Transient and asymptotic behavior of synchronization processes in assembly-like queues

Abstract

This paper analyzes the synchronization process of an assembly-like queueing system in which two distinct types of items/customers arrive at separate buffers, according to independent Poisson processes, so as to be synchronized into pairs at a synchronization node. Once a pair is synchronized it then queues up for service from a single server on a first-in-first-out basis as pairs. It is assumed that the service times of pairs are exponentially distributed and that the system has infinite capacity. Despite their practical significance, such queueing systems have not been adequately treated in the literature due to their transience or null recurrence. We first investigate the transient and asymptotic properties of the synchronization process' first two moments, both analytically and numerically. Motivated by the observed asymptotic behavior, we then propose an M/M/1 approximation to describe the behavior of such assembly-like queueing systems. Finally, a numerical study of the proposed approximation reveals that it performs sufficiently well for practical applications.

Modeling Urban Taxi Services with E-Hailings: A Queueing Network Approach

“…The approximation has been validated by simulation demonstrating small differences in system performance metrics between the simulation and approximation, less than 1% in most cases and no more than 3% under the condition of heavy traffic (i.e. large min(λ P , λ V )/µ), as well as equal arrival rates of two distinct flows [34]. For completeness, we present several key properties of S t .…”

“…Let X P t and X V t be the number of passengers and vehicles in corresponding buffers at time t. S t = min(X P t , X V t ) is number of the synchronized pairs of passengers and vehicles at the virtual buffer. Prior studies [34,35] have explored the transient and asymptotic behaviors of S t and proved that the synchronized flow S t converges to a Poisson process both analytically and numerically. The literature further yields a M/M/1 approximation for SM/M/1, with the arrival rate min(λ P , λ V ) and service rate µ > min(λ P , λ V ) shown in the right side of Fig.6.…”

“…For completeness, we present several key properties of S t . The details in derivations and proofs can be found in [34,35].…”