2.1. The three fixed finite time $t$ pdfs in the extreme renewal process

Figure 1 shows a sketch of an extreme renewal process where the inter-renewal times are i.i.d. Pareto(II) r.v.s (random variables) $\{ Z_{n}\}_{n=1,2,\ldots }$, showing the variates of interest: excess, age, and total life with respect to fixed finite time $t$.

2.2. The Pareto(II) probability distribution for inter-renewals of the extreme renewal process ${\{Z_{n}\}_{n=1,2,\ldots }}$

Consider the Pareto(II) inter-renewals of the extreme renewal process. Denote the cdf, ccdf (complementary cdf), and pdf of each inter-renewal in $\{Z_{n}\}_{n=1,2,\ldots }$ by: $B(x)$; $\bar {B}(x) := 1-B(x)$, $0\leq x\lt\infty$; and $b(x)$, $0\lt x\lt\infty,$ respectively. The inter-renewals $Z_{ {n}} \overset {d}{=} Z \overset {d}{=} {\rm Pareto(II)}(x,\alpha ),\ 0\leq x\lt\infty,$ $\alpha \in (0,1]$, where

(1)\begin{equation} \left.\begin{array}{l} B(x) = 1-( 1+x)^{-\alpha },\quad 0\leq x\lt\infty, \\ \bar{B}(x) =1-B(x)=( 1+x)^{-\alpha },\quad 0\leq x\lt\infty, \\ b(x)=\dfrac{d}{dx}B(x)=\alpha ( {1+x})^{-{\alpha -1} },\quad 0\lt x\lt\infty, \end{array}\right\} \end{equation}

where $B(x)\equiv P(Z\leq x)$. Then, $Z$ is heavy-tailed [Reference Sigman17]. Since $\alpha \in (0,1]$, $Z$ has no mean, implying that the limiting pdfs of excess, age, and total life do not exist.

We will use the shape parameter $\alpha =0.5$ throughout for illustrative purposes. For any value of $\alpha \in (0,1]$, the analysis would be similar.

Consider the 1-dimensional illustration of the extreme renewal process $\{Z_{n}\}_{n=1,2,\ldots }$ in Figure 1. The fixed finite time point of interest is $t\gt0$. Also, $N(t)$ := number of renewals in the interval $(0,t)$. Let $N_{t}$ := number of renewals required to first exceed $t$, that is, $N_{t} := \min \{n|\sum _{{i=1}}^{{n}}Z_{{i}} \gt t\}$. Then, $N_{t}$ is a stopping time for $\{Z_{n}\}_{n=1,2,\ldots }$. Also, $N_{t}= N(t)+1$, and $E[ N_{t}] = E[ N(t)] +1 \equiv M(t)+1$, where $M(t)\equiv E[ N(t)]$ is the “renewal function” (see, e.g., [Reference Karlin and Taylor11] pp. 167–169).

In $\{Z_{n}\}_{n=1,2,\ldots }$, the fixed finite time $t$ pdfs of excess, age, and total life exist [Reference Brill4]. But the corresponding limiting pdfs of excess, age, and total life as $t\rightarrow \infty$, do not exist because $Z_{n}$ has no mean.

2.3. The truncated Pareto(II) probability distribution in the companion renewal process

Define a companion renewal process $\{Z_{K\!,n}^{\textrm {TR}}\}_{{n=1,2,\ldots }}$ related to, but distinct from $\{ Z_{n}\}_{n=1,2,\ldots }$, where the inter-renewals are: $Z_{K\!,n}^{\textrm {TR}}\overset {d}{=}Z_{K}^{\textrm {TR}}\overset {d}{=} \textrm {right-truncated Pareto(II)}(x,\alpha ),\ 0\lt x\lt K$. The superscript “$\textrm {TR}$” indicates that $K := \textrm {right truncation point of}\ Z$. In $\{Z_{K\!,n}^{\textrm {TR}}\}_{{n=1,2,\ldots }}$, the limiting pdfs of excess, age, and total life, as $t\rightarrow \infty$, do exist because $E[ Z_{K}^{\textrm {TR}}]\lt\infty$ (finite). Importantly, the limiting pdfs of excess, age, and total life in $\{Z_{K\!,n}^{\textrm {TR}}\}_{{n=1,2,\ldots }}$ have relatively simple formulas (Section 2.5).

We use the limiting pdfs of excess, age, and total life in $\{Z_{K\!,n}^{\textrm {TR}}\}_{{n=1,2,\ldots }}$ to approximate the analytic fixed finite time $t$ pdfs of interest (large $t$) in the extreme renewal process $\{ Z_{n}\}_{n=1,2,\ldots }$. No computations are required since the theoretical formulas of the limiting pdfs are well known. This approximation method saves a great deal of computer time required for the computational derivation of the analytic fixed finite time $t$ pdfs, for even a single large $t$.

Substituting from formula (1) in Section 2.2, we obtain the truncated cdf $B_{K}^{\textrm {TR}}(x)$, ccdf $\overline {B_{K}^{\textrm {TR}}}(x)$, and pdf $b_{K}^{\textrm {TR}}(x)$, of $Z_{K}^{\textrm {TR}}$ to be, respectively:

(2)\begin{equation} \left.\begin{array}{l} B_{K}^{{\rm TR}}(x)=\dfrac{B(x)}{B(K)}=\dfrac{1-( 1+x)^{-\alpha }}{ 1-( 1+K)^{-\alpha }},\quad 0\lt x\lt K, \\ \overline{B_{K}^{{\rm TR}}}(x) =1-B_{K}^{{\rm TR}}(x)=1-\dfrac{1-( 1+x)^{-\alpha }}{1-( 1+K)^{-\alpha }},\quad 0\lt x\lt K, \\ b_{K}^{{\rm TR}}(x)=\dfrac{d}{dx}B_{K}^{{\rm TR}}(x)=\dfrac{\alpha ( {1+x})^{-{\alpha -1}}}{1-( 1+K)^{-\alpha }},\quad 0\lt x\lt K\text{.} \end{array}\right\} \end{equation}

2.4. The expected value of the right-truncated Pareto(II) at level $K$

For all $\alpha \in ( 0,1]$, $Z_{K}^{\textrm {TR}}$ has a finite mean which is given in formulas (3), when using $\overline {B_{K}^{\textrm {TR}}}(x)$ in (2).

(3)\begin{equation} \left.\begin{aligned} E(Z_{K}^{{\rm TR}}) & =\int_{x=0}^{K}\overline{B_{K}^{{\rm TR}}}(x)\,dx=\int_{x=0}^{K}\left( 1-\dfrac{1-( 1+x)^{-\alpha }}{1-( 1+K)^{-\alpha }} \right) dx \\ & = \int_{x=0}^{K}\left( \dfrac{( 1+x)^{-\alpha }-( 1+K)^{-\alpha }}{1-( 1+K)^{-\alpha }}\right) dx\\ & = K-\dfrac{( -\alpha +1) K-( 1+K)^{-\alpha +1}+1}{ ( -\alpha +1) ( 1-( 1+K)^{-\alpha }) },\quad \text{if }0\lt\alpha \lt1; \\ E(Z_{K}^{{\rm TR}}) & =\left(1+\dfrac{1}{K}\right)\ln (1+K)-1,\quad \text{if }\alpha =1. \end{aligned}\right\} \end{equation}

In (3), $0\lt E(Z_{K}^{\textrm {TR}})\lt K$ (finite) when $K\gt0$. (Note: When $\alpha =1$, $\inf E(Z_{K}^{\textrm {TR}})=\inf _{K\downarrow 0}E(Z_{K}^{\textrm {TR}})=0$). Thus, the limiting pdfs of excess, age, and total life in $\{Z_{K\!,n}^{\textrm {TR}}\}_{n=1,2,\ldots }$ exist as time $\rightarrow \infty$ (Section 2.5).

We explore using the limiting pdfs of $\{Z_{K\!,n}^{\textrm {TR}}\}_{{n=1,2,\ldots }}$ to approximate the analytic fixed finite time $t$ pdfs of excess, age, and total life for large $t$ in the extreme renewal process $\{Z_{n}\}_{{n=1,2,\ldots }}$. When constructing the renewal process $\{Z_{K\!,n}^{\textrm {TR}}\}_{{n=1,2,\ldots }}$ we will need to determine the value of a right truncation point $K$ such that the limiting pdfs of excess, age, and total life approximate the corresponding analytic fixed finite time $t$ pdfs in $\{Z_{n}\}_{n=1,2,\ldots }$ on a nontrivial subset of the interval $( 0,K)$ (Section 4). The quality of the approximation will be measured by an $L_{1}$ distance measure between the analytic fixed finite time $t$ pdfs in $\{Z_{n}\}_{n=1,2,\ldots }$ and the corresponding (approximating) limiting pdfs in $\{Z_{K\!,n}^{\textrm {TR}} \}_{{n=1,2,\ldots }}$ (Section 5).

2.5. The limiting pdfs of excess, age, and total life in the renewal process with right-truncated inter-renewals ${\{Z_{K\!,n}^{\textrm {TR}}\}_{{n=1,2,\ldots }}}$

Denote the limiting excess, age, and total life by $\gamma _{K}^{\textrm {TR}}$, $\delta _{K}^{\textrm {TR}}$, and $\beta _{K}^{\textrm {TR}}$, respectively; with corresponding pdfs $f_{\gamma _{{K}}}^{\textrm {TR}}(x)$, $0\lt x\lt K$; $f_{\delta _{K}}^{\textrm {TR}}(x)$, $0\lt x\lt t$; $f_{\beta _{K}}^{\textrm {TR}}(x)$, $0\lt x\lt K$, where $K\gt t$. We use the well-known formulas for the limiting pdfs of a standard renewal process where the inter-renewals have a finite mean (e.g., formulas (6.2), (6.5) and (6.6), pp. 193–194, in [Reference Karlin and Taylor11]) and substitute from the formulas in (2). This gives for the renewal process $\{Z_{K\!,n}^{\textrm {TR}}\}_{{n=1,2,\ldots }}$:

(4)\begin{align} f_{\gamma_{{K}}}^{{\rm TR}}(x)& =\frac{1}{E[ Z_{K\gamma }^{{\rm TR}}] } \overline{B_{K_{\gamma }}^{{\rm TR}}}(x)=\frac{1}{E[ Z_{K\gamma }^{{\rm TR}}] }\left( 1-\frac{1-( 1+x)^{-\alpha }}{1-(1+K_{\gamma })^{-\alpha }}\right),\quad x\in ( 0,K_{\gamma }), \ K_{\gamma }\gt t, \end{align}

(5)\begin{align} f_{\delta_{K}}^{{\rm TR}}(x)& =\frac{1}{E[ Z_{K_{\delta }}^{{\rm TR}}] } \overline{B_{K_{\delta }}^{{\rm TR}}}(x)=\frac{1}{E[ Z_{K_{\delta }}^{{\rm TR}} ] }\left( 1-\frac{1-( 1+x)^{-\alpha }}{1-( 1+K_{\delta })^{-\alpha }}\right),\quad x\in ( 0,K_{\delta }),\ K_{\delta }=t, \end{align}

(6)\begin{align} f_{\beta_{K}}^{{\rm TR}}(x)& =\frac{1}{E[ Z_{K_{\beta }}^{{\rm TR}}] } xb_{K_{\beta }}^{{\rm TR}}(x)=\frac{1}{E[ Z_{K_{\beta }}^{{\rm TR}}] }x\left( \frac{\alpha ( {1+x})^{-{\alpha -1}}}{1-( 1+K_{\beta })^{-\alpha }}\right),\quad x\in ( 0,K_{\beta }),\ K_{\beta }\gt t, \end{align}

where $E[ Z_{K\zeta }^{\textrm {TR}}]$ is given in (3) on replacing $K$ by $K_{\zeta }$, where $\zeta =\gamma$, $\delta$, or $\beta$.

We can quickly check formulas (4)–(6) using the elementary renewal theorem (e.g., [Reference Ross16] pp. 432–433) and the renewal reward theorem (e.g., [Reference Tijms21] p. 33ff). Note that the sample paths of all three regenerative processes have the same rate out of level $0$ (see Figure 2), and have the same expected cycle, for example, $E [ Z_{K_{\zeta }}^{\textrm {TR}}]$. That is, by the elementary renewal theorem the rate out of level $0$ is $1/E[ Z_{K\zeta }^{\textrm {TR}}]$, $\zeta =\gamma$, $\delta$. $\beta$ and all three regenerative cycles are equal.

Figure 2.

In figure $Z_{n} \overset {d}{=}Z_{K_{ \varsigma }}^{\textrm {TR}}$, $\varsigma = \gamma, \delta, \beta$. $\gamma _{Z}(t)$ (slope $-1$) := limiting excess. $\delta _{Z}(t)$ (slope $+1$) := limiting age. $\beta _{Z}(t)$ (slope $0$) := limiting total life. Expected regenerative cycles $E[ Z_{K \zeta }^{\textrm {TR}}]$ are all equal.

Clarifying formula (4), the limiting pdf of excess at time $t$ (same as the limiting pdf of remaining service time at instant $t$ (see Figure 2)). The expected number of upcrossings of arbitrary state-space level $x$, ($0\lt x\lt K$) in a regenerative cycle is equal to $1$ with probability $\overline {B_{K_{\gamma }}^{\textrm {TR}}}(x)$ (probability that an upward jump representing a service time overshoots level $x$). By the renewal reward theorem (first two equalities immediately below) and the basic level crossing theorem (third equality immediately below [Reference Brill3]):

$$\frac{E[ \text{No. of }x\text{-upcrossings in cycle}] }{E[ \text{cycle}] }=\frac{\overline{B_{K_{\gamma }}^{{\rm TR}}}(x)}{E[ Z_{K_{\gamma }}^{{\rm TR}}] }=\lim_{s\rightarrow \infty }\frac{\mathcal{D}_{s}(x)}{s}=f_{\gamma_{K}}^{{\rm TR}}(x),\quad 0\lt x\lt K_{\gamma },$$

where $\mathcal {D}_{s}(x)$ is the number of sample-path downcrossings of level $x$ during a time interval $( 0,s)$.

The age at time $t$ is the time that a customer has been in service at instant $t$ (Figure 2). The sample path of the regenerative process for limiting age at $t$ overlays the sample path of the limiting excess at time $t$. Using the renewal reward theorem and basic LC theorem similarly as for the limiting excess at time $t$ yields the formula

$$\frac{\overline{B_{K_{\delta }}^{{\rm TR}}}(x)}{E[ Z_{K_{\delta }}^{{\rm TR}}] }=\lim_{s\rightarrow \infty }\frac{\mathcal{U}_{s}(x)}{s}=f_{\delta_{K}}^{{\rm TR}}(x),\quad 0\lt x\lt K_{\delta},$$

where $\mathcal {U}_{s}(x)$ is the number of sample path upcrossings of level $x$ during time interval $( 0,s)$.

The limiting total life at time $t$ is equal to limiting excess $+$ limiting age. It is constant during the service time of the customer in service at time $t$. It is also a regenerative process (Figure 2). The sample path of limiting total life overlays the sample paths of the limiting excess and limiting age. The renewal reward theorem gives:

$$\frac{\overline{B_{K_{\beta }}^{{\rm TR}}}(x)}{E[ Z_{K_{\beta }}^{{\rm TR}}] } =\lim_{t\rightarrow \infty }\frac{\mathcal{U}_{t}(x)}{t}=\lim_{t\rightarrow \infty }\frac{\mathcal{D}_{t}(x)}{t}=\int_{y=x}^{K}\frac{1}{y} f_{\beta_{K}}^{{\rm TR}}(y)\,dy,\quad 0\lt x\lt K_{\beta }.$$

Taking $d/dx$ in the left-most and right-most terms in the immediately-above formula gives

$$-\frac{b_{K_{\beta }}^{{\rm TR}}(x)}{E[ Z_{K_{\delta }}^{{\rm TR}}] }={-}\frac{1 }{x}f_{\beta_{K}}^{{\rm TR}}(x),\quad 0\lt x\lt K_{\beta },$$

or

$$f_{\beta_{K}}^{{\rm TR}}(x)=\frac{xb_{K_{\beta }}^{{\rm TR}}(x)}{E[ Z_{K_{\delta }}^{{\rm TR}}] },\quad 0\lt x\lt K_{\beta }.$$

3.1. Obtaining an integral equation for the key limiting mixed pdf ${\{ \pi _{{\bf RG},0}^{(t)},f_{{\bf RG}}^{(t)}(x)\}_{0\lt x\lt t}}$ in the underlying regenerative process ${\{ X_{{\bf RG}}(s)\}_{s\geq 0}}$

Applying the stochastic level crossing method in Sections 3.2.1–3.2.2, p. 198, in Brill [Reference Brill4] (or Section 10.3 in [Reference Brill5]), leads to the Volterra integral equation of the second kind for $f_{\textrm {RG}}^{{(}t{)}}(x)$:

(9)\begin{equation} f_{{\rm RG}}^{({t})}(x)=\pi_{{\rm RG},0}^{(t)}\alpha (1+x)^{{-\alpha -1} }+\alpha \int_{{y=0}}^{{x}}(1+x-y)^{{-\alpha -1} }f_{{\rm RG}}^{({t})}(y)\,dy,\quad 0\lt x\lt t,\ \alpha \gt0, \end{equation}

and the normalizing condition

(10)\begin{equation} \pi_{{\rm RG},0}^{(t)}+\int_{0}^{t}f_{{\rm RG}}^{({t})}(x)\,\textit{dx}=1. \end{equation}

Formulas (3.13) and (3.14) in Brill [Reference Brill4] connect the solution of Eqs. (9) and (10) to the classical renewal function $M(x)$ (e.g., [Reference Karlin and Taylor11] p. 169):

(11)\begin{equation} \pi_{{\rm RG}}^{(t)}=\frac{1}{1+M(t)},\quad f_{{\rm RG}}^{(t)}(x)=\frac{ M^{\prime }(x)}{1+M(t)},\quad 0\lt x\lt t, \end{equation}

where $M(t)$ is equal to a series of self-convolutions of the inter-renewal cdf $B(x)$. This series may be time-consuming computationally, and tedious to derive, especially for large $t$.

Therefore, we use a relatively straightforward numerical procedure (Section 3.2) to obtain a computational solution for the pdf $f_{\textrm {RG}}^{({t})}(x)$, $0\lt x\lt t$ by solving integral Eq. (9). This is followed by the application of Eq. (10) to obtain a computational solution for $\pi _{\textrm {RG},0}^{(t)}$. (As a by-product, this also gets a computational solution for $M(t)$ by virtue of formula (11).)

This proposed numerical procedure used here is sufficient to gain insight into the approximation of the analytic fixed finite time $t$ pdfs of excess, age, and total life in the renewal process $\{ Z_{n}\}_{n=1,2\ldots }$ for large $t$ by applying the corresponding computational limiting pdfs in the companion renewal process $\{Z_{K\!,n}^{\textrm {TR}} \}_{{n=1,2,\ldots }}$.

Section 5 considers a plausible $L_{1}$ measure of the distance (discrepancy) between these two pdfs for each time-$t$ random variable, and gives a computed numerical value of the corresponding measure of distance.

(The literature contains various numerical solution techniques to solve Volterra integral equations of the second kind, e.g., [Reference Brunner6,Reference Press and Teukolsky14]—and references therein.)

3.2. Details about solving for the limiting mixed pdf ${\{ \pi _{{\bf RG},0}^{(t)},f_{{\bf RG},}^{(t)}(x)\}_{x\in (0,t)}}$ of the regenerative process ${\{ X_{{\bf RG}}(s)\}_{s\geq 0}}$

We solve Eq. (9) with the normalizing condition (10) to derive $\{ \pi _{\textrm {RG},0}^{(t)},f_{\textrm {RG},}^{(t)}(x)\}_{x\in (0,t)}$ using directly the Riemann–Stieltjes definition of an integral on a finite interval (e.g., [Reference Apostol1] p. 141). The resulting numerical solution is a step function with respect to a preassigned partition of state-space interval $(0,t)$ with a norm $h\gt0$. To get a good computational solution for $\{ \pi _{\textrm {RG},0}^{(t)},f_{\textrm {RG}}^{(t)}(x)\}_{x\in (0,t)}$, we choose small $h$ such that $t = Nh$ where $N$ is a positive integer.

3.2.2. Details about computer program to compute limiting mixed pdf $\{ \pi _{\textrm {RG},0}^{(t)},f_{\textrm {RG},}^{(t)}(x)\}_{x\in (0,t)}$ of the regenerative process $\{ X_{\textrm {RG}}(s)\}_{s\geq 0}$

Let $f_{\textrm {RG}\ast }^{(t)}(x) ={f_{\textrm {RG}}^{({t})}(x)}/{\pi _{\textrm {RG},0}^{(t)}}$, $x\in (0,t)$. This transforms Eq. (9) into the following integral equation for $f_{\textrm {RG}\ast }^{({t})}(x)$:

(12)\begin{equation} f_{{\rm RG}\ast }^{({t})}(x)=\alpha (1+x)^{{-\alpha -1}}+\alpha \int_{{y=0}}^{{x}}(1+x-y)^{{-\alpha -1}}f_{{\rm RG}\ast }^{( {t})}(y)\,dy,\quad 0\lt x\lt t. \end{equation}

The computation consists of the following five steps.

• Step 1. Compute and store in computer memory:

  (13)\begin{equation} b(ih):=\alpha (1+ih)^{{-\alpha -1}},\quad i=0,\ldots,N,\ h\gt0. \end{equation}

• Step 2. Using Eq. (12), start the computation with $i=0$, that is,

  (14)\begin{equation} f_{{\rm RG}\ast }^{({t})}(x)=b(0)+0=\alpha,\quad 0\lt x\lt h. \end{equation}
  Compute for $i = 1,\ldots,N$,
  (15)\begin{align} f_{{\rm RG}\ast }^{(t)}(ih)& =\alpha (1+ih)^{{-\alpha -1} }+\int_{y=0}^{ih}\alpha (1+ih-y)^{{-\alpha -1}}f_{{\rm RG}\ast }^{(t)}(y)\,dy \nonumber\\ & =\alpha (1+ih)^{{-\alpha -1}}+\sum_{j=1}^{i}\int _{y=(j-1)h}^{jh}\alpha (1+ih-y)^{{-\alpha -1}}f_{{\rm RG}\ast }^{(t)}(y)\,dy,\quad i=1,\ldots,N. \end{align}
  where $f_{\textrm {RG}\ast }^{(t)}(y) = f_{\textrm {RG}\ast }^{(t)}(( i-1) h)$, $( i-1) h\lt y\lt ih,$ $i = 1,\ldots,N$.

  This gives the values of $f_{\textrm {RG}\ast }^{(t)}(ih),$ $i =1,\ldots,N$.

• Step 3. From Step 2, construct the step function

  (16)\begin{equation} f_{{\rm RG}\ast }^{(t)}(x)=f_{{\rm RG}\ast }^{(t)}((i-1)h),\quad x\in ( ( i-1) h,ih) ,\quad i=1,\ldots,N. \end{equation}

• Step 4. Compute the estimation for

  (17)\begin{equation} \int_{x=0}^{t}f_{{\rm RG}\ast }^{(t)}(x)\,dx\approx h\sum_{i=1}^{N}f_{{\rm RG}\ast }^{(t)}(ih). \end{equation}

• Step 5. Denote the computational solution for $\{ \pi _{\textrm {RG},0}^{(t)},f_{\textrm {RG}}^{(t)}(x)\}_{x\in (0,t)}$ by $\{ \widehat {\pi _{\textrm {RG},0}^{(t)}},\widehat {f_{\textrm {RG}}^{(t)}}(x)\}_{x\in (0,t)}$. Then,

  (18)\begin{align} \widehat{f_{{\rm RG}}^{(t)}}(x)& =\widehat{\pi_{{\rm RG},0}^{(t)}} \widehat{f_{{{\rm RG}}\ast }^{(t)}}(x),\quad 0\lt x\lt t, \\ \widehat{\pi_{{\rm RG},0}^{(t)}}+\int_{0}^{t}\widehat{f_{{\rm RG}}^{(t)}}(x)\,dx& = \widehat{\pi_{{\rm RG},0}^{(t)}}+\widehat{\pi_{{\rm RG},0}^{(t)}}\int_{0}^{t}\widehat{ f_{{\rm RG}\ast }^{(t)}}(x)\,dx=1 \nonumber \end{align}
  (19)\begin{align} & \approx \widehat{\pi_{{\rm RG},0}^{(t)}}+\widehat{\pi_{{\rm RG},0}^{(t)}} h\sum_{i=1}^{N}\widehat{f_{{\rm RG}\ast }^{(t)}}(ih)\,dx=1, \end{align}
  giving
  (20)\begin{equation} \widehat{\pi_{{\rm RG},0}^{(t)}}\approx \frac{1}{1+h\sum_{i=1}^{N}\widehat{ f_{{\rm RG}\ast }^{(t)}}(ih)}. \end{equation}

To make the computational solutions concrete, we assume, in addition to $\alpha = 0.5$, $t= 400$, that $h = 0.1$, $N = 4000$.

In the sequel, we denote pdfs and related quantities which we derive computationally, by using a “hat”, for example, “$\widehat {\textrm {pdf}}$”. Quantities which are known or derived from theory will not have a “hat”.

The computational mixed pdf $\{ \widehat {\pi _{\textrm {RG},0}^{(t)}},\widehat { f_{\textrm {RG}}^{(t)}}(x)\} _{0\lt x\lt t}$ is plotted in Figure 4.

Figure 4.

Computational mixed limiting pdf of the regenerative process $\{\widehat {\pi _{\textrm {RG},0}^{(t)}}, \widehat {f_{\textrm {RG}}^{(t)}(x)}\}_{{x\in (0,t)}}$ (see Eq. (9)). $\alpha =0.5$, $t=400$; $\widehat {\pi _{\textrm {RG},0}^{(t)}}= 0.072996.$

The computational solutions can be improved in various ways, for example, making the norm size $h$ smaller and the corresponding $N$ larger; using trapezoidal areas instead of rectangular areas in intervals $( (i-1)h,ih)$, $i = 1,\ldots,N$; etc. (see [Reference Brunner6]). Section 4 displays the analytic fixed finite time $t$ pdfs of excess, age, and total life in the extreme renewal process $\{ Z_{n}\}_{n=1,2\ldots }$, with the corresponding limiting pdfs of the companion renewal process $\{Z_{K\!,n}^{\textrm {TR}}\}_{{n=1,2,\ldots }}$, given in Section 2.5. (Note: The limiting pdfs are known from theory and do not require any extra computation here.)

3.3. Obtaining integral expressions for the analytic fixed finite time $t$ pdfs in the extreme renewal process ${\{ Z_{n}\}_{n=1,2,\ldots },0\lt t\lt\infty}$

Using the computational solutions for the mixed pdf $\{ \widehat {\pi _{\textrm {RG},0}^{(t)}},\widehat {f_{\textrm {RG}}^{(t)}}(x)\}_{0\lt x\lt t}$ of the regenerative process $\{ X_{\textrm {RG}}(s)\}_{s\geq 0}$, we create integral expressions of the analytic fixed finite time $t$ pdfs in $\{ Z_{n}\}_{n=1,2\ldots }$.

Denote the analytic fixed finite time $t$ excess, age, and total life by $\gamma _{t}$, $\delta _{t}$, and $\beta _{t}$, respectively, with corresponding pdfs: $\widehat {f_{\gamma _{t}}}(x),$ $0\lt x\lt\infty$; $\{ \widehat {\pi _{\delta _{t}}},\widehat {f_{\delta _{t}}}(x)\}_{0\lt x\lt t}$ (Note: the atom $\widehat {\pi _{\delta _{t}}}\approx P(\delta _{t}=t)$; the computational pdf of total life is denoted by $\widehat { f_{\beta _{t}}}(x)$, $0\lt x\lt\infty$.)

For each large $t\gt0$, evaluating the pdf $\{ \widehat {\pi _{\textrm {RG},0}^{(t)}},\widehat {f_{\textrm {RG}}^{(t)}}(x)\}_{0\lt x\lt t}$ may require a time-consuming computational solution of a Volterra integral equation of the second kind, due to the nature of the Pareto(II) distribution. Increasing values of $t$ require corresponding increasing computation times.)

Formulas (21)–(23) are obtained from formulas derived in Brill [Reference Brill4]. For example, formula (4.4a) in [Reference Brill4] gives

(21)\begin{equation} f_{\gamma_{t}}(x)=b(t+x)+\int_{y=0}^{t}b(t+x-y)\frac{f_{{\rm RG}}^{( t) }(y)}{\pi_{{\rm RG},0}^{(t)}}\,dy,\quad 0\lt x\lt\infty. \end{equation}

Formula (4.11) in [Reference Brill4] gives

(22)\begin{equation} f_{\delta_{t}}(x)=\bar{B}(x)\frac{f_{{\rm RG}}^{(t)}(t-x)}{\pi_{{\rm RG},0}^{(t)}} ,\quad 0\lt x\lt t;\quad \pi_{\delta_{t}}=\bar{B}(t). \end{equation}

The formula for $f_{\beta _{t}}(x),0\lt x\lt\infty$, is given in formula (4.13) in [Reference Brill4], and also in formula (23) for convenience.

(23)\begin{equation} f_{\beta_{t}}(x)=\left\{\begin{array}{l} \displaystyle b(x)\int_{y=t-x}^{t}\dfrac{f_{{\rm RG}}^{(t)}(y)}{\pi_{{\rm RG}}^{(t)},_{0}}\,dy,\quad 0\lt x\lt t, \\[6pt] \displaystyle b(x)\left( 1+\int_{y=t-x}^{t}\dfrac{f_{{\rm RG}}^{(t)}(y)}{\pi_{{\rm RG},0}^{(t)}} \,dy\right) ,\quad t\leq x\lt\infty. \end{array}\right. \end{equation}

The formulas (21)–(23), and those for $\bar {B}(x)$ and $b(x)$ in formula (1), are valid when the inter-renewals are $\overset {d}{=}$ Pareto(II) variate with shape parameter $\alpha \in ( 0,1]$.

Remark 1 Formula (22) shows that $\delta _{t}$ has an atom at level $t$ with probability $\pi _{\delta _{t}} = ( 1+t)^{-\alpha }$. Formula (23) shows that $f_{\beta _{t}}(x)$ has a discontinuity at level $x=t$ of size

(24)\begin{equation} f_{\beta_{t}}(t^{+})-f_{\beta_{t}}(t^{-})=b(t)=\alpha (1+t)^{-\alpha -1}. \end{equation}

We obtain computational results for the analytic fixed finite time $t$ pdfs in formulas (21)–(23) by substituting in them the computational step function $\{ \widehat {\pi _{\textrm {RG},0}^{(t)}}, \widehat {f_{\textrm {RG}}^{(t{)}}}(x)\} _{0\lt x\lt t}$ given in formulas (18) and (20) for the pdf of the regenerative process $\{ \widehat {\pi _{\textrm {RG},0}^{(t)}},\widehat {f_{\textrm {RG}}^{(t {)}}}(x)\}_{0\lt x\lt t}$, where $t \in (0,\infty )$.

The computational analytic fixed finite time $t$ pdfs in $\{ Z_{n}\}_{n=1,2,\ldots }: \widehat {f_{\gamma _{t}}}(x)$, $x\gt0$, $\{ \widehat {\pi _{\delta _{t}}},\widehat {f_{\delta _{t}}}(x)\}_{0\lt x\lt t}$, and $\widehat {f_{\beta _{t}}}(x)$, $x\gt0$, are specified directly in terms of $\{ \widehat {\pi _{\textrm {RG},0}^{(t)}},\widehat {f_{\textrm {RG}}^{(t {)}}}(x)\} _{0\lt x\lt t}$ in $\{ X_{\textrm {RG}}(s)\}_{s\geq 0}$ (Section 3.2.2). Examples of these estimated analytic fixed finite time $t$ pdfs are plotted in Figures 5, 6, and 7, respectively.

Figure 5.

$\widehat {f_{ \gamma _{t}}}(x)$, $0\lt x\lt800$, $\alpha =0.5$, $t=400$, $h=0.1$, $N=4000$.

Figure 6.

$\{ \widehat {\pi _{\delta _{t}}},\widehat {f_{\delta _{t}}}(x)\} _{0\lt x\lt t}$, $\alpha =0.5$, $t=400$, $h=0.1$, $N=4000$. $\widehat {\pi _{\delta _{t}}}=0.049938$.

Figure 7.

$\widehat {f_{\beta _{t}}}(x)$, $0\lt x\lt800$, $\alpha =0.5$, $t=400$, $h=0.1$, $N=4000$. Discontinuity at time point $t$ is equal to $b(t)=\alpha (1+t)^{-\alpha -1}$.

We give further discussion on Figures 6 and 7 in Section 5.

4.1. Selecting right truncation points of the inter-renewals in the renewal process ${\{Z_{K\!,n}^{\textrm {TR}}\}_{{n=1,2,\ldots }}}$

We derive the right truncation points of the Pareto(II) variate, $K_{\gamma }$, $K_{\delta,}$, and $K_{\beta }$, which are useful for approximating the fixed finite $t$ pdfs of excess, age, and total life, respectively. The resulting right-truncated inter-renewals, denoted by $Z_{K_{\gamma }}^{\textrm {TR}}$, $Z_{K_{\delta,}}^{\textrm {TR}}$, and $Z_{K_{\beta }}^{\textrm {TR}}$, respectively, have pdfs given by formulas (4)–(6) on interval $( 0,K_{\zeta })$, $\zeta =\gamma, \delta, \beta$. The $Z_{K_{\zeta }}^{\textrm {TR}}$s in the renewal process $\{Z_{K_{\varsigma },n}^{\textrm {TR}}\}_{n=1,2,\ldots }$have the same shape parameter $\alpha$ as the Pareto(II) variate in the original extreme renewal process of interest $\{ Z_{n}\} _{n=1,2\ldots }$.

4.2. Details for approximating the analytic fixed finite time pdfs of ${\gamma _{t}}$, ${\delta _{t}}$, and ${\beta_{t}}$

4.2.1. Details for approximating the pdf of $\gamma _{t}$

Letting $x\downarrow 0$ in formula (21) gives

(25)\begin{equation} f_{{\gamma }_{t}}(0^{+})=\alpha (1+t)^{-\alpha -1}+\alpha \int_{{y=0}}^{{t}}(1+t-y)^{{-\alpha -1}}\dfrac{ {f}_{{\rm RG}}^{(t)}{(y)}}{{\pi }_{{\rm RG},{0}}^{({t})}}\,dy,\quad 0\lt t\lt\infty. \end{equation}

Since $\overline {B_{K_{\gamma }}^{\textrm {TR}}}(0) = 1$ in (4), we equate the values of formula (25) and (4) at $x=0^{+}$ and solve for $K_{\gamma }$ in the equation

(26)\begin{equation} f_{\gamma_{t}}(0^{+})=\frac{1}{E[ Z_{K\gamma }^{{\rm TR}}] }, \end{equation}

where $E[ Z_{K\gamma }^{\textrm {TR}}]$ is given in formula (3). The solution of (26) is $K_{\gamma }$ for the inter-renewals in $\{Z_{K\!,n}^{\textrm {TR}}\}_{{n=1,2,\ldots }}$. We shall use the Policy expressed by Eq. (26) for the approximation procedure because it is plausible and of theoretical interest.

When $t = 400$ (considered here as a large $t$) and $\alpha =0.5$, Eq. (26) gives $K_{\gamma } = 3877.5672$ and $E(Z_{K}^{\textrm {TR}}) = 61.2781$. The right truncation point $K_{\gamma }$ is significantly greater than $t=400$. Pdf $\widehat {f_{\gamma _{t}}}(x)$ versus pdf $f_{\gamma _{K}}^{\textrm {TR}}(x)$ is plotted in Figure 8. The closeness of the two pdfs appears to be good.

Figure 8.

Analytical time $t$ pdf of excess $\widehat {f_{\gamma _{t}}}(x)$ (solid blue line) versus approximating pdf $f_{\gamma _{K}}^{\textrm {TR}}(x)$ (red line), $0\lt x\lt800$. $\alpha =0.5$, fixed finite $t=400$.

4.2.2. Details for approximating the pdf of $\delta _{t}$

From the discussion in Section 2.5 and Figure 2, we have

(27)\begin{equation} E[ Z_{K_{\delta }}^{{\rm TR}}] =E[ Z_{K_{\beta }}^{{\rm TR}}] =E [ Z_{K_{\gamma }}^{{\rm TR}}]. \end{equation}

Note that $K_{\delta }=K_{\gamma }=3877.5672$. In pdf, $\{ \widehat {\pi _{\delta _{t}}},\widehat {f_{\delta _{t}}}(x)\}_{0\lt x\lt t}$, $\widehat { \pi _{\delta _{t}}}\gt0$ is a small probability when $t$ is large. The absolutely continuous part of the pdf $\widehat {f_{\delta _{t}}}(x)$ versus $f_{\delta _{K}}^{\textrm {TR}}(x)$, $0\lt x\lt t$ ($t=400)$, is plotted in Figure 9. The closeness between the two pdfs appears to be good on the nontrivial interval $( 0,200)$, the discrepancy increases rapidly on $( 200,400).$

Figure 9.

Analytic time $t$ pdf of $\widehat {f_{\delta _{t}}}(x)$ (solid blue line) versus $f_{\delta _{K}}^{\textrm {TR}}(x)$ (red line), $0\lt x\lt400$. $\alpha =0.5$, fixed finite $t=400$.

4.2.3. Details for approximating the pdf of $\beta _{t}$

Using an equation similar to Eq. (26) and letting $x=0$ gives “$0=0$”, which does not provide an equation directly for $K_{\beta }$. Hence, we will use a plausible $K_{\beta }\gt0$. We now compare analytic pdf $\widehat {f_{{\beta }_{ {t}}}}(x)$ with the corresponding approximating pdf $f_{\beta _{K}}^{\textrm {TR}}(x)$. When $t = 400$ and $\alpha =0.5$, we use $K_{{\beta }} = 3877.5672$; giving $E [Z_{{K}_{{\beta }}}] = 61.2781$, the same as for $K_{\gamma }$ and for $K_{\delta }$. Pdf $\widehat { f_{{\beta }_{t}}}(x)$ versus limiting pdf $f_{{K}_{ {\beta }}}(x)$ is plotted in Figure 10.

Figure 10.

$\widehat {f_{\beta _{t}}}(x)$ (blue line) versus $f_{\beta _{K}}^{\textrm {TR}}(x)$ (red line), $0\lt x\lt800$. $\alpha =0.5$, fixed finite $t=400$.

5.1. The metric ${\vert \widehat {f_{ \zeta _{t}}}(x)-f_{\zeta _{K}}^{\textrm {TR}}(x)\vert}$, ${\zeta =\gamma,\delta,\beta}$

We use a measure of distance (discrepancy) motivated by the areas between the fixed time $t$ pdfs $\widehat {f_{\zeta _{t}}}(x)$ and the approximating pdfs $f_{\zeta _{K}}^{\textrm {TR}}(x)$. The areas between the pdfs must be in the interval $(0,2)$ since the areas below each pdf is equal to $1$. The $L_{1}$ measure between the pdfs accounts for all $x\in ( 0,K_{\zeta })$, $\zeta = \gamma,\delta, \beta$. Denote this measure by

(28)\begin{equation} \rho (\widehat{f_{\zeta_{t}}},f_{\zeta_{K}}^{{\rm TR}})=\frac{1}{2} \int_{x=0}^{K_{\zeta }}\vert \widehat{f_{\zeta_{t}}}(x)-f_{\zeta_{K}}^{{\rm TR}}(x)\vert \,dx,\quad \zeta =\gamma,\delta,\beta , \end{equation}

where

(29)\begin{equation} 0\leq \rho (\widehat{f_{\zeta_{t}}},f_{\zeta_{K}}^{{\rm TR}})\lt1. \end{equation}

The measure $\rho (\widehat {f_{\zeta _{t}}}$, $f_{\zeta _{K}}^{\textrm {TR}})=0$ iff $\widehat {f_{\zeta _{t}}}(x)=f_{\zeta _{K}}^{\textrm {TR}}(x)$ for all $x\in ( 0,K_{\zeta })$. Moreover, $\rho (\widehat {f_{\zeta _{t}}},f_{\zeta _{K}}^{\textrm {TR}})\lt1$. The less the distance $\rho (\widehat {f_{\zeta _{t}}},f_{\zeta _{K}}^{\textrm {TR}})$, the better is the approximation (i.e., $\rho ( \widehat {f_{\zeta _{t}}},f_{\zeta _{K}}^{\textrm {TR}})$ is nearer to $0$). The greater the distance $\rho (\widehat {f_{\zeta _{t}}},f_{\zeta _{K}}^{\textrm {TR}})$, the worse is the approximation (i.e., $\rho (\widehat {f_{\zeta _{t}}},f_{\zeta _{K}}^{\textrm {TR}})$ is nearer to $1$).

(Other measures of discrepancy between pdfs are given on p. 35 and in Section 3.7 in [Reference Silverman19].)

5.2. Example values of the measure ${\rho (\widehat {f_{\zeta _{t}}},f_{\zeta _{K}}^{\textrm {TR}})}$

In the three examples here, the shape parameter of the Pareto(II) distribution is $\alpha =0.5$.