Florian Simatos

A queueing system for modeling a file sharing principle

We investigate in this paper the performance of a simple file sharing principle. For this purpose, we consider a system composed of N peers becoming active at exponential random times; the system is initiated with only one server offering the desired file and the other peers after becoming active try to download it. Once the file has been downloaded by a peer, this one immediately becomes a server. To investigate the transient behavior of this file sharing system, we study the instant when the system shifts from a congested state where all servers available are saturated by incoming demands to a state where a growing number of servers are idle. In spite of its apparent simplicity, this queueing model (with a random number of servers) turns out to be quite difficult to analyze. A formulation in terms of an urn and ball model is proposed and corresponding scaling results are derived. These asymptotic results are then compared against simulations.

Load balancing via random local search in closed and open systems

In this paper, we analyze the performance of random load resampling and migration strategies in parallel server systems. Clients initially attach to an arbitrary server, but may switch servers independently at random instants of time in an attempt to improve their service rate. This approach to load balancing contrasts with traditional approaches where clients make smart server selections upon arrival (e.g., Join-the-Shortest-Queue policy and variants thereof). Load resampling is particularly relevant in scenarios where clients cannot predict the load of a server before being actually attached to it. An important example is in wireless spectrum sharing where clients try to share a set of frequency bands in a distributed manner. We first analyze the natural Random Local Search (RLS) strategy. Under this strategy, after sampling a new server randomly, clients only switch to it if their service rate is improved. In closed systems, where the client population is fixed, we derive tight estimates of the time it takes under RLS strategy to balance the load across servers. We then study open systems where clients arrive according to a random process and leave the system upon service completion. In this scenario, we analyze how client migrations within the system interact with the system dynamics induced by client arrivals and departures. We compare the load-aware RLS strategy to a load-oblivious strategy in which clients just randomly switch server without accounting for the server loads. Surprisingly, we show that both load-oblivious and load-aware strategies stabilize the system whenever this is at all possible. We further demonstrate, using large-system asymptotics, that the average client sojourn time under the load-oblivious strategy is not considerably reduced when clients apply smarter load-aware strategies.

The weak convergence of regenerative processes using some excursion path decompositions

We consider regenerative processes with values in some Polish space. We define their \epsilon-big excursions as excursions e such that f(e)>\epsilon, where f is some given functional on the space of excursions which can be thought of as, e.g., the length or the height of e. We establish a general condition that guarantees the convergence of a sequence of regenerative processes involving the convergence of \epsilon-big excursions and of their endpoints, for all \epsilon in a countable set whose closure contains 0. Finally, we provide various sufficient conditions on the excursion measures of this sequence for this general condition to hold and discuss possible generalizations of our approach to processes that can be written as the concatenation of i.i.d. paths.

Occupancy schemes associated to Yule processes

An occupancy problem with an infinite number of bins and a random probability vector for the locations of the balls is considered. The respective sizes of the bins are related to the split times of a Yule process. The asymptotic behavior of the landscape of the first empty bins, i.e. the set of corresponding indices represented by point processes, is analyzed and convergences in distribution to mixed Poisson processes are established. Additionally, the influence of the random environment, the random probability vector, is analyzed. It is represented by two main components: an independent, identically distributed sequence and a fixed random variable. Each of these components has a specific impact on the qualitative behavior of the stochastic model. It is shown in particular that, for some values of the parameters, some rare events, which are identified, determine the asymptotic behavior of the average values of the number of empty bins in some regions.

Tightness for processes with fixed points of discontinuities and applications in varying environment

We establish a sufficient condition for the tightness of a sequence of stochastic processes. Our condition makes it possible to study processes with accumulations of fixed times of discontinuity. Our motivation comes from the study of processes in varying or random environment. We demonstrate the usefulness of our condition on two examples: Galton Watson branching processes in varying environment and logistic branching processes with catastrophes.

Introduction: Special Issue: ACM SIGMETRICS 2015

The six papers in this special issue are based on a selection of top theoretical papers from the ACM SIGMETRICS International Conference on Measurement and Modeling of Computer Systems, which took place in Portland, Oregon, in June 2015. This is the premier conference on the measurement and modelling of computer systems, and its technical program featured papers that covered both theory and applications from a wide variety of areas, including algorithms, communication networks, dynamics and control, optimization, performance analysis, resource allocation and scheduling, and stochastic modeling, among others. The program consisted of 40 papers selected from over 200 submissions from all over the world. We seek with this special issue to highlight some of the recent theoretical work on the mathematical analysis and modeling of computer systems. The paper of Gardner et al. studies the use of redundant requests to reduce latency caused by queueing delays. It brings together a well-motivated problem, a simple algorithm and sophisticated analysis to make the argument for the benefits of replication, by presenting an exact analysis of a fairly broad class of such queueing models. A related queueing model corresponds to splitting a task rather than replicating it; the chunks are submitted to different servers, and the last of these to complete (rather than the first) determines the service time of the task. This is the classical fork-join queueing model, and is studied in the paper of Rizk, Poloczek and Ciucu, who obtain stochastic bounds on delays in such a system. The paper of Shah and de Veciana considers the performance analysis of queueing systems employing various fair scheduling policies, such as alpha-fairness or balanced

Special Issue: ACM SIGMETRICS 2014

The five papers in this special issue are based on a selection of top theoretical papers from the ACM SIGMETRICS International Conference on Measurement and Modeling of Computer Systems, which took place at Austin, Texas, in June 2014. As the premier conference on the measurement and modeling of computer systems, the technical program featured papers that covered both theory and applications from a wide variety of areas, including algorithms, communication networks, dynamics and control, optimization, performance analysis, resource allocation and scheduling, and stochastic modeling, among others. The program consisted of 40 papers selected from more than 237 international submissions by a technical program committee of well-established researchers from all over the world. We seek with this special issue to highlight some of the recent theoretical work on the mathematical analysis and modeling of computer systems. The paper of Larranaga, Ayesta, and Verloop deals with a resource allocation problem in a multi-class server with convex holding costs and user impatience under the average cost criterion. Although the optimal policy has a complex dependency on all the input parameters and state information, the authors derive simple index policies obtained by solving a relaxed version of the optimal stochastic control problem and combining results from restless multi-armed bandits and queueing theory.

State space collapse for critical multistage epidemics

We study a multistage epidemic model which generalizes the SIR model and where infected individuals go through K ≥ 1 stages of the epidemic before being removed. An infected individual in stage k ∈ {1, …, K} may infect a susceptible individual, who directly goes to stage k of the epidemic; or it may go to the next stage k + 1 of the epidemic. For this model, we identify the critical regime in which we establish diffusion approximations. Surprisingly, the limiting diffusion exhibits an unusual form of state space collapse which we analyze in detail.

Stability properties of linear file-sharing networks: invited presentation, extended abstract

File-sharing networks are distributed systems used to disseminate information among a subset of the nodes of the Internet. The general principle is the following: once a node of the system has retrieved a file it becomes a server for this file. The advantages of this scheme are numerous, and it has been used for some time now in peer-to-peer systems such as BitTorrent or Emule.

Study of a stochastic model for mobile networks

Recent wireless technologies have triggered interest in a new class of stochastic networks, called mobile networks in the technical literature. In contrast with Jackson networks where users move upon completion of service at some node, in these mobile networks, transitions of customers within the network occur independently of the service received. Moreover, at any given time, each node capacity is divided between the users present, whose service rate thus depends on the capacity and on the state of occupancy of the node. Once his initial service requirement has been fulfilled, a customer definitively leaves the network. In [3], complex capacity sharing policies are considered, but in the simplest setting, which will be of interest to us, nodes implement the Processor-Sharing discipline by dividing their capacity equally between all the users present. Previous works [3, 6] have mainly focused on determining the stability region of such networks, and it has been commonly observed that the users’ mobility represents an opportunity for the network to increase this region. Indeed, because of their mobility, users offer a diversity of channel conditions to the base stations (in charge of allocating the resources of the nodes), thus allowing them to select the users in the most favorable state. Such a scheduling strategy is sometimes referred to as an opportunistic scheduling strategy, see [2] and the references therein for more details.