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Short Communications
On the output stream and the joint distribution of sojourn times in a multiphase system with identical service
O. P. Vinogradov M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
Abstract:
We consider a multiphase queueing system consisting of
$(k+1)$ servers, in which the service time
$T_{n,i}$ of the
$n$th customer at the
$i$th server possesses the property $\mathbf{P}\{T_{n,1}=T_{n,2}=\dots=T_{n,k+1}=T_n\}=1$. It is proved that when the number of servers grows to infinity, then, under some conditions, the output stream of the system converges to a renewal process. In the case of the Poisson input stream and general service time we find, for finite
$k$, the distribution of the interdeparture times. Let
$U_i$ be the sojourn time of customer at the
$i$th server under steady-state conditions
$(n\to\infty)$ and let
$V_k=\sum_{i=2}^{k+1}U_i$. We find the distributions of variables
$V_{k-1}$ and
$V_k$, and the distributions of sojourn times at each server.
Keywords:
multiphase queueing systems with identical service, waiting time, sojourn time, renewal process, output stream. Received: 30.06.1993
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