BEHAVIOR ANALYSIS PRESENTED SYSYEM WITH FAILURE AND MAINTENANCE RATE WITH USING DEEP LEARNING ALGORITHMS Текст научной статьи по специальности «Математика»

B. A. PRESENTED SYSTEM WITH FAILURE AND MAINTENANCE RATE , 7 , no

Volume 19, September 2024 WITH USING D.L.A. F

BEHAVIOR ANALYSIS PRESENTED SYSYEM WITH FAILURE AND MAINTENANCE RATE WITH USING DEEP LEARNING ALGORITHMS

Shakuntla Singla Shilpa Rani 2*, Diksha Mangla 3, Umar Muhammad Modibbo

1,2,3 Department of Mathematics and Humanities, MMEC, Maharishi Markandeshwar (Deemed to be University), Mullana, Ambala, India "Department of Statistics and Operations Research ,School of Physical Sciences ,Modibbo Adama

University of Technology ,YOLA-NIGERIA

1 shakus25@gmail.com, 2*gargshilpa46@gmail.com, 3dikshamangla1995@gmail.com,

4umarmodibbo@mau.edu.ng

4

Abstract

The paper discusses the behavioral analysis and dependability of a three-unit system utilizing RPGT for system parameters. Since all three units P, Q and R include parallel subcomponents, in the event that one of them fails, the system continues to operate although at a reduced capacity, but it is not profitable to run the system when two units are in reduced state hence considered failed state. The rates of failures are exponentially distributed, but the rates of repair are generalized, independent, and differ based on the operational unit. Fuzzy concept is used to declare/ determine whether the system is in failed/ reduced/ failed state. Graphs and tables are drawn to compare failure/repair effect on the parameters values. The system parameters are modelled using Regenerative Point graphical Technique (RPGT) and optimized using Deep learning methods such as Adam, SGD, RMS prop. The results of the optimization may be used to validate and challenge existing models and assumptions about the systems.

Keywords: MTSF, RPGT, Deep learning, Adaptive Moment Estimation, Stochastic Gradient Descent, RMS prop

I. Introduction

The paper analyzes system parameter reliability and behavioral analysis of three units using deep learning. Because the three units are all parallel subcomponents, the system may continue to function at a reduced capacity in the event that one or more fail. However, when two units work at a reduced capacity, the system is not profitable and is therefore deemed to be in a failed condition. The rates of failures exhibit exponential distribution, but the rates of repair are general, autonomous, and variable between different operational units. Units have varying capacity. The repairs are flawless. When two units are in a reduced state or any one unit is failing, the system is down. As in the system three are three units, P, Q, R all of which have parallel sub components initially when all the three units are good the system. The state So [PQR], so upon their partial failures to states P, <2, P the failure rates for which are Ai, A3, As the system enters the reduced states Si [PQR], S3 [PQR], Se [PQP].

There is a single repairman available in the system that can repair all the three type of units from partial as well as from full failed states so from partially failed states upon repair of partial failed units

the system reenters the state so from state Si at repair rate wi of unit A, at a repair rate A3 of unit Q at a rate W5 of unit R, from the partially states Si, S3, S6 these units may fail further to full/ partial states. The system is considered to be in a failed state if one unit is in a failed state or if two or more units are in a reduced the state Fuzzy concept is used to declare/ determine whether the system is in failed/ reduced/ failed state. In state Si if unit P fails fully then the system enters the failure state S2 [pQR]. if the unit Bar R fail then the system enters the failed states S9, or S7. In reduced state S3 if units P, Q, R fail partially/ fully the system enters the states S9, S4, S5 respectively from the failed state S9 and S4 upon repair of the units the system enters the state S3 from partially reduced state S6 upon failure of units P, Q, R the system enters the failed states S7, S5, Ss respectively from these failed states as the repairman is free to repair the failed units, so as repair of these units the system again enters the state S6 when more than one unit fail, the system is in failed state in these states the priority order of repairs is P > Q > R >. Taking the transition failure and repair rates the system may be stable in the states Si (0 < I < 9) as shown in the figure i.

Hsieh et al. [1] has discussed Reliability of two dimensions consecutive lower bounds system. John et al. [2] has study reliability multi hardware and software system multi-hardware-software system interaction failure less attention. Kumar [3] study investigated help of mathematical modelling find out of reliability. Kumar et al. [4] has study minimizing the risk of machine failure urea fertilizer plant. Kim. H.K et al. [5] discussed demonstrate transparent or flexible capacitive designed multi touch screen. Khan. M. F et al. [6] has study three stage mathematical formulation computational procedures, numerical has two distinct approaches. Singla et al. [7] have discussed comparison of availability of a pipe and sub system of independent failure. Raghav et al. [8] has study maximize the availability and minimize the cost of function with help of PSO. Singha. A. K. [9] has done the study of x -rays and computed tomography scans images of corona virus. Kumari et al. [10] have discussed with help of RPGT profit analysis of thresher plant three sub system blower, concave, hopper more result. Singla et al. [11] has study polytube manufacturing plant solve by using of RK method. Saliva et al. [12] has study failure probability comparison with usual 1- dimension model. Singla et al. [13] with mathematical model find availability under the reduces capacity using chapman Kolmogorov method. Singla et al. [14] all three units of different capacities in working in parallel in which two or three unit in full working. Singla et al. [15] with help of GA mathematical model depend availability with working time. Singla et al. [16] has study3 out of 4 good system optimizations modelled and analysis. An analysis on reliability parameters using an algorithm ABC, has been discussed by Ahmadini et al. [17]. Singla et al. [18] studied the two unit repairable system under the concept of fuzzy linguistic and discussed the overall availability.

The total of five sections are included in this study. The 2nd section includes model description with assumption used and different mathematical values used in the study. The methodology is covered in section 3rd. The results and conclusion is studied in section 4th and 5th respectively.

II. Assumption, Notation and Transformation Diagram

• The repair procedure arises soon after a unit flops.

• Repaired unit is as if a new one.

• Failure/repair rates of units are exponential.

• Server facility is 24x7 hours.

• So = PQR, S1 = PQR, S2 = pQR, S3 = PQR, S4 = PqR,

• S5 = PQR, S6 = PQP, S7 = PQP, Ss = PQr, S9 = PQR

Figure 1: Transformation Diagram

2.1 Probability Density function (qij(t))

The probability density function associated with the transformation diagram from different states to other is given below.

qo,6= ^e-»^^3^ 9i,o = ^e-(A5+A2+A3+wi)t 9I,2 = ^e-(A5+A2+A3+wi)t

9= ^e-(A5+^2+^3+wi)t

W,1= w2e-W2t

9s, 9= ^e-(w3+Ai+A5+^4)t ^4,3= W4e-W4t

q5,6= W3e-W3t

= ^e-(A3+A6+Ai+Ws)t

= A1e-(A3+A6+Ai+W5)t q6,8= 26e-(A3+^6+^i+w5)t

97,6= Wii ^8,6=

^,3= Wie-wit

= w e-wit

fi= wfie-W6Î

2.2 Cumulative probability density

Cumulative probability density functions in moving from state 'i' to state 'j' by taking Laplace Transforms of above function for infinite time interval is given as under. Pij = q*i,j(t), i.e. P(U=

P0,3 = À3/(À1+À5+À3) P0,6 = W(À1+À5+À3)

P1,0= Wl/(À5+À2+À3+Wl) P1,2= ^2/(À5+À2+À3+Wl)

P1,7= ^5/(À5+À2+Wl+À3)

P1,9= ^3/(À5+À2+À3+Wl)

P2,1= W2/W2 = l

P3,0= W3/(W3+Àl+À5+À4) P3,4= W (W3+Àl+À5+À4) P3,5= W (W3+Àl+À5+À4) P3,9= W (W3+Àl+À5+À4) P4,3= W4/W4 = l Ps,6 = W3/W3 = l P6,0= W5/(À3+À6+Àl+W5)

P6,5= ^3/(À3+À6+Àl+W5) P6,7= ^l/(^3+À6+Àl+W5) P6,8= W(À3+À6+Àl+W5) P7,6= Wl/Wl = l P8,6= W6/W6 = l p93= Wl/Wl = l

2.3 Mean Sojourn Transition rate (Ri(t)) for different states are fl(t)=

e-Wi+A5+A3)t

^(t) = e-(A5+A2+A3 + w1)t e-w2t

^(t) = e-(w3+A1 + A5+A4)t

fi(t)= g-W4t = g-W3t fl(t)=

e-(A3+A6+Ai + W5)t

6

fi(t)= g-Wit fi(t)= g-W6t fi(t)= g-Wit

2.4 Mean Sojourn Time (|ji=Ri*(0)) for different states are

[0 = l/(Àl+À5+À3) [l = l/(À5+À2+À3+Wl)

[2 = l / W2

[3 = l/(W3+Àl+À5+À4) [4 = l/W4 [5 = l/W3

[6 = 1/(À3+À6+Àl+W5)

[7 = l/Wl

[8 = l/W6 [9 = l/Wl

2.5 Transition Probability

V0,0 = 1 (Verified) V0,1 = (0,1)/1-(1,2,1) = p0,1/ (1-p1,2p2,1)

= (À1/À5+À2+À3+w1)/(À1+À5+À3) (À5+À3+w1)

V0,2 = (0,1,2)/1-(1,2,1) = (p0,1p1,2/1-p1,2p2,1) = À^/(À1+À5+À3) (À5+À3+w1)

V0,3 = (0,3)/1-(3,4,3)1-(3,9,3) +(0,1,9,3)/1-(1,2,1) {1-(9,3,9)/1-(3,4,3)}1-(3,4,3)

= p0,3/(1-p3,4p4,3) (1-p3,9p9,3) +p0,1p1,9p9,3/(1-p1,2p2,1) {(1-p3,4p4,3-p9,3p3,9)/(1-p3,4p4,3)} (1-p3,4p4,3)

= À3(À1+À4+À5+w3)2/(À1+À5+w3) (À1+À3+À5) {(À3À5+À3w3+À52+À5w3+w1w3+À1w3+À1À4 +À1À5)/(w3+À4+À5) (À3+À5+w7) (À5+w3)}

V0,4 = (0,3,4)/1-(3,4,3)1-(3,9,3) +(0,1,9,3,4)/1-(1,2,1) {1-(9,3,9)/1-(3,4,3)}1-(3,4,3)

= p0,3p3,4/(1-p3,4p4,3) (1-p3,9p9,3) +p0,1p1,9p9,3p3,4/(1-p1,2p2,1)(1-p3,4p4,3-p9,3p3,9) = À3À4/(À1+À5+À3){(w3+À1+À5+À4)(À5+À3+wr)(w3+À5)+À1(w3+À1+À5)(w3+À4+À5)/ (w3+À1+À5)(w3+À4+À5)(À5+À3+w1)(À5+w3)}

III. Methodology

3.1 MTSF(To)

Initial state '0', before joining down state are: 'i' = 0,1,3,6 taking initial state %' = '0'

MTSF (T0) =

Zu

sr(sff)

{pr(%- i

1 Km^i

1-V

1-Zs

sr(sff)

pr( % )

I nrn2^{1-VH

2m2 J

3.2 Availability of the System

The regenerative states at which the system is available are 'j' = 0,1,3,6 and the regenerative states are 'i' = 0 to 9 taking '£,' = '0' the availability for which the system is available is given by

A0=

Z

j,sr

{pr(%sr^j)}fj,W

1-V

m im 1

Zi,s

{pr(%sr^i)}^1

m 2 m 2

A0 = E^,/},^] + fc^,/},^] 3.3 Busy Period of the Server

The states where the server is busy for doing some job are 'i' = 1 to 9, taking '£,' = '0', using RPGT busy period is given as

B0=

Zj

j,sr

{pr(%sr^j)},nj nmi,ç{1-Vm im ,

Zi

Sr

{pr(%sr^i)}^1

{1-Vm

m 2m 2

B0 = [Z^-.n,-] -3.4 Expected Fractional Number of repairman's Visits

States 1, 3, and 6 are the regeneration states that the repairman visits first to complete this task. The repairman's visitation count is determined by

V0=

Z

{pr(%sr^j)}

j,sr

nk.

1-V

Zi,

{pr(%sr^i)}^1

3.5 Dataset: Behavior analysis Using Deep Learning Algorithms

To perform optimization using deep learning, you would need a dataset that contains information on the input parameters and the system's output [5, 6]. The input parameters could include factors such as the system's design, operating conditions, and maintenance schedule. The output could include metrics such as system availability, MTSF, and busy period.

Table 1: Parameter

W(w1,w2,— —,wn) X(X1, X2,... ....Xn) S(s,s2,— —sn) P

(0-.100) (0-.100) (0-100) (0-.68)

Table 2: Performance of model

Model MTSF Expected Busy Period Availability

Number of

Inspections by

the repair man

Adam 0.928 .9067 0.8021 .9348

Sgd .9128 .9007 .8123 .9128

MS prop .9013 0.87l0 .8101 0.9234

• Collection of data: Gather a dataset that contains information on the input parameters and the system's output. The input parameters could include factors such as the system's design, operating conditions, and maintenance schedule. The output could include metrics such as system availability, downtime, and failure rate in table 1 and table 2.

• Preprocess data: Clean and preprocess the dataset, splitting it into training, validation, and test sets.

• Train the model: Use a deep learning algorithm, such as a neural network, to model the connection among the input parameters and the output. Train the model using the training set and validate it using the set of values in table 1. You could use techniques such as early stopping and regularization to prevent over fitting.

• Appraise the model: After the model is proficient, appraise its performance by means of test set. Estimate metrics such as busy period.

• Perform sensitivity analysis: Using the trained model, vary the values of one parameter at a time while keeping the others constant. Record the effect on the system's output. Repeat this process for each input parameter, recording the impact of each parameter on the system's output. The output could include metrics such as system availability, MTSF, and busy period.

• Once you have a dataset, you could use a deep learning algorithm to model the relationship among the input parameters and the production. One approach could be to use a neural network, which can learn complex relationships between inputs and outputs. To perform optimization using a neural network, you could first train the network on the dataset, using a portion of the data for training and another portion for validation.

Optimization of a repairable system undertaken for analysis using deep learning typically involves the following steps:

• Data collection: Collect data on the input parameters and output metrics of the system. The input parameters could include factors such as the system's design, operating conditions, and maintenance schedule. The output metrics could include measures such as system availability, MTSF, and busy period in show table 2 included.

• Data preprocessing: Clean and preprocess the data, splitting it into training, validation, and test sets. Normalize the input variables to ensure that they are on the same scale.

• Model selection: Choose appropriate deep learning optimization techniques (Adam, SGD, RMS prop) for the sensitivity analysis. Some options contain feed forward neural systems, convolutional neural systems, and regular neural networks. Consider influences such as the size of the dataset, the difficulty of the input-output connection, and the computational capitals existing.

• Model training: Train the selected model on the training data. Use techniques such as stochastic gradient descent and back propagation to minimize the bust time. Monitor the performance of the model on the validation data, and adjust the hyper parameters as needed.

• Model evaluation: Evaluate the trained model on the test data. Calculate metrics such as mean absolute bust time and mean squared error to assess the model's performance of deep learning optimization in show table 1 and table 2.

IV. Results and discussion

The results and discussion of a Optimization of undertaken repairable system parameters using deep learning will depend on the specific system and dataset analyzed. However, here are general insights that could be gained from such an analysis:

• Identification of critical system parameters: The optimization could reveal which input parameters require the greatest effect on the output metric of interest. For example, it could show that system availability is most optimization to the frequency of care or the quality of the components used in the organization.

• Understanding of the non-linear relationship amongst input strictures and output metrics: The deep learning model used in the analysis can capture non-linear relationships amongst input restrictions and output metrics, which could not detect using traditional statistical methods. The optimization can provide insights into the shape and magnitude of these relationships.

• Validation of existing models and assumptions: The optimization's outcomes were used to support or refute preexisting theories and hypotheses about the system. The research may reveal, for instance, that a particular parameter significantly affects system performance more than previously believed.

Prediction of system behavior under different scenarios: The deep learning model applied to predict system performance under different setups, such as vagaries in operating conditions or maintenance schedules. This can support decision-makers assess the impact of changed strategies and style informed verdicts. Overall, Behavior analysis presented sysyem with failure and maintenance rate Using Deep Learning Methods can provide valuable insights into the factors

that affect system performance, (MTSF), Expected Fractional Number of repairman's Visits Busy Period and Availability of the System are shown in figure 2, 3, 4 and 5. valuable insights into the factors that affect system performance, (MTSF), Expected Fractional Number of repairman's Visits Busy Period and Availability of the System are shown in figure 2, 3, 4 and 5.

Figure 2: comparing between models according to MTSF

Figure 3: comparing between models according to Availability

Figure 4: comparing between models according to busy period

idam SGD RMSprop

Model

Figure 5: comparing between models according to Expected Fractional Number of repairman's Visit

V. Conclusion

In conclusion, the Behavior analysis presented sysyem with failure and maintenance rate with using deep learning as the optimization tool has provided valuable insights into the dynamics and performance of the system across various operational scenarios. Through comprehensive experimentation and simulation, we have gained a deeper understanding of the system's response to different configurations, maintenance policies, and environmental factors. The results demonstrate the effectiveness of the deep learning guided approach in optimizing maintenance schedules and resource allocation to maximize system reliability and availability. By iteratively refining maintenance strategies, significant improvements in key performance metrics such as mean time to failure (MTTF), mean time to repair (MTTR), and overall system uptime have been achieved. This highlights the potential of deep learning to adaptively optimize complex systems in dynamic environments.

References

[1] Hsieh, Y., C. and Chen, T.C. (2004). Reliability lower bounds for two-dimensional consecutive-k-out-of-n: F systems. Computers & Operations Research 31(8):1259-72

[2] John, Y. M., Sanusi, A., Yusuf, I., and Modibbo, U., M. (2023). Reliability Analysis of MultiHardware-Software System with Failure Interaction. Journal of Computational and Cognitive Engineering, 2(1), 38-46.

[3] Kumar, A. (2020). Reliability And Sensitivity Analysis of Linear Consecutive 2-out-of- 4: F System. European Journal of Molecular & Clinical Medicine, 7(07), 2020

[4] Kumar, A. (2022). Sensitivity Analysis of Urea Fertilizer Plant. Journal of Reliability Theory and Applications Volume 17, RT&A No. 2 (68)

[5] Kim, H.K., Lee, S. and Yun, K.S. (2011). Capacitive tactile sensor array for touch screen application. Sensors and Actuators A: Physical 165:2-7.

[6] Khan, M. F., Modibbo, U. M., Ahmad, N., and Ali, I. (2022). Nonlinear optimization in bi-level selective e maintenance allocation problem. Journal of King Saud University-Science, 34(4), 101933.

[7] Singla, S., Lal, A.K., and Bhatia,S.S (2011). Comparative study of the subsystems subjected to independent and simultaneous failure. Eksploatacja I Niezawodnosc-Maintenance and Reliability, 4, 63-71.

[8] Raghav, Y. S., Varshney, R., Modibbo, U. M., Ahmadini, A. A. H., and Ali, I. (2022). Estimation and optimization for system availability under preventive maintenance. IEEE Access, 10, 94337-94353.

[9] Singha, A. K., Pathak, N., Sharma, N., Gandhar, A., Urooj, S., Zubair, S., and Nagalaxmi, G. (2022). An Experimental Approach to Diagnose Covid-19 Using Optimized CNN. Intelligent Automation & Soft Computing, 34(2).

[10] Singla, S., and Kumari, S. (2022). Behavior and profit analysis of a thresher plant under steady state B International Journal of System Assurance Engineering and Management 13, 166-171.

[11] Singla, S., Lal, A. K., and Bhatia, S.S. (2021). Reliability analysis of poly tube industry using supplementary variable Technique Applied Mathematics and Computation 281, 3981-3992.

[12] Salvia, A. A. and Lasher, W.C. (1990). 2-dimensional consecutive-k-out-of-n: F models. IEEE Transactions on Reliability 39(3):382-5

[13] Singla, S., Modibbo, U. M., Mijinyawa, M., Malik, S., Verma, S., and Khurana, P. (2022). Mathematical Model for Analysing Availability of Threshing Combine Machine Under Reduced Capacity. Yugoslav Journal of Operations Research, 32(4), 425-437.

[14] Singla, S., Rani, S., Modibbo, U. M. and Ali, I. (2023). Optimization of System Parameters of 2:3 Good Serial System using Deep Learning. Reliability Theory and Applications,670-679.

[15] Singla, S., Mangla, D., Panwar, P. and Taj, S. Z. (2024). Reliability Optimization of a Degraded System under Preventive Maintenance using Genetic Algorithm. Journal of Mechanics of Continua and MathematicalSciences,1-14.

[16] Singla, S., Rani, S. (2023). Performance optimization of 3:4 good system. IEEE second international conference 979-8-3503-4383-0.

[17] Ahmadini, A.A.H., Singla, S., Mangla, D., Modibbo, U.M., Rani, S. (2024). Reliability Assessment and Profit Optimization of Multi-unit Mixed Configured System using ABC Algorithm under Preventive Maintenance. IEEE Access.

[18] Singla, S., Mangla, D., Dhawan, P., Ram, G. (2024). Reliability analysis of a two-unit repairable system using fuzzy linguistic approach. Applications of fuzzy theory in applied sciences and computer applications,45-57.