Sergey Foss

For a stochastically monotone Markov chain taking values in a Polish space, we present a number of conditions for existence and for uniqueness of its stationary regime, as well as for closeness of its transient trajectories. In particular, we generalise a basic result by Bhattacharya and Majumdar (Random Dynamical Systems: Theory and Applications (2007) Cambridge Univ. Press) where a certain form of mixing, or splitting condition was assumed uniformly over the state space. We do not rely on continuity properties of transition probabilities.

This paper defines a new model which incorporates three key ingredients of a large class of wireless communication systems: 1) spatial interactions through interference, (1) dynamics of the queueing type, with users joining and leaving, and 2) carrier sensing and collision avoidance as used in, e.g., WiFi. In systems using (2), rather than directly accessing the shared resources upon arrival, a customer is considerate and waits to access them until nearby users in service have left. This new model can be seen as a missing piece of a larger puzzle that contains such dynamics as spatial birth-and-death processes, the Poisson-Hail model, and wireless dynamics as key other pieces. It is shown that, under natural assumptions, this model can be represented as a Markov process on the space of counting measures. The main results are then two-fold. The first result is on the shape of the stability region and, more precisely, on the characterization of the critical value of the arrival rate that separates stability from instability. The second result is of a more qualitative or perhaps even ethical nature. There is evidence that for natural values of the system parameters, the implementation of sensing and the delayed access for collision avoidance can stabilize a system that would be unstable if immediate access to the shared resources would be granted. In other words, for these parameters, renouncing greedy access makes sharing sustainable, whereas indulging in greedy access kills the system.

This paper concerns the recurrence structure of the infinite server queue, as viewed through the prism of the maximum dater sequence, namely the time to drain the current work in the system as seen at arrival epochs. Despite the importance of this model in queueing theory, we are aware of no complete analysis of the stability behavior of this model, especially in settings in which either or both the inter arrival and service time distributions have infinite mean. In this paper, we fully develop the analog of the Loynes construction of the stationary version in the context of stationary ergodic inputs, extending earlier work of E. Altman [1], and then classify the Markov chain when the inputs are independent and identically distributed. This allows us to classify the chain, according to transience, recurrence in the sense of Harris, and positive recurrence in the sense of Harris. We further go on to develop tail asymptotics for the stationary distribution of the maximum dater sequence, when the service times have tails that are asymptotically exponential or Pareto, and we contrast the stability theory for the infinite server queue relative to that for the single server queue.

The problem of monitoring rare events in distributed systems arises in a wide range of applications, including environmental monitoring, industrial automation, and smart infrastructures. In such systems, devices operate independently and without coordination, making random multiple access the only practical and effective approach for utilizing the shared communication channel. We study a rare-event monitoring system consisting of a set of devices and a base station, where devices report events to the base station via a random multiple access scheme. The model accounts for channel noise, which may lead to message loss even in the absence of transmission collisions. Event occurrences are modeled as a family of independent two-state Markov chains. We analyze the impact of repeated transmissions on system performance, aiming to determine the optimal number of retransmissions under selected efficiency criteria and to characterize its dependence on system parameters. Specifically, we investigate (i) the maximum probability that a message from a fixed device is successfully delivered to the base station, and (ii) the maximum frequency at which the base station successfully receives updates about the entire system. For each criterion, we derive explicit expressions as functions of the number of users, the frequency of new messages, and the channel noise. The optimal number of retransmissions is then obtained by optimizing these expressions. By explicitly incorporating channel noise—a factor often neglected but crucial in practice—we show that repeated transmissions naturally arise as part of the optimal design. Our numerical results further support the theoretical analysis, showing that optimal retransmission strategies can substantially enhance overall system performance.

We consider a rare event monitoring system consisting of a set of devices and a base station, where devices transmit information about rare events to the base station using a random multiple access scheme. We introduce a model in which the presence of noise in the multiple access channel can cause message loss even in the absence of transmission collisions. The occurrence of events is modeled by a family of independent two-state Markov chains (with states 0 and 1). We analyze how repeated transmissions affect system performance. Two efficiency criteria are proposed and studied: the maximum probability that a message about an event from a fixed device is successfully delivered to the base station and the maximum frequency at which the base station successfully receives updates about the entire system. For each criterion, we determine the optimal number of retransmissions as a function of the system parameters.

Consider the following migration process based on a closed network of N queues with . The analysis of the Kolmogorov equations of this SIS model reduces to that of a wave-type PDE for which we have found no explicit solution. This plain SIS model is one among several companion stochastic processes that exhibit both random migration and contagion. Two of them are discussed in the present paper as they provide variants to the plain SIS model as well as some bounds and approximations. These two variants are the departure-on-change-of-state (DOCS) model and the averaged-infection-rate (AIR) model, which both admit closed-form solutions. The AIR system is a classical mean-field model where the infection mechanism based on the actual population of infected customers is replaced by a mechanism based on some empirical average of the number of infected customers in all stations. The latter admits a product-form solution. DOCS features accelerated migration in that each change of SIS state implies an immediate departure. This model leads to another wave-type PDE that admits a closed-form solution. In this text, the main focus is on the closed stochastic networks and their limits. The open systems consisting of a single station with Poisson input are instrumental in the analysis of the thermodynamic limits and are also of independent interest. This class of SIS dynamics has incarnations in virtually all queueing networks of the literature.