1. Introduction

In most of the studies relating to life testing and reliability, the lifetime is treated as continuous and accordingly continuous probability distributions are proposed as models. David & Epstein[4] have investigated exponential distribution in the life experiments. Interest in estimation is very important for statistical inference of random phenomena and time to failure is treated as random phenomena in reliability theory. There is extensive literature on reliability modeling, inference relating to reliability characteristics. However, there is very little discussion in assessing and estimating system performance measures like availability, frequency of failures etc. Vesely[10] discussed the Binomial failure rate model. Atwood[1], Atwood and Steverson[2] and Meachum and Atwood[7] have applied the Binomial failure rate model for Common cause shock failures to the data associated with nuclear power plants. Chari et al[3] discussed the CCS failures to arrive the expressions for reliability. Lin et al[6] derived the exact maximum likelihood estimates of the parameters included in the lives distribution of components with constant failure rates in a series system. Sarhan and EI-Bassiouny[9] used M L estimation approach to derive the estimators for the parameters included in a parallel system under the assumption that the life times of the system’s components

This paper presents, Maximum likelihood estimation approach to assess the performance of the redundant systems. In this connection, we considered three component identical system which is affected by human errors and Common cause shock failures. The M L estimates are proposed for reliability indices such as availability function, frequency of failure function in the case of series and parallel systems. Also the proposed estimates are assessed based on simulated samples.

2. Notations

λ_i, λ_c & λ_h : the failure rates of individual, CCS failures and human errors respectively.

p₁, p₂ & p₃: the chance of individual, CCS failures and

human errors respectively.

µ_0, µ₁ & µ₂: service rates.

time-dependent system availability for seriesconfiguration in the case of CCS failures as well as human errors.

: M L estimate of time-dependent system availability for series configuration in the case of CCS failures as well as human errors.

: time-dependent system availability forparallel configuration in the case of CCS failures as well as human errors.

: M L estimate of time-dependent system availability for parallel configuration in the case of CCS failures as well as human errors.

: steady-state system availability for series configuration in the case of CCS failures as well as human errors.

: M L estimate of steady-state system availa-bility for series configuration in the case of CCS failures as well as human errors.

: steady-state system availability for parallel

configuration in the case of CCS failures as

well as human errors.

: M L estimate of steady-state system availa-bility for parallel configuration in the case of CCS failures as well as human errors.

: steady-state frequency of failure for series configuration in the case of CCS failures as well as human errors.

: M L estimate of steady-state frequency of failure for series configuration in the case of CCS failures as well as human errors.

: steady-state frequency of failure for parallel configuration in the case of CCS failures as well as human errors.

: M L estimate of steady-state frequency of failure for parallel configuration in the case of CCS failures as well as human errors.

: sample means of the occurrence of individual, CCS failures and human errors respectively.

sample mean of service time of the components.

: sample estimates of individual failure rate, CCS failure rate and human errors respectively.

: sample estimate of service time of the components.

n: sample size.

N: number of simulated samples.

M S E: mean square error.

3. Assumptions

(i) The system has three identical components, which are stochastically independent.

(ii) The system is affected by individual, CCS failures as well as human errors.

(iii) The components in the system will fail singly at the constant rate λ_i and failure probability is p₁

(iv) The components may fail due to CCS failures as well as human errors at the constant rates λ_c and λ_h with failure probabilities p₂ and p₃ s.t p₁+p₂+p₃ = 1.

(v) The occurrences of all failures follow an exponential distribution.

(vi) The failed components are serviced singly and service time follows an exponential distribution with rate of service ‘µ’.

4. The Model

Under the assumptions given in section 3, we have formulated a Markov graph which represents a three component identical system given in Fig.4.1 to derive the availability function and the frequency of encountering the different states in the presence of individual failures, human errors and CCS failures. The numerals in Fig.4.1 denote the state numbers. The quantities appeared in Fig.4.1 are defined as

(1)

(2)

Figure 4.1. Markov graph for availability and frequency of failure functions of 3-component identical system with individual, CCS failures and human error

Using the Laplace transformation, the set of equations stated in (2) can be solved with given initial conditions, i.e at t=0, P₀ (t) = 1 and P₁ (t) = P₂ (t) = P₃ (t) = 0 and the solution is

(3)

(4)

(5)

(6)

Where the quantities ℓ₁, ℓ₂, τ₁, τ₂, τ₃, ψ₁, ψ₂, ψ₃, γ₁, γ₂ & γ₃ are defined as follows

(7)

(8)

(10)

Here

Where υ₁, υ₂ & υ₃ are defined as follows

υ₁ = (μ₀+μ₁+μ₂+λ₀+λ₁+λ₂+λ₁₂+λ₁₃)

υ₂ = (μ₀ λ₂+λ₁λ₂+λ₁μ₂+λ₀μ₁+μ₁λ₁₂+λ₁λ₁₂+λ₁₂μ₂+μ₀μ₂+λ₁₃ μ₂+λ₁₂ λ₂+λ₁ λ₁₃ + μ₀ λ₁₂+μ₀ μ₁+μ₁λ₁₃+λ₀ μ₂+μ₁μ₂ + λ₀ λ₂ + λ₀ λ₁ + μ₀ λ₁₃ + λ₁₃ λ₂)

υ₃ = (λ₁₃ μ₀μ₁ + λ₀λ₁λ₂ + λ₁₃μ₀λ₂ + μ₂λ₁λ₁₃ + λ₁₃μ₀λ₂+μ₀μ₁μ₂ +λ₁₂μ₀μ₁+μ₀μ₂λ₁₂+λ₀λ₁μ₂ + λ₁₂λ₁λ₂ +μ₁μ₂λ₁₃+μ₂λ₁λ₁₂+λ₀μ₁μ₂+μ₁μ₂λ₁₂+λ₁₃λ₁λ₂+μ₂μ₀λ₁₃)

And the quantities λ_0, λ_1, λ₂, λ_12, λ₁₃, μ₀, μ₁ & μ₂ as given in (1) are to be substitute in equations (7) – (10)

5. Maximum Likelihood (M L) Estimation of the System Availability and Frequency of Failures

This section discusses the maximum likelihood estimation approach for estimating the reliability measures like system availability and frequency of failures of a 3-component identical system under the influence of individual, CCS failures as well as human errors. The M L estimates are proposed for series and parallel systems.

Let be a sample of ‘n’ number of times between individual failures which will obey exponential law.

Let be a sample of ‘n’ number of times between CCS failures which follow exponential as well.

Let be a sample of ‘n’ number of times between human errors failures which follow exponential as well.

Let be a sample of ‘n’ number of times repair of the components with exponential population law.

are the maximum likelihood estimates of individual failure rate (λ_i), CCS failure rate (λ_c), human errors rate (λ_h) and repair rate ‘µ’ of the system respectively.

Where,

and

are the sample estimates of the rate of individual failure times, rate of CCS failure times, rate of human error times and rate of repair times of the components respectively.

5.1. Estimation of Time-dependent System Availability

The M L estimates of time-dependent availability function for series and parallel systems are obtained in this section.

5.1.1. Series System

For a series system, state 1 itself is a absorbing state and hence no transition is allowed from state 1 to state 2 and state 2 to state 3. Hence λ₁=λ₂=0. Therefore, the time-dependent system availability function for a series system is given by

(11)

Where

φ₁ =[ (γ₁³ + 6μ γ₁² + 11μ² γ₁ + 6μ³) / γ₁ (γ₁ – γ₂) (γ₁ – γ₃) ]

φ₂ =[ (γ₂³ + 6μ γ₂² + 11μ² γ₂ + 6μ³) / γ₂ (γ₁ – γ₂) (γ₂ – γ₃) ]

φ₃ =[ (γ₃³ + 6μ γ₃² + 11μ² γ₃ + 6μ³) / γ₃ (γ₁ – γ₃) (γ₂ – γ₃) ]

and

(12)

Here

= (2/3) (m₁² –3 m₂)^1/2

α = sin^-1(– 4q/ ³)/3

q = m₃ – (m₁ m₂)/3 +2 m₁³/27

Where m₁, m₂ & m₃ are defined as follows

m₁ = (6μ + 3λ_i p₁ + λ_c p₂ + λ_h p₃)

m₂ = (15λ_i p₁ μ + 6λ_c p₂ μ +6λ_h p₃ μ +11μ²)

m₃ = (11λ_h p₃ μ² + 6μ³ +11λ_c p₂ μ²+18 λ_i p₁ μ²)

The expression given in (11) agrees with the result already developed by Sagar et al[8]. Therefore, the maximum likelihood estimate of the time-dependent system availability function of the series system in the presence of individual, CCS failures as well as human errors can be obtained by

(13)

And

(14)

and are sample estimates given in “section 5”.

5.1.2. Parallel System

The time-dependent system availability function of theparallel system can be obtained by

(15)

where

Қ₁ =[(γ₁³ + γ₁² ℓ₁ + τ₁ γ₁ + ψ₁) / γ₁ (γ₁ – γ₂) (γ₁ – γ₃ )]

Қ₂ =[(γ₂³ + γ₂² ℓ₁ + τ₁ γ₂ + ψ₁) / γ₂ (γ₁ – γ₂) (γ₂ – γ₃ )]

Қ₃ =[(γ₃³ + γ₃² ℓ₁ + τ₁ γ₃ + ψ₁) / γ₃ (γ₁ – γ₃) (γ₂ – γ₃ )]

Ұ₁ =[(γ₁² ℓ₂ + γ₁ τ₂ + ψ₂) / γ₁ (γ₁ – γ₂) (γ₁ – γ₃)]

Ұ₂ =[(γ₂² ℓ₂ + γ₂ τ₂ + ψ₂) / γ₂ (γ₁ – γ₂) (γ₂ – γ₃)]

Ұ₃ =[(γ₃² ℓ₂ + γ₃ τ₂ + ψ₂) / γ₃ (γ₁ – γ₃) (γ₂ – γ₃)]

ф₁ =[(γ₁ τ₃ + ψ₃) / γ₁ (γ₁ – γ₂) (γ₁ – γ₃)]

ф₂ =[(γ₂ τ₃ + ψ₃) / γ₂ (γ₁ – γ₂) (γ₂ – γ₃)]

ф₃ =[(γ₃ τ₃ + ψ₃) / γ₃ (γ₁ – γ₃) (γ₂ – γ₃)]

Қ₄ = – ψ₁ / (γ₁ γ₂ γ₃)

Ұ₄ = – ψ₂ / (γ₁ γ₂ γ₃)

ф₄ = – ψ₃ / (γ₁ γ₂ γ₃)

The quantities ℓ₁, ℓ₂, τ₁, τ₂, τ₃, ψ₁, ψ₂, ψ₃, and γ₁, γ₂, γ₃ associated with Қ₁, Қ₂, Қ₃, Қ₄, Ұ₁, Ұ₂, Ұ₃, Ұ₄, ф₁, ф₂, ф₃, ф₄ are defined in (7), (8), (9) & (10) respectively.

The time-dependent system availability function given in (15) agrees with the result already developed by Sagar et al[8]. Therefore, the maximum likelihood estimate of the time-dependent system availability function of the parallel system in the presence of individual, CCS failures as well as human errors can be obtained by

(16)

Where

And the quantities are defined as follows

(17)

Here

Where

And

(18)

(19)

(20)

where are sample estimates given in “section 5”.

5.2. Estimation of Steady-State System Availability

The maximum likelihood estimates of steady-state system availability functions for series and parallel systems are obtained in this section.

5.2.1. Series System

The steady-state availability of series system can be obtained by using the final value theorem of Laplace transformation.

(21)

Where ψ₁, γ₁, γ₂ & γ₃ are defined in (9) and (12).

The above expression given in (21) agrees with the result already developed by Sagar et al[8]. Therefore, the maximum likelihood estimate of steady-state system availability function of series system in the presence of individual, CCS failures as well as human errors can be obtained by

(22)

Where are defined in (14)

5.2.2. Parallel System

The steady-state availability function of parallel system can be obtained by using the final value theorem of Laplace transformation.

(23)

Where ψ₁, ψ₂, ψ₃ and γ₁, γ₂ , γ₃ are defined in (9) and (10)

The above expression given in (23) agrees with the result already developed by Sagar et al[8]. Therefore, the maximum likelihood estimate of steady-state system availability function of parallel system in the presence of individual, CCS failures as well as human errors can be obtained by

(24)

Where and are defined in (17) and (20)

5.3. Estimation of Frequency of Failures

The maximum likelihood estimates of the frequency of failure functions for series and parallel systems in the presence of individual, CCS failures and human errors are obtained in this section.

5.3.1. Series System

The frequency of failure function for series system is given by

(25)

Where ψ₁ and ₁, ₂, ₃ are given in (9) and (12)

The above expression given in (25) agrees with the result already developed by Sagar et al[8]. Therefore, the M L estimate of the frequency of failure function for series system is given by

(26)

Where are defined in (14)

5.3.2. Parallel System

The frequency of failure function for parallel system is given by

(27)

Where ₁, ₂ &₃ are given in (10)

The above expression given in (27) agrees with the result already developed by Sagar et al[8]. Therefore, the maximum likelihood estimate of the frequency of failure function of parallel system is given by

(28)

Where are defined in (17)

6. Simulation and Results

The M L estimates of system availability (time-dependent & steady-state) and frequency of failure functions were not identified with exact or analytical form of probability density function since they are complex functions of the sample information. As such, the probability density function of the estimates is not identified analytically. Hence an attempt is made to establish the validity and to find precision of the estimates of the above measures; Monte-Carlo simulation is used.

For a range of specified values of the rates of individual (λ_i), CCS failures (λ_c), human errors (λ_h) and service rate (µ) for the samples of sizes n=5(5)30 were simulated in each case with N=10,000(20,000)90,000 in order to evolve mean square error (MSE) in each case and given in numerical illustration. It is interesting to note that estimation gives a very close estimate in the case of very small samples of size n=5. This shows that M L approach and estimators are quite useful in estimating reliability and availability measures.

Table 6.1. Time-dependent system availability function for three component identical Series System with i = 0.05; c = 0.04; h = 0.03; µ = 5; p1 = 0.5; p2 = 0.25; p3 = 0.25; t = 2

Table 6.2. Time-dependent system availability function for three component identical Parallel System with i = 0.05; c = 0.04; h = 0.03; µ = 5; p1 = 0.5; p2 = 0.25; p3 = 0.25; t = 2

Table 6.3. Frequency of failure function for three component identical Series System with i = 0.05; c = 0.04; h = 0.03; µ = 5; p1 = 0.5; p2 = 0.25; p3 = 0.25

Table 6.4. Frequency of failure function for three component identical Parallel System with i = 0.05; c = 0.04; h = 0.03; µ = 5; p1 = 0.5; p2 = 0.25; p3 = 0.25

7. Conclusions

The maximum likelihood estimates of system availability (time-dependent and steady-state) and frequency of failure functions[Av(t), Av(∞), f(T)] were developed for three-component identical system with CCS failures as well as human errors. The estimates are proposed for all the measures said above both for series and parallel systems. We presented numerical simulation studies in order to establish empirically, the validity of the proposed M L estimates of the above measures in the absence of analytical approach. From the simulation results, we observed that

(i) The point estimates become more accurate when the sample size is large.

(ii) Each of MSE decreases with increasing the sample size.

Therefore, this paper suggest that the use of Maximum likelihood estimation approach is found satisfactory for estimation process of some important reliability measures of the system performance, since estimation gives a very close estimate of the above measures even in the case of very small samples of size n=5.