On the Integrodifferential Equation of Takacs. I

2008

Queueing Syst

We analyze tail asymptotics of a two-node tandem queue with spectrallypositive Lévy input. A first focus lies in the tail probabilities of the type, for α ∈ (0, 1) and x large, and Q i denoting the steadystate workload in the ith queue. In case of light-tailed input, our analysis heavily uses the joint Laplace transform of the stationary buffer contents of the first and second queue; the logarithmic asymptotics can be expressed as the solution to a convex programming problem. In case of heavy-tailed input we rely on sample-path methods to derive the exact asymptotics. Then we specialize in the tail asymptotics of the downstream queue, again in case of both light-tailed and heavy-tailed Lévy inputs. It is also indicated how the results can be extended to tandem queues with more than two nodes.

“…[37], whereas the total queue, Q 1 + Q 2 is distributed as a single queue with service rate c 2 , i.e., sup σ >0 (A(−σ, 0) − c 2 σ ), cf. [3,22,33,39].…”

Section: Proposition 32mentioning

confidence: 99%

Asymptotic analysis of Lévy-driven tandem queues

Lieshout

2008

Queueing Syst

“…Now let Z = {Z t : t ∈ R + }, where Z t = (Z 1 t , Z 2 t ). Note that the rate function I Z corresponding to Z is given by I Z (x) = I c1 (x 1 )+I c2 (x 2 ), where I c1 and I c2 are given by (14). Finally, define a new process W = {W t : t ∈ R + } via (W …”

Section: 1mentioning

confidence: 99%

Logarithmic Asymptotics for Multidimensional Extremes Under Nonlinear Scalings

Kosinski

Journal of Applied Probability

2015

Abstract. Let W = {W n : n ∈ N} be a sequence of random vectors in R d , d ≥ 1. This paper considers the logarithmic asymptotics of the extremes of W , that is, for any vector q > 0 in R d , we findWe follow the approach of the restricted large deviation principle introduced in Duffy et al. [7]. That is, we assume that, for every q ≥ 0, and some scalings {an}, {vn}, 1 vn log P (W n/an ≥ uq) has a, continuous in q, limit J W (q). We allow the scalings {an} and {vn} to be regularly varying with a positive index. This approach is general enough to incorporate sequences W , such that the probability law of W n/an satisfies the large deviation principle with continuous, not necessarily convex, rate functions. The formula for these asymptotics agrees with the seminal papers on this topic [3,6,7,9].

“…Furthermore, it holds that the total service capacity (at a constant rate of 1 per unit time) is used as long as there is any work present, which entails that the system is work-conserving. According to Reich's formula (Reich 1958), the steady-state overall-workload representation therefore reads…”

Section: Preliminariesmentioning

confidence: 99%

A fluid model for a relay node in an ad hoc network: the case of heavy-tailed input

Bekker

2008

Math Meth Oper Res

Relay nodes in an ad hoc network can be modelled as fluid queues, in which the available service capacity is shared by the input and output. In this paper such a relay node is considered; jobs arrive according to a Poisson process and bring along a random amount of work. The total transmission capacity is fairly shared, meaning that, when n jobs are present, each job transmits traffic into the queue at rate 1/(n + 1) while the queue is drained at the same rate of 1/(n + 1). Where previous studies mainly concentrated on the case of exponentially distributed job sizes, the present paper addresses regularly varying jobs. The focus lies on the tail asymptotics of the sojourn time S. Using sample-path arguments, it is proven that P {S > x} behaves roughly as the residual job size, i.e., if the job sizes are regularly varying of index −ν, the tail of S is regularly varying of index 1 − ν. In addition, we address the tail Part of the research was conducted while R. Bekker was affiliated to CWI. Hans van den Berg (Telecom ICT) is acknowledged for bringing this model under our attention.