956).CrossRefGoogle Scholar
[4]Luchak, G., The Continuous Time Solution of the Equations of the Single Channel Queue with a General Class of Service-time Distributions by the Method of Generating Functions, J.R.S.S. (B), 20, 176–181 (1958).Google Scholar
[5]Whittaker, E. T. and Watson, G. N., A Course of Modern Analysis (C.U.P.), 4th ed., 1952.Google Scholar
[6]Bahadur, R. R., Some Approximations to the Binomial Distribution Function A.M.S., 31 (1), 43–54 (1960).Google Scholar
[7]Foster, F. G., Queues with Batch Arrivals I, Acta Math. Hungarica, 12, 1–10 (1961).Google Scholar