Join the shortest queue among $k$ parallel queues: tail asymptotics of its stationary distribution

TL;DR

This paper analyzes the tail behavior of the stationary distribution in a join the shortest queue system with $k$ heterogeneous servers, proving the geometric decay rate of the minimum queue length and its relation to the traffic intensity.

Contribution

It establishes the exact geometric tail asymptotics for the minimum queue length in a heterogeneous $M/M$-JSQ system, extending known results beyond the homogeneous case.

Findings

Tail of the minimum queue length is exactly geometric.

Decay rate is the $k$-th power of the traffic intensity.

Uses QBD process and reflecting random walk formulations.

Abstract

We are concerned with an $M/M$-type join the shortest queue ($M/M$-JSQ for short) with $k$ parallel queues for an arbitrary positive integer $k$, where the servers may be heterogeneous. We are interested in the tail asymptotic of the stationary distribution of this queueing model, provided the system is stable. We prove that this asymptotic for the minimum queue length is exactly geometric, and its decay rate is the $k$-th power of the traffic intensity of the corresponding $k$ server queues with a single waiting line. For this, we use two formulations, a quasi-birth-and-death (QBD for short) process and a reflecting random walk on the boundary of the $k+1$-dimensional orthant. The QBD process is typically used in the literature for studying the JSQ with 2 parallel queues, but the random walk also plays a key roll in our arguments, which enables us to use the existing results on tail asymptotics for the QBD process.