Asymptotic stationarity of queues in series and the heavy traffic approximation

W. Szczotka
F. P. Kelly

Annals of Probability 18 (1990) 1232-1248.

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 ${\bf w}=\{\tilde w_k,k\geq 1\}$ and ${\bf W}=\{\tilde W_k,k\geq 1\}$ are asymptotically stationary in the same sense. Moreover, their stationary representations are given and the heavy traffic approximation of that stationary representation is given.

AMS 1980 subject classifications. 60K25, 60K20.

Keywords and phrases. Tandem queue, asymptotic stationarity, stationary representation, heavy traffic approximation, diffusion approximation.

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