Compound geometric approximation under a failure rate constraint

We consider compound geometric approximation for a nonnegative, integervalued random variable W . The bound we give is straightforward but relies on having a lower bound on the failure rate of W . Applications are presented to M/G/1 queuing systems, for which we state explicit bounds in approximations for the number of customers in the system and the number of customers served during a busy period. Other applications are given to birth-death processes and Poisson processes.