Asymptotic Stationarity of Queues in Series and the Heavy Traffic Approximation

Abstract

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 $R^{m + 1}$. The first $m$ coordinates of the $k$th element of the generic sequence represent the service times of the $k$th unit in $m$ single server queues, respectively, and the $(m + 1)$th coordinate represents the interarrival time between the $k$th and $(k + 1)$th units to the tandem queue. The sequences of vectors $\tilde{w}_k = (w_k(1), w_k(2),\ldots, w_k(m))$ and $\tilde{W}_k = (W_k(1), W_k(2),\ldots, W_k(m))$, where $w_k(i)$ represents the waiting time of the $k$th unit in the $i$th queue and $W_k(i)$ represents the sojourn time of the $k$th 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 $\mathbf{w} = \{\tilde{w}_k, k \geq 1\}$ and $\mathbf{W} = \{\tilde{W}_k, k \geq 1\}$ are asymptotically stationary in the same sense. Moreover, their stationary representations are given and the heavy traffic approximation of that stationary representation is given.