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Heavy-Traffic Limits for a Many-Server Queueing Network with Switchover

Part of: Operations research and management science Special processes Limit theorems

Published online by Cambridge University Press:  04 January 2016

Guodong Pang  and
David D. Yao
Guodong Pang*
Affiliation:
Pennsylvania State University
David D. Yao*
Affiliation:
Columbia University
*
∗ Postal address: The Harold and Inge Marcus Department of Industrial and Manufacturing Engineering, Pennsylvania State University, University Park, PA 16802, USA. Email address: gup3@psu.edu
∗∗ Postal address: Department of Industrial Engineering and Operations Research, Columbia University, New York, NY 10027-6699, USA. Email address: yao@columbia.edu
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Abstract

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We study a multiclass Markovian queueing network with switchover across a set of many-server stations. New arrivals to each station follow a nonstationary Poisson process. Each job waiting in queue may, after some exponentially distributed patience time, switch over to another station or leave the network following a probabilistic and state-dependent mechanism. We analyze the performance of such networks under the many-server heavy-traffic limiting regimes, including the critically loaded quality-and-efficiency-driven (QED) regime, and the overloaded efficiency-driven (ED) regime. We also study the limits corresponding to mixing the underloaded quality-driven (QD) regime with the QED and ED regimes. We establish fluid and diffusion limits of the queue-length processes in all regimes. The fluid limits are characterized by ordinary differential equations. The diffusion limits are characterized by stochastic differential equations, with a piecewise-linear drift term and a constant (QED) or time-varying (ED) covariance matrix. We investigate the load balancing effect of switchover in the mixed regimes, demonstrating the migration of workload from overloaded stations to underloaded stations and quantifying the load balancing impact of switchover probabilities.

Keywords

Many-server queuemulticlassMarkoviantime-varying arrivalstate-dependent switchovercustomer abandonmentheavy trafficQED regimeED regimemixed regimefluid limitdiffusion limitmultidimensional (piecewise-linear) Ornstein‒Uhlenbeck process

MSC classification

Primary: 60K25: Queueing theory
Secondary: 60F17: Functional limit theorems; invariance principles 90B22: Queues and service
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 45 , Issue 3 , September 2013 , pp. 645 - 672
Copyright
© Applied Probability Trust 

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