Two Parallel Queues Created by Arrivals with Two Demands II

We consider 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 rate $\alpha ,\beta $, and $1 < \alpha \leqq \beta $, which insures stability of the queue. Let $X_1 ,X_2 $ be the respective lengths of the $\alpha $- and $\beta $-queues, and $P_{mn} = P [ X_1 = m,X_2 = n ]$ at equilibrium. In a previous paper with the same title we obtained a formula for the generating function $P( z,w ) = \sum p_{mn} z^m w^n $. We use this to derive the asymptotic behavior of $p_{mn}$ as $m,n \to \infty $. The asymptotic results are employed to study the interdependence of $X_1 ,X_2$. We derive limit laws for the expectation and distribution for either of these random variables conditioned on the other.