Power and Effectiveness in Queueing Systems

Abstract

In a series of papers, Kleinrock proposed a performance metric called power for stable queueing systems in equilibrium, which captures the tradeoff every queue makes between work done and time needed to. In particular, he proved that in the M / GI / 1 family of models, the maximal power is obtained when the mean number of customers in the system is exactly one (a keep the pipe empty property). Since then, this metric has been used in different works, in the analysis of communication systems. In this paper we add some results about power, showing in particular that the concept extends naturally to Jackson product form queueing networks, and that, again, the keep the pipe empty property holds. We also show that this property does not hold in general for single-node models of the GI / GI / 1 type, or when the storage capacity of the system is finite. The power metric takes into account the cost in time to provide service, but not the cost associated with the server itself. For this purpose, we define a different metric, which we call effectiveness, whose aim is to also include this aspect of systems. Both metrics coincide on single server models, but differ on multi-server ones, and on networks. We provide arguments in support of this new metric and, in particular, we show that in the case of a Jackson network, the keep the pipe empty property again holds.