Reibman, A. and Trivedi, K. S. (1989) Transient analysis of cumulative measures of Markov model behavior. Commun. Stat.---Stoch. Models, 5, 683--710.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....a numerical stable implementation and utilizing the BMAP for traffic modeling is an open research problem. 3 In this paper, we introduce an efficient and numerical stable method for estimating the parameters of a BMAP with the EM algorithm. We show how the randomization technique [9] [14], 17] and a stable calculation of Poisson jump probabilities [8] can effectively be utilized for the computation of the time dependent conditional expectation of a continuous time Markov chain (CTMC) required by the E step of the EM algorithm. In fact, we present efficient computational formulas ....

....rate matrices D(0) D(M) Based on this parameter estimation, such a customized BMAP constitutes an aggregated IP traffic model considering both packet interarrival times and packet lengths. 2. 2 The Randomization Technique Randomization (also called uniformization or Jensen s method [9] [14], 14a] has proven to be an effective numerical method for computing transient measures of CTMCs involving matrix exponentials as introduced in Eq. 1) Reibman and Trivedi showed that randomization constitutes the method of choice for non stiff and mildly stiff CTMCs. For transient analysis of ....

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A. Reibman and K.S. Trivedi, Transient Analysis of Cumulative Measures of Markov Model Behavior, Comm. in Statistics: Stochastic Models 5, 683-710, 1989.

....lie in the irregular, dynamically growing data structure of the CTMC whose size and shape cannot be estimated in advance. After the CTMC is generated, the transient numerical solver computes its state probabilities for the desired time points. Our algorithms use the uniformization method [19, 17, 18]. Uniformization is a very popular transient solution method as it shows good convergence behavior in many cases, is memory efficient and easy to implement. The last step in the quantitative analysis process is the back transformation of the results to the net level. For this purpose stochastic ....

....needs one mutex variable per B tree node which is O(jMj=b) on an average (with b average node size) but no parentpointers have to be maintained. State numbers are augmented by the thread number. 3 Transient Numerical Solution Now as the CTMC is generated, instantaneous [19] and cumulative [18] transient measures are computed for one or a series of user defined time points. processes are called threads in the SM context Susann Allmaier and David Kreische For the basic theoretical considerations we switch from regarding the CTMC as a graph (which is nothing else than a sparse matrix ....

[Article contains additional citation context not shown here]

A.L. Reibman and K.S. Trivedi. Transient analysis of cumulative measures of Markov model behaviour. Stochastic Models, 5(4):683--710 1989.

....developing a numerical stable implementation and utilizing the BMAP for traffic modeling is an open research problem. In this paper, we introduce an efficient and numerical stable method for estimating the parameters of a BMAP with the EM algorithm. We show how the randomization technique [6] [11] and a stable calculation of Poisson jump probabilities [5] can effectively be utilized for the computation of the time dependent conditional expectation of a continuous time Markov chain (CTMC) required by the E step of the EM algorithm. In fact, we present efficient computational formulas for ....

....matrices DD 0 05 0 5 , # M . Based on this parameter estimation, such a customized BMAP constitutes an aggregated IP traffic model considering both packet interarrival times and packet lengths. B. The Randomization Technique Randomization (also called uniformization or Jensen s method [6] [11]) has proven to be an effective numerical method for computing transient measures of CTMCs involving matrix exponentials as introduced in Eq. 1) Reibman and Trivedi showed that randomization constitutes the method of choice for non stiff and mildly stiff CTMCs. For transient analysis of stiff ....

[Article contains additional citation context not shown here]

A. Reibman and K.S. Trivedi, Transient Analysis of Cumulative Measures of Markov Model Behavior, Comm. in Statistics: Stochastic Models 5, 683-710, 1989.

....growing data structure of the CTMC whose size and shape cannot be estimated in advance. 2 Susann Allmaier and David Kreische After the CTMC is generated, the transient numerical solver computes its state probabilities for the desired time points. Our algorithms use the uniformization method [19, 17, 18]. Uniformization is a very popular transient solution method as it shows good convergence behavior in many cases, is memory efficient and easy to implement. The last step in the quantitative analysis process is the back transformation of the results to the net level. For this purpose stochastic ....

....needs one mutex variable per B tree node which is O(jMj=b) on an average (with b average node size) but no parent pointers have to be maintained. State numbers are augmented by the thread number. 3 Transient Numerical Solution Now as the CTMC is generated, instantaneous [19] and cumulative [18] transient measures are computed for one or a series of user defined time points. 2 processes are called threads in the SM context Susann Allmaier and David Kreische For the basic theoretical considerations we switch from regarding the CTMC as a graph (which is nothing else than a sparse ....

[Article contains additional citation context not shown here]

A.L. Reibman and K.S. Trivedi. Transient analysis of cumulative measures of Markov model behaviour. Stochastic Models, 5(4):683--710, 1989.

....chain is stiff. So, the computation of the distribution of the interval availability remains very expensive in time. This is why there is still an interest in computing just the expectation of this random variable. This expectation has been obtained by means of the uniformization technique in [5] [6]. In this paper we develop a new method to compute the expectation of the interval availability. It is based on the technique of the uniformized power and the starting PI n 1014 4 Hascam ABDALLAH, Raymond MARIE, Bruno SERICOLA point is the work performed in [7] The interest of this new approach ....

A. Reibman and K. Trivedi. Transient Analysis of Cumulative Measures of Markov Model Behavior. Commun. Statist.-Stochastic Models, 5(4):683--710, 1989. PI n 1014 18 Hascam ABDALLAH, Raymond MARIE, Bruno SERICOLA

....Grassman [Gra87] considers the computation of the means and variances of time averages in Markovian environments. Smith et al. S 88] propose cumulative measures for studying computer system performance metrics such as accumulated reward and useful work accomplished . Trivedi and Reibman [RT89] evaluate two different techniques for computing time averaged measures such as interval availability and accumulated reward. Our focus in this section will be on such cumulative measures as they directly reflect the user oriented QOS criteria in which we are interested. We propose two basic ....

....the value 1 if the QOS criteria is not met over the block of packets (k 1 ; k 2 ) We then define the block QOS as, AB (n; l) P M k=1 B ( k Gamma1)l 1;kl) M (7) where M = b n l c. Time Average The time average is defined as A T (t) R t 0 f(X(s) ds t : 8) Several researchers [RT89, S 88, Gra87] have investigated in detail the efficient computation of the time averages for Markovian queueing systems and hence our treatment here is brief. To illustrate the usefulness of the time average consider the following definition of f( f(x) 1 x 0; 0 Otherwise: 9) ....

[Article contains additional citation context not shown here]

Andrew Reibman and Kishore Trivedi. Transient analysis of cumulative measures of markov models. Communications in Statistics-Stochastic Models, 5:683--710, 1989.

....Grassman [Gra87] considers the computation of the means and variances of time averages in Markovian environments. Smith et al. S 88] propose cumulative measures for studying computer system performance metrics such as accumulated reward and useful work accomplished . Trivedi and Reibman [RT89] evaluate two different techniques for computing time averaged measures such as interval availability and accumulated reward. Our focus in this section will be on such cumulative measures as they directly reflect the user oriented QOS measures in which we are interested. We propose two basic ....

....over shorter time scales of burst and block lengths. Hence, it seems natural and more meaningful to define the QOS over these time scales rather than over the entire connection. Time Average The time average is defined as a random variable, A T (t) R t 0 f(X(s) ds t (9) Several researchers [RT89, S 88, Gra87] have investigated in detail the efficient computation of the time averages for Markovian queueing systems and hence our treatment here is brief. We, however, do consider it briefly and provide some numerical examples. To illustrate the usefulness of the time average consider the ....

[Article contains additional citation context not shown here]

Andrew Reibman and Kishore Trivedi. Transient analysis of cumulative measures of markov models. Communications in Statistics-Stochastic Models, 5:683--710, 1989.

....with p(0) 3) Semi symbolic solutions for this system of differential equations is computationally expensive [68] and therefore only applicable for small models. For larger models, numerical techniques are used. One can either use Runge Kutta based methods, possibly adapted for stiff systems [69, 71], or randomization (also called uniformization) 31, 32, 58, 63, 86, 87] For the computation of E[Y (t) similar techniques can be employed. For the CDFs of the cumulative measures various specialized algorithms exist [22, 26, 66, 67] For acyclic Markov chains recursive algorithms exist. For ....

A.L. Reibman, K.S. Trivedi, "Transient Analysis of Cumulative Measures of Markov Model Behavior", Stochastic Models 5(4), pp.683--710, 1989.

....process from the transient arrival rate process. 5 Transient Cummulative Measures in Markovian Systems So far we have been successful in computing the transient probability distribution and the transient average loss and departure intensities. These are only instantaneous (or point ) measures ([32]) Often, one is more interested in the transient analysis of cumulative or integral measures of Markov behaviour. These measures may include moments of the average proportion of time that the buffer is full in a finite time interval or moments of the accumulated losses or departures that occur in ....

....by the integration of the quantities that we have already obtained in the previous 2 sections. They may also directly be computed by the numerical inversion of the Laplace Transforms of their time dependent equations. The computation of cumulative measures has been considered in the past in [27] [32], 33] and [34] In [27] the authors define new QoS metrics based on the cummulative measures in Markovian systems of the type M=G=1=K and develop approximations for their computation. In [32] and [33] the authors use the methods of uniformization and differential equation solution for systems ....

[Article contains additional citation context not shown here]

A. Reibman and K. Trivedi, "Transient Analysis of Cumulative Measures of Markov Model Behaviour," Stochastic models, vol. 5(4), 1989, pp. 683-710.

....performance issues in transient state. In this work, we assume that arrival processes are stationary. For a nonstationary process, both steady state and transient state are meaningless. Previous work in transient analysis of queueing systems usually focuses on Markov models, for example, see [1, 17]. Due to complexity involved in analysis, one may have to resort to simulation. Nagarajan and Kurose have examined transient loss performance defined on an interval basis for packet voice connections in high speed networks by simulation [13] They observed that in transient state, voice ....

A. Reibman and K. Trivedi, "Transient analysis of cumulative measures of Markov models", Communications in Statistics-Stochastic Models, vol. 5, pp. 683-- 710, 1989.

....the error tolerance can be pre computed. This is one of the advantages of using UNIFORMIZATION. This method can easily handle large state spaces and is numerically stable but not ecient for sti problems [35] Computation of the cumulative probabilities for the CTMC is also based on UNIFORMIZATION [36]. The initial probability distribution for the CTMC a ects the results of transient analysis. If the initial marking of the GSPN is tangible, the underlying CTMC contains a state corresponding to this marking. In this case, the initial probability for this state is 1:0 and all the other states ....

A. L. Reibman and K. S. Trivedi. Transient analysis of cumulative measures of Markov model behavior. Stochastic Models. To appear.

....error tolerance can be pre computed. This is one of the advantages of using UNIFORMIZATION. This method can easily handle large state spaces and is numerically stable but not efficient for stiff problems [35] Computation of the cumulative probabilities for the CTMC is also based on UNIFORMIZATION [36]. The initial probability distribution for the CTMC affects the results of transient analysis. If the initial marking of the GSPN is tangible, the underlying CTMC contains a state corresponding to this marking. In this case, the initial probability for this state is 1:0 and all the other states ....

A. L. Reibman and K. S. Trivedi. Transient analysis of cumulative measures of Markov model behavior. Stochastic Models. To appear.

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Reibman, A. and Trivedi, K. S. (1989) Transient analysis of cumulative measures of Markov model behavior. Commun. Stat.---Stoch. Models, 5, 683--710.

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A. Reibman and K. Trivedi. Transient Analysis of Cumulative Measures of Markov Model Behavior. Commu. Statist.-Stochastic Models, 5(4):683--710, 1989.

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