Question

Scenario: There is only one teller working at a bank. The teller takes an average of 3 minutes to service a customer. Assume that the time the teller takes to service a customer can be represented as an Exponential Probability distribution.
Customers arrive at the teller line at the average rate of 1every 10 minutes. Their arrival pattern follows a Poisson distribution.
Of 200 customers who get serviced by the teller, how many can expect to be serviced in more than 6 minutes?:

Answer 1

Given :

Teller takes an average of 3 minutes to service a customer.

Let x be the number of customers taken by teller.

x follow exponential distribution

Therefore λ = 1 / 3

Cumulative distribution function of exponential distribution : P( X ≤ x ) = 1 - e^-λx

We have to find P( x> 6 ) = 1 - P( x ≤ 6 )

P( x ≤ 6 ) = 1 - e^-(6/3)   = 1 - e^-2 = 0.8647

P( x> 6 ) = 1 - P( x ≤ 6 ) = 1 - 0.8647

P( x > 6) = 0.1353

Therefore number of customers expect to be serviced in more than 6 minutes = 200*P( x > 6) = 200*0.1353 =27.06 ~ 27

Therefore approximately 27 customers are expect to be serviced in more than 6 minutes