Computer Server Down Probability

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SUMMARY

The discussion centers on calculating the probability of a computer server being down and undergoing repairs over a 7-day period, given its operational and repair time distributions. The server operates for an exponentially distributed time of Exp(0.2) days before requiring repairs that also follow an exponential distribution of Exp(0.5) days. Participants suggest utilizing Markov chains and infinitesimal generators to derive the probabilities of the server being fixed at least once and its operational status on specific days.

PREREQUISITES
  • Understanding of exponential distributions, specifically Exp(0.2) and Exp(0.5).
  • Familiarity with Markov chains and their applications in probability theory.
  • Knowledge of infinitesimal generators in the context of stochastic processes.
  • Basic concepts of conditional probability and its implications in future events.
NEXT STEPS
  • Study the application of Markov chains in reliability engineering.
  • Learn how to compute probabilities using infinitesimal generators in stochastic processes.
  • Explore the properties of exponential distributions and their role in modeling system failures.
  • Investigate conditional probability techniques in the context of future event predictions.
USEFUL FOR

Mathematicians, data scientists, reliability engineers, and anyone involved in modeling system performance and downtime probabilities.

iikii
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Moved from a technical forum, so homework template missing
So the problem asks:

A computer server runs smoothly for Exp(0.2) days and then takes Exp(0.5)days to fix. The server is running fine on Monday morning, t=0. Find the probability that the server was fixed at least once (i.e. at least one complete repair was done) in the next 7 days and the probability that the server was down on Friday t = 4 given that it was running next Sunday t = 6. (W are conditioning on the `future'. )

So do I have to use Markov chain to do this problem? If so, do I need to find out the innitesimal generators to find the probability? I appreciate your insights!
 
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iikii said:
So the problem asks:

A computer server runs smoothly for Exp(0.2) days and then takes Exp(0.5)days to fix. The server is running fine on Monday morning, t=0. Find the probability that the server was fixed at least once (i.e. at least one complete repair was done) in the next 7 days and the probability that the server was down on Friday t = 4 given that it was running next Sunday t = 6. (W are conditioning on the `future'. )

So do I have to use Markov chain to do this problem? If so, do I need to find out the innitesimal generators to find the probability? I appreciate your insights!

Try it and see for yourself. Of course, if you are studying Markov processes in the course, that would be a pretty strong hint.
 

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