Donald L. Iglehart

Multiple channel queues in heavy traffic. II: sequences, networks, and batches

Abstract : Sequences of queueing facilities with r parallel arrival channels and s parallel service channels are studied under the conditions of heavy traffic: the associated sequences of traffic intensities approaching a limit greater than or equal to one. Weak convergence is obtained for sequences of random functions induced in D(0,1) by the basic queueing processes. Sequences of queueing systems in heavy traffic which are networks of the facilities described above are also investigated. Furthermore, customers are allowed to arrive and be served in batches.

Weak convergence in queueing theory

In the last ten years the theory of weak convergence of probability measures has been used extensively in studying the models of applied probability. By far the greatest consumer of weak convergence has been the area of queueing theory. This survey paper represents an attempt to summarize the experience in queueing theory with the hope that it will prove helpful in other areas of applied probability. The paper is organized into the following sections: queues in light traffic, queues in heavy traffic, queues with a large number of servers, continuity of queues, rates of convergence, and special queueing models. CONTINUITY OF QUEUES; DIFFUSION APPROMIXATIONS; FUNCTIONAL CENTRAL LIMIT THEOREMS; INVARIANCE PRINCIPLE; QUEUEING THEORY; QUEUES IN HEAVY TRAFFIC; QUEUES IN LIGHT TRAFFIC; QUEUES WITH MANY SERVERS; WEAK CONVERGENCE

Approximations for the repairman problem with two repair facilities, I: No spares

Abstract : The report considers a generalization of the repairman models treated in a previous report. Here the system studied consists of n operating units, M(n) spare units, and two repair facilities. The operating units are subject to failures of two types: minor and major. Minor failures are sent to a local repair facility and major failures to a central repair facility. Once a unit is repaired it is returned to the spare pool is n units are operating, otherwise it goes directly into operation. The stochastic processes representing the number of units waiting or undergoing repair at each of the two repair facilities is a finite state, continuous-parameter Markov chain, which attains a stationary or steady-state distribution after a long time has elapsed. The method developed in this paper provides a readily calculated approximation when n is large, along with results which allow for reliable prediction of system performance. The principal concern in this paper is to understand how system performance is effected by the many system parameters.