Shot-noise queueing models

Abstract

We provide a survey of so-called shot-noise queues: queueing models with the special feature that the server speed is proportional to the amount of work it faces. Several results are derived for the workload in an M/G/1 shot-noise queue and some of its variants. Furthermore, we give some attention to queues with general workload-dependent service speed. We also discuss linear stochastic fluid networks, and queues in which the input process is a shot-noise process.

Optimal dividend strategies for a catastrophe insurer

“…In recent years, such models have been studied for various purposes, see e.g. Dassios & Zhao [15], Macci & Torrisi [22], Jang & Oh [20] and Pojer & Thonhauser [24,25] in an insurance context, Boxma & Mandjes [13] for a related model in queueing and Schmidt [28] for applications in finance.…”

“…By definition, u m (x, λ) is a weighted average of values of u in A m = [x, x + 1/m] × [λ, λ + 1/m]; (a), (b) and (d) follow by standard techniques, because u m ≥ u and u is absolutely continuous in (0, ∞) × (λ, ∞) and satisfies the growth condition (13) (see for instance [30]). From Equation (11) we have that u x ≥ 1 a.e., also since for all (x, λ) ∈ D, L(u) (x, λ) ≤ 0 we have…”

“…Proposition 4.2. The optimal value function V is the smallest viscosity supersolution of (11) among those that are non-increasing in λ and satisfy the growth condition (13).…”

“…Corollary 6.7. The G δ -optimal value function V δ : G δ ×[λ, ∞) → R can be characterized as the smallest supersolution of the discrete HJB equation (23) with growth condition (13). Definition 6.1.…”

Functional equations with multiple recursive terms

“…2. The model under consideration is the D M /G/1 shot-noise queue; we refer to [7] for a recent survey on shot-noise queueing models. The D M /G/1 shot-noise queue is a single server queue, in which the successive interarrival times A 1 , A 2 , .…”