An analytical investigation of the behavior of a priority queuing system with a mix of correlated train and uncorrelated batch arrivals

In this paper, we present an exact transient and steady-state analysis of a discrete-time queuing system with two head-of-line (HOL) priority queues and a mix of correlated and uncorrelated arrivals. The arrival process to the high priority queue is correlated and consists of a train of a fixed number of cells, while the low priority traffic consists of batch arrivals that are independent and identically distributed from slot-to-slot. In the first part of the paper, we derive an expression for the functional equation describing the transient evolution of this priority queuing system. This functional equation is then manipulated and transformed into a mathematical tractable form, which allows us to derive the transient joint probability generating function (pgf) of the system. From this transient pgf, time-dependent performance measures such as transient probability of empty queue, transient mean of buffer occupancy and instantaneous packet overflow probabilities can be derived. By applying the final-value theorem, the corresponding exact expressions for the steady-sate marginal are pgfs of the system contents derived. Finally, we illustrate our solution technique with some numerical examples, whereby we demonstrate the negative effect of correlation (in the high-priority queue) on the performance of the low-priority queue. The proposed approach which is purely based on probability generating functions is entirely analytical and does not require any matrix concepts. The paper presents new results on the transient and steady-state performance analysis of priority queues.