2008-12-01

# Transient asymptotics of Lévy-driven queues

## Publication

### Publication

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.

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CWI | |

CWI. Probability, Networks and Algorithms [PNA] | |

Organisation | Stochastics |

Dȩbicki, K., Es-Saghouani, A., & Mandjes, M. (2008). Transient asymptotics of Lévy-driven queues. CWI. Probability, Networks and Algorithms [PNA]. CWI. |