Exact and ordinary lumpability in finite Markov chains is considered. Both concepts naturally define an aggregation of the Markov chain yielding an aggregated chain that allows the exact determination of several stationary and transient results for the original chain. We show which quantities can be determined without an error from the aggregated process and describe methods to calculate bounds on the remaining results. Furthermore, the concept of lumpability is extended to near lumpability yielding approximative aggregation.
We present new algorithms for the solution of large structured Markov models whose in nitesimal generator can be expressed as a Kronecker expression of sparse matrices. We then compare them with the shu e-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 modi ed 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 di erent sparsity assumptions. A nontrivial example is studied to illustrate the complexity of the implemented algorithms.
Bisimulation is a well known equivalence relation for discrete event systems and has been extended to probabilistic and stochastic systems. This paper introduces a general definition of bisimulation which can be applied to finite automata where weights and labels are assigned to transitions. It is shown that this general view contains several known bisimulations as special cases and defines naturally equivalences for different classes of models. Apart from the well known forward bisimulation, also backward bisimulation is introduced and it is shown that both types of bisimulation preserve different types of results. Furthermore it is shown that forward and backward bisimulation are congruences according to commonly known composition operations for automata.
The representation of general distributions or measured data by phase-type distributions is an important and non-trivial task in analytical modeling. Although a large number of different methods for fitting parameters of phase-type distributions to data traces exist, many approaches lack efficiency and numerical stability. In this paper, a novel approach is presented that fits a restricted class of phase-type distributions, namely mixtures of Erlang distributions, to trace data.For the parameter fitting an algorithm of the expectation maximization type is developed. The paper shows that these choices result in a very efficient and numerically stable approach which yields phase-type approximations for a wide range of data traces that are as good or better than approximations computed with other less efficient and less stable fitting methods. To illustrate the effectiveness of the proposed fitting algorithm, we present comparative results for our approach and two other methods using six benchmark traces and two real traffic traces as well as quantitative results from queueing analysis.
Keywords:Performance and dependability assessment/analytical and numerical techniques, design of tools for performance/dependability assessment, traffic modeling, hyper-Erlang distributions.
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