3,099 research outputs found
Waiting times in queueing networks with a single shared server
We study a queueing network with a single shared server that serves the
queues in a cyclic order. External customers arrive at the queues according to
independent Poisson processes. After completing service, a customer either
leaves the system or is routed to another queue. This model is very generic and
finds many applications in computer systems, communication networks,
manufacturing systems, and robotics. Special cases of the introduced network
include well-known polling models, tandem queues, systems with a waiting room,
multi-stage models with parallel queues, and many others. A complicating factor
of this model is that the internally rerouted customers do not arrive at the
various queues according to a Poisson process, causing standard techniques to
find waiting-time distributions to fail. In this paper we develop a new method
to obtain exact expressions for the Laplace-Stieltjes transforms of the
steady-state waiting-time distributions. This method can be applied to a wide
variety of models which lacked an analysis of the waiting-time distribution
until now
A Tandem Fluid Network with L\'evy Input in Heavy Traffic
In this paper we study the stationary workload distribution of a fluid tandem
queue in heavy traffic. We consider different types of L\'evy input, covering
compound Poisson, -stable L\'evy motion (with ), and
Brownian motion. In our analysis we separately deal with L\'evy input processes
with increments that have finite and infinite variance. A distinguishing
feature of this paper is that we do not only consider the usual heavy-traffic
regime, in which the load at one of the nodes goes to unity, but also a regime
in which we simultaneously let the load of both servers tend to one, which, as
it turns out, leads to entirely different heavy-traffic asymptotics. Numerical
experiments indicate that under specific conditions the resulting simultaneous
heavy-traffic approximation significantly outperforms the usual heavy-traffic
approximation
Linear Stochastic Fluid Networks: Rare-Event Simulation and Markov Modulation
We consider a linear stochastic fluid network under Markov modulation, with a
focus on the probability that the joint storage level attains a value in a rare
set at a given point in time. The main objective is to develop efficient
importance sampling algorithms with provable performance guarantees. For linear
stochastic fluid networks without modulation, we prove that the number of runs
needed (so as to obtain an estimate with a given precision) increases
polynomially (whereas the probability under consideration decays essentially
exponentially); for networks operating in the slow modulation regime, our
algorithm is asymptotically efficient. Our techniques are in the tradition of
the rare-event simulation procedures that were developed for the sample-mean of
i.i.d. one-dimensional light-tailed random variables, and intensively use the
idea of exponential twisting. In passing, we also point out how to set up a
recursion to evaluate the (transient and stationary) moments of the joint
storage level in Markov-modulated linear stochastic fluid networks
Random matrices, non-colliding processes and queues
This is survey of some recent results connecting random matrices,
non-colliding processes and queues.Comment: To appear in Seminaire de Probabilites XXXV
Networks of Server Queues with Shot-Noise-Driven Arrival Intensities
We study infinite-server queues in which the arrival process is a Cox process
(or doubly stochastic Poisson process), of which the arrival rate is given by
shot noise. A shot-noise rate emerges as a natural model, if the arrival rate
tends to display sudden increases (or: shots) at random epochs, after which the
rate is inclined to revert to lower values. Exponential decay of the shot noise
is assumed, so that the queueing systems are amenable for analysis. In
particular, we perform transient analysis on the number of customers in the
queue jointly with the value of the driving shot-noise process. Additionally,
we derive heavy-traffic asymptotics for the number of customers in the system
by using a linear scaling of the shot intensity. First we focus on a one
dimensional setting in which there is a single infinite-server queue, which we
then extend to a network setting
Separation of timescales in a two-layered network
We investigate a computer network consisting of two layers occurring in, for
example, application servers. The first layer incorporates the arrival of jobs
at a network of multi-server nodes, which we model as a many-server Jackson
network. At the second layer, active servers at these nodes act now as
customers who are served by a common CPU. Our main result shows a separation of
time scales in heavy traffic: the main source of randomness occurs at the
(aggregate) CPU layer; the interactions between different types of nodes at the
other layer is shown to converge to a fixed point at a faster time scale; this
also yields a state-space collapse property. Apart from these fundamental
insights, we also obtain an explicit approximation for the joint law of the
number of jobs in the system, which is provably accurate for heavily loaded
systems and performs numerically well for moderately loaded systems. The
obtained results for the model under consideration can be applied to
thread-pool dimensioning in application servers, while the technique seems
applicable to other layered systems too.Comment: 8 pages, 2 figures, 1 table, ITC 24 (2012
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