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Chapter 3: Problem 191

Four identical computer servers operate a Web site. 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 all four computers fail?

Short Answer

Expert verified
The mean time until all four servers fail is 40,000 requests.

Step by step solution

01

Understand the Problem

We have four identical servers, where only one is actively used at a time. The probability of a failure for each request is given as \( p = 0.0001 \). We need to find the mean time until all servers fail.
02

Define the System

Since each server has its failure probabilities independently and identically distributed, we'll treat this problem under a sequence of Bernoulli trials, where the event of failure is the 'success' of a Bernoulli trial.
03

Calculate the Mean Time for One Server to Fail

The mean number of requests until failure of one server follows a geometric distribution. For a geometric distribution with probability \( p \), the mean is given by \( \frac{1}{p} \). So, \( \frac{1}{0.0001} = 10,000 \).
04

Calculate the Total Mean Time for All Servers to Fail

Each server fails after 10,000 requests on average, independently of the others. Thus, for all four servers in sequence, we expect \( 4 \times 10,000 = 40,000 \) requests until all servers have failed.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Bernoulli trials
Bernoulli trials are a fundamental concept in probability theory. Imagine an experiment that results in a binary outcome. This means there are only two possible outcomes鈥攖ypically labeled as "success" or "failure." Each trial is identical and independent, meaning the outcome of any one trial does not affect another.
  • Each server's response can be seen as a Bernoulli trial.
  • The trial has been defined such that a server failure is considered a "success" for our mathematical treatment (despite it being a failure in real life).
In our problem, if we consider a server failing upon a website request, we are looking at a series of Bernoulli trials. Here, the probabilistic "success" is a failure with a given probability of the server going down when a request is made.
probability of failure
The probability of failure is a crucial figure when assessing risks and outcomes over multiple trials. In a Bernoulli trial situation, this is the likelihood that a failure occurs on any single experiment or attempt.
  • In our exercise, each server has a failure probability of 0.0001 per request.
  • This number is small, indicating that failures are rare, but not impossible over many trials.
The small size of this probability means that most requests will not result in a failure, but it also sets the scene for calculating the mean time until eventual failures happen over prolonged use.
mean time calculation
To calculate the mean time until a certain event happens, such as server failure, it is common to turn to the concept of the mean in geometric distribution. The geometric distribution is key in predicting when the first "success" will occur in a sequence of Bernoulli trials.
  • The mean number of trials until the first failure can be determined by taking the reciprocal of the failure probability: \( \frac{1}{p} \).
  • For the given problem, the mean time until a single server fails is \( \frac{1}{0.0001} = 10,000 \).
Thus, it takes on average 10,000 requests for a single server to fail. If we are considering all four servers independently, the mean time calculation becomes a cumulative sum of each individual mean failure time.
independent trials
Independent trials ensure that the outcome of one trial does not affect another. This is a critical aspect when modeling situations using Bernoulli or other related distributions. In the context of server operations, it implies one server鈥檚 failure does not influence another server's chance of failing.
  • This independence allows each server to be considered separately when estimating failure times.
  • Therefore, the mean number of requests until all four servers fail can be calculated by multiplying the individual expectations: \( 4 \times 10,000 = 40,000 \).
Understanding independence helps describe the resilience and operability of systems over time, especially in technology landscapes where reliability is crucial.

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Most popular questions from this chapter

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