Question

Students arrive at the Adminsitrative Office at an average of one every 24 minures, and their requests take on average 12 minutes to be processed. The service counter is staffed by only one clerk, Judy Gumshoes, who works eight hours per day. Assume Poisson arrivals and exponential service times.a) What percentage of time is Judy Idle? (round to 2 decimal Places)b) How much time on average, does a student spend waiting in line? (round your answer to the nearest whole number)c) How long is the waiting line on average? (2 decimal places)d) What is the probability that an arriving student just before entering the Administrative Services Office) will find at least one other student waiting in line? (4 decimal places)

Answer 1

Solution and Explanation:

arrival time = 24 per minute, processing time = 12 minutes, number of servers = 1 ( work for 8 hours a days)

a. arrival rate = 60 divide 24 = 2.5 customer per hour, service rate = 60 divide 12 = 5 customers per hour

therefore, the probability that the operator will be busy = 2.5 divide 5 = 0.5

thus, the percentage of idle time = 1 minus 2.5 divide 5 = 0.50

Thus, the percenatge idle time is 50%

b. the average spent by a student waiting in a line = $2.5 / 5(5-2.5)$

= 0.20 hours.

Thus, on an average a student spent 0.20 hours waiting in line.

c. the average number in waiting line = 0.5 customers

d. the probability for the given condition to happen is 0.25, 25%