# Verification and Control of Turn-Based Probabilistic Real-Time Games

## TL;DR

This paper introduces a new model for turn-based probabilistic timed multi-player games, enabling quantitative verification of probabilistic, real-time, multi-agent systems with applications in security and scheduling.

## Contribution

It extends probabilistic timed automata to multi-player settings with nondeterminism and real-time clocks, providing methods to compute probabilities and expected costs for verification.

## Key findings

- Successfully computed probabilities and costs in case studies.
- Extended verification techniques to multi-player probabilistic real-time systems.
- Demonstrated applicability in security and task scheduling scenarios.

## Abstract

Quantitative verification techniques have been developed for the formal analysis of a variety of probabilistic models, such as Markov chains, Markov decision process and their variants. They can be used to produce guarantees on quantitative aspects of system behaviour, for example safety, reliability and performance, or to help synthesise controllers that ensure such guarantees are met. We propose the model of turn-based probabilistic timed multi-player games, which incorporates probabilistic choice, real-time clocks and nondeterministic behaviour across multiple players. Building on the digital clocks approach for the simpler model of probabilistic timed automata, we show how to compute the key measures that underlie quantitative verification, namely the probability and expected cumulative price to reach a target. We illustrate this on case studies from computer security and task scheduling.