Moment properties and long-range dependence of queueing processes

We develop confidence estimation of moments and autocovariance function of the workload processes in a tandem queueing system where service time has finite moments of first three orders. Provided stability condition is satisfied, we use regenerative simulation to construct both point and confidence estimates of the overflow probability. For infinite buffer system overflow may be related to re-routing traffic for the sake of preserving bandwidth, and for finite buffer systems connected with estimation of QoS. An important aim of the research is to confirm both divergence of the autocovariance function under these moment conditions, and applicability of regenerative simulation for the estimation mentioned above.

In general, such research turns out to be difficult (or impossible) to realize because divergence of covariance function means long-range dependence, which in turns typically corresponds to infinite variance of the corresponding random variables. But latter precludes a possibility to use regenerative central limit theorem for confidence estimation.

At that we extend simulation experiments outside the limit of previous study which has been performed only for workload processes in both stations in tandem. Moreover, we present simulation results related to verification of the asymptotic behavior of the stationary workload tail probability in the case when tail of service time at second station is heavier than the tail in first station of the network.