Renewal processes versus Poisson processes

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Problem 5.37 from Gallager R.G. "Stochastic Processes: Theory for Applications" [Gal13]More problems in Chapter 5 "Renewal processes"

(a)(b)(c)(d)

Let $\left\{ N_{1}(t); t>0\right\}$ and $\left\{ N_{2}(t); t>0\right\}$ be independent renewal counting processes. Assume that each has the same CDF $\mathsf{F}(x)$ for interarrival intervals and assume that a density $f(x)$ exists for the interarrival intervals.

(a)

Is the counting process $\left\{ N_{1}(t)+N_{2}(t); t>0\right\}$ a renewal counting process? Explain.

(b)

Let $Y(t)$ be the interval from $t$ until the first arrival (from either process) after $t$ . Find an expression for the CDF of $Y(t)$ in the limit $t \rightarrow \infty$ (you may assume that time averages and ensemble averages are the same).

(c)

Assume that a reward $R$ of rate 1 unit per second starts to be earned whenever an arrival from process 1 occurs and ceases to be earned whenever an arrival from process 2 occurs. Assume that $\lim_{t \rightarrow \infty }(1/t) \int_{0}^{t} R(\tau ) \, d\tau$ exists with probability 1 and find its numerical value.

(d)

Let $Z(t)$ be the interval from $t$ until the first time after $t$ that $R(t)$ (as in (c)) changes value. Find an expression for $\mathbb {E}\left[Z(t)\right]$ in the limit $t \rightarrow \infty$ .

Problem 10.3.1 from Grimmett G.R., Stirzaker D.R. "Probability and Random Processes" [GrSt20]More problems in Section 10.3 "Excess life"

Suppose that the distribution of the excess lifetime $E(t)$ does not depend on $t$ . Show that the renewal process is a Poisson process.

Problem 10.6.9 from Grimmett G.R., Stirzaker D.R. "Probability and Random Processes" [GrSt20]More problems in Section 10.6 "Problems"

Show that Poisson processes are the only renewal processes with non-arithmetic interarrival times having the property that the excess lifetime $E(t)$ and the current lifetime $C(t)$ are independent for each choice of $t$ .

Problem 10.6.10 from Grimmett G.R., Stirzaker D.R. "Probability and Random Processes" [GrSt20]More problems in Section 10.6 "Problems"

Let $N_{1}$ be a Poisson process, and let $N_{2}$ be a renewal process which is independent of $N_{1}$ with non-arithmetic interarrival times having finite mean. Show that $N(t) = N_{1}(t)+N_{2}(t)$ is a renewal process if and only if $N_{2}$ is a Poisson process.

Problem 8.3.5 from Grimmett G.R., Stirzaker D.R. "Probability and Random Processes" [GrSt20]More problems in Section 8.3 "Renewal processes"

Prove Theorem (8.3.5): Poisson processes are the only renewal processes that are Markov chains.

Problem 3.3.1 from Жуковский М.Е., Родионов И.В., Шабанов Д.А. "Основы теории случайных процессов" [ЖРШ16]More problems in Chapter 3.3 "Processes with Independent Increments. The Poisson Process"

Let $\left(X_{t}, t \geq 0\right)$ be a renewal process built from $\left\{ \xi_{n}, n \in \mathbb {N}\right\}$ . Is it true that the process $X_{t}$ always has independent increments?

Problem 3.3.4 from Жуковский М.Е., Родионов И.В., Шабанов Д.А. "Основы теории случайных процессов" [ЖРШ16]More problems in Chapter 3.3 "Processes with Independent Increments. The Poisson Process"

(a)(b)(c)

Let $\left(\xi_{n}, n \in \mathbb {N}\right)$ be independent exponential random variables with parameter $\lambda , S_{n}=\xi_{1}+\ldots +\xi_{n}$ , and let $N=\left(N_{t}, t \geqslant 0\right)$ be the renewal process built from them (a Poisson process of intensity $\lambda$ ). For each $t>0$ denote $V_{t}=S_{N_{t}+1}-t$ (the "overshoot") and $U_{t}=t-S_{N_{t}}$ (the "undershoot").

(a)

Compute the probability $\mathbb {P}\left(V_{t}>v, U_{t}>u\right)$ .

(b)

Prove that $V_{t}$ and $U_{t}$ are independent and that $V_{t} \sim \operatorname {Exp}(\lambda )$ .

(c)

Compute the distribution function of $U_{t}$ and $\mathbb {E}\left[U_t\right]$ .

Problem 3.9.2 from Жуковский М.Е., Родионов И.В., Шабанов Д.А. "Основы теории случайных процессов" [ЖРШ16]More problems in Chapter 3.9 "Continuous-Time Markov Chains"

Is every renewal process a Markov chain?

Problem 10.82 from Прохоров А.В., Ушаков В.Г., Ушаков Н. "Задачи по теории вероятностей" [ПУУ86]More problems in Chapter 10 "Random Processes"

Prove that a random flow is Poisson with intensity $\lambda$ if and only if it is a recurrent flow with $A(t) = 1-e^{-\lambda t}, t \geqslant 0$ .

References

[Gal13] Gallager R.G. Stochastic Processes: Theory for Applications. Cambridge University Press, 2013.

[GrSt20] Grimmett G.R., Stirzaker D.R. Probability and Random Processes. 4th ed., Oxford University Press, 2020.

[ЖРШ16] Жуковский М.Е., Родионов И.В., Шабанов Д.А. Основы теории случайных процессов. МФТИ, 2016.

[ПУУ86] Прохоров А.В., Ушаков В.Г., Ушаков Н. Задачи по теории вероятностей. Наука. Главная редакция физико-математической литературы, 1986.