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randy patton wrote at 2013-07-14 08:52:18
This question needs to be answered using the results of queueing theory. The question uses the terminology of queueing theory so it is assumed that the questioner has access to these results. In particular, Little's Law is key,

mean number of units in system = (arrival rate of units)x(mean time of a unit in system)

E(n) = λE(Ts), where Ts = waiting time (in queue) plus service time = sojourn time.

The system described has exponential arrival and processing times (Poisson processes) with one processor and so has a Kendall representation of M/M/1.

From the given information, λ = 3/hr and μ = (1/(10 minutes/unit) = 6/hr.

i) The utilization factor is ρ = λ/λ = 3/6 = 1/2. Note that we need ρ < 1 or else the system overloads.

ii) The probability of finding n people in the system is given by P(n) = (1-ρ)ρ^n, so for 2 people, we have P(2) = 1/8.

iii) Expected number of units in the queue is E(Pq) = ρ^2/(1-ρ) = 1/2.

iv) For the mean waiting time in the system, I assume you mean the mean waiting time for a customer in the queue (as opposed to the sojourn time = waiting time in queue + processing time); this is given from iii) by Little's Law

expected number of units in the queue = (arrival rate)x(mean number of people in queue)

E(Pq) = λE(Wq)  or  E(Wq) = (1/μ)/(1-ρ) = (1/6)hr = 10 minutes.

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Scott A Wilson


I can answer any question in general math, arithetic, discret math, algebra, box problems, geometry, filling a tank with water, trigonometry, pre-calculus, linear algebra, complex mathematics, probability, statistics, and most of anything else that relates to math. I can also say that I broke 5 minutes for a mile, which is over 12 mph, but is that relevant?


Experience in the area; I have tutored people in the above areas of mathematics for over two years in I have tutored people here and there in mathematics since before I received a BS degree back in 1984. In just two more years, I received an MS degree as well, but more on that later. I tutored at OSU in the math center for all six years I was there. Most students offering assistance were juniors, seniors, or graduate students. I was allowed to tutor as a freshman. I tutored at Mathnasium for well over a year. I worked at The Boeing Company for over 5 years. I received an MS degreee in Mathematics from Oregon State Univeristy. The classes I took were over 100 hours of upper division credits in mathematical courses such as calculus, statistics, probabilty, linear algrebra, powers, linear regression, matrices, and more. I graduated with honors in both my BS and MS degrees. Past/Present Clients: College Students at Oregon State University, various math people since college, over 7,500 people on the PC from the US and rest the world.

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Master of Science at OSU with high honors in mathematics. Bachelor of Science at OSU with high honors in mathematical sciences. This degree involved mathematics, statistics, and computer science. I also took sophmore level physics and chemistry while I was attending college. On the side I took raquetball, but that's still not relevant.

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I earned high honors in both my BS degree and MS degree from Oregon State. I was in near the top in most of my classes. In several classes in mathematics, I was first. In a class of over 100 students, I was always one of the first ones to complete the test. I graduated with well over 50 credits in upper division mathematics.

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