arXiv subjects

Lukasz Kruk

Publications and source records attributed to Lukasz Kruk.

3 recordsLinked to original sources

arXiv2019

Continuity and monotonicity of solutions to a greedy maximization problem

Motivated by an application to resource sharing network modelling, we consider a problem of greedy maximization (i.e., maximization of the consecutive minima) of a vector in $R^n$, with the admissible set indexed by the time parameter. The structure of the constraints depends on the underlying network topology. We investigate continuity and monotonicity of the resulting maximizers with respect to time. Our results have important consequences for fluid models of the corresponding networks which are optimal, in the appropriate sense, with respect to handling real-time transmission requests.

math.OC↗

arXiv2007

An explicit formula for the Skorokhod map on $[0,a]$

The Skorokhod map is a convenient tool for constructing solutions to stochastic differential equations with reflecting boundary conditions. In this work, an explicit formula for the Skorokhod map $Γ_{0,a}$ on $[0,a]$ for any $a>0$ is derived. Specifically, it is shown that on the space $\mathcal{D}[0,\infty)$ of right-continuous functions with left limits taking values in $\mathbb{R}$, $Γ_{0,a}=Λ_a\circ Γ_0$, where $Λ_a:\mathcal{D}[0,\infty)\to\mathcal{D}[0,\infty)$ is defined by \[Λ_a(ϕ)(t)=ϕ(t)-\sup_{s\in[0,t]}\biggl[\bigl(\ phi(s)-a\bigr)^+\wedge\inf_{u\in[s,t]}ϕ(u)\biggr]\] and $Γ_0:\mathcal{D}[0,\infty)\to\mathcal{D}[0,\infty)$ is the Skorokhod map on $[0,\infty)$, which is given explicitly by \[Γ_0(ψ)(t)=ψ(t)+\sup_{s\in[0,t]}[-ψ(s)]^+.\] In addition, properties of $Λ_a$ are developed and comparison properties of $Γ_{0,a}$ are established.

math.PR↗

arXiv2004

Earliest-deadline-first service in heavy-traffic acyclic networks

This paper presents a heavy traffic analysis of the behavior of multi-class acyclic queueing networks in which the customers have deadlines. We assume the queueing system consists of J stations, and there are K different customer classes. Customers from each class arrive to the network according to independent renewal processes. The customers from each class are assigned a random deadline drawn from a deadline distribution associated with that class and they move from station to station according to a fixed acyclic route. The customers at a given node are processed according to the earliest-deadline-first (EDF) queue discipline. At any time, the customers of each type at each node have a lead time, the time until their deadline lapses. We model these lead times as a random counting measure on the real line. Under heavy traffic conditions and suitable scaling, it is proved that the measure-valued lead-time process converges to a deterministic function of the workload process.

math.PR↗