Takis Konstantopoulos

Burst reduction properties of the leaky bucket flow control scheme in ATM networks

The leaky bucket is a simple flow control scheme for ATM networks. An arriving cell can be transmitted only if it finds a token in the token buffer, in which case it is transmitted instantaneously by consuming a token. If the token buffer is empty, the cell has to wait until the generation of a new token. For purposes of analysis the authors assume an infinite cell buffer. The control parameter is the token buffer size C. The authors examine the burstiness of the output how as a function of C and show that the burstiness increases with C. In particular the output flow is always less bursty than the input flow. This monotonicity simplifies optimal choice of the token buffer size. The result is true for fairly arbitrary input flows and deterministic token generation times. >

Modeling file-sharing with BitTorrent-like incentives

We propose a new model for file-sharing peer-to-peer (P2P) networks that mimics the incentives provided by the popular BitTorrent system. In it, larger files are split into chunks and a peer can download or swap only one chunk at a time. We propose a Markov chain model in continuous time that resembles a stochastic epidemic/coagulation model. We prove that the Markov chain is approximated by a differential equation which, by itself, can give some rough information about the performance of the system. Finally, using this model, we explore the performance of BitTorrent-like incentives for an open system with peer departures and arrivals and a single file (torrent) with two chunks.

Avoiding overages by deferred aggregate demand for PEV charging on the smart grid

We model the aggregate overnight demand for electricity by a large community of (possibly hybrid) plug-in electric vehicles (PEVs) each of whose power demand follows a prescribed profile and is interruptible. The community is served by a regional electrical utility which is assumed to purchase electricity from a state/national distribution grid according to a flat-rate Φ per kilowatt-unit-time up to a threshold L, and thereafter overage (demand >; L) charges π >; Φ are leveed per kilowatt-unit-time. Rather than a spot-price system for household consumers (which would necessarily need to be operated by automated means overnight when most consumers sleep), the “grid” (regional utility) is “smart” in that it monitors its total load and, when overages threaten, can reduce load by signaling certain consumers to interrupt charging and defer their charging load by one unit of time. In this paper, we model the uninterrupted load by a Gaussian process which we justify by means of a functional central limit theorem (FCLT). This limiting Gaussian process is the arrival process of a discrete-time queue which is used to model the (partially) interrupted and deferred load over a finite time-horizon. We can then compute the mean amount of overage at the end of this time horizon (say at 6 AM when charging is to be completed ahead of the morning commute).

Response to Prof. Baccelli's lecture on Modelling of Wireless Communication Networks by Stochastic Geometry

The history of science teaches us that, occasionally, certain applied problems give rise to new ways of thinking, forcing scientists to revisit the way they do things. Thomas Kuhn [1] argues that scientific theories do not evolve smoothly but are due to paradigm shifts, i.e. due to certain events resulting in the changing of intellectual circumstances and possibilities. A classic example is the circumstances that led to what we know as the theory of functions and the theory of sets (Cantor’s ordinal numbers). They both have arisen in response to Fourier’s introduction of trigonometric series in order to understand the solution to the problem of propagation of heat: indeed, how is it possible for a discontinuous function (a cold rod brought into contact with a hot one) to become, immediately, a smooth one? Fourier series were created to answer this problem; the notion of function was generalized as a response to this; and ordinal numbers were created to understand the notion of infinite sum of functions [2]. Professor Baccelli, in his lecture, made us aware of a number of practical and theoretical problems arising in the area of wireless communications. In my opinion, these types of problems require novel uses of mathematical methods and, conversely, can lead to interesting mathematical developments. We were told that the way to think about coverage, connectivity, communication and routing issues is via the lens of stochastic geometry, an already important sub-field of probability theory. In a world where everybody wants to communicate with everybody else everywhere and at all times, it is clear that good mathematical methods for tackling practical problems are very much needed. What is also clear is that these particular problems can give rise to new mathematics. I would like to point out some research results and open questions related to Prof. Baccelli’s lecture. Suppose that communicating or sensing devices are placed at the points of a spatially homogeneous Poisson process in R with intensity λ. Let ξ = (ξ(t), t ≥ 0) be a stochastic process with values in R and with almost surely continuous paths. Each device has a sensing radius equal to r and performs an independent stochastic motion with law that of ξ . Let K ⊂ R be a fixed compact set and let SK be the time required for one of the moving devices to detect K . (A device positioned at point x ∈ R at time t detects a set K if the ball B(x, r) with radius r centred at x intersects K .) Then

Limit theorems for a random directed slab graph

We consider a stochastic directed graph on the integers whereby a directed edge between i and a larger integer j exists with probability pj i depending solely on the distance between the two integers. Under broad conditions, we identify a regenerative structure that enables us to prove limit theorems for the maximal path length in a long chunk of the graph. The model is an extension of a special case of graphs studied in [18]. We then consider a similar type of graph but on the ‘slab’ Z × I, where I is a finite partially ordered set. We extend the techniques introduced in the in the first part of the paper to obtain a central limit theorem for the longest path. When I is linearly ordered, the limiting distribution can be seen to be that of the largest eigenvalue of a |I| × |I| random matrix in the Gaussian unitary ensemble (GUE).

ANALYSIS OF STOCHASTIC FLUID QUEUES DRIVEN BY LOCAL-TIME PROCESSES

We consider a stochastic fluid queue served by a constant rate server and driven by a process which is the local time of a reflected Lévy process. Such a stochastic system can be used as a model in a priority service system, especially when the time scales involved are fast. The input (local time) in our model is typically (but not necessarily) singular with respect to the Lebesgue measure, a situation which, in view of the nonsmooth or bursty nature of several types of Internet traffic, is nowadays quite realistic. We first discuss how to rigorously construct the (necessarily) unique stationary version of the system under some natural stability conditions. We then consider the distribution of performance steady-state characteristics, namely, the buffer content, the idle period, and the busy period. These derivations are much based on the fact that the inverse of the local time of a Markov process is a Lévy process (a subordinator), hence making the theory of Lévy processes applicable. Another important ingredient in our approach is the use of Palm calculus for stationary random point processes and measures.

A Stochastic Epidemiological Model and a Deterministic Limit for BitTorrent-Like Peer-to-Peer File-Sharing Networks

We propose a stochastic model for a file-sharing peer-to-peer network which resembles the popular BitTorrent system: large files are split into chunks and a peer can download or swap from another peer only one chunk at a time. We exhibit the fluid and diffusion limits of a scaled Markov model of this system and look at possible uses of them to draw practical conclusions.

Functional approximation theorems for controlled renewal processes

We prove a functional law of large numbers and a functional central limit theorem for a controlled renewal process, that is, a point process which differs from an ordinary renewal process in that the ith interarrival time is scaled by a function of the number of previous i arrivals. The functional law of large numbers expresses the convergence of a sequence of suitably scaled controlled renewal processes to the solution of an ordinary differential equation. Likewise, the functional central limit theorem establishes that the error in the law of large numbers converges weakly to the solution of a stochastic differential equation. Our proofs are based on martingale and time-change arguments.

Laplace transform asymptotics and large deviation principles for longest success runs in Bernoulli trials

The longest stretch

Integral representation of Skorokhod reflection

L(n)