[inigyou]

To explain this: if the system is 95% utilized, and a new request comes in, there's a 95% chance the system is already busy and the request has to wait. But after the currently in-progress request finishes, there's still a 95% chance the system is busy (with another request that was already queued behind it) and request X has to wait. After that one finishes, same thing. On average, request X has to wait for about 20 other requests at 95% utilisation - or 5 requests at 80% utilisation - or 10000 requests at 99.99% utilisation. And that's just the mean, not percentiles.

[thaumasiotes]

Your math doesn't make any sense.

[inigyou]

Which part? 95% utilisation means at any random point in time there's a 95% chance the system is utilized and a 5% chance it is not utilized at that moment, so on average you have to try 20 times to find an unutilised moment?

[thaumasiotes]

No, none of that is true.

Think about a secretarial pool with five secretaries. The pool is at 95% utilization. This might mean that one of the secretaries is at 100% utilization and the other four are at 93.75%. It might mean that all of the secretaries are at 95%. It might mean that four of them are at 100% and one is at 75%. It might mean something else.

Going back to that third case, we can easily see that with the system at 95% utilization, at any given moment in time there is a 75% chance that the system is busy and unable to begin a new task, and a 25% chance that the system has capacity to begin a new task right away.

In the full-average case where all five secretaries are at 95% utilization, the odds that the system has spare capacity at any given moment are 22.62% (= 1 - 0.95 ^ 5), not too different from the case where four secretaries are kept busy 100% of the time.