I am attempting to calculate the the probability of a customer joining a queue by using the Erlang C curve in an M/M/m queue, but I am struggling to get to the final answer.

In the question I am working on, customers arrive at a rate of 120 customers per hour. Dealing with each customer takes on average 3 minutes. When all serving assistants are occupied, customers are placed in a holding system. There are 10 serving assistants.

I understand that the following parameters are as follows:

Arrival rate, $$λ = 2$$ Service rate, $$µ = \frac{1}3$$ Number of servers, $$m = 10$$ And thus the offered load, $$a = \frac{λ}µ = 6\:Erlangs$$

The solution provided to this question is given exactly as seen below, but I am unsure as to how to get to this final value:

"and from the Erlang C curve, it follows that the probability of waiting is about 0.10. "

I have tried to submit the above values into the Erlang C formula to get to the same answer, to no success. My attempt, assuming that this is how to get to the probability of waiting using the Erlang C curve:

$$C(m, a) = P_0(\frac{a^m}{m!(1-\frac{a}m)}) $$

We were told that care should be exercised when C(m,a) was computed for large values of m because overflow may occur, so could avoid this by computing the logarithm of C(m,a): $$⇒ lnC(m,a) = lnP_0 +mln(a) -ln(m!) - ln(1-\frac{a}m)$$ $$= ln(0.4) + 10ln(6)-ln(10!)-ln(1-\frac{6}{10})$$ $$= 2.81$$

I calculated the the first P0 term in the above log computation (I think that this is the probability that the system is empty) as follows: $$P_0 = 1-\rho = 1 - \frac{λ}µ=0.4$$

To summarise:

  1. Using the parameters: arrival rate, service rate, number of servers, and thus offered load, how can the probability of a customer joining a queue be calculated "from the Erlang C Curve"?

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