Math Problems Vault

Problem 10.4.1

Find the distribution of the excess lifetime for a renewal process each of whose interarrival times is the sum of two independent exponentially distributed random variables having respective parameters $\lambda$ and $\mu$ . Show that the excess lifetime has mean

\frac{1}{\mu }+\frac{\lambda e^{-(\lambda +\mu ) t}+\mu }{\lambda (\lambda +\mu )}

?

Problem 10.4.2

(a)(b)(c)

Let $f$ be the density function and $F$ the distribution function of the interarrival times $\left\{ X_{i}: i \geq 2\right\}$ of a stationary renewal process, and $\mu$ their common mean. Let $C, D, E$ be the current, total, and excess lifetimes in equilibrium.

?

(a)

Show that $C$ has density function $h(x) = (1-F(x)) / \mu$ for $x > 0$ .

(b)

Show that $D$ has the size-biased (or length-biased) density function $g(y) = (y / \mu ) f(y)$ .

(c)

Let $U$ be uniformly distributed on $(0,1)$ and independent of $D$ . Show that $U D$ has the same distribution as both $C$ and $E$ . Explain why this should be so.

Problem 10.4.3

(a)(b)(c)

Let $m$ be the renewal function of an ordinary renewal process $N$ whose interarrival times have finite mean.

?

(a)

Show that the mean number of renewals in the interval $(a, b]$ is no larger than $1+m(b-a)$ .

(b)

Show that there exists $A > 0$ such that $m(t) \leq A(1+t)$ for $t \geq 0$ .

(c)

Suppose the interarrival times $X_{r}$ are non-arithmetic with finite variance and mean $\mu$ . Show that

m(t)-\frac{t}{\mu } \rightarrow \frac{\sigma ^{2}-\mu ^{2}}{2 \mu ^{2}} \quad \text{ as } t \rightarrow \infty

[Hint: Use coupling and Blackwell's renewal theorem (10.2.5).]