Mean Waiting Time in Large-Scale and Critically Loaded Power of d Load Balancing Systems

Abstract

Mean field models are a popular tool used to analyse load balancing policies. In some exceptional cases the waiting time distribution of the mean field limit has an explicit form. In other cases it can be computed as the solution of a differential equation or a recursive relation. In our paper [6] we study the limit of the mean waiting time E[W&955;] as the arrival rate &955; approaches 1 for a number of load balancing policies when job sizes are exponential with mean 1 (i.e. when the system gets close to instability). As E[W_&955;] diverges to infinity, we scale with -log(1-&955;) and present a method to compute the limit lim&955;-> 1- -E[W&955;]/log(1-&955;). We show that this limit has a surprisingly simple form for the load balancing algorithms considered. In addition, we propose an alternate scaling -log(p&955;) instead of -log(1-&955;), where p&955; is adapted to the policy at hand. This scaling allows us to obtain good approximations of the mean waiting time when &955; is close to one or zero.