Clumping of Losses and Long-Delays in a Queue

Clumping of Losses and Long-Delays in a Queue
Two of the most widely used performance measures for queueing systems are the fraction of lost customers (in finite-buffer queues) and the probability of experiencing a long delay (in the infinite buffer context). However, in many applications the Quality of Service (QoS) experienced by users depends strongly on the "clumpiness" of losses or long delays; for example, in video applications over the Internet, losing one isolated packet out of every 10^7 is very different from losing a group of 10^3 nearly consecutive packets out of every 10^10 served. Surprisingly, the literature on such loss-clumps is quite limited. We fill the gap in this literature by studying the distribution of the loss-clumps in finite-buffer queues and that of exceedance-clumps in infinite-buffer queues. We are particularly interested in the impact of long-range dependence and heavy tails on these distributions, because of the relevance of such traffic models to various applications settings. We provide conditional limit theorems for the distribution and dynamics of the clump-size for several stylized traffic models that offer insight into both the clump-magnitude and the associated dynamics that are most likely to lead to large clumps.
As suggested above, the clumpiness of the loss process is of fundamental interest in the context of of Internet traffic modeling. In the traffic modeling community there is a standing need for a quantifiable measure of the "burstiness" of a traffic stream. We argue that burstiness is related to the clumpiness of the loss process, and that this relation can be useful in the design of a quantifiable measure of burstiness.
Our work is partly motivated by the problem of predicting congestion in Internet routers. Next-generation switch-routers will likely have the capability to monitor their own performance, predict upcoming congestion, and warn their neighbors or end systems so that preventive actions can be taken. We discuss the problem of predicting congestion at the router level, and some of the issues that make it challenging both theoretically and in practice.