MONOTONICITY RESULTS FOR QUEUES WITH DOUBLY STOCHASTIC POISSON ARRIVALS - ROSS CONJECTURE

In this paper, we compare queueing systems that differ only in their arrival processes, which are special forms of doubly stochastic Poisson (DSP) processes. We define a special form of stochastic dominance for DSP processes which is based on the well-known variability or convex ordering for random variables. For two DSP processes that satisfy our comparability condition in such a way that the first process is more 'regular' than the second process, we show the following three results: (i) If the two systems are DSP/Gl/1 queues, then Ef(V(1)) less-than-or-equal-to Ef(V(2)) for all f increasing convex, with V(i), i = 1 and 2, representing the workload (virtual waiting time) in system. (ii) If the two systems are DSP/M(k)/1 --> /M(k)/1 -->...-->/M(k)/1 tandem systems, with M(k) representing an exponential service time distribution with a rate that is increasing concave in the number of customers, k, present at the station, then Ef(Q(1)) less-than-or-equal-to Ef(Q(2)) for all f increasing convex, with Q(i), i = 1 and 2, being the total number of customer in the two systems. (iii) If the two systems are DSP/M(k)/1/N systems, with N being the size of the buffer, then P(B)(1) less-than-or-equal-to P(B)(2), where P(B)(i) denotes the blocking (loss) probability of the two systems. A model considered before by Ross (1978) satisfies our comparability condition; a conjecture stated by him is shown to be true.