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I'm trying to find the probability of $n$ customers in a queue of a $M/M/1$ model. I only have the formula for $n$ customers in the system that includes queue and service. The formula is

$$P_n=\left(1-\frac{\lambda}{\mu}\right)\left(\frac{\lambda}{\mu}\right)^n$$

Where $\lambda$ is the average rate of arrival and $\mu$ is the average service rate.

Thanks.

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If total number of customers in the system $X=n$ and $n\geq 1$, then total number of customers in the queue equals to $Y=X-1$. If $X=0$ then $Y=0$ too. Set $\rho=\frac\lambda\mu$.

Then $$\mathbb P(Y=0)=\mathbb P(X=0)+\mathbb P(X=1)=(1-\rho)+\rho(1-\rho)=1-\rho^2.$$

And for $n\geq 1$ $$ \mathbb P(Y=n)=\mathbb P(X=n+1)=(1-\rho)\rho^{n+1} $$

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