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1. Willow Brook National Bank operates a driveup teller window that allows customers to compiete bank transactions without getting out of their cars. Dn weekday mornings, arrivals to the driveup teller window occur at random- Assume that Poisson probability distribution with an arrival rate of 24 customers per hour or {1.4 customer per minute can be used to describe the arrival pattern. Assume further that the service times for the driveup teller follow an exponential probability distribution with a service rate of 36 customers per hour, or {1.5 customer per minute- Use the singlechannel drive up bani: teller operation to determine the toltowing operating characteristics for the system: The probabity that no customers are in the system The average number of customers waiting The average number of customers in the system The average time a customer spends waiting The average time a customer spends in the system The probability that aniving custa mers will have to wait for service The probability of 4 customers in the system The average anival time in minutes of customers The average service time in minutes of the diveup teller assessor-re 2. A Florida coastal community experiences a population increase during the winter months with seasonal residents arriving from northern states and Canada. The post office counter has three work stations. The service rate of each postal clerk is 0.75 customer per minute. The anticipated arrival rate is 1.2 customers per minute. Assume that customer arrivals follow a Poisson probability distribution, with an arrival and that service times follow an exponential probability distribution. Determine the following operating characteristics for the system: a. The probability that no customers are in the system b. The average number of customers waiting C. The average number of customers in the system d. The average time a customer spends waiting e . The average time a customer spends in the system f. The probability that arriving customers will have to wait for service 9. The probability of 5 customers in the system3. (a-b). The Gorman Manufacturing Company must decide whether to manufacture a component part at its Milan, Michigan, plant or purchase the component part from a supplier. The resulting profit is dependent upon the demand for the product. The state-of-nature probabilities are P(s,) = 0.35, P(s2) = 0.35, and P($3) = 0.30. The following payoff table shows the projected profit (in thousands of dollars): State of Nature Low Demand Medium Demand High Demand Decision Alternative $2 Manufacture, -20 40 100 Purchase. d2 10 45 70 a. Use expected value to recommend a decision. b. Use EVPI to determine whether Gorman should attempt to obtain a better estimate of demand.3. (c-f). The Gorman Manufacturing Company must decide whether to manufacture a component part at its Milan, Michigan, plant or purchase the component part from a supplier. The resulting profit is dependent upon the demand for the product. State of Nature Low Demand Medium Demand High Demand Decision Alternative $1 Manufacture, d -20 40 100 Purchase. da 10 45 70 The relevant conditional probabilities are as follows: P(F S, ) = 0.10 P(U | $1) = 0.90 P(F | $2) = 0.40 P(U | 52) = 0.60 P(F $3) = 0.60 P(U $3) - 0.40 C. A test market study of the potential demand for the product is expected to report either a favorable (F) or unfavorable (U] condition. i. Compute the Posterior Probabilities by completing the table Ti. What is the probability that the market research report will be favorable? III. What is the probability that the market research report will be unfavorable? iv. Draw the decision tree leading to market study/ prediction of favorable condition. V. Draw the decision tree leading to market study/ prediction of unfavorable condition. vi. Draw the decision tree incorporating the 2 conditions. d. What is Gorman's optimal decision strategy? e. What is the expected value of the market research information? f. What is the efficiency of the information?4. A large corporation collected data on the reasons both middle managers and senior managers leave the company. Some managers eventually retirer but others leave the company prior to retirement for personal reasons including more attractive positions with other rms. Assume that the following matrix of oneyear transition probabilities applies with the four states of the Markov process being retirement, leaves priorto retirement for personal reasons, stays as a middle manager: stays as a senior manager. lemu erlrle Seller W fer-lull hung-t Mel-[er Mum I Ill} am um um Ifm Dl] I till] on: 0.00 Huddle "mice on: om om Isle SEIHI' Hunger um em on! [1.33 a. What states are considered absorbing states? Why? b. Compute the matrix NR: c. What percentage of the current middle mmagers wil eventually retire from the cornpanyir d. What percentage of the senior managers wiil leave the comp-my for personal reasore? e. The company currently has 920 managers: 640 middle managers and 2&0 SEHTGF managers. l. How many of these managers will eventually retire from the company? | ii. How my will leave the company for personal reasons