Higher Order Moments and Conditional Asymptotics of the Batch Markovian Arrival Process

Abstract

We consider the number of arrivals in a Batch Markovian Arrival Process (BMAP) and derive matrix analytic expressions for its moments of arbitrary order. These expressions consist of decomposition formulas connected to the semigroup structure of the moments, forward and backward differential equations, and recursive as well as direct integral formulas. This extends earlier work by Narayana and Neuts on the first two factorial moment matrices. We next turn to the terminating BMAP, i.e., a BMAP with an absorbing state in which no arrivals occur. We consider the asymptotic behavior of the moments conditional on the process not yet having terminated. We show that the conditional mean and variance possess affine asymptotics and derive the coefficients explicitly. Finally, we discuss how parts of our work also apply to the more general class of Rational Arrival Processes (RAPs).

Keywords:

• Asymptotic analysis
• Markovian arrival processes
• Moments
• Rational arrival processes

AMS Subject Classification:

• Primary 60K15
• Secondary 60G55

ACKNOWLEDGMENTS

This research was supported by the SLIP research school under the Danish Network for Fisheries and Aquaculture Research (www.fishnet.dk). Bo Friis Nielsen also wants to acknowledge the support by the Technical University of Denmark for providing the opportunity of a sabbatical leave, the Technical Research Council of Denmark for the support under grant no. 26-02-0155, and the Department of Statistics University of California at Berkeley for the hospitality of hosting him during his sabbatical leave. The authors thank Kasper Kristensen for helpful discussions.