Optimal server selection in a queueing loss model with heterogeneous exponential servers, discriminating arrivals, and arbitrary arrival times

Abstract

We consider a multiple server queueing loss system where the service times of server i are exponential with rate μ_i, where μ_i decreases in i. Arrivals have associated vectors (X₁, . . ., X_n) of binary variables, with X_i = 1 indicating that server i is eligible to serve that arrival. Arrivals finding no idle eligible servers are lost. Letting I_j be the indicator variable for the event that the jth arrival enters service, we show that, for any arrival process, the policy that assigns arrivals to the smallest numbered idle eligible server stochastically maximizes the vector (I₁, . . ., I_r) for every r if the eligibility vector of arrivals is either (a) exchangeable, or (b) a vector of independent variables for which P(X_i = 1) increases in i.