Transient Asymptotics of Lévy-Driven Queues

Abstract

With (Qt)t denoting the stationary workload process in a queue fed by a Lévy input process (Xt)t, this paper focuses on the asymptotics of rare event probabilities of the type P(Q0 > pB, QTB > qB) for given positive numbers p and q, and a positive deterministic function TB. We first identify conditions under which the probability of interest is dominated by the ‘most demanding event’, in the sense that it is asymptotically equivalent to P(Q > max{p, q}B) for large B, where Q denotes the steady-state workload. These conditions essentially reduce to TB being sublinear (i.e. TB/B → 0 as B → ∞). A second condition is derived under which the probability of interest essentially ‘decouples’, in that it is asymptotically equivalent to P(Q > pB)P(Q > qB) for large B. For various models considered in the literature, this ‘decoupling condition’ reduces to requiring that TB is superlinear (i.e. TB / B → ∞ as B → ∞). This is not true for certain ‘heavy-tailed’ cases, for instance, the situations in which the Lévy input process corresponds to an α-stable process, or to a compound Poisson process with regularly varying job sizes, in which the ‘decoupling condition’ reduces to TB / B2 → ∞. For these input processes, we also establish the asymptotics of the probability under consideration for TB increasing superlinearly but subquadratically. We pay special attention to the case TB = RB for some R > 0; for light-tailed input, we derive intuitively appealing asymptotics, intensively relying on sample path large deviations results. The regimes obtained can be interpreted in terms of the most likely paths to overflow.