1. Introduction

Recent technological developments have given rise to the design of many complex systems containing several subsystems to perform different operations in various fields such as defence, industry and engineering systems. Because of the varied nature, these problems have attracted the attention of systems engineers and applied probabilists.

Repairable systems were studied in the past with reference to the evaluation of their performance in terms of reliability and availability. Work (Claasen, S.J., Joubert & Yadavalli) [2] have considered a two-unit standby system with non-instantaneous switch-over and "dead time" and obtain exact confidence limits for the steady-state availability of system. In the work (Chandrasekhar, Natarajan & Yadavalli) [1] studied the two unit standby system and obtain exact confidence limits for the steady-state availability of system, when the failure rate of an operative unit is constant and the repair time of the failed unit follows a two stage Erlang distribution. The stochastic analysis of a non-identical two-unit parallel system with common-cause failure by graphical evaluation and review technique (GERT) considered by (Sridharan & Kalyani) [6]. The cost-benefit analysis of a two-unit cold standby system with two types of repair- minor (regular) and major (expert) are considered by [5, 9] studied the optimal system for series systems with mixed standby components. In the work (Wang, Liu & Pearn) [4] studied the availability analysis of three different series system configurations with warm standby components and general repair times. Work (Wang & Kuo) [3] has considered the reliability and availability characteristics of four different series system configurations with mixed standby.

The purpose of the present paper is to study reliability analysis of a two-compressor arranging in parallel and find an estimator and asymptotic confidence interval for steady-state availability and mean time to failure of the system.

2. Description of the System

For the sake of discussion, we consider the system consists of two-compressor (A, B) (Fig.1) being part of the refrigeration system of an ammonia storage facility. Its function is to condense ammonia vapors coming from the tank, to maintain tank pressure at normal level, which is close to atmospheric pressure. After compression, condensation and expansion liquid cryogenic ammonia returns to the tank. We assume that two-compressor (non-identical) are arranged in parallel. In the beginning, the failure rate of a compressor A (B) is a constant with parameter If compressor A (B) failed, the failure rate of B (A) is a constant with parameter The repair time distribution of two-compressor (A, B) is a two-stage Erlang distribution with parameter . Respectively, there is only one repair facility where the service discipline is FCFS. Whenever one of these compressors is operating online only and the other fails, the failed compressor goes to the repair. In the first stage, the repairing process of the compressor is started but it doesn’t complete, while the repairing process is completed in the second stage. Each compressor is assumed to be as new after repairment.

Figure 1. Two-compressor system of a refrigeration system

Since an Erlang distribution can be considered as the distribution of the sum of two independent and identically distributed exponential random variables, the stochastic process describing the behavior of the system is a Markov Process.

Let be the probability that the system is in state at time . The infinitesimal generator of the Markov Process is given below.

(1)

We assume that initially both the compressors are operable and obtain the measures of system performance.

3. System Availability

The system availability is the probability that the system operates within the tolerances at a given instant of time and is obtained as follows. Include the diagram for the infinitesimal generator for the system of ODEs for probabilities ; eq. (1) and eqs. (2) to (10).

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

At time and all the other initial condition probabilities are equal to zero.

Taking Laplace transforms on both the sides of the differential equations given above, solving for Laplace transforms and inverting, we get .

(11)

(12)

(13)

(14)

(15)

(16)

(17)

(18)

(19)

Where

and are the roots of the following equation:

Since and correspond to system up-states, the system availability is given by

(20)

3.1. The Steady-state Availability

The steady-state availability of the system is given by

(21)

(22)

where

and,

4. Reliability

The following differential equations associated with the system up states are obtained:

(23)

(24)

(25)

(26)

(27)

At time and all the other initial condition probabilities are equal to zero. By solving the above equations using Laplace transforms and inverting, we get . Then the system reliability is given by:

(28)

(29)

where,

and are the roots of the following equation:

(30)

Now, we calculate the mean time to failure of the system by using the relation

(31)

where,

and

The variance of time to failure of the system is given by

(32)

is the derivative of with respect to s.

(33)

where,

5. Special Case Model

When two units are independent (i.e. ).

5.1. Availability of the System

We have

(34)

(35)

(36)

(37)

(38)

(39)

(40)

(41)

(42)

At time and all the other initial condition probabilities are equal to zero.

Taking Laplace transforms on both the sides of the differential equations given above, solving for Laplace transforms and inverting, we get

(43)

(44)

(45)

(46)

(47)

(48)

(49)

(50)

(51)

where,

,

and are the roots of the following equation:

Since and correspond to system up-states, the system availability is given by

(52)

5.1.1. The Steady-state Availability

The steady-state availability of the system is given by

(53)

(54)

where,

5.2. Reliability Analysis

The following differential equations associated with the system up states are obtained:

(55)

(56)

(57)

(58)

(59)

At time and all the other initial condition probabilities are equal to zero. By solving the above equations using Laplace transforms and inverting, we get . Then the system reliability is given by:

(60)

where,

, and are the roots of the following equation:

(61)

Now, we calculate the mean time to failure of the system by using the relation

(62)

where,

, and

The variance of time to failure of the system is given by

(63)

where is the derivative of with respect to s.

(64)

where

In the following sections, we obtain CAN estimator, a asymptotic confidence interval for the steady state availability of the system and an estimator of the system reliability.

6. Confidence Interval for Steady-State Availability of the System

Let be random samples of size each drawn from different exponential populations with first (A) compressor’s failure rate second (B) compressor’s failure rate The failure of compressor B changes the parameter of exponential lifetime of A from to , while the failure of component A changes the parameter of exponential lifetime of B from to

If is the parameter of the exponential distribution, then an estimate can be found for either or for the parameter, which is equal to the mean value of the time of failure. For the analysis, let

The maximum likelihood estimator (MLE) of is given by Similarly and are the MLE’s of and respectively.

Moreover, Let be random samples of size each drawn from different Erlang populations with first unit’s repair rate and second unit’s repair rate If µ₁ is the parameter of the Erlangian distribution, then an estimate can be found for either µ₁, or for the parameter , which is equal to the mean value of the time of the failure time. For the analysis, let .

The maximum likelihood estimator (MLE) of is given by Similarly is the MLE’s of .

Hence, the MLE of

(65)

where

It should be noted that is real valued differential function in and Now consider the following application of the multiplicative central limit theorem (Rao, 1973), it follows that

where

and the dispersion matrix is given by

According to center limit theorem, see (Rao, 1973), as ,

we have

where

Consequently is estimator of :

Let be the estimator of obtained by replacing by a consistent estimator namely.

By Slutskey’s theorem

That is,

Where is obtained from normal tables. Hence, a asymptotic confidence interval for is given by

7. Confidence Limits Interval for Mean Time to Failure of the Systems

Since, the maximum likelihood estimator (MLE) of is given by Similarly and are the MLE’s of and respectively. Also, the maximum likelihood estimator (MLE) of is given by similarly are the MLE’s of .

Hence, the MLE of

(66)

where

and,

It should be noted that is real valued differential function in and Now consider the following application of the multiplicative central limit theorem (Rao, 1973), it follows that

where

and the dispersion matrix is given by

According to center limit theorem, see (Rao, 1973), as ,

we have

where

Consequently is estimator of

Let be the estimator of obtained by replacing by a consistent estimator namely.

By Slutskey’s theorem

That is,

Where is obtained from normal tables. Hence, a asymptotic confidence interval for is given by .

8. Confidence Interval for Steady-State Availability of the Special Case Model

Let be random samples of size each drawn from different exponential populations with first unit’s failure rate and second unit’s failure rate . If α₁ is the parameter of the exponential distribution, then an estimate can be found for either α₁, or for the parameter , which is equal to the mean value of the time of the failure time. For the analysis, let .

The maximum likelihood estimator (MLE) of is given by Similarly is the MLE’s of .

Moreover, Let be random samples of size each drawn from different Erlang populations with first unit’s repair rate and second unit’s repair rate . If µ₁ is the parameter of the Erlangian distribution, then an estimate can be found for either µ₁, or for the parameter , which is equal to the mean value of the time of the failure time. For the analysis, let

The maximum likelihood estimator (MLE) of is given by Similarly is the MLE’s of .

Hence, the MLE of

(67)

where

and,

It should be noted that is real valued differential function in and Now consider the following application of the multiplicative central limit theorem (Rao, 1973), it follows that

where

and the dispersion matrix is given by

According to center limit theorem, see (Rao, 1973), as ,

we have

where

Consequently is estimator of :

Let be the estimator of obtained by replacing by a consistent estimator namely.

By Slutskey’s theorem

That is,

Where is obtained from normal tables. Hence, a asymptotic confidence interval for is given by

9. Confidence Limits Interval for Mean Time to Failure of the Special Case Model

Since, the maximum likelihood estimator (MLE) of is given by similarly are the MLE’s of . also, the maximum likelihood estimator (MLE) of is given by similarly are the MLE’s of .

Hence, the MLE of

(68)

where

and,

It should be noted that is real valued differential function in and Now consider the following application of the multiplicative central limit theorem (Rao, 1973), it follows that

where

and the dispersion matrix is given by

According to center limit theorem, see (Rao, 1973), as ,

we have

where

Consequently is estimator of :

Let be the estimator of obtained by replacing by a consistent estimator namely.

By Slutskey’s theorem

That is,

Where is obtained from normal tables. Hence, a asymptotic confidence interval for is given by .

10. Study of System Behavior through Graphs

We plot the MTTF and steady-state availability for the two models (system model and special case model), against and . These curves are shown in Figures 2 –3.

Figure 2. S.S. Availability vs. Repair rate(A) Repair rate(B)

From Figure (2) we conclude that, as the repair time rate is increase and the repair time rate is increase, the steady-state availability is increases. When the steady-state availability of special case is greater than the system steady-state availability

Figure 3. MTTF vs. Repair rate(A) Repair rate(B)

In Figure (3) it is seen that, as the repair time rate is increase and the repair time rate is increase, the mean time to failure of the system is increases. The mean time to failure of special case (i.e. ) is better than The mean time to failure of the system

Therefore, for achieving high reliability of the system, we recommend that the two-compressor are independent (i.e. the failure rate of compressor is constant), this is clear in the graphs.