The Orange Blossom Marathon takes place in Orlando, Florida, each December. The organizers of this race are trying to solve a problem that occurs at the finish line each year. Thousands of runners take part in this race. The fastest runners finish the 26-mile course in just over two hours, but the majority of the runners finish about 1 1/2 hours later. After runners enter the finish area, they go through one of four finish chutes where their times and places are recorded. (Each chute has its own queue.) During the time in which the majority of the runners finish the race, the chutes become backlogged and significant delays occur. The race organizers want to determine how many chutes should be added to eliminate this problem. At the time in question, runners arrive at the finish area at a rate of 50 per minute according to a Poisson distribution, and they randomly select one of the four chutes. The time required to record the necessary information for each finishing runner at any chute is an exponentially distributed random variable with a mean of four seconds.
a. On average, how many runners arrive at each chute per minute?
b. Under the current arrangement with four chutes, what is the expected length of the queue at each chute?
c. Under the current arrangement, what is the average length of time a runner waits before being processed?
d. How many chutes should be added if the race organizers want to reduce the queue time at each chute to an average of five seconds?