Approximate steady-state analysis of large Markov models based on the structure of their decision diagram encoding

We propose a new approximate numerical algorithm for the steady-state solution of general structured ergodic Markov models. The approximation uses a state-space encoding based on multiway decision diagrams and a transition rate encoding based on a new class of edge-valued decision diagrams. The new method retains the favorable properties of a previously proposed Kronecker-based approximation, while eliminating the need for a Kronecker-consistent model decomposition. Removing this restriction allows for a greater utilization of event locality, which facilitates the generation of both the state-space and the transition rate matrix, thus extends the applicability of this algorithm to larger and more complex models.

[1]  Andrew S. Miner.  Efficient solution of GSPNs using canonical matrix diagrams , 2001, Proceedings 9th International Workshop on Petri Nets and Performance Models.

[2]  Marco Ajmone Marsan,et al.  Modelling with Generalized Stochastic Petri Nets , 1995, PERV.

[3]  Gianfranco Ciardo,et al.  Logical and Stochastic Modeling with SMART , 2003, Computer Performance Evaluation / TOOLS.

[4]  William J. Stewart,et al.  Introduction to the numerical solution of Markov Chains , 1994 .

[5]  Yukio Takahashi,et al.  SNS 1.0: Synchronized Network Solver , 1996 .

[6]  Gianfranco Ciardo,et al.  Exploiting interleaving semantics in symbolic state-space generation , 2007, Formal Methods Syst. Des..

[7]  Kishor S. Trivedi,et al.  A Decomposition Approach for Stochastic Reward Net Models , 1993, Perform. Evaluation.

[8]  Gianfranco Ciardo,et al.  Saturation Unbound , 2003, TACAS.

[9]  Tiziano Villa,et al.  Multi-valued decision diagrams: theory and applications , 1998 .

[10]  Gianfranco Ciardo,et al.  Saturation-Based Symbolic Reachability Analysis Using Conjunctive and Disjunctive Partitioning , 2005, CHARME.

[11]  Enric Pastor,et al.  Traversal Techniques for Concurrent Systems , 2002, FMCAD.

[12]  Gianfranco Ciardo,et al.  Logic and stochastic modeling with S m A r T , 2006, Perform. Evaluation.

[13]  Masahiro Fujita,et al.  Multi-Terminal Binary Decision Diagrams: An Efficient Data Structure for Matrix Representation , 1997, Formal Methods Syst. Des..

[14]  Andrew S. Miner,et al.  Saturation for a General Class of Models , 2004, IEEE Transactions on Software Engineering.

[15]  Oded Maler,et al.  On the Representation of Probabilities over Structured Domains , 1999, CAV.

[16]  G. De Micheli,et al.  Computer-Oriented Formulation of Transition-Rate Matrices via Kronecker Algebra , 1981, IEEE Transactions on Reliability.

[17]  Gianfranco Ciardo,et al.  The saturation algorithm for symbolic state-space exploration , 2006, International Journal on Software Tools for Technology Transfer.

[18]  Peter Buchholz,et al.  Complexity of Memory-Efficient Kronecker Operations with Applications to the Solution of Markov Models , 2000, INFORMS J. Comput..

[19]  Robert K. Brayton,et al.  Algorithms for discrete function manipulation , 1990, 1990 IEEE International Conference on Computer-Aided Design. Digest of Technical Papers.

[20]  Randal E. Bryant,et al.  Graph-Based Algorithms for Boolean Function Manipulation , 1986, IEEE Transactions on Computers.

[21]  Andrew S. Miner,et al.  Implicit GSPN reachability set generation using decision diagrams , 2004, Perform. Evaluation.

[22]  Jordi Cortadella,et al.  Petri Net Analysis Using Boolean Manipulation , 1994, Application and Theory of Petri Nets.

[23]  Gianfranco Ciardo,et al.  Symbolic State-Space Generation of Asynchronous Systems Using Extensible Decision Diagrams , 2009, SOFSEM.

[24]  Gianfranco Ciardo,et al.  Saturation: An Efficient Iteration Strategy for Symbolic State-Space Generation , 2001, TACAS.

[25]  Gianfranco Ciardo,et al.  Using Edge-Valued Decision Diagrams for Symbolic Generation of Shortest Paths , 2002, FMCAD.

[26]  Paulo Fernandes,et al.  Efficient descriptor-vector multiplications in stochastic automata networks , 1998, JACM.

[27]  Gianfranco Ciardo,et al.  Using the exact state space of a Markov model to compute approximate stationary measures , 2000, SIGMETRICS '00.