A Heavy Traffic Limit Theorem for Workload Processes with Heavy Tailed Service Requirements

A system with heavy tailed service requirements under heavy load having a single server, has an equilibrium waiting time distribution which is approximated by the Mittag{Leeer distribution. This fact is understood by a direct analysis of the weak convergence of a sequence of negative drift random walks with heavy right tail and the associated all time maxima of these random walks. This approach complements the recent transform view of Boxma and Cohen 4]. 1. Introduction. Heavy traac limit theorems were devised to study the behavior of complex networks. For economic reasons, these systems are typically heavily loaded. If one wants to estimate the performance of such a system, a direct simulation may not be eecient, because the system parameters are near the boundary of the parameter set which makes the system stable. This means long excursions are likely. A heavy traac limit theorem studies a sequence of systems under normalization when system parameters approach the stability boundary. Original work assumed the component random variables of the model all had nite variance. who with coworkers started the subject of diiusion process approximations (see 9]), which is still a subject of active research. This classical work on heavy traac approximations has little relevance to recent work in communication networks, which explains self similarity of network traac by means of on/oo models having innnite variance and heavy tailed transmission time distributions. A recent stimulating paper by Boxma and Cohen 4] studies the stationary waiting time for the GI/G/1 queue under the assumptions that the system is under heavy traac and the service distribution has innnite variance, while the interarrival distribution tail is of smaller order than the service time distribution tail. Assumptions and methodology use Laplace transform techniques. We demonstrate that one can analyze such systems without transforms and this direct attack illuminates and simpliies the various assumptions that must be made to achieve the heavy traac limit theorem. The setup is as follows. We assume we have a sequence of Lindley queues (see 1, 27]) indexed by k and the delay process of the kth Lindley queue is given by W (k) 0 = 0; W (k) n+1 = (W (k) n + (k)