On the steady-state queue size distribution of the discrete-time Geo/G/1 queue with repeated customers

In this paper, we study the steady-state queue size distribution of the discrete-time Geo/G/1 retrial queue. We derive analytic formulas for the probability generating function of the number of customers in the system in steady-state. It is shown that the stochastic decomposition law holds for the Geo/G/1 retrial queue. Recursive formulas for the steady-state probabilities are developed. Computations based on these recursive formulas are numerically stable because the recursions involve only nonnegative terms. Since the regular Geo/G/1 queue is a special case of the Geo/G/1 retrial queue, the recursive formulas can also be used to compute the steady-state queue size distribution of the regular Geo/G/1 queue. Furthermore, it is shown that a continuous-time M/G/1 retrial queue can be approximated by a discrete-time Geo/G/1 retrial queue by dividing the time into small intervals of equal length and the approximation approaches the exact when the length of the interval tends to zero. This relationship allows us to apply the recursive formulas derived in this paper to compute the approximate steady-state queue size distribution of the continuous-time M/G/1 retrial queue and the regular M/G/1 queue.