Article ID: | iaor2017475 |
Volume: | 63 |
Issue: | 7 |
Start Page Number: | 549 |
End Page Number: | 561 |
Publication Date: | Oct 2016 |
Journal: | Naval Research Logistics (NRL) |
Authors: | Kim Sunkyo |
Keywords: | queues: applications, markov processes, matrices, networks, scheduling, simulation |
A Markovian arrival process of order n, MAP(n), is typically described by two n × n transition rate matrices in terms of 2 n 2 − n rate parameters. While it is straightforward and intuitive, the Markovian representation is redundant since the minimal number of parameters is n2 for non‐redundant MAP(n). It is well known that the redundancy complicates exact moment fittings. In this article, we present a minimal and unique Laplace‐Stieltjes transform (LST) representations for MAP(n)s. Even though the LST coefficients vector itself is not a minimal representation, we show that the joint LST of stationary intervals can be represented with the minimum number of parameters. We also propose another minimal representation for MAP(3)s based on coefficients of the characteristic polynomial equations of the two transition rate matrices. An exact moment fitting procedure is presented for MAP(3)s based on two proposed minimal representations. We also discuss how MAP(3)/G/1 departure process can be approximated as a MAP(3). A simple tandem queueing network example is presented to show that the MAP(3) performs better than the MAP(2) in queueing approximations especially under moderate traffic intensities.