This one is an example from the book A First Course in Stochastic Models by H.C. Tijms (Chapter 2: Renewal-Reward Processes).
Consider a periodic-review inventory system for which the demands for a single product in the successive weeks $t = 1, 2, \dots$ are independent random variables having a common continuous distribution. Let $X_i$ be the demand in the $ith$ week, $i=1, 2, \dots$. Then $1+N(u)$ is the number of weeks until depletion of the current stock $u$.
I try to understand why the number of weeks until depletion by:
- Let $u$ be current stock, so $u$ should be a non-negative integer number.
- The total number of this kind of product that have been delivered till week $n$ is $S_n = \sum_{i=1}^n X_i$.
- Then $N(u)$ is the number of weeks until $S_n \leq u$.
I cannot derive the result $1 + N(u)$. Also, why does the demand $X_i$ have the continuous distribution since it receives discrete values (or do I misunderstand something there)?
A product such as oil may have a continuous demand, for example.
Since $N(u)=\sup\{n>0:S_n\leqslant u\}$ and $\mathbb P(S_n=u)=0$ (due to $S_n$ being a continuous random variable), here $N(u)$ is the week before depletion of stock, hence the $1+N(u)$.