Inequalities for workload process in queues with NBU/NWU input

Abstract

We give a simple proof of the well-known property PASTA for the workload and queue size process in the queueing systems with Poisson input. The proof is based on a relation connecting the workload process at an arbitrary instants and the arrival instants of the customers and, in particular, yields famous Pollaczeck-Khintchine equality. It is then shown that this equality transforms to the corresponding inequality when the interarrival time has New-Better-than-Used (NBU) or New-Worse-than-Used (NWU) distribution. Then the inequalities for the stationary workload are extended to classical multiserver system and multiserver model with simultaneous service, describing the modern high performance cluster. The analysis is illustrated by simulation of the single-server and cluster models with Weibull interarrival time, covering NBU, NWU and Poisson inputs. The obtained results are of practical interest for Quality-of-Service estimation of modern high-performance computing systems.