Rate Matrix

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Krisztina László - One of the best experts on this subject based on the ideXlab platform.

Attila Domján - One of the best experts on this subject based on the ideXlab platform.

Erik Geissler - One of the best experts on this subject based on the ideXlab platform.

Tien Van - One of the best experts on this subject based on the ideXlab platform.

  • An enhanced algorithm to solve multiserver retrial queueing systems with impatient customers
    Computers & Industrial Engineering, 2013
    Co-Authors: Tien Van, Nam H., Jie Zhang
    Abstract:

    The homogenization of the state space for solving retrial queues refers to an approach, where the performance of the M/M/c retrial queue with impatient customers and c servers is approximated with a retrial queue with a maximum retrial Rate restricted beyond a given number of users in the orbit. As a consequence, the stationary distribution can be obtained by the Matrix-geometric method, which requires the computation of the Rate Matrix. In this paper, we revisit an approach based on the homogenization of the state space. We provide the exact expression for the conditional mean number of customers based on the computation of the Rate Matrix R with the time complexity of O(c). We develop simplified equations for the memory-efficient implementation of the computation of the performance measures. We construct an efficient algorithm for the stationary distribution with the determination of a threshold that allows the computation of performance measures with a specific accuracy.

  • An efficient method to compute the Rate Matrix for retrial queues with large number of servers
    Applied Mathematics Letters, 2010
    Co-Authors: Tien Van, Ram Chakka
    Abstract:

    Abstract The approximate solution technique for the main M / M / c retrial queue based on the homogenization of the model employs a quasi-birth–death (QBD) process in which the maximum retrial Rate is restricted above a certain level. This approximated continuous-time Markov chain (CTMC) can be solved by the Matrix-geometric method, which involves the computation of the Rate Matrix R . This paper is motivated by two observations. Firstly, retrial queues for the performability analysis of telecommunication systems often involve the number of servers in the order of several hundreds of thousands. Secondly, there are no workable solutions till now for systems with such large number of servers, due to ill-conditioning or prohibitively large computation times. Our paper is the first to tackle the problem of large number of servers, very efficiently, in the homogenized M / M / c retrial queue which has paramount applications in networks. We present an efficient algorithm with the time complexity of only O ( c ) to compute the Rate Matrix R .

  • Comments on “multi-server system with single working vacation”
    Applied Mathematical Modelling, 2009
    Co-Authors: Tien Van
    Abstract:

    Abstract Lin and Ke consider the M/M/R queue with working vacation [Chuen-Horng Lin, Jau-Chuan Ke, Multi-server system with single working vacation, Appl. Math. Modell. (2008), doi:10.1016/j.apm.2008.10.006], and derive a computable explicit form for Rate Matrix R of the geometric approach and the stationary probabilities of the queue. However, it contains some errors concerning the terminology, notations and the final form of Rate Matrix R . This note shows that the classical Quasi-birth-death (QBD) formulation of the M/M/R queue with working vacation naturally leads to the infinitesimal generator Matrix of the QBD process and the probability interpretation of matrices involving in the infinitesimal generator Matrix.

Rob Knight - One of the best experts on this subject based on the ideXlab platform.

  • comparison of methods for estimating the nucleotide substitution Matrix
    BMC Bioinformatics, 2008
    Co-Authors: Maribeth Oscamou, Gavin A Huttley, Manuel E Lladser, Daniel Mcdonald, Rob Knight
    Abstract:

    The nucleotide substitution Rate Matrix is a key parameter of molecular evolution. Several methods for inferring this parameter have been proposed, with different mathematical bases. These methods include counting sequence differences and taking the log of the resulting probability matrices, methods based on Markov triples, and maximum likelihood methods that infer the substitution probabilities that lead to the most likely model of evolution. However, the speed and accuracy of these methods has not been compared. Different methods differ in performance by orders of magnitude (ranging from 1 ms to 10 s per Matrix), but differences in accuracy of Rate Matrix reconstruction appear to be relatively small. Encouragingly, relatively simple and fast methods can provide results at least as accuRate as far more complex and computationally intensive methods, especially when the sequences to be compared are relatively short. Based on the conditions tested, we recommend the use of method of Gojobori et al. (1982) for long sequences (> 600 nucleotides), and the method of Goldman et al. (1996) for shorter sequences ( 2000 nucleotides) at the expense of substantially longer computation time. The availability of methods that are both fast and accuRate will allow us to gain a global picture of change in the nucleotide substitution Rate Matrix on a genomewide scale across the tree of life.

  • using the nucleotide substitution Rate Matrix to detect horizontal gene transfer
    BMC Bioinformatics, 2006
    Co-Authors: Micah Hamady, M D Betterton, Rob Knight
    Abstract:

    Background: Horizontal gene transfer (HGT) has allowed bacteria to evolve many new capabilities. Because transferred genes perform many medically important functions, such as conferring antibiotic resistance, improved detection of horizontally transferred genes from sequence data would be an important advance. Existing sequence-based methods for detecting HGT focus on changes in nucleotide composition or on differences between gene and genome phylogenies; these methods have high error Rates. Results: First, we introduce a new class of methods for detecting HGT based on the changes in nucleotide substitution Rates that occur when a gene is transferred to a new organism. Our new methods discriminate simulated HGT events with an error Rate up to 10 times lower than does GC content. Use of models that are not time-reversible is crucial for detecting HGT. Second, we show that using combinations of multiple predictors of HGT offers substantial improvements over using any single predictor, yielding as much as a factor of 18 improvement in performance (a maximum reduction in error Rate from 38% to about 3%). Multiple predictors were combined by using the random forests machine learning algorithm to identify optimal classifiers that sepaRate HGT from non-HGT trees. Conclusion: The new class of HGT-detection methods introduced here combines advantages of phylogenetic and compositional HGT-detection techniques. These new techniques offer order-ofmagnitude improvements over compositional methods because they are better able to discriminate HGT from non-HGT trees under a wide range of simulated conditions. We also found that combining multiple measures of HGT is essential for detecting a wide range of HGT events. These novel indicators of horizontal transfer will be widely useful in detecting HGT events linked to the evolution of important bacterial traits, such as antibiotic resistance and pathogenicity.