Suppose ${X_i}{i = 1}^\infty$ are i.i.d random variables, such that $P(X_1 > 0) = 1$. Then the corresponding renewal process is $\nu(t) = \max{n \in \mathbb{N} | \Sigma{i = 1}^n X_i \leq t}$. Here $t \in \mathbb{R}_+$.
Questions tagged [renewal-processes]
137 questions
2
votes
0 answers
Renewal Process where $N(t) + 1 \sim Geometric(exp(-t))$
I have the following question in my homework:
$N(t)$ is a renewal process where $L(N(t) + 1) = \operatorname{Geometric}(\exp(-t))$. Find $E[T_1]$ and $E[T_2]$.
However, I'm not sure where to start in order to solve this question. I do know…
1
vote
0 answers
Asymptotic Expression for Number of Renewals
Consider a renewal process with the lifetimes $X_1,X_2,\ldots$ having the continuous uniform distribution $\mathrm{Unif(1.5,4)}$. Determine the asymptotic expression for the expected number of renewals up to time $t$:
My attempt:
Using the formula…
waterr
- 67
1
vote
1 answer
Renewal Process - Inventory problem
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,…
k2pctdn
- 39
0
votes
0 answers
Limiting Mean Excess Life of Renewal Proces
Consider a renewal process with the lifetimes $X_1,X_2,\ldots$ having the probability density function $f(x)=0.125\cdot (4-x)\;$ for $\;0
waterr
- 67
0
votes
0 answers
Two-dimensional renewal equations — a confusing condition
Suppose that $(X_n)_{n=0}^∞$, $(Y_n)_{n=0}^∞$ satisfy the following discrete renewal-style equations for all $n \in \mathbb N$:
$$ \begin{align}
X_n &= \sum_{i=1}^n \, (α_i\, X_{n-i} + β_i\,Y_{n-i}) \tag{1}\\
Y_n &= \sum_{i=1}^n \,…
Good Boy
- 2,210
0
votes
0 answers
Distribution of ages with exponentially distributed resets?
Suppose there is a population of trees, and in each period, each tree faces a probability $p$ of getting cut down.
If a tree is cut down, its height is reset to zero (it doesn't die, this is merely a setback for it); if a tree is not cut down, then…
wintergreen_plaza
- 115
- 6