2,840 research outputs found
Fixed points for multi-class queues
Burke's theorem can be seen as a fixed-point result for an exponential
single-server queue; when the arrival process is Poisson, the departure process
has the same distribution as the arrival process. We consider extensions of
this result to multi-type queues, in which different types of customer have
different levels of priority. We work with a model of a queueing server which
includes discrete-time and continuous-time M/M/1 queues as well as queues with
exponential or geometric service batches occurring in discrete time or at
points of a Poisson process. The fixed-point results are proved using
interchangeability properties for queues in tandem, which have previously been
established for one-type M/M/1 systems. Some of the fixed-point results have
previously been derived as a consequence of the construction of stationary
distributions for multi-type interacting particle systems, and we explain the
links between the two frameworks. The fixed points have interesting
"clustering" properties for lower-priority customers. An extreme case is an
example of a Brownian queue, in which lower-priority work only occurs at a set
of times of measure 0 (and corresponds to a local time process for the
queue-length process of higher priority work).Comment: 25 page
Lattice path counting and the theory of queues
In this paper we will show how recent advances in the combinatorics of lattice paths can be applied to solve interesting and nontrivial problems in the theory of queues. The problems we discuss range from classical ones like M^a/M^b/1 systems to open tandem systems with and without global blocking and to queueing models that are related to random walks in a quarter plane like the Flatto-Hahn model or systems with preemptive priorities. (author´s abstract)Series: Research Report Series / Department of Statistics and Mathematic
Stability conditions for a decentralised medium access algorithm: single- and multi-hop networks
We consider a decentralised multi-access algorithm, motivated primarily by
the control of transmissions in a wireless network. For a finite single-hop
network with arbitrary interference constraints we prove stochastic stability
under the natural conditions. For infinite and finite single-hop networks, we
obtain broad rate-stability conditions. We also consider symmetric finite
multi-hop networks and show that the natural condition is sufficient for
stochastic stability
Modeling bursts and heavy tails in human dynamics
Current models of human dynamics, used from risk assessment to
communications, assume that human actions are randomly distributed in time and
thus well approximated by Poisson processes. We provide direct evidence that
for five human activity patterns the timing of individual human actions follow
non-Poisson statistics, characterized by bursts of rapidly occurring events
separated by long periods of inactivity. We show that the bursty nature of
human behavior is a consequence of a decision based queuing process: when
individuals execute tasks based on some perceived priority, the timing of the
tasks will be heavy tailed, most tasks being rapidly executed, while a few
experiencing very long waiting times. We discuss two queueing models that
capture human activity. The first model assumes that there are no limitations
on the number of tasks an individual can hadle at any time, predicting that the
waiting time of the individual tasks follow a heavy tailed distribution with
exponent alpha=3/2. The second model imposes limitations on the queue length,
resulting in alpha=1. We provide empirical evidence supporting the relevance of
these two models to human activity patterns. Finally, we discuss possible
extension of the proposed queueing models and outline some future challenges in
exploring the statistical mechanisms of human dynamics.Comment: RevTex, 19 pages, 8 figure
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