Overview
One way to extend the inventory models is to assume that the demand for purchasing product is random. This means that the firm will not know ahead of time how much inventory to purchase in order to cover demand over the next period.
See Stochastic Demand Modeling for information about assigning and fitting a distribution to the demand variable.
Probability of Missing Inventory
Once demand has been modeled as a statistical distribution, one can compute the probability that demand will exceed inventory through the cumulative distribution funciton,
{% \alpha = F_N (inventory, \mu, \sigma) %}
where {% \alpha %} is the probability, {% \mu %} is the average demand, {% \sigma %} is the standard deviation and {% inventory %} is the current inventory level.
Likewise, one can compute the amount of inventory to purchase to cover the next periods demand within a given probability. (i.e. how much inventory to purchase so that there is only a 5% chance of running out over the period.)
{% purchase = F_N^{-1} (\alpha, \mu, \sigma) %}
Correlated Demand
The above formula assumes that demand from one period to the next are independent (or uncorrelated). In practice, this may be a bad assumption. A simple correction to the variance of the normal distribution includes the correlation.