Two stage sampling procedures to estimate the parameter λ of a Poisson process by a bounded length confidence interval with specified confidence coefficient are compared. At each stage, one can either fix the parametertand count the events occurring in the time interval (0;t] or fix the number of events to be observed and measure the waiting time. All combinations of these two approaches are considered. Expected total durations are given for each case and it is concluded that at any stage, counting events in fixed time intervals is preferred.