B.L. Fox and P.W. Glynn. Computing Poisson probabilities. Comm. of the ACM 31(4): 440-445, 1988.  Home/Search   Document Not in Database   Summary   ACM   TOC   Related Articles   Check

....i , i.e. i ( t) is the probability that a Poisson process with rate generates i events in an interval of length t. A straightforward computation of i ( t) is numerically stable only for relatively small values of t, for arbitrary values of t, the approach of Fox and Glynn [26] is frequently employed, which gives left and right truncation points 0 6 l r, such that i ( t) 0 for i l and i r. 9 Let p = 0 be the initial vector. Then t is given by: t = i ( t) 4) where p = p p Q (i 0) and max x2f0; jT jg ....

....potentials are computed by successive matrix vector multiplication [S 1 ] such that q[S 1 ] The in nite summation is truncated as soon as at least one of the following truncation (termination) criteria holds. First, the derivation of the Poisson distribution by the approach of [26] provides only a nite number of values, so the right truncation point r imposed by [26] limits the summation in a natural way. Since for i r; i ( t) 0 no further summation is necessary. Secondly, if the iteration reaches a x point at some step j, i.e. x (j 1) x , then we need not ....

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B.L. Fox and P.W. Glynn. Computing Poisson probabilities. Comm. ACM, 31:440-445, 1988.

....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 for the E step of the EM algorithm and show how to utilize the ....

....(5) Since the probability mass function of the Poisson distribution thins for growing qt, round off errors for large qt may affect the computation of Poisson probabilities. Thus, the randomization technique is enhanced by a stable calculation of Poisson probabilities proposed by Fox and Glynn [8]. Given an error tolerance , the computational complexity of a dense sparse implementation of the randomization method is given by O(N 2 .qt) and O (q. qt) respectively, where q denotes the number of nonzero entries in the generator matrix Q. Furthermore, the randomization technique is an ....

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B.L Fox, and P.W. Glynn, Computing Poisson Probabilities, Comm. of the ACM 31, 440-445, 1988.

....truncation does not effect the number of matrix vector multiplications that must occur and only reduces the amount of summation necessary. It can then be seen, with kxk1 = maxfjx i jg, that A n ffl (2. 7) since all elements of n are in [0; 1] Fox and Glynn [5] have given a method to find N sl and N sr given a predefined ffl, and t, and using this, SU allows solution of (t) to any desired accuracy ffl 0. 2.2 Adaptive Uniformization In adaptive uniformization, the uniformization rate can change with the number of jumps, or current epoch, and ....

....ACE O(N a ) Unstable Uniformization (LU) O(N a NBr ) Stable Poisson Probabilities (SU) O( N sr ) Stable Table 3.1: Complexity and Stability of Methods 28 3. 2 Poisson Probability Calculation For both SU and LU, Poisson probabilities are calculated using the method of Fox and Glynn [5]. Starting with the mode, which is assigned a weight with the intent to avoid underflow, the remaining weights between the left and right truncation points are found recursively. These weights are then divided by a constant ff to approximate the probabilities. In [5] ff = W where W is the sum ....

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B.L. Fox and P.W. Glynn, "Computing Poisson Probabilities," Comm. ACM 31, pp. 440-445, 1988.

....is carried out in an iterative manner per individual state s starting from the initial distribution (s; 0) The pseudo code of this algorithm is presented in Fig. 1. Here, and in the subsequent algorithms in this paper, the Poisson probabilities are computed using the Fox Glynn algorithm [15] that avoids over ow for large q t. The overall time complexity of this procedure is O(N q t M ) where q is the uniformisation rate of the CTMC at hand, t the time bound of the until formula, N the number of states and M the number of non zero entries in R. This follows directly from the fact ....

B.L. Fox and P.W. Glynn. Computing Poisson probabilities. Comm. of the ACM, 31(4): 440-445, 1988.

....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 the E step of the EM algorithm and show how to utilize the EM ....

....A . 5) Since the probability mass function of the Poisson distribution thins for growing qt, round off errors for large qt may affect the computation of Poisson probabilities. Thus, the randomization technique is enhanced by a stable calculation of Poisson probabilities proposed by Fox and Glynn [5]. Given an error tolerance H, the computational complexity of a sparse implementation of the randomization method is given by OqtK 0 5 where K denotes the number of nonzero entries in the generator matrix Q. Furthermore, the randomization technique is suitable for calculating the conditional ....

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B.L Fox, and P.W. Glynn, Computing Poisson Probabilities, Comm. of the ACM 31, 440-445, 1988.

....given by the user. Letting r max = max i2 Omega r i , the required N is N = min n m : r max 1 X k=m 1 e Gammat (t) k k ffl o : Stable and efficient computation of the Poisson probabilities e Gammat (t) k =k is a delicate issue but several good alternatives are available [1, 8, 9, 13]. Defining the vector q(k) P [ b X k = i] i2 Omega , q(k) k 0 can be obtained from q(0) using q(k 1) T = q(k) T P, where P = P i;j ) i;j2 Omega is the transition probability matrix of b X . The computational cost of standard randomization is roughly proportional to N . For large ....

B. L. Fox and P. W. Glynn, "Computing Poisson probabilities," Communications of the ACM , vol. 31, 1988, pp. 440--445.

....t) i i P i , which can be rewritten as #(#, t) # # i=0 PP (i) # i , where PP (i) e qt (qt) i i is the i th Poisson probability with parameter qt, and # i = # i 1 P and # 0 = #. The Poisson probabilities can be computed in a stable way with the algorithm of Fox and Glynn [14]. There is no need to compute explicit powers of the matrix P. Furthermore, since the terms in the summation are all between 0 and 1, the number of terms to be taken given a required accuracy, can be computed a priori. For large values of qt, this number is of order O(qt) Notice, however, that ....

B.L. Fox and P.W. Glynn. Computing Poisson probabilities. Comm. of the ACM 31(4): 440--445, 1988.

.... (q t) i i P i ; which can be rewritten as ( t) 1 X i=0 PP (i) i ; where PP (i) e q t (q t) i i is the i th Poisson probability with parameter qt, and i = i 1 P and 0 = The Poisson probabilities can be computed in a stable way with the algorithm of Fox and Glynn [14]. There is no need to compute explicit powers of the matrix P. Furthermore, since the terms in the summation are all between 0 and 1, the number of terms to be taken given a required accuracy, can be computed a priori. For large values of qt, this number is of order O(qt) Notice, however, that ....

B.L. Fox and P.W. Glynn. Computing Poisson probabilities. Comm. of the ACM 31(4): 440-445, 1988.

.... = N X n=0 PP (n) n) 9) So, the computed result ER c (t) for the expected reward at moment t is then ER c (t) X i2S c i (t)r i : 10) Finally, we note that the Poisson probabilities PP (n) n = 0; 1; N; can be computed in a stable manner as formalized by Fox and Glynn [5], see also [7] We will now illustrate how the computational SU scheme can be formulated as a probabilistic evaluation method. To this end, we discuss SU for the example Markov chain given in Figure 1. The Markov model represents an extended machine repairman model consisting of two components. ....

....the fastest access during the step wise computations. As the size of P can change during the computations, this data structure is made up of dynamically adjustable linked lists as well. The Poisson probabilities are calculated only once, and in a stable way, using the approach of Fox and Glynn [5]. The state space generator is currently only virtually present. What actually is implemented is that the overall state space is generated using the SPNP package [1] however, access to the state space and the generator matrix Q is done in such a way that, from the viewpoint of the uniformization ....

B.L. Fox, P.W. Glynn, "Computing Poisson Probabilities," Communications of the ACM 31, pp.440--445, 1988.

....method. In particular, we will show in Section V that the computed transient state probabilities are lower than the actual state probabilities, with an arbitrary small error. We also discuss the influence of computing Poisson probabilities on the error bound when using the Fox Glynn algorithm [9]. The Fox Glynn algorithm is enhanced in the Appendix to improve its error bound properties, and the thus obtained results can be used to more precisely determine a bound on the error in any Fox Glynn based implementation of SU and AU. Finally, having presented the theoretical properties of ....

....the literature are sometimes imprecise in that the computation of the jump Poisson probabilities is not considered, we derive accurate expressions for ffl a and ffl s in the next two subsections. A Expression for ffl s Most recent implementations of SU use an algorithm developed by Fox and Glynn [9] to compute the required Poisson probabilities (see [3, 10] for variations that have been used) In the Fox Glynn algorithm, the relative values of the needed Poisson probabilities (those between L s and R s ) are computed exactly, but absolute values of the Poisson probabilities require ....

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B. L. Fox and P. W. Glynn, "Computing Poisson probabilities," Communications of the ACM, vol. 31, pp. 440--445, 1988.

.... Gammaqt (qt) k k , and the right truncation bound R(qt; ffl) as a function of qt and of the error tolerance ffl. Sparse data structures are used for all operations. The vectors p k are computed by iterative vector matrix multiplications and the Poisson probabilities by the method described in [5]. In order to compute the transient probabilities at multiple time instants, as it is required for the computation of the time dependent curves, two variants of the algorithm are possible. Assume that the stepsize is fixed and given by h. Variant a) Storage of the Matrix Exponential. In this ....

B. L. Fox, P. W. Glynn. Computing Poisson Probabilities. Comm. of the ACM, 31 (1988) 440--445.

....well. The main problem with the randomization method is the computation of Poisson probabilities for stiff differential equation systems. Therefore, we adopted a similar refined randomization method based on a numerically stable subroutine for the computation of Poissonprobabilities as proposed in [4] and has been used by several other tool developers [2, 20] 3 ANALYSIS OF THE UNDERLYING CTMC 7 3.2 Specification of Measures The result of numerical analysis is the state probability distribution vector. This is usually a too detailed measure and not well suited for getting insight into the ....

B.L. Fox and P.W. Glynn. Computing Poisson Probabilities. Communications of the ACM, 31(4):440--445, 1988.

....based on the transformation of Q to the stochastic matrix of an embedded discrete time Markov chain. Among others, this approach was adopted also by Lindemann [27] who presented a refined randomization technique based on a numerically stable algorithm for the computation of Poisson probabilities [12]. Unfortunately, in addition to largeness another well known problem arises, especially in performability models, namely the problem of having to solve a stiff differential equation system. Typically, such systems are caused by transition rates that differ in many orders of magnitude. The above ....

B.L. Fox and P.W. Glynn. Computing Poisson Probabilities. Communications of the ACM, 31(4):440--445, 1988.

....on the transformation of Q into a stochastic matrix of the embedded discrete time Markov chain. Among others, this approach was adopted also by Lindemann [30] who presented a refined randomization technique based on a numerically stable algorithm for the computation of Poisson probabilities [13]. Unfortunately, in addition to largeness another well known problem arises, especially in performability models, namely the problem of having to solve a stiff differential equation system. Typically, stiffness is caused by transition rates that differ in many orders of magnitude. The above ....

B.L. Fox and P.W. Glynn. Computing Poisson Probabilities. Comm. of the ACM, 31(4):440--445, 1988.

....a Poisson variate greater efficiency results from stratifying the (Poisson) distribution corresponding to N and then generating stratified versions of N on appropriate runs. To stratify, we need to be able to compute Poisson probabilities accurately and efficiently; for that, see Fox and Glynn [9]. Except possibly in the right tail, given the stratum we generate N by inversion or by the alias method; in those strata, this adds respectively one or two dimensions to X . In the right tail, one could use some (tailored) rejection algorithm (for example, the clever method of Hormann and ....

....we need error bounds on tail masses so we know how far outwards from the mode to go while capturing a user specified percentage of the total probability mass. As far as we know, finding reasonably tight easily computed bounds is a research problem. It is analogous to one solved by Fox and Glynn [9] for computing Poisson probabilites. One reason the problem seems hard (or, at least, messy) is that the tails are the place where the normal approximation is the worst. The problem is compounded, because the error bounds must also take account of the tail mass neglected when computing Poisson ....

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FOX, B. L. and GLYNN, P. W. Computing Poisson probabilities, Comm. ACM 31(1988), 440-445. Generating Poisson processes 18

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B.L. Fox and P.W. Glynn. Computing Poisson probabilities. Comm. of the ACM 31(4): 440-445, 1988.

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B. L. Fox and P. W. Glynn, "Computing Poisson Probabilities," Comm. of the ACM, vol. 31, pp. 440--445, 1988.

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B.L. Fox and P.W. Glynn. Computing Poisson probabilities. Comm. of the ACM, 31(4): 440-445, 1988.

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B. Fox and P. Glynn. Computing Poisson probabilities. Communications of the ACM, 31(4):440-445, 1988.

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Fox, B. L. and Glynn, P. W. Computing Poisson probabilities. Communications of the ACM, 31(4):440-445, 1988.

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B. Fox and P. Glynn. Computing Poisson probabilities. Communications of the ACM, 31(4):440-445, 1988.

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Fox, B. L. and Glynn, P. W. Computing Poisson probabilities. Communications of the ACM, 31(4):440--445, 1988.

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B.L. Fox and P.W. Glynn. Computing Poisson Probabilities. Comm. of the ACM, 31(4):440--445, 1988.

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