Jorma T. Virtamo

An elementary approach to a Girsanov formula and other analytical results on fractional Brownian motions

The Radon-Nikodym derivative between a centred fractional Brownian motion Z and the same process with constant drift is derived by finding an integral transformation which changes Z to a process with independent increments. A representation of Z through a standard Brownian motion on a finite interval is given. The maximum-likelihood estimator of the drift and some other applications are presented.

Spatial node distribution of the random waypoint mobility model with applications

The random waypoint model (RWP) is one of the most widely used mobility models in performance analysis of ad hoc networks. We analyze the stationary spatial distribution of a node moving according to the RWP model in a given convex area. For this, we give an explicit expression, which is in the form of a one-dimensional integral giving the density up to a normalization constant. This result is also generalized to the case where the waypoints have a nonuniform distribution. As a special case, we study a modified RWP model, where the waypoints are on the perimeter. The analytical results are illustrated through numerical examples. Moreover, the analytical results are applied to study certain performance aspects of ad hoc networks, namely, connectivity and traffic load distribution.

A queueing analysis of max-min fairness, proportional fairness and balanced fairness

We compare the performance of three usual allocations, namely max-min fairness, proportional fairness and balanced fairness, in a communication network whose resources are shared by a random number of data flows. The model consists of a network of processor-sharing queues. The vector of service rates, which is constrained by some compact, convex capacity set representing the network resources, is a function of the number of customers in each queue. This function determines the way network resources are allocated. We show that this model is representative of a rich class of wired and wireless networks. We give in this general framework the stability condition of max-min fairness, proportional fairness and balanced fairness and compare their performance on a number of toy networks.

The superposition of variable bit rate sources in an ATM multiplexer

When variable-bit-rate sources are multiplexed in an asynchronous transfer mode (ATM) network, there arise queues with a particular form of correlated arrival process. Such queues are analyzed by exploiting a result expressing the distribution of work in system of the G/G/1 queue originally derived by V.E. Benes (1963). A simple alternative demonstration of this result is analyzed and extended to the case of fluid input systems. The result is applied first to a queue where the arrival process is a superposition of periodic sources (the Sigma D/sub i//D/1 queue), and then to a variable-input-rate constant-output-rate fluid system. The latter is shown to model the so-called burst component of the considered superposition queuing process. The difference between this and the real queue, the cell component, can be evaluated by means of the results obtained for the Sigma D/sub i//D/1 queue. The relative importance of these two components is explored with reference to the particular case of a superposition of on/off sources. >

The superposition of periodic cell arrival streams in an ATM multiplexer

The authors consider the queue arising in a multiservice network using ATM (asynchronous transfer mode) when a superposition of periodic streams of constant-length cells is multiplexed on a high-speed link. An exact closed formula is derived for the queue length distribution in the case where all streams have the same period, and tight upper and lower bounds are obtained on this distribution when the periods are different. Numerical results confirm that the use of a Poisson approximation (i.e. the M/D/1 queue) can lead to a significant overestimation of buffer requirements, particularly in the case of heavy loads. Buffer requirements for a mixture of different period streams can be accurately estimated from the upper bound on the queue length distribution. For given load, requirements increase with the number of long-period (i.e. low-bit rate) sources. The results are deduced from a novel characterization of the single-server constant service time queue, which should be useful in other applications. >

Random waypoint mobility model in cellular networks

In this paper we study the so-called random waypoint (RWP) mobility model in the context of cellular networks. In the RWP model the nodes, i.e., mobile users, move along a zigzag path consisting of straight legs from one waypoint to the next. Each waypoint is assumed to be drawn from the uniform distribution over the given convex domain. In this paper we characterise the key performance measures, mean handover rate and mean sojourn time from the point of view of an arbitrary cell, as well as the mean handover rate in the network. To this end, we present an exact analytical formula for the mean arrival rate across an arbitrary curve. This result together with the pdf of the node location, allows us to compute all other interesting measures. The results are illustrated by several numerical examples. For instance, as a straightforward application of these results one can easily adjust the model parameters in a simulation so that the scenario matches well with, e.g., the measured sojourn times in a cell.

Broadband Network Traffic

From the Publisher: This coherently structured and authorative text is the final report of the European Action COST 242, devoted to advancing research in the field of multiservice network design and performance evaluation. The excellent results presented are largely based on some 240 action reports compiled in cooperation between researchers from academia and professionals from industry and discussed during the COST 242 meetings.

Optimal Degree Distribution for LT Codes with Small Message Length

Fountain codes provide an efficient way to transfer information over erasure channels. We give an exact performance analysis of a specific type of fountain codes, called LT codes, when the message length N is small. Two different approaches are developed. In a Markov chain approach the state space explosion, even with reduction based on permutation isomorphism, limits the analysis to very short messages, N < 4. An alternative combinatorial method allows recursive calculation of the probability of decoding after N received packets. The recursion can be solved symbolically for values of N < 10 and numerically up to N ap30. Examples of optimization results give insight into the nature of the problem. In particular, we argue that a few conditions are sufficient to define an almost optimal LT encoding.

A recursive formula for multirate systems with elastic traffic

We present a recursive formula for evaluating the per-flow throughput in a multirate system with elastic traffic, which is the analogue of the well-known Kaufman-Roberts formula used to evaluate the blocking probability in a multirate system with circuit traffic. Such a system is representative of a link in a packet-switched network like the Internet where data flows are not blocked in case of congestion but experience lower bit rates. The maximum bit rate of each flow typically corresponds to the speed of the user access line such as the DSL access lines.

Fluid queue driven by anM/M/1 queue

A fluid queue receiving its input from the output of a precedingM/M/1 queue is considered. The input can be characterized as a Markov modulated rate process and the well known spectral decomposition technique can be applied. The novel features in this system relate to the nature of the spectrum, which is shown to be composed of a continuous part and one or two discrete points depending on whether the load of the fluid queue is less or greater than the output to input rate ratio. Explicit expressions of the generalized eigenvectors are given in terms of Chebyshev polynomials of the second kind, and the resolution of unity is determined. The solution for the buffer content distribution is obtained as a simple integral expression. Numerical examples are given.