158 research outputs found
Stability of multi-dimensional birth-and-death processes with state-dependent 0-homogeneous jumps
We study the positive recurrence of multi-dimensional birth-and-death
processes describing the evolution of a large class of stochastic systems, a
typical example being the randomly varying number of flow-level transfers in a
telecommunication wire-line or wireless network.
We first provide a generic method to construct a Lyapunov function when the
drift can be extended to a smooth function on , using an
associated deterministic dynamical system. This approach gives an elementary
proof of ergodicity without needing to establish the convergence of the scaled
version of the process towards a fluid limit and then proving that the
stability of the fluid limit implies the stability of the process. We also
provide a counterpart result proving instability conditions.
We then show how discontinuous drifts change the nature of the stability
conditions and we provide generic sufficient stability conditions having a
simple geometric interpretation. These conditions turn out to be necessary
(outside a negligible set of the parameter space) for piece-wise constant
drifts in dimension 2.Comment: 18 pages, 4 figure
Heavy-traffic analysis of the maximum of an asymptotically stable random walk
For families of random walks with we consider their maxima . We
investigate the asymptotic behaviour of as for
asymptotically stable random walks. This problem appeared first in the 1960's
in the analysis of a single-server queue when the traffic load tends to 1 and
since then is referred to as the heavy-traffic approximation problem. Kingman
and Prokhorov suggested two different approaches which were later followed by
many authors. We give two elementary proofs of our main result, using each of
these approaches. It turns out that the main technical difficulties in both
proofs are rather similar and may be resolved via a generalisation of the
Kolmogorov inequality to the case of an infinite variance. Such a
generalisation is also obtained in this note.Comment: 9 page
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
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
Stability of a Markov-modulated Markov Chain, with application to a wireless network governed by two protocols
We consider a discrete-time Markov chain , , where
the -component forms a Markov chain itself. Assume that is
Harris-ergodic and consider an auxiliary Markov chain whose
transition probabilities are the averages of transition probabilities of the
-component of the -chain, where the averaging is weighted by the
stationary distribution of the -component.
We first provide natural conditions in terms of test functions ensuring that
the -chain is positive recurrent and then prove that these conditions
are also sufficient for positive recurrence of the original chain .
The we prove a "multi-dimensional" extension of the result obtained. In the
second part of the paper, we apply our results to two versions of a
multi-access wireless model governed by two randomised protocols.Comment: 23 page
The end time of SIS epidemics driven by random walks on edge-transitive graphs
Network epidemics is a ubiquitous model that can represent different
phenomena and finds applications in various domains. Among its various
characteristics, a fundamental question concerns the time when an epidemic
stops propagating. We investigate this characteristic on a SIS epidemic induced
by agents that move according to independent continuous time random walks on a
finite graph: Agents can either be infected (I) or susceptible (S), and
infection occurs when two agents with different epidemic states meet in a node.
After a random recovery time, an infected agent returns to state S and can be
infected again. The End of Epidemic (EoE) denotes the first time where all
agents are in state S, since after this moment no further infections can occur
and the epidemic stops.
For the case of two agents on edge-transitive graphs, we characterize EoE as
a function of the network structure by relating the Laplace transform of EoE to
the Laplace transform of the meeting time of two random walks. Interestingly,
this analysis shows a separation between the effect of network structure and
epidemic dynamics. We then study the asymptotic behavior of EoE (asymptotically
in the size of the graph) under different parameter scalings, identifying
regimes where EoE converges in distribution to a proper random variable or to
infinity. We also highlight the impact of different graph structures on EoE,
characterizing it under complete graphs, complete bipartite graphs, and rings
Large deviations for random walks under subexponentiality: the big-jump domain
For a given one-dimensional random walk with a subexponential
step-size distribution, we present a unifying theory to study the sequences
for which as
uniformly for . We also investigate the stronger "local"
analogue, . Our
theory is self-contained and fits well within classical results on domains of
(partial) attraction and local limit theory. When specialized to the most
important subclasses of subexponential distributions that have been studied in
the literature, we reproduce known theorems and we supplement them with new
results.Comment: Published in at http://dx.doi.org/10.1214/07-AOP382 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
- …