A STUDY ON A TWO UNIT PARALLEL SYSTEM WITH ... - Statistica

STATISTICA, anno LXVII, n. 1, 2007

A STUDY ON A TWO UNIT PARALLEL SYSTEM WITH ERLANGIAN REPAIR TIME S.N. Singh, S.K. Singh

1. INTRODUCTION Introduction of the redundancy, repair maintenance and the preventive maintenance are some of the well known methods by which the reliability of a system can be improved. Parallel systems have been extensively studied by several authors in the past. A detailed list of the work on two unit system is complied by (Osaki and Nakagawa 1976). It can be shown that any failure or repair time distribution can be approximated arbitrarily closely by a general erlang distribution (Cox 1970). In the present paper, an attempt has been made to study a two unit parallel system with erlangian distribution for the repair time. For convenience, an erlangian distribution with two stages has been considered. Most of the studies on several systems are confined in obtaining expressions for various measures of system performance but the associated inference problems are not considered. In 1994, P. Chandrasekhar and R. Natarajan studied a two unit parallel system and obtained exact confidence limits for the steady state availability of the system when the failure rate of an operable unit is a constant and the repair time of the failed unit is a two stage erlangian distribution. In general, the failure time and repair time are independent random variables and hence there is a need to study a model by relaxing this imposed condition. Hence an attempt is made in this paper to study a two unit parallel system , wherein the failure rate of a unit is a constant and the repair time distribution is a two stage erlangian distribution under the assumption that an operable unit will not fail, while the other unit is in the second stage of repair. Besides obtaining expressions for the system reliability, MTBF, availability and steady state availability, we obtain a CAN estimator and an asymptotic confidence interval for the steady state availability of the system and the MLE of the system reliability. The model and assumptions are given in the next section. 2. MODEL AND ASSUMPTIONS The system under consideration is a two unit parallel system with a single repair facility. We have precisely the following assumptions:

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S.N. Singh, S.K. Singh

(i) The units are similar and statistically independent. Each unit has a constant failure rate, say λ . (ii) There is only one repair facility and the repair time distribution is a two stage erlangian distribution with probability density function (p.d.f.) given by

g ( y ) = (2 µ )2 e −2 µ y , 0 < y < ∞ , µ > 0

(1)

(iii) Eeach unit is new after repair. (iv) Switch is perfect and the switchover is instantaneous. (v) An operable unit will not fail, while the other unit is in the second stage of repair. 3. ANALYSIS OF THE SYSTEM To analyse the behaviour of the system, we note that at any time t, the system will be found in any one of the following mutually exclusive and exhaustive states:

S0 : Both the units are operating. S1 : One unit is operating and the other is in the first stage of repair. S2 : One unit is operating and the other is in the second stage of repair.

S3 : One unit is in the first stage of repair and the other is waiting for repair. S4 : One unit is in the second stage of repair and the other is waiting for repair. Since an erlangian distribution is the distribution of the sum of two independent and identically distributed exponential random variables, the stochastic process describing the behaviour of the system is a markov process. Let pi ( t ) , i = 0,1, 2, 3, 4 be the probability that the system is in state Si at time t. Clearly the infinitesimal generator of the markov process is given by

S0 ⎛ 2 λ 0 −µ 0 0 ⎞ ⎜ S1 ⎜ −2λ ( λ + µ ) 0 0 − µ ⎟⎟ 0 0 ⎟ −µ µ Q = S2 ⎜ 0 ⎜ ⎟ S3 ⎜ 0 0 0 ⎟ µ −λ S4 ⎜⎝ 0 0 0 − µ µ ⎟⎠

(1)

It may be noted that states S0 , S 1 and S2 are system up-states, whereas S 3 and S4 are system down states. We assume that initially, both the units are operable the state transition diagram is given in fig. 1.

71

A study on a two unit parallel system with erlangian repair time

S2 Fr2 , N 0

λ

µ

S0

µ

S1

2λ

S4

µ

Fr2 , Fwr

Fr1 , N 0

N 0, N 0

λ µ S3 Fr1 , Fwr

Upstate

Down state

Figure 1 – State transition diagram.

SYMBOLS USED FOR THE STATES

N 0 : Unit is under operation Fri

: Failed unit is under phase i th ( i = 1, 2) repair.

Fwr : Failed unit is waiting for repair. We obtain the measures of system performance as follows: 3.1 Reliability The system reliability R ( t ) is the probability of failure free operation of the system in [0,t]. To derive an expression for the reliability of the system, we restrict the transitions of the Markov process to the system up-states namely S0 , S1 and S 2 . Using the infinitesimal generator given in (3.1), pertaining to these up-states and standard probabilistic arguments, we derive the following system of differential-difference equations: p0/ ( t ) = − 2λ p0 ( t ) + µ p2 ( t )

(1)

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S.N. Singh, S.K. Singh

p1/ ( t ) = − ( λ + µ ) p1 ( t ) + 2λ p0 ( t )

(2)

p2/ ( t ) = − µ p2 ( t ) + µ p1 ( t )

(3)

Let pi* ( s ) be the laplace transform of pi ( t ) , i = 0, 1,2. Taking laplace transform on both the sides of diffrential-difference equations given above, solving for, pi* ( s ) , i = 0, 1, 2 and inverting, we get pi ( t ) , i = 0, 1,2. Then the system reliability is given by R(t) =

3

∑

[(α i + 3λ + µ )(α i + µ ) + 2λµ ] e αi t

i =1

3

(α i − α j ) ∏ j =1

(4)

j ≠i

Where α1, α2 and α3 are the roots of the cubic equation s 3 + (3λ + µ )s 2 + (2λ 2 + 5λµ + µ 2 )s + 2λ 2 µ 3.2 Mean time before failure (MTBF) The system mean time before failure is given by MTBF = p1* (0) + p2* (0) + p3* (0) =

(5λ + µ ) 2λ 2

(1)

3.3 System availability The system availability A(t) is the probability that the system operates within the tolerances at a given instant of time t and is obtained as follows: Using the infinitesimal generator given in (3.1), we obtain the following system of differential-difference equations p0′ ( t ) = − 2λ p0 ( t ) + µ p2 ( t )

(1)

p1′ ( t ) = − ( λ + µ ) p1 ( t ) + 2λ p0 ( t ) + µ p4 ( t )

(2)

p2′ ( t ) = − µ p2 ( t ) + µ p1 ( t )

(3)

p3′ ( t ) = − µ p3 ( t ) + λ p1 ( t )

(4)

p4′ ( t ) = − µ p4 ( t ) + µ p3 ( t )

(5)

Solving the above system of equations given in (3.3.1)-(3.3.5) using the fact that 4

∑ pi ( t ) = 1 , we obtain i =0

A study on a two unit parallel system with erlangian repair time

p0 ( t ) =

5

∑

(α i + µ ) [ (α i + λ + µ )(α i + µ ) µ − λµ 2 ] e αi t 3

∏ (α i − α j )

i =1

73

(6)

j =1 j ≠i

p1 ( t ) = 2 λ

(α i + µ )3

5

∑ i =1

e αi t

5

(α i − α j ) ∏ j =1

(7)

j ≠i 5

p2 ( t ) = 2 λ µ ∑ i =1

(α i + µ )2 5

e αi t

∏ (α i − α j )

(8)

j =1 j ≠i

p3 ( t ) = λ 2

5

∑ i =1

(α i + µ )3 5

e αi t

(9)

e αi t

(10)

(α i − α j ) ∏ j =1 j ≠i

p4 ( t ) = λ 2

5

∑ i =1

(α i + µ )3 5

∏ (α i − α j ) j =1 j ≠i

Where α1, α2 , α3 , α4 and α5 are the roots of the equation

s 5 + (3λ + 4 µ ) s 4 + [4 λµ + 3µ 2 + (2λ + µ )( λ + 3µ )] s 3 +[( λ + µ )µ 2 + (2λ + µ )µ (3µ + 2λ ) + 2λµ ( λ + 3µ ) − 2λ 2 ]s 2 + µ[(2λ + µ )( λ + µ )µ + 2λµ (3µ + 2λ ) − 4 λ 2 ]s +2λµ 2 [ µ ( λ + µ ) − λ ] Since s0, s1 and s2 correspond to system up-states, the availability of the system is given by A(t) = p0 ( t ) + p1 ( t ) + p2 ( t ) 3.4 Steady state availability The system steady state availability is given by

(11)

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S.N. Singh, S.K. Singh

A∞ = lim A( t ) t →∞

= lim sA * ( s ) s →0

=

(1)

( µ 3 + 4λ ) ( λ + µ )(2λ + µ ) + 6λµ 2

In the following sections, we obtain a CAN estimator, a 100 (1-α)% asymptotic confidence interval for steady state availability of the system and MLE of the system reliability. 4. CONFIDENCE INTERVAL FOR STEADY STATE AVAILABILITY OF THE SYSTEM Let X 1 , X 2 , X 3 ,... X n be a random sample of times to failure of the unit with p.d.f. given by f ( x ) = λ e −λ x , 0 < x < ∞ , λ > 0

(1)

Let Y1 , Y2 , Y3 ,... Yn be a random sample times to repair with the p.d.f. as in (2.1). 1 ⎛Y ⎞ 1 and E ⎜ ⎟ = where X and Y are the sample λ ⎝2⎠ µ means of time to failure and time to repair of the unit respectively. It can be Y 1 are the maximum likelihood estimators (MLEs) of and shown that X and 2 λ 1 1 1 respectively.let θ1 = and θ 2 = . Clearly , the steady state availability given λ µ µ in (3.4.1) reduces to

It is clear that E( X ) =

A∞ =

θ1θ 2 (θ1 + 4θ 23 ) (θ1 + θ 2 )(2θ 2 + θ1 ) + 6θ1

(2)

Hence the mle of A∞ is given by ˆ = A ∞

X Y [2 X + Y 3 ] 2[(2 X + Y )( X + Y ) + 12 X ]

(3)

ˆ is a real valued function in X and Y which is also It may be noted that A ∞ differentiable. Now consider the following application of multivariance central limit theorem. See (Rao 1974). Suppose T1′ , T2′,... are independent and identically distributed k-dimensional random variables such that

A study on a two unit parallel system with erlangian repair time

75

Tn′ = ( T1n , T2n ,........, Tkn ), n = 1, 2, 3,.... Having the first and second order moments E( Tn ) = µ and D( Tn ) = ∑ . Define the sequence of random variables Tn = ( T1n , T2n ,...., Tkn ), n = 1, 2, 3,....

where Tn = then,

1 n ∑ Tij , n j =1

i =1,2,.......k j =1,2,.....,n

d

n ( Tn − µ ) → N (0, ∑ ) as n → ∞

by applying the above application of multivariate central limit theorem, it readily follows that ⎡⎛ Y n ⎢⎜ X , 2 ⎣⎝

⎤ d ⎞ ( θ , θ ) − ⎥ → N (0, ∑ ) as n → ∞ 1 2 ⎟ ⎠ ⎦

Where the dispersion matrix ∑ = ((σ ij ))2 x 2. . again, from (RAO, 1974), we have d

ˆ − A ) → N (0, σ 2 (θ )) as n → ∞ n( A ∞ ∞ where θ = (θ1 , θ 2 ) and 2

⎛ ∂A ⎞ σ (θ ) = ∑ ⎜ ∞ ⎟ σ ii i =1 ⎝ ∂θ i ⎠ 2

2

2[2θ 2{(θ1 + θ 2 )(θ1 + 2θ 2 + 6θ1 )}(θ1 + 2θ 23 )2 − (2θ1 + 3θ 2 + 6)]2 θ12 =

+[(θ1 + θ 2 )(θ1 + 2θ 2 + 6θ1 )θ1 (θ1 + 16θ 23 ) − {θ1θ 2 (θ1 + 4θ 23 )(3θ1 + 4θ 2 )]2 θ 22 [(θ1 + θ 2 )(θ1 + 2θ 2 + 6θ1 )]2

ˆ is a CAN estimator of A . hence A ∞ ∞ Let σ 2 (θˆ ) be the estimator of σ 2 (θ ) obtained by replacing θ by a consistent ⎛ Y estimator θˆ namely θˆ = ⎜ X , 2 ⎝

⎞ ⎟. ⎠

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S.N. Singh, S.K. Singh

Let θˆ 2 = σ 2 (θˆ ) since σ 2 (θ ) is a continious function of θ , σˆ 2 is a consistent p

estimator of σ 2 (θ ) i.e., σˆ 2 → σ 2 as n → ∞ by slutskey theorem, we have ˆ −A ) d n( A ∞ ∞ → N (0,1) ˆ σ ˆ −A ) ⎡ ⎤ n( A ∞ ∞ < kα ⎥ = (1 − α ), i.e. P ⎢ −kα < σˆ ⎢⎣ 2 ⎦ 2⎥ kα is obtained from normal tables hence, a 100(1-α)% asymptotic confidence 2

ˆ ±k . σ interval for A∞ is givn by A α ∞ n 2 5. MLE OF SYSTEM RELIABILITY Y 1 1 are the MLEs of and respectively, by applying the 2 λ µ method given in (ZACKS, 1992) , we obtain the MLE of system reliability as

Since X and

3 [αˆ XY + 3Y + 2 X )(αˆ i Y + 2) + 4 X ] α∧i t Rˆ ( t ) = ∑ i e 3 2 i =1 XY ∏ (α i − α j )

(1)

j =1 j ≠i

where αˆ1 , αˆ 2 and αˆ 3 are the roots of the cubic equation ( X Y )2 s 3 + X Y (4 X + 3Y ) s 2 + (4 X 2 + 10 X Y + 2Y 2 ) s + 4 Y = 0 Department of Mathematics University of Transket, South Africa

S.N. SINGH

School of Studies in Statistics Ravishankar University, India

S.K. SINGH

REFERENCES

(1994), Confidence limits for steady state availiability of a parallel system, “Micronelectron. Reliab.”, 34, No. 11, pp. 1847-1851. D.R. COX (1970), “Renewal Theory”, Methuen & Co. Ltd. P. CHANDRASHEKHAR, R. NATARAJAN

A study on a two unit parallel system with erlangian repair time

77

(1990), Confidence limits for system steady state availiability, “Pak. J. Statis.”, 6(2)A, pp. 189-196. S. OSAKI, T. NAKAGAWA (1976), Bibliography for reliability and availability of stochastic systems, “IEEE Transactions on Reliab.”, R25, pp. 284-287. C. RADHAKRISHNA RAO (1974), Linear statistical inference and its application, First Wiley Eastern Reprint. S. ZACKS (1972), Introduction to Reliability Analysis, Springer Verlag, New York. ABU-SALIH MOHAMMED, N. ANAKERH NUHAN, SALAHUDDIN AHMED MOHAMMED

SUMMARY

A study on a two unit parallel system with erlangian repair time A two unit parallel system where in the failure rate of a unit is a constant and the repair time distribution is a two stage erlangian distribution is considered. Measures of system performance such as reliability, MTBF ,system availability and steady state availability are derived. Also, a consistent asymptotically normal (CAN) estimator, a 100(1-α) % asymptotic confidence interval for the steady state availability of the system and the maximum likelihood estimator (MLE) of the system reliability are obtained.