Many of a bank's customers use its automatic teller machine to transact business after normal banking...

 

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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 81 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 3 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 0.50 c. Determine the average number of customers waiting to use the machine. (Round your answer to 2 decimal places.) Average number customers hint #0 Guided Example Solution [Single Server, Exponential Service Time, M/M/1] We must determine and first (we will use hours): = (1 customer/3 minutes) x (60 minutes/hour) = 20 customers/hour = (3600 seconds/hr) / (100 seconds/customer) = 36 customers/hour a) Average time spent in the system (waiting + transaction): L= 22 (-) W 9 = W s = / Wa 202 = = 0.694customers 36(36-20) = 0.035hrs 0.694 20 1 1 + == 0.035 + = 0.035 0.028 = 0.063hrs 36 McGraw-Hill Education Guided Example Solution = 20 customers/hour = 36 customers/hour c. The average number waiting to use the machine. La 22 202 = = 0.694customers (-) 36(36-20) McGraw-Hill Education Guided Example Solution = 20 customers/hour = 36 customers/hour b. The probability that a customer will not have to wait upon arriving Po = 1 = = 1 - McGraw-Hill Education 20 = 0.444 == 36 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 81 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 3 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 0.50 c. Determine the average number of customers waiting to use the machine. (Round your answer to 2 decimal places.) Average number customers hint #0 Guided Example Solution [Single Server, Exponential Service Time, M/M/1] We must determine and first (we will use hours): = (1 customer/3 minutes) x (60 minutes/hour) = 20 customers/hour = (3600 seconds/hr) / (100 seconds/customer) = 36 customers/hour a) Average time spent in the system (waiting + transaction): L= 22 (-) W 9 = W s = / Wa 202 = = 0.694customers 36(36-20) = 0.035hrs 0.694 20 1 1 + == 0.035 + = 0.035 0.028 = 0.063hrs 36 McGraw-Hill Education Guided Example Solution = 20 customers/hour = 36 customers/hour c. The average number waiting to use the machine. La 22 202 = = 0.694customers (-) 36(36-20) McGraw-Hill Education Guided Example Solution = 20 customers/hour = 36 customers/hour b. The probability that a customer will not have to wait upon arriving Po = 1 = = 1 - McGraw-Hill Education 20 = 0.444 == 36