So on average, there is 1 television set in the shop?

In summary, the repairman in this scenario fixes broken televisions with a mean repair time of 20 minutes and a Poisson arrival rate of 12 sets per working day. The fraction of time he has work to do is 2/3, and the mean put-through time of a television is 2/3 minutes. On average, there is 1 television set in his shop at a time.
  • #1
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Homework Statement


A repairman fixes broken televisions. The repair time is exponentially distributed with a mean of 20 minutes. Broken television sets arrive at his shop according to a Poisson process with arrival rate 12 sets per working day. (8 hours).
(i) What is the fraction of time that the repairman has work to do?
(ii) What is the mean put-through time (waiting time plus repair time) of a television?
(iii) How many television sets, on average, in his shop?

2. The attempt at a solution
Mean arrival rate = 12/8 hours = 12/480 minutes = 0.025 minutes
Mean servicing time = 20 minutes

So:
[tex]\lambda = 3/2[/tex] = Arrival Rate
[tex]\mu = 1/(1/3) = 3[/tex] = Service Rate
[tex]\rho = (3/2)/(3) = 1/2[/tex] = Traffic Density

(i) - Fraction of time that the repairman has work to do.
I'm basing all this on my assumption that it's an M\M\1 queuing system. So the fraction of time which repairman has no work to do is 1 - the average time per customer, T?

[tex]T = N/\lambda = 1/(\mu - \lambda)[/tex] so, 1/(3 - 3/2) = 2/3? So 1/3 of his time he has no work to do?

(ii) - Mean throughput time:
Average waiting time:
[tex]\rho/(\mu(1-\rho)) + 1/\mu[/tex]

(iii) - [tex]\rho/1-\rho[/tex] = [tex](0.5)/(0.5)[/tex] Which is 1...
 
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  • #2
I think your theory is incorrect. You seem to be using formulae for an infinite and continuous queue.
 
  • #3
Ok, thanks for the advice.

Any ideas on what formulas I should be using?
 

What does "on average" mean in this context?

In this context, "on average" means that if you were to take the total number of television sets in all the shops and divide it by the number of shops, the resulting number would be 1.

How was the average calculated?

The average was calculated by adding up the total number of television sets in all the shops and dividing it by the number of shops.

What is the significance of there being 1 television set on average?

The significance of there being 1 television set on average is that it gives us an idea of how many television sets are typically found in a shop. This can help us understand the demand for televisions and the competitiveness of the market.

Are there any outliers or exceptions to this average?

There may be outliers or exceptions to this average, as the number 1 is simply an average and not a strict rule. Some shops may have more or fewer television sets, but on average, there is 1 television set in a shop.

What other factors may affect the number of television sets in a shop?

Other factors that may affect the number of television sets in a shop include the size and location of the shop, the target market of the shop, and the current demand for televisions in that area. Additionally, promotions, sales, and stock availability may also play a role in the number of television sets in a shop.

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