Transient Asymptotics of Lévy-Driven Queues

Abstract

With (Q t ) t denoting the stationary workload process in a queue fed by a Lévy input process (X t ) t , this paper focuses on the asymptotics of rare event probabilities of the type P(Q 0 > pB, Q T B > qB) for given positive numbers p and q, and a positive deterministic function T B . 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 T B being sublinear (i.e. T B /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 T B is superlinear (i.e. T B / 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 T B / B 2 → ∞. For these input processes, we also establish the asymptotics of the probability under consideration for T B increasing superlinearly but subquadratically. We pay special attention to the case T B = 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.