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?
Best Answer
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?