Example: (Ross, p.338 #48(a)). Consider an $n$ -server parallel queueing system where customers arrive according to a Poisson process with rate $\lambda,$ where the service times are exponential random variables with rate $\mu,$ and where any arrival finding all servers busy immediately departs without receiving any service. If an arrival finds all servers busy,

Question: find the expected number of busy servers found by the next arrival.

It is easy to show that if we had a single server, and at $t=0$, it was busy, than the probability that the customers find the server busy is $\frac{\lambda}{\mu + \lambda}$. Hence $$T_1 := \text{# of busy servers} = 1 * \frac{\lambda}{\mu + \lambda} + 0*(1-\frac{\lambda}{\mu + \lambda}) = \frac{\lambda}{\mu + \lambda}.$$

Given that we have $n$ out of $n$ busy servers at $t=0$ which are independent from each other, the expected number of busy servers should be $$n* \frac{\lambda}{\mu + \lambda},$$ but in the lecture notes where I got this example, the author first finds $T_1$, then considers a $k$-server system with all servers busy and separates the cases according to whether first a customers arrives or at least one of servers finishes it's job. Then it tries to form a recursive relation between $T_i's$, and obtains quite a complex answers.

However, I cannot understand what is wrong with my answers. After all, we throw $n$ independent coins, each of which having probability $p$ to get heads, then we would expect to see $p*n$ heads. But, in this case, for some reason the auhors thinks that this logic does not work. Why?

I just made small simulation of the above system, with $n = 10$ and #of simulations = 100000. As far as numbers tell, the expected number of busy servers found by the customers is quite close to

$$\sum_{i=1}^n \frac{u_i}{u_i+\lambda}.$$ and this is the answer the author gives

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Your approach and result are correct. You mention independence twice. The linearity of expectation does not require independence; your approach and result would be correct even without it. Since you didn’t give any details about the author’s more complicated solution, there’s not much more to say.

  • $\begingroup$ but the simulation does not agree with my answer? $\endgroup$ – Our Mar 7 '20 at 17:28
  • $\begingroup$ see the updated main post also. $\endgroup$ – Our Mar 7 '20 at 17:30
  • $\begingroup$ @onurcanbektas: I did see the update, but didn't understand it, since it contains variables $u_i$ that you didn't introduce. $\endgroup$ – joriki Mar 7 '20 at 21:54
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    $\begingroup$ @onurcanbektas: There's little point in repeating that without replying to my question above about the result of the simulation. In any case, what do you think is more likely – that there's a bug in your simulation (or perhaps you're misinterpreting its results), or that you, I, and Ross in the solution manual all made the same mistake? $\endgroup$ – joriki Mar 7 '20 at 22:45
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    $\begingroup$ There was indeed a bug in the code; thanks a lot for the help. $\endgroup$ – Our Mar 8 '20 at 11:22

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