We have developed a new approximate probabilistic model-checking method for untimed properties in probabilistic systems, expressed in a probabilistic temporal logic (PCTL, CSL). This method, in contrast to the existing ones, does not require the untimed until properties to be bounded a priori, where the bound refers to the number of discrete steps in the system required to verify the until property. The method consists of two phases. In the first phase, a suitable system- and property-dependent bound k 0 is obtained automatically. In the second phase, the probability of satisfying the k 0 -bounded until property is computed as the estimate of the probability of satisfying the original unbounded until property. Both phases require only verification of bounded until properties, which can be effectively performed by simulation-based methods. We prove the correctness of the proposed two-phase method and present its optimized implementation in the widely used PRISM model-checking engine. We compare this implementation with sampling-based model-checking techniques implemented in two tools: PRISM and MRMC. We show that for several models these existing tools fail to compute the result, while the two-phase method successfully computes the result efficiently with respect to time and space.

In a number of applications, the underlying stochastic process is modeled as a finite-state discrete-time Markov chain that cannot be observed directly and is represented by an auxiliary process. The maximum a posteriori (MAP) estimator is widely used to estimate states of this hidden Markov model through available observations. The MAP path estimator based on a finite number of observations is calculated by the Viterbi algorithm, and is often referred to as the Viterbi path. It was recently shown in, and, (see also and) that under mild conditions, the sequence of estimators of a given state converges almost surely to a limiting regenerative process as the number of observations approaches infinity. This in particular implies a law of large numbers for some functionals of hidden states and finite Viterbi paths. The aim of this paper is to provide the corresponding large deviation estimates.

We study the problem of statistical model checking of probabilistic systems for PCTL unbounded until property PJoinp(Æ1UÆ2) (where Join |X| {<, d, >, e}) using the computation of P d 0(Æ1UÆ2). The approach is first proposed by Sen et al. in CAV'05 but their approach suffers from two drawbacks. Firstly, the computation of Pd0Æ1UÆ2) requires for its validity, a user-specified input parameter ´2 which the user is unlikely to correctly provide. Secondly, the validity of computation of Pd0Æ1UÆ2) is limited only to probabilistic models that do not contain loops. We present a new technique which addresses both problems described above. Essentially our technique transforms the hypothesis test for the unbounded until property in the original model into a new equivalent hypothesis test for bounded until property in our modified model. We empirically show the effectiveness of our technique and compare our results with those using the method proposed by Sen et al.

Modeling the available bandwidth of a path using a known stochastic process is one possible method for estimating future available bandwidth along the path without explicit support from network routers. Our two hypotheses for the stochastic process are as follows. First, an auto-regressive integrated moving-average process (ARIMA) is a suitable model for the available bandwidth over time of a path. Second, the available bandwidth over time of a path can be modeled as a self-similar process. We verify both hypotheses using R statistical software and available bandwidth data sets published by Stanford Linear Accelerator Center (SLAC). Our results indicate that the available bandwidth over time of an end-to-end path can be modeled as fractional Gaussian Noise (FGN) and seasonal fractional ARIMA (SFARIMA) processes. On the other hand, we found that an ARIMA process is not a good model for available bandwidth over time of an end-to-end path.