Question

Assignment 7a (GBUS303)                                    Name:

Willow Brook National bank operates a drive-up teller window that allows customers to complete bank transactions without getting out of their cars. On weekday mornings, arrivals to the drive-up teller window occur at random with a Poisson distribution with arrival rate of 20 customers per hour. (All applicable formulas are required)

1). how much is the mean number of arrivals per minute? the arrival rate λ.

Assume that the service times for the drive-up teller follow an exponential probability distribution with a service rate of 40 customers per hour.

2) how much is the mean number of customers that can be served per minute? the service rate μ.  

3) The probability that no customers are in the system

4) The average number of customers waiting

5) The average number of customers in the system

6)the average time a customer spends waiting

7) The average time a customer spends in the system

8) The probability that arriving customers will have to wait for service.

9) The probability that 3 customers in the system.

Answer 1

Answer:

a)

Given,

Arrival rate = = 20 customer/hour

b)

Service rate = = 40 customers/hour

c)

Probability of no customers in queue = 1 - /

substitute values

= 1 - 20/40

= 1 - 0.5

Po = 0.5

d)

The average number of customers waiting = ^2 / ( - )

substitute values

= 20^2 / (40(40 - 20))

= 400/800

= 0.5

e)

The average number of customers in the system = /( - )

substitute values

= 20/(40 - 20)

= 20/20

= 1

f)

The average time a customer spends waiting = /(( - ))

substitute values

= 20/(40(40-20))

= 0.025 hrs

= 1.5 min

g)

The average time a customer spends in the system = 1/( - )

substitute values

= 1 / (40-20)

= 1/20

= 0.05 hrs

= 3 min

h)

The probability that arriving customers will have to wait for service = /

substitute values

= 20/40

= 0.5

i)

The probability that 3 customers in the system = (/)^3 * Po

substitute values

= (20/40)^3*0.5

= 0.0625