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]
148 questions
8
votes
2 answers
An example of divergent discrete renewal process
Let us consider two sequences $(a_n)$ and $(b_n)$ of nonnegative real numbers, such that :
$$\sum_{n=0}^{+\infty} a_n=1$$
$$\sum_{n=0}^{+\infty} n\,a_n=+\infty$$
$$\sum_{n=0}^{+\infty} b_n<+\infty$$
and define a sequence $(u_n)$ by the recurrence…
Adren
- 8,184
7
votes
1 answer
Formula for the variance of a renewal process
Let $N(t)$ be a renewal process, with a sequence of IID inter-arrival times $X_{1}, X_{2}, \dots$ having finite second moment: $EX_{i}^{2} < \infty$.
How would I show that $$\mathrm{Var}N(t)= 2 \int^{t}_{0} m(t-s) \cdot m'(s)ds + m(t) -…
christoph
- 335
6
votes
1 answer
When Superposition of Two Renewal Processes is another Renewal Process?
When superposition of two renewal processes is another renewal process?
If you merge (superpose) two Poisson processes with parameters $\lambda_1$ and $\lambda_2$, the outcome is another Poisson process with parameter $\lambda_1+\lambda_2$. But,…
Susan_Math123
- 2,850
4
votes
0 answers
Property of a Markov renewal process
Definition of a a Markov renewal process:
Let the states of a process be denoted by the set $E= \{0,1,2, \dots\}$ and let the transitions of the process occur at epochs $t_0 =0,t_1,t_2, \dots$. Let $X_n$ denote the transition occurring at epoch…
Mimimi
- 437
4
votes
1 answer
Negative moment of the number of arrivals in a renewal process
Suppose $\{X_1, \ldots, X_n,\ldots\}$ are i.i.d. non-negative random variables with $\mathbb{E}[X_1]= \mu <\infty$, and $N(t):= \sup\{k: \sum_{i=1}^k X_i < t\}$. Notice that $X_i$ may not have higher order moments. The classical renewal theorem…
shong
- 187
4
votes
1 answer
Splitting a renewal process
This is a follow-up question of the question "When superposition of two renewal processes is another renewal process?".
How can we split a renewal process $P$ into a renewal process $P_1$ and another process $P_2$ (not necessarily renewal), where…
Susan_Math123
- 2,850
3
votes
1 answer
Estimate for Overshoot
Suppose $X_1,X_2,\dots,$ are i.i.d random variables with mean $\mu\in(0,\infty)$ and finite variance. Define the stopping time $N=\min\{ n: \sum_{i=1}^n X_i \geq B\}$. Why it is true that $E \sum_{i=1}^N X_i = B + O(1)$?
I read about this statement…
Percy W
- 386
- 1
- 9
3
votes
1 answer
Compute the renewal function when the interarrival distribution $F$ is such that $1-F(t) = pe^{-\mu_1t} + (1-p)e^{-\mu_2t}$
As described in title, the problem is to compute the renewal function with interarrival distribution $F$,
$$
1-F(t) = pe^{-\mu_1t} + (1-p)e^{-\mu_2t}
$$
I tried to compute renewal function M(t) with renewal equation,
$$
M(t) = F(t) + \int_0^t M(t-y)…
Chia
- 1,057
3
votes
2 answers
Wrong expected value definition in book?
I am currently hospitalised and reading a queueing theory book. I encountered in a proof this, and I fail to understand how this is true:
$$E[R_j]=\int_0^\infty{P(R_j>u)du}$$
Other than the fact that $R_j$ is a random variable defined in…
3
votes
0 answers
I used to think a Poisson process is the only point process with constant event rate.
It's not hard to show that the exponential distribution is the only inter-arrival distribution with a constant event rate. And if you consider the distribution of the number of events falling into an interval, you get a Poisson distribution, with…
Rohit Pandey
- 7,547
3
votes
2 answers
Long run percentage of customers who wait for a bus less than x units of time if customers arrive according to a homogenous Poisson process?
Assume that customers arrive to a bus stop according to a homogenous Poisson process with rate $\alpha$ and that the arrival process of buses is an independent renewal process with interarrival distribution $F$. What is the long run percentage of…
TOMILO87
- 540
3
votes
0 answers
Renewal Process with continuous interarrival times of finite expectations: prove $E[S_{N(t)+1}^2]=E[X_1^2](m(t)+1) - 2E[X_1] \int_{0}^{t} m(x) dx$
Consider a renewal process $\{N(t), t ≥ 0\}$ whose interarrival times $\{X_i\}$ are $IID$ continuous random variables.
Assume that $E[X_1^2]$ is finite (so $E[X_i^2]$ is finite in general).
Prove that $E[S_{N(t)+1}^2]=E[X_1^2](m(t)+1) - 2E[X_1]…
John
- 1,978
3
votes
1 answer
An Indentity of Poisson process
Question:
Suppose buses arrive at a bus stop according to a Poisson process $N_t$ with parameter
. Given a fixed $t > 0$. The time of the last bus before t is $S_{N_t}$ , and the time of the
next bus after $t$ is $S_{N_{t+1}}$. Show the following…
anmo
- 85
3
votes
1 answer
What is the expected length of the interval that time $t$ belongs to?
Consider two Artillery Pieces ($X$ and $Y$) that have been firing forever.
Artillery Piece $X$ fires according to a Poisson process with rate $x$.
Artillery Piece $Y$ fires according to a Poisson process with rate $y$.
Let us fix time $t$ to be $7$…
podge cassidy
- 143
3
votes
1 answer
Renewal for Levy Processes
Suppose $X(t)$ is a Levy process with almost surely positive increments (for all $t_1 < t_2$ $P(X(t_1) < X(t_2)) = 1$)
Define
$$\nu X(t) := \sup \{\tau \in \mathbb{R_+}| X(\tau) < t\}$$
It is not hard to see, that $\nu X$ is also a stochastic…
Chain Markov
- 16,012