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The rate of service is exponential and the service rate is 12 customers served per hour. The arrival of customers is in a Poisson distribution at the rate of 30 per hour. There are 3 servers and the average waiting time is 7 minutes. How would I go about doing this? Could really use some help!

Edit - I've also calculated the probability that an arriving customer has to wait as 0.7. Not sure how to calculate the probability regarding waiting time though.

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What you want is not the total time a customer spends in the system, but just in the line. Do you know the formula for the queue waiting time $\mathcal{W_q}$ distribution of an M/M/3 queue? Its a pretty common formula in queue theory textbooks. Its cumbersome to write, so ill jsut link you to it, see Slide 35 of this ppt. And p.13 of this

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  • $\begingroup$ I know the formula for the average waiting time in a queue, that is Wq. I even know Pw, the probability that a customer would have to wait. But I'm not sure how to calculate the probability that a customer would have to wait for a/greater than/lesser than a certain time limit. Yeah, what I want is the total time a customer spends waiting in the line before he is served. Formula's not given in my text book. $\endgroup$ Nov 4, 2013 at 15:35
  • $\begingroup$ See the links I sent you, $P(\mathcal{W_q}>t)$ is in there. Should help you out a lot. $\endgroup$
    – user76844
    Nov 4, 2013 at 15:46
  • $\begingroup$ Just did. Thanks! $\endgroup$ Nov 4, 2013 at 15:49

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