# Poisson subtraction distribution

We have $k$ servers. Servers drops occurs according to Poisson distribution with average of $\alpha$ every second, servers repairs occurs according to Poisson distribution with average of $\beta$ every second. What's the probability, that after $t$ seconds, $x$ servers are standing?

Here's what I've tried:

Let $Z_1\sim\mathrm{Poisson}(\alpha\cdot t)$ and $Z_2\sim\mathrm{Poisson}(\beta\cdot t)$ describing the number of falls and repairs until time $t$ ,respectively. Then, We're looking for $P(Z_1-Z_2) = k-x$. Here I got stuck since we have infinite sum of events that need to be considered (Each 2 positive integers $a,b$ such that $a-b=k-x$ ).

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The inconsistent grammatical number in "servers drops occurs" and "servers repair occurs" makes it ambiguous whether these drops and repairs occur at a rate of $\alpha$ and $\beta$ per second, respectively, for each server individually or for the server park as a whole. Also it needs to be clarified whether a server drop can affect a server that's already out, and if so, whether it then needs two repairs or just one. – joriki Dec 15 '12 at 14:12
@joriki The drops/repairs occur for the whole park. Server drop can not effect a server that's already out. – Shmoopy Dec 15 '12 at 14:16
That seems to be inconsistent with the fact that the drops and repairs occur according to Poisson distributions; those distributions assign non-zero probability to cases in which there are $k+1$ server drops before there are any repairs, but that's not compatible with your above comment. – joriki Dec 15 '12 at 14:19
Actually my above comment assumes independence between the drops and the repairs; if that wasn't meant to be implied (you should specify it anyway for completeness), the probability couldn't be determined without knowing the joint distribution. – joriki Dec 15 '12 at 14:21
@joriki Drops can only occur if at least one server is active, and repairs can only occur if at least one server is down. – Shmoopy Dec 15 '12 at 14:21