Summary: We investigate busy period and queue length (steady-state and transient) of a single-server Poisson queue with delayed-service, to set the tone for more complicated models that will appear later. We will analyse this model by considering $M/G/1$ as well as the use of differential difference equations approximating a non-Markovian system. We obtain the distribution of the length of a busy period, steady-state mean and distribution of the queue length, Laplace transform of the probability of the system being empty and Laplace transform of the generating function of the distribution of the transient queue length.