# Question 1. Part a. Using the formula given and substit

Added on - 03 Nov 2021

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Question 1

Part a

Using the formula given and substituting in for different values of beta and t; we are able to see

that the number of critical failures is the same across the time interval (5, 15) for beta equivalent

to 1. The expected number of critical failures decrease as beta tends to 0 andtime increases from

5 to 15. Also,the expected number of critical failures increase for beta increase for B>1,as time

increases from 5 to 15.

Part b

To get the expected number critical failure we find the sum of the intensity function for beta=0.5

and time=5, 6, 7,..., 15. Hence:

E(t)=∑t=6

15

(0.5βt(β−1)),forβ=0.5

E(t)=(0.5β5(β−1))+(0.5β6(β−1))+...+(0.5β15(β−1)),forβ=0.5

E(t)=0.91

The probability is given by

Exactly 2 hence x=2

Interval (5, 15)

Lambda= 0.5

p(x;λ)=¿λxe−λ

x!

p(2;0.5)=¿0.52e−0.5

2!

p(2;0.5)=¿0.075816

Part C

Part a

Using the formula given and substituting in for different values of beta and t; we are able to see

that the number of critical failures is the same across the time interval (5, 15) for beta equivalent

to 1. The expected number of critical failures decrease as beta tends to 0 andtime increases from

5 to 15. Also,the expected number of critical failures increase for beta increase for B>1,as time

increases from 5 to 15.

Part b

To get the expected number critical failure we find the sum of the intensity function for beta=0.5

and time=5, 6, 7,..., 15. Hence:

E(t)=∑t=6

15

(0.5βt(β−1)),forβ=0.5

E(t)=(0.5β5(β−1))+(0.5β6(β−1))+...+(0.5β15(β−1)),forβ=0.5

E(t)=0.91

The probability is given by

Exactly 2 hence x=2

Interval (5, 15)

Lambda= 0.5

p(x;λ)=¿λxe−λ

x!

p(2;0.5)=¿0.52e−0.5

2!

p(2;0.5)=¿0.075816

Part C

The cumulative distribution function is given by

F(x;λ)=∑x=0

∞λxe−λ

x!

And the probability density function

p(x;λ)=¿λxe−λ

x!

They both have elements of the exponential distribution and it's important to note that Poisson

distribution is a subset of binomial distribution.

Question 2

Part a

(i)

Birth takes place when the process moves from state t to t+1 (the customers enter the store and

join the line) and dead takes place when the process moves from state t to t-1(the customer exists

the store because he finds a given number of people in the line). The process is subject to birth

rate (rate of customer entry into store), and death rate (rate of customer leaving).

(ii)

The customers arrive at rate 20 per hour hence 1 person per 3 minutes

The service time per customer is 3 minutes

Hence, the probability that the first customer would have left the queuing system before the

second customer arrives is one or 100%.

Part b

(i)

F(x;λ)=∑x=0

∞λxe−λ

x!

And the probability density function

p(x;λ)=¿λxe−λ

x!

They both have elements of the exponential distribution and it's important to note that Poisson

distribution is a subset of binomial distribution.

Question 2

Part a

(i)

Birth takes place when the process moves from state t to t+1 (the customers enter the store and

join the line) and dead takes place when the process moves from state t to t-1(the customer exists

the store because he finds a given number of people in the line). The process is subject to birth

rate (rate of customer entry into store), and death rate (rate of customer leaving).

(ii)

The customers arrive at rate 20 per hour hence 1 person per 3 minutes

The service time per customer is 3 minutes

Hence, the probability that the first customer would have left the queuing system before the

second customer arrives is one or 100%.

Part b

(i)

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