HEAVY-TAILED DEPENDENT QUEUES IN HEAVY TRAFFIC
W³adys³aw Szczotka
Wojbor A. Woyczyński
Abstract: The paper studies G/G/1 queues with heavy-tailed probability distributions of the
service times and/or the interarrival times. It relies on the fact that the heavy traffic limiting
distribution of the normalized stationary waiting times for such queues is equal to the
distribution of the supremum where is a Lévy process.
This distribution turns out to be exponential if the tail of the distribution of interarrival times
is heavier than that of the service times, and it has a more complicated non-exponential shape
in the opposite case; if the service times have heavy-tailed distribution in the domain of
attraction of a one-sided -stable distribution, then the limit distribution is Mittag-Leffler’s.
In the case of a symmetric -stable process the Laplace transform of the distribution of
the supremum is also given. Taking into account the known relationship between the
heavy-traffic-regime distribution of queue length and its waiting time, asymptotic results for
the former are also provided. Statistical dependence between the sequence of service
times and the sequence of interarrival times, as well as between random variables
within each of these two sequences, is allowed. Several examples are provided.
2000 AMS Mathematics Subject Classification: 60K25, 60F17, 60G10, 60E07.
Key words and phrases: Lévy process, stable distribution, stationary process, heavy
traffic, queueing systems, stationary waiting time, stationary queue length, weak
convergence.