Consider the following migration process based on a closed network of N queues with $K_N$ customers. Each station is a $\cdot$/M/$\infty$ queue with service (or migration) rate $\mu$. Upon departure, a customer is routed independently and uniformly at random to another station. In addition to migration, these customers are subject to a susceptible–infected–susceptible (SIS) dynamics. That is, customers are in one of two states: I for infected, or S for susceptible. Customers can swap their state either from I to S or from S to I only in stations. More precisely, at any station, each susceptible customer becomes infected with the instantaneous rate $\alpha Y$ if there are Y infected customers in the station, whereas each infected customer recovers and becomes susceptible with rate $\beta$. We let N tend to infinity and assume that $\lim_{N\to \infty} K_N/N= \eta $, where $\eta$ is a positive constant representing the customer density. The main problem of interest concerns the set of parameters of such a system for which there exists a stationary regime where the epidemic survives in the limiting system. The latter limit will be referred to as the thermodynamic limit. We use coupling and stochastic monotonicity arguments to establish key properties of the associated Markov processes, which in turn allow us to give the structure of the phase transition diagram of this thermodynamic limit with respect to $\eta$. 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.

We consider a friendship model in which each member of a community has a latent value such that the probability that any two individuals are friends is a function of their latent values. We consider such questions as does information that i and j are both friends with k make it more likely that i and j are themselves friends. Among other things, we show that for fixed sets S and T, the more friends that i has in S, then the stochastically more friends i has in T. We consider how a variation of the friendship paradox applies to our model. We also study the distribution of the number of friendless individuals in the community and derive a bound on the total variation distance between it and a Poisson with the same mean.

In large storage systems, files are often coded across several servers to improve reliability and retrieval speed. We study load balancing under the batch sampling routeing scheme for a network of n servers storing a set of files using the maximum distance separable (MDS) code (cf. Li (2016)). Specifically, each file is stored in equally sized pieces across L servers such that any k pieces can reconstruct the original file. When a request for a file is received, the dispatcher routes the job into the k-shortest queues among the L for which the corresponding server contains a piece of the file being requested. We establish a law of large numbers and a central limit theorem as the system becomes large (i.e. n → ∞), for the setting where all interarrival and service times are exponentially distributed. For the central limit theorem, the limit process take values in ℓ2, the space of square summable sequences. Due to the large size of such systems, a direct analysis of the n-server system is frequently intractable. The law of large numbers and diffusion approximations established in this work provide practical tools with which to perform such analysis. The power-of-d routeing scheme, also known as the supermarket model, is a special case of the model considered here.

Let X = {Xt: t ≥ 0} be a stationary piecewise continuous Rd-valued process that moves between jumps along the integral curves of a given continuous vector field, and let S ⊂ Rd be a smooth surface. The aim of this paper is to derive a multivariate version of Rice's formula, relating the intensity of the point process of (localized) continuous crossings of S by X to the distribution of X0. Our result is illustrated by examples relating to queueing networks and stress release network models.

We give a simple and direct treatment of insensitivity in stochastic networks which is quite general and provides probabilistic insight into the phenomenon. In the case of multi-class networks, our results generalise those of Bonald and Proutière (2002), (2003).

We consider a general control problem for networks with linear dynamics which includes the special cases of scheduling in multiclass queueing networks and routeing problems. The fluid approximation of the network is used to derive new results about the optimal control for the stochastic network. The main emphasis lies on the average-cost criterion; however, the β-discounted as well as the finite-cost problems are also investigated. One of our main results states that the fluid problem provides a lower bound to the stochastic network problem. For scheduling problems in multiclass queueing networks we show the existence of an average-cost optimal decision rule, if the usual traffic conditions are satisfied. Moreover, we give under the same conditions a simple stabilizing scheduling policy. Another important issue that we address is the construction of simple asymptotically optimal decision rules. Asymptotic optimality is here seen with respect to fluid scaling. We show that every minimizer of the optimality equation is asymptotically optimal and, what is more important for practical purposes, we outline a general way to identify fluid optimal feedback rules as asymptotically optimal. Last, but not least, for routeing problems an asymptotically optimal decision rule is given explicitly, namely a so-called least-loaded-routeing rule.

We consider a migration process whose singleton process is a time-dependent Markov replacement process. For the singleton process, which may be treated as either open or closed, we study the limiting distribution, the distribution of the time to replacement and related quantities. For a replacement process in equilibrium we obtain a version of Little's law and we provide conditions for reversibility. For the resulting linear population process we characterize exponential ergodicity for two types of environmental behaviour, i.e. either convergent or cyclic, and finally for large population sizes a diffusion approximation analysis is provided.

The standard uniformization technique for continuous-time Markov chains is generalized to non-exponential stochastic networks.