Distributions Arising From Birth and Death Processes at Equilibrium and Their Extensions

Abstract

The goal of this project, is to demonstrate the use of special functions; in this case - hypergeometric function in statistics. We start by deriving the basic di erence di erential equations for birth and death processes at equilibrium and solving it iteratively using di erent values of ln and mn. The solution of the basic di erence - di erential equations are applied and hence obtain distributions to the: (i) Growth models; (ii) Waiting time problems and (iii) Queuing processes as special cases of Birth and Death processes at equilibrium. The basic di erence di erential equations are also expressed as ratio of polynomials and the equations are solved to obtain probability distributions in terms probability generating function technique and hypergeometric functions. Birth and death processes at equilibrium and their extensions based on recursive models of Pn+1 as a function of Pn and Pnð€€€1; Katz, Crow-Bardwell, Panjer’s and Kemp’s families of recursive models as a ratio of polynomials Pn+1 Pn = Q(n) R(n) ; Kapur’s recursive model as a ratio of polynomials Pi Pið€€€1 . Note that, Kapur (1978a) generalized birth and death processes are expressed in terms of generalized hypergeometric functions at equilibrium as ratio of polynomials given; Pi Pið€€€1 = lið€€€1 mi ; mi 6= 0 where various cases of li and mi are solved. A con uent hypergeometric series distribution is constructed using Kummer’s series is used as a tool to construct hypergeometric function from a ratio of polynomials. Some special cases and properties of the distributions arising from these processes are discussed.