A Fluid Model for an Overloaded Bipartite Queueing System with Heterogeneous Matching Utility

66 Pages Posted: 21 Oct 2016 Last revised: 27 Aug 2018

See all articles by Yichuan Ding

Yichuan Ding

Desautels Faculty of Management, McGill University

S. McCormick

University of British Columbia (UBC)

Mahesh Nagarajan

University of British Columbia (UBC) - Sauder School of Business

Date Written: August 22, 2018

Abstract

We consider a bipartite queueing system (BQS) with multiple types of servers and customers, where different customer-server combinations may generate different utilities. Whenever a server is available, it serves the customer with the highest index, which is the sum of a customer's waiting index and the matching index. We call this an {\em M W} index. We assume that the waiting index is an increasing function of a customer's waiting time and the matching index depends on both the customer's and the server's types. We develop a fluid model to approximate the behavior of such a BQS system, and show that the fluid limit process can be computed over any finite horizon. We develop an efficient algorithm to check whether a steady state of the fluid process exists or not. When a steady state exists, the algorithm also computes one efficiently. We prove that there can be at most one steady state, and that the fluid limit process converges to the steady state under mild conditions. These results enable a policy designer to predict the behavior of a BQS when using an M W index, and to choose an indexing formula that optimizes a given set of performance metrics. We derive a closed-form M W index that optimizes the steady-state performance according to some well-known efficiency and fairness metrics.

Keywords: Bipartite Queueing System, Min Cost Max Flow, Nested Cuts for Parameterized Network, Value-based Routing, Public Housing Assignment, Scarce Resource Allocation

Suggested Citation

Ding, Yichuan and McCormick, S. and Nagarajan, Mahesh, A Fluid Model for an Overloaded Bipartite Queueing System with Heterogeneous Matching Utility (August 22, 2018). Available at SSRN: https://ssrn.com/abstract=2854492 or http://dx.doi.org/10.2139/ssrn.2854492

Yichuan Ding (Contact Author)

Desautels Faculty of Management, McGill University ( email )

1001 Sherbrook St W
Montreal, Quebec H3A 1G5
Canada
5143982482 (Phone)

HOME PAGE: http://https://www.mcgill.ca/desautels/yichuan-daniel-ding

S. McCormick

University of British Columbia (UBC) ( email )

2329 West Mall
Vancouver, British Columbia BC V6T 1Z4
Canada

Mahesh Nagarajan

University of British Columbia (UBC) - Sauder School of Business ( email )

2053 Main Mall
Vancouver, BC V6T 1Z2
Canada

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