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A new look at transient versions of Little's law, and M/G/1 preemptive last-come-first-served queues

Part of: Special processes Stochastic processes Operations research and management science

Published online by Cambridge University Press:  14 July 2016

Brian H. Fralix  and
Germán Riaño
Brian H. Fralix*
Affiliation:
EURANDOM and Eindhoven University of Technology
Germán Riaño*
Affiliation:
Universidad de los Andes, Colombia
*
∗Current address: Department of Mathematical Sciences, Clemson University, O-110 Martin Hall, Box 340975, Clemson, SC 29634, USA. Email address: bfralix@clemson.edu
∗∗Current address: Strategic Operations Research Team, Kimberly-Clark, Latin America Operations, Bogotá, Colombia.
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Abstract

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We take a new look at transient, or time-dependent Little laws for queueing systems. Through the use of Palm measures, we show that previous laws (see Bertsimas and Mourtzinou (1997)) can be generalized. Furthermore, within this framework, a new law can be derived as well, which gives higher-moment expressions for very general types of queueing system; in particular, the laws hold for systems that allow customers to overtake one another. What is especially novel about our approach is the use of Palm measures that are induced by nonstationary point processes, as these measures are not commonly found in the queueing literature. This new higher-moment law is then used to provide expressions for all moments of the number of customers in the system in an M/G/1 preemptive last-come-first-served queue at a time t > 0, for any initial condition and any of the more famous preemptive disciplines (i.e. preemptive-resume, and preemptive-repeat with and without resampling) that are analogous to the special cases found in Abate and Whitt (1987c), (1988). These expressions are then used to derive a nice structural form for all of the time-dependent moments of a regulated Brownian motion (see Abate and Whitt (1987a), (1987b)).

Keywords

Little's lawpreemptive queuetransient momentregulated Brownian motion

MSC classification

Secondary: 60K25: Queueing theory 90B22: Queues and service 60G55: Point processes
Type
Research Article
Information
Journal of Applied Probability , Volume 47 , Issue 2 , June 2010 , pp. 459 - 473
Copyright
Copyright © Applied Probability Trust 2010 

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