Abstract:
A multiphase queueing system is considered. The service time of the nth arrival at the $i$th server is $T_{n,i} $ and $\mathbf{P}\{ T_{n,1} = \cdots = T_n \} = 1$, where $\{ T_n \} $ are independent identically distributed random variables with an arbitrary common distribution. Let $U_l (n)$ be the time spent by the $l$ arrival at the $l$th server. Some algebraic properties of the sequence $\{ U_l (n)\} (l \ge 2)$ are cleared up. In the case of Poisson input flow, the distributions of some characteristics of the system are obtained, as well as a number of limit theorems for the situation where the number of servers grows infinitely.
Keywords:multiphase queueing systems with identical service times, tandem queues, waiting time, sojourn time, recurrence relation, limit theorem, regularly varying function, Tauberian theorem.
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