Matched Queues with Matching Batch Pair (m, n)

TL;DR

This paper models a complex bilateral matching queue with batch matching and customer impatience, analyzing its stability, queue lengths, and wait times using a novel bidirectional QBD process, with applications in various practical systems.

Contribution

It introduces a new matched queue model with batch matching and impatience, and develops a bidirectional QBD process for detailed analysis, extending prior queueing models.

Findings

System stability conditions derived.

Average queue lengths computed.

Expected sojourn times determined.

Abstract

In this paper, we discuss an interesting but challenging bilateral stochastically matching problem: A more general matched queue with matching batch pair (m, n) and two types (i.e., types A and B) of impatient customers, where the arrivals of A- and B-customers are both Poisson processes, m A-customers and n B-customers are matched as a group which leaves the system immediately, and the customers' impatient behavior is to guarantee the stability of the system. We show that this matched queue can be expressed as a novel bidirectional level-dependent quasi-birth-and-death (QBD) process. Based on this, we provide a detailed analysis for this matched queue, including the system stability, the average stationary queue lengthes, and the average sojourn time of any A-customer or B-customer. We believe that the methodology and results developed in this paper can be applicable to dealing with more general matched queueing systems, which are widely encountered in various practical areas, for example, sharing economy, ridesharing platform, bilateral market, organ transplantation, taxi services, assembly systems, and so on.