Simulate Nested Logit Errors

I'm trying to simulate data that can be estimated using a nested logit model. If epsilon is distributed nested logit with a nesting structure $B_1, ..., B_K $, and nesting parameters $\lambda_1, ... \lambda_K$ then the CDF of epsilon is

I am looking for a way of simulating random deviates from this distribution in R. The package evd does not allow the user to specify a particular nesting structure. Clearly, the inverse CDF method won't work here, either.

Any advice would be greatly appreciated.