52 research outputs found
Concentration of measure and mixing for Markov chains
We consider Markovian models on graphs with local dynamics. We show that,
under suitable conditions, such Markov chains exhibit both rapid convergence to
equilibrium and strong concentration of measure in the stationary distribution.
We illustrate our results with applications to some known chains from computer
science and statistical mechanics.Comment: 28 page
On the maximum queue length in the supermarket model
There are queues, each with a single server. Customers arrive in a
Poisson process at rate , where . Upon arrival each
customer selects servers uniformly at random, and joins the queue at a
least-loaded server among those chosen. Service times are independent
exponentially distributed random variables with mean 1. We show that the system
is rapidly mixing, and then investigate the maximum length of a queue in the
equilibrium distribution. We prove that with probability tending to 1 as
the maximum queue length takes at most two values, which are
.Comment: Published at http://dx.doi.org/10.1214/00911790500000710 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
A fixed-point approximation for a routing model in equilibrium
We use a method of Luczak (arXiv:1212.3231) to investigate the equilibrium
distribution of a dynamic routing model on a network. In this model, there are
nodes, each pair joined by a link of capacity . For each pair of nodes,
calls arrive for this pair of endpoints as a Poisson process with rate
. A call for endpoints is routed directly onto the link
between the two nodes if there is spare capacity; otherwise two-link paths
between and are considered, and the call is routed along a path with
lowest maximum load, if possible. The duration of each call is an exponential
random variable with unit mean. In the case , it was suggested by Gibbens,
Hunt and Kelly in 1990 that the equilibrium of this process is related to the
fixed points of a certain equation. We show that this is indeed the case, for
every , provided the arrival rate is either sufficiently
small or sufficiently large. In either regime, we show that the equation has a
unique fixed point, and that, in equilibrium, for each , the proportion of
links at each node with load is strongly concentrated around the th
coordinate of the fixed point.Comment: 33 page
Balanced routing of random calls
We consider an online network routing problem in continuous time, where calls
have Poisson arrivals and exponential durations. The first-fit dynamic
alternative routing algorithm sequentially selects up to random two-link
routes between the two endpoints of a call, via an intermediate node, and
assigns the call to the first route with spare capacity on each link, if there
is such a route. The balanced dynamic alternative routing algorithm
simultaneously selects random two-link routes, and the call is accepted on
a route minimising the maximum of the loads on its two links, provided neither
of these two links is saturated. We determine the capacities needed for these
algorithms to route calls successfully and find that the balanced algorithm
requires a much smaller capacity. In order to handle such interacting random
processes on networks, we develop appropriate tools such as lemmas on biased
random walks.Comment: Published at http://dx.doi.org/10.1214/14-AAP1023 in the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Central limit approximations for Markov population processes with countably many types
When modelling metapopulation dynamics, the influence of a single patch on the metapopulation depends on the number of individuals in the patch. Since there is usually no obvious natural upper limit on the number of individuals in a patch, this leads to systems in which there are countably infinitely many possible types of entity. Analogous considerations apply in the transmission of parasitic diseases. In this paper, we prove central limit theorems for quite general systems of this kind, together with bounds on the rate of convergence in an appropriately chosen weighted norm
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