Singular perturbation for a two-class Processor-Sharing queue with impatience

Abstract

A two-class Processor-Sharing queue with one impatient class is studied. Local exponential decay rates for its stationary distribution (N, M) are established in the heavy traffic regime where the arrival rate of impatient customers grows proportionally to a large factor A. This regime is characterized by two time-scales, so that no general Large Deviations result is applicable. In the framework of singular perturbation methods, we instead assume that an asymptotic expansion of the solution of associated Kolmogorov equations exists for large A and derive it in the form P(N = Ax, M = Ay) ~ g(x,y)/A exp(-A H(x,y)) for x > 0 and y > 0 with explicit functions g and H. This result is then applied to the model of mobile networks proposed in a previous work and accounting for the spatial movement of users. We give further evidence of a unusual growth behavior in heavy traffic in that the stationary mean queue length E(N') and E(M') of each customer-class increases proportionally to E(N') ~ E(M') ~ -log(1-rho) with system load rho tending to 1, instead of the usual 1/(1-rho) growth behavior.