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Queueing Theory : Swimmers and Exponential Distribution

Queueing Theory Question 1

An average of 10 people per hour arrive (inter-arrival times are exponential) intending to swim laps at the local YMCA. Each intends to swim an average of 30 minutes. The YMCA has 3 lanes open for lap swimming. If one swimmer is in a lane, he or she swims up and down the right side of the lane. If 2 swimmers are in a lane, each swims up and down one side of the lane. Swimmers always join the lane with the fewest number of swimmers. If all 3 lanes are occupied by 2 swimmers, a prospective swimmer becomes disgusted and goes running.

a. What fraction of the time will 3 people be swimming laps?
b. On the average, how many people are swimming laps in the pool?
c. How many lanes does the YMCS need to allot to lap swimming to ensure that at most 5% of all prospective swimmers will become disgusted and go running?
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Queuing Theory Question 2

An average of 140 people per year applies for public housing in Boston. An average of 20 housing units /yr becomes available. During a given year, there is a 10% chance that a family on the waiting list will find private housing and remove themselves from the list. Assume that all relevant random variables are exponentially distributed.

a. On the average, how many families will be on the waiting list?
b. On the average, how much time will a family spend on the list before obtaining housing (either public or private)?
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Solution Summary

Queueing theory is investigated. The solution is detailed and well presented.

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