Filtering and the EM-algorithm for the Markovian arrival process

In this paper, we deal with the so-called Markovian Arrival process (MAP). An MAP is thought of as a partially observed Markov process, so that the ExpectationMaximization (EM) algorithm is a natural way to estimate its parameters. Then, non-linear filters of basic statistics related to the MAP must be computed. The forward-backward principle is the basic way to do it. Here, bearing in mind a filterbased formulation of the EM-algorithm proposed by Elliott, these filters are shown to be the solution of non-linear stochastic differential equations (SDEs) which allows a recursive computation. This is well suited for processing large data sets. We also derive linear SDEs or Zakai equations for the so-called unnormalized filters. Key-Words Hidden Markov process, Point process, Innovations method, Zakai’s filter, Queuing theory, Software reliability