A. Bobbio and M. Telek. A benchmark for PH estimation algorithms: result for acyclic-PH. Stochastic models, 10(661677) , 1994.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....distribution can be approximated by phase type distributions, the price for introducing phase type distribution is an enlargement of the state space. In recent years, several classes of SPN models have been elaborated which incorporate some non exponential characteritics in their definition [16, 7, 26, 3, 6]. 5.4.2 Fluid Stochastic Petri Nets [14] Fluid Stochastic Petri Nets (FSPN) extends the SPNs by introducing real (positive) tokens to special continuous places. The places are partitioned into a set of discrete places containing an integer number of tokens and a set of fluid (or continuous) ....

A. Bobbio, M. Telek, " A Benchmark for PH Estimation Algorithm: Results for AcyclicPH ", Stochastic Models, vol. 10, pp. 661-667, 1994.

No context found.

A. Bobbio and M. Telek. A benchmark for PH estimation algorithms: results for Acyclic-PH. Stochastic Models, 10:661--677, 1994.

No context found.

A. Bobbio and M. Telek. A benchmark for PH estimation algorithms: results for Acyclic-PH. Stochastic Models, 10:661--677, 1994.

No context found.

A. Bobbio and M. Telek. A benchmark for PH estimation algorithms: results for Acyclic-PH. Stochastic Models, 10:661--677, 1994.

....and the column vector contain the rates from each state into the absorbing state. The distribution function and the pdf of a PH with the above parameters are: # ( # (2) where is a column vector with 1 in its each position. Studies show ([22] [24] that most distributions can be accurately approximated by an appropriately chosen Phase Type distribution. Moreover, if measurement data is available for a random variable with unknown distribution, a properly chosen PH can be fitted according the data series and this PH approximates the ....

A. Bobbio and M. Telek, "A benchmark for PH estimation algorithms: results for Acyclic-PH," Stochastic Models, vol. 10, pp. 661--677, 1994.

....discrete [Neuts, 1975] phase type (PH) distributions were formalized already in the 1970s. Since then, many research activities and application oriented work have been devoted to this field of stochas tic modeling with emphasis on the continuous PH distributions (e.g. Aldous and Shepp, 1987, Bobbio and Telek, 1994, O Cinneide, 1990, Neuts, 1992, Johnson and Taaffe, 1990] In recent years, discrete PH distributions have attracted increasing attention, because their relation to physical observations and their usefulness in the numerical solution of non Markovian processes have been oh served. ....

A. Bobbio and M. Telek, "A benchmark for PH estimation algorithms: Results for acyclic-PH", Commun. Statist.- Stochastic Models, 10(3):661-677, 1994.

....is determined by the initial probability vector # and the infinitesimal generator matrix # and a column vector # . Matrix # contains the rates among the non absorbing states of the Markov chain and the column vector # contain the rates from each state into the absorbing state. Studies show ([21] [23] that most distributions can be accurately approximated by an appropriately chosen Phase Type distribution. Moreover, if measurement data is available about a random variable with unknown distribution, a properly chosen PH can be fitted according the data series and this PH approximates the ....

A. Bobbio and M. Telek, "A benchmark for PH estimation algorithms: results for Acyclic-PH," Stochastic Models, vol. 10, pp. 661--677, 1994.

.... has motivated the present paper whose aim is to investigate more closely the properties of the DPH family and to provide results that can be pro tably exploited for the implementation of an algorithm to estimate the model parameters given an assigned distribution or a set of experimental points [4]. The DPH representation of a given distribution function is, in general, non unique [13] and non minimal. Hence, we rst explore a subclass of the DPH class for which the representation is an acyclic graph (Acyclic DPH ADPH) and we show that, similarly to the continuous case [8] the ADPH class ....

....case [3] while a novel time domain version is provided. It is shown that the time domain algorithm is easier to implement and more stable. The algorithm is then tested on a benchmark of 10 di erent continuous distributions that have been already utilized for a similar study in the continuous case [4]. However, since a continuous distribution needs to be discretized in order to feed the tting algorithm, the role of the discretization interval on the performance of the algorithm and on the goodness of the t is extensively discussed. The structure of the paper is as follows. Section 2 ....

[Article contains additional citation context not shown here]

A. Bobbio and M. Telek. A benchmark for PH estimation algorithms: results for Acyclic-PH. Stochastic Models, 10:661-677, 1994.

....1 Introduction Continuous [13] and discrete [15] phase type (PH) distributions were formalized already in the 1970s. Since then, many research activities and application oriented work have been devoted to this eld of stochastic modeling with emphasis on the continuous PH distributions (e.g. [1, 5, 16, 14, 10]) In recent years, discrete PH distributions have attracted increasing attention, because their relation to physical observations and their usefulness in the numerical solution of non Markovian processes have been observed. Approximating general distributions possibly given only partially in ....

A. Bobbio and M. Telek. A benchmark for PH estimation algorithms: results for Acyclic-PH. Stochastic Models, 10:661-677, 1994.

....to be approximated is less than 1=n, should satisfy the following relation (see Theorem 3) X) E(X) 8) Let X be a Lognormal r.v. with parameters (1; 0:2) whose mean is E(X) 1 and cv (X) 0:0408 (this distribution is the distribution L3 taken from the benchmark examined in [5, 4], hence we refer to it as L3) Table 1 reports the lower and upper bounds of , with n = 2; 4; 8; 12, computed from (8) and (7) The cdf and pdf of the approximating CPH and DPH distributions of order n = 10, with different scale factors , are presented in Figure 6. When considering the ....

....a low coefficient of variation by means of a DPH distributions. In this subsection, we investigate the optimal value of when fitting distributions with a high coefficient of variation. Let X be a Lognormal r.v. with parameters (1; 1:8) this is the distribution L1 taken from the benchmark in [5, 4]) For X we have E(X) 1 and cv (X) 24:534. Figure 8 shows the measure of the goodness of fit as a function of for various orders n (the cases when the number of phases are greater than 2 result in practically the same goodness of fit) The distance measures D decreases as 0 indicating ....

[Article contains additional citation context not shown here]

A. Bobbio and M. Telek. A benchmark for PH estimation algorithms: results for Acyclic-PH. Stochastic Models, 10:661--677, 1994.

....numerical techniques for MAP D 1( K) queues (e.g. 14] were integrated into the proposed decomposition methodology, but are not treated in this paper due to limited space. Canonical PH type tting If c 2 X 0:5, moment tting can be performed by means of a canonical PH type distribution [3] of order 2 (i.e. m = 2) Its representation ( T ) is given by = p; 1 p) and T = 1 1 0 2 : 2) Note that in the canonical form 2 1 . By xing p and solving the explicit equations of the rst two moments of the canonical PH(2) type distribution for the rates ....

A. Bobbio and M. Telek. A benchmark for PH estimation algorithms: Results for acyclic-PH. Commun. Statist.{Stochastic Models, 10(3):661-677, 1994.

....the original non Markovian marking process is approximated by means of a CTMC defined over an augmented state space. According to the definitions given in Section 2, the expansion technique can be realized by assigning to each transition a continuous Phase type (PH) distributed random variable [96, 12]. The merit of this approach is the flexibility in modeling any combination of prd and prs memory policies and any number of concurrent or conflicting transitions with generally distributed firing times. Moreover, the expansion technique can be easily implemented by a computer program, starting ....

....variables and generating the expanded CTMC obtained by combining the reachable states with all the phases of the PH distributions associated with the enabled transitions. An overview of the methods and tools available to estimate the parameters of a PH distribution from a given cdf can be found in [12]. The expansion algorithm can be performed automatically by a computer program, and is driven by the execution policy associated with different transitions [48] The result of the expansion algorithm is that each marking of the original PN is blown into a macrostate in the new state space. When ....

[Article contains additional citation context not shown here]

A. Bobbio and M. Telek. A benchmark for PH estimation algorithms: results for Acyclic-PH. Stochastic Models, 10:661--677, 1994.

....dominant and other methods can outperform it. The need to compare the properties of di erent tting approaches was recognized a decade ago, and a set of tests was de ned during the workshop on Fitting Phase type distributions, Aalborg, Denmark, organized by S. Asmussen in February 1991. In [4] the proposed set of tests was evaluated using the MLAPH method and some new measures were proposed to be considered as well. In [9] a wider set of tting methods was compared and their tting measures were evaluated. Some of the consequences are quite natural. The methods that intend to minimize ....

....section gives the conclusion. Several numerical results are given in the Appendix. 2 Fitting parameters and distance measures Participants of the Aalborg workshop proposed a set of parameters to measure the goodness of Phase type tting methods. The original set of parameters was extended in [4] and the weakness of some measures proposed at the workshop was reported as well. Later on the following set of (non negative) parameters was commonly used (e.g. in [9] 1. Relative error in the 1st moment: e 1 = j c 1 ( F ) c 1 (F ) j = c 1 (F ) 2. Relative error in the 2nd moment: e ....

[Article contains additional citation context not shown here]

A. Bobbio and M. Telek. A benchmark for PH estimation algorithms: results for Acyclic-PH. Stochastic Models, 10:661-677, 1994.

....activities. Moreover, if the random variables of the system to be modeled are really of PH type, the PHSPN provides exact results. Otherwise, a preliminary step is needed in which the random times of the system are approximated by PH random variables resorting to a suitable estimation technique [8]. The expansion of the state space is, of course, a cause of non negligible diculties, since it worsens the problem of the exponential growth of the 23 state space both with the model complexity, and with the order of the PH distribution assigned to each transition. The MRSPN model, combining ....

A. Bobbio and M. Telek. A benchmark for PH estimation algorithms: results for Acyclic-PH. Stochastic Models, 10:661-677, 1994.

No context found.

A. Bobbio and M. Telek. A benchmark for PH estimation algorithms: results for Acyclic-PH. Stochastic Models, 10:661--677, 1994.

....the original non Markovian marking process is approximated by means of a CTMC de ned over an augmented state space. According to the de nitions given in Section 2, the expansion technique can be realized by assigning to each transition a continuous Phase type (PH) distributed random variable [102, 16]. The merit of this approach is the exibility in modeling any combination of prd and prs memory policies and any number of concurrent or con icting transitions with generally distributed ring times. Moreover, the expansion technique can be easily implemented by a computer program, starting from ....

....variables and generating the expanded CTMC obtained by combining the reachable states with all the phases of the PH distributions associated with the enabled transitions. An overview of the methods and tools available to estimate the parameters of a PH distribution from a given cdf can be found in [16]. The expansion algorithm can be performed automatically by a computer program, and is driven by the execution policy associated with di erent transitions [53] The result of the expansion algorithm is that each marking of the original PN is blown into a macrostate in the new state space. When ....

[Article contains additional citation context not shown here]

A. Bobbio and M. Telek. A benchmark for PH estimation algorithms: results for Acyclic-PH. Stochastic Models, 10:661-677, 1994.

....until a Markov chain reaches an absorbing state. So a PH is determined by the infinitesimal generator matrix and the initial distribution of the describing Markov chain. The use of PH distribution has two advantages. One is that PH distributions are widely use to approximate other distributions [15], or to fit a distribution to a series of statistical data according to the moments of the series. The other advantage is that while it makes the use of any distributions possible it still allows us to exploit the memoryless property of the exponential distribution. Since by PH distributions any ....

A. Bobbio and M. Telek, "A benchmark for PH estimation algorithms: results for Acyclic-PH," Stochastic Models, vol. 10, pp. 661--677, 1994.

....all phase type distributed [31] this approach gives the exact solution. The analysis method is composed by the following steps: Step 1: Approximate firing time distribution of all the timed transitions by a Phase type distribution (an overview of approximation methods and tools can be found in [6]) Step 2: Based on the net description, the phase type model and the memory policy of the transitions compose the expanded state model of the stochastic process which is a CTMC over the state space S Theta T 1 Theta : Theta Tn , where S is the set of reachable tangible markings and T g is ....

A. Bobbio and M. Telek. A benchmark for PH estimation algorithms: results for Acyclic-PH. Stochastic Models, 10:661--677, 1994.

No context found.

A. Bobbio and M. Telek. A benchmark for PH estimation algorithms: result for acyclic-PH. Stochastic models, 10(661677) , 1994.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC