holding times

[floydeon]

Please can someone explain the mathematical definition given for the residual holding time in the CR. {R_s>w,X_s=i}={X_u=i,s<u<s+w}. I understand the concept but the definition given seems a bit confusing.

Also would i be correct in saying that the current holding time plus the residual holding time has an exponential distribution in a time-homogeneous Markov jump process since it is simply the total holding time?

 

[Goku]

Cool

Please can someone explain the mathematical definition given for the residual holding time in the CR. {R_s>w,X_s=i}={X_u=i,s<u<s+w}. I understand the concept but the definition given seems a bit confusing.
QUOTE]

Hi Floydeon,

I can answer the 1st part of the question only. What it means is that:
the event consisting of the X_s = i and R_s > w is equivalent to the event of consisting of X_u = i for all u in (s,s+w) (strictly speaking speaking I think X_u = i for all u in [s,s+w] but let's not get into that). I think it makes sense if you just think of it as following: To be in state i at time s, and to be in that state for a period time greater than w, I have to be in state i for the entire period [s,s+w].

I hope you understand the 1st part better now.

Goku

 

[floydeon]

thanks goku..you must be a super saiyan when it comes to CT4. that does make sense although i would be happier if the definition was {R_s=w,X_s=i}={X_u=i,s<u<s+w}, but sure its not worth getting upset about.

 

[Goku]

Gee thanks Floydeon, however I very much doubt I'll ever reach that level! Anyway, as regards your preference "{R_s=w,X_s=i}={X_u=i,s<u<s+w}", from my understanding of statistics and probability, that event occurs with probability zero. Well, it's because R_s has a continuous state space and therefore has a corresponding probability density function associated to it. Kinda like how f(x) is a density for a Normal distribution. So if indeed we wanted to find P[X = a] in this case, we would have to integrate the pdf from a to a, which is just zero.

However a discrete distribution is associated with a probability mass function and is characterized by a 'jumping' non-continuous distribution. An example of that is N(t) where N(t) is Poi(lambda*t). Hence it is permissible to write and speak of P[N(t) = a].

So to conclude, to have X_u = i, s<u<s+w, we require the event {R_s > w,X_s=i} to occur.

Hope that answers you query.

Ka-me-ha-me-ha!