## Using the exact state space of a Markov model to compute approximate stationary measures (2000)

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Venue: | Proc. 2000 ACM SIGMETRICS Conf. on Measurement and Modeling of Computer Systems |

Citations: | 19 - 9 self |

### BibTeX

@INPROCEEDINGS{Miner00usingthe,

author = {Andrew S. Miner},

title = {Using the exact state space of a Markov model to compute approximate stationary measures},

booktitle = {Proc. 2000 ACM SIGMETRICS Conf. on Measurement and Modeling of Computer Systems},

year = {2000},

pages = {207--216},

publisher = {ACM Press}

}

### Years of Citing Articles

### OpenURL

### Abstract

We present a new approximation algorithm based on an exact representation of the state space S, using decision diagrams, and of the transition rate matrix R, using Kronecker algebra, for a Markov model with K submodels. Our algorithm builds and solves K Markov chains, each corresponding to a different aggregation of the exact process, guided by the structure of the decision diagram, and iterates on their solution until their entries are stable. We prove that exact results are obtained if the overall model has a product-form solution. Advantages of our method include good accuracy, low memory requirements, fast execution times, and a high degree of automation, since the only additional information required to apply it is a partition of the model into the K submodels. As far as we know, this is the first time an approximation algorithm has been proposed where knowledge of the exact state space is explicitly used. 1.

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Citation Context ..., for example, it might suggest that certain events can occur while they have probability zero, or it might negatively aect the quality of the results. Recent advances in decision diagram techniques [=-=1; 2; 19; 20; 23]-=- and the use of Kronecker algebra [3; 4; 5; 11; 13; 14; 21; 24] can be used to store the state space and the transition rate matrix of a huge Markov model very eciently, but the size of the probabilit... |

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Citation Context ..., for example, it might suggest that certain events can occur while they have probability zero, or it might negatively aect the quality of the results. Recent advances in decision diagram techniques [=-=1; 2; 19; 20; 23]-=- and the use of Kronecker algebra [3; 4; 5; 11; 13; 14; 21; 24] can be used to store the state space and the transition rate matrix of a huge Markov model very eciently, but the size of the probabilit... |

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Citation Context ...can occur while they have probability zero, or it might negatively aect the quality of the results. Recent advances in decision diagram techniques [1; 2; 19; 20; 23] and the use of Kronecker algebra [=-=3; 4; 5; 11; 13; 14; 21; 24]-=- can be used to store the state space and the transition rate matrix of a huge Markov model very eciently, but the size of the probability vector still precludes us from obtaining an exact solution fo... |

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Citation Context ...rithm has been proposed where knowledge of the exact state space is explicitly used. 1. INTRODUCTION Fixed-point approximation algorithms for the solution of large Markov models have been widely used =-=[7; 12; 15; 16; 17; 22; 25; 26; 27]. The-=- idea is to \decompose" the original model into submodels and then somehow use the results obtained from the solution of the submodels to update their rate parameters, until the iteration converg... |

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Citation Context ...can occur while they have probability zero, or it might negatively aect the quality of the results. Recent advances in decision diagram techniques [1; 2; 19; 20; 23] and the use of Kronecker algebra [=-=3; 4; 5; 11; 13; 14; 21; 24]-=- can be used to store the state space and the transition rate matrix of a huge Markov model very eciently, but the size of the probability vector still precludes us from obtaining an exact solution fo... |

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Citation Context ..., for example, it might suggest that certain events can occur while they have probability zero, or it might negatively aect the quality of the results. Recent advances in decision diagram techniques [=-=1; 2; 19; 20; 23]-=- and the use of Kronecker algebra [3; 4; 5; 11; 13; 14; 21; 24] can be used to store the state space and the transition rate matrix of a huge Markov model very eciently, but the size of the probabilit... |

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Citation Context ...an occur while they have probability zero, or it might negatively affect the quality of the results. Recent advances in decision diagram techniques [1; 2; 19; 20; 23] and the use of Kronecker algebra =-=[3; 4; 5; 11; 13; 14; 21; 24]-=- can be used to store the state space and the transition rate matrix of a huge Markov model very efficiently, but the size of the probability vector still precludes us from obtaining an exact solution... |

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Citation Context ...an occur while they have probability zero, or it might negatively affect the quality of the results. Recent advances in decision diagram techniques [1; 2; 19; 20; 23] and the use of Kronecker algebra =-=[3; 4; 5; 11; 13; 14; 21; 24]-=- can be used to store the state space and the transition rate matrix of a huge Markov model very efficiently, but the size of the probability vector still precludes us from obtaining an exact solution... |

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Citation Context ...can occur while they have probability zero, or it might negatively aect the quality of the results. Recent advances in decision diagram techniques [1; 2; 19; 20; 23] and the use of Kronecker algebra [=-=3; 4; 5; 11; 13; 14; 21; 24]-=- can be used to store the state space and the transition rate matrix of a huge Markov model very eciently, but the size of the probability vector still precludes us from obtaining an exact solution fo... |

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Citation Context ...two can indeed be shown to be equal with a derivation based on stratication by queue lengths, along the lines of the classical proof of the \ ow equivalent server" for product form queueing netwo=-=rks [6]. Con-=-cluding this section, we stress that the relevance of the fact that our algorithm gives exact results for this class of models is that it provides us with an additional \sanity check" about the r... |

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Citation Context ...1 2 0 1 0 2 1 one B(p) p (b) Figure 1: Sets encoded by decision diagrams. now brie y describe the relevant denitions and terminology, more detailed information and further references can be found in [=-=8; 19]-=-; a few illustrative examples are given at the end of this section. In our particular application, a decision diagram is a directed acyclic graph whose nodes are organized in levels ranging from K, at... |

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Citation Context ...rithm has been proposed where knowledge of the exact state space is explicitly used. 1. INTRODUCTION Fixed-point approximation algorithms for the solution of large Markov models have been widely used =-=[7; 12; 15; 16; 17; 22; 25; 26; 27]. The-=- idea is to \decompose" the original model into submodels and then somehow use the results obtained from the solution of the submodels to update their rate parameters, until the iteration converg... |

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Citation Context ...rithm has been proposed where knowledge of the exact state space is explicitly used. 1. INTRODUCTION Fixed-point approximation algorithms for the solution of large Markov models have been widely used =-=[7; 12; 15; 16; 17; 22; 25; 26; 27]. The-=- idea is to \decompose" the original model into submodels and then somehow use the results obtained from the solution of the submodels to update their rate parameters, until the iteration converg... |

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Citation Context ...isualize such an \unreduced decision diagram" as a tree where the nodes at level k have anywhere between one and nk pointers to nodes at level k 1. This is the multilevel data structure introduce=-=d in [-=-10] to store S. For example, the open queueing network of Fig. 2(a) would have a decision diagram with 1, n4 = 4, n4 n3 = 12, and n4 n3 n2 = 24 nodes at level 4, 3, 2, and 1, respectively, while th... |

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Citation Context ...Pentium II 400Mhz machine, with 384Mbytes of RAM, with the Linux operating system, and without the use of virtual memory. The exact results used to evaluate our approximation were obtained with SMART =-=[9]-=-. In each case, we report the relative error in aggregated measures, such as the mean number of tokens in a given place or the throughput of transitions. In all cases, CTMCs were solved using Gauss-Se... |

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Citation Context ...ester Research Assignment from the College of William and Mary, and by a travel grant from Italy's Consiglio Nazionale delle Ricerche. exact result. Proofs of convergence are usually dicult to obtain =-=[18]. A f-=-urther problem not often mentioned is that, when submodels are combined, \spurious" states might be introduced, which do not exist in the exact model. This can be a problem, as, for example, it m... |

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