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Surabhi Sengar RT&A, No 4 (59) Sensitivity analysis with wait in line theory_Volume , December 2020
Sensitivity Analysis of a Complex Engineering System with the Application of Wait in Line Theory
Surabhi Sengar
Pimpri Chinchwad college of Engineering, Pune, India sursengar@gmail.com
Abstract
In this bloodthirsty scenario of competition, rapid and cost-effective production is a key obligation for endurance. To attain this objective the thought of production area is fetching people now days. In this sector, non- identical equipment is arranged rationally to execute desired procedure to convert unprocessed materials into the processed material. Sometimes during the manufacturing, the problem of waiting line arises because of some unpredictable reasons, so here this paper reveals the same problem with its effects on system reliability and availability, also system sensitive nature is analyzed with respect to unexpected failures. Supplementary variable technique and copula method is used to solve the system.
Keywords: Markov processes, steady state behavior, supplementary variable technique, reliability, wait in line etc.
1. Introduction
Wait in line theory has an important role in almost all analysis of repair services. Wait in line models are very helpful tool for calculating the performance of various repair systems like automated systems, production systems, computer systems, telecommunication systems, networking systems like computer networking or communication networking, and flexible manufacturing systems. Conventional wait in models forecast the system performance on the basis of the assumption that all service facilities offer failure-free service
With reference to the above facts, here we have discussed the behavior analysis of an engineering system having three-subsystems 1, 2 and 3, connected in series, having waiting line for repair. First subsystem has two units one is key unit and another is active superfluous. Second subsystem has one key Unit and another is cold superfluous while Third subsystem consists of two units' connected in parallel arrangement. The system can completely fail due to failure of any of the subsystems [3]. In the beginning when the system starts working, the key units of subsystem 1 and 2 and both units of the subsystem 3 are fully operational. When the main units of the subsystems 1 and 2 fail, the supporting units are switched on automatically and failed units are sent for repair to repairing section [1, 2]. Here, a realistic situation is taken into the consideration that when main units and supporting units of both subsystems 1 and 2 are failed and they are sent for repair to repairing section, they are in waiting line because repairmen is busy somewhere else. In this case a wait in line is there at repair section. So the main concern of the study is that all the four units are waiting for repair [4]. Transition state diagram is shown by Figures 1. Table 1 shows the state specification of the system.
Surabhi Sengar
Sensitivity analysis with wait in line theory
Assumptions
• In the beginning the system is in good operating state.
• All Subsystems are connected in series.
• System has two states only good and failed not degraded.
• Catastrophic failure is also responsible for system failure in the study also they require constant and exponential repair. So, copula technique is used for finding probability distribution [5].
• Repair facility which follows general time distribution is there for the service of both the subsystems of unit 3 and also failure are exponential in both cases.
• For the subsystems 1 and 2 failure and repairs both are exponential.
Table 1: State specification of the system
States Description System State
S0 The system is in good working state G
S1 The system is in working state when key unit is failed. G
S2 The system is in failed state because of failure of superfluous unit. F
S3 When all four units are in waiting at repair section, system is in failed state. F
S4 The system is in working state when superfluous unit of subsystem 1 is failed. G
S5 The system is in failed state due to the failure of key unit of subsystem 1. F
S6 The system is in working condition when key unit subsystem 2 is failed. G
S7 The system is in failed state when superfluous unit of subsystem 2 is failed. F
S8 The system is in operable condition when key unit of subsystem 3 failed. G
S9 The system is in failed state from the state S8 due to failure of superfluous unit of subsystem 3. FR
S10 The system is in operable condition when superfluous unit of subsystem 3 is failed. G
S11 The system is in failed state from the state S10 due to failure of key unit of subsystem 3. FR
S12 The system is in failed state from the state S1 due to failure of subsystem 3. FR
S13 The system is in failed state from the state S6 due to failure of subsystem 3. FR
S14 System is failed state because of catastrophic failure. FR
G: Good state; F: Failed State; FR= Failed state and under repair.
Pr 2. Notations Probability
Po(t) Pr (at time t system is in good state S0)
Pi (t) t}, Pr {the system is in failed state due to the failure of the ith subsyste where i=2, 5, 7, 14. m at time
Failure rates of subsystems, where i=a1, a2, b1, b2, a, 02, CSF.
W Arrival rate of all four units of subsystems 1 and 2 to the repair section named as a1, a2, b1, b2.
H Repair rate of unit's a1, a2, b1, b2.
k (k) General repair rate of ith system in the time interval (k, k+A), where i= a, c2, (names for
the units of subsystem 3) CSF and k=v, g, r, l.
PÄt )
to
Pr (at time t there is a queue (a1, a2, b1, b2) in the maintenance section due _servicing of some other unit and all four machines are waiting for repair.
P (j, k, t )
unit
Pr (at time t system is in failed state due to the failure of jth unit when kth has been already failed, where i=9, 11. j=g, v. and k=v, g._
K1, K2
Profit cost and service cost per unit time respectively.
Let U e and U 2
Hougaard family of copula is given as ^CSF
u 2 =Фа8Е (l )
then the expression for joint probability according to Gumbel-</>CSF (l) = exp[ld+ (lGg^csF (l)f f ]
M
( Pl0 (t) \
ф^
A.
M
фсSF (l)
Фс (r )
-► Pl2 (Г, t)
^2
Sl2
Figure 1: Transition state diagram
3. Formulation of the mathematical model
The following differential equations have been obtained by considering limiting procedures and different probability constraints which satisfying the model:
d + + \ + \ + \ + \ + \ + ^CSF
P0 (t) =jv (i)P (t)di + (t>ci P (t)+^ Pw (t) +
— + Xa +Ac
5t a2 c
]$cSP (l )Pu(l, t )dl
0
œ
Pi(t ) = \Po(t ) + !*c (r P(r, t)dr
0
5t
+ W
P2(t ) = XaP(t )
5 5.
— + —+ (m + W)
5t 5i
P (t) = W[P2 (t) + P5 (t) + P6 (t) + P (t)] +
WW Y e
3 n-W
"5 , — + 4
5t ai
+ W
5
5t
"5 , 5 + ^ +W
+ W
5
5t
— + 4>c (v) + Xc
5t V^V , c2
5 5 , , N 5t 5g
Jt+$c2( g) + \
5 5
— + — + $c (v)
5t 5v c
P4(t ) = XaP0(t ) P(t ) = \P*(t )
œ
P6(t) = \Po(t) +!$c (rP(r, t)dr
0
P(t )=\P6(t )
œ
Ps(t) = XcPo(t) (g)Pn(g, v, t)dg
0
P9( g, v, t ) = o
œ
Pio(t) Po(t) + K (v)Pn(v,g,t)dv
0
Pii(v, g, t ) = o
5 5
+ + (r)
5t 5r
5 5 , , N
— + — + &C (r)
5t 5r
5 5 ,
+ ^ + (l )
5t 5l
P\2 (r, t ) = 0
P13(r, t ) = 0 Pu(h t ) = o
Boundary Conditions:
P (i = 0, t) = w[ P2 (t) + P (t) + P (t) + P (t)] p(0, t) = AcP0(t)
... (1) ... (2) ... (3)
... (4) ... (5)
... (6) ... (7) ... (8)
. (9) ... (10) ... (11) . (12)
... (13) ... (14) ... (15)
... (16) ... (17)
œ
0
6
P9(0,V,t) = \P8(t) ... (18)
Po(0,t) = AcPo(t) ... (19)
Pii(0, g, t ) = \Pw(t ) . (20)
P12(0,t) = 4P(t) ... (21)
P¡3 (0, t) = ¿CPe(t ) . (22)
Pl4 (0, t ) = ^CSFP0(t ) .•• (23)
Initial condition: P (0) = 1, otherwise zero.
Solving equations (1) through (15) by taking Laplace transform and by using initial and boundary conditions we obtained following probabilities of system is in up) and down states at time t,
Pup = P0 (s) + Pi (S ) + P4 (s) + P6 (s) + P8 (s) + Pi0 (s)
1
-[ 1
Л
Л
Л
K(s) [s + Л +Лc-ЛcSфc(s)] [s + Л] [s + ЛЬ2 +у-Л^фс (s)]
Л
Л
[s + Л + Фс, (v) - Л SфCг (s)] [s + ЛСх + фС2 (g) - ЛС1 Stcl (s)] Pdown = P2(s) + P5(s) + P7 (s) + P9(s) + Pli (s) + P\2 (s) + Pl3 (s) + Pl4 (s)
]
••• (24)
A, A,
l
+
A, A2
l
[s + щ][s + Л^ +Л -Ac(s)] K(s) [s + щ][s + Ла1 ] K(s)
+
A, \
AA d (s)
[s + Ac2 +фСх (v) -Лс$фсг (s)] K (s)
AAD (s) l
+
+
[s + щ][s + Ab2 + w-ЛcSфc (s)] K(s)
ac\ A DФЧ (s)__\_
[s + ЛСх +фСг (g) - AS»] K(s)
A AD (s) l
+
[s + Ла2 + Ac - Ac Sфс (s)] K(s) [s + ЛК + щ-Ac Sфс (s)] K(s)
^sf^f (s)
+
+
K (s)
where,
K(s)=s+A +A +A +л +Ac +A2 +acsf-щ{[
• (25)
Aal A2
A, Aa2 +-l-2-+ -
Л
+
[s + щ][s + A + AC -ЛCSФc (s)]
]DM(s)
Ab Ab
bl b2
[s + щ][s + A] [s + ЛК +щ-ЛcSфc (s)] [s + щ][s + ЛК +щ-ЛcSфc (s)]
3 A Ф(v) A фС,( g)
} -
(s+щ4 [s+a + ф (v) - A sФСг (s)] [s+A + фСг (g) - A sфcl(s)]
- ЛCSF~SФcsF (s)
M (s) = щ{[
A, Aa2
Aa, Aa, ■ +-l-2-+ ■
Л
[s + щ][s + Aa2 +Ac -ЛcSфc (s)] [s + щ][s + Aa] [s + Л^ +щ-ЛcSфc (s)]
+
+
+
+
2
+
l
l
Surabhi Sengar RT&A, No 4 (59) Sensitivity analysis with wait in line theory_Volume , December 2020
\ \ irw„\ , , V3
+--=-]D (s) + + ^--}
[s + vls + \ + w-^cStc (s)] (s + W)
1 - S M(s)
... (26)
DM(s) = ... (27)
5 + y
$csp(l) — exp[ls + (logKcsF(l))e fe] ... (28)
Also,
P up (5) + P down (5) —
5 .•• (29)
Steady state behavior of the system By Abel's lemma we have,
lim{sF (5)} — lim F (t)
.5—0 t—
In equations (24) and (25) we get,
Pup (5) — [1 + —^ + +-b-+ —+ —... (30)
() K(0)1 vK2 K K +y-K K(v) K(gy )
- 1 K K K Kb K K M6 K K M6
Pdown — —K+m (0) + +-bK-+ + +
K(°) y y y(\ +y-Kc) K(v) Kc2 (g)
K K +TC+y~K + KcsFMKcsp] . (31)
where,
M (0) = limM (s)
1 - S t (s)
... (32)
M^ = lim
5 ... (33)
Sfi (5) = ••• (34)
Sensitivity analysis:
Here we have done sensitivity analysis of the system for catastrophic failure rate and key unit of subsystem 1.
S = -t*exp (-(Lcsf + La1 + La2 + Lb1 + Lb2 + Lc1 + Lc2)*t)
5. Results and Discussion
Here reliability, availability and sensitivity analysis with respect to catastrophic failure and key unit of subsystem 1 is done for the considered system by employing Supplementary variables technique and Copula methodology. The Figure 2 shows the movement of reliability of the system against time for fixed values of failure and repair rates. From the graph we conclude that the reliability of the system reduces hastily with passage of time because of waiting line for repair in the repair section.
Figure 3 talks about the availability of the system which says that the availability reduces approximately in a constant manner as time increases.
The sensitivities of the system reliability R (t) with respect to system parameters like catastrophic failure and key unit of subsystem 1 are shown in figures-4 and 5. It can easily be observed that there is very negligible impact of both the parameters on system sensitivity.
Surabhi Sengar
SENSITIVITY ANALYSIS WITH WAIT IN LINE THEORY
Figure 2: Reliability against time
Figure 3: Availability against time
Figure 4: Sensitivity Analysis for catastrophic failure
Figure 5: Sensitivity Analysis for key unit of system1
Surabhi Sengar
SENSITIVITY ANALYSIS WITH WAIT IN LINE THEORY
References
[1] Barlow, R. E. and Proschan, F. (1975). Statistical Theory of Reliability and Life Testing: Probability models, New York: Holt, Rinehart and Winston.
[2] Brown, M. and Proschan, F. (1983), Imperfect repair. Journal of Applied Probability: 20: 851859.
[3] Liebowitz, B. R. (1966), Reliability considerations for a two-element redundant system with generalized repair times", Operation Research: 14, 233-241.
[4] Surabhi and Singh (2014), Reliability Analysis of an Engine Assembly Process of Automobiles with Inspection Facility. Mathematical Theory and Modelling: 4(6):153-164.
[5] Sengar, Surabhi and Singh, S. B. (2012), Operational Behaviour and Reliability Measures of a Viscose Staple Fibre Plant Including Deliberate Failures, International journal of reliability and applications: Vol. 13, No.1, pp.1-17.
Received: August 09, 2020 Accepted: Oktober 12, 2020
