Abstract
We consider a parallel queueing model which has k identical servers. Assume that customers arrive from outside according to a Poisson process and join the shortest queue. Their service times have an i.i.d . exponential distribution, which is referred to as an M/MJSQ with k parallel queues. We are interested in the asymptotic behavior of the stationary distribution for the shortest queue length of this model, provided the stability is assumed. For this stationary distribution, it can be guessed conjectured that the tail decay rate is given by the k-th power of the traffic intensity of the corresponding M/M/k queue with a single waiting line. We prove this fact by obtaining the exactly geometric asymptotics. For this, we use two formulations. One is a quasi-birth-and-death (QBD for short) process which is typically used, and the other is a reflecting random walk on the boundary of the k + 1-dimensional orthant which is a key for our proof.
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