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The probability of large queue lengths and waiting times in a heterogeneous multiserver queue I: Tight limits

Part of: Limit theorems Markov processes Special processes

Published online by Cambridge University Press:  01 July 2016

John S. Sadowsky  and
Wojciech Szpankowski
John S. Sadowsky*
Affiliation:
Arizona State University
Wojciech Szpankowski*
Affiliation:
Purdue University
*
* Postal address: Department of Electrical Engineering, Arizona State University, Tempe, AZ 85287–5706, USA.
** Postal address: Department of Computer Science, Purdue University, West Lafayette, IN 47907, USA.
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Abstract

We consider a multiserver queuing process specified by i.i.d. interarrival time, batch size and service time sequences. In the case that different servers have different service time distributions we say the system is heterogeneous. In this paper we establish conditions for the queuing process to be characterized as a geometrically Harris recurrent Markov chain, and we characterize the stationary probabilities of large queue lengths and waiting times. The queue length is asymptotically geometric and the waiting time is asymptotically exponential. Our analysis is a generalization of the well-known characterization of the GI/G/1 queue obtained using classical probabilistic techniques of exponential change of measure and renewal theory.

Keywords

QUEUEING THEORYHARRIS RECURRENT MARKOV CHAINSFOSTER–LYAPUNOV THEORYMARKOV ADDITIVE PROCESSESRENEWAL THEORY

MSC classification

Primary: 60K25: Queueing theory
Secondary: 60F10: Large deviations 60J05: Discrete-time Markov processes on general state spaces 60K15: Markov renewal processes, semi-Markov processes 60K20: Applications of Markov renewal processes (reliability, queueing networks, etc.)
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 27 , Issue 2 , June 1995 , pp. 532 - 566
Copyright
Copyright © Applied Probability Trust 1995 

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Footnotes

Supported by the National Science Foundation (NCR-9003007).

Supported by the National Science Foundation (NCR-9206315 and CCR-9201078), AFOSR (90–0107), and by the National Library of Medicine (R01 LM05118).

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