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 $\alpha ,\beta $ and $1 < \alpha \leqq \beta $, which implies stability of the queue. The equations for the equilibrium probabilities $p_{ij} = P$ (i customers in $\alpha $-queue, j customers in $\beta $-queue) are converted into a functional equations for $P(z,w) = \sum p_{ij} z^i w^j $, which exhibits a relation between $P(z,0)$, $P(0,w)$ on the portion $| z |$, $| w |\leqq 1$ of $S = \{ (z,w):(1 + \alpha + \beta )zw - \alpha w - \beta z - z^2 w^2 = 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 $p_{ij} $ as either i or $j \to \infty $.