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

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 - 1 log E exp ∑ i c i X i n ( T ) , where $$c_i>0$$ c i > 0 and $$T>0$$ T > 0 are constants, $$X^n_i$$ X i n 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 $$n\rightarrow \infty $$ n → ∞ , of a priority policy, provided that $$c_i$$ c i 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.