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I am preparing for an exam and got stuck. The problem is as follows: In a gas station there is one gas pump. Cars arrive at the gas station according to a Poisson proces, rate $\frac{1}{3}$. An arriving car finding n cars at the station immediately leaves with probability $q_n = n/4$, and joins the queue with probability $1 − q_n$, n = 0, 1, 2, 3, 4. Cars are served in order of arrival. The service time is $Exponential(\frac{1}{3}$).

Now I am asked to calculate the sojourn time of cars deciding to take gas at the station: I calculated this using Little, and conditioning on the size of the queue using PASTA, and found $\frac{384}{103}$ both times. However, the answer is supposed to be $\frac{384}{71}$. I'm off by a factor, but I don't understand how I should get to this answer.

I already calculated the stationairy distribution of the number of cars in the system (correctly), but I don't see how to continue from here to get the correct answer.

I would appreciate any and all help, thanks in advance!

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Solving the detailed balance equations $\pi_iP_{i,i+1}=\pi_{i+1}P_{i+1,i}$, $i=0,1,2,3$ along with $\sum_{i=0}^4\pi_i=1$ yields the stationary distribution $$ \pi = \left(\frac{32}{103},\frac{32}{103},\frac{24}{103},\frac{12}{103},\frac3{103} \right). $$ The long-run average size in system is given by $$ L = \sum_{i=0}^4 i\pi_i = \frac{32}{103}+ 2\cdot\frac{24}{103}+3\cdot\frac{12}{103}+4\cdot \frac3{103}=\frac{128}{103}. $$ The effective arrival rate to the system is less than $\frac13$ due to balking; we compute it by $$ \lambda_{\mathrm{eff}} = \frac13\left(\pi_0 + \frac34\pi_1+\frac12\pi_2+\frac14\pi_3\right) = \frac{71}{309}. $$ The mean sojourn time is then given by Little's Law: $$ W = \frac L{\lambda_{\mathrm{eff}}} = \frac{128}{103}\cdot\frac{309}{71} = \frac{384}{71}. $$

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