We show that if an input process ζ to a queue is asymptotic stationary in some sense, satisfies a condition AB and some other natural conditions, then the output processes (w, ζ) and (w, q,ζ) are asymptotic stationary in the same sense. Here, w and q are the waiting time and queue length processes, respectively.

We derive an asymptotic relation for the expected value of the stopping time , where Sk is a random walk with negative drift.

We consider networks with a very large or infinite number of nodes, linked by cable channels. The request which comes to a node is ordered to occupy a certain route of successive channels. The functioning of the system is regulated by the reserving of channels in order of the arrivals of the requests. Under some general conditions the existence of an ergodic region for such networks is proved. The practical value of the result lies in the fact that these conditions do not depend on the size of the graph.

The queueing system considered in this paper consists of r independent arrival channels and s independent service channels, where, as usual, the arrival and service channels are independent. In the queueing system, each server of the system has his own queue and arriving customers join the shortest line in the system. We give functional central limit theorems for the stochastic processes characterizing this system after appropriately scaling and translating the processes in traffic intensity ρ > 1.

A queueing theoretic model of an assembly operation is introduced. The model, consisting of K ≧ 2 renewal input processes and a single server, is a multiple input generalization of the GI/G/1 queue. The server requires one input item of each type k = 1,…, K for each of his services. It is shown that the model is inherently unstable in the following sense. The associated vector waiting time process Wn cannot converge in distribution to a non-defective limit, regardless of how well balanced the input and service processes may be. Limit theorems are developed for appropriately normalized versions of Wn under the various possible load conditions. Another waiting time process, equivalent to that in a single-server queue whose input is the minimum of K renewal processes, is also identified. It is shown to converge in distribution to a particular limit under certain load conditions.

In the paper a single server semi-Markov queue is considered. The analysis is based on a fundamental matrix identity due to H. D. Miller. A natural method for the solution of semi-Markov queues is indicated; use is also made of a duality relation, which is discussed in the paper.