Performance Evaluation of Stochastic Bipartite Matching Models

TL;DR

This paper analyzes a stochastic bipartite matching model with multi-class customers and servers, deriving new formulas for key performance metrics and providing insights into system behavior.

Contribution

It introduces a new compact expression for the normalization constant, waiting probabilities, and mean waiting times in the bipartite matching model.

Findings

Derived a manageable formula for the normalization constant.

Provided explicit expressions for waiting probabilities.

Presented numerical examples illustrating the model's behavior.

Abstract

We consider a stochastic bipartite matching model consisting of multi-class customers and multi-class servers. Compatibility constraints between the customer and server classes are described by a bipartite graph. Each time slot, exactly one customer and one server arrive. The incoming customer (resp. server) is matched with the earliest arrived server (resp. customer) with a class that is compatible with its own class, if there is any, in which case the matched customer-server couple immediately leaves the system; otherwise, the incoming customer (resp. server) waits in the system until it is matched. Contrary to classical queueing models, both customers and servers may have to wait, so that their roles are interchangeable. While (the process underlying) this model was already known to have a product-form stationary distribution, this paper derives a new compact and manageable expression for the normalization constant of this distribution, as well as for the waiting probability and mean waiting time of customers and servers. We also provide a numerical example and make some important observations.