A new non-parametric estimator for instant system availability

Abstract

Instant availability of a repairable system is a very important measure of its performance. Among the extensive literature in system availability of the steady state, which is the limit of instant availability as time approaches infinity, many methods and approaches have been explored. However, less has been done on instant system availability owing to its theoretical and computational challenges. A new non-parametric estimator of instant availability is proposed. This estimator is both asymptotically consistent and efficient in numerical computation. Multiple numerical simulations are presented to demonstrate the performance of the new estimator.

Introduction

Let $\{(U_i,D_i),i\ge 1\}$ be an alternative renewal process, where $U_i,i\ge 1$ are lifetimes of the system with common CDF $F\left(x\right)$, $D_i,i\ge 1$ are the repair times of the system with common CDF $G\left(x\right)$, and $(U_i,D_i),i\ge 1$ are iid. Suppose both $F\left(x\right)$ and $G\left(x\right)$ are continuous on the common support set $[0,\infty )$.

Define the instant availability $A\left(t\right)$ of the system above by

$$A\left(t\right)=\mathbb{P}\left(\text{the system functions properly at time }t\right).$$

From Karlin and Taylor (1975), Barlow and Proschan (1975), and Ross (1983), it is known that $A\left(t\right)$ is the unique solution of the renewal equation:

$$A\left(t\right)=\overline{F}\left(t\right)+{\int }_0^{t}A(t-s)dH\left(s\right),$$

where $\overline{F}\left(t\right)=1-F\left(t\right)$ and $H\left(t\right)=(F\ast G)\left(t\right)={\int }_0^{t}F(t-x)dG\left(x\right)={\int }_0^{t}G(t-x)dF\left(x\right)$ is the value of the convolution of the functions $F(\cdot )$ and $G(\cdot )$ at $t$.

The instant availability $A\left(t\right)$ can be also expressed in the following explicit form:

$$A\left(t\right)=\overline{F}\left(t\right)+{\int }_0^t\overline{F}(t-s)dM\left(s\right),$$

where the renewal function $M\left(t\right)$ is given as

$$M\left(t\right)=\sum _{k=1}^{\infty }{H}^{\left(k\right)}\left(t\right),$$

and $H^{\left(k\right)}\left(t\right)$ is the $k$th fold convolution of $H(\cdot )$, i.e., $H^{\left(k\right)}\left(t\right)$ is defined inductively by $H^{\left(2\right)}\left(t\right)=(H\ast H)\left(t\right)$ and $H^{\left(k\right)}\left(t\right)=(H^{(k-1)}\ast H)\left(t\right)$, $k\ge 2$. For the details of the derivation of (1), (2), the readers are referred to Karlin and Taylor (1975), Barlow and Proschan (1975), and Ross (1983).

The closed-form expression of $A\left(t\right)$ does not exist in general. In a recent work of the authors (Huang and Mi, 2013), this issue was addressed and a number of useful properties of $A\left(t\right)$ were discovered. In another paper (Huang and Mi, 2012), the authors proposed a numerical method to obtain the approximate value of $A\left(t\right)$ when both $F(\cdot )$ and $G(\cdot )$ are known.

If $F(\cdot )$ and $G(\cdot )$ belong to certain parametric families, one can first try to estimate the pertinent parameters for estimating $F(\cdot )$ and $G(\cdot )$, and then get an estimate of $A\left(t\right)$ by using (1) or (2). However, in many situations, the parametric forms of $F(\cdot )$ and $G(\cdot )$ are either unknown or the knowledge about them rather limited. Thus, nonparametric methods become necessary.

Gamiz et al. (2011) proposed a kernel estimator for $A\left(t\right)$ as follows. Let $\{(u_i,d_i),1\le i\le n\}$ be the available observations of $(U_i,D_i),1\le i\le n$. Let $W_i\left(t\right),j=1,2$ be the exponential distribution functions given by

$$W_j\left(t\right)=1-e^{-t∕{\lambda }_j},t\ge 0,j=1,2.$$

$W_1(\cdot )$ was used as the kernel to estimate $F\left(t\right)$, i.e., 

$${\hat{F}}_n(t,h_1)=\frac{1}{n}\sum _{i=1}^n{W}_1\left(\frac{t-u_i}{h_1}\right),t\ge 0.$$

Similarly, $W_2(\cdot )$ was used as the kernel to estimate $G\left(t\right)$, i.e., 

$${\hat{G}}_n(t,h_2)=\frac{1}{n}\sum _{i=1}^n{W}_2\left(\frac{t-d_i}{h_2}\right),t\ge 0.$$

In (3), (4), $h_1,h_2>0$ are the pre-selected bandwidth parameters. With both ${\hat{F}}_n(t,h_1)$ and ${\hat{G}}_n(t,h_2)$, $H^{\left(k\right)}$ and $M\left(t\right)$ were further estimated by

$${\hat{H}}_n^{\left(k\right)}(t,\mathbf{h})=\frac{1}{n^{2k}}\sum _{\stackrel{i_1,\dots ,i_k}{j_1,\dots ,j_k}}^n\left[W_1\left(\frac{t-u_{i_1}}{h_1}\right)\ast \cdots \ast W_1\left(\frac{t-u_{i_k}}{h_1}\right)\right]\ast \left[W_2\left(\frac{t-d_{j_1}}{h_2}\right)\ast \cdots \ast W_2\left(\frac{t-d_{j_k}}{h_2}\right)\right],$$

and

$${\hat{M}}_n(t;\mathbf{h})=\sum _{k=1}^{\infty }{\hat{H}}_n^{\left(k\right)}(t;\mathbf{h})\approx \sum _{k=1}^{k_0}{\hat{H}}_n^{\left(k\right)}(t;\mathbf{h}),$$

where $k_0$ is an appropriately selected integer and $\mathbf{h}=(h_1,h_2)$. With exponential kernels $W_1(\cdot )$ and $W_2(\cdot )$, (5) can be simplified to

$${\hat{H}}_n^{\left(k\right)}(t,\mathbf{h})=\frac{1}{n^{2k}}\sum _{\stackrel{i_1,\dots ,i_k}{j_1,\dots ,j_k}}^{n}Gamm{a}_1^{\left(k\right)}\left(\frac{t-\sum _{l=1}^k{u}_{i_l}}{h_1}\right)\ast Gamm{a}_2^{\left(k\right)}\left(\frac{t-\sum _{m=1}^k{d}_{j_m}}{h_2}\right),$$

where $Gamm{a}_j^{\left(k\right)}$ is the gamma distribution function with pdf

$$\frac{1}{(k-1)!{\lambda }_j^k}{t}^{k-1}{e}^{-t∕{\lambda }_j},j=1,2.$$

Using the formulae (6), (7), (2) from above, Gamiz et al. were able to propose the following estimator for $A\left(t\right)$

$${\hat{A}}_n(t,\mathbf{h})=\left[1-{\hat{F}}_n(t,h_1)\right]+\left(1-{\hat{F}}_n(t,h_1)\right)\ast {\hat{M}}_n(t,\mathbf{h}),$$

in (2011) (Gamiz et al., 2011).

In their approach, the computation of ${\hat{H}}_n^{\left(k\right)}(t,\mathbf{h})$ for each $1\le k\le k_0$ is very time-consuming. A great number of convolutions are computed which leads to considerable errors. Thus, the estimation (8) is not accurate enough.

To overcome these challenges, we propose a new kernel estimator for the instant system availability $A\left(t\right)$. In Section 2, we propose and discuss this new estimator. Two lemmas for the new estimator’s properties are derived in Section 3. In Section 4, we prove the asymptotic properties of this new estimator. In Section 5, we conduct numerical studies for the method proposed, and conclude our work with remarks and possible future directions.