A. van de Liefvoort

Approximation models of feed-forward G/G/1/N queueing networks with correlated arrivals

In this paper we develop approximation models for feed-forward networks of G/G/1/N queues. We use Linear Algebra Queueing Theory (LAQT) techniques to create reduced state space representations for the queue departure processes. Reduced state space departure processes are presented where the first three moments and the correlation decay are mapped to a two state process. A three state process is also presented matching the first five moments and the first three lag autocorrelations. Numerical examples of end-to-end performance for high-speed communications networks with correlated arrival traffic are presented. The results are compared with simulation models and other approximation methods.

Solutions of M/G/1//N-type Loops with Extensions to M/G/1 and GI/M/1 Queues

Closed form solutions of the joint equilibrium distribution of queue sizes are derived for a large class of M/G/1//N queues, i.e., any closed loop of two servers in which one is exponential (but possibly load dependent), and the other has a probability density function which has a rational Laplace transform and a queueing discipline that is FCFS. The class of G/M/k//N queues are included as special cases of load-dependent servers. The solutions are presented in terms of vectors and matrices whose size depends only on the distribution of the general server and not on the number of customers in the system. Efficient algorithms are outlined, and expressions for various system measurements are presented. Depending on the relative service rates of the two servers, solutions for both the M/G/1 and GI/M/1 open queues are derived as limiting cases of the M/G/1//N system. All results are contrasted with existing formulas.

Performance analysis of a digital link with heterogeneous multislot traffic

Presents a unified model to compute various performance measures when Poissonian and non-Poissonian (renewal) multi-slot traffic streams are offered to a digital link in a (double) loss system. the authors represent the non-Poissonian arrival process by a matrix-exponential distribution, requiring only that the inter-arrival distribution has a rational Laplace transform. Several distributions are considered as the non-Poissonian traffic. The resulting model uses matrix algebraic techniques only, thus not requiring any complex and/or tedious transform techniques. The authors also incorporate various control policies in their modeling framework using acceptance functions. Through computational studies, they conclude that the second and the third moments of the non-Poissonian traffic have significant impact on various performance measures. >

Approximation models of wireless cellular networks using moment matching

In this paper we present an analytical model for micro- and pico-cell wireless networks for any arbitrary topology in a high mobility feedforward environment. We introduce an approximation technique which uses a single-cell decomposition analysis which incorporates moment matching of handoff processes into the cell. The approximation technique can provide close approximations for non-Poisson arrival traffic and it is easily parallelized. Performance measures such as new calls blocked, handoff calls lost, and forced termination are derived for any general independent call arrival distribution in a heterogeneous traffic environment. We produce some numerical examples for some simple topologies with varying mobility for several call arrival distributions and compare our results to those from simulation studies.

An iterative algorithm for the Reeve's puzzle

A divide-and-conquer approach for the presumed minimal solution to the Reves puzzle, the Tower of Hanoi with four pegs, leads to a simple structure of the solution. In particular, it allows an easy, iterative algorithm-for the three-peg problem-to be extended to four pegs.

Correlation properties of the token leaky bucket departure process

In this paper we focus on the behavior of the leaky bucket mechanism with respect to second-order statistics of the departure process under various correlated cell arrival and deterministic token arrival distributions. We derive expressions for the lag-k auto-correlation of the inter-departure times for the leaky bucket cell departure stream and we produce numerical examples for the lag-k auto-correlation of the leaky bucket inter-departure times for several different cell arrival distributions exhibiting both positive and negative auto-correlations.

A new approach for the performance analysis of a single-bus multiprocessor system with general service times

The linear algebraic approach to queuing theory is applied to analyze the performance of a typical single-bus multiprocessor system. This system can be modeled as an M/G/1/N queuing system with load-dependent arrivals. The method presented requires only that the nonexponential service time distribution for the system be a matrix-exponential, that is, one with a rational Laplace transform. Using linear algebraic techniques, expressions are obtained for the performance characteristics of interest, such as the processing power for the multiprocessor system. The algorithm does not rely on root finding and can be implemented using symbolic programming techniques. The explicit closed-form expression for the processing power is presented for some special cases. >

A case study on evaluating the benefits of MPLS traffic engineering through constraint-based routing and network controls

In this paper, we attempt to understand the benefits of MPLS traffic engineering through a case study. Specifically, we do a comparative study of MPLS traffic engineering in the presence of dynamic constraint-based routing compared to destination-based routing. For this case study, we use a model derived from an actual service provider network. Different control schemes that can be deployable for MPLS traffic engineering are also considered. This case study shows that services can indeed obtain benefits from constraint-based routing and traffic engineering controls.

A linear algebraic approach to queueing theory

A survey of the basic formulas of queueing theory is presented, indicating their limitations in dealing with non-exponential service time distributions and non-steady-state behavior. The authors then describe a linear algebraic formulation which is a complete procedure for dealing with non-exponential servers. The formulation is invariant to a class of similarity transformations and thus does not depend on the phases used to describe the servers. The authors show how to treat M/G/1, G/M/1 and G/G/1 queues using this formalism. Finally, they show how transient properties of queueing systems can be calculated, using the same mathematical objects which were used for analyzing their steady-state properties.<<ETX>>

A spectral approach to compute performance measures in a correlated single server queue

The coupling matrix was introduced in [8] to compute the performance measures of a PH/PH/1 single server queue. This matrix was extended in [1, 2] to include arrival and service processes that are possibly serially correlated processes, although the service process remains independent of the arrival process and all marginal distributions are matrix exponential, and this current paper is an extended abstract of [2]. The coupling matrix is constructed from the arrival and the service distributions without any computational effort, and the performance measures (such as waiting times and queue length distributions) are derived directly from its spectrum.