Probability: supply and demand

[bigplanet401]

Homework Statement

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?

Homework 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
$$\mathbb{P} \, [D = k] = \frac{\lambda^{-k} e^{-\lambda}}{k!}$$

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

$$\mathbb{E}\, <s> = \frac{\lambda}{m} - s<br /> </s>$$

 

[bigplanet401]

bump

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

P[Shortage within lead time] =
$$\int_0^\infty d\tau \, \,m e^{-m\tau} \times \frac{(\lambda \tau)^s e^{-\lambda \tau}}{s!}\\<br /> =\frac{m \lambda^s}{(m + \lambda)^{s+1}}$$

I think this is in the right direction, but it doesn't have the right limiting behavior for
$$m \rightarrow 0$$. Help!