The Marginal Distributions of a Crossing Time and Renewal Numbers Related with Two Poisson Processes are as Ph-Distributions

In this paper we consider, how to find the marginal distributions of crossing time and renewal numbers related with two poisson processes by using probability arguments. The obtained results show that the one-dimension marginal distributions are N+1 order PH-distributions.

Obviously, this is a negative binomial distribution.Thus the probability generating function is given by and Newton's binomial formula see Ref [1,9] the mean and variance of the distribution is a rational function of s, it may be written following form See Ref. [2,10] where Taking the inverse of the L-transform one can get the density function The mean and variance of the distribution are It is a geometric distribution.Therefore the probability generating function of the distribution is

One Dimension Marginal Distributions as Ph-Distributions.
A particular instance of the method of supplementary variables is known as the method of phases and involves ideas of remarkable simplicity which were first proposed by A. K. Erlang.He observed that gamma distributions, whose shape parameter is a positive integer, may be considered as the probability distributions of sums of independent, negative exponential random variables.In this manner a number of highly useful results for renewal processes of Erlang type can be derived from those of the much simpler poisson process.
The basic idea of Erlang, which ultimately rests on the memoryless property of the negative exponential distribution, has been applied and extended by many authors.
Most useful elementary applications are discussed in the monograph by D. R. Cox [4], no attempt will be made to survey the uses of the method of phases in the existing literature, but we must draw attention to the paper by D. R. Cox [3] which introduces interesting Notion of complex-valued probabilities in an attempt to find phase representations for all probability distributions on the positive real line which have rational Laplace-stieltjes transforms.Many open questions raised there have essentially remained un answered up to this time, and other related to the numerical use and fitting of such distributions are deserving of much further investigation.
The analytical and computational simplifications resulting from the method of phases are clear.They permeate the discussions of a large number of paper in the theory of queues and have recently been exploited in the construction of algorithms for certain single server queues [8].One may anticipate that the method of phases will permit the algorithmic solution of a growing number of system of queues, as well as an exact numerical investigation of the Lesstractable priority queues.Instances of such results may be found in Ref. [6,7].Theorem 3.1 N T ξ is a (N+1) order continuous PH-distribution with represent (α ,T) where α = 0), , 0, 0, (1, Proof The definition and property of PH-distribution see ref [5].For continuous PH-distribution its L-transform is  Proof The conclusion may be similarly verified as theorem 3.2.Because negative binomial distribution is the N-fold convolution of geometric distribution it is also may be obtained from the convolution property of PH-distribution. Assume that i X , j Y are mutually independent.In the case of F(t) and G(t) are exponentially distributed with parameter λ and µ that the obtained results are (N+1) order PH-distributions2.Probability ArgumentsIn this section, first of all one can find out the marginal distribution of random variables