Discrete-time queues are frequently used in telecommunication networks. Thi
s paper proposes a new method which is based only on probabilistic argument
s. It derives the distributions of numbers of customers in the discrete-tim
e systems Geom/G/1 and Geom/G/1/M. Though the derivation does not involve u
se of the transforms, the transforms may be obtained, if desired. Another a
dvantage of this is that the numerical results obtained are stable as there
are no negative signs involved in summations. Further, the method can be e
asily used to solve more complex problems in discrete- and continuous-time
queues.