Heavy-traffic analysis of the maximum of an asymptotically stable random walk

For families of random walks $\{S_k^{(a)}\}$ with $\mathbf E S_k^{(a)} = -ka < 0$ we consider their maxima $M^{(a)} = \sup_{k \ge 0} S_k^{(a)}$. We investigate the asymptotic behaviour of $M^{(a)}$ as $a \to 0$ 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 $\lambda$ 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 $\lambda < 2/5$. 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 $2/5$. (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 $2/5$.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 $(X^t,Y^t)$, $t=0,1,2,...$, where the $X$-component forms a Markov chain itself. Assume that $(X^t)$ is Harris-ergodic and consider an auxiliary Markov chain ${\hat{Y}^t}$ whose transition probabilities are the averages of transition probabilities of the $Y$-component of the $(X,Y)$-chain, where the averaging is weighted by the stationary distribution of the $X$-component. We first provide natural conditions in terms of test functions ensuring that the $\hat{Y}$-chain is positive recurrent and then prove that these conditions are also sufficient for positive recurrence of the original chain $(X^t,Y^t)$. 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