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Jobs are submitted for a computer according to a Poisson process with a rate λ
(jobs / hour).

Determine the probability that no job is lost if the computer crashes in the:
in the first 'a' minutes 
and 
in the last 'b' minutes of one hour.

(a job is lost if the computer is not working at the moment it is submitted).

I tried to modify λ to match the 'a' and the first few minutes 'b' past, but ended up losing. But I could see a few things: The problem whether the probability of:

1) No job lost if the computer crashes in the first 'a' minutes;

2) No job lost if the computer crashes in the last 'b' minutes;

Really, I'm confused, can someone help me?

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1 Answer 1

up vote 2 down vote accepted

1)

Let's define $X$ to be the number of jobs submitted to the computer in 1 hour. $X$ is Poisson with mean $\lambda$.

Because of the memoryless property of a Poisson event, it doesn't matter which $a$ minutes timeframe was taken to measure. First, convert $a$ mins to hours like so:

$$a \text{ mins} = \frac{a}{60} \text{hours}$$

Number of jobs lost in $a$ minutes will follow a Poisson distribution, let's call it $A$ with its parameter, let's call it $\lambda_A$ to be:

$$\begin{align*} \lambda_A &=\frac{a}{60} \times \lambda \\&= \frac{a\lambda}{60} \end{align*} $$

Plug this into the formula for Poisson for $P(A=0)$ and you will be fine.

2)

This is a little bit trickier. Let's call this parameter $\lambda_B$. This time period can be defined as

$$ \lambda_B = \frac{60-b}{60} \text{ hours} $$

Do the same thing and you should get your correct answer in terms of $\lambda$.

I'll give you another hint - the probability that there are no jobs received in a 1 hour period is

$$P(X=0) = \frac{\lambda^0e^{-\lambda}}{0!}$$

Recall that this $X$ is actually the $X$ referred to at the beginning of my answer.

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Thank you! I was thinking that solution maybe diferent because this problem has two time periods: first 'a' minutes and last 'b' minutes. This solution seems like that one that I write for another problem. math.stackexchange.com/questions/305626/…. Thank you, you rocks!! –  Richard Feb 17 '13 at 11:51
    
If I multiply the result, will I find the intersection? the probability that no job is lost in the first 'a' minutes AND in the last 'b' minutes? –  Richard Feb 17 '13 at 11:54

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