Tomasz Nycz

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.

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.

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.

Modelling Dynamics of TCP Flows in Very Large Network Topologies

The article presents an approach to numerical modeling of dynamics of flows in wide area TCP/IP computer networks with the use of SAP HANA in-memory database. The aim is to explore the possibility of transforming a broadly known modeling method—fluid-flow approximation—into database language, overcoming this way the need to develop dedicated solutions and to transmit the results from the application to the database. We implemented the model logic into SQL procedures and performed mathematical calculations for an exemplary vast topology. The experiments show that the database engine may be used to perform all model computations.

An Analysis of the Extracted Parts of Opte Internet Topology

The paper defines a method to extract a part of a graph, which corresponding to a network (Internet) topology covering a certain area. The method refers to data gathered within Opte Project and describing the Internet topology. The extracted parts of graphs have been analyzed in terms of the number of neighbours, longest path length and existence of cycles. Then the resulting topologies are used to model transient behaviour of wide area networks with the use of fluid-flow approximation.

Traffic Engineering: Erlang and Engset Models Revisited with Diffusion Approximation

Classical Erlang and Engset formulae determining the availability of channels, loss probability, and characteristics of overflow traffic are still used in telecommunications. Moreover, they are also interesting for traffic management in mobile networks and in Internet. They are based on the assumption of Poisson flows and exponentially distributed time of connections. By means of diffusion approximation queuing models, we extend these results to the case of general distributions and transient state analysis.

A Numerical Comparison of Diffusion and Fluid-Flow Approximations Used in Modelling Transient States of TCP/IP Networks

The paper presents a comparison of two approaches to queuing models used in performance evaluation of computer networks: diffusion approximation and fluid-flow approximation. Both methods are well known and are used alternatively in similar problems but the differences between their results and the errors they introduce when applied to TCP driven time dependent flows were not sufficiently investigated.

Queue with limited volume, a diffusion approximation approach

The article presents a di_usion approximation model applied to investigate queues with _nite capacity. Usually it is assumed that the size of a queue is limited by the maximum number of customers allowed to the system. Here, we assume that the size of customers is random (i.e. we consider batch arrivals with an arbitrary batch size distribution) and the constraint is given by the total volume of the queue. It is more adequate if we consider the queues of packets in e.g. IP routers where the volume of a bu_er where packets are stored is limited and the size of packets is variable.

Modelling Transient States in Queueing Models of Computer Networks: A Few Practical Issues

The article summarises author’s experience in two problems related to the use of queueing models in performance evaluation of computer networks: modelling transient states of queues and computations for queueing network models having large number of nodes. Both issues are not well represented in classical queueing theory, yet important to applications, because the observed traffic is time dependant and network topologies that should be considered become larger and larger. The article discusses two approaches: diffusion approximation and fluid-flow approximation that can cope with much larger models that are attainable with the use of Markov chains.