Results 1  10
of
13
Moment characterization of matrix exponential and Markovian arrival processes
 ANN OPER RES
"... This paper provides a general framework for establishing the relation between various moments of matrix exponential and Markovian processes. Based on this framework we present an algorithm to compute any finite dimensional moments of these processes based on a set of required (low order) moments. Th ..."
Abstract

Cited by 7 (2 self)
 Add to MetaCart
This paper provides a general framework for establishing the relation between various moments of matrix exponential and Markovian processes. Based on this framework we present an algorithm to compute any finite dimensional moments of these processes based on a set of required (low order) moments. This algorithm does not require the computation of any representation of the given process. We present a series of related results and numerical examples to demonstrate the potential use of the obtained moment relations.
Trace Data Characterization and Fitting for Markov Modeling
 Elsevier Performance Evaluation
, 2010
"... We propose a trace fitting algorithm for Markovian Arrival Processes (MAPs) that can capture statistics of any order of interarrival times between measured events. By studying real traffic and workload traces often used in performance evaluation studies, we show that matching higher order statistica ..."
Abstract

Cited by 6 (3 self)
 Add to MetaCart
We propose a trace fitting algorithm for Markovian Arrival Processes (MAPs) that can capture statistics of any order of interarrival times between measured events. By studying real traffic and workload traces often used in performance evaluation studies, we show that matching higher order statistical properties, in addition to first and second order descriptors, results in increased queueing prediction accuracy with respect to algorithms that only match the mean, the coefficient of variation, and the autocorrelations of the trace. This result supports the approach of modeling traces by the interarrival time process instead of the counting process that is more frequently used in previous work. We proceed by first characterizing the general properties of MAPs using a spectral approach. Based on this result, we show how different MAPs can be combined together using Kronecker products to define a larger MAP with predefined properties of interarrival times. We then devise an algorithm that is based on this Kronecker composition and can accurately fit data traces. This MAP fitting algorithm uses nonlinear optimization that can be customized to fit an arbitrary number of moments and to meet the desired costaccuracy tradeoff. Numerical results of the fitting algorithm on real data, such as the Bellcore Aug89 trace and a Seagate disk drive trace, indicate that the proposed fitting technique achieves increased prediction accuracy with respect to other stateoftheart fitting methods.
An empirical comparison of MAP fitting algorithms.
 In Proceedings of MMB & DFT
, 2010
"... Abstract The paper presents an empirical comparison of different methods to fit the parameters of a MAP according to the quantities derived from three different real traces. The results indicate that for two of the three traces an adequate fitting with low order MAPs is possible whereas almost all ..."
Abstract

Cited by 3 (3 self)
 Add to MetaCart
(Show Context)
Abstract The paper presents an empirical comparison of different methods to fit the parameters of a MAP according to the quantities derived from three different real traces. The results indicate that for two of the three traces an adequate fitting with low order MAPs is possible whereas almost all approaches failed for the third trace. Apart form this the question for the best approach for fitting MAPs is still open although there seems to be a tendency that the most costly EM algorithms provide the best fitting results.
On capturing dependence in point processes: Matching moments and other techniques
, 2010
"... Providing probabilistic analysis of queueing models can be difficult when the input distributions are nonMarkovian. In response, a plethora of methods have been developed to approximate a general renewal process by a process with the time between renewals being distributed as a phase type random va ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
(Show Context)
Providing probabilistic analysis of queueing models can be difficult when the input distributions are nonMarkovian. In response, a plethora of methods have been developed to approximate a general renewal process by a process with the time between renewals being distributed as a phase type random variable, which allows the resulting queueing models to become analytically or numerically tractable. However, from previous studies on the manufacturing sector, and more recently in analysis of telecommunications systems, assumptions of independence do not always hold and efforts have been made to approximate nonrenewal processes with Markovian Arrival Processes. In this paper we survey techniques for deriving the appropriate parameters of a Markovian process to accurately capture relevant characteristics of the original point process.
KPCToolbox: Best Recipes for Automatic Trace Fitting Using Markovian Arrival Processes
"... We present the KPCToolbox, a library of MATLAB scripts for fitting workload traces into Markovian Arrival Processes (MAPs) in an automatic way based on the recently proposed Kronecker Product Composition (KPC) method. We first present detailed sensitivity analysis that builds intuition on which tra ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
(Show Context)
We present the KPCToolbox, a library of MATLAB scripts for fitting workload traces into Markovian Arrival Processes (MAPs) in an automatic way based on the recently proposed Kronecker Product Composition (KPC) method. We first present detailed sensitivity analysis that builds intuition on which trace descriptors are the most important for queueing performance, stressing the advantages of matching higherorder correlations of the process rather than higherorder moments of the distribution. 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). The KPCToolbox then automatically derives a MAP that captures accurately the most essential features of the trace. Extensive experimentation illustrates the effectiveness of the KPCToolbox in fitting traces that are welldocumented in the literature as very challenging to fit, showing that the KPCToolbox offers a simple and powerful solution to fitting accurately trace data into MAPs. We provide a characterization of moments and correlations that can be fitted exactly by KPC, thus proving the wider applicability of the method compared to small order MAPs. We also consider the fitting of phasetype (PHtype) distributions, which are an important specialization of MAPs that are useful for describing traces without correlations in their time series. We illustrate that the KPC methodology can be easily adapted to PHtype fitting and present experimental results on networking and disk drive traces showing that the KPCToolbox can also match accurately higherorder moments of the interarrival times in place of correlations.
ANALYSIS OF BMAP VACATION QUEUE AND ITS APPLICATION TO IEEE 802.16E SLEEP MODE
"... (Communicated by the associate editor name) Abstract. The paper deals with the continuoustime BMAP/G/1 queue with multiple vacations and with its application to IEEE 802.16e sleep mode. The lengths of the vacation periods have general distribution and they depend on the number of preceding vacation ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
(Show Context)
(Communicated by the associate editor name) Abstract. The paper deals with the continuoustime BMAP/G/1 queue with multiple vacations and with its application to IEEE 802.16e sleep mode. The lengths of the vacation periods have general distribution and they depend on the number of preceding vacations (dependent multiple vacation). We give the expressions for the vector generating function of the stationary number of customers and its mean. Moreover we obtain new formulas for the vector LaplaceStieljes transform of the stationary virtual waiting time and for its first two moments in case of FirstCome FirstServe scheduling. We apply this vacation model to the IEEE 802.16e sleep mode mechanism, and we evaluate its performance as a function of the traffic intensity and the traffic correlation parameter. We give an example for determining the best sleep mode parameters for a simple optimization criteria and we also develop a cost model for the more general case. For traffic modeling we use a twophase Markovian Arrival Process, which is appropriate to model a fairly general correlated traffic. 1. Introduction. Queueing
ON THE IDENTIFIABILITY OF THE TWOSTATE BMAP
, 2012
"... The capability of modeling nonexponentially distributed and dependent interarrival times as well as correlated batches makes the Batch Markovian Arrival Processes (BMAP) suitable in different reallife settings as teletraffic, queueing theory or actuarial contexts. An issue to be taken into accoun ..."
Abstract
 Add to MetaCart
(Show Context)
The capability of modeling nonexponentially distributed and dependent interarrival times as well as correlated batches makes the Batch Markovian Arrival Processes (BMAP) suitable in different reallife settings as teletraffic, queueing theory or actuarial contexts. An issue to be taken into account for estimation purposes is the identifiability of the process. This is an open problem concerning BMAPrelated processes. This paper explores the identifiability issue of the twostate BMAP noted BMAP2(k), where k is the maximum batch arrival size. It is proven that for k = 2 the process cannot be identified, under the assumptions that both the interarrival times and batches sizes are observed. Additionally, a method to obtain an equivalent BMAP2(k), to a given one is provided.
PhaseType Distribution (PH)
"... Main topics covered in this tutorial: • Phasetype (PH) distributions • Moment matching • Markovian arrival processes (MAP) • Interarrival process fitting Important topics not covered in this tutorial: • Queueing applications, matrixgeometric method,... • NonMarkovian workload models (e.g., Paret ..."
Abstract
 Add to MetaCart
(Show Context)
Main topics covered in this tutorial: • Phasetype (PH) distributions • Moment matching • Markovian arrival processes (MAP) • Interarrival process fitting Important topics not covered in this tutorial: • Queueing applications, matrixgeometric method,... • NonMarkovian workload models (e.g., Pareto, matrix exponential process, ARMA processes, fBm, wavelets,...) • MaximumLikelihood (ML) methods, EM algorithm,...
Fitting SecondOrder Acyclic Marked Markovian Arrival Processes
"... Abstract—Markovian Arrival Processes (MAPs) are a tractable class of pointprocesses useful to model correlated time series, such as those commonly found in network traces and system logs used in performance analysis and reliability evaluation. Marked MAPs (MMAPs) generalize MAPs by further allowing ..."
Abstract
 Add to MetaCart
Abstract—Markovian Arrival Processes (MAPs) are a tractable class of pointprocesses useful to model correlated time series, such as those commonly found in network traces and system logs used in performance analysis and reliability evaluation. Marked MAPs (MMAPs) generalize MAPs by further allowing the modeling of multiclass traces, possibly with crosscorrelation between multiclass arrivals. In this paper, we present analytical formulas to fit secondorder acyclic MMAPs with an arbitrary number of classes. We initially define closedform formulas to fit secondorder MMAPs with two classes, where the underlying MAP is in canonical form. Our approach leverages forward and backward moments, which have recently been defined, but never exploited jointly for fitting. Then, we show how to sequentially apply these formulas to fit an arbitrary number of classes. Representative examples and tracedriven simulation using storage traces show the effectiveness of our approach for fitting empirical datasets. KeywordsMulticlass workload, point process, dependence I.
Contents lists available at ScienceDirect Performance Evaluation
"... journal homepage: www.elsevier.com/locate/peva A joint moments based analysis of networks of MAP/MAP/1 queues ✩ ..."
Abstract
 Add to MetaCart
(Show Context)
journal homepage: www.elsevier.com/locate/peva A joint moments based analysis of networks of MAP/MAP/1 queues ✩