CLUSTER MODELING OF LINDLEY PROCESS WITH APPLICATION TOQUEUING

Consider a GI/GI/1 queue and let Wn be the waiting time of the nth customer. Here some basic characteristics of clusters of extreme values of the Lindley's recursion, which describes the evolution of Wn, are being investigated. A cluster of extreme values is a series of Wn, which are successive exceedances of a fixed threshold. Closed-form expressions for the two distributions related to the cluster size and the inter-cluster time are obtained: the distribution of the cluster size, which appears after the first customer arrived to the empty system; the distribution of the time (measured in terms of number of customers) between the cluster, formed by the first customer arrived to the empty system, and the next cluster. Some insights into the impact of the system's load on the cluster and inter-cluster distributions is being demonstrated through the M/M/1 queue.