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A Discrete Parametric Markov-Chain Model of a Two Non-Identical Units Warm Standby Repairable System with
Two Types of Failure
Pradeep Chaudhary, Anika Sharma, Rakesh Gupta
Department of Statistics Ch. Charan Singh University, Meerut - 250004 (India) E-mail: [email protected]; [email protected]; [email protected]
Abstract
The paper deals with the stochastic analysis of two non-identical units (unit-1 and unit-2). Initially, one unit is operative and other is kept into warm standby. Each unit of the system has two possible modes-Normal (N) and Total Failure (F). A single repairman is always available with the system to repair a failed unit. The operative unit is non-repairable, hence upon failure it goes for replacement. The system failure occurs when both the units are in total failure mode. Failure and repair times of a unit are taken as independent random variables of discrete nature having geometric distributions with different parameters.
Keywords: Transition probabilities, mean sojourn time, geometric distribution, regenerative point technique, reliability, MTSF, availability, expected busy period of repairman, net expected profit.
1. Introduction
The Two non-identical units warm standby system have been widely studied in the literature of reliability as they are frequently used in modern business and industries. It is obvious that the standby unit is switched to operate when the operating unit fails and the switching device which is used to put the standby unit into operation may be perfect at the time of need. Some authors including [5, 9, 10 and 13] analyzed two unit warm standby and two non-identical units warm standby redundant system models using different concepts. All the above system models have been analyzed by considering continuous distributions of all the random variables involved.
In many realistic situations, some writers [4 and 7] analyzed a two identical unit and two non-identical units cold standby system with two types of failure and later on [3] analyzed two non-identical units parallel system subject to two types of failure and correlated life times. Some authors [11, 12] analyzed the deferent concepts of assumptions. So in case of discrete random variable, discrete distribution is considered to be appropriate for obtaining the effectiveness of different reliability measures.
In the area of reliability using discrete distribution had given their ideas by analyzing two non-identical unit parallel system with geometric failure and repair time distribution. Since there is
always a possibility for failure of any system during in its operative conditions in different measures. So to detect the type of failure inspection is very much required which had been always ignored by the researchers, whether using continuous or discrete distributions.
This system model is based on discrete parametric Markov-chain. Moreover, [1, 2, 6 and 8] introduced the concept of discrete parametric Markov-chain in analyzing the system models in the field of reliability modeling. The following economic related measures of system effectiveness are obtained by using regenerative point technique-
i. Transition probabilities and mean sojourn times in various states.
ii. Reliability and mean time to system failure.
iii. Point-wise and steady-state availability of the system during time (0, t-1).
iv. Expected busy period of repairman during time (0, t-1).
v. Net expected profit incurred by the system during a finite and steady-state are obtained.
2. System Description and Assumptions
1. The system comprises of two non-identical units. Initially, one unit is operative and other is kept into warm standby.
2. Each unit of the system has two modes- Normal (N) and total failure (F).
3. A single repairman is always available with the system to repair a failed unit.
4. The operative unit is non-repairable, hence upon failure it goes for replacement.
5. The system failure occurs when both the units are in total failure mode.
6. The repaired unit works as good as new.
7. Failure and repair times of the units follow independent geometric distributions with different parameters.
3. Notations and States of the System
3.1 Notations :
pqf : p.m.f. of failure time of type-1 and type-2 respectively for i=1,2,3 and pi + qi = 1. rsf : p.m.f. of repair time by repairman of type-1 and type-2 respectively for i=1, 2 and
r + si = 1.
p'q'x : p.m.f. of repair time of first unit; p + q = 1.
8,0 : probability that the replacement of a second unit respectively; 9+9 = 1 qM (u), QM (u) : p.m.f. and C.d.f. of one step or direct transition time from state S, to Sj. p : steady state transition probability from state Si to Sj.
py= Qij H
Zi (t) : probability that the system sojourn in state Si up to epoch (t-1). ^ : Mean sojourn time in state S .
* h : symbol and dummy variable used in geometric transform e. g.
GT [j t )] = q*( h ) = £ htqij (t)
t=0
3.2 Symbols for the states of the system
Nj : unit-i is in normal mode(N) and operative; i=1,2 Ni : unit-i is in normal mode(N) and warm standby.; i=1,2
F / Fr : unit-2 is in total failure (F) mode and under replacement/waits for replacement. F1 / : unit-1 is in total failure (F) mode and under repair. The transition diagram of the system model is shown in Figure. 1.
TRANSITION DIAGRAM
O
: Up State
□
Failed State
S3
0 : Regenerative Point Figure.1
^^ : Non-Regenerative Point
With the help of above symbols the possible states of the system are:
So-(N0,NWs), S =(Fr\N0), S2 =(Fr,N0)
S3 = (No,FR), S4 = (F\FR), S5 = (f^Fr)
The states S0,S1;S2>S are up states; S4,S5 are failed states.
4. Transition Probabilities and Sojourn Times
Let Qjj (t) be the probability that the system transits from state St to Sj during time interval (0, t) i.e., if T is the transition time from state S to Sj then
Qj(t ) = P L jt J
By using simple probabilistic arguments we have,
Q01 ( t )=T
pm2
Q03 ( t )=T
- qq^ p'q^
qq1q2 pp2^
1 -(qq^ )t+
1 -(qq^2 )t+
î < \t-1 -(q q1q2 )
Q02 (0 = -^- 1 -(q'q^ ) 1 - qqq - v 7
-qq^ Q04 (t )=y
p p1q2
-qq^
/ ' \t-1-(qq^2 )
Q05 ( t ) = r
1- q q^
Q13(t) = T^[1 -(sa)t+1 ] ,
1 - s1q3L J
Q10 (t^T^[1 -fr*)t+1 ]
1 s1q3
Q" ( ' >■ i-pt ['-(s1q- r ]
S
S
4
5
Q20 (t) = [l-(s2Q3)+1 ],
1 - s2q3 [ J
Q25 (t )=rrV [1 )t+1 ], 1 s->q -i
Q23 (t) = [1 -(s2q3)t+1 ]
1 - s2q3[ ]
Q30 (t )= 1 -(e)t+
Q43 (t) = [1 -(s1 )t+1 ], Q53 (t) = [1 "(s2 )t+1 ] (1-14)
The steady state transition probabilities from state Si to Sj can be obtained from (1-14) by taking t ^■x , as follows:
P01 =
P05 =
P20 =
P1qq2 1 - q'q^
p'p2q1 1 - q'q^
r2q3 1 - s2q3,
P02 =
P10 =
P23 =
P2q q1 1 - q'q^
1 - s1q3,
r2P3 1 - s2q3,
P03 = P13 = P25 =
P q^ 1- q'q^
r1P3 1 - s1q3,
s2P3 1 - s2q3
P04 =
P14 =
P P1q2 1 - q'q^
s1P3
1 - sA
We observe that the following relations hold-
P30 = P43 = P53 =1, P01 + P02 + P03 + P04 + P05 =1
P10 + P13 + P14 =1, P20 + P23 + P25 =1,
(15-18)
5. Mean Sojourn Time Let T be the sojourn time in state Si (i=0-5) then mean sojourn time in state Si is given by
x
^ = E(Ti) = £ P [T ^ t-1]
t=1
In particular,
Vo= qq1q2 , , ^2 = " s2q3
, ' ' T1 - , / t 2 1
1 - qq^2 1 - s1q3 1 - s2q3
V3 =-, ^4 = -, = -1 (19-24)
u r, r
6. Methodology for Developing Equations
In order to obtain various interesting measures of system effectiveness we developed the recurrence relations for reliability, availability and busy period of repairman as follows-
6.1 Reliability of the system
Here we define Ri(t) as the probability that the system does not fail up to epochs 0, 1, 2,.., (t-1) when it is initially started from up state S . To determine it, we regard the failed states S4,S5 as absorbing state. Now, the expression for R;(t); i=0, 1, 2, 3; we have the following set of convolution equations.
t-1 t-1 R0 (t) = qtq1q2 +^01 (u) R1 (t -1 - u ) + Zq<* (u) R2 (t-1 - u)
u=0 u=0
= Z (t) + q01 (t -1) ©R1 (t -1) + q02 (t -1)©R2 (t -1) + q03 (t -1)©R3 (t -1)
Similarly,
R (t) = Z, (t) + q,o (t - l)©Ro (t -1) + q,3 (t -1)©R3 (t -l)
R2 (t) = Z2 (t) + q2o (t - l)©Ro (t -1) + q23 (t -1^ (t -l)
R3 (t) = Z3 (t) + q3o (t -1) ©Ro (t -1) (25-28)
Where,
Z (t) = Sj1q3 , Z2 (t) = s2q3 , Z3 (t) = q4
6.2 Availability of the System
Let Aj (t) be the probability that the system is up at epoch (t-1), when it initially started from state S . Then, by using simple probabilistic arguments, as in case of reliability the following recurrence relations can be easily developed for A (t); i=0 to 5.
t-1 t-1 t-1
Ao (t) = q tqtq2 + £q„ (u)©A (t-1 -u) + £qo2 (t -(t -1) + £qo3 (t -1)©A (t -1)
u=o u=o u=o
t-1 t-1 +£ qo4 (t -1) ©A4 (t -1)+£ qo5 (t -1) ©a5 (t -1)
u=o u=o
= Zo (t) + qM (t -1) ©A (t -1) + qo2 (t -1) ©A2 (t -1) + qo3 (t -1) ©A3 (t -1)
+ qo4 (t -1)©A (t -1) + qo5 (t -1)©A5 (t-1)
Similarly,
A1 (t ) = Z (t) + q1o (t-1)©Ao (t-1) + qB (t-1)©A3 (t-1) + q14 (t-1)©A (t-1) A2 (t) = Z2 (t) + q2o (t -1) ©Ao (t -1) + q23 (t -1) ©A3 (t -1) + q25 (t -1) ©A5 (t -1) A3 (t) = Z3 (t) + q3o (t -1)©A (t -1) A4 (t) = q43 (t-1)©A (t-1)
A5 (t) = q53 (t-1)©A3 (t-1) (29-34)
Where the values of Zi (t); i=0 to 3 are same as given in section 6.1.
6.3 Busy Period of Repairman
Let B( (t) and BR (t) be the probability that the repairman is busy in the repair and replacement of a failed unit at epoch t-1, when it initially started from state St. Then, by using simple probabilistic arguments, as in case of reliability the following recurrence relations can be easily developed for B4 (t) and BR (t); i=0 to 5.
Bro (t) = qo1 (t-1)©Br (t-1) + qo2 (t -1)©B2 (t-1) + qo3 (t-1)©B' (t -1) + qo4 (t -1)©Br4 (t -1)
+ qo5 (t- 1)©b; (t-1) Br (t) = Z (t) + q№ (t -1) ©Bo (t -1) + qB (t -1) ©B3 (t -1) + qM (t -1) ©B4 (t -1) B2 (t) = Z2 (t) + q2o (t -1)©Bo (t -1) + q23 (t -1)©B3 (t -1) + q^ (t -1)©B5 (t -1) B(( t ) = q3o (t-1)©Br0( t-1) Br4(t) = Z4 (t) + q43 (t-1)©B((t -1)
Bo rt) = Z5 (t) + q53 (t -1) ©Bo (t -1) (35-40)
Where,
The values of Z1 (t) and Z2 (t) are same as given in section 6.1, Z4 (t) = s, and Z5 (t) = st,.
BR (t) = q„i (t -1) ©BR (t -1) + q02 (t -1) ©B2r (t -1) + q03 (t -1) ©BR (t -1) + q04 (t -1) ©B4R (t -1)
+ qo5 (t - 1)©BR (t -1)
BR (t) = q,o (t-1)©BR (t-1) + q,3 (t -1)©BR (t-1) + qM (t -1)©BR (t-1) BR (t) = q2o (t -1) ©BR (t -1) + q23 (t -1) ©BR (t -1) + q25 (t -1) ©BR (t -1) BR Rt)= Z3 (t) + q3o (t-1)©BR Rt-1) B4 ( t ) = q43 ( t -1)©BR ( t -1)
BR (t) = q5R (t-1)©BR (t-1) (41-46)
Where,
The value of Z3 (t) is same as given in section 6.1.
7. Analysis of Reliability and MTSF
Taking geometric transform of (25-28) and simplifying the resulting set of algebraic equations for
*
Ro(h) we get
40 (h) =
N1 ( h ) D1 (h )
(47)
Where,
N1 (h) = Z0 + hq0X + hq02Z2 +|_h2q*01qi*3 + h2q*02q*23 + M0R J Z3
/-i\ 1 1 2 * * 1 2 * * 1 2 * * 1 3 * ** -, 3 * * *
D1 (h) =1 -h q01q№ -h q02q20-h q03q30-h q^q^ -h q02q23q30
Collecting the coefficient of ht from expression (47), we can get the reliability of the system R0 (t). The MTSF is given by-
E (T) = lim £ htR ( t ) =
t=1
N0) _1 D1 (1)
(48)
Where,
N1 (1) = V0 + P01Y1 + P02Y2 +[P01P13 + P02P23 + P03 ] V3 D1 (1)=1- P01P10- P02P20- P03- P01P13- P02P23
8. Availability Analysis
On taking geometric transform of (29-34) and simplifying the resulting equations for we get, N2 (h)
A ( h ) =
D2 (h)
(49)
Where,
N2 (h) =
Z0 - hq01 - hq02 - hq03 - hq*04 - hq05
Z* 1 0 - hq*3 - hq*4 0
z2 0 1 - hq*23 0 - hq2s
7* Z3 0 0 1 0 0
0 0 0 - hq43 1 0
0 0 0 - hq53 0 1
and
D2 (h) =
1 - hq01 - hq02 - hq*03 - hq04 - hq*05
hq*o 1 0 - hq*3 - hq*4 0
hq2o 0 1 - hq*23 0 - hq^5
hq3o 0 0 1 0 0
0 0 0 - hq*43 1 0
- ^5:
The steady state availabilities of the system due to operation of unit -
N2 (h)
A0 = lim A (t) = lim (1- h)
D2 (h)
But D (h) at h=1 is zero, therefore by applying L. Hospital rule, we get N2 (1)
Ao =-
D2 (1)
(50)
Where,
N2 C1) = Vo + P01V1 + P02V2 +[PoiPl3 + PoiPl4 + P02P23 + P02P25 +P03 + Po4 + Po5 ] V3
and
D2 (1) = Vo + P01V1 + P02V2
Now the expected uptime of the system due to operative unit upto epoch (t-1) are given by K.P (t) = ZAo (x)
x=0
So that
A0 (h)
H*p (h ) =
(1 - h)
(51)
0
0
0
0
1
9. Busy Period Analysis
On taking geometric transforms of (35-40) and (41-46), simplifying the resulting equations, we get
B0*( h ) = M) and BR*( h ) = (52-53)
0 w D (h) D2 (h) v ;
Where,
"Vt/I\ r-w* t * r-w* -t * r-w* t 2 * * r-r* t * r-w* t 2 * * r-r* t *
N3 (h) = Zihqo1 + z2HG+Z4h qo1qi4 + z4HM + Z5h qc^ + ZsH)5
and
-vt/i\ 1 2 * * 3*** . 2 * * i3* * * 1* i 2 * * i2**l
n4 (h) = Z3 q^qu + h qo^^ + h qo2q23 + h qo2q25q53 + hqo3 +h qo4q43 + h qo5q53 J and D2 (h) is same as in availability analysis.
In the long run the respective probabilities that the repairman is busy in the repair and replacement of a failed unit are given by-
B0 = limBO (t) = lim(1 - h)N3-(h) and BR = limBR (t) = lim(1 - h)^^
0 t^» °V h^ 'd2 (h) 0 t^» °V h^ >d2 (h)
But D (h) at h=1 is zero, therefore by applying L. Hospital rule, we get
B =- N4) and BR =- M) (54-55)
0 d; (1) 0 d; (1) ( )
Where,
N3 (1) = P01V1 + P02V2 +(P01P14 + P04 H +(P02P25 + P05 )V5
and
N4 (!) = Y3 [P01P13 + P01P14 + P02P23 + P02P25 + P03 + P04 + P05 ] and D2 (1) is same as in availability analysis.
Now the expected busy period of the repairman in repair of a failed unit up to epoch (t-1) are respectively given by-
^ (t) = ZB0 (x) and (t)=£B0r (x) (56-57)
x=0 x=0
10. Profit Function Analysis
We are now in the position to obtain the net expected profit incurred up to epoch (t-1) by considering the characteristics obtained in earlier section. Let us consider, K0 = revenue per-unit time by the system due to operative unit. K = cost per-unit time when repairman is busy in the repair of failed unit. K2 = cost per-unit time when repairman is busy in the replacement of a failed unit. Then, the net expected profit incurred up to epoch (t-1) is given by,
P(t) = K0^p (t)-KlMb (t)-^R (t) (58)
The expected profit per unit time in steady state is given by-
P = limp(t) = lim (1 - h)2 P* (h)
t t h v ' v '
n U2A0(h) ( ,ob;*(h) 2BQ*(h)
=Kolm(1 - h) (i-h)- Kilim(1 - h) -(1"-hy- K2lim(1-h) -(Thy
= KqAQ - K1BO - K2BO (59)
11. Graphical Representation
The curves for MTSF and profit function have been drown for different values of failure parameters. Fig.2 depicts the variation in MTSF with respect to failure rate ( p ) for different values
of repair rate (p2) of a unit and constant repair rate (p ) when values of other parameters are kept fixed as p3 = 0.001, r = 0.5, r2 = 0.7 and 9 = 0.01. From the curves we conclude that expected life of the system decrease with increase in P . Further, increases as the values of P and P' increases.
Similarly, Fig.3 reveals the variations in profit (P) with respect to p for varying values of p2 and p', when other parameters are kept fixed as p3 = 0.01, r, = 0.92, r2 = 0.99 and 9 = 0.01, K0 = 100, K1 = 100, K2 = 400 and K3 = 300 . From the figure it is clearly observed from the smooth curves, that the system is profitable if the value of parameter P is greater than 0.2, 0.33 and 0.5 respectively for p2 = 0.4, 0.6 and 0.8 for fixed value of p' = 0.15 . From dotted curves, we conclude that system is profitable only if value of parameter P is greater than 0.27, 0.39 and 0.6 respectively for p2 = 0.4, 0.6 and 0.8 for fixed value of p' = 0.3 .
Behavior of MTSF with respect to p1, p2 and p'
P1
Figure.2
Behavior of Profit (P) with respect to p1, p2 and p'
—•— p2=0.4,p'=0.15 —•— p2=0.4,p'=0.3
12. Conclusions
1. It is indicated in fig.2 that we can easily obtain the upper limit of " p1" to achieve at least a particular value of MTSF. As an illustration to get at least MTSF 16.8 unit, the failure rate " p " must be less than 0.24, 0.56 and 0.79 respectively for repair rate p2 = 0.01, 0.03 and 0.05 when
activation rate is kept fixed as p' = 0.93. Similarly, when p' = 0.99 is kept fixed as " p1" must be less than 0.43, 0.68 and 0.87 corresponding to p2 = 0.01, 0.03 and 0.05. 2. In fig. 3 it is reveled from the smooth curves, that the system is profitable if the value of parameter p1 is greater than 0.2, 0.33 and 0.5 respectively for p2 = 0.4, 0.6 and 0.8 for fixed
value of p' = 0.15 . From dotted curves, we conclude that system is profitable only if value of parameter p is greater than 0.27, 0.39 and 0.6 respectively for p2 = 0.4, 0.6 and 0.8 for fixed value of p' = 0.3 .
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