# PDF of processors

(See attached file for full problem description with full equations)

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1. In a computing facility there are two types of processors: type A and type B. Suppose random variable X represents the processing time of a job. With type A processor, the processing time has PDF:

and with type B processor it is:

Also suppose that 60% of the processors in the facility are of type A. When a job arrives, a processor is assigned to it at random and independent of whether it is processing another job or not. Assume there can be infinite queue for any processor.

a. Find the PDF of the processing time of X.

b. Find the probability that a given job is completed in less than 20 time units.

c. Given that the job is completed in 5 units of time, what is the probability that it was processed by a type A processor.

2. Suppose a lot contains parts manufactured by companies A, B, and C with relative populations (in the lot) of 50%, 30%, and 20% respectively. The "lifetime" of a part (in hours) is represented by the random variable X. According to the data supplied by manufacturers, the lifetime of the parts obey the following PDF:

with =50, 70, and 100 for companies A, B, and C respectively.

a. Find the PDF of the lifetime of a part randomly selected from the lot.

b. Find the probability that a randomly selected part is still functioning 120 hours after it is put into operation.

c. Given that a randomly selected part is still functioning after 120 hours, what is the probability that it is manufactured by A?

d. Given that a randomly selected part is still functioning after 120 hours, what is the probability that it is manufactured by B?

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#### Solution Preview

Please see the attached file.

1. In a computing facility there are two types of processors: type A and type B. Suppose random variable X represents the processing time of a job. With type A processor, the processing time has PDF:

and with type B processor it is:

Also suppose that 60% of the processors in the facility are of type A. When a job arrives, a processor is assigned to it at random and independent of whether it is processing another job or not. Assume there can be infinite queue for any processor.

a. Find the PDF of the processing time of X.

Denote by the processing time of a job by type A processor. Denote by the processing time of a job by type B processor. Then

.

Note: We assume that a job can be done only by A or B not both. So, the least time is X; the same for question 2.

Now we want to find CDF of X.

Since are independent, we have

Note that

...................(1)

Similarly, we have

................(2)

So,

So, the pdf is ...

#### Solution Summary

Two questions relating to the processing time of two types of processors based on their probability density functions.