On the non-Markovian multiclass queue under risk-sensitive cost

Abstract

This paper studies a control problem for the multiclass G/G/1 queue for a risk-sensitive cost of the form $$n^{-1}\log E\exp \sum _ic_iX^n_i(T)$$n-1logEexpźiciXin(T), where $$c_i>0$$ci>0 and $$T>0$$T>0 are constants, $$X^n_i$$Xin denotes the class-i queue length process, and the numbers of arrivals and service completions per unit time are of order n. The main result is the asymptotic optimality, as $$n\rightarrow \infty $$nźź, of a priority policy, provided that $$c_i$$ci are sufficiently large. Such a result has been known only in the Markovian (M/M/1) case. The index which determines the priority is explicitly computed in the case of Gamma-distributed interarrival and service times.