Waiting-time distribution for a finite-capacity single-server queue with constant service and vacation times

We consider a finite-capacity single-server queue with constant service and vacation times, which is seen in the time division multiple access (TDMA) scheme. First we derive the probability that j customers remain in the queue when a test customer arrives. Using this probability we then evaluate the probability that the test customer who arrives during the vacation or service time has to wait in the queue for longer than a given time. From these results, we obtain the waiting time distribution for the customer arriving at an arbitrary time. We also show a practical application to wireless TDMA communications systems.