The polling system with a stopping server

This paper analyzes the polling system in which, unlike nearly all previous studies, the server comes to a stop when the system is empty rather than continuing to cycle. The possibility of server stopping permits a rich variety of alternatives for server behavior; the paper considers three stopping rules, governing server behavior when the system is emptied, and two starting rules, governing server behavior when an arrival occurs to an idle system. The Laplace-Stieltjes transforms and means for the waiting time and server return time (the interval from an arrival at an unserved queue until the server returns to that queue) are determined. For the special case of zero changeover times and strictly cyclic service, explicit results are obtained.