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 $a-b=k-x$ ).