Some New Results for the M/M/1 Queue

New results are obtained for the single server queue with Poisson arrivals and exponential service times. A closed form solution involving only finite sums is obtained for the probability that exactly i arrivals and j services occur over a time interval of length t in a queueing system that is idle at the beginning of the interval. Since many applications of queueing theory involve queues which are emptied and restarted periodically and thus not susceptible to analysis using the well-known equilibrium results, there are many potential applications for results obtained. Some ways in which the results obtained may be applied are discussed.