This paper studies generalized M/G/1 queues in which the first N customers of each busy period receive exceptional services. Applying the supplementary variable approach, we derive the recursion formulas to obtain the generating function of the stationary queue length distribution given that n customers have been served since the beginning of current busy period. Furthermore, we present a computationally tractable scheme which recursively determines the moments of the queue length distribution and the sojourn time distribution. Special cases are treated in detail. Numerical examples are also provided.
