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

We consider two parallel queues, each with independent Poisson arrival rates, that are tended by a single server. The exponential server devotes all of its capacity to the longer of the queues. If both queues are of equal length, the server devotes ν of its capacity to the first queue and the remaining 1−ν to the second. We obtain the marginal probabilities, both exactly and asymptotically, and establish several conditional limit laws, which show how many customers may be in the first queue, given a large number in the second, etc. Numerical studies are used to back up the asymptotic analysis.