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Applied Math Seminar
Rare Events and Tail Behavior for Queues
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We will discuss two different sets of results for the single-server queue. Our first set of results is intended to add insight into the conditional behavior of such queues at epochs when customers experience unusually long waiting times (in the context of an infinite buffer system) or are blocked (in the context of a finite buffer system). We seek to understand issues related to the conditional dynamics of the system that lead to such events, as well as the distribution of the corresponding "clump size" (e.g. Do losses typically occur as large clumps infrequently or as small clumps frequently?) Our second set of results relates to the transition from "heavy traffic" to "heavy tails" that arises in a heavily loaded queue with heavy-tailed processing times. It is known, on the one hand, that heavily loaded queues have an exponential steady-state waiting time distribution in diffusion scale, whereas customers in queues with heavy-tailed processing times experience waiting times that are heavy-tailed. We provide a precise mathematical description of the transition from "heavy traffic" to "heavy tails". |