[Math] Weighted average of two frequencies

averageprobabilityprobability theorystatistics

This should be really simple but I'm getting stuck and I'm probably extremely dumb..

I know that a machine receives two kind of parts:

Type 1 -> with frequency 50 per week, each one is processed for 20 minutes
Type 2 -> with frequency 100 per week, each one is processed for 8 minutes

It was told me to consider (in working hours) 1 week = 2400 minutes.

I have to calculate the average arrival rate of the machine and the average service time of the machine.

Average arrival rate of the machine = 50 pieces per week + 100 pieces per week = 150 pieces per week, this is simple since each piece counts as an arrival (they're all equal)

I'm confused with the Average service time per piece of the machine…

I tried:

the machine can serve 2400/20 type 1 pieces per week = 120
and 2400 / 8 type 2 pieces per week = 300

so average_service_rate = (120*50 + 300*100) / 150 = 240, but this is wrong according to my solution (which is 200)…

if I use service times it works:
time per each type 1 piece: 20 min
time per each type 2 piece: 8 min
average_time_per_piece = (20*50 + 8*100)/150 = 12 min, i.e. 12/2400 week
so 1/averaget_time_per_piece = average_service_rate = 2400/12 = 200. And this is the right value.

My question is: why my first approach with the average frequency didn't work? What's wrong with it?

Best Answer

Your original wrong answer might be illustrated by

  • the question of the average speed of a person who travels at 20 seconds per kilometre for 50 seconds and then at 8 seconds per kilometre for another 100 seconds,

compared with

  • the question of the average speed person who travels at 20 seconds per kilometre for 50 kilometres and then at 8 seconds per kilometre for another 100 kilometres.

Average speed is total distance divided by total time so:

  • For the first person the average speed is $\dfrac{\frac{50}{20} + \frac{100}{8}}{50 + 100} = \frac{1}{10}$ kilometres per second.

  • For the second person the average speed is $\dfrac{50 + 100}{50 \times 20 + 100 \times 8} = \frac{1}{12}$ kilometres per second.

Multiply both answers by $2400$ seconds per week if it helps.

In your original question you were asked for the second method (the weights given are items not times) but initially answered using the first.

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