Customers arrive at a bank teller machine at the rate one every three minutes. Each customer spends an average of two minutes at the teller machine. The arrival rate and the service rate are approximated by Poisson and negative exponential distributions respectively. Determine the following:
a – utilization of the teller machine
b – average number of customers in line
c – average number of customers in the system
d – average time customers spend in line
e – average time customers spend in the system
f – probability of three customers in the system
g – probability of two or more customers in the system.

Arrival rate

1 every 3 minutes

1 minute = 1/3 = 0.33

60 minutes = 0.33*60 = 19.8 = 20 per hour

λ = 20 per hour

Service Rate

1 every 2 minutes

1 minutes = 1/2 = 0.5

60 minutes = 0.5*60 = 30 per hour

µ = 30 per hour

a. Utilization of Teller Machine

P = λ / µ

P = 20/30

P = 66.67%

b. Average number of customers in line

Lq = pL = (λ/µ) (λ/µ- λ)

= (20 / 30) (20 / 30 - 20)

= 20/30 * 20 / 10

= 1.33 customers

c. Average number of customers in the system

L = (λ/µ- λ)

= 20 / 30 - 20

= 20 / 10

= 2 customers

d. Average time customer spends in line

Wq = λ/[µ*(µ- λ)]

= 20 / [30 * (30-20)]

= 20 / 30 * 10

= 0.06667 hours or 4 minutes

e. Average time customers spend in the system

W = 1/(µ- λ)

= 1 / 30 - 20

= 1/10

= 0.10 hours or 6 minutes

f. Probability that there are 3 customers in the system

Pn = (1-p)*p^n

= (1 - 20/30) * (20/30)^3

= 0.3333 * 0.296296

= 0.09876

g. Probability that there are two or more customers in the system

= 1 - P(0) - P(1)

= 1 - (1 - 20/30) * (20/30)^0 - (1 - 20/30) * (20/30)^1

= 1 - 1/3 - 2/9

= 4/9

= 0.4444

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