Modelling resource contention in multi-robot task allocation problems with uncertain timing

This paper proposes an analytical framework for modelling resource contention in multi-robot systems, where the travel times and task durations are uncertain. It uses several approximation methods to quickly and accurately calculate the probability distributions describing the times at which the tasks start and finish. Specific contributions include a method for calculating the probability of a set of independent normally distributed random events occurring in a given order, an upper bound on that probability, and a method for calculating the most likely and $n$-th most likely orders of occurrence for a set of independent normally distributed random events that have equal standard deviations. The complete framework is shown to be much faster than a Monte Carlo approach for the same accuracy in two multi-robot task allocation problems. This is a general framework that is agnostic to the optimisation method and objective function used, and is applicable to a wide range of robotics and non-robotics problems.