Exponential limiting distributions in queueing systems with deadlines

This paper contains some general theorems on the limiting distribution of a certain random variable arising in the context of recurrent events. is especially meaningful to the investigation of discrete-time queueing systems subjected to service time deadlines. It may be viewed as a sum of mutually independent conditional random variables having the same distribution. It is assumed that depends on a parameter T and tends to an unconditional random variable for Under some weak conditions concerning the conditional probability generating function , it follows that is approximately exponentially distributed with parameter , , which tends to infinity for Provided here are uniform asymptotic expansions for the appropriate probabilities and for all moments, also.