On the Optimal Design of a Bipartite Matching Queueing System

59 Pages Posted: 24 Mar 2019 Last revised: 11 Nov 2019

See all articles by Philipp Afeche

Philipp Afeche

University of Toronto - Rotman School of Management

Rene Caldentey

University of Chicago - Booth School of Business

Varun Gupta

University of Chicago - Booth School of Business

Date Written: March 1, 2019

Abstract

We consider a multi-class multi-server queueing system and study the problem of designing an optimal matching topology (or service compatibility structure) between customer classes and servers under a FCFS-ALIS service discipline. Specifically, we are interested in finding matching topologies that optimize --in a Pareto efficiency-- sense the trade-off between two competing objectives: (i) minimizing customers' waiting time delays and (ii) maximizing matching rewards generated by pairing customers and servers. Our analysis of the problem is divided in three main parts.

First, under heavy-traffic conditions, we show that any bipartite matching system can be partitioned into a collection of complete resource pooling (CRP) subsystems, which are interconnected by means of a direct acyclic graph (DAG). We show that this DAG together with the aggregate service capacity on each CRP component fully determine the vector of steady-state waiting times. In particular, we show that the average (scaled) steady-state delay across all customer classes is asymptotically equal to the number of CRP components divided by the total system capacity.

Second, since computing matching rewards under a FCFS-ALIS service discipline is computationally infeasible as the number of customer classes and servers grow large, we propose a quadratic programming (QP) formulation to approximate matching rewards. We show that the QP formulation is exact for a number of instances of the problem and provides a very good approximation in general. Extensive numerical experiments show that in over 98% of problem instances the relative error between the exact rewards and the QP approximate rewards is less than 2%.

Lastly, combining our characterization of average delays in terms of the number of CRP components and the quadratic programming formulation to compute matching rewards, we propose a mixed-integer linear program (MILP) that can be used to find the set of matching topologies that define the Pareto frontier of reward-delay pairs.

Keywords: Queueing, Bipartite Matching, Quadratic Programming

Suggested Citation

Afeche, Philipp and Caldentey, Rene and Gupta, Varun, On the Optimal Design of a Bipartite Matching Queueing System (March 1, 2019). Available at SSRN: https://ssrn.com/abstract=3345302 or http://dx.doi.org/10.2139/ssrn.3345302

Philipp Afeche

University of Toronto - Rotman School of Management ( email )

105 St. George Street
Toronto, Ontario M5S 3E6 M5S1S4
Canada
416-978-1591 (Phone)

HOME PAGE: http://www.rotman.utoronto.ca/facbios/viewFac.asp?facultyID=philipp.afeche

Rene Caldentey (Contact Author)

University of Chicago - Booth School of Business ( email )

5807 S. Woodlawn Avenue
Chicago, IL 60637
United States

HOME PAGE: http://www.chicagobooth.edu/faculty/directory/c/rene-caldentey

Varun Gupta

University of Chicago - Booth School of Business ( email )

5807 S. Woodlawn Avenue
Chicago, IL 60637
United States

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