Properties of Markovian arrival processes

Second, the set of MAPs is closed under some operations. In particular, a superposition of two independent MAPs is a MAP.

Third, the average rate of events and the marginal distribution of the inter-event time of a MAP have simple mathematical expressions.

*The marginal distribution of the inter-event time of the above MAP is a PH(
) distribution,
where
*

The variability of a point process can be characterized by the covariance
of the inter-event times or the index of dispersion for intervals.
The index of dispersion for intervals, IDI(), of a MAP is defined
by

where is the sum of the first inter-event times of the MAP. Observe that IDI() is the ratio of the variance of to the corresponding variance of a Poisson process with the same rate. The MAP have a convenient mathematical expressions for the covariance and the index of dispersion for intervals.

*The IDI() of the above MAP is given by
*

Finally, a MAP can be translated into a counting process, i.e., the number of events by time , .
The MAP has a simple -transform of , and this leads
to a convenient mathematical expression for the index of dispersion for counts,
which characterizes the variability of a MAP via the variability of .
The index of dispersion for counts, IDC(), of the MAP is defined by

Observe that IDC() is the ratio of the variance of to the corresponding variance of a Poisson process with the same rate.

*The IDC() of the above MAP is given by
*