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April 2008 Q10 - Notation

B

buddingactuary

Member
Hi

I am having difficulty understanding the notation here. The question mentions that P ON (t) means probability of customer being online and P OFF (t) to mean pr of customer being offline. In that case, what does P' OFF (t) mean? what are the states under consideration for this differential equation in part (ii) of the question?

it looks like P'OFF (t) considers -
P' OFF OFF (t) = P OFF,OFF (t)* pr of OFF to OFF state in short time dt + P OFF, ON (t) * pr of ON to OFF state in short time dt
= P OFF, OFF(t)*(-0.8)+P OFF ON (t)*0.8
But the answer is not making sense to me as there is 0.2 in multiplication with P OFF OFF (t) as per the answer by the examiner. Can someone explain how this is working?
 
I guess, I understand what is going on now... P OFF OFF in short time dt is negative of the transition rate from state OFF to ON which is (-0.2)
so the answer becomes
P' OFF OFF (t) = P OFF, OFF(t)*(-0.2)+P OFF ON (t)*0.8
= 0.8 P ON (t) - 0.2 P OFF (t)
I still don't understand how to make up with these different notations in the exam... consistency in the exam papers will be so much of a help to students!
 
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