Gamma variate ratio distribution with application to CDMA performance analysis

An expression for the probability density function (pdf), f_R (r), is derived for the ratio, R, of gamma variates of the form R = X₁/(a₁X₁ + a₂X₂), where a₁ and a₂ are constants and X₁ and X₂ are independent gamma distributed random variables. This distribution arises naturally in the performance analyses of wireless communication systems experiencing Nakagami fading. It is shown that f_R(r) reduces to the standard beta distribution and the beta prime distribution in special cases. Expressions for the moments, the moment generating function (MGF) and the cumulative distribution function (cdf) of R are obtained in terms of the hypergeometric function. Finally, the use of f_R(r) is illustrated by considering the problem of deriving outage probabilities and throughput bounds for a CDMA system employing adaptive modulation and coding (AMC)