Storage capacity of a dam with gamma type inputs

Summary

Consider mutually independent inputsX ₁,...,X _n onn different occasions into a dam or storage facility. The total input isY=X ₁+...+X _n. This sum is a basic quantity in many types of stochastic process problems. The distribution ofY and other aspects connected withY are studied by different authors when the inputs are independently and identically distributed exponential or gamma random variables. In this article explicit exact expressions for the density ofY are given whenX ₁,...,X _n are independent gamma distributed variables with different parameters. The exact density is written as a finite sum, in terms of zonal polynomials and in terms of confluent hypergeometric functions. Approximations whenn is large and asymptotic results are also given.