DES - Definition of 'Poisson Clock Structure'

Poisson Clock Structure

A Poisson clock structure is a stochastic clock structure $F=\left\{F_e:e\in \mathcal{E}\right\}$ where all the distributions $F_e$ are exponential. i.e. for all $e\in \mathcal{E}$ :

$$F_e(t)=\{\begin{array}{cl}1-e^{-\lambda et}& \text{ if }t⩾0\\ 0& \text{ otherwise. }\end{array}{\lambda }_e>0$$

The Poisson Clock Structure holds an important property due to the memory-less property of the exponential distribution:

$$\forall e\in \Gamma (x)V_e\sim \mathrm{Exp}\left(\frac{1}{{\lambda }_e}\right)⟹Y_e\sim \mathrm{Exp}\left(\frac{1}{{\lambda }_e}\right)$$

Where:

• $V_e$ is the total lifetime of event $e$
• $Y_e$ is the remaining lifetime of event $e$

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Demonstration by induction:

1. At initialization ($k=0$):

$$Y_{e,0}=V_{e,1}\text{for all}e\in \Gamma (x_0)$$

$\Rightarrow$ Since we know from the model:

$$V_{e,1}\sim \mathrm{Exp}\left(\frac{1}{{\lambda }_e}\right)⟹Y_{e,0}\sim \mathrm{Exp}\left(\frac{1}{{\lambda }_e}\right)$$

For $k=0$, the distribution of the residual lifetime is equal to the distribution of the total lifetime $\Rightarrow$ So the statement is true for $k=0$

2. We assume that the statement is true for $k-1$,

$$Y_{e,k-1}\sim \mathrm{Exp}\left(\frac{1}{{\lambda }_e}\right)$$

And then we prove that it is true for $k$.

$$Y_{e,k}\sim \mathrm{Exp}\left(\frac{1}{{\lambda }_e}\right)$$

So, we divide the possible cases:

• Case $i$. Event $e$ is activated in state $X_k$, so it started and it can occur.

$$Y_{e,k}=V_{e,N_e,k}$$

$\Rightarrow$ The $k$-th residual lifetime of event $e$ is equal to the total life time from the clock sequence of event $e$. And since, $V_{e,N_e,k}$ is for hypothesis an exponential distribution, then $Y_{e,k}$ is also an exponential distribution with the same $\lambda$

$$\begin{array}{rl}\text{since}{V}_{e,N_e,k}\sim & \mathrm{Exp}\left(\frac{1}{{\lambda }_e}\right)\\ \text{then,}{Y}_{e,k}\sim & \mathrm{Exp}\left(\frac{1}{{\lambda }_e}\right)\end{array}$$

• Case $ii$. Event $e$ “continues” in state $X_k$: event $e$ does not occur but another event happens before $e$ So there exist an event $a$ which residual lifetime is less the the residual lifetime of event $e$.

So the $k$-th residual lifetime of event $e$ is: $Y_{e,k}=Y_{e,k-1}-Y_{a,k-1}$ So:

$$\begin{array}{rl}P\left(Y_{e,k}\le t\mid Y_{e,k-1}>Y_{a,k-1}\right)& =\\ & =P\left(Y_{e,k-1}-Y_{a,k-1}\le t\mid Y_{e,k-1}>Y_{a,k-1}\right)\\ & =P\left(Y_{e,k-1}\le Y_{a,k-1}+t\mid Y_{e,k-1}>Y_{a,k-1}\right)\\ & \text{< using the memory less property >}\\ & =P\left(Y_{e,k-1}\le t\right)\end{array}$$

The probability of event $e$ at state $k$ to happen is the same as the probability of event $e$ at state $k-1$

Now this is always true if we also suppose that event $e$ at state $k$ is not killed, and its ${\lambda }_e$ doesn’t change.