Optimal arrival rate in a simple queueing system

Over the last decade increasing attention has been devoted to introducing optimization into queueing models. The emphasis has predominantly been upon choosing facilities to minimize cost of servicing a given demand. Recently there has been some discussion of optimal balking rules and charges in order to influence demand. Now the concept of pricing to determine the optimal level of congestion (e.g. on roads) has long been familiar to economists. However, such economic analyses have not been couched in terms of queueing theory nor do recent texts in operational research reflect the economic approach. This paper therefore gives a concise derivation and discussion of the pricing principle in terms of one of the simplest queueing models. A graphical interpretation is also available which should aid in the intuitive understanding of the more sophisticated models referred to. The general implication of these analyses is that, just as providing ‘ excess ” capacity is worth while if it costs less than the cost of delays avoided, so it is also worth while to turn away arrivals or reduce usage as long as the benefit thus sacrificed is more than offset by reduction in cost of delay.