I have been trying to wrap my head around an M/M/s queue problem but I can't seem to understand what's going on.
The problem is queuing at a gas station and I have done some research about the M/M/s queue model but can't seem to figure out how it works.
Here is what I know:
Expected vehicles in 1 hour is 70
Average service rate of 13.6 vehicles per hour per fueling position
10 fueling positions
So this tells me:
$\lambda$ = 70
$\mu$ = 13.6
$c$ = 10
I know how to get $P_0$ and $P_n$ but how do I solve for average customers in system, average customers in queue, average time spent in system, average time waiting in line?
Let the number of jobs in the system be $0,1, \ldots$. What you need is stationary distribution as $t \to \infty, \ \pi_k$. If the probability to observe $k$ jobs in the system at time $t$ is $P(X_t=k), \ \pi_k = \lim_{t \to \infty} P(X_t=k)$. This is found using Kolmogorov forward equations: $$ \pi_k = p \pi_{k-1}+q \pi_{k+1} $$ (keep in mind $p+q=1$). Once you have derived the expression for $\pi_k$, the definition of expectation of $X$: $$ \mathbf{E}X=\sum_{k=0}^{\infty}k \pi_k $$ and this will be the mean number of jobs in the system. For the mean waiting time you should use Little's formula: $L= \lambda W$. Can you handle from here?