Some limit theorems of the queueing theory.~II. (Many channels systems)

A. A. Borovkov

Teoriâ veroâtnostej i ee primeneniâ, Tome 10 (1965) no. 3, pp. 409-436

Résumé

Queneing systems with many input and output channels are considered. By the channel with index $j$, $1\le j\le m$, $m\ge1$, groups of calls of the random size $\eta_1^{(j)},\eta_2^{(j)},\dots$ arrive at the instants $a^{(j)}$, $a^{(j)}+\tau^{(j)}$, $a^{(j)}+\tau_1^{(j)}+\tau_2^{(j)},\dots$. The arrived calls fall in the general queue. The service is made by $M-m\ge1$ servers and described in the similar way: the server with index $j$, $m+1\le j\le M$, begins to work at the instant $a^{(j)}$ and can serve $-\eta^{(j)}\ge0$ calls during random time $\tau^{(j)}$. The sequences $\tau_1^{(j)},\tau_2^{(j)},\dots$; $\eta_1^{(j)},\eta_2^{(j)},\dots$; $j=\overline{1,M}$ “managing” the system are mutually independent sequences of independent identically distributed random variables. Every server can work in one of the two ways: either the service starts at instans $a^{(j)}$, $a^{(j)}+\tau^{(j)}$, $a^{(j)}+\tau_1^{(j)}+\tau_2^{(j)},\dots$ regardless of the presence of the queue or the service begins only when there is at least one call. In this paper all the limit distribution laws of the length of the queue $\theta(T)$ at instant $T\to\infty$ are found for the described systems working in heavy traffic when $$ \delta=-\sum_{j=1}^M\frac{\mathbf M\eta^{(j)}}{\mathbf M\tau^{(j)}}\to0. $$ Theorem. Let $$ \sigma^2(\delta)=\sum_{j=1}^M\biggl[\frac{\mathbf D\eta^{(j)}}{\mathbf M\tau}+\frac{(\mathbf M\eta^{(j)})^2\mathbf D\tau^{(j)}}{(\mathbf M\tau^{(j)})^3}\biggr]\to\sigma^2>0\text{ as }\delta\to0 $$ and let for some $\gamma>0$ the moments $\mathbf M\tau^{(j)}$, $\mathbf M\bigl|\frac{\tau^{(j)}-\mathbf M\tau^{(j)}}{\sqrt{\mathbf D\tau^{(j)}}}\bigr|^{2+\gamma}$, $\frac{\mathbf M|\eta^{(j)}-\mathbf\eta^{(j)}|^{2+\gamma}}{\mathbf D\eta^{(j)}}$ be bounded uniformly in $\delta$. Let further the initial conditions $\theta(0)$, $a^{(1)},\dots,a^{(M)}$ satisfy the following requirements $$ (\delta T)^{-1}\max_j\biggl(\theta(0),\frac{a^{(j)}}{\mathbf M\tau^{(j)}}\biggr)\underset{\mathbf P}\to0\text{ if }\delta\sqrt T\ge1 $$ and $$ T^{1/2}\max_j\biggl(\theta(0),\frac{a^{(j)}}{\mathbf M\tau^{(j)}}\biggr)\underset{\mathbf P}\to0\text{ if }\delta\sqrt T1 $$ A. If $\delta\sqrt T\to U$, $-\infty\le U\le\infty$ \begin{gather*} \lim_{\delta\to0}\mathbf P\biggl(\theta(T)\frac x{|\delta|}\mid\theta(0),a^{(1)},\dots,a^{(M)}\biggr)= \\ =\mathbf P\biggl(\omega(t)\frac{x+t\operatorname{sign}\delta}\sigma,\ 0\le t\le U^2\biggr). \end{gather*} B. If $\delta\sqrt T\to0$ $$ \lim_{\delta\to0}\mathbf P(\theta(T)\sqrt T\mid\theta(0),a^{(1)},\dots,a^{(M)})=\sqrt{\frac2\pi}\int_0^{x/\sigma}e^{-t^2/2}\,dt. $$ C. If $\delta\sqrt T\to-\infty$ $$ \lim_{\delta\to0}\mathbf P(\theta(T)-\delta T+x\sqrt T\mid\theta(0),a^{(1)},\dots,a^{(M)})=\frac1{\sqrt2\pi}\int_{-\infty}^{x/\sigma}e^{-t^2/2}\,dt. $$ Here $\omega(t)$ is the standard Brownian motion process. The similar assertion holds true for the waiting time.