I have a queue system, a classic system with an input generator, a queue and a servant.
- The servant is a $M$-servant with a certain serving rate $\mu$.
- The queue can contain an infinite number of elements.
- The input process is the one that has some particular characteristics.
The input process
The input process behaves in two different ways:
- Normally it behaves like an $M$-process with rate $\lambda$.
- However when the queue has more than $m$ elements, the process provides no input!
Where $m \in \mathbb{N}$ and $m \geq 0$.
Question
How to model this system? How can I build the underlying Markov Chain with such an input process?
Edit
I also need to consider another version of the problem. The input process is not considered $M$ anymore but some kind of Deterministic. It means that when it is active, it will generate $w \geq 0$ elements all at once that get into the queue. How to handle this?
What progress have you made so far yourself? If this question is homework please label it as such.
The system you are dealing with is (in Kendall's notation) an M/M/1/$m+1$ queue, with the following state space. Can you see why?
In states $1, 2, \ldots, m$ arrivals occur at rate $\lambda$, but as you stated in the description when there are more than $m$ jobs in the system there are no further arrials. Services always occur at rate $\mu$, and I've assumed that job arrivals will restart when the number of jobs in the system is no longer greater than $m$.
What performance metrics do you need to compute for the system? Can you see why we don't need to consider infinitely many states, even though you mentioned an infinite state space in your specification of the problem?