ABSTRACT
Shanthikumar ( 1 985) who called it the generalized phase-type (GPH). Ott ( 1987) also studied this
PH-distributions appeared in literature much earlier (Neuts, 1975). Defining a PH-distribution by
GIIMII type (Neuts, 1 98 1 ) and the MIGII type (Neuts, 1 989), in the general framework of the matrix
PH-distribution; (2) a discrete PH mixture of the successive convolutions of a continuous PH distribution is still a PH-distribution; but (3) infinite mixtures of PH-distributions are generally not
Chaudhry tackled the GJIGII model from a different angle (see Chaudhry, 1 992 and its
Shi ( 1 985, 1994) studied counting processes in continuous time Markov chains and vector Markov
processes, in particularly, the number of state transitions from one subset A to another subset B. He obtained a general formula to calculate the transition frequency. When B is an absorbing set, this formula gives the density function of the absorbing time. It was further observed that: ( 1 ) When the Markov chain is finite, the density function is PH-type; (2) When the Markov chain is countable, the
transform is irrational (p2 1 5, Kleinrock, 1976), it is clearly not a PH-distribution (see O'Cinneide,
1 990). What are the properties of the SPH class? How can these properties be used to solve queueing and other stochastic model problems systematically? These are the two issues we set out to address in
and the system time) distribution of the GI I PH 1 1 queue is SPH (theorem 5.2); (6) the busy period distribution of the PHIPHII queue is SPH (theorem 5 .3); and (7) every non-negative discrete random variable corresponds to a discrete IPH distribution, i.e. we can construct a discrete countable absorbing
Markov chain for any non-negative discrete random variable (remark 2.3).
