Many of a bank\'s customers use its automatic teller machine to transact busines

Question

Many of a bank's customers use its automatic teller machine to transact business after normal banking hours. During the early evening hours in the summer months, customers arrive at a certain location at the rate of one every other minute. This can be modeled using a Poisson distribution. Each customer spends an average of 97 seconds completing his or her transactions. Transaction time is exponentially distributed a. Determine the average time customers spend at the machine, including waiting in line and completing transactions. (Do not round intermediate calculations. Round your answer to the nearest whole number.) Average time 11 minutes b. Determine the probability that a customer will not have to wait upon arriving at the automatic teller machine. (Round your answer to 2 decimal places.) Probability c. Determine the average number of customers waiting to use the machine. (Round your answer to 2 decimal places.) Average number 19 % 11 customers

Explanation / Answer

Arrival rate of customers ( basis one every other minute ) = a = 30 per hour

Service rate of customers = s = 3600 / 97 = 37.11 per hour

Average time customer spends at the machine including waiting in line and completing transactions

= a / s x ( s -a ) + 1/s   hour

= 30/ 37.11 x ( 37.11 – 30)     + 1/37.11

= 30 / 37.11 x 7.11 + 1/37.11

= ( 30 + 7.11) / ( 37.11 x 7.11)

= 37.11/ ( 37.11 x 7.11)

= 1/7.11 hours

= 8.438 minutes ( 8 minutes rounded to nearest whole number )

AVERAGE TIME =8 MINUTES

Probability that a customer will not have to wait upon arrival at automatic teller machine

= PO

= ( 1 – a/s )

= 1 – 30/37.11

= 1 – 0.808

= 0.192

PROBABILITY = 0.19

Average number of customers waiting to use the machine

= a^2/S x ( S – a)

= 30 x 30 / 37.11 x ( 37.11 – 30)

= 30 x 30 / 37.11 x 7.11

= 3.411 ( 3 rounded to nearest whole number )

AVERAGE NUMBER = 3 CUSTOMERS

AVERAGE TIME =8 MINUTES

Related Questions

Navigate

• Browse (All)
• Browse M
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.