Properties of the geometric and related processes

Some properties of the geometric process are studied along with those of a related process which we propose to call the α-series process. It is shown that the expected number of counts at an arbitrary time does not exist for the decreasing geometric process. The decreasing version of the α-series process does have a finite expected number of counts, under certain conditions. This process also has the same advantages of tractability as the geometric process; it exhibits some properties which may make it a useful complement to the increasing geometric process. In addition, it may be fitted to observed data as easily as the geometric process. Applications in reliability and stochastic scheduling are considered in order to demonstrate the versatility of the alternative model.