Resource Sharing for Efficiency in Traffic Systems

Abstract

Experience has shown that efficiency usually increases when sep arate traffic systems are combined into a single system. For example, if Group A contains 10 trunks and Group Β 8 trunks, there should be fewer blocked calls if A and Β are combined into a single group of 18 trunks. It is intuitively clear that the separate systems are less efficient because a call can be blocked in one when trunks are idle in the other. Teletraffic engineers and queuing theorists widely accept such efficiency principles and often assume that their mathematical proofs are either trivial or already in the literature. This is not the case for two fundamental problems that concern combining blocking systems (as in the example above) and combining delay systems. For the simplest models, each problem reduces to the proof of an inequality involving the corresponding classical Erlang function. Here the two inequalities are proved in two different ways by exploiting general stochastic comparison concepts: first, by monotone likelihood-ratio methods and, second, by sample-path or "coupling" methods. These methods not only yield the desired inequalities and stronger compar isons for the simplest models, but also apply to general arrival processes and general service-time distributions. However, it is as sumed that the service-time distributions are the same in the systems being combined. This common-distribution condition is crucial since it may be disadvantageous to combine systems with different servicetime distributions. For instance, the adverse effect of infrequent long calls in one system on frequent short calls in the other system can outweigh the benefits of making the two groups of servers mutually accessible. I. I N T R O D U C T I O N A N D S U M M A R YFrom extensive experience in teletraffic engineering, it is well known that congestion can often be reduced by sharing resources. The block- 9ing probability in a loss system and the average waiting time in a delay system are usually much less when separate facilities serving separate streams of traffic are combined to serve all the streams together. Alternatively, for a given level of congestion, fewer facilities are usually required to serve the streams together. Sometimes such results are trivial: Whenever the combined system may be managed as if it were in fact separate systems, the optimal performance of the combined system is at least as good as that of the separate systems. However, such management is not allowed in the models treated here. In any case, the efficiency of shared resources is certainly a fundamental principle of teletraffic engineering. The purpose of this paper is to establish versions of this efficiency principle mathematically. Our first two results verify conjectures by Arthurs and Stuck. 1 T o state our first result, let L(s, λ μ) denote the stationary loss or overflow rate in an M/M/s loss system (no waiting room) with s servers, arrival rate λ, and individual service rate μ. (See Kleinrock 2 for background on the queuing models.) It is well known that L(s...

On Pooling in Queueing Networks

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“…. The Erlang-B formula plays an important role in many problems of teletraffic theory and this is probably the reason why it has been the subject of intensive study, as shown in references [17] [18] [19] [20] [26]. In several teletraffic studies the need arose to extend the definition of Erlang-B function to non-integer values of x by using its analytical extension, ascribed to R. Fortet [27, pag.…”

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