The superposition of two PH-renewal processes
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A large variety of stochastic models may be viewed as systems of two interacting renewal processes. Unless at least one of these is a Poisson process, the analytic properties of the model are difficult to study and are often intractable. It is clear that the greater analytic tractability of such models as the M/G/1 and GI/M/1 queues is due to the memory-less property of the exponential distribution. This tractability is gained, however, at the price of a severe distributional assumption. A useful compromise is found in the introduction of the probability distributions of phase type (PH-distributions). For these, a formalism of matrix manipulations has been developed which extends the elementary properties of the exponential distribution, yet is general and versatile enough to cover the probability distributions needed in a large variety of modelling problems.
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