## Complexity of memory-efficient Kronecker operations with applications to the solution of Markov models (2000)

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Venue: | INFORMS J. Comp |

Citations: | 65 - 19 self |

### BibTeX

@ARTICLE{Buchholz00complexityof,

author = {Peter Buchholz and Gianfranco Ciardo and Susanna Donatelli and Peter Kemper},

title = {Complexity of memory-efficient Kronecker operations with applications to the solution of Markov models},

journal = {INFORMS J. Comp},

year = {2000},

volume = {2000},

pages = {12--3}

}

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### Abstract

We present new algorithms for the solution of large structured Markov models whose infinitesimal generator can be expressed as a Kronecker expression of sparse matrices. We then compare them with the shuffle-based method commonly used in this context and show how our new algorithms can be advantageous in dealing with very sparse matrices and in supporting both Jacobi-style and Gauss-Seidel-style methods with appropriate multiplication algorithms. Our main contribution is to show how solution algorithms based on Kronecker expression can be modified to consider probability vectors of size equal to the "actual" state space instead of the "potential" state space, thus providing space and time savings. The complexity of our algorithms is compared under different sparsity assumptions. A nontrivial example is studied to illustrate the complexity of the implemented algorithms. Continuous time Markov chains (CTMCs) are an established technique to analyze the performance, reliability, or performability of dynamic systems from a wide range of application areas. CTMCs are usually specied in a high-level modeling formalism, then a software tool is employed to generate the state space and generator matrix of the underlying CTMC and compute the stationary

### Citations

584 |
Introduction to the Numerical Solution of Markov Chains
- Stewart
- 1994
(Show Context)
Citation Context ... from a Jacobi or Gauss-Seidel iteration step and 0 < ω < 2. Apart from the above classical stationary techniques, “projection techniques” have recently become very popular for the analysis of CTMCs (=-=Stewart 1994-=-). These approximate the exact solution using a sequence of vectors on a subspace with a smaller dimension (often the so-called Krylov subspace L = span{π · Q, . . . , π · Q m } is used). Different pr... |

342 | Modeling with Generalized Stochastic Petri Nets
- Marsan, Balbo, et al.
- 1995
(Show Context)
Citation Context ...proach further since its time complexity is at least as high as that of the algorithms we present and events requiring no time, e.g., immediate transitions in GSPNs (Ajmone Marsan et al. 1984, Ajmone =-=Marsan et al. 1995-=-), cause additional overhead, by requiring the exploration of a path, not a single event, to generate a single entry. 5.2 Algorithms Pot-Sh-JCB and Act-Sh-JCB The shuffle-based algorithms in Figure 11... |

259 | A class of generalized stochastic Petri nets for the performance evaluation of multiprocessor systems
- Marsan, Conte, et al.
- 1984
(Show Context)
Citation Context ... we do not consider this approach further since its time complexity is at least as high as that of the algorithms we present and events requiring no time, e.g., immediate transitions in GSPNs (Ajmone =-=Marsan et al. 1984-=-, Ajmone Marsan et al. 1995), cause additional overhead, by requiring the exploration of a path, not a single event, to generate a single entry. 5.2 Algorithms Pot-Sh-JCB and Act-Sh-JCB The shuffle-ba... |

140 |
On the stochastic structure of parallelism and synchronization models for distributed algorithms
- Plateau
- 1985
(Show Context)
Citation Context ... from a high-level model structured into submodels. The method has been applied to several high-level formalisms where models are described in a compositional way (Donatelli 1993, 1994, Kemper 1996a, =-=Plateau 1985-=-). Solution methods exploiting a Kronecker structure are iterative but they differ from conventional iterative techniques in how they perform the required vector-matrix multiplications. A first approa... |

107 |
Stochastic automata network for modeling parallel systems
- Plateau, Atif
- 1991
(Show Context)
Citation Context ...a structured model M described as the parallel composition of a set of K submodels M[1,K]: each submodel Mk is described as a stochastic automaton in the stochastic automata network (SANs) framework (=-=Plateau and Atif 1991-=-), or as a generalized stochastic Petri net (GSPN) in the superposed GSPNs (SGSPNs) framework (Donatelli 1994, Kemper 1996a). The interactions among submodels can be either “simultaneous jumps”, when ... |

92 | Efficient descriptor-vector multiplications in stochastic automata networks
- Fernandes, Plateau, et al.
- 1998
(Show Context)
Citation Context ... (provided ˆx is not used elsewhere), since the algorithms overwrite only those values of ˆx which are not used any further or are currently stored in z. The complexity of Pot-Sh (see Buchholz, 1994; =-=Fernandes et al., 1998-=-) can be rewritten as: � K� O ¯nk · η[A k � � K� η[A ] = O n · k � ] = O (n · K · α) . k=1 Hence, Pot-Sh is faster than a multiplication using explicit storage if and only if k=1 nk n · K · α < n · α ... |

84 |
Automated Generation and Analysis of Markov Reward Models using Stochastic Reward Nets
- Ciardo, Blakemore, et al.
- 1992
(Show Context)
Citation Context ...ributions can also be used for other analyses such as the computation of the expected time spent in transient states up to absorption in absorbing CTMCs and the transient analysis of arbitrary CTMCs (=-=Ciardo et al. 1993-=-). Direct solution methods such as the well-known Gaussian elimination are not applicable for the solution of (1), since their fill-in results in excessive memory requirements. Iterative techniques ba... |

70 |
Kronecker products and shuffle algebra
- Davio
- 1981
(Show Context)
Citation Context ...w on, the suffix “+” denotes the version for the simpler case of a product k+1 where all matrices are the identity except one). Pot-Sh considers the matrices Ak sequentially, exploiting the equality (=-=Davio 1981-=-): K� A k=1 k K� = k=1 S T (n k 1 ,nK k+1 ) · (I¯nk ⊗ Ak ) · S (n k 1 ,n K k+1 ), (4) where S(a,b) ∈ {0, 1} a·b×a·b is the matrix describing an (a, b) perfect shuffle permutation: � 1 (S(a,b))i,j = 0 ... |

68 |
Superposed generalized stochastic Petri nets: definition and efficient solution
- Donatelli
(Show Context)
Citation Context ...ibed as a stochastic automaton in the stochastic automata network (SANs) framework (Plateau and Atif 1991), or as a generalized stochastic Petri net (GSPN) in the superposed GSPNs (SGSPNs) framework (=-=Donatelli 1994-=-, Kemper 1996a). The interactions among submodels can be either “simultaneous jumps”, when changes of state in two or more submodels must happen in a synchronous manner, e.g., “synchronizing events” i... |

62 | Numerical analysis of superposed GSPNs
- Kemper
- 1995
(Show Context)
Citation Context ...ices resulting from a high-level model structured into submodels. The method has been applied to several high-level formalisms where models are described in a compositional way (Donatelli 1993, 1994, =-=Kemper 1996-=-a, Plateau 1985). Solution methods exploiting a Kronecker structure are iterative but they differ from conventional iterative techniques in how they perform the required vector-matrix multiplications.... |

59 | A methodology for solving Markov models of parallel systems - Plateau, Fourneau - 1991 |

56 |
Superposed stochastic automata: a class of stochastic Petri nets with parallel solution and distributed state space
- Donatelli
- 1993
(Show Context)
Citation Context ...oducts of smaller matrices resulting from a high-level model structured into submodels. The method has been applied to several high-level formalisms where models are described in a compositional way (=-=Donatelli 1993-=-, 1994, Kemper 1996a, Plateau 1985). Solution methods exploiting a Kronecker structure are iterative but they differ from conventional iterative techniques in how they perform the required vector-matr... |

47 |
Dynamic Probabilistic Systems, Volume II: Semi-Markov and Decision Processes
- Howard
- 1971
(Show Context)
Citation Context ...d generator matrix of the underlying CTMC and compute the stationary 1sprobability vector, from which most quantities of interest can be obtained as a weighted sum by using “reward rates” as weights (=-=Howard 1971-=-). Although the mapping of a high-level model onto the CTMC and the computation of the stationary distribution are conceptually simple, practical problems arise due to the enormous size of CTMCs model... |

45 | M.: On the use of Kronecker operators for the solution of generalized stochastic Petri nets
- Ciardo, Tilgner
- 1996
(Show Context)
Citation Context ...s manner, e.g., “synchronizing events” in SANs or “synchronizing transitions” in SGSPNs, or rate dependencies, e.g., “functional rates” in SANs or a simpler type of product-form dependency in SGSPNs (=-=Ciardo and Tilgner 1996-=-). Rate dependencies are discussed in Section 4.3. We define • ˆ T k , the set of states of submodel Mk when considered in isolation. | ˆ T k | = nk. • ˆ T = ˆ T 1 × . . . × ˆ T K , the potential stat... |

43 | The numerical solution of stochastic automata networks - Stewart, Atif, et al. - 1995 |

38 | Storage alternatives for large structured state spaces
- Ciardo, Miner
- 1997
(Show Context)
Citation Context ...states that precede i in lexicographic order (Ψ(i) = null if i �∈ T ). To focus on the reachable states alone, we need to: • Generate T . Efficient algorithms for the generation of T can be found in (=-=Ciardo and Miner 1997-=-, Kemper 1996b) • Ensure that only AT ,T contributes to the value of y. If A is one of the matrices whose sum constitutes ˆ R, then Ai,j = 0 whenever i ∈ T and j �∈ T , that is, starting from a reacha... |

35 | Smart: Simulation and markovian analyzer for reliability and timing
- Ciardo, Miner
- 1996
(Show Context)
Citation Context ... the Power method. The proposed algorithms have been implemented in SupGSPN, a numerical solution package contained in the APNN-toolbox (Bause et al. 1998), for the Petri net formalism, and in SMART (=-=Ciardo and Miner 1996-=-), for arbitrary discrete-state formalisms. The reduced memory requirements allows us to solve very large Markov models (over 10 7 states) on a modern workstation in a matter of hours. Acknowledgments... |

34 |
A chaotic asynchronous algorithm for computing the fixed point of a nonnegative matrix of unit spectral radius
- Lubachevsky, Mitra
- 1986
(Show Context)
Citation Context ...riant of Gauss-Seidel in conjunction with an on-the-fly approach for a set of modeling formalisms including GSPNs, stochastic activity networks, and stochastic reward nets. A similar idea is also in (=-=Lubachevsky and Mitra 1986-=-). However, we do not consider this approach further since its time complexity is at least as high as that of the algorithms we present and events requiring no time, e.g., immediate transitions in GSP... |

32 | On-The-Fly” Solution Techniques for Stochastic Petri Nets and Extensions - Deavours, Sanders - 1997 |

28 | Numerical solution methods based on structured descriptions of Markovian models, in Computer performance evaluation - Buchholz - 1991 |

26 |
Hierarchical structuring of superposed GSPNs
- Buchholz
- 1997
(Show Context)
Citation Context ...rom projection techniques, aggregation/disaggregation is a promising way of speeding up the convergence of iterative techniques. An approach in the context of Kronecker-based analysis is proposed in (=-=Buchholz 1997-=-a). As the most time-consuming step is still the computation of vector-matrix products, the algorithms presented in this paper can be applied. An alternative approach to realize Gauss-Seidel and block... |

25 | A toolbox for functional and quantitative analysis of deds
- Bause, Buchholz, et al.
- 1998
(Show Context)
Citation Context ...ration purposes only. Since K is model-dependent, a recursive implementation is required. To improve performance, we implemented this recursion iteratively with dynamicallyallocated arrays of size K (=-=Bause et al. 1998-=-). 4.3 Functional Rates In the SANs proposed by Fernandes, Plateau, and Stewart (1998), rates of a submodel can be a function of the state of other submodels. These are called functional rates and can... |

23 | An Efficient Disk-Based Tool for Solving Very Large Markov Models - Deavours, Sanders - 1997 |

22 |
Reachability Analysis Based on Structured Representations
- Kemper
- 1996
(Show Context)
Citation Context ...ices resulting from a high-level model structured into submodels. The method has been applied to several high-level formalisms where models are described in a compositional way (Donatelli 1993, 1994, =-=Kemper 1996-=-a, Plateau 1985). Solution methods exploiting a Kronecker structure are iterative but they differ from conventional iterative techniques in how they perform the required vector-matrix multiplications.... |

20 | A class of hierarchical queueing networks and their analysis, Queueing Syst
- BUCHHOLZ
- 1994
(Show Context)
Citation Context ...nd ˆy at the end (provided ˆx is not used elsewhere), since the algorithms overwrite only those values of ˆx which are not used any further or are currently stored in z. The complexity of Pot-Sh (see =-=Buchholz, 1994-=-; Fernandes et al., 1998) can be rewritten as: � K� O ¯nk · η[A k � � K� η[A ] = O n · k � ] = O (n · K · α) . k=1 Hence, Pot-Sh is faster than a multiplication using explicit storage if and only if k... |

19 | Structured analysis approaches for large Markov chains
- Buchholz
- 1999
(Show Context)
Citation Context ...torization of Q cannot be used in our case, as they would destroy the Kronecker representation of Q. Preliminary results on preconditioners that can be represented as sum of Kronecker products exist (=-=Buchholz 1999-=-, Stewart 1994) but, as stated on p. 490 of (Stewart 1994): “Much more research needs to be conducted into finding preconditioning techniques that can be applied to SAN descriptors without the need to... |

11 | On-the- solution techniques for stochastic Petri nets and extensions - Deavours, Sanders - 1997 |

10 | Iterative methods based on splittings for stochastic automata networks - Uysal, Dayar - 1998 |

8 | Kronecker products and shue algebra - Davio - 1981 |

7 | Numerical Issues for Stochastic Automata Networks
- Fernandes, Plateau, et al.
- 1996
(Show Context)
Citation Context ...tions, f(e, i[1,K]) = � K k=1 fk(e, ik), are naturally managed by the Kronecker representation considered so far. We can always achieve this “product form” by merging submodels into larger submodels (=-=Fernandes et al. 1996-=-), although, in the worst case, we might end up with a single submodel. Alternatively, we could simply store the entries corresponding to f(e, i[1,K]), for 21seach reachable state i[1,K] and for any e... |

6 |
An aggregation-disaggregation algorithm for stochastic automata networks
- Buchholz
- 1997
(Show Context)
Citation Context ...rom projection techniques, aggregation/disaggregation is a promising way of speeding up the convergence of iterative techniques. An approach in the context of Kronecker-based analysis is proposed in (=-=Buchholz 1997-=-a). As the most time-consuming step is still the computation of vector-matrix products, the algorithms presented in this paper can be applied. An alternative approach to realize Gauss-Seidel and block... |

6 | An ecient disk-based tool for solving very large Markov models - Deavours, Sanders - 1997 |

6 | A toolbox for functional and quantitative analysis of - BAUSE, BUCHHOLZ, et al. - 1998 |

2 |
Superposition of generalized stochastic Petri nets and its impact on performance analysis
- Kemper
- 1996
(Show Context)
Citation Context ...ices resulting from a high-level model structured into submodels. The method has been applied to several high-level formalisms where models are described in a compositional way (Donatelli 1993, 1994, =-=Kemper 1996-=-a, Plateau 1985). Solution methods exploiting a Kronecker structure are iterative but they differ from conventional iterative techniques in how they perform the required vector-matrix multiplications.... |

1 | On the use of Kronecker operators for the solution of generalized stochastic Petri nets - Buchholz, Donatelli, et al. - 1996 |