# Question

In the Blue Chip Life Insurance Company, the deposit and withdrawal functions associated with a certain investment product are separated between two clerks, Clara and Clarence. Deposit slips arrive randomly (a Poisson process) at Clara’s desk at a mean rate of 16 per hour. Withdrawal slips arrive randomly (a Poisson process) at Clarence’s desk at a mean rate of 14 per hour. The time required to process either transaction has an exponential distribution with a mean of 3 minutes. To reduce the expected waiting time in the system for both deposit slips and withdrawal slips, the actuarial department has made the following recommendations: (1) Train each clerk to handle both deposits and withdrawals, and (2) put both deposit and withdrawal slips into a single queue that is accessed by both clerks.

(a) Determine the expected waiting time in the system under current procedures for each type of slip. Then combine these results to calculate the expected waiting time in the system for a random arrival of either type of slip.

(b) If the recommendations are adopted, determine the expected waiting time in the system for arriving slips.

(c) Now suppose that adopting the recommendations would result in a slight increase in the expected processing time. Use the Excel template for the M/M/s model to determine by trial and error the expected processing time (within 0.001 hour) that would cause the expected waiting time in the system for a random arrival to be essentially the same under current procedures and under the recommendations.

(a) Determine the expected waiting time in the system under current procedures for each type of slip. Then combine these results to calculate the expected waiting time in the system for a random arrival of either type of slip.

(b) If the recommendations are adopted, determine the expected waiting time in the system for arriving slips.

(c) Now suppose that adopting the recommendations would result in a slight increase in the expected processing time. Use the Excel template for the M/M/s model to determine by trial and error the expected processing time (within 0.001 hour) that would cause the expected waiting time in the system for a random arrival to be essentially the same under current procedures and under the recommendations.

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