If items are known to arrive at a certain fixed rate, then the probability of any particular number of arrivals during an interval fits a Poisson distribution.
However, this assumes that the mean number of arrivals is constant and does not change.
We can imagine a system in which the mean number of arrivals is changing. For example, imagine that every day we count the number of arrivals that day and this is all of our information:
53, 97, 107, 60, 203, 42, 19, 127 ...
The mean constantly changes. Furthermore, if we imagine that more recent arrivals are more indicative of future trends, then we can imagine a changing weighted average or fading window.
So, there is a need to do two things: (1) determine the degree to which more recent arrival counts have a greater effect on the probability of the next day's arrival count probability distribution, and (2) then use this information to mathematically describe the resulting distribution and mass function.
How can this be done?
As a simple possible answer to this question, which may not be the best, one approach I considered is just to use a fading window. So in this case we weight the mean by a two-parameter function and then try each type of growth pattern: linear, geometric and and exponential. Each relation has two parameters. For example, for the linear model there is an offset and a slope. For example, one instance might be an offset of 2 days and a slope of 1/3. So, given the example sequence, the mean would be computed as:
(127 + 19 + 42 * 2/3 + 203 * 1/3 ) / ( 1 + 1 + 2/3 + 1/3 ) = 80.5556
So, every possible slope and offset is tried on each instance of the historical data series (3-dimensions). The process is repeated for geometric and exponential fading functions. By this means, the best fitting fading function is discovered. Then, using this function a mean is computed and the Poission distribution is used as normal to predict the probability of the arrivals for the next day.