Hostname: page-component-848d4c4894-x24gv Total loading time: 0 Render date: 2024-05-18T01:47:32.840Z Has data issue: false hasContentIssue false
  • English
  • Français

Transient solution to the time-dependent multiserver Poisson queue

Part of: Markov processes Special processes

Published online by Cambridge University Press:  14 July 2016

B. H. Margolius
B. H. Margolius*
Affiliation:
Cleveland State University
*
∗Postal address: Department of Mathematics, 1515 Rhodes Tower, 2121 Euclid Avenue, Cleveland, OH 44114-2214, USA. Email address: b.margolius@csuohio.edu
Rights & Permissions [Opens in a new window]

Abstract

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.

We derive an integral equation for the transient probabilities and expected number in the queue for the multiserver queue with Poisson arrivals, exponential service for time-varying arrival and departure rates, and a time-varying number of servers. The method is a straightforward application of generating functions. We can express pĉ−1(t), the probability that ĉ − 1 customers are in the queue or being served, in terms of a Volterra equation of the second kind, where ĉ is the maximum number of servers working during the day. Each of the other transient probabilities is expressed in terms of integral equations in pĉ−1(t) and the transition probabilities of a certain time-dependent random walk. In this random walk, the rate of steps to the right equals the arrival rate of the queue and the rate of steps to the left equals the departure rate of the queue when all servers are busy.

Keywords

Transient probabilitytime-dependent Poisson arrival ratetime-dependent exponential service ratetime-dependent queuemultiserver queuesingle-server queuemodified Bessel function

MSC classification

Primary: 60K25: Queueing theory
Secondary: 60J35: Transition functions, generators and resolvents
Type
Research Papers
Information
Journal of Applied Probability , Volume 42 , Issue 3 , September 2005 , pp. 766 - 777
Copyright
© Applied Probability Trust 2005 

References

Breuer, L. (2001). The periodic BMAP/PH/c queue. Queueing Systems 38, 67–76.Google Scholar
Heyman, D. P. and Whitt, W. (1984). The asymptotic behavior of queues with time-varying rates. J. Appl. Prob. 21, 143–156.Google Scholar
Parthasarathy, P. R. and Sharafali, M. (1989). Transient solution to the many-server Poisson queue: a simple approach. J. Appl. Prob. 26, 584–594.Google Scholar
Zhang, J. and Coyle, E. J. (1991). The transient solution of time-dependent M/M/1 queues. IEEE Trans. Inf. Theory 37, 1690–1696.CrossRefGoogle Scholar