On the time reversal of Markovian Arrival Processes

We study the point process obtained by reversing time in a stationary Markovian Arrival Process (MAP). That process is also a MAP. We show that the most frequently used classical statistical descriptors of point processes are insensitive to the orientation of the time-axis. Therefore they fail to distinguish between a MAP and its reverse. That is the case for the second order descriptors of the counting and interval processes. Actually, for a MAP and its reverse the marginal distributions of the counting and interval processes agree. Using simple examples, we demonstrate that the behavior of two queues, one with a MAP and the other with its reverse as input streams, can be very different. This, in spite of the agreement of most of the standard descriptors. These findings illustrate the limitations of most of the standard descriptors of point processes in predicting queueing behavior. Quantification of non-reversibility could lead to new informative descriptors.