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Expected time to reach state k-Ch4

A

Actuary@22

Very Active Member
Hi
I have a doubt in the concept Expected time to reach state k starting from state i in Ch 4 page 31 as per acted 2019 material.

Eg There are 3 states i=1,2,3 and transitions are possible from i to i +1,i to i -1 and even from i to i .
We want to calculate the average time for an object currently in state 1 to move to state 3.
So in the formula mentioned below will we also include j=1 or not?Pls explain.

m1=1/lambda1 + summation of j=1,2 (mew 1j/lamda1 )*mj
 
Hi

I'm not sure what formula you are looking at as I don't have immediate access to the 2019 course notes, but the latest course notes and the formula on page 38 of the tables explicitly exclude the current state i and the target state k from the summation.

In your example, let's say we are in state 1 and we are interested in the expected time to reach state 3. We would then get:

m_1 = 1 / lambda_1 + mu_12 / lambda_1 * m_2

We don't need to consider state 1 as an intermediate state because this is a continuous time process, and the 1/ lambda_1 term represents the expected time until the first jump out of state 1, which is necessarily to a different state.

We also don't need to consider state 3 as an intermediate state, since this is the target state and hence m_3 = 0.

Hope this helps,

Dave
 
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