A queueing theoretical proof of increasing property of Polya frequency functions
Let \(f_1, \dots, f_n\) be the Polya frequency counting density functions of order 2 \((\text{PF}_2\) functions) with supports \(E_j = [a_j, b_j]\), \(0 \leq a_j \leq b_j < \infty\), or \(E_j = [a_j, \infty)\), \(j = 1, \dots, n\), and \(X_1, \dots, X_n\) be independent integer-valued r.v.'s having these densities. Let \(S = X_1 + \cdots + X_n\) and \(\varphi\) be an increasing real-valued function defined on \(N^n\). The basic result is the queueing theoretical proof of the fact that \(E (\varphi (X_1, \dots, X_n) \mid S = s)\) is increasing in almost all \(s \in [\sum^n_1 a_j, \sum^n_1 b_j + 1)\). This fact (under a more general setup) has been proven by \textit{B. Efron} [Ann. Math. Stat. 36, 272-279 (1965; Zbl 0134.36704)]. Some applications to queueing networks are given.
Polya frequency counting density functions
scientific article; zbMATH DE number 885158
scientific article