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A newspaper uses one ton of newsprint every day. It buys its newsprint from a local distributor. This ditributor supplies the newsprint in one-ton rolls at the cheapest price, but unfortunatley its deliveries are erratic. Hence, the number of rolls of newsprint in the newpaper warehouse varies randomly. If on a particular day, there are no rolls in the warehouse, the newspaper buys and uses a rold from an emergency supplier. Let $Y_n$ be the number of rolls received from the local distributor on day $n$. The supply arrives in the evening, whereas the one tone is consumed in the morning. Let $X_n$ be the number of rolls in the warehouse at the beginning of day $n$ before the one ton demand has to be satisfied and the supply for day $n$ has arrived. Thus $X_{n=1} = X_n - 1 + Y_n$ if $X_n>0$ and $X_{n+1} = X_n$ is $X_n = 0$.

Suppose that the random variables $\{Y_n, n \geq 0\}$ are idependent of $X_{0}$. Besides, suppose that they are naturally independent and identically distributed and $P(Y_n = k) = a_k, k = 0,1,...$.

  • Suppose $a_k = 0$ for $k \geq 2$ and $X_0 \in \{0, 1\}$, what values can the process $X_n$ take in this situation?

From my transition matrix, I said if we are only looking at the values for when $i, j$ are $0$ and $1$, we would get the little $2 \times 2$ matrix of the whole one step transition matrix and so this would just be

$$\pmatrix{a_0 & a_1 \\ a_0 & a_1}$$

and so the only values that $X_n$ can take in this situation are $a_0$ and $a_1$. But in the answers it says it should be $0$ and $1$. Why is this?

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If you just look at your recursions then this should yield the solution. If $X_0=0$ then $X_n = 0$ for all $n$ -> let's call this the absorbtion.

So we analyze the case that $X_0=1$. Then $$ X_{n+1} = X_n - 1 + Y_n $$ Consider $n=0$ if $Y_0 = 0$ then $X_1 = 1 - 1 + 0 = 0$ and we are caught in $0$ due to the absorption. If $Y_0=1$ then $X_1 = 1 - 1 + 1 = 1$ and we are back in the case $X_1 = 1$. Because $Y$ can only take the values $0$ or $1$ (all other probabilities are zero). Then we can continue this reasoning for the transition from $X_1$ to $X_2$ and finally for $X_n,n\ge 0$ this concludes the proof.

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  • $\begingroup$ An when you consider the transition matrix the $a_{i,j}$ are the probabilities - not the values of $X$. The values are ${0,1}$ otherwise you would have a bigger matrix. The dimension is given by the number of states. $\endgroup$ – Ric Jan 24 '13 at 13:53

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