# From one-dimensional to two-dimensional Markov chains

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.

• How does service work in this queueing system? Is it like the regular M/M/1 queue where only the first customer is experiencing service and subsequent customers are served in the order in which they arrived (skipping any customers whose patience expired and who left the queue)? If that is the case then your model state needs to include more information, not just how many customers of each type you have in the queue, but also their position. Once a customer reaches the front of the queue its service rate will depend on what type of customer it is. – Gareth Sep 8 '15 at 22:32
• Hi. Actually I ended up with the same thing. Order of the customers also matters and it makes the problem very complicated. So is it possible to model it with Markov chains? – alamaranka Sep 9 '15 at 19:01
• I can't see an elegant way to describe the state space of the problem. You could make an approximation using processor sharing discipline rather than first come first served and then the problem can be described using a two dimensional state space as you have above. – Gareth Sep 9 '15 at 22:27
• I searched a little bit about processor sharing queues. In fact there is no queue such systems. An arrived customer gets the service immediately. Would you give me some key ideas how to implement this to my problem? In particular I couldn't figure out how to select the correct service time? Thank you for time. – alamaranka Sep 9 '15 at 23:28
• Processor sharing (en.wikipedia.org/wiki/Processor_sharing) is when the service capacity of the system ($\mu$) is shared between all of the jobs in the queue, rather than being focussed just on the first customer as in first-come first-served (en.wikipedia.org/wiki/First-come,_first-served) – Gareth Sep 10 '15 at 21:18