The use of the distributional Little’s law in the computational analysis of discrete-time GI/G/1 and GI/D/c queues

In this paper, we first establish a discrete-time version of what is called the distributional Little’s law, a relation between the stationary distributions of the number of customers in a system (or queue length) and the number of slots a customer spends in that system (or waiting time). Based on this relation, we then present a simple computational procedure to obtain the queue-length distribution of the discrete-time GI/G/1 queue from its waiting-time distribution, which is readily available by various existing methods. Using the same procedure, we also obtain the queue-length distribution of the discrete-time multi-server GI/D/c queue in a unified manner. Sample numerical examples are also given.