I am learning the basics for queuing theory for a M/M/1 queue. One thing I do not understand is to do with utilization factor $ρ$.
Assuming that:
$λ$ = arrival rate, exponentially distributed
$μ$ = service rate, exponentially distributed
$ρ$ = utilization factor
Therefore $ρ = \dfrac{λ}{μ} < 1$.
$ρ$ needs to be less than one for it to work, but why? What if a have a problem where $λ$ is greater than $μ$ therefore $ρ$ > 1. For example $λ = 4$ and $μ = 2$ then $p = 2$. What to do then?
The problem is that there is no equilibrium in such a situation. Customers are going into the system faster (e.g., $\lambda = 4$) than they can be served (e.g., $\mu = 2$), so the queue increases without bound. All of the usual formulas that go with the M/M/1 system (average queue length, average waiting time, etc.) apply only to systems in equilibrium.