Waiting time distribution of a FIFO/LIFO Geo/D/1 queue

We consider a Geo/D/1 queue operating under a hybrid FIFO/LIFO discipline and obtain the waiting distribution. This model is a discrete version of a special case of the model earlier considered by Doshi. The number of customers in the system at steady state is independent of the service discipline and knowing that a deterministic service time in discrete time can be represented as a phase distribution we study the system as a Geo/Ph/1 queue. We present a simpler approach for studying this special case by using the matrix-geometric method to obtain the steady state distribution of this system and obtaining the waiting time by studying the system as an absorbing Markov chain. Computed example values of this solution illustrate significant changes in the maximum traffic intensity meeting a specified waiting time probability quantile as the FIFO waiting space varies. These differences are significant in favor of the FIFO/LIFO strategy when the server is fast, in the sense that many tasks can be served while mean delay remains less than a specific waiting time.