Abstract
We study the occurrence of large queue lengths in the GI / GI / d queue with heavy-tailed Weibull-type service times. Our analysis hinges on a recently developed sample path large-deviations principle for Lévy processes and random walks, following a continuous mapping approach. Also, we identify and solve a key variational problem which provides physical insight into the way a large queue length occurs. In contrast to the regularly varying case, we observe several subtle features such as a non-trivial trade-off between the number of big jobs and their sizes and a surprising asymmetric structure in asymptotic job sizes leading to congestion.
Highlights
The queue with multiple servers, known as the G I /G I /d queue, is a fundamental model in queueing theory
This has led to lines of research that focus on approximations, either considering heavily loaded systems [3,4] or investigating the frequency of rare events, for example the probability of a long waiting time or large queue length
The literature on this topic is extensive, as there is an explicit connection between waiting times and first passage times of random walks; a textbook treatment can be found in [7]
Summary
The queue with multiple servers, known as the G I /G I /d queue, is a fundamental model in queueing theory. In the 2009 Erlang centennial conference, Sergey Foss posed the question “how many big service times are needed to cause a large waiting time to occur, if the system is in steady state?” He noted that even a physical or heuristic treatment has been absent. We do not make these claims rigorous for γ = ∞ (which requires an interchange of limits argument beyond the scope of the paper), it makes a clear suggestion of what the tail behavior of the steady-state queue length should be This can be related to the steady-state waiting-time distribution, and the original question posed by Foss, using distributional Little’s law.
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