The accumulation of beneficial mutations and convergence to a Poisson process

We consider a model of a population with fixed size N, which is subjected to an unlimited supply of beneficial mutations at a constant rate μ_N. Individuals with k beneficial mutations have the fitness (1+s_N)^k. Each individual dies at rate 1 and is replaced by a random individual chosen with probability proportional to its fitness. We show that when μ_N≪1/(NlogN) and N^−η≪s_N≪1 for some η<1, the fixation times of beneficial mutations, after a time scaling, converge to the times of a Poisson process, even though for some choices of s_N and μ_N satisfying these conditions, there will sometimes be multiple beneficial mutations with distinct origins in the population, competing against each other.