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Alka Chaudhary, Shivali Sharma RT&A, No.3 (74)
A TWO NON-IDENTICAL UNIT PARALLEL SYSTEM Volume 18, September 2023
A TWO NON-IDENTICAL UNIT PARALLEL SYSTEM WITH PRIORITY IN REPAIR
Alka Chaudhary, Shivali Sharma
Principal, KLPG College, Meerut, 250002 Department of Statistics, Meerut College, Meerut, 250004 [email protected], [email protected]
Abstract
The paper deals with a system composed of two-non identical units (unit-1 and unit-2). Initially both the units are arranged in parallel configuration. Each unit has two possible modes- Normal (N) and Total Failure (F). The first unit gets priority in repair. System failure occurs when both the units stop functioning. A single repairman is always available with the system to repair a totally failed unit and repair discipline is first come, first served (FCFS).If during the repair of a failed unit the other unit also fails, then the later failed unit waits for repair until the repair of the earlier failed unit is completed. The repair times of both the units are exponential distribution with different parameters. Each repaired unit works as good as new. Using regenerative point technique, various important measures of system effectiveness have been obtained.
Keywords: Transition probabilities, mean sojourn time, reliability, MTSF, availability, expected busy period of repairman, net expected profit.
1. Introduction
Reliability is an important concept in the planning design and operation stages of various complex systems. Reliability is a significant area that is accepting awareness internationally and it is crucial for actual usage and care of any industrial system. It requires technical Knowledge for growing system effectiveness by decreasing the frequency of failure and reducing the worth of maintenance. Chaudhary and Tyagi [3] analyzed a two non-identical unit parallel system with two types of failure. Pundir et al. [7] analyzed a two non-identical unit parallel system with priority in repair. Chaudhary et al. [5] analyzed two non-identical unit warm standby repairable system with two types of failure. Saxena et al. [9] analyzed two unit parallel system with working and rest time of repairman.
A single repairman is always available with the system to repair a totally failed unit and repair discipline is first come, first served (FCFS). Chaudhary and Masih [1, 4] analyzed a two non-identical unit. Saini et al. [8], Chaudhary and Sharma [2] and Dabas et al. [6] analyzed a two non-identical unit system models assuming two modes- Normal mode and total failure mode of each unit and analyzed parallel system with priority in repair. System failure occurs when both the units stop functioning. The first unit gets priority in repair.
By using regenerative point technique, the following measures of system effectiveness are
obtained-
i. Transition probabilities and mean sojourn times in various states.
ii. Reliability and mean time to system failure (MTSF).
iii. Point-wise and steady-state availabilities of the system as well as expected up time of the system during time interval (0, t).
iv. Expected busy period of repairman in the repair of unit-1 and unit-2 during time interval (0, t).
v. Net expected profit earned by the system in time interval (0, t).
2. System Description and Assumptions
1. The system comprises of two non-identical units (unit-1 and unit-2). Initially, both the units work in parallel configuration.
2. Each unit of the system has two possible modes-Normal (N) and total failure (F).
3. The first unit gets priority in repair.
4. System failure occurs when both the units stop functioning.
5. A single repairman is always available with the system to repair a totally failed unit and repair discipline is first come, first served (FCFS).
6. If during the repair of a failed unit the other unit also fails, then the later failed unit waits for repair until the repair of the earlier failed unit is completed.
7. The repair time of both the units is exponential distribution with different parameters. Each repaired unit 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-
N1O,N2O : Unit-land Unit-2 is in N-mode and operative in parallel.
F1r ,F2r : Unit-1 and Unit-2 is in F-mode and under repair.
F2w : Unit-2 is in F-mode and under waiting 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 S1 from S2 are non-regenerative, whereas all the other entrance epochs into the states of the systems are regenerative.
The other notations used are defined as follows:
E : Set of regenerative states.
aj, a 2 : Constant Failure rate of Unit-1 and Unit- 2.
ßj, ß 2 : Constant Repair rate of Unit-1 and Unit- 2.
G (•) : CDF of time to repair and its repair is continued to state Si
H (•) : General Distribution of Unit-2
* : Symbol for Laplace Transform i.e. g*(s) = J estq1J(t)dt
~ : Symbol for Laplace Stieltjes Transform i.e. Q1J(s) =J e stdQ1J(t)t
t
© : Symbol for ordinary convolution i.e. A(t)©B(t)=J A(u)B(t-u)du
0
TRANSITION DIAGRAM
So
Si
S4 S3
^^^ : Up State j j : Failed State • : Regenerative Point ^^ : Non Regenerative Point
Figure 1: Exponential Model
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 = {S0toS4} .The various measures of system effectiveness are obtained in terms of steady-state transition probabilities as follows:
-(a, +a2 )tdt = a1
P01 _Ja^™ldt =
a1 + a2
-(«1 +a2 )tdt _ a2
P02 _ _
a1 +a2
p,o ={pie-p,tG (t) dt = l - G (p,) P20 =jp2e-P2tH (t) dt = l- H (p2)
P24 =J e-P2tdH (t ) = H (p2) P42 = JP,e-Pltdt = l = P32
The two step transition probability (Steady State) is given by
p(2)_(l - e-ftt )J e-ßJudG (u )_ G (ßj)
It can be easily verified that,
P01 + P02 _ 1
(3) 1 P10 + P12 _1
P32 _ P42 _1
P20 + P24 _1
(1)
(2)
(3)
(4)
+The limits of integration are 0 to <» whenever they are not mentioned.
S
2
0
5. Mean Sojourn Time The mean sojourn time y1 in state S1 is defined as the expected time taken by the system in state S1 before transiting into any other state. If random variable U1 denotes the sojourn time in state S1 then,
V = J P [U1 > t ]dt
Therefore, its values for various regenerative states are as follows-
Vo =J e-(aj+a2 )tdt = ?—(5) (a1 +a 2 )
So that,
Vi =J e~ßlt G(t) dt (6)
V2 = J eßltH (t) dt (7)
V 4 =J e-ßltdt = ß- (8) J ßi
6. Analysis of Characteristics
6.1. Reliability and MTSF
Let Rj (t) be the probability that the system operates during (0, t) given that at t=0 system starts from Sj e E .To obtain it we assume the failed states S2 and S4 as absorbing. By simple probabilistic arguments, the value of R0 (t) in terms of its Laplace Transform (L.T.) is given by
r0 (s)= Z0 + q0lZ i+q02Z2 (9)
0 V / \/
1 - q01q10- q02q20
We have omitted the argument's from q*(s) and Z*(s) for brevity. Z*(s); i = 0, 1, 2 are the L. T. of
_ e ' Z1 (t)_J e-fttG(t)dt, Z2 (t)_J e-P ^h (t) dt
Taking the Inverse Laplace Transform of (9), one can get the reliability of the system when system initially starts from stateS0 . The MTSF is given by,
E(To) _JRo (t) _ UrnR 0 (s) _ +Po1^ +Po2^2 (10)
J 1 - P01P10 - P02P20
6.2. Availability Analysis
Let Aj (t) be the probability that the system is up at epoch t, when initially it starts operation from state Sj 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. ^ N1 (s)
A0 (11)
Where,
N1(s) _ Z0 f1 - q*24q*42 ]+Ziq01 f1 - q*24q42 ]+Z2
and
* (3)* * q01qi2 + q02
Alka Chaudhary, Shivali Sharma RT&A, No.3 (74) A TWO NON-IDENTICAL UNIT PARALLEL SYSTEM_Volume 18, September 2023
** **/i * * \ * / * (3)* * \ . .
D,(s) =q24q42- q,oqo, - q24q42)-q2o (qoiq12 + qo2) (12) Where, Zi (t), i=0,1,2 are same as given in section 6.1. The steady-state availability of the system is given by
A0 = lim A0 (t) = lim s AO (s) (13)
We observe that D, (0) = 0
Therefore, by using L. Hospital's rule the steady state availability is given by N (s) N,
A0 = lim-^^ = -,- (14)
0 s-0Dx (s) D, V '
Where,
Nl = V0 [l - P24P42 ] + VlP0l [l - P24P42 ] + V2 and
Dl = V0P20 +VlP0lP20 +V2 (!-Pl0P0l) + V4P24 (!-Pl0P0l ) (15)
The expected up time of the system in interval (0, t) is given by
t
^up (t) = J A0 (u ) du
(3)
P0lPl2 + P02
0
A* (s)
So that, ^ (s) = (16)
6.3. Busy Period Analysis
Let B| (t) and B2 (t) be the respective probabilities that the repairman is busy in the repair of unit-
1 failed due to first repair with priority of unit-1 and unit-2 failed due to second repair 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 two probabilities in terms of their L. T. i.e. Bi1* (s) and B2* (s) as follows-
Bi*(s) = N2(s), B2*(s) = ^
Di (s) iW Di (s)
Where,
AT t \ 7* * /l * M , * (3)* * , * * \
n2 (s) = Zl q0l (l - q42q24)+Z4 (q0lql2 q24 + q02q24)
and
q(3)
N3(s) = Z2 (q0lq(2)* + q02) and D, (s) is same as defined by the expression (12) of section VI(II). The steady state results for the above two probabilities are given by-
(17-18)
B0 = lim s B0* (s) = N2 \ Dl and B2 = lim s B0* (s) = N3 \ Dl (19-20)
s^0 s^0
Where,
N2 (0) = VlP0l (l - P24 ) + V4 (P0lPl2)P24 + P02P24 ) (21)
N3 (0)=V2 (P0lPl2)+ P02) (22)
and Dl is same as given in the expression (15) of section 6.2.
The expected busy period in repair of unit-1 failed due to first repair with priority of unit-1 and unit-2 failed due to second repair during time interval (0, t) are respectively given by-
lb (t )_J B0 ( u ) du
and
(t )_J b2 (u) du
So that,
Bq" (s)
and
if(s)_
B2"(s )
(23-24)
0
0
s
s
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)
_ K°^up (t)- K^b (t)- K2|b (t) (25)
Where, K° is the revenue per- unit up time by the system during its operation. K1 and K2 are the amounts paid to the repairman per-unit of time when the system is busy in repair of unit-1 failed due first repair with priority of unit-1 and unit-2 failed due to second repair respectively. The expected total profit incurred in unit interval of time is P _ K°A° -K1B° -K2B2
7. Particular Case Let G (t) _ Xe-Xt, H (t) _ |e-|t
In view of above, the changed values of transition probabilities and mean sojourn times.
__P_ _ P2 _ I
p10 n » ' p20 n ' p24 "
P1 + X P2 +I P2 +I
(3) _ X _ 1 _ 1
p12 P1+X' 11 P1+X' 12 P2 + I
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. a1 for three different values of failure parameter a2 =0.1, 0.5, 0.9 and two different values of repair parameter P1 =0.01, 0.7 while the other parameters are P2 =0.99, | = 0.01, X = 0.06. It is clearly observed from Fig. 2 that MTSF increases uniformly as the value of a2 and P1 increase and it decrease with the increase in a1. Further, to achieve MTSF at least 10 units we conclude for smooth curves that the values of a1 must be less than 0.18, 0.29 and 0.49 respectively for a2 =0.1, 0.5, 0.9 when P1 =0.01. Whereas from dotted curves we conclude that the values of a1 must be less than 0.15, 0.22 and 0.39 for a2 =0.1, 0.5, 0.9 when P1 =0.7.
Similarly, Fig.3 reveals the variations in profit (P) with respect to a1 for three different values of a2 = 0.4, 0.6, 0.8 and two different values of P1 =0.03, 0.2, when the values of other parameters P2 =0.01, | = 0.09, X = 0.6, K0=80, K1=125 and K2=175. Here also the same trends in respect of a1, a2 and P1 are observed in case of MTSF. Moreover, we conclude from the smooth curves that the system is profitable only if a1 is less than 0.20, 0.39 and 0.79 respectively for a2 = 0.4, 0.6, 0.8 when
P, =0.03. From dotted curves, we conclude that the system is profitable only if al is less than 0.13,
0.27 and 0.58 respectively for a2 = 0.4, 0.6, 0.8 when P, =0.2.
--■•—a2=0.1, Pi=0.01 —■— az=0.5, Pi=0.01
30
* \ \ \
—A—a2=0.9, Pi=0.01
25
a2=0.1, Pi=0.7 a2=0.5, Pi=0.7 a2=0.9, Pi=0.7
№ A
m H
20
15 ' 10 5
0
0.1
0.2
~1-1-1-r
0.3 0.4 0.5 0.6
T-1-1
0.7 0.8 0.9
Figure 2: Behaviour of MTSF w.r.t. at for different values of a2 and Pj
to O 0£ e*
80
60
40
20
--a2=0.4, Pi=0.03 --a2=0.6, Pi=0.03 --a2=0.8, Pi=0.03
■a2=0.4, Pi=0.2 •a2=0.6, Pi=0.2 ■a2=0.8, Pi=0.2
-20
-40
-60
Figure 3: Behaviour of PROFIT (P) w.r.t. at for different values of a2 and Pj
0
References
[1] Chaudhary, P. and Masih, S. (2020). Analysis of a standby system with three modes of priority unit and correlated failure and repair times. Statistics and Reliability Engineering, 7(1):123-130.
[2] Chaudhary, P. and Sharma, A. (2022). A two non-identical unit parallel system with priority in repair and correlated life times. Reliability: Theory & Applications, 1(67):113-122
[3] Chaudhary, P. and Tyagi, L. (2021). A two-non-identical unit parallel system subject to two types of failure and correlated lifetimes. Reliability: Theory & Applications, 2(62):247-258.
[4] Chaudhary, P., Masih, S. and Gupta, R. (2022). Parallel system with repair and post repair policies of a failed unit and correlated life times. Reliability: Theory & Applications, 3(69):42-51.
[5] Chaudhary, P., Sharma, A. and Gupta, R. (2022). A Discrete Parametric markov chain model of a two non-identical unit warm standby repairable system with two types of failure. Reliability: Theory & Applications, 2(68):21-30.
[6] Dabas, N., Rathee, R. and Sheoran, A. (2023). Reliability Analysis of Parallel system using priority to PM over inspection. Reliability: Theory & Applications, 1(72):329-339.
[7] Pundir, P.S., Patawa, R. and Gupta, P. K. (2018). Stochastic Outlook of two non-identical unit parallel system with priority in repair. Cogent Mathematics and Statistics, 5(1467208): 1-18.
[8] Saini, M., Devi, K. and Kumar, A. (2021). Stochastic Modelling of a two non-identical redundant system with priority in repair activities. Thailand Statistician, 19(1):154-161.
[9] Saxena, V., Gupta, R. and Singh, B. (2023). Classical and Bayesian stochastic analysis of a two unit parallel system with working and rest time of repairman. Reliability: Theory & Applications, 1(72):43-55.
