I have a $M/M/1$ queueing system that is described below:
- There are two types of customers in the system with different arrival rates, $\lambda_{sg}$ and $\lambda_{sb}$.
- Service rate is $\mu$.
- Type $sg$ customers abandon the queue if waiting time exceeds the patience time $\tau$.
- Patience time is exponentially distributed with rate $\xi=\frac{1}{\tau}$.
To model this system, first I tried one-dimensional Markov chains. You can find the chain in the following image:
However, since only type $sg$ customers abandon the queue, it is better to use two-dimensional Markov chains to model the system.
Following Markov chain is my first try, but it is not true.
The reason is that this two-dimensional Markov chain considers two different queue for two different customers. However there is only one queue for both customers.
The question is:
How to revise this chain to model the system defined above?
Thanks.