During a 3h shift, an employee estimates that the probability for him to drink a cup of coffee rises at a constant rate $\omega=0.8/hour$. Given that the rate is time-independent, he expects an average of 2.4 cups consumed in the whole shift. No cup has been consumed at the beginning of the shift ($P_n(t=0)=\delta_{n 0}$).

I want to model this problem by considering Markovian time evolution for the probabilities in order to keep track of fluctuactions.

This is my attemp:

Probability vector whose components $p_i$ give the probability that $i$ have been consumed after the 3h shift at a given time $t>0$.

$$ P(t)=\begin{bmatrix} p_{0}(t) \\ p_{1}(t)\\ p_{2}(t)\\ p_{3}(t)\\ \end{bmatrix} $$

The transition matrix, in left stochastic notation, with elements given by:

$$(W)_{ij}=w_{ij}-\delta_{ij}\sum_{k=0}^{3}w_{kj}, w_{ii}=0$$

where $w_{ij}$ are the non-negative rates for transition from state $j$ to state $i$ per unit time. In this case, the transition rate from having $j$ cups of coffe to $i$ cups.

In this problem I have the following transition rates: $$w_{ij}=0 \hspace{1cm} j>i \\ w_{i+1,i}=0.8\\ w_{i+2,i}=0.8/2=0.4\\ w_{i+3,i}=0.8/3=0.27$$\

I'm considering that there is not possibility to "untake" a cup a coffe, $w_{ij}=0$ for $j>i$ and that if we have 1 cup of coffe, the probability to take 2 more cups of coffe in the next hour ($w_{i+2,i}\rightarrow w_{31}$) will be half the probability to take 1 cup. I don't know if this assumption makes sense, but I couldn't think of anything better with the information "the probability for him to drink a cup of coffee rises at a constant rate $\omega=0.8/hour$" I have been given.

I end up with this transition matrix: $$ M= \left( {\begin{array}{cc} -1.47 & 0 & 0 & 0 \\ 0.8 & -1.2 & 0 & 0 \\ 0.4 & 0.8 & -0.8 & 0 \\ 0.27 & 0.4 & 0.8 & 0 \\ \end{array} } \right) $$

Now the next step would be to solve the master equation: $$\frac{d}{dt}P(t)=W·P(t)$$

Nevertheless, I'm not very confident with the transition matrix I got because I am not sure the transitions rates I wrote make sense. Any comments on this? Would you model it in a different way?


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