Question

If the average service rate is 12 minutes per customer, and assuming the negative exponential distribution is used to describe the randomness of the service time distribution, then determine the probability that the service time will be less than or equal to 10 minutes.

a. 0.624
b. 0.565
c. 0.283
d. 0.000

Answer 1

Answer: b. 565

Probability that service time will be less than or equal to 10 minutes is P = 0.5654.

Explanation:

Data Given are as follows,

Service time
`T = 10 minutes`

Average service rate
`\beta = (1)/(12) per minute`

First of all, as here average service rate is given as 12 minutes per customer

Using concept of Queuing Theory, as it is case of probability of service time less than or equal to T.

In addition, it is given that negative exponential distribution is assumed to describe the randomness of the service time distribution.

So it is given by,

`P = 1 - e^ {-\beta * T}`

where P = probability that the service time will be less than T

β = average service rate in per minutes

T = service time in minutes

It can be solved by

`P = 1 - e^ {-\beta * T}`

`P = 1 - e^ {-((1)/(12) ) * (10)}`

`P = 1 - e^ {-0.8333}`

`P = 1 - \frac{1}{e^ {0.8333}}`

`P = 1 - (1)/(2.3)`

`P = 1 - 0.4345`

`P = 0.5654`

Hence, from above analysis, it an be finally concluded that probability that service time will be less than or equal to 10 minutes is P = 0.5654.