How often is the server overloaded?

Question

A web server is either healthy or overloaded each minute, and the next minute depends only on the current one: - If healthy now, it becomes overloaded next minute with probability 0.1 (and stays healthy with probability 0.9). - If overloaded now, it becomes healthy next minute with probability 0.4 (and stays overloaded with probability 0.6). In the long run, what fraction of minutes is the server overloaded?

Accepted Answer

We want the steady long-run shares (\piH, \piO) that stop changing minute to minute. In steady state the flow into "overloaded" equals the flow out. Minutes entering overloaded: \piH 	imes 0.1. Minutes leaving overloaded: \piO 	imes 0.4. Balance them: \piH 	imes 0.1 = \piO 	imes 0.4 \implies \piH = 4\,\piO. Since the shares add to 1: 4\piO + \piO = 1 \implies \piO = 	frac15. So the server is overloaded one fifth of the time, or 20\%. Using the two-state shortcut for the overloaded state: it is entered with probability b = 0.1 and left with probability a = 0.4, so its share is \frac{b}{a+…

How often is the server overloaded?

Your answer

Discussion

Learn the concepts

Related questions