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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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    @joriki I guess I wasn't clear enough. The rate in which servers drop is $min(\alpha,x)$ where $x$ is the number of servers standing. The rate in which servers are repaired is $min(\beta,x)$ where x is the number of inactive servers. $Z_1,Z_2$ are very much dependent.2012-12-15

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