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Tollbooth tandem queues with infinite homogeneous servers

Part of: Operations research and management science Special processes

Published online by Cambridge University Press:  30 March 2016

Xiuli Chao ,
Qi-Ming He  and
Sheldon Ross
Xiuli Chao*
Affiliation:
University of Michigan
Qi-Ming He*
Affiliation:
University of Waterloo
Sheldon Ross*
Affiliation:
University of Southern California
*
∗Postal address: Department of Industrial and Operations Engineering, University of Michigan, Ann Arbor, MI 48109, USA. Email address: xchao@umich.edu
∗∗Postal address: Department of Management Sciences, University of Waterloo, 200 University Avenue West, Waterloo, ON N2L 3G1, Canada. Email address: q7he@uwaterloo.ca
∗∗∗Postal address: Daniel J. Epstein Department of Industrial and Systems Engineering, University of Southern California, Los Angeles, CA 90089, USA. Email address: smross@usc.edu
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Abstract

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In this paper we analyze a tollbooth tandem queueing problem with an infinite number of servers. A customer starts service immediately upon arrival but cannot leave the system before all customers who arrived before him/her have left, i.e. customers depart the system in the same order as they arrive. Distributions of the total number of customers in the system, the number of departure-delayed customers in the system, and the number of customers in service at time t are obtained in closed form. Distributions of the sojourn times and departure delays of customers are also obtained explicitly. Both transient and steady state solutions are derived first for Poisson arrivals, and then extended to cases with batch Poisson and nonstationary Poisson arrival processes. Finally, we report several stochastic ordering results on how system performance measures are affected by arrival and service processes.

Keywords

Tollbooth tandem queuedeparture delaydeparture-delayed customer

MSC classification

Primary: 60K25: Queueing theory
Secondary: 90B22: Queues and service
Type
Research Papers
Information
Journal of Applied Probability , Volume 52 , Issue 4 , December 2015 , pp. 941 - 961
Copyright
Copyright © Applied Probability Trust 2015 

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