Waiting Time in an MSP Queue with Active Management

We study waiting times in a queue with active management and correlated job/packet sizes, which induce correlated service times. In the transient case, formulae for the distribution tail, the probability density, and the expected virtual waiting time at any time t, are found. Then, grounded on these results, stationary versions of the probability density and the expected value are obtained. The correlation of service times resulting from correlated job sizes is modeled through an MSP (Markovian service process). Theoretical results are reinforced by numerical examples, in which we examine the impact of symmetric positive and negative correlation of service times, and the impact of symmetric weak and strong active management, on transient and stationary waiting times. We also compare the effects of these factors on the waiting time with their effects on the queue length. In these examples, we can see a surprisingly large expected virtual waiting time, much greater than the product of the expected service time and the queue length. This effect is observed for both weak and strong management functions when the correlation is positive, but it vanishes when a symmetric negative correlation is applied. We also observe a weaker effect of active management on virtual waiting times than that of service time correlation, as well as a weaker impact of active management on virtual waiting time densities than on queue length distributions.