A transient symmetry analysis for the M/M/1/k queue

We develop new techniques involving group symmetries and complex analysis to obtain exact solutions for the transition probabilities of the M/M/1/k queueing process. These methods are based on the underlying Markovian structure of these random processes and do not involve any generating functions, Laplace transforms, or advanced special functions. Our techniques exploit the intrinsic group symmetries for both the state spaces and the matrix generators of the Markov processes related to the M/M/1/k queue. These results complement and extend the previous transient solutions given by Takács (Introduction to the theory of queues. University texts in the mathematical sciences, Oxford University Press, New York, 1962). Much of the inspiration for this work comes from viewing this queueing process as a fundamental Markovian model for the dynamics of a bike sharing station. The exact transient analysis for a related stopped version of this process can be used to address fundamental decision-making issues for managing bike-sharing services.