Some extremal problems in the queueing theory

B. A. Rogozin

Teoriâ veroâtnostej i ee primeneniâ, Tome 11 (1966) no. 1, pp. 161-169

Résumé

The simplest queueing systems are considered. It is supposed that the periods of time between two succesive arrivals of the calls $\tau_1,\tau_2,\dots,\tau_n,\dots$ as well as the service times $\eta_1,\eta_2,\dots,\eta_n,\dots$ are independent identically distributed random variables, with $\eta_1,\eta_2,\dots,\eta_n$ being independent of $\tau_1,\tau_2,\dots,\tau_n,\dots$. 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 $x_1\ge0$, $x_2\ge0$.