FITTING TRAFFIC TRACES WITH DISCRETE CANONICAL PHASE TYPE DISTRIBUTIONS AND MARKOV ARRIVAL PROCESSES

Recent developments of matrix analytic methods make phase type distributions (PHs) and Markov Arrival Processes (MAPs) promising stochastic model candidates for capturing traffic trace behaviour and for efficient usage in queueing analysis. After introducing basics of these sets of stochastic models, the paper discusses the following subjects in detail: (i) PHs and MAPs have different representations. For efficient use of these models, sparse (defined by a minimal number of parameters) and unique representations of discrete time PHs and MAPs are needed, which are commonly referred to as canonical representations. The paper presents new results on the canonical representation of discrete PHs and MAPs. (ii) The canonical representation allows a direct mapping between experimental moments and the stochastic models, referred to as moment matching. Explicit procedures are provided for this mapping. (iii) Moment matching is not always the best way to model the behavior of traffic traces. Model fitting based on appropriately chosen distance measures might result in better performing stochastic models. We also demonstrate the efficiency of fitting procedures with experimental result

Fitting simulation input models for correlated traffic data

The adequate representation of input models is an important step in building valid simulation models. Modeling independent and identically distributed data is well established in simulation, but for some application areas like computer and communication networks it is known, that the assumption of independent and identically distributed data is violated in practice and that for example interarrival times or packet sizes exhibit autocorrelation over a large number of lags. Moreover, it is known that negligence of these correlations can result in a serious loss of validity of the simulation model. Although different stochastic processes, which can model these autocorrelations, like e.g. Autoregressive-To-Anything (ARTA) processes and Markovian Arrival Processes (MAPs), have been proposed in the past and more recently fitting algorithms to set the parameters of these processes such that they resemble the behavior of observations from a real system have been developed, the integration of correlated processes into simulation models is still a challenge. In this work ARTA processes are extended in several ways to account for the requirements when simulating models of computer and communication systems. In a first step ARTA processes are extended to use an Autoregressive Moving Average (ARMA) process instead of a pure Autoregressive (AR) base process to be able to capture a large number of autocorrelation lags, while keeping the model size small. In a second step they are enabled to use the flexible class of acyclic Phase-type distributions as marginal distribution. To support the usage of these novel processes in simulation models a fitting algorithm is presented, software for fitting and simulating these processes is developed and the tools are integrated into the toolkit ProFiDo, which provides a complete framework for fitting and analyzing different stochastic processes. By means of synthetically generated and real network traces it is shown that the presented stochastic processes are able to provide a good approximation of the marginal distribution as well as the correlation structure of the different traces and result in a compact process description

Matching marginal moments and lag autocorrelations with MAPs

This paper presents a procedure that constructs a Markovian Arrival Process (MAP) based on the mean, the squared coefficient of variation and the lag-1 autocorrelation of the inter-arrival times. This method always provides a valid MAP without posing any restrictions on the three input parameters. Besides matching these three parameters, it is possible to match the third moment of the inter-arrival times and the decay of the autocorrelation function as well, if they fall into the given (very wide) bounds

Aggregate matrix-analytic techniques and their applications

The complexity of computer systems affects the complexity of modeling techniques that can be used for their performance analysis. In this dissertation, we develop a set of techniques that are based on tractable analytic models and enable efficient performance analysis of computer systems. Our approach is three pronged: first, we propose new techniques to parameterize measurement data with Markovian-based stochastic processes that can be further used as input into queueing systems; second, we propose new methods to efficiently solve complex queueing models; and third, we use the proposed methods to evaluate the performance of clustered Web servers and propose new load balancing policies based on this analysis.;We devise two new techniques for fitting measurement data that exhibit high variability into Phase-type (PH) distributions. These techniques apply known fitting algorithms in a divide-and-conquer fashion. We evaluate the accuracy of our methods from both the statistics and the queueing systems perspective. In addition, we propose a new methodology for fitting measurement data that exhibit long-range dependence into Markovian Arrival Processes (MAPs).;We propose a new methodology, ETAQA, for the exact solution of M/G/1-type processes, (GI/M/1-type processes, and their intersection, i.e., quasi birth-death (QBD) processes. ETAQA computes an aggregate steady state probability distribution and a set of measures of interest. E TAQA is numerically stable and computationally superior to alternative solution methods. Apart from ETAQA, we propose a new methodology for the exact solution of a class of GI/G/1-type processes based on aggregation/decomposition.;Finally, we demonstrate the applicability of the proposed techniques by evaluating load balancing policies in clustered Web servers. We address the high variability in the service process of Web servers by dedicating the servers of a cluster to requests of similar sizes and propose new, content-aware load balancing policies. Detailed analysis shows that the proposed policies achieve high user-perceived performance and, by continuously adapting their scheduling parameters to the current workload characteristics, provide good performance under conditions of transient overload

Moment Matching-Based Distribution Fitting with Generalized Hyper-Erlang Distributions

This paper describes a novel moment matching based fitting method for phase-type (PH) distributions. A special sub-class of phase-type distributions is introduced for the fitting, called generalized hyper-Erlang distributions. The user has to provide only two parameters: the number of moments to match, and the upper bound for the sum of the multiplicities of the eigenvalues of the distribution, which is related to the maximal size of the resulting PH distribution. Given these two parameters, our method obtains all PH distributions that match the target moments and have a Markovian representation up to the given size. From this set of PH distributions the best one can be selected according to any distance function