We consider the Markov processes never exiting a subspace A of the state space E or in another words, Markov processes conditioned to stay in the subspace A. We show how the knowledge of the exact asymptotics of the tail distribution of the exit time helps to find the suitable exponential martingale, which in turn, serves for the change of measure. Under the new probability measure the process is the sought for never exiting one the subspace A. Using the theory, we find formulas for the extended generator of a PDMP under the new probability measure. A detailed analysis is given for the workload in the M/G/1 queue conditioned to stay positive. We also find the exact asymptotics for the busy period in the GI/G/1 queue. Of course we must first markovize the workload process by the attaching the second component of the remaining arrival time process.