A computer system carries out tasks submitted by two users. Time is divided into slots. A slot...

Question:

A computer system carries out tasks submitted by two users. Time is divided into slots. A slot can be idle, with probability PI = 1/6, and busy with probability PB = 5/6. During a busy slot, there is probability P1/B = 2/5 (respectively, P2/B = 3/5) that a task from user 1 (respectively, 2) is executed. We assume that events related to different slots are independent.

a) Find the probability that a task from user 1 is executed for the first time during the 4th slot.

b) Given that exactly 5 out of the first 10 slots were idle, find the probability that the 6th idle slot is slot 12. c) Find the expected number of slots up to and including the 5th task from user 1.

d) Find the expected number of busy slots up to and including the 5th task from user 1.

e) Find the PMF, mean, and variance of the number of tasks from user 2 until the time of the 5th task from user 1.

Probability Mass Function:

The Probability Mass Function is the probability distribution function of the random variables taking up discrete values where the expected value of the random variable can be calculated using its mass function itself.

Answer and Explanation: 1

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(a)

If a task from user 1 is executed for the first time during the fourth slot, it means that it will not get executed for the first three slots.

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Probability Density Function | Formula, Properties & Examples

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Chapter 22 / Lesson 8
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Learn to define a probability density function. Discover the probability density function formula. Learn how to find the probability density function. See examples.


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