A Comparison of Sequential and GPU Implementations of Iterative Methods to Compute Reachability Probabilities∗

We consider the problem of computing reachability probabilities: given a Markov chain, an initial state of the Markov chain, and a set of goal states of the Markov chain, what is the probability of reaching any of the goal states from the initial state? This problem can be reduced to solving a linear equation A · x = b for x, where A is a matrix and b is a vector. We consider two iterative methods to solve the linear equation: the Jacobi method and the biconjugate gradient stabilized (BiCGStab) method. For both methods, a sequential and a parallel version have been implemented. The parallel versions have been implemented on the compute unified device architecture (CUDA) so that they can be run on a NVIDIA graphics processing unit (GPU). From our experiments we conclude that as the size of the matrix increases, the CUDA implementations outperform the sequential implementa-tions. Furthermore, the BiCGStab method performs better than the Jacobi method for dense matrices, whereas the Jacobi method does better for sparse ones. Since the reachability probabilities problem plays a key role in probabilistic model checking, we also compared the implementations for matrices obtained from a probabilistic model checker. Our experiments support the conjecture by Bosnacki et al. that the Jacobi method is superior to Krylov subspace methods, a class to which the BiCGStab method belongs, for probabilistic model checking.