Computing the Value of the Maximum Likelihood Estimate of the Parameter of a Posisson Distribution Truncated below One

In attempting to predict the number of demands occurring in a given interval of time for supplies in military inventory systems, it is useful to examine probability models for the distribution of demands for individual items of supply. A first approach in model building for this problem has been to assume that the distribution of demands for a given item is a Poisson distribution, with parameter. However, when Poisson distributions have been fitted to available data, it has been found in many cases that this model tends seriously to underestimate the frequency of periods in which no demands occur. A more general approach is to postulate the existence of two random mechanisms, one governing the occurrence or non-occurrence of demands in the given time period, and the other governing the specific number of demands made, given that demands occur.