Asymptotically Optimum Arrangements for a Special Class of Normed Spaces

Abstract

A nonclassical queuing-theory problem with calls arising in a space is considered. Stations must be placed to minimize the service time for arising calls. The service time is an increasing function that depends on the distance between a call and a station. The time spent to overcome the same distance frequently depends on the direction of motion. In this case, a metric that considers the nonequivalence of coordinates of the space in order must be chosen to construct an adequate mathematical model. Optimum arrangements of stations can be found for problems of this kind only in exceptional situations. However, an asymptotical solution to the problem can be found that is acceptable from a practical viewpoint. An algorithm is given for constructing asymptotically optimum arrangements.