Neve Commercial Bank is the only bank in the town of York, Pennsylvania. On a ty

Question

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). Please show the work because I need to understand it and show what I did to receive credit.

Explanation / Answer

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Arrival rate = Lamda = 10 customers per hour

teller = 1 = server = C = 1

Service rate = Miu = 4 minutes per customer

4 min – 1 customer

60 min – 1*60/4= 15 customers per hour

Utilization rate = chi = Lambda/Miu = 10/15 = 0.66667

Ls = Chi / (1-Chi) = 0.66667/(1-0.666667) = 2

Lq = Ls – Chi = 2 – 0.6666667 = 1.333333

Ws = Ls/Lambda = 2/10 = 0.2

Wq = Lq/Lambda = 1.333333/10 = 0.1333333

a) Average time in the line = Wq = A customer waits for 0.13333 hours = (7+1) MINUTES IN THE queue

b) Average number in the line = Lq = 1.3333 customers waiting in the queue

c) Average time in the system = Ws = A customer spends on average 0.2 hours = 12 minutes in the system

d) Average number in the system = Ls = 2 customers in the system

e) Probability that the bank is empty = P0 = QTPMMS_PrEmpty(10,15,1) = 1 – chi = 1 – 0.66667 = 0.33333

f) After adding a second teller, number of servers = C = 2

Formulae are different:

P0 = Sigma n runs from 0 to C-1 (( Chi^n) /(n!) + Chi^C ) / (C!*(1-Chi/C)) ) ^ -1

Lq = C Chi Pc / (C-Chi)^2

Ls = Lq + Chi

Wq = Lq/Lambda

Ws = Wq + 1/Miu

The QTP add on for excel can be used to calculate the above values:

The function syntax is:

QTPMMS_L(arrival rate, service rate, Number of servers)

f. a) Average time in the line = Wq =QTPMMS_Wq(10,15,2) = 0.00833333 hours = 0.5 minutes – a great reduction than (7+1) minutes with a single teller - with 2 tellers, customers wait for just half a minute

service time = total time – wait time = 4.5 minutes – 0.5 minutes = 4 minutes – this is in line with the 4 minutes of service time mentioned in the question

f. b) Average number in the line = Lq = QTPMMS_Lq(10,15,2) = 0.08333333

f. c) Average time in the system = Ws =

=QTPMMS_W(10,15,2) = 0.075 hours = 4.5 minutes – this is an improvement over the 12 minutes in the single teller case – customer spends lesser time in the system

f. d)

Average number in the system = Ls = Length of the system = number of customers in the system both being served and waiting = Ls = QTPMMS_L(10,15,2) = 0.75

f. e) Probability that the bank is empty = P0 = 0.5 = QTPMMS_PrEmpty(10,15,2)

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12-(7+1)=4 minutes to serve one customer

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