Math Problems Vault

Problem 10.5.1

If $X(t)$ is an irreducible positive recurrent Markov chain, and $u(\cdot )$ is a bounded function on the integers, show that

\frac{1}{t} \int _{0}^{t} u(X(s)) d s \xrightarrow {\text{ a.s. }} \sum _{i \in S} \pi _{i} u(i)

where $\mathbf{\pi }$ is the stationary distribution of $X(t)$ .

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Problem 10.5.2

Let $M(t)$ be an alternating renewal process, with interarrival pairs $\left\{ X_{r}, Y_{r}: r \geq 1\right\}$ . Show that

\frac{1}{t} \int _{0}^{t} I_{\left\{ M(s) \text{ is even }\right\} } d s \xrightarrow {\text{ a.s. }} \frac{\mathbb {E}\left[X_{1}\right]}{\mathbb {E}\left[X_{1}\right]+\mathbb {E}\left[Y_{1}\right]} \quad \text{ as } t \rightarrow \infty

Is this limit valid for an arbitrary joint distribution of independent pairs $\left(X_{i}, Y_{i}\right)$ ?

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Problem 10.5.3

Let $C(s)$ be the current lifetime (or age) of a renewal process $N(t)$ with a typical interarrival time $X$ . Show that

\frac{1}{t} \int _{0}^{t} C(s) d s \xrightarrow {\text{ a.s. }} \frac{\mathbb {E}\left[X^{2}\right]}{2 \mathbb {E}\left[X\right]} \quad \text{ as } t \rightarrow \infty

Find the corresponding limit for the excess lifetime.

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Problem 10.5.4

Let $j$ and $k$ be distinct states of an irreducible discrete-time Markov chain $X$ with stationary distribution $\pi$ . Show that

\mathbb {P}\left(T_{j} < T_{k} \mid X_{0} = k\right) = \frac{1 / \pi _{k}}{\mathbb {E}\left[T_{j} \mid X_{0} = k\right]+\mathbb {E}\left[T_{k} \mid X_{0} = j\right]}

where $T_{i} = \min \left\{ n \geq 1: X_{n} = i\right\}$ is the first passage time to the state $i$ . [Hint: Consider the times of return to $j$ having made an intermediate visit to $k$ .]

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Problem 10.5.5

Use the result of Exercise (10.5.2) to show that the limiting distribution function as $t \rightarrow \infty$ of the total life $D(t)$ of a renewal process is

F_{D}(y) = \int _{0}^{y} \frac{1}{\mu } x f(x) d x

where $f$ is the density function of a typical interarrival time $X$ , and $\mu = \mathbb {E}\left[X\right]$ . The integrand is called the size-biased (or length-biased) density for $X$ .

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Problem 10.5.6

(a)(b)

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(a)

Let $X$ be exponentially distributed with parameter $\lambda$ . Show that

\mathbb {E}\left[\min \left\{ X, d\right\} \right] = \frac{1}{\lambda }\left(1-e^{-\lambda d}\right), \quad d \geq 0

(b)

John's garage offers him the choice of two tyre replacement plans.

The garage undertakes to replace all the tyres of his car at the normal price whenever one of the tyres requires replacing.
The garage undertakes to replace all the tyres of his car at $5 \%$ of the normal price two years after they have last been replaced. However, if any tyre needs replacing earlier, the garage will then replace all the tyres at a price that is $5 \%$ higher than the normal price. Assuming that a new tyre has an exponentially distributed lifetime with mean 8 years, determine the long run average cost per year under the two options. Which option should John choose if his car is new?

Problem 10.5.7

A machine $M$ is repaired at time $t = 0$ , and its uptime after any repair is exponentially distributed with parameter $\lambda$ , at which point it breaks down (assume the usual independence). Following any repair at time $T$ , say, it is inspected at times $T, T+m, T+2 m, \ldots$ , and instantly repaired if found to be broken (the inspection schedule is then restarted). Show that the long run proportion of time that $M$ is working (the 'uptime ratio') is $m^{-1} \int_{0}^{m} e^{-\lambda x} d x$ .

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