The representation of general distributions or measured data by phase-type distributions is an important and non-trivial task in analytical modeling. Although a large number of different methods for fitting parameters of phase-type distributions to data traces exist, many approaches lack efficiency and numerical stability. In this paper, a novel approach is presented that fits a restricted class of phase-type distributions, namely mixtures of Erlang distributions, to trace data. For the parameter fitting an algorithm of the expectation maximization type is developed. The paper shows that these choices result in a very efficient and numerically stable approach which yields phasetype approximations for a wide range of data traces that are as good or better than approximations computed with other less efficient and less stable fitting methods. To illustrate the effectiveness of the proposed fitting algorithm, we present comparative results for our approach and two other methods using six benchmark traces and two real traffic traces. 1.

Markovian models provide a convenient way of evaluating the performance of network traffic since their queueing analysis enjoys established theoretical results and efficient solution algorithms [1]. Although unable to directly generate traffic with long-range dependent (LRD) behavior, Markovian models can approximate

We present the KPC-Toolbox, a collection of MATLAB scripts for fitting workload traces into Markovian Arrival Processes (MAPs) in an automatic way. We first present detailed sensitivity analysis that builds intuition on which trace descriptors are most important for queueing. This sensitivity analysis stresses the importance of matching higher-order correlations (i.e., joint moments) of the process inter-arrival times rather than higher order moments of the distribution and provides guidance on the relative importance of different descriptors on queueing. Given that the MAP parameterization space can be very large, we focus on first determining the order of the smallest MAP that can fit the trace well, using the Bayesian Information Criterion (BIC) for determining the best order-accuracy tradeoff. Having determined the order of the target MAP, the KPC-Toolbox automatically derives a MAP that captures accurately the most essential features of the trace. Extensive experimentation illustrates the effectiveness of the KPC-Toolbox in fitting traces that are well-documented in the literature as very challenging to fit, showing that the KPC-Toolbox provides a simple and powerful solution to fitting accurately trace data into MAPs. 1

This paper presents closed-form expressions to exactly match three moments and a correlation parameter into a Markovian arrival process of second order (MAP(2)), whose marginal distribution is a mixture of two exponentials. Mixtures of exponential distributions are popular in distribution fitting. In the two-dimensional setting, we study to which extent correlations can be introduced in a sequence of intervals following such a distribution. Besides the known moment bounds, exhaustive bounds for the single correlation parameter are given explicitly (in terms of the first three moments) for the first time. This allows one to quickly check the feasibility of the correlation parameter before model construction or to modify an infeasible parameter set in order to obtain a valid (but approximate) MAP(2) representation.

The representation of general distributions or measured data by phase-type distributions is an important and non-trivial task in analytical modeling. Although a large number of different methods for fitting parameters of phase-type distributions to data traces exist, many approaches lack efficiency and numerical stability. In this paper, a novel approach is presented that fits a restricted class of phase-type distributions, namely mixtures of Erlang distributions, to trace data. For the parameter fitting an algorithm of the expectation maximization type is developed. The paper shows that these choices result in a very efficient and numerically stable approach which yields phase-type approximations for a wide range of data traces that are as good or better than approximations computed with other less efficient and less stable fitting methods. To illustrate the effectiveness of the proposed fitting algorithm, we present comparative results for our approach and two other methods using six benchmark traces and two real traffic traces as well as quantitative results from queueing analysis. Keywords: Performance and dependability assessment/analytical and numerical techniques, design of tools for performance/dependability assessment, traffic modeling, hyper-Erlang distributions.

In this paper, we define and study a new class of capacity planning models called MAP queueing networks. MAP queueing networks provide the first analytical methodology to describe and predict accurately the performance of complex systems operating under bursty workloads, such as multi-tier architectures or storage arrays. Burstiness is a feature that significantly degrades system performance and that cannot be captured explicitly by existing capacity planning models. MAP queueing networks address this limitation by describing computer systems as closed networks of servers whose service times are Markovian Arrival Processes (MAPs), a class of Markov-modulated point processes that can model general distributions and burstiness. In this paper, we show that MAP queueing networks provide reliable performance predictions even if the service processes are bursty. We propose a methodology to solve MAP queueing networks by two state space transformations, which we call Linear Reduction (LR) and Quadratic Reduction (QR). These transformations dramatically decrease the number of states in the underlying Markov chain of the queueing network model. From these reduced state spaces, we obtain two classes of bounds on arbitrary performance indexes, e.g., throughput, response time, utilizations. Numerical experiments show that LR an QR bounds achieve good accuracy. We also illustrate the high effectiveness of the LR and QR bounds in the performance analysis of a real multi-tier architecture subject to TPC-W workloads that are characterized as bursty. These results promote MAP queueing networks as a new robust class of capacity planning models.

Burstiness and temporal dependence in service processes are often found in multi-tier architectures and storage devices and must be captured accurately in capacity planning models as these features are responsible of significant performance degradations. However, existing models and approximations for networks of first-come firstserved (FCFS) queues with general independent (GI) service are unable to predict performance of systems with temporal dependence in workloads. To overcome this difficulty, we define and study a class of closed queueing networks where service times are represented by Markovian Arrival Processes (MAPs), a class of point processes that can model general distributions, but also temporal dependent features such as burstiness in service times. We call these models MAP queueing networks. We introduce provable upper and lower bounds for arbitrary performance indexes (e.g., throughput, response time, utilization) that we call Linear Reduction (LR) bounds. Numerical experiments indicate that LR bounds achieve a mean accuracy error of 2%. The result promotes LR bounds as a versatile and reliable bounding methodology of the performance of modern computer systems.

This paper shows how to construct a Markovian arrival process of second order from information on the marginal distribution and on its autocorrelation function. More precisely, closed-form explicit expressions for the MAP(2) rate matrices are given in terms of the first three marginal moments and one parameter that characterizes the behavior of the autocorrelation function. Besides the permissible moment ranges, which were known before, also the necessary and sufficient bounds for the correlation parameter are computed and shown to depend on a free parameter related to equivalent acyclic PH(2) representations of the marginal distribution. We identify the choices for the free parameter that maximize the correlation range for both negative and positive correlation parameters.

The problem we deal with is the analysis of a class of large structured Markov chains. In particular we assume that the whole state space can be partitioned into disjoint sets (called macro states) in which the process corresponds to the parallel execution of independent jobs. Petri nets and process algebras with phase type (PH) distributed execution times give rise to this kind of model. These models are subject to the phenomenon of state space explosion. It is known that the infinitesimal generator of such models can be handled in a memory efficient way by storing only the “structure ” of the infinitesimal generator as Kronecker expressions or decision diagrams. Less is known instead on how to perform the analysis of the model in a memory efficient manner because in case of most of the available methods the vector of transient or steady state probabilities are stored in an explicit manner. In this paper we consider the calculation of measures connected to the probability that the process passes through a given series of macrostates. We show that such measures can be calculated in a memory efficient manner by Laplace transform techniques. The method is illustrated by numerical examples.

This paper defines Structured Markovian Arrival ... blocks each being represented by a random variable specifying the duration of staying in that block. Leaving a block indicates an arrival event of the SMAP. The routing between blocks is governed by a stochastic matrix Q. It is shown that the joint moments of the SMAP can be directly determined from the moments of the block random variables and routing matrix Q, if Q is doubly stochastic. The characteristics of the SMAP can be computed very efficiently if Q is in addition circulant. Furthermore we show that for given block random variables the determination of a routing matrix Q and thus the fitting of the SMAP essentially results in solving a set of linear equations.