Asymptotic Stationarity of Queues in Series and the Heavy Traffic Approximation

A tandem queue with m single server stations and unlimited interstage storage is considered. Such a tandem queue is described by a generic sequence of nonnegative random vectors in Rm + 1. The first m coordinates of the kth element of the generic sequence represent the service times of the kth unit in m single server queues, respectively, and the $(m + 1)$th coordinate represents the interarrival time between the kth and $(k + 1)$th units to the tandem queue. The sequences of vectors w̃ₖ = (wₖ(1), wₖ(2),…, wₖ(m)) and W̃ₖ = (Wₖ(1), Wₖ(2),…, Wₖ(m)), where wₖ(i) represents the waiting time of the kth unit in the ith queue and Wₖ(i) represents the sojourn time of the kth unit in the first i queues, are studied. It is shown that if the generic sequence is asymptotically stationary in some sense and it satisfies some natural conditions then w = ̃ₖ, k ≥ 1\ and W = ̃ₖ, k ≥ 1\ are asymptotically stationary in the same sense. Moreover, their stationary representations are given and the heavy traffic approximation of that stationary representation is given.