Maximum-entropy distributions of nonnegative random variables with conditional-additive structure

Abstract

The principle of maximum entropy (ME) is used to obtain distributions of a nonnegative random variable X with conditional-additive structure: for each value of a finite control factor ηX is a sum of components whose conditional expectations (given the value of η) are known. We show that the ME distribution of X is a convex combination of convolutions of geometric sequences or exponential distributions for integer or continuous X, respectively, and find the distribution of η. In the integer case, we apply the method of “matrix unfolding” to the probability generating function of X and show that the probabilities can be blocked in rows that form a general matrix-geometric sequence. Studying an M/G/ l queueing system where the service time has conditional-additive structure, we show that the stationary queue-length probabilities can be blocked in a general matrix-geometric sequence of rows and evaluated by simple real-arithmetic matrix computations