Information Theory of the Discriminatory Processor Sharing Queues

The discriminatory processor sharing queues with multiple classes of customers (abbreviated as DPS queues) are an important but difficult research direction in queueing theory, and it has many important practical applications in the fields of, such as, computer networks, manufacturing systems, transportation networks, and so forth. Recently, researchers have carried out some key work for the DPS queues. They gave the generating function of the steady-state joint queue lengths, which leads to the first two moments of the steady-state joint queue lengths. However, using the generating function to provide explicit expressions for the steady-state joint queue lengths has been a difficult and challenging problem for many years. Based on this, this paper applies the maximum entropy principle in the information theory to providing an approximate expression with high precision, and this approximate expression can have the same first three moments as those of its exact expression. On the other hand, this paper gives efficiently numerical computation by means of this approximate expression, and analyzes how the key variables of this approximate expression depend on the original parameters of this queueing system in terms of some numerical experiments. Therefore, this approximate expression has important theoretical significance to promote practical applications of the DPS queues. At the same time, not only do the methodology and results given in this paper provide a new line in the study of DPS queues, but they also provide the theoretical basis and technical support for how to apply the information theory to the study of queueing systems, queueing networks and more generally, stochastic models.