Transient asymptotics of Lévy-driven queues

With (Qt)t denoting the stationary workload process in a queue fed by a L´evy 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, q, and a positive determinstic 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 B large, where Q denotes the steady-state workload. These conditions essentially reduce to TB being sublinear (i.e., TB/B ! 0 as B ! 1) - 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 B large. For various models considered in the literature this ‘decoupling condition’ reduces to requiring that TB is superlinear (i.e., TB/B ! 1as B ! 1). Notable exceptions are two ‘heavy-tailed’ cases, viz. the situations in which the L´evy input process corresponds to an -stable process, or to a compound Poisson process with regularly varying job sizes, in which the ‘decoupling condition’ reduces to TB/B2 ! 1. 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 have appealing interpretations in terms of most likely paths to overflow

Some Asymptotic Results for the Transient Distribution of the Halfin-Whitt Diffusion Process

We consider the Halfin-Whitt diffusion process $X_d(t)$, which is used, for example, as an approximation to the $m$-server $M/M/m$ queue. We use recently obtained integral representations for the transient density $p(x,t)$ of this diffusion process, and obtain various asymptotic results for the density. The asymptotic limit assumes that a drift parameter $\beta$ in the model is large, and the state variable $x$ and the initial condition $x_0$ (with $X_d(0)=x_0>0$) are also large. We obtain some alternate representations for the density, which involve sums and/or contour integrals, and expand these using a combination of the saddle point method, Laplace method and singularity analysis. The results give some insight into how steady state is achieved, and how if $x_0>0$ the probability mass migrates from $X_d(t)>0$ to the range $X_d(t)<0$, which is where it concentrates as $t\to\infty$, in the limit we consider. We also discuss an alternate approach to the asymptotics, based on geometrical optics and singular perturbation techniques.Comment: 43 pages and 8 figure