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bigplanet401
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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 [tex]\lambda[/tex] 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
[tex]
\mathbb{P}\, [T=t] = m e^{-mt} \, .
[/tex]
Let D be the daily demand. Then D has density function
[tex]
\mathbb{P} \, [D = k] = \frac{\lambda^{-k} e^{-\lambda}}{k!}
[/tex]
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
[tex]
\mathbb{E}\,
[/tex]
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