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 Teor. Veroyatnost. i Primenen.: Year: Volume: Issue: Page: Find

 Teor. Veroyatnost. i Primenen., 1966, Volume 11, Issue 1, Pages 161–169 (Mi tvp576)

Short Communications

Some extremal problems in the queueing theory

B. A. Rogozin

Novosibirsk

Abstract: The simplest queueing systems are considered. It is supposed that the periods of time between two succesive arrivals of the calls $\tau_1,\tau_2,…,\tau_n,…$ as well as the service times $\eta_1,\eta_2,…,\eta_n,…$ are independent identically distributed random variables, with $\eta_1,\eta_2,…,\eta_n$ being independent of $\tau_1,\tau_2,…,\tau_n,…$.
In the case of queueing systems it is established that when the usual conditions are satisfied, the distribution of $\tau_1$ is fixed and $\mathbf E\eta_1=\alpha$, the greatest lower bound of the expectation of the limit distribution of the waiting time $\mathbf EW$ is attained on the distribution $\mathbf P\{\eta_1=\alpha\}=1$. The similar question concerning $\mathbf EW$ is considered when the distribution of $\eta_1$ is fixed and $\mathbf E\tau_1=\beta$. Besides in the same situation an upper estimate for $\mathbf EW$ is given.
In the case of systems with losses of calls it is established that the extrema of the probability to be served when the distribution of $\tau_1$ is fixed and $\mathbf E\eta_1=\alpha$ is attained on, the distributions of $\eta_1$ such that $\mathbf P\{\eta_1=x_1\}+\mathbf P\{\eta_2=x_2\}=1$ for some $x_1\ge0$, $x_2\ge0$.

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English version:
Theory of Probability and its Applications, 1966, 11:1, 144–151

Bibliographic databases:

Citation: B. A. Rogozin, “Some extremal problems in the queueing theory”, Teor. Veroyatnost. i Primenen., 11:1 (1966), 161–169; Theory Probab. Appl., 11:1 (1966), 144–151

Citation in format AMSBIB
\Bibitem{Rog66} \by B.~A.~Rogozin \paper Some extremal problems in the queueing theory \jour Teor. Veroyatnost. i Primenen. \yr 1966 \vol 11 \issue 1 \pages 161--169 \mathnet{http://mi.mathnet.ru/tvp576} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=193689} \zmath{https://zbmath.org/?q=an:0147.36805} \transl \jour Theory Probab. Appl. \yr 1966 \vol 11 \issue 1 \pages 144--151 \crossref{https://doi.org/10.1137/1111011}