Asymptotic optimality of a fixed priority rule for a queueing problem in the large deviation regime∗

A multi-class M/M/1 system, with service rate μin for class-i customers, is considered with the risk-sensitive cost criterion n−1 logE exp ∑ i ciXi(T ), where ci > 0, T > 0 are constants, Xi(t) denotes the class-i queue-length at time t assuming the system starts empty. It is shown that, as n → ∞, asymptotic optimality is achieved by the service policy that prioritizes the classes in decreasing order of the index μi(1− e−ci). This index, noticed first in [1], can thus be thought of as a modification of the well-known cμ rule for risk-sensitive cost. AMS subject classifications: 60F10, 60K25, 49N70, 93E20