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Matrix-analytic models and their analysis. (English) Zbl 0959.60085
A phase-type distribution (PHT) is a distribution of the lifetime of a terminating Markov process with finite set of states and time homogeneous transition rates. Probability characteristics of such distributions are described in terms of $$\exp(Tx)$$, where $$T$$ is the matrix of transition rates. Using the Markov representation of PHT the author evaluates renewal densities, ladder height distributions and some queueing theory characteristics for PHT-distributions in terms of integrals of $$\exp(Tx)$$ by $$x$$. This theory is connected with Markovian arrival processes, which are, roughly speaking, Poisson processes with intensity modulated by a hidden Markov chain (with possible additional arrivals at the moments of jumps of this chain). It is shown that the set of PHT is dense in the class of all distributions on $$(0,+\infty)$$. The problem of statistical fitting of data distribution by appropriate PHT is considered. An EM-type algorithm is proposed to estimate the best matrix $$T$$. Simulation results are presented. The conclusion is that well-behaved distributions are easily fitted by PHT of Markov process with 3-6 states. But distributions with sleepy density increases or decreases need much more states.

##### MSC:
 60K25 Queueing theory (aspects of probability theory) 62G05 Nonparametric estimation 60E05 Probability distributions: general theory
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