Resource sharing for efficiency in traffic systems

Experience has shown that efficiency usually increases when separate traffic systems are combined into a single system. For example, if Group A contains 10 trunks and Group B 8 trunks, there should be fewer blocked calls if A and B 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 comparisons for the simplest models, but also apply to general arrival processes and general service-time distributions. However, it is assumed 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 service-time 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.