Delays for last-come first-served service and the busy period

For Poisson input to a single server, it is shown that the stationary delay distribution function for last-come first-served service is equal to the distribution function for the busy period (the interval of time during which the server is continuously busy) only for exponential distribution of service time. A similar argument shows that the identity persists for a group of fully accessible servers, each with exponential distribution of service times — a result which has been a curiosity for some time. Finally, the delay distribution for last-come first-served service and a single server with constant service time is derived.