A Web site is operated by four identical computer servers....

A Web site is operated by four identical computer servers. Only one is used to operate the site; the others are spares that can be activated in case the active server fails. The probability that a request to the Web site generates a failure in the active server is $0.0001 .$ Assume that each request is an independent trial. What is the mean time until failure of all four computers?

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Geometric Distribution

The geometric distribution models the number of independent trials needed for the first success (or, in reliability terms, the occurrence of a failure event) when each trial has the same probability of success. In the context of server failures, each request represents a trial with a fixed failure probability, and the mean number of trials until failure is given by the reciprocal of that probability.

Linearity of Expectation

Linearity of expectation is a fundamental principle in probability which states that the expected value of a sum of random variables is equal to the sum of their individual expected values, regardless of whether they are independent. In problems involving redundant systems, such as with multiple spare servers, the overall expected time until system failure can be computed as the sum of the expected times until each individual failure.

Redundant Systems in Reliability Theory

Redundant systems are designed to improve reliability by incorporating spare components that can take over when the active component fails. In reliability theory, the use of redundancy is key to extending the overall system lifetime. The system is considered operational until all redundant components have failed, and the analysis typically involves calculating the aggregate reliability or the expected time to total failure.

Please give Ace some feedback