In this video, the professor embarks on finding the equilibrium solution to an M/M/c queueing model, with the condition that $\displaystyle\frac{\lambda}{c\mu}<1$, where:
- Customers arrive into the system as a Poisson process with rate $\lambda$
- Service times by the $c$ servers are exponentially distributed with parameter $\mu$.
I want to interpret the expression $$\displaystyle\frac{\lambda}{c\mu}<1.$$
To start with, I tried interpreting the inequality $\frac{\lambda}{\mu}<c$, in a sense that the arrival rate mustn't be significantly high, so that the system can converge to a steady state.
Another approach was to think that $\mu>\frac{\lambda}{c}$, as in the departure rate should be significant enough not to cause an overload to the system and have it converge to a steady state at $t=\infty$.
But why does the term $c$ assume such an important role here? What I understand is that as the number of servers increases, the capability of the system to handle larger inflow of customers increases, so that the arrival rate can increase without causing the system to explode in the longer run, using the expression $\frac{\lambda}{\mu}<c$.
I would really appreciate a clearer view into the interpretation of the expression as applied to a queueing system.
Best Answer
Intuitively I think about it like this; the 'rate in' is $\lambda$, and the maximum possible 'rate out' is $c\mu$ ($c$ servers working at rate $\mu$). So if $\lambda > c\mu$, we have more people coming in than we can reasonably serve, and the length of the queue goes to infinity.