I'm stuck in a problem on markov jump process:
A gas station receives cars at a rate of 20 vehicles per hour, the station has only one gas pump. If the pump is empty, it receives one client, if a new client comes and find that the pump is occupied by a client he waits. Every client takes around 6 minutes to fill his vehicle (1/10 th of an hour). If two clients are at the pump, one filling and the other one waiting, new clients would leave the station directly.
Solution:
Modeling this problem we have the states are the number of clients at the pump at any instant $t$ so we would increment by one with a rate of 20, and decrement by one (leave the station) by a rate of 10 (because it takes in average 1/10th of hour to fill the vehicle).
So the Jump rate matrix for the chain with the following states $\{0,1,2\}$ is:
$$\begin{bmatrix}-20 & 20 & 0 \\10 & -30 & 20 \\ 0 &10 &-10 \end{bmatrix}$$
Solving for the stationary distribution we find:
$$(u_1,u_2,u_3) = (1/7,2/7,4/7) $$
Now I'm asked to find the average number of clients served by the gas station every hour. The answer should be 50/7 but I can't find how to get this.
The limiting mean number of customers served per hour is simply the limiting fraction of time that a customer is being served times the number of customers that would be served if the server was always busy:
$$ \left(1-\frac17\right)\cdot10 = \frac{60}7. $$