Mean fixation time estimates in constant size populations

We consider a population consisting of $N$ particles each of which some type is ascribed to. All particles die at the integer time moments and produce a random amount of particles of the same type as the parent. Moreover, the population retains its size $N$ and the random vectors defining the number of offsprings of each particle have exchangeable distributions. We obtain several upper bounds for the expectation of the variable equal to the number of the generation when all particles in the population become single-type or almost single-type. Here we fix an arbitrary initial configuration of particles according to types.