Transient behaviour of queueing systems with correlated traffic

In this paper, we present the time-dependent solutions of various stochastic processes associated with a finite Quasi-Birth-Death queueing system. These include the transient queueing solutions, the transient departure and loss intensity processes and certain transient cumulative measures associated with the queueing system. The focus of our study is the effect of the arrival process correlation on the queueing system before it reaches steady-state. With the aid of numerous examples, we investigate the strong relationship between the time scales of variation of the arrival process and those of the transient queueing, loss and departure processes. These time-dependent solutions require the computation of the exponential of the stochastic generator matrix G which may be of very large order. This precludes the use of known techniques to solve the matrix exponential such as the eigenvalue decomposition of G. We present a numerical technique based on the computation of the Laplace Transform of the matrix exponential which may then be numerically inverted to obtain the time-dependent solutions. In this paper, we also propose new QoS metrics based on transient measures and efficient techniques for their computation.