On the Time Reversal of Markovian Arrival Processes

Abstract

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.

Key Words:

• Markovian arrival processes
• Reversibility
• Matrix-analytic methods
• Algorithmic probability
• Arrival process descriptors

AMS Subject Classification:

• Primary 60J25
• Secondary 65U05

Acknowledgments

This research of M. F. Neuts was supported in part by NSF Grant Nr. DMI-9306828 and was partially undertaken during his visit to the Technical University of Denmark in Fall 1997. M. F. Neuts acknowledges the hospitality and support of the Danish Technical University where part of this research was undertaken during a visit in 2001 were he collaborated with his coauthors. Allan T. Andersen acknowledges financial support from the Technical Research Council of Denmark under grant no. (9500947).