The derivation of sojourn time distribution is easily understood in M/M/1 queues in terms of numbers of service completions, using the random observer property. The same approach does not generalise directly to the M/G/1 queue because of a subtle dependence between the random variables involved, and an entirely different approach is usually taught in most courses on queueing theory. The M/M/1 approach to the M/G/1 case is applied by accounting for the dependence explicitly. The method then extends simply to M/G/1 queues with priority classes. Although the results themselves are not new, it is believed that the approach used is illuminating, constructive to consistent teaching of the subject and facilitates a concise treatment of priority queues.
Information from pubs.doc.ic.ac.uk/teaching-m-g-1.