On the Adan-Weiss Loss Model having Skill-Based Servers and Longest Idle Assignment Rule

We consider a queueing loss system with heterogeneous skill based servers with arbitrary distributions. We assume Poisson arrivals, with each arrival having a vector indicating which of the servers are eligible to serve it. Arrivals can only be assigned to a server that is both idle and eligible. We assume arrivals are assigned to the idle eligible server that has been idle the longest and derive, up to a multiplicative constant, the limiting distribution for this system. We show that the limiting probabilities of the ordered list of idle servers depend on the service time distributions only through their means. Moreover, conditional on the ordered list of idle servers, the remaining service times of the busy servers are independent and have their respective equilibrium service distributions. We also provide an algorithm using Gibbs sampler Markov Chain Monte Carlo method for estimating the limiting probabilities and other desired quantities of this system.