Neve Commercial Bank is the only bank in the town of York, Pennsylvania. On a typical Friday, an average of 10 customers per hour arrive at the bank to transact business. There is currently one teller at the bank, and the average time required to transact business is 4 minutes. It is assumed that service times may be described by the exponential distribution. If a single teller is used, find:a) The average time in the line.b) The average number in the line.c) The average time in the system.d) The average number in the system.e) The probability that the bank is empty.f) CEO Benjamin Neve is considering adding a second teller (who would work at the same rate as the first) to reduce the waiting time for customers. A single line would be used, and the customer at the front of the line would go to the first available bank teller. He assumes that this will cut the waiting time in half. If a second teller is added, find the new answers to parts (a) to (e).

Answer:a 3minutes, b 33 people, c 2minutes, d 2minutes e 0.04979, f 2minutes New answers to a -e a 7minutes, b 66people, c 1/2 minutes, d 1/2minutes e 0.14936

Explanation:

Arrival rate = 10 customer per hour , Service rate = 60/4 = 15

ti = Arrival rate /service rate ( service rate - Arrival rate )

= 10/15 ( 15 - 10)

= 10/15 (5)

= 0.6 × 5 = 3.3

= 3minutes Approximately

b ti = Arrival rate ^2/service rate (service rate - Arrival rate )

ti = 10^2/15 (15-10)

=100/15 (5)

= 6.66 × 5

= 33.3

= 33 people

c ts = Arrival rate /(service rate - Arrival rate )

= 10/(15-10)

= 10/5

= 2minutes

d ts = Arrival rate /service rate -Arrivalrate

= 10/15-10

= 10/5

=2minutes

e since arrival and services rate is either exponential or poison Using poison distribution formula e-m^r/r!

Where p (r) = probability of event r, m = the item observed, e = a constant with value 2.71828

e P (0) = (2.7182^-3 (3)^0/0!

=1/(2.71828)^3

= 0.04979

f standard deviation = Arrival rate /service

=10/2 =5

√5

= 2.23

= 2minutes approximately

New answers to a -e

Arrival rate = 10 customer per hour, service rate = 60÷2 = 30

a ti = Arrival rate /service rate (service rate - Arrival rate )

= 10/30 (30-10)

= 1/3 (20)

= 6.6 Approximately 7minutes

b ti = Arrival rate ^2/service rate (service rate - Arrival rate )

10^2/30 (20)

= 3.3 × 20 = 66people

c ts = Arrival rate /(service rate -Arrivalrate )

= 10/(30 -10)

= 10/20

= 1/2

d ts = Arrival rate /service rate -Arrival rate

= 10/30-10

= 10/20

= 1/2

e P (1) = (2.71828)^-3(3)^1/1!

=1/(2.71828)^3 × (3)^1/1!

=1/20.08550 ×3

= 3/20.08550

= 0.14936