Ordinary CLT and WLLN Versions of L = λW

P. W. Glynn and W. Whitt

Mathematics of Operations Research, Vol.13,  674-692 (1988)

The familiar queueing principle expressed by the formula L = λW (Little's law) can be interpreted as a relation among strong laws of large numbers (SLLNs). Here we prove central-limit-theorem (CLT) and weak-law-of-large-numbers (WLLN) versions of L = λW. For example, if the sequence of ordered pairs of interarrival times and waiting times is strictly stationary and satisfies a joint CLT, then the queue-length process also obeys a CLT with a related limiting distribution. In a previous paper we proved a functional-central-limit-theorem version of L = λW, without stationarity, by very different arguments. The two papers highlight the differences between establishing ordinary limit theorems and their functional-limit-theorem counterparts.