An average of 125 packets of information per minute arrive at an internet router. It takes an average of 0.002 seconds to process a packet of information. The router is designed to have a limited buffer to store waiting messages. Any message that arrives when the buffer is full is lost to the system. Assuming that interarrival and service times are exponentially distributed, how big a buffer size is needed to ensure that at most 1 in a million messages is lost?
13) Two one-man barber shops sit side by side in Dunkirk Square. Each can hold a maximum of 4 people, and any potential customer who finds a shop full will not wait for a haircut. Barber 1 charges $11 per haircut and takes an average of 12 minutes to complete a haircut. Barber 2 charges $5 per haircut and takes an average of 6 minutes to complete a haircut. An average of 10 potential customers per hour arrive at each barber shop. Of course, a potential customer becomes an actual customer only if he finds that the shop is not full. Assuming that interarrival times and haircut times are exponential, which barber will earn more money?
14) A supermarket is trying to decide how many cash registers to keep open. Suppose an average of 18 customers arrive each hour, and the average checkout time for a customer is 4 minutes. Interarrival times and service times are exponential, and the system may be modeled as an M/M/s/GD// queuing system. It costs $20 per hour to operate a cash register, and a cost of $25 is assessed for each minute the customer spends in the cash register area. How many registers should the store open?
15) A small bank is trying to determine how many tellers to employ. The total cost of employing a teller is $100 per day, and a teller can serve an average of 60 customers per day. An average of 50 customers per day arrive at the bank, and both service times and interarrival times are exponential. If the delay cost per customer-day is $100, how many tellers should the bank hire?
16) In this problem, all interarrival and service times are exponential.
a. At present, the finance department and the marketing department each have one typist. Each typist can type 25 letters per day. Finance requires that an average of 20 letters per day be typed, and marketing requires that an average of 15 letters per day be typed. For each department, determine the average length of time elapsing between a request for a letter and completion of the letter.
b. Suppose that the two typists were grouped into a typing pool; that is, each typist would be available to type letters for either department. For this arrangement, calculate the average length of time between a request for a letter and completion of the letter.
c. Comment on the results of parts (a) and (b).
d. Under the pooled arrangement, what is the probability that more than 0.200 day will elapse between a request for a letter and completion of the letter?
