  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 exact and fast approximation
methods for calculating the probability of a set of independent normally
distributed random events occurring in a given order, 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, and a method for approximating the conditional probability
distributions of the events given a specific order of the events. The complete
framework is shown to be faster than a Monte Carlo approach for the same
accuracy in two multi-robot task allocation problems. In addition, the
importance of incorporating uncertainty is demonstrated through a comparison
with a deterministic method. This is a general framework that is agnostic to
the optimisation method and objective function used, and is applicable to a
wide range of problems.
