  We provide a probabilistic analysis of the banker algorithm when transition
probabilities may depend on time and space. The transition probabilities
evolve, as time goes by, along the trajectory of an ergodic Markovian
environment, whereas the spatial parameter just acts on long runs. Our model
appears as a new (small) step towards more general time and space dependent
protocols. Our analysis relies on well-known results in stochastic
homogenization theory and investigates the asymptotic behaviour of the rescaled
algorithm as the total amount of resource available for allocation tends to the
infinity. In the two dimensional setting, we manage to exhibit three different
possible regimes for the deadlock time of the limit system.
