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ON THE TIME-DEPENDENT OCCUPANCY DISTRIBUTION OF THE G/G/1 QUEUING SYSTEM

Published online by Cambridge University Press:  16 February 2009

Jorge Limón–Robles  and
Martin A. Wortman
Jorge Limón–Robles
Affiliation:
Industrial and Systems Engineering, Instituto Tecnológico y de Estudios Superiores de Monterrey, C.P. 64849, Monterrey, N.L., México E-mail: jorge.limon@itesm.mx
Martin A. Wortman
Affiliation:
Department of Industrial and Systems Engineering, Texas A&M University, College Station, TX 77843–3131 E-mail: wortman@tamu.edu
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Abstract

This article offers an approach for studying the time-dependent occupancy distribution for a modest generalization of the GI/G/1 queuing system in which interarrival times and service times, although mutually independent, are not necessarily identically distributed. We develop and explore an analytical model leading to a computational approach that gives tight bounds on the occupancy distribution. Although there is no general closed-form characterization of probability law dynamics for occupancy in the GI/G/1 queue, our results offer what might be termed “near-closed-form” in that accurate plots of the transient occupancy distribution can be constructed with an insignificant computational burden. We believe that our results are unique; we are unaware of any alternative analytical approach leading to a numerical characterization of the time-dependent occupancy distribution for the G/G/1 queuing systems considered here.

Our analyses employ a marked point process that converges to the occupancy process at any fixed time t; it is shown that this process forms a Markov chain from which the transient occupancy law is available. We verify our analytical approach via comparison with the well-known closed-form expressions for time-dependent occupancy distribution of the M/M/1 queue. Additionally, we suggest the viability of our approach, as a computational means of obtaining the time-dependent occupancy distribution, through straightforward application to a Gamma[x]/Weibull/1 queuing system having batch arrivals and batch job services.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 23 , Issue 2 , April 2009 , pp. 261 - 280
Copyright
Copyright © Cambridge University Press 2009

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