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Probability: supply and demand

  1. Feb 24, 2009 #1
    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 [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?

    2. Relevant 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]

    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

    [tex]
    \mathbb{E}\, = \frac{\lambda}{m} - s
    [/tex]
     
    Last edited: Feb 24, 2009
  2. jcsd
  3. Feb 24, 2009 #2
    bump

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

    P[Shortage within lead time] =
    [tex]
    \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}}
    [/tex]

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