A convexity property of the Poisson distribution and its application in queueing theory (Q910801)

The authors prove that the Poisson distribution belongs to a class of distributions with the property of convex ratios (ratio convexity property), which for all \(x\geq 0\), \(n\geq 2\), means that \[ S_{n- 2}(x)/S_{n-1}(x)+S_ n(x)/S_{n+1}(x)\leq 2S_{n-1}(x)/S_ n(x), \] where \(S_ n(x)=\sum^{n}_{j=0}x^ j/j!\). Two applications of the inequality in queueing theory are discussed. The first application concerns the bound on the mean queue length in a one-line queueing system with an unbounded queue and the second is in the proof of monotonicity and convexity of the mean number of customers in the stages of the closed queueing system.