# Probability: supply and demand

1. Feb 24, 2009

### bigplanet401

1. The problem statement, all variables and given/known data
A replenishment order is placed to raise the stock level of a given product. The current stock level is s units. The lead time of the replenishment order is a continuous random variable having an exponential distribution with a mean of 1/m days. Customer demand for the product occurs according to a Poisson process with an average demand of $$\lambda$$ units per day. Each customer asks for one unit of the product. What is the probability of a shortage occurring during the replenishment lead time and what is the expected value of the total shortage?

2. Relevant equations
The lead time has density function
$$\mathbb{P}\, [T=t] = m e^{-mt} \, .$$
Let D be the daily demand. Then D has density function
$$\mathbb{P} \, [D = k] = \frac{\lambda^{-k} e^{-\lambda}}{k!}$$

3. The attempt at a solution
I'm not sure how to combine this information to get the probability of a shortage. Do you have to use the law of total probability or generating functions? My guess for the second part is

$$\mathbb{E}\, = \frac{\lambda}{m} - s$$

Last edited: Feb 24, 2009
2. Feb 24, 2009

### bigplanet401

bump

I did some more work on it, and tried the following:

$$\int_0^\infty d\tau \, \,m e^{-m\tau} \times \frac{(\lambda \tau)^s e^{-\lambda \tau}}{s!}\\ =\frac{m \lambda^s}{(m + \lambda)^{s+1}}$$
$$m \rightarrow 0$$. Help!