On the use of second-order descriptors to predict queueing behavior of Markovian arrival processes

The contributions of this paper are the following: We derive a formula for the IDI (Index of Dispersion for Intervals) for the Markovian Arrival Process (MAP). We show that two-state MAPs with identical fundamental rate, IDI and IDC (Index of Dispersion for Counts), define interval stationary point processes that are stochastically equivalent; this is true for the time stationary point processes they define too. Special cases of the two-state MAP are frequently used as source models in the literature. The result shows that, fitting to the rate, IDC and IDI of a source completely determine the interval stationary and time stationary behavior of the two-state model. We give various illustrative numerical examples on the merits in predicting queueing behavior on the basis of first- and second-order descriptors by considering queueing behavior of MAPs with constant fundamental rate and IDC, repectively, constant fundamental rate and IDI. Disturbing results are presented on how different the queueing behavior can be with these descriptors fixed. Even MAPs with NO correlations in the counting process, i.e., IDC(t) = 1 are shown to have very different queueing behavior.