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 α ,β and 1 < α β, which implies stability of the queue. The equations for the equilibrium probabilities pᵢⱼ = P (i customers in α-queue, j customers in β-queue) are converted into a functional equations for P(z,w) = ∑ pᵢⱼ zⁱ wʲ, which exhibits a relation between $P(z,0)$, $P(0,w)$ on the portion $| z |$, | w | 1 of S = \ (z,w):(1 + α + β )zw - α w - β 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 pᵢⱼ as either i or j → ∞.