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 $n$ queues, each with a single server. Customers arrive in a Poisson process at rate $\lambda n$, where $0<\lambda<1$. Upon arrival each customer selects $d\geq2$ 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 $n\to\infty$ the maximum queue length takes at most two values, which are $\ln\ln n/\ln d+O(1)$.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 $n$ nodes, each pair joined by a link of capacity $C$. For each pair of nodes, calls arrive for this pair of endpoints as a Poisson process with rate $\lambda$. A call for endpoints $\{u,v\}$ is routed directly onto the link between the two nodes if there is spare capacity; otherwise $d$ two-link paths between $u$ and $v$ 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 $d=1$, 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 $d \ge 1$, provided the arrival rate $\lambda$ 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 $j$, the proportion of links at each node with load $j$ is strongly concentrated around the $j$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 $d$ 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 $d$ 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 $\ell_1$ norm

The supermarket model with bounded queue lengths in equilibrium

In the supermarket model, there are n queues, each with a single server. Customers arrive in a Poisson process with arrival rate λn , where λ=λ(n)∈(0,1) . Upon arrival, a customer selects d=d(n) servers uniformly at random, and joins the queue of a least-loaded server amongst those chosen. Service times are independent exponentially distributed random variables with mean 1. In this paper, we analyse the behaviour of the supermarket model in the regime where λ(n)=1−n−α and d(n)=⌊nβ⌋ , where α and β are fixed numbers in (0, 1]. For suitable pairs (α,β) , our results imply that, in equilibrium, with probability tending to 1 as n→∞ , the proportion of queues with length equal to k=⌈α/β⌉ is at least 1−2n−α+(k−1)β , and there are no longer queues. We further show that the process is rapidly mixing when started in a good state, and give bounds on the speed of mixing for more general initial conditions

Extinction times in the subcritical stochastic SIS logistic epidemic

Many real epidemics of an infectious disease are not straightforwardly super- or sub-critical, and the understanding of epidemic models that exhibit such complexity has been identified as a priority for theoretical work. We provide insights into the near-critical regime by considering the stochastic SIS logistic epidemic, a well-known birth-and-death chain used to model the spread of an epidemic within a population of a given size $N$. We study the behaviour of the process as the population size $N$ tends to infinity. Our results cover the entire subcritical regime, including the "barely subcritical" regime, where the recovery rate exceeds the infection rate by an amount that tends to 0 as $N \to \infty$ but more slowly than $N^{-1/2}$. We derive precise asymptotics for the distribution of the extinction time and the total number of cases throughout the subcritical regime, give a detailed description of the course of the epidemic, and compare to numerical results for a range of parameter values. We hypothesise that features of the course of the epidemic will be seen in a wide class of other epidemic models, and we use real data to provide some tentative and preliminary support for this theory.Comment: Revised; 34 pages; 6 figure

Risk management for traffic safety control

This paper offers a range of modelling ideas and techniques from mathematical statistics appropriate for analysing traffic accident data for the East region operation of CLP Power Hong Kong Limited and for the Hong Kong population in general. We further make proposals for alternative ways to record and collect data, and discuss ways to identify the major contributing factors behind accidents. We hope that our findings will enable the design of effective accident prevention strategies for CLP

Concentration of measure and mixing of Markov chains

M. J