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July, 1990 Asymptotic Stationarity of Queues in Series and the Heavy Traffic Approximation
W. Szczotka, F. P. Kelly
Ann. Probab. 18(3): 1232-1248 (July, 1990). DOI: 10.1214/aop/1176990744


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.


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W. Szczotka. F. P. Kelly. "Asymptotic Stationarity of Queues in Series and the Heavy Traffic Approximation." Ann. Probab. 18 (3) 1232 - 1248, July, 1990.


Published: July, 1990
First available in Project Euclid: 19 April 2007

zbMATH: 0726.60092
MathSciNet: MR1062067
Digital Object Identifier: 10.1214/aop/1176990744

Primary: 60K25
Secondary: 60K20

Keywords: asymptotic stationarity , diffusion approximation , heavy traffic approximation , stationary representation , tandem queue

Rights: Copyright © 1990 Institute of Mathematical Statistics


Vol.18 • No. 3 • July, 1990
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