Ferhan Pekergin

Diffusion Approximation Model of Multiserver Stations with Losses

The article presents a diffusion approximation model of a G/G/N/N station -N parallel servers without queueing. Diffusion approximation allows us to include in queueing models fairly general assumptions. First of all it gives us a tool to consider in a natural way transient states of queues, which is very rare in classical queueing models. Then we may consider input streams with general interarrival time distributions and servers with general service time distributions. Single server models may be easily incorporated into a network of queues. Here, we apply the diffusion approximation formalism to study transient behaviour of G/G/N/N station and use it to construct a model of a typical call centre and to study the sliding window mechanism, a popular Call Admission Control (CAC) algorithm.

Diffusion Approximation Model for the Distribution of Packet Travel Time at Sensor Networks

We propose a model to estimate the probability density function of the distribution of a packet travel time in a multihop wireless sensor network. The model is based on diffusion approximation and it takes into consideration the heterogeneity of the propagation medium and of the distribution of relay nodes. The modeled system parameters, e.g. the packet loss probability or the network topology, may depend on time.

Diffusion approximation as a modelling tool

Diffusion theory is already a vast domain of knowledge. This tutorial lecture does not cover all results; it presents in a coherent way an approach we have adopted and used in analysis of a series of models concerning evoluation of some traffic control mechanisms in computer, especially ATM, networks. Diffusion approximation is presented from engineers point of view, stressing its utility and commenting numerical problems of its implementation. Diffusion approximation is a method to model the behavior of a single queueing station or a network of stations. It allows one to include in the model general sevice times, general (also correlated) input streams and to investigate transient states, which, in presence of bursty streams (e.g. of multimedia transfers) in modern networks, are of interest.

Stability and dynamics of TCP-NCR(DCR) protocol in presence of UDP flows

The fluid-flow approximation models investigate with much success the dynamics and stability of TCP/RED connections. Their main assumption is that the fluctuations of variables characterizing the behaviour of the connections are relatively small, that enables the linearization of model and the use of traditional control analysis tools to obtain such measures as Bode gain, phase margins, tracking error or delay margin. In this article, preserving linear fluid-flow model, we propose its extension to the case when a network is composed of wired and wireless part. We consider a variant of TCP algorithm (TCP-DCR or its new version TCP-NCR) and fluid-flow differential equations representing the size of congestion window, mean queue at the bottleneck router and loss probability at a RED queue are supplemented with terms representing constant loss probability due to transmission in wireless part and probability that a fraction of these errors is recovered by a link level mechanism. The decrease of congestion window due to TCP mechanism is delayed to allow the link protocol to deal with the errors. The model considers the presence of uncontrollable UDP flows.

A diffusion approximation model for wireless networks based on IEEE 802.11 standard

The article presents an analytical model of wireless networks using the IEEE 802.11 protocol to access the transport medium. The model allows to determine such key factors of the quality of service as transmission delays and losses. The model is based on diffusion approximation approach which was proposed three decades ago to model wired networks. We show that it can be adapted to take into consideration the input streams with general interarrival time distributions and servers with general service time distributions. The diffusion approximation has been chosen because of fairly general assumptions of models based on it, hard to be represented in Markov models. A queueing network model can have an arbitrary topology, the intensity of transmitted flows can be represented by non-Poisson (even self-similar) streams, the service times at nodes can be defined by general distributions. These assumptions are important: because of the CSMA/CA algorithm, the overall times needed to sent a packet are far from being exponentially distributed and therefore the flows between nodes are non-Poisson. Diffusion approximation allows us also to analyse the of transient behaviour of a network when traffic intensity is changing with time.

A saturated linear dynamical network for approximating maximum clique

We use a saturated linear gradient dynamical network for finding an approximate solution to the maximum clique problem. We show that for almost all initial conditions, any solution of the network defined on a closed hypercube reaches one of the vertices of the hypercube, and any such vertex corresponds to a maximal clique. We examine the performance of the method on a set of random graphs and compare the results with those of some existing methods. The proposed model presents a simple continuous, yet powerful, solution in approximating maximum clique, which may outperform many relatively complex methods, e.g., Hopfield-type neural network based methods and conventional heuristics.

Analytical and Numerical Means to Model Transient States in Computer Networks

Transient queue analysis is needed to model the influence of time-dependent flows on congestion in computer networks. It may be applied to the networks performance evaluation and the analysis of the transmissions quality of service. However, the exact queuing theory gives us only few practically useful results, concerning mainly M/M/1 and M/M/1/N queues. The article presents potentials of three approaches: Markovian queues solved numerically, the diffusion approximation, and fluid-flow approximation. We mention briefly a software we implemented to use these methods and summarise our experience with it.

Transient States of Priority Queues - A Diffusion Approximation Study

The article presents a diffusion approximation model applied to investigate the behaviour of priority queues. Diffusion approximation allows us to include in queueing models fairly general assumptions. First of all it gives us a tool to consider in a natural way transient states of queues, which is vary rare in classical queueing models. Then we may consider input streams with general interarrival time distributions and servers with general service time distributions. Single server models may be easily incorporated into the network of queues. Here, we apply the diffusion approximation formalism to study transient and steady-state behavior of G/G/1 and G/G/1/N priority preemptive models. The models can be easily converted to nonpreemptive queueing discipline. Also the introduction of self-similar traffic is possible. The models may be useful in performance evaluation of mechanisms to differentiate the quality of service e.g. in WiMAX, metro networks, etc.

Diffusion models to study nonstationary traffic and cell loss in ATM networks

We study the effects of nonstationary traffic patterns in a network of ATM nodes. Dynamic behaviour of ATM networks is of interest due to the highly nonhomogenous nature of the load: periods of basic activities are interleaved with bursty periods of demands. The models frequently used to predict transient behaviour of these networks are based on fluid approximation. Usually they assume Poisson arrivals and consider only mean values of queues. Here, we propose a diffusion model which takes into account general input process and allows us to study the dynamics of nonstationary traffic along virtual path, to approximate transient distributions of queues and transient distributions of response times of one or several nodes. It also permits the estimation of time-varying loss rates due to limited capacity of buffers.

A Diffusion Approximation Model of Web Servers

We present a model of Web servers in terms of G/G/m/N service system with autocorrelated (self-similar) input traffic. Diffusion approximation enables introduction of time-varying input and the analysis of transient states.