Copyright © 2013 Charles Knessl and Haishen Yao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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 of its capacity to the first queue and the remaining to the second. We obtain exact integral representations for the joint probability distribution of the number of
customers in this two-node network. Then we evaluate this distribution
in various asymptotic limits, such as large numbers of customers
in either/both of the queues, light traffic where arrivals are infrequent,
and heavy traffic where the system is nearly unstable.