Queueing systems fed by many exponential on-off sources: an infinite-intersection approach

摘要

In queueing theory, an important class of events can be written as 'infinite intersections'. For instance, in a queue with constant service rate c, busy periods starting at 0 and exceeding L > 0 are determined by the intersection of the events ∩{sub}(t∈0,L{Q{sub}0 = 0, A{sub}t > ct}, i.e., queue Q{sub}t is empty at 0 and for all t ∈ 0, L the amount of traffic A, arriving in (0, t) exceeds the server capacity. Also the event of exceeding some predefined threshold in a tandem queue, or a priority queue, can be written in terms of this kind of infinite intersections. This paper studies the probability of such infinite intersections in queueing systems fed by a large number n of i.i.d. traffic sources (the so-called 'many-sources regime'). If the sources are of the exponential on-off type, and the queueing resources are scaled proportional to n, the probabilities under consideration decay exponentially; we explicitly characterize the corresponding decay rate. The techniques used stem from large deviations theory (particularly sample-path large deviations).