Stochastic decomposition of the M/M/1 queue with environment dependent working vacation

We consider an M/M/1 queue with n types of working vacation. After a non zero busy period if the server finds the system empty, it opts for one of the n types of working vacation depending on the environment. On vacation completion epoch finding an empty system, it remains in the respective vacation. We demonstrate the stochastic decomposition structure of the queue length and waiting time of this M/M/1 queue and obtain the distribution of the additional queue length and additional delay.