Nelson Antunes

Stochastic networks with multiple stable points

This paper analyzes stochastic networks consisting of a set of finite capacity sites where different classes of individuals move according to some routing policy. The associated (non-reversible) Markov jump processes are analyzed under a thermodynamic limit regime, i.e. when the networks have some symmetry properties and when the number of nodes goes to infinity. A metastability property is proved: under some conditions on the parameters, it is shown that, in the limit, several equilibrium points coexist for the empirical distribution. The key ingredient of the proof of this property is a dimension reduction achieved by the introduction of two energy functions and a convenient mapping of their local minima and saddle points. Cases with a unique equilibrium point are also presented.

Analysis of Loss Networks with Routing

This paper analyzes stochastic networks consisting of finite capacity nodes with different classes of requests which move according to some routing policy. The Markov processes describing these networks do not have, in general, reversibility properties so that the explicit expression of their invariant distribution is not known. A heavy traffic limit regime is considered: the arrival rates of calls as well as the capacities of the nodes are proportional to a factor going to infinity. It is proved that, in the limit, the associated rescaled Markov process converges to a deterministic dynamical system with a unique equilibrium point characterized by a non-standard fixed point equation.

Metastability of CDMA cellular systems

In this paper, it is shown that the coexistence of a variety of different traffics in third generation cellular networks may lead to a very undesirable behavior of the whole network: a metastability property. When this property holds, the state of the network fluctuates on a very long time scale between different set of states. These long oscillations of the network make impossible to predict the average performances of some of the key characteristics of the connections, such as the handoff blocking rate or the probability of call blocking. As a consequence, the quality of service provided by such a network can be guaranteed only by, sometimes poor, lower bounds. Experiments of a UMTS network with this behavior are presented and the analysis of a corresponding simplified mathematical model is developed. The practical implications in the design of radio resource management for CDMA cellular networks are discussed.

Traffic capacity of large WDM passive optical networks

As passive optical networks (PON) are increasingly deployed to provide high speed Internet access, it is important to understand their fundamental traffic capacity limits. The paper discusses performance models applicable to wavelength division multiplexing (WDM) EPONs and GPONs under the assumption that users access the fibre via optical network units equipped with tunable transmitters. The considered stochastic models are based on multiserver polling systems for which explicit analytical results are not known. A large system asymptotic, mean-field approximation is used to derive closed form solutions of these complex systems. Convergence of the mean field dynamics is proved in the case of a simple network configuration. Simulation results show that, for a realistic sized PON, the mean field approximation is accurate.

Perturbation analysis of a variable M/M/1 queue: a probabilistic approach

In this paper, motivated by the problem of the coexistence on transmission links of telecommunications networks of elastic and unresponsive traffic, we study the impact on the busy period of an M/M/1 queue of a small perturbation in the service rate. The perturbation depends upon an independent stationary process (X(t)) and is quantified by means of a parameter ε ≪ 1. We specifically compute the two first terms of the power series expansion in ε of the mean value of the busy period duration. This allows us to study the validity of the reduced service rate approximation, which consists in comparing the perturbed M/M/1 queue with the M/M/1 queue whose service rate is constant and equal to the mean value of the perturbation. For the first term of the expansion, the two systems are equivalent. For the second term, the situation is more complex and it is shown that the correlations of the environment process (X(t)) play a key role.

Stability of multi-server polling system with server limits

We consider a multi-server polling system with server limits, that is the number of servers that can attend a queue simultaneously is limited. Stability conditions are available when service policies are unlimited. The definition of stability conditions when both server limits and limited service policies apply remains an open problem. We postulate a conjecture for the stability condition in this case that is supported by our simulation results. The study of this particular variant of the multi-server polling system is motivated by the performance evaluation of next generation passive optical access networks.

GATE-Driven Dynamic Wavelength and Bandwidth Allocation for WDM EPONs

Passive optical networks are increasingly used for access to the Internet and it is important to understand the performance of future long-reach, multi-channel variants. In this paper we discuss requirements on upstream dynamic wavelength and bandwidth allocation (DWBA) and propose a simple novel algorithm that is considerably more efficient than classical approaches. We demonstrate that the algorithm emulates a multi-server polling system and deduce capacity formulas that are valid for general traffic processes.

Inverting sampled ADSL traffic

On the basis of a reference model for ADSL traffic on an IP backbone link, established in an earlier study, we show that it is possible to infer the characteristics of long flows by performing a deterministic 1/N packet sampling. By using the fact that the number of active long flows can be represented by means of the number of customers in an M/G//spl infin/ queue with Weibullian service times, we derive some probabilistic properties of the sampled data. These properties are then used to infer the characteristics of the original flows. The method is illustrated by considering an actual traffic trace captured in the France Telecom IP backbone network. Experimental data show that the method proves quite efficient.

An intergrated traffic model for multimedia wireless networks

Abstract Multimedia wireless networks are seen today as one of the key factors for the success of the global communication infrastructure in the near future. Such networks will have to handle a range of heterogeneous traffic classes with different QoS requirements. Their design, planning and control must be supported by suitable traffic models capable of dealing with a new set of constraints where QoS management and mobility play an important role. This paper proposes a traffic model for a cellular multimedia wireless network characterized by the integration of mobility and traffic management aspects. User mobility is modeled through a Markov renewal process, which allows for non-exponential cell residence times and may restrict the user mobility to existent paths in the system. A Markov-modulated fluid process is used to describe the changes in the bandwidth requirements of each mobile over the duration of its calls, including the periods where the mobile is inactive (without call). Based on the proposed model, the number of mobiles per class of traffic in a cell and the handoff processes are characterized. System performance results, such as new and handoff call blocking probabilities, for network planning are derived. Also, the distribution of the required capacity in a short time interval for network control is obtained. Simulations were carried out to validate the analytical results. The comparisons have shown that the integrated model may be regarded as a good basis to build useful teletraffic engineering tools for multimedia wireless networks.

Probabilistic sampling of finite renewal processes

Consider a finite renewal process in the sense that interrenewal times are positive i.i.d. variables and the total number of renewals is a random variable, independent of interrenewal times. A finite point process can be obtained by probabilistic sampling of the finite renewal process, where each renewal is sampled with a fixed probability and independently of other renewals. The problem addressed in this work concerns statistical inference of the original distributions of the total number of renewals and interrenewal times from a sample of i.i.d. finite point processes obtained by sampling finite renewal processes. This problem is motivated by traffic measurements in the Internet in order to characterize flows of packets (which can be seen as finite renewal processes) and where the use of packet sampling is becoming prevalent due to increasing link speeds and limited storage and processing capacities.