Two M/M/1 queues with transfers of customers

We study a system consisting of two M/M/1 queues with transfers of customers. In that system, when the difference of the queue lengths reaches L (> 0), a batch of (0 < K< L) customers is transferred from the longer to the shorter queue. A quasi birth-and-death Markov process describes the queueing process of the two queues. By using the mean-drift method, a simple condition for system stability is found. For the stationary distribution of the queueing process, a matrix geometric solution is obtained. It is shown that the stationary distribution of the total number of customers in the system decays exponentially. The decay rate is found explicitly. Results on the busy periods of the queueing system are also obtained. By using these theoretical results, we numerically explore the optimal design of such queueing systems in terms of customer transfer rates, the batch size of transfer, throughputs, arrival rates, and service rates. In particular, we observe that a balanced queueing system has a total transfer rate that is either the smallest or close to the smallest. We also observe that for system optimization with respect to system descriptors such as the total transfer rate, the mean total queue length, or the system idle probability, the choice of the batch size K has much to do with the difference of the relative traffic intensities of the two M/M/1 queues.