The Single Server Queue: Heavy Tails and Heavy Traffic

Abstract

Recently, there has been much interest in the behavior of queues with heavy-tailed service time distributions. This interest has been triggered by a large number of traffic measurements on modern communication network traffic. These measurements and their statistical analysis suggest that modern communication traffic often possesses the properties of self-similarity and long-range dependence. The first issue that we consider in this chapter is whether other service disciplines besides first come-first served (FCFS) may lead to a less detrimental waiting time behavior. The second issue under consideration in this chapter is the heavy traffic behavior of a queue with heavy-tailed service time distribution. The third issue under consideration is the convergence of the workload process in the GI / G / 1 queue with heavy-tailed interarrival and/or service time distribution. This chapter is organized in the following way. We discuss the relation between the tail behavior of service times and waiting times, for the M / G / 1 queue with service disciplines FCFS, PS, and LCFS-PR. For the same three M / G / 1 variants, we present heavy traffic limit theorems for waiting times, in the case of a regularly varying service time distribution with an infinite variance. In the LCFS-PR case, the sojourn time distribution coincides with the M / G / 1 busy period distribution. The heavy traffic behavior of the latter distribution is investigated in detail; the tail behavior of the limiting distribution is studied next. Heavy traffic limit theorems for the waiting time in the GI / G / 1 queue with the FCFS service discipline are presented. The input process and the workload process are studied for the GI / G / 1 queue with heavy-tailed interarrival and/or service time distribution. The heavy traffic behavior of those processes is characterized. Conclusions and some topics for further research are presented.

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