Why is this Markov chain irreducible?

Question

I am studying Markov chains at the moment and came across this question on an old exam :

"A call centre of an insurance company is interested in modeling in the working process of its employees from one minute to the next. They observe that if an employee is waiting for a call, ten in 40% of the cases, (s)he will be responding to a call the next minute. Employees that are on the phone will in 40% of the cases conclude their phone call to start to work on a follow-up related to this phone-call. During this period they are not available for receiving a phone call. Employees will finish the paperwork the next minute with probability 0.2, in which case they are again available for receiving a phone-call. We assume that this process is a Markov chain.

Why is this Markov chain irreducible?

Can someone explain?

Answer 1

A Markov chain is said to be irreducible if it is possible to get to any state from any state. Here there are $3$ states:

1. Waiting for a call;
2. On a call;
3. Doing paper work related to call.

The description says that:

• $1\rightarrow 2$: $\;\;\operatorname{Pr}(2|1)=0.4$;
• $2\rightarrow3$: $\;\;\operatorname{Pr}(3|2)=0.4$;
• $3\rightarrow1$: $\;\;\operatorname{Pr}(1|3)=0.2$.

Clearly, starting in one state you could get to any of the other ones.