Queueing Systems: Theory and Applications
Proceedings of the 3rd International Conference on Performance Evaluation Methodologies and Tools
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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 $$\bigcap_{t\in[0,L]}\{Q_0=0,\;A_t ct\}$$ , i.e., queue Q t is empty at 0 and for all t驴 [0, L] the amount of traffic A t 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).