On the Nonsymmetric Longer Queue Model II: Marginal and Conditional Probabilities

Abstract

queue, with 0 < ν < 1. We let (N1, N2) be the numbers of customers in the two parallel queues, and let p(m, n) = Prob[N1 = m, N2 = n] be the joint queue length distribution in the steady state. The two arrival rates are λ1 and λ2, the exponential server works at rate μ, and ρ1 = λ1/μ, ρ2 = λ2/μ. If N1 > N2 (N1 < N2), the server works on the first (second) queue, but if N1 = N2, the server works at rate μν on the first queue and rate μ(1 − ν)