I'm trying to understand queueing systems and I found some notes online. They define $\lambda$ as the mean arrival rate, and $\mu$ as the mean service rate (the average number of customers who can be served by a single service station per unit time). The example they give is

Customers arrive at a bank at a rate of 30 per hour. Arrivals are random and service time is exponential, so that the $M/M/1$ model applies. The clerk’s service time is 90 seconds. Therefore $\lambda = 30$ and $\mu = 45$, and $\rho = \lambda/\mu = 30/45$.

I don't understand where the $\mu = 45$ came from. Can some please explain. Thank you.

  • $\begingroup$ As you mentioned $\mu$ is mean service rate. The clerk's service time is 90, but in average, it's 45 $\endgroup$ – Rowan Dec 5 '15 at 9:32

There are $60\times 60 = 3600$ seconds in an hour.

So I would have thought that the service rate was $\mu = \dfrac{3600}{90}= 40$ customers per hour, and $\dfrac{\lambda}{\mu}=\dfrac{30}{40}=\dfrac34$.

  • $\begingroup$ Yes, that's what I was thinking too. I also got $\mu = 40$. Now I'm fairly convinced it was a typo. $\endgroup$ – David Dec 5 '15 at 9:44

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