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## An empirical comparison of MAP fitting algorithms. (2010)

Venue: | In Proceedings of MMB & DFT |

Citations: | 3 - 3 self |

### Citations

3099 |
Numerical Recipes in C++: The Art of Scientific Computing
- Press, Teukolsky, et al.
- 2005
(Show Context)
Citation Context ...ed such that the autocorrelations ρ = (ρ1, · · · , ρn) of the MAP (D0,D1) (cf. Eq. 6) approximate the autocorrelations ρ = (ρ1, · · · , ρn) that have been estimated from the trace, i.e. we have to solve the following minimization problem: min D1:D1≥0, D1 1I=−D0 1I, πMD1=π ( n∑ i=1 (βi|ρi − ρi|) ) (11) where the βi are weights which again may be used to privilege lower lag autocorrelations. In this paper, we use a slightly modified approach of the two step algorithm presented in [13]. For minimizing Eq. 11 we use the Nelder-Mead algorithm [16]. An implementation can for example be found in [21]. For a MAP of order n we have n2 variables from matrix D1 and Nelder-Mead requires n2 + 1 initial solutions D (i) 1 , i = 1, · · · , n+ 1. The first initial solution is the MAP representation of the given PH distribution (π,D0), i.e. D (1) 1 = (−D0 1I)π. The possible range for other valid initial solutions is bounded by the constraints on the row sums (−D0 1I = D1 1I) and on the steady-state vector (πP = π). Let x be a vector that contains the first row of matrix D1 in positions 1, · · · , n, the second row in positions n + 1, · · · , 2n etc. Then we can define a linear system of equations us... |

2546 |
A simplex method for function minimization
- Nelder, Mead
- 1965
(Show Context)
Citation Context ...zero entries is maximized. Then matrix D1 is generated such that the autocorrelations ρ = (ρ1, · · · , ρn) of the MAP (D0,D1) (cf. Eq. 6) approximate the autocorrelations ρ = (ρ1, · · · , ρn) that have been estimated from the trace, i.e. we have to solve the following minimization problem: min D1:D1≥0, D1 1I=−D0 1I, πMD1=π ( n∑ i=1 (βi|ρi − ρi|) ) (11) where the βi are weights which again may be used to privilege lower lag autocorrelations. In this paper, we use a slightly modified approach of the two step algorithm presented in [13]. For minimizing Eq. 11 we use the Nelder-Mead algorithm [16]. An implementation can for example be found in [21]. For a MAP of order n we have n2 variables from matrix D1 and Nelder-Mead requires n2 + 1 initial solutions D (i) 1 , i = 1, · · · , n+ 1. The first initial solution is the MAP representation of the given PH distribution (π,D0), i.e. D (1) 1 = (−D0 1I)π. The possible range for other valid initial solutions is bounded by the constraints on the row sums (−D0 1I = D1 1I) and on the steady-state vector (πP = π). Let x be a vector that contains the first row of matrix D1 in positions 1, · · · , n, the second row in positions n + 1, · · · , 2n etc... |

1769 | Wide-Area Traffic: The Failure of Poisson Modeling
- Paxson
- 1995
(Show Context)
Citation Context ...inimization problem is not a simple least squares problem. Depending on the number of lag-k autocorrelations that are considered the approach takes between some seconds and few minutes, e.g. to fit the first 30 lags for LBL-TCP-3 with a MAP of order 5 the algorithm required less than 10 seconds, for the first 100 lags it took 2 minutes. 7 4 Experimental Results To compare the different fitting algorithms we use three different traces. The trace BC-pAug89 contains a million packet arrivals observed at the Bellcore Morristown Research and Engineering facility in August 1989. The trace LBL-TCP-3 [20] contains two hours of TCP traffic from the Lawrence Berkeley Laboratory and was recorded in January 1994. Both traces are taken from the Internet Traffic Archive [1]. The third trace TUDo contains the interarrival times of one million packets that have been measured from the Squid proxy server at the Computer Science Department of TU Dortmund in 2006. We fitted MAPs of different order (from n = 2 to n = 6) with the three fitting approaches from Sec. 3. Since fitting according to joint moments and autocorrelations both require a given distribution that is fitted in the first step, we used Gfit... |

924 |
Solving Least-Squares Problems
- Lawson, Hanson
- 1974
(Show Context)
Citation Context ... βk,l µk,l νk,l − βk,l )2 (10) βk,l are some weights which allow one to discriminate higher order joint moments. In our experiments we present later, all weights are set to 1. However, if the resulting MAP cannot match the required moments adequately, it is often appropriate to set the weights such that lower order joint moments get a higher weight, e.g., by choosing βk,l = 2−(k−1)(l−1). In this case, lower order joint moments are often matched exactly or almost exactly with the price of a bad fit for higher order moments. The least squares solution can be computed with available algorithms [15]. The major advantage of joint moment fitting is the efficiency. I.e., to fit the joint moments νk,l with 1 ≤ k, l ≤ 3 for LBL-TCP-3 with a MAP of order 5 requires less than 1 second which is negligible compared to the fitting times of EM-algorithms. 3.3 Fitting of Autocorrelations The approach for the fitting of autocorrelation works similarly to joint moment fitting. In a first step, the initial probability vector π and the matrix D0 are determined by a PH fitting algorithm like [7] or [25] and are transformed such that the number 6 of non zero entries is maximized. Then matrix D1 is generat... |

894 |
Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach
- Neuts
- 1981
(Show Context)
Citation Context ...ntically and independently distributed are not sufficient to describe real behavior, instead stochastic processes have to be used to model the distribution and the autocorrelation structure. Markovian arrival processes (MAPs) [17] are stochastic processes which can be applied to capture a wide range of different stochastic behaviors and can be used in queuing network models as arrival or service processes. Queuing networks with MAPs can be analyzed numerically by solving the global balance equations [23], if the state space is not too large, they can be analyzed with matrix analytical methods [18], if they are of the MAP/MAP/m type, they may as well be analyzed approximately [9] or by simulation. To capture real behavior by MAPs, the parameters of a MAP have to be fitted according to some trace resulting from observations or measured behavior. The fitting problem of MAPs is a nonlinear optimization problem which becomes even more complex since the matrix representation of MAPs is redundant [24] and a canonical representation is only available for MAPs of order two [3]. Different fitting approaches have been proposed in the literature which all have their pros and cons. The most general... |

750 |
Introduction to the numerical solution of Markov chains
- Stewart
- 1994
(Show Context)
Citation Context ...t many real processes include some correlation which implies that random variables that are identically and independently distributed are not sufficient to describe real behavior, instead stochastic processes have to be used to model the distribution and the autocorrelation structure. Markovian arrival processes (MAPs) [17] are stochastic processes which can be applied to capture a wide range of different stochastic behaviors and can be used in queuing network models as arrival or service processes. Queuing networks with MAPs can be analyzed numerically by solving the global balance equations [23], if the state space is not too large, they can be analyzed with matrix analytical methods [18], if they are of the MAP/MAP/m type, they may as well be analyzed approximately [9] or by simulation. To capture real behavior by MAPs, the parameters of a MAP have to be fitted according to some trace resulting from observations or measured behavior. The fitting problem of MAPs is a nonlinear optimization problem which becomes even more complex since the matrix representation of MAPs is redundant [24] and a canonical representation is only available for MAPs of order two [3]. Different fitting appro... |

128 |
Fitting Phase-type distributions via the EM algorithm
- Asmussen, Nerman
- 1991
(Show Context)
Citation Context ...ture real behavior by MAPs, the parameters of a MAP have to be fitted according to some trace resulting from observations or measured behavior. The fitting problem of MAPs is a nonlinear optimization problem which becomes even more complex since the matrix representation of MAPs is redundant [24] and a canonical representation is only available for MAPs of order two [3]. Different fitting approaches have been proposed in the literature which all have their pros and cons. The most general approach is to find a MAP that maximizes the likelihood according to the available trace. The EM algorithm [2] can be used for this purpose and many specific variants of the algorithm for MAP fitting are available [4, 5, 14, 22]. However, EM algorithms have several disadvantages since they have a slow convergence, may converge towards local minima and require a huge effort that grows linearly in the length of a trace. Since for MAP fitting very long 2 traces are required to adequately match the autocorrelation structure, in practice, EM approaches are not sufficient to obtain good fitting result with an acceptable effort. Alternative approaches first derive some quantities from a trace, like higher or... |

74 |
A versatile markovian point process
- Neuts
- 1979
(Show Context)
Citation Context ...en although there seems to be a tendency that the most costly EM algorithms provide the best fitting results. 1 Introduction In stochastic modeling, the appropriate representation of arrival and service processes is of major importance to build realistic models. It turns out that many real processes include some correlation which implies that random variables that are identically and independently distributed are not sufficient to describe real behavior, instead stochastic processes have to be used to model the distribution and the autocorrelation structure. Markovian arrival processes (MAPs) [17] are stochastic processes which can be applied to capture a wide range of different stochastic behaviors and can be used in queuing network models as arrival or service processes. Queuing networks with MAPs can be analyzed numerically by solving the global balance equations [23], if the state space is not too large, they can be analyzed with matrix analytical methods [18], if they are of the MAP/MAP/m type, they may as well be analyzed approximately [9] or by simulation. To capture real behavior by MAPs, the parameters of a MAP have to be fitted according to some trace resulting from observati... |

33 | A minimal representation of Markov arrival processes and a moments matching method, Performance Evaluation 64
- Telek, Horváth
- 2007
(Show Context)
Citation Context ...ses. Queuing networks with MAPs can be analyzed numerically by solving the global balance equations [23], if the state space is not too large, they can be analyzed with matrix analytical methods [18], if they are of the MAP/MAP/m type, they may as well be analyzed approximately [9] or by simulation. To capture real behavior by MAPs, the parameters of a MAP have to be fitted according to some trace resulting from observations or measured behavior. The fitting problem of MAPs is a nonlinear optimization problem which becomes even more complex since the matrix representation of MAPs is redundant [24] and a canonical representation is only available for MAPs of order two [3]. Different fitting approaches have been proposed in the literature which all have their pros and cons. The most general approach is to find a MAP that maximizes the likelihood according to the available trace. The EM algorithm [2] can be used for this purpose and many specific variants of the algorithm for MAP fitting are available [4, 5, 14, 22]. However, EM algorithms have several disadvantages since they have a slow convergence, may converge towards local minima and require a huge effort that grows linearly in the l... |

32 | Phfit: A general Phase-type fitting tool
- Horváth, Telek
- 2002
(Show Context)
Citation Context ...orithm is rather inefficient but if one restricts the class of PH distributions, much more efficient variants can be defined. In [25] an EM algorithm which fits the parameters of a generalized Erlang distribution is shown to be rather efficient. We will use this approach as a first step for MAP fitting. Alternatively, one may fit the PH distribution according to the moments of the trace. In this case acyclic phase type distributions are used since for this subclass a canonical representation exists. Methods for moment fitting which we also apply as a first step for MAP fitting are proposed in [7, 12]. 2.3 Basic Definitions and Results for MAPs A MAP [17] of order n is a stochastic process defined by two n × n matrices (D0,D1) where D0 is as defined for a PH distribution above and D1 ≥ 0 such that Q = D0 +D1 and Q1I = 0. Matrix D0 contains the rates of internal transitions 4 without an arrival and matrix D1 contains the rates of transitions generating an arrival. We assume that Q is an irreducible generator matrix. Define P = −D−10 D1 as the transition matrix of the embedded discrete time Markov chain after an arrival. The stationary vector πP = π, π 1I = 1 includes the distribution just a... |

32 | Modeling IP traffic using the batch Markovian arrival process.
- Klemm, Lindemann, et al.
- 2003
(Show Context)
Citation Context ...m observations or measured behavior. The fitting problem of MAPs is a nonlinear optimization problem which becomes even more complex since the matrix representation of MAPs is redundant [24] and a canonical representation is only available for MAPs of order two [3]. Different fitting approaches have been proposed in the literature which all have their pros and cons. The most general approach is to find a MAP that maximizes the likelihood according to the available trace. The EM algorithm [2] can be used for this purpose and many specific variants of the algorithm for MAP fitting are available [4, 5, 14, 22]. However, EM algorithms have several disadvantages since they have a slow convergence, may converge towards local minima and require a huge effort that grows linearly in the length of a trace. Since for MAP fitting very long 2 traces are required to adequately match the autocorrelation structure, in practice, EM approaches are not sufficient to obtain good fitting result with an acceptable effort. Alternative approaches first derive some quantities from a trace, like higher order moments, joint moments or lag-k autocorrelations and then fit the parameters of a MAP according to these quantitie... |

30 | Markovian modeling of real data traffic: Heuristic phase type and MAP fitting of heavy tailed and fractal like samples.
- Horvath, Telek
- 2002
(Show Context)
Citation Context ...some measured quantities are matched. Usually these quantities result from a trace which is an observation of some real behavior. From a trace different quantities like moments, joint moments, lag-k coefficients of autocorrelation or values of the empirical distribution function or density can be computed. Since a trace is only a sample of some real behavior, all values are estimates. The goal of a fitting approach is to find a PH distribution that matches the quantities of the trace as good as possible. A large number of fitting methods for PH distributions exist, an overview can be found in [11]. We only outline a few approaches which we later use as a first step for MAP fitting. In general one can distinguish between fitting methods that work on the whole trace and those that try to match some quantities derived from the trace. Methods of the former type usually maximize the likelihood value which is defined for a trace t1, . . . tm as L(D0,π)(t1, . . . , tm) = m∏ k=1 πetkD0 (−D0 1I) . (4) Maximization is done with the so called EM algorithm [2]. However, the general variant of this algorithm is rather inefficient but if one restricts the class of PH distributions, much more efficie... |

29 |
An EM algorithm for estimation in Markov-modulated Poisson processes
- Rydén
- 1996
(Show Context)
Citation Context ...m observations or measured behavior. The fitting problem of MAPs is a nonlinear optimization problem which becomes even more complex since the matrix representation of MAPs is redundant [24] and a canonical representation is only available for MAPs of order two [3]. Different fitting approaches have been proposed in the literature which all have their pros and cons. The most general approach is to find a MAP that maximizes the likelihood according to the available trace. The EM algorithm [2] can be used for this purpose and many specific variants of the algorithm for MAP fitting are available [4, 5, 14, 22]. However, EM algorithms have several disadvantages since they have a slow convergence, may converge towards local minima and require a huge effort that grows linearly in the length of a trace. Since for MAP fitting very long 2 traces are required to adequately match the autocorrelation structure, in practice, EM approaches are not sufficient to obtain good fitting result with an acceptable effort. Alternative approaches first derive some quantities from a trace, like higher order moments, joint moments or lag-k autocorrelations and then fit the parameters of a MAP according to these quantitie... |

26 |
A MAP fitting approach with independent approximation of the inter-arrival time distribution and the lag-correlation. In
- Horvath, Telek, et al.
- 2005
(Show Context)
Citation Context ...ce, like higher order moments, joint moments or lag-k autocorrelations and then fit the parameters of a MAP according to these quantities. This implies that fitting becomes independent of the trace length. As shown in [24], a non redundant MAP of order n which is characterized by 2n2 − n free parameters is completely determined by n2 parameters, e.g., by the first 2n− 1 moments and (n− 1)2 joint moments. Thus, one may fit a MAP according to the empirical moments and joint moments of a trace as done in [7]. Other approaches use the lag-k autocorrelation instead of the joint moments for fitting [10, 13]. However, all these approaches have their limitations since in practice n2 parameters of a trace hardly define a MAP. In [7] we used least square fitting to obtain the nearest MAP of order n according to some measured moments and joint moments. It turns out that it is hard to fit even approximately in the range of n2 parameters of a real trace with a MAP of order n. Another problem which is also considered in [7] is the reliability of quantities derived from a trace. In general, a trace is only a sample of the behavior of a system such that the quantities computed from the trace are only esti... |

23 |
Characterization of Phase-type distributions
- O’Cinneide
- 1990
(Show Context)
Citation Context ...lly, we present basic results for MAPs. 3 2.1 Basic Definitions and Results for PH Distributions A PH distribution [18] of order n is defined by a non-singular n × n matrix D0 with D0(i, j) ≥ 0 for i 6= j, D0(i, i) ≤ − ∑n j=1,j 6=iD0(i, j) and a row vector π with π(i) ≥ 0 and π 1I = 1 where 1I is the unit column vector of length n. Let M = (−D0)−1, the so called moment matrix. The distribution function, density and the moments of a random variable X with a PH distribution (D0, π) are given by FX(t) = 1− πetD0 1I (1) fX(t) = πe tD0(−D0 1I) (2) µk = E(X k) = k!π (M) k 1I . (3) It has been shown [19] that every non negative random variable with a continuous density that is non-zero in (0,∞) can be approximated arbitrarily close by a PH distribution. 2.2 Fitting Methods for PH distributions The task of fitting PH distributions is to choose the parameters of a PH distribution in such a way that some measured quantities are matched. Usually these quantities result from a trace which is an observation of some real behavior. From a trace different quantities like moments, joint moments, lag-k coefficients of autocorrelation or values of the empirical distribution function or density can be com... |

21 | A novel approach for phase-type fitting with the EM algorithm.
- Thummler, Buchholz, et al.
- 2006
(Show Context)
Citation Context ...hes which we later use as a first step for MAP fitting. In general one can distinguish between fitting methods that work on the whole trace and those that try to match some quantities derived from the trace. Methods of the former type usually maximize the likelihood value which is defined for a trace t1, . . . tm as L(D0,π)(t1, . . . , tm) = m∏ k=1 πetkD0 (−D0 1I) . (4) Maximization is done with the so called EM algorithm [2]. However, the general variant of this algorithm is rather inefficient but if one restricts the class of PH distributions, much more efficient variants can be defined. In [25] an EM algorithm which fits the parameters of a generalized Erlang distribution is shown to be rather efficient. We will use this approach as a first step for MAP fitting. Alternatively, one may fit the PH distribution according to the moments of the trace. In this case acyclic phase type distributions are used since for this subclass a canonical representation exists. Methods for moment fitting which we also apply as a first step for MAP fitting are proposed in [7, 12]. 2.3 Basic Definitions and Results for MAPs A MAP [17] of order n is a stochastic process defined by two n × n matrices (D0,D... |

19 | KPC-Toolbox: Simple Yet Effective Trace Fitting Using Markovian Arrival Processes. In
- Casale, Zhang, et al.
- 2008
(Show Context)
Citation Context ...ce, like higher order moments, joint moments or lag-k autocorrelations and then fit the parameters of a MAP according to these quantities. This implies that fitting becomes independent of the trace length. As shown in [24], a non redundant MAP of order n which is characterized by 2n2 − n free parameters is completely determined by n2 parameters, e.g., by the first 2n− 1 moments and (n− 1)2 joint moments. Thus, one may fit a MAP according to the empirical moments and joint moments of a trace as done in [7]. Other approaches use the lag-k autocorrelation instead of the joint moments for fitting [10, 13]. However, all these approaches have their limitations since in practice n2 parameters of a trace hardly define a MAP. In [7] we used least square fitting to obtain the nearest MAP of order n according to some measured moments and joint moments. It turns out that it is hard to fit even approximately in the range of n2 parameters of a real trace with a MAP of order n. Another problem which is also considered in [7] is the reliability of quantities derived from a trace. In general, a trace is only a sample of the behavior of a system such that the quantities computed from the trace are only esti... |

18 |
An EM-algorithm for MAP fitting from real traffic data.
- Buchholz
- 2003
(Show Context)
Citation Context ...m observations or measured behavior. The fitting problem of MAPs is a nonlinear optimization problem which becomes even more complex since the matrix representation of MAPs is redundant [24] and a canonical representation is only available for MAPs of order two [3]. Different fitting approaches have been proposed in the literature which all have their pros and cons. The most general approach is to find a MAP that maximizes the likelihood according to the available trace. The EM algorithm [2] can be used for this purpose and many specific variants of the algorithm for MAP fitting are available [4, 5, 14, 22]. However, EM algorithms have several disadvantages since they have a slow convergence, may converge towards local minima and require a huge effort that grows linearly in the length of a trace. Since for MAP fitting very long 2 traces are required to adequately match the autocorrelation structure, in practice, EM approaches are not sufficient to obtain good fitting result with an acceptable effort. Alternative approaches first derive some quantities from a trace, like higher order moments, joint moments or lag-k autocorrelations and then fit the parameters of a MAP according to these quantitie... |

16 |
An EM algorithm for batch Markovian arrival processes and its comparison to a simpler estimation procedure. Annals OR,
- Breuer
- 2002
(Show Context)
Citation Context |

12 | A markovian canonical form of second-order matrix-exponential processes
- Bodrog, Heindl, et al.
(Show Context)
Citation Context ...lobal balance equations [23], if the state space is not too large, they can be analyzed with matrix analytical methods [18], if they are of the MAP/MAP/m type, they may as well be analyzed approximately [9] or by simulation. To capture real behavior by MAPs, the parameters of a MAP have to be fitted according to some trace resulting from observations or measured behavior. The fitting problem of MAPs is a nonlinear optimization problem which becomes even more complex since the matrix representation of MAPs is redundant [24] and a canonical representation is only available for MAPs of order two [3]. Different fitting approaches have been proposed in the literature which all have their pros and cons. The most general approach is to find a MAP that maximizes the likelihood according to the available trace. The EM algorithm [2] can be used for this purpose and many specific variants of the algorithm for MAP fitting are available [4, 5, 14, 22]. However, EM algorithms have several disadvantages since they have a slow convergence, may converge towards local minima and require a huge effort that grows linearly in the length of a trace. Since for MAP fitting very long 2 traces are required to ... |

7 | A heuristic approach for fitting MAPs to moments and joint moments.
- Buchholz, Kriege
- 2009
(Show Context)
Citation Context ...ult with an acceptable effort. Alternative approaches first derive some quantities from a trace, like higher order moments, joint moments or lag-k autocorrelations and then fit the parameters of a MAP according to these quantities. This implies that fitting becomes independent of the trace length. As shown in [24], a non redundant MAP of order n which is characterized by 2n2 − n free parameters is completely determined by n2 parameters, e.g., by the first 2n− 1 moments and (n− 1)2 joint moments. Thus, one may fit a MAP according to the empirical moments and joint moments of a trace as done in [7]. Other approaches use the lag-k autocorrelation instead of the joint moments for fitting [10, 13]. However, all these approaches have their limitations since in practice n2 parameters of a trace hardly define a MAP. In [7] we used least square fitting to obtain the nearest MAP of order n according to some measured moments and joint moments. It turns out that it is hard to fit even approximately in the range of n2 parameters of a real trace with a MAP of order n. Another problem which is also considered in [7] is the reliability of quantities derived from a trace. In general, a trace is only a... |

7 |
A Two-Step EM Algorithm for MAP Fitting.
- Buchholz, Panchenko
- 2004
(Show Context)
Citation Context ...1 1I− 1 2µ−21 π(−D0)−1(−D0)−1 1I− 1 (6) and the joint density of the first m interarrival times is defined as f(τ1, . . . , τm) = π ( m∏ i=1 eτiD0D1 ) 1I . (7) Fitting methods as introduced in the subsequent section try to approximate the empirical measures of a trace by a MAP. As for fitting PH distributions either the complete trace may be used resulting in the maximization of the likelihood L(D0,D1)(t1, . . . , tm) = π ( m∏ k=1 etkD0D1 ) 1I . (8) or some derived quantities like joint moments or lag-k autocorrelations may be used for fitting. One approach which has been applied successfully [7, 8, 13] is to separate distribution and dependency fitting. In a first step, a PH distribution is generated that captures the distribution of the elements in the trace and in a second step the distribution is expanded into a MAP by considering the dependencies in the trace. This expansion implies that matrix D0 remains unchanged and D1 is chosen such that −D0 1I = D1 1I and πMD1 = π which puts 2n constraints for the elements of D1. 3 MAP Fitting Approaches 3.1 Expectation Maximization We begin with a brief look on EM algorithms for MAP fitting and refer for the details of the approaches to the litera... |

3 |
Equivalence transformations for acyclic phase type distributions.
- Buchholz, Kriege
- 2009
(Show Context)
Citation Context ...onvergence. 3.2 Fitting of Joint Moments If fitting of the distribution and the autocorrelation structure are done separately, then the matrix D0 and vector π result from distribution fitting. Since we use the moment fitting approach from [7] or the EM algorithm of [25] for distribution fitting, the result is in both cases an acyclic PH distribution with an upper triangular matrix D0. Acyclic PH distributions of order n have n(n+1)/2+(n−1) free parameters but only 2n−1 parameters are necessary to characterize the distribution such that different representations of the same distribution exist. [6] summarizes three methods to perform equivalence transformations that generate different acyclic representations of the same distribution. For MAP fitting the number of non zero entries in π and D0 1I has to be maximized to maximize the number of possible non zero entries in D1. However, even with this goal the transformation is non unique and different approaches may be tried. Define vk = πMk+1 and wk = Mk 1I, then µk,l = k! l! v kD1w l . (9) Now assume that J is a set of joint moments that should be matched by the MAP and let for (k, l) ∈ J νk,l be the joint moments of the trace. Then the fo... |

3 |
MPA-AMVA: Approximate mean value analysis of bursty system.
- Casale, Smirni
- 2009
(Show Context)
Citation Context ... instead stochastic processes have to be used to model the distribution and the autocorrelation structure. Markovian arrival processes (MAPs) [17] are stochastic processes which can be applied to capture a wide range of different stochastic behaviors and can be used in queuing network models as arrival or service processes. Queuing networks with MAPs can be analyzed numerically by solving the global balance equations [23], if the state space is not too large, they can be analyzed with matrix analytical methods [18], if they are of the MAP/MAP/m type, they may as well be analyzed approximately [9] or by simulation. To capture real behavior by MAPs, the parameters of a MAP have to be fitted according to some trace resulting from observations or measured behavior. The fitting problem of MAPs is a nonlinear optimization problem which becomes even more complex since the matrix representation of MAPs is redundant [24] and a canonical representation is only available for MAPs of order two [3]. Different fitting approaches have been proposed in the literature which all have their pros and cons. The most general approach is to find a MAP that maximizes the likelihood according to the available... |