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A Two Non-Identical Unit Parallel System with Repair and Post Repair Policies of a Failed Unit and Correlated Lifetimes
Pradeep Chaudhary, Surbhi Masih, Rakesh Gupta
Department of Statistics Ch. Charan Singh University, Meerut-250004(India) [email protected]; [email protected]; [email protected]
Abstract
The paper deals with the analysis of a system model consisting of two non-identical units arranged in a parallel configuration. If a unit fails it goes to repair. After its repair, the repaired unit is sent for post repair to complete its repair. A single repairman is always available with the system to repair a failed unit and for post repair of repaired unit. A post repaired unit always works as good as new. Failure time of both the units is assumed to be correlated random variables having their joint distribution as bivariate exponential (B.V.E.). The repair time distribution of both the units are taken as general with different c.d.fs whereas the post repair time distribution of both the units are taken as exponential with different parameters.
Keywords: Transition probabilities, mean sojourn time, bi-variate exponential distribution, regenerative point, reliability, MTSF, availability, expected busy period of repairman, net expected profit.
1. Introduction
Undoubtedly, various actual systems in the field of manufacture, shipment, computation etc. are factitious / unnatural and manufactured by human while there are various natural systems also exist which illustrates that all the systems are manufactured by human and nature and can be simple or complicated. Chopra and Ram [5] studied a two non-identical unit parallel system with two types of failure - common cause failure and partial failure. A repairman is not always available with the system to repair a failed unit i.e. whenever a unit fails, a repairman is called to visit the system and he takes some significant amount of time to reach at the system. This time is known as the waiting time for repairman and during this time the failed unit waits for repair. Pundir et al. [14] investigated a two non-identical unit parallel system where priority is given to first unit in repair. Chandra et al. [4] analysed two different system models in which one consist of two identical units in parallel whereas the other composed of two non-identical units in parallel. They have obtained the reliability characteristics by applying the Semi Markov Process and Regenerative Point Technique. In the above papers, the authors considered a two identical / non-identical unit parallel system. The concept of repair and post repair is not considered in all the above system models.
Goel et al. [6] studied a two unit redundant system in which one unit is operative and the other is a warm standby which replaces the operative failed unit instantaneously. After repair of the operative failed unit it is sent for inspection to decide whether the repaired unit is perfect or not. If the repaired unit is found to be imperfect, it is sent for post repair. Goel et al. [7] investigated a stochastic model of a two unit warm standby system with 'n' failure modes of each unit. Before starting the repair, the failed unit is examined for the type of fault, which decides the failure mode. This process takes a significant random amount of time. After repair, the unit is inspected to decide whether the repair is perfect or not. If the repair is found to be imperfect, the unit is sent for post repair. Agarwal and Mahajan [1] analysed a two-unit degrading system model with repair. After each repair, the unit is tested to see whether the repair meets certain pre-defined specifications. If it does, the unit is put to operation, otherwise it goes to post repair. Pandey et al. [13] discussed a two non-identical unit system with two types of repair, the internal and the external. The external repair is called only when the internal staff fails to do the job. In the case of external repair, there is a provision of inspection, wherein if the repair is found unsatisfactory, it is sent for post repair. Agnihotri et al. [2] analysed a system model consisting of two non-identical parallel units. They have assumed that the repaired unit goes for inspection to decide whether the repair is perfect or not. If the repair is found imperfect then it is sent for post repair. Agnihotri and Satsangi [3] analysed a system model consisting of two non-identical parallel units, in which the one unit gets the priority over the other for repair, inspection and post repair. Mokaddis et al. [12] investigated a two dissimilar unit cold standby redundant system with random interchange of the units. In this system it is assumed that the failure, repair, post repair, interchange of units and inspection times are stochastically independent random variables, each having an arbitrary distribution. The system is analysed by the Semi-Markov Process technique. In all the above system models, the concept of repair and post repair is used. After repair of the operative failed unit, it is sent for inspection to decide whether the repaired unit is perfect or not. If the repaired unit is found to be imperfect then the unit is sent for post repair. The above authors have assumed that time to failure of both the units are uncorrelated random variables.
Gupta and co-workers [8, 9] studied two unit complex / duplicate system by assuming different presumptions. In both these system models the failure and repair times are taken as correlated random variables. The joint distribution of failure and repair times is considered as bivariate exponential in both the models. Gupta and co-workers [10, 11] have also studied two unit active redundant systems by assuming different assumptions. Considering the lifetimes of the units as correlated random variable i.e. the joint distribution of lifetimes of the units is taken as bivariate exponential.
The objective of this paper is devoted to raise the idea of repair and post repair in two non-identical units parallel system assuming that the lifetimes of the units are correlated random variables having their joint distribution as bivariate exponential with joint p.d.f. as follows-
is the modified Bessel function of type-I and order zero.
By using regenerative point technique, the following measures of system effectiveness are obtained-
ii
Transient-state and steady-state transition probabilities. Mean sojourn time in various regenerative states. Reliability and mean time to system failure (MTSF).
iv. Point-wise and steady-state availabilities of the system as well as expected up time of the system during time interval (0, t).
v. The expected busy period of repairman in time interval (0, t).
vi. Net expected profit earned by the system in time interval (0, t) and in steady-state.
2. System Description and Assumptions
1. The system consists of two non-identical units- unit-1 and unit-2. Initially, both the units are operative in parallel configuration.
2. Each unit has two possible modes- Normal (N) and Total failure (F).
3. When a unit fails it goes to repair. After its repair, the repaired unit is sent for post
repair.
4. A single repair facility is always available with the system to repair a failed unit and for post repair of repaired unit.
5. A post repaired unit always works as good as new.
6. The repair / post repair discipline is FCFS.
7. Failure time of both the units are assumed to be correlated random variables having
their joint distribution as bivariate exponential (B.V.E.) with density function as follows-
f(xj,x2) = axa2(1-r)e-aiX-aiXl I0(^axa2rxxx2); xx,x2,ax,a2 >0; 0<r <1
where Io (z ) = X ^
k=o (k!)
8. The repair time distribution of both the units are taken as general with different c.d.fs whereas the post repair time distribution of both the units are taken as exponential with different parameters.
9. The system failure occurs when both the units are in total failure mode.
10. A repaired unit always works as good as new.
3. Notations and States of the System We define the following symbols for generating the various states of the system-
no , N2: Unit-1 and Unit-2 in normal (N) mode and operative.
F1, Fr2 : Unit-1 and Unit-2 is in failure (F) mode and under repair.
Fpr, Fp2r : Unit-1 and Unit-2 is in failure (F) mode and under post repair.
F1, F2 : Unit-1 and Unit-2 is in failure (F) mode and waits for repair.
Considering the above symbols in view of assumptions stated in section-2, the possible states of the system are shown in the transition diagram represented by Figure. 1. It is to be noted that the epochs of transitions into the state S3 from Sj, S4 from Sj, S5 from S2 and S6 from S2 are non-regenerative, whereas all the other entrance epochs into the states of the systems are regenerative.
ß2
ßl
y X1X2
F1 F2 G2 (.) - - Fl F2 r w'r r
w > pr
- -
F1,F2 Gl (.) F1 ,F2
r ' w pr w
0
: Up State
□
: Failed State
» : Regenerative Point
Non-Regenerative Point
Figure 1: Transition diagram
The other notations used are defined as follows: E :
X (i = 1,2) f ( x1,x2 )
Set of regenerative states.
Random variables representing the failure time of unit-1 in N-mode and unit-2 respectively for i = 1, 2. Joint p.d.f. of (xj,x2 ).
f(x1;x2) = a1a2 (1-r) e
xl5x2,al5a2 > 0 ; 0 < r < 1
-a^-a2x2
I (2a1a2rx1x2 ) ;
(z/2)
2k
gi ( x )
k1 (x^X2 = x2 ) :
k2 (x2 I X1 = x1 ) :
K (• | x )
ß1, ß2
G1 I") ,G2 I") jj)
where, Io (z ) = Y- 2
k=S (k!)2
Marginal p.d.f. of Xi = x
= ai (1 -) e-ai(1-r )x Conditional p.d.f. of X^X2 = x.
= a1e (a*Xl+a2rx)I0 (a1 a2rxxx) Conditional p.d.f. of X2|X1 = x.
= a2e+a1rx)lo a1 a2rxx2 )
Conditional c.d.f. of Xi|Xj = x, i ^ j; i, j = 1,2. Constant post repair rate of unit-1 and unit-2 respectively. c.d.f. of repair time of unit-1 and unit-2 respectively.
p.d.f. of transition time from state S; to Sj and S; to Sj via Sk.
Sk .
Pü'Pf
ijx'^ljlx
©
Steady-state transition probabilities from state Sl to Sj and Sl to Sj via
Steady-state transition probabilities from state Sl to Sj and Sl to Sj via Sk when it is known that the unit has worked for time x before its failure. fSymbol for Laplace Transform i.e. q*(s) = J e-stqlj (t) dt Symbol for Laplace Stieltjes Transform i.e. Q^ (s) = J e-stdQlj (t)
Symbol for ordinary convolution i.e. t
A (t) ©B (t) = J A (u )B (t - u ) du
*
4. Transition Probabilities and Sojourn Times
Let X(t) be the state of the system at epoch t, then {X (t) ;t > 0} constitutes a continuous parametric Markov-Chain with state space E = {S0 to S5}. The various measures of system effectiveness are obtained in terms of steady-state transition probabilities and mean sojourn times in various states. First we obtain the direct conditional and unconditional transition probabilities in terms of
a,
a,
a0
a, =-
a, =
a2 =-
1 _-'
a, +12 a2 + ß1
ai + P2
as follows-
p01 = fa1 (1-r ) e-{ai(1-r ^ (1-r)!ldt = ^
J a +1
Similarly,
and
a0
a2 = ■
a2 +
a, + a2
P02 =-
a-,
Pfoîx = Jßie-1|x ) jjeb1dGl(u )j dv,
P(4)x = i Gi (u ) ^ ( ulx ) ,
P72 = Jßie-Pltdt = 1, p8i = ^e^dt = 1 fV 1 P(i7|x = Ie-PlV ip! ( - )j dK2 ( v|x )
(5) = ^20|x
jß2e(v|x) jeß2dG2 (u) dv
P23x = i e-p2V {2 ( u)|^ ( vlx ), p26sîx = i (u ) ^ ( ulx )
The unconditional transition probabilities with correlation coefficient from some of the above conditional transition probabilities can be obtained as follows:
Pi0} = J P(i0|xg1 ( x ) dx = { p!30|x{a 1(1 - r)e-a1(1-rîx}dx
Similarly,
pl7)= J" Pi7|x {a id - De-ai(1-r)x}dx,
p S = i P 20|x {a 2(1 - r)e-a2(1-r)x}dx,
-a2(1-r)x
p268)= i p ipa 2(1 - r)
It can be easily verified that,
}dx = P26
P01+P02 =1, p1O + P17 + Pr/ =1,
p(7} = f ^{«id - r)e-a1(1-r)x}dx = p14 P (258)= J" P (28|x{a 2(1 - r)e-a2(1-r)x}dx
(5 ) (5) (6) , P20 + P28 + P28 = 1, P72 = P51 = 1
(1-5)
fThe limits of integration are 0 to oo whenever they are not mentioned.
5. Mean Sojourn Times
The mean sojourn time in state S; is defined as the expected time taken by the system in state S; before transiting into any other state. If random variable U; denotes the sojourn time in state S; then,
M/i=JP[Ui>t]dt
Therefore, its values for various regenerative states are as follows-
= fg-K+^Xi-r)1^.
1
(oij +a2)(l-r)
f or, oo / \J
-(o^u+cqrx) ^ iala2I"XUJ
è (.,o
so that, = J\|/1|xg1(x)dx = J\|/1|xa1(l-r)e_a^1_r^xdx
dt
-du
OO f \j ^
y2|x =/G2(t)K1(t|x)dt=jG2(t)
It j=o (j!)
so that, \|/2 = J\|/2|xg2 (x)dx = J\|/2|xa2 (l-r)e_a^1_r^xdx
V3|x = Je^K^x)*
dt
that, =i^3|xgi(x)dx = ^-
(l-ra2')
= J" (t)dt
VS|I=Je^tK1(t|x)dt
so that, v|/5 = |m'5|xÖ2 (x)dx =
12
v|/6 =jG2(t)dt 7 J ßi
! «/(i-r)
(l-nx/j
(6)
(7)
(8)
(9) (10)
(11)
(12)
(13)
(14)
6. Analysis of Characteristics
6.1. Reliability and MTSF
Let R; (t)be the probability that the system operates during (0, t) given that at t = 0 system starts from S; e E. To obtain it we assume the failed states S4, S6, S7 and S8 as absorbing. By simple probabilistic arguments, the value of R0 (t) in terms of its Laplace Transform (L.T.) is given by
Z* * I rr* * T* J * I P7* * T* |
*,, 0 + q0i (Zi + qi3Z3)+q02 (Z2 + q25Z5)
R 0 (s ) =---7TT-h---(15)
0 W 1 * (3)* * (5)*
1 - q0lqi0 - q02q20
We have omitted the argument 's' from q*(s) and Z*(s) for brevity. Z*(s); 1 = 0,1,2,3,5 are the L. T. of
Z0 (t) = e-(a+a)(1-r)t , Z1 (t) = G1 (t) j j a2e-(a2z+arx)l0 (^/a^)dzjg (x)dx
Z2 (t) = G2 (t)^a^™^ (24a1a2rxz)dz|g2 (x)dx Z3 (t) = J e-bt j J a2e-(a2z+a1'x)I0 (2^00^) dz|g1 (x) dx
Z5 (t) = j e-bzt |¥f a1 e-(a z+a2ix) I0 (2^/0102^) dz|g2 (x) dx
Taking the Inverse Laplace Transform of (15), one can get the reliability of the system when system initially starts from state S0. The MTSF is given by,
„/Tv fv /A. r ^0 + P01 (y1 + P13V3) + P02 (V2 + P25V5) n „
E(T0) = J R0 (t)dt = ¡1®0R0 (s) =---(if-(5)-1 (16)
s® 1- p01p10 - p02p20
6.2. Availability Analysis
Let Ai (t) be the probability that the system is up at epoch t, when initially it starts operation from state Si e E. Using the regenerative point technique and the tools of Laplace transform, one can obtain the value of A0 (t) in terms of its Laplace transforms i.e. A0 (s) given as follows-
A0(s > = N^ ^
where,
(V i * * * * \ * / * * * * \ / * * * \ / * * * * \ / * * * \
s) = (1 - qi7q72q28q*1) Zo+(q°1 + qt^sqsi) (Zi + q^Z3)+^^172 + q^) (Z + q25Z5)
and
,, * * * * / * * * * \ (3)* / * * * * \ (5)*
Di (s) =1 - ql7qo2q28qol- (q0i+qo2q28q°i) qV -(qoiq^q^ + q02)qV where, Ai (t), i = 0,1,2,3,5 are same as given in section 6.1. The steady-state availability of the system is given by A0 = lim A0 (t) = llm sA* (s)
t®<» s ®0
We observe that D1 (0 ) = 0
Therefore, by using L. Hospital's rule the steady state availability is given by
(18)
(19)
A0 = lim
_n1(s) = n d( s) d;
s®0
where,
N1 =(1-p17P28 ) V0 +(p01 + P02P28 )(y + ^3 ) + (P01P17 + P02 )(y + P25V5 )
and
D =(1- p17P28 ) y 0 + I1 - P02P(20 ) {(W + ^3 + 1^4 ) + ft 7^7 }
+ I1 - p01p(0)){(y( + p(5V5 + p(6y6 ) + p28y8 } (21)
The expected up time of the system in interval (0, t) is given by t
Kup (t )= j A0 (U) dU
0
So that, ^ (s ) = ^ s
(22)
6.3. Busy Period Analysis
Let b1 (t), B( (t), B3 (t) and b4 (t) be the respective probabilities that the repairman is busy in the repair of unit-1 and unit-2 and in the post repair of unit-1 and unit-2 at epoch t, when initially the system starts operation from state Si e E. Using the regenerative point technique and the tools of L.T., one can
obtain the values of above four probabilities in terms of their L.T. i.e. b1* (s), B(* (s), B3* (s) and B4* (s) as follows-
B1- (s) = N# Bf (s) = NS, Bf (s) = NS and B4* (s) = M) (23-26) D1 (s) D1 (s ) D1 (s ) D1 (s)
where,
n2 (s) = (q01 + q02q(8q* 1) (Z* + q*4Z*) N3(s) = (q^^ + q*2) (Z( + q*26Z*) N4(s) = q*3 (q01+q0(q*28q* 1)Z* + (q01q*7 + q02q*28q*1q*7)Z*
N5(s) = q**5 (q01q*7q7(+q0() Z5 +(q01q*7q72q*28 + q0(q*28)Z*
and D1 (s) is same as defined by the expression (18) of section 6.2. Also Z4, Z6, z7 and Z* are the L. T. of
Z4 (t ) = G1 (t), Z6 (t ) = G( (t), Z7 (t ) = e-b1t, Z8 (t ) = e-b(t The steady state results for the above three probabilities are given by-
BO = limsB0 (s ) = N2/D1, B( = N3/D{, B^ = N4/D{ and b4 = N5/d; (27-30)
s®0
N2 = (1 - p02P((50) ) ( V + P14V4 )
N3 =(1 - p0ipi30))(y 2 + p26 y 6 )
N4 = (1 - p02p ((50)) (pi3 y3 + pi7 y7 )
N5 =(1 - p0ipi0))(p25 y5 + P28 y8 ) and DJ is same as given in the expression (21) of section 6.2.
The expected busy period in repair of unit-1 and unit-2 and post repair of unit-1 and unit-2 during time interval (0, t) are respectively given by-
t t t t Mb (t ) = j B0 (u) du, ^ (t ) = j B2 (u) du, ^ (t ) = i B0 (u ) du and ^ (t ) = j B4 (u) du
0 0 0 0
So that,
£ (s) = B0* (s)/s, ^T (s) = B2* (s)/s, ^2* (s) = B0* (s )/s and tf (s) = B0* (s)/s
(31-34)
6.4. Profit Function Analysis
The net expected total cost incurred in time interval (0, t) is given by
P (t) = Expected total revenue in (0, t) - Expected cost of repair in (0, t)
= ^up (t) - KX (t) - ^2 (t) - K3^b (t) - K4^2 (t) (35)
Where, K0 is the revenue per- unit up time by the system during its operation. K1, K2, K3 and K4 are the amounts paid to the repairman per-unit of time when he is busy in the repair of unit-1 and unit-2 and in the post repair of unit-1 and unit-2 respectively. The expected total profit incurred in unit interval of time is
P = K0 A0 " K1B0 " K2B0 " K3B0 " K4B0 (36)
7. Particular Case
When the time of satisfactory repair of unit-1 and unit-2 also follow exponential with p.d.fs as follows-g!(t) = , g2(t) = 12e-M
The Laplace Transform of above density function are as given below-
g*(s) = G 1(s) =
I1
s + 11
g*(s) = G 2 (s) =
s +
Here, G 1(s) and G2(s) are the Laplace-Stieltjes Transforms of the c.d.fs G1(t) and G2(t) corresponding to the p.d.fs g1(t) and g2(t).
In view of above, the changed values of transition probabilities and mean sojourn times.
p10J = 1 -
l1 a2(1 - r) p1 a 2' (1 - r)
(l-P1) (1 -ra2) (I1 -P1) (1 -ra2')'
p(4) = a 2,(1 - r)
"17 /i ff\'
(1 - r«2)
p£' =
¥1 =
y = 7", k1
(l2 -b2 ) 1
a2(1 - r) + V 1
a; (1 - r) a"(1 - r) (1 - ra;) (1 - raD
(3)
p17 =
11
p20 = 1 -
(1 -b1)
It
a2(1 - r) a2'(1 - r)
(1 - ra2) (1 -ra2') a; (1 - r) p2 «''(1 - r)
pV28 =
y 2 =
(12-p2) (1 - ra') (12 -b2) (1 - ra1I a1'(1 - r) (1 - r aD 1
V6 =7-
k2
a1(1 - r) + 12 1
8. Graphical Study of Behaviour and Conclusions
For a more clear view of the behaviour of system characteristics with respect to the various parameters involved, we plot curves for MTSF and profit function in Fig. 2 and Fig. 3 w.r.t. </., for three different values of correlation coefficient r=0.16, 0.26 and 0.35 and two different values of repair parameter =0.41 and 0.5 while the other parameters are kept fixed as = 0.15, a2 = 0.05, |i, = 0.35 and p2 = 0.01 . From the curves of Fig. 2, we observe that MTSF increases uniformly as the values of r and increase and it decreases with the increase in a,. Further, to achieve MTSF at least 20 units we conclude from smooth curves that the value of </., must be less than 0.2, 0.15 and 0.11 respectively for r = 0.16 , 0.26 , 0.35 when = 0.5. Whereas from dotted curves we conclude that the value of (/., must be less than 0.2, 0.16, 0.12 for r = 0.16 , 0.26 , 0.35 when ^ = 0.41 to achieve at least 12 units of MTSF.
Similarly, Fig. 3 reveals the variations in profit (P) w.r.t. otj for varying values of r and , when the values of other parameters are kept fixed as 'k2 =0.1, a2 =0.05, |i, =0.35, P2 =0.3,K0 =800, Kj = 600, K2 = 500, K3 = 400 and K4 = 300. Here also the same trends in respect of </.,, r and /., are observed as in case of MTSF. Moreover, we conclude from the smooth curves that the system is profitable only if (/., is less than 0.75, 0.5 and 0.37 respectively for r = 0.25, 0.35, 0.45 when )., = 0.5 . From dotted curves, we conclude that the system is profitable only if (/., is less than 0.2, 0.15 and 0.12 respectively for r = 0.25, 0.35, and 0.45 when = 0.4 -
0
0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1
Figure 2: Behaviour of MTSF w.r.t. <Xi for different values of r and Aj
9. Acknowledgment
"Author Surbhi Masih thanks to University Grants Commission, Maulana Azad National Fellowship for providing funding support during my Ph.d Course."
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