Maximizing the Conditional Expected Reward for Reaching the Goal

The paper addresses the problem of computing maximal conditional expected accumulated rewards until reaching a target state (briefly called maximal conditional expectations) in finite-state Markov decision processes where the condition is given as a reachability constraint. Conditional expectations of this type can, e.g., stand for the maximal expected termination time of probabilistic programs with non-determinism, under the condition that the program eventually terminates, or for the worst-case expected penalty to be paid, assuming that at least three deadlines are missed. The main results of the paper are (i) a polynomial-time algorithm to check the finiteness of maximal conditional expectations, (ii) PSPACE-completeness for the threshold problem in acyclic Markov decision processes where the task is to check whether the maximal conditional expectation exceeds a given threshold, (iii) a pseudo-polynomial-time algorithm for the threshold problem in the general (cyclic) case, and (iv) an exponential-time algorithm for computing the maximal conditional expectation and an optimal scheduler.Comment: 103 pages, extended version with appendices of a paper accepted at TACAS 201

Energy loss in perturbative QCD

We review the properties of energetic parton propagation in hot or cold QCD matter, as obtained in recent works. Advances in understanding the energy loss - collisional and radiative - are summarized, with emphasis on the latter: it features very interesting properties which may help to detect the quark-gluon plasma produced in heavy ion collisions. We describe two different theoretical approaches, which lead to the same radiated gluon energy spectrum. The case of a longitudinally expanding QCD plasma is investigated. The energy lost by a jet with given opening angle is calculated in view of making predictions for the suppression (quenching) of hard jet production. Phenomenological implications for the difference between hot and cold matter are discussed. Numerical estimates of the loss suggest that it may be significantly enhanced in hot compared to cold matter.Comment: 49 pages latex file with 11 embedded PS figures. Uses ar.sty (included), one equation revised. submitted to Annual Review of Nuclear and Particle Scienc

The Complexity of Graph-Based Reductions for Reachability in Markov Decision Processes

We study the never-worse relation (NWR) for Markov decision processes with an infinite-horizon reachability objective. A state q is never worse than a state p if the maximal probability of reaching the target set of states from p is at most the same value from q, regard- less of the probabilities labelling the transitions. Extremal-probability states, end components, and essential states are all special cases of the equivalence relation induced by the NWR. Using the NWR, states in the same equivalence class can be collapsed. Then, actions leading to sub- optimal states can be removed. We show the natural decision problem associated to computing the NWR is coNP-complete. Finally, we ex- tend a previously known incomplete polynomial-time iterative algorithm to under-approximate the NWR

Distributed Synthesis in Continuous Time

We introduce a formalism modelling communication of distributed agents strictly in continuous-time. Within this framework, we study the problem of synthesising local strategies for individual agents such that a specified set of goal states is reached, or reached with at least a given probability. The flow of time is modelled explicitly based on continuous-time randomness, with two natural implications: First, the non-determinism stemming from interleaving disappears. Second, when we restrict to a subclass of non-urgent models, the quantitative value problem for two players can be solved in EXPTIME. Indeed, the explicit continuous time enables players to communicate their states by delaying synchronisation (which is unrestricted for non-urgent models). In general, the problems are undecidable already for two players in the quantitative case and three players in the qualitative case. The qualitative undecidability is shown by a reduction to decentralized POMDPs for which we provide the strongest (and rather surprising) undecidability result so far

Value Iteration for Long-run Average Reward in Markov Decision Processes

Markov decision processes (MDPs) are standard models for probabilistic systems with non-deterministic behaviours. Long-run average rewards provide a mathematically elegant formalism for expressing long term performance. Value iteration (VI) is one of the simplest and most efficient algorithmic approaches to MDPs with other properties, such as reachability objectives. Unfortunately, a naive extension of VI does not work for MDPs with long-run average rewards, as there is no known stopping criterion. In this work our contributions are threefold. (1) We refute a conjecture related to stopping criteria for MDPs with long-run average rewards. (2) We present two practical algorithms for MDPs with long-run average rewards based on VI. First, we show that a combination of applying VI locally for each maximal end-component (MEC) and VI for reachability objectives can provide approximation guarantees. Second, extending the above approach with a simulation-guided on-demand variant of VI, we present an anytime algorithm that is able to deal with very large models. (3) Finally, we present experimental results showing that our methods significantly outperform the standard approaches on several benchmarks

Optimizing Performance of Continuous-Time Stochastic Systems using Timeout Synthesis

We consider parametric version of fixed-delay continuous-time Markov chains (or equivalently deterministic and stochastic Petri nets, DSPN) where fixed-delay transitions are specified by parameters, rather than concrete values. Our goal is to synthesize values of these parameters that, for a given cost function, minimise expected total cost incurred before reaching a given set of target states. We show that under mild assumptions, optimal values of parameters can be effectively approximated using translation to a Markov decision process (MDP) whose actions correspond to discretized values of these parameters

Percentile Queries in Multi-Dimensional Markov Decision Processes

Markov decision processes (MDPs) with multi-dimensional weights are useful to analyze systems with multiple objectives that may be conflicting and require the analysis of trade-offs. We study the complexity of percentile queries in such MDPs and give algorithms to synthesize strategies that enforce such constraints. Given a multi-dimensional weighted MDP and a quantitative payoff function $f$, thresholds $v_i$ (one per dimension), and probability thresholds $\alpha_i$, we show how to compute a single strategy to enforce that for all dimensions $i$, the probability of outcomes $\rho$ satisfying $f_i(\rho) \geq v_i$ is at least $\alpha_i$. We consider classical quantitative payoffs from the literature (sup, inf, lim sup, lim inf, mean-payoff, truncated sum, discounted sum). Our work extends to the quantitative case the multi-objective model checking problem studied by Etessami et al. in unweighted MDPs.Comment: Extended version of CAV 2015 pape

Explicit Model Checking of Very Large MDP using Partitioning and Secondary Storage

The applicability of model checking is hindered by the state space explosion problem in combination with limited amounts of main memory. To extend its reach, the large available capacities of secondary storage such as hard disks can be exploited. Due to the specific performance characteristics of secondary storage technologies, specialised algorithms are required. In this paper, we present a technique to use secondary storage for probabilistic model checking of Markov decision processes. It combines state space exploration based on partitioning with a block-iterative variant of value iteration over the same partitions for the analysis of probabilistic reachability and expected-reward properties. A sparse matrix-like representation is used to store partitions on secondary storage in a compact format. All file accesses are sequential, and compression can be used without affecting runtime. The technique has been implemented within the Modest Toolset. We evaluate its performance on several benchmark models of up to 3.5 billion states. In the analysis of time-bounded properties on real-time models, our method neutralises the state space explosion induced by the time bound in its entirety.Comment: The final publication is available at Springer via http://dx.doi.org/10.1007/978-3-319-24953-7_1

An iterative decision-making scheme for Markov decision processes and its application to self-adaptive systems

Software is often governed by and thus adapts to phenomena that occur at runtime. Unlike traditional decision problems, where a decision-making model is determined for reasoning, the adaptation logic of such software is concerned with empirical data and is subject to practical constraints. We present an Iterative Decision-Making Scheme (IDMS) that infers both point and interval estimates for the undetermined transition probabilities in a Markov Decision Process (MDP) based on sampled data, and iteratively computes a confidently optimal scheduler from a given finite subset of schedulers. The most important feature of IDMS is the flexibility for adjusting the criterion of confident optimality and the sample size within the iteration, leading to a tradeoff between accuracy, data usage and computational overhead. We apply IDMS to an existing self-adaptation framework Rainbow and conduct a case study using a Rainbow system to demonstrate the flexibility of IDMS