Stochastic comparisons for fork-join queues with exponential processing times

Abstract

Consider a fork-join queue, where each job upon arrival splits into k tasks and each joins a separate queue that is attended by a single server. Service times are independent, exponentially distributed random variables. Server i works at rate , where μ is constant. We prove that the departure process becomes stochastically faster as the service rates become more homogeneous in the sense of stochastic majorization. Consequently, when all k servers work with equal rates the departure process is stochastically maximized.