220 research outputs found
Analysis of stochastic fluid queues driven by local time processes
We consider a stochastic fluid queue served by a constant rate server and
driven by a process which is the local time of a certain Markov process. Such a
stochastic system can be used as a model in a priority service system,
especially when the time scales involved are fast. The input (local time) in
our model is always singular with respect to the Lebesgue measure which in many
applications is ``close'' to reality. We first discuss how to rigorously
construct the (necessarily) unique stationary version of the system under some
natural stability conditions. We then consider the distribution of performance
steady-state characteristics, namely, the buffer content, the idle period and
the busy period. These derivations are much based on the fact that the inverse
of the local time of a Markov process is a L\'evy process (a subordinator)
hence making the theory of L\'evy processes applicable. Another important
ingredient in our approach is the Palm calculus coming from the point process
point of view.Comment: 32 pages, 6 figure
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
Many-Sources Large Deviations for Max-Weight Scheduling
In this paper, a many-sources large deviations principle (LDP) for the
transient workload of a multi-queue single-server system is established where
the service rates are chosen from a compact, convex and coordinate-convex rate
region and where the service discipline is the max-weight policy. Under the
assumption that the arrival processes satisfy a many-sources LDP, this is
accomplished by employing Garcia's extended contraction principle that is
applicable to quasi-continuous mappings.
For the simplex rate-region, an LDP for the stationary workload is also
established under the additional requirements that the scheduling policy be
work-conserving and that the arrival processes satisfy certain mixing
conditions.
The LDP results can be used to calculate asymptotic buffer overflow
probabilities accounting for the multiplexing gain, when the arrival process is
an average of \emph{i.i.d.} processes. The rate function for the stationary
workload is expressed in term of the rate functions of the finite-horizon
workloads when the arrival processes have \emph{i.i.d.} increments.Comment: 44 page
Stability conditions for a discrete-time decentralised medium access algorithm
We consider a stochastic queueing system modelling the behaviour of a
wireless network with nodes employing a discrete-time version of the standard
decentralised medium access algorithm. The system is {\em unsaturated} -- each
node receives an exogenous flow of packets at the rate packets per
time slot. Each packet takes one slot to transmit, but neighboring nodes cannot
transmit simultaneously. The algorithm we study is {\em standard} in that: a
node with empty queue does {\em not} compete for medium access; the access
procedure by a node does {\em not} depend on its queue length, as long as it is
non-zero. Two system topologies are considered, with nodes arranged in a circle
and in a line. We prove that, for either topology, the system is stochastically
stable under condition . This result is intuitive for the circle
topology as the throughput each node receives in a saturated system (with
infinite queues) is equal to the so called {\em parking constant}, which is
larger than . (The latter fact, however, does not help to prove our
result.) The result is not intuitive at all for the line topology as in a
saturated system some nodes receive a throughput lower than .Comment: 22 page
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