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

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

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Article ID: iaor20132907
Volume: 74
Issue: 2
Start Page Number: 303
End Page Number: 332
Publication Date: Jun 2013
Journal: Queueing Systems
Authors: , ,
Abstract:

We are concerned with an M / M equ1 ‐type join the shortest queue ( M / M equ2 ‐JSQ for short) with k equ3 parallel queues for an arbitrary positive integer k equ4 , 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 equ5 th power of the traffic intensity of the corresponding k equ6 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 equ7 ‐dimensional orthant. The QBD process is typically used in the literature for studying the JSQ with two 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.

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