Two Parallel Queues Created by Arrivals with Two Demands I

We analyze the double queue that arises when arriving customers simultaneously place two demands handled independently by two servers. It is assumed that the customer arrivals form a Poisson process with mean 1, the servers have exponential service times with rates, and 1 <, which implies stability of the queue. The equations for the equilibrium probabilities p₈₉ = P (i customers in -queue, j customers in -queue) are converted into a functional equations for P (z, w) = p₈₉ zⁱ wʲ, which exhibits a relation between P (z, 0), P (0, w) on the portion | z |, | w | 1 of S = \ (z, w): (1 + +) zw - w - z - z² w² = 0\. S is a Riemann surface of genus 1 which is parametrized by a pair of elliptic functions z = z (t), w = w (t). The functional equation for P (z, w) is converted into a set of conditions on P (z (t), 0), P (0, w (t) ), which in turn lead to the determination of P (z, w). From this, one obtains asymptotic formulas for p₈₉ as either i or j.