A System Dynamics Model for Transient Availability Modeling of Repairable Redundant Systems

Transcription

1 International Journal of Performability Engineering Vol., No. 3, May 05, pp RAMS Conultant Printed in India A Sytem Dynamic Model for Tranient Availability Modeling of Repairable Redundant Sytem M. S. Rao and Vallayil N. A. Naikan GMR Intitute of Technology, Rajam, INDIA Reliability Engineering Centre, Indian Intitute of Technology, Kharagpur, INDIA Received on April, 04, revied on Augut 04, 04 Abtract: In thi paper, a novel Markov ytem dynamic MSD imulation frame work for the availability modeling of a repairable ytem i developed. The propoed method combine the Markov approach with ytem dynamic imulation approach for availability modeling. Thi approach will have the advantage of both Markov a well a Sytem dynamic methodologie. Initially, the propoed framework i illutrated for a two component active redundant ytem with repair with contant failure and repair rate. There after, tranient availability modeling with time varying failure and repair rate i performed for the ame ytem. It i worth mentioning here that finding the teady tate condition and tranient availability modeling are extremely difficult or impoible uing traditional approache. The reult of MSD imulation are compared with that obtained by traditional Markov method for validation. Thi ha hown that MSD imulation i a good alternative to the traditional mathematically intenive method. Keyword: MSD imulation, tranient availability modeling, Markov approach, repairable ytem, ytem dynamic.. Introduction Tranient availability modeling i an important proce ued to evaluate and improve the effectivene of any indutrial plant, where mot of the machine are repairable ytem. The complexity of the modern engineering ytem beide the need for realitic conideration when modeling their availability render analytical method very difficult to be ued. There i extenive literature on availability characteritic of repairable ytem with two or three component under varying aumption on the failure and repair []-[6]. In thee paper, author ued either Laplace tranform method or Lagrange method to olve Chapman Kolmogorov differential equation aociated with a particular problem. It ha been oberved that thee method involve complex computation and it i very difficult to compute availability of the ytem by thee method. Among everal available method, Markov method i widely ued for reliability/availability modeling. Markov chain model provide accurate long run availability and failure characterization calculation and can ometime be olved analytically, but are hard to formulate and involve high computational effort particularly a the number of tate grow large. The olution procedure of thee model i alo mathematically intenive. The exiting literature how that if the failure time and/or the repair time ditribution are not exponential, the analytical expreion for the availability become difficult. Simulation ha been ued a a powerful tool for modeling and analyi of ytem availability. It i ued to repreent the dynamic behavior of ytem in the mot realitic ene. Therefore, a imulation technique i required for etimating the availability [7-8]. Many reearcher have been earching for alternate methodologie for more practical and realitic availability modeling. The preent work propoe a hybrid approach *Correponding author riniva.m@gmrit.org 03

2 04 M. S. Rao and Vallayil N. A. Naikan called a Markov Sytem Dynamic MSD approach for availability modeling. The propoed methodology i illutrated for a two component active redundant ytem with repair. However thi methodology can be extended eaily for more complex multi tate ytem. The ret of thi paper i organized a follow. Section propoe a Markov ytem dynamic approach for availability modeling of a two component active redundant ytem with repair with contant failure and repair rate. Section 3 preent the reult obtained from availability modeling with contant failure and repair rate. Section4 decribe the tranient availability modeling uing propoed approach. Finally, Section5 conclude the paper.. Propoed Model In order to model the availability of redundant ytem with repair, the required aumption hould be mentioned. The detail of the ytem under tudy and it availability modeling aumption are tated with an illutrated example. Two eparate approache are contructed to model the availability of the ytem,. The firt one i the Markov approach and the econd one i the propoed MSD approach. Conequently, to mitigate the limitation of the Markov approach [9], the MSD model ha imulated and compared the reult with contant failure and repair rate for availability modeling. Thereafter, tranient availability modeling i performed on the ame ytem, conidering time varying failure and repair rate.. Conventional Markov Approach A two unit active redundant ytem i conidered to decribe the approach. A per the conventional Markov approach, the following aumption are ued to analyze the ytem under conideration and for the contruction of it tate tranition diagram. The two component active redundant ytem can be repaired only after both unit have failed. Only one unit can be repaired at a time. The unit have contant failure and repair rate. The tate tranition diagram for Markov analyi i hown in Figure. In thi diagram, tate indicate that both the unit are in the operating mode, tate indicate that the redundant unit i in operating mode and the active unit in the failed tate, the tate 3 repreent a failure of the redundant unit and tate 4 indicate that both unit are failed and thereafter ent for the repair. After the contruction of tate tranition diagram, the remaining tage of traditional Markov analyi conit of computation of the tate probabilitie by olving the reultant Chapman Kolmogorov differential equation. Thi i highly mathematically intenive exercie the intereted reader can refer [3] for thi olution for availability modeling of the ytem. After the rigorou mathematical treatment uch a by taking Laplace tranform both ide to the reultant Chapman Kolmogorov differential equation and thereafter olving by ue of Cramer rule, the following final equation are obtained which can be ued to compute the tate probabilitie.

3 A Sytem Dynamic Model for Tranient Availability Modeling of Repairable Redundant Sytem 05 [ ] [ ] _ = p [ ] [ ] _ = p [ ] [ ] 3 _ = p 3 [ ] [ ] 4 _ = p 4 By taking the invere Laplace tranform of the above equation,,,, 4 3 t P and t P t P t P value can be calculated. But, tranforming back to the time domain by taking invere Laplace tranform both ide to the above equation i very difficult and tediou a decribed by the everal author [-6]. Due to thi difficulty, they performed teady tate olution for thi type of ytem auming that the teady tate i reached at time infinity. Generally, the teady tate availability analyi of ytem i evaluated by auming time equal to infinity in thee Kolmogorov ytem of differential equation. However, thi analyi doe not give the exact time at which a ytem reache it teady tate. In many practical ituation we require to find the time at which our ytem reache teady tate condition for planning maintenance activitie. The propoed MSD method i capable of finding thi teady tate point and calculate the time dependent availability At very eaily. Thi approach i dicued in the following ection. Figure : Sytem State Tranition Diagram

4 06 M. S. Rao and Vallayil N. A. Naikan. MSD Approach The origin of the propoed approach: Simulation ha been ued a an approximation tool to remedy the limitation of analytical Markov chain. It ha been proved by the author [9]-[0] that the tationary, continuou time Markov model are algebraically equivalent to linear ytem dynamic model. The intereted reader can refer the quoted reference. From thi, a ytem dynamic repreentation of Markov model open up the poibility of numerical olution and of avoiding the tedium of analytical olution. The approach of ytem dynamic wa created and developed in the late 950 by a group of reearcher led by Forreter at the Maachuett Intitute of Technology MIT, Cambridge, MA []. It i a methodology which build on information feedback theory, which provide ymbol for mapping ytem in term of diagram and equation, and a programming language for conducting computer imulation. Another advantage of ytem dynamic modeling i that it i eay to experiment with alternative value of parameter. Finally, the teady tate olution for thee problem can be obtained eaily by the inpection of the flow diagram []-[3]. The preent work propoe MSD approach for availability modeling of a two component active redundant ytem with repair a decribed in the following ection..3 Availability Modeling of Redundant Sytem with Repair uing MSD Approach The propoed modeling and analyi methodology [9-0] conit the following tage. Stage : Contruct a State Tranition Diagram In thi tage, the tate tranition diagram of the conidered ytem a per the Markov approach given in ection. i taken a input for the propoed MSD imulation modeling. Thi propoed modeling ha performed in the following tage. Stage : Data Collection. The propoed MSD methodology tart after identification of the ytem tate a per the tate tranition diagram. Conider a two component active redundant ytem that can be repaired only after both unit have failed. Only one unit can then be repaired at a time. The unit have contant failure rate of per month and 4 per month repectively. The mean repair time to complete both repair MTTR i of 0 day /3 month. It i important to mention here that the failure and repair rate of the ytem need to be etablihed from the failure and maintenance data by parameter etimation technique a uual. Now it i required to olve for the ytem availability. Thi illutrative example i a imilar to the ytem under tudy i.e., a two component active redundant ytem with repair. The data regarding the contant failure and repair rate of thi illutrative example ha been ued in the propoed MSD imulation model. Stage 3: Building the Comprehenive MSD Simulation Model. The next tage in the modeling proce i to convert the tate tranition diagram of the ytem in to the rate and level diagram. The tate tranition diagram Figure of the two component active redundant ytem with repair i now converted into a comprehenive MSD model. Thi i preented in Figure.

5 A Sytem Dynamic Model for Tranient Availability Modeling of Repairable Redundant Sytem 07 Figure : A Comprehenive Markov Sytem Dynamic Model In the availability model depicted in Figure, the four tate of the two component active redundant ytem with repair are indicated with level variable P, P, P 3, P 4 and the tate tranition rate are indicated with rate variable R, R 3, R 34, R 4, R 4 with the correponding tranition rate,,, indicate failure rate of the firt and econd unit repectively and indicate repair rate. The initial value of ytem availability a indicated at the level variable P i aumed a unity. The effect of failure rate i to decreae the ytem availability and that of repair rate i to increae the ame. The tate probabilitie can be computed for all the four tate by running thi model over a period of time and oberving the tate variable. The rate variable are influenced by the repective auxiliary variable, i.e., failure rate and repair rate of the primary unit and the redundant unit along the entire miion or operating time. In thi example, it can be een that the ytem i available for operation when it i in any of the tate,, or 3. The tate 4 i the ytem down tate. That mean the um of tate probabilitie P, P, P 3 give the value of ytem availability at any time. Stage 4: MSD Simulation: The next tage of MSD approach i to imulate the comprehenive MSD model of the ytem which i developed in the previou tage. Thi imulation give the required tate probabilitie of the ytem for it availability modeling. The following algorithm explain the propoed Markov Sytem dynamic imulation tep. Propoed Algorithm: Step: The value of tranition rate,, and the time interval dt are taken a input. Alo the total time T i taken a input, i.e., the time for which the ytem ha to be imulated. It i important to note that maller the value of dt, more accurate will be the computed value of the tate probabilitie. Step: Initially et P equal to one. Step3: Set P i equal to zero for i=, 3, 4. Step4: A conditional loop i formed with the condition, t<t.

6 08 M. S. Rao and Vallayil N. A. Naikan Step5: In each execution of the loop, the time i increaed by dt, i.e., t =t dt. So, the loop will continue till time T with each tep taken at time difference dt. Step6: A aumed P initially ha probability unity, and the ytem fail when it become zero. Run a conditional loop a long a the condition i.e., the probability of P >0 i atified. Step7: Within the loop all the rate variable are calculated. The probabilitie of the tate are calculated firt and the outflow rate are calculated according to the logic a follow. P t = P t-dt R 4 -R -R 3 * dt P t = P t-dt R --R 4 * dt P 3 t = P 3 t-dt R 3 -R 34 * dt P 4 t = P 4 t-dt R 34 R 4 -R 4 * dt Step8: Then all the required value are diplayed and the required graph can be drawn by uing thee value to tudy the dynamic behavior. Stage 5: The Model Experimentation By implementing thi algorithm, the model experimentation i performed. The ytem ha been imulated by imultaneouly conidering all the four tate. The imulation reult clearly indicate that the probability of operating tate of the ytem decreae and reache it teady tate with increae in time due to primary and redundant unit failure. And alo the ytem will be in operating tate if one of it unit i in operating tate. The ytem reache it failed tate when both of it unit fail. And the um of P, P, and P 3 tate probabilitie will give the ytem availability. 3. Reult obtained from Availability Modeling with Contant Failure and Repair Rate So far in the literature, teady tate availability modeling of ytem i evaluated by auming time equal to infinity in the Kolmogorov ytem of differential equation. However, thi analyi doe not give the exact time at which a ytem reache it teady tate. In many practical ituation we require to find the time at which our ytem reache teady tate condition for planning maintenance activitie. The propoed MSD method in thi work i capable of finding thi teady tate point very eaily. The propoed approach for the illutrative example how thi teady tate point, interval and teady tate availabilitie along with the dynamic behavior of the ytem. The reult obtained are preented in Table which clearly indicate that the imulated reult very cloely match with computed value. It i found that the ytem reache it teady tate at.45 month and it teady tate availability i It i clear from the reult that the propoed Markov ytem dynamic MSD modeling frame work can be ued a an alternative approach to analyze the availability of maintainable ytem. Moreover, the MSD model developed in thi paper can alo be ued when the failure and repair rate are time varying function.

7 A Sytem Dynamic Model for Tranient Availability Modeling of Repairable Redundant Sytem 09 Table: Sytem Availability by MSD and Markov Approach with Contant Failure and Repair Rate Time month Availability by MSD Approach A 0 Availability by Markov Approach A m Tranient Availability Modeling with Time Varying Failure and Repair Rate The MSD model a hown in the Figure ha been imulated by auming a hypothetical cae in which the time to failure i following Weibull ditribution with parameter β =., θ = 00, β = 3.9, θ = 90 and time to repair are following Weibull ditribution with parameter β 3 =.9, θ3 = 50 a the input. The imulation i continued till the ytem reache a teady tate. The reult of imulation are preented in Table which how the time dependent probabilitie of the ytem at different tate. It i found that the ytem tranient availability declining from to 0 within 0 to day with the above time varying failure and repair rate a input value. It i found that the ytem reache it teady tate at day and it teady tate availability i In conventional method it i extremely difficult to obtain uch time dependent tate probabilitie and to find out when the ytem reache it teady tate, when the failure and repair rate are time varying function. Table : Sytem State Probability and Tranient Availability with Time Varying Failure and Repair Rate Tranient Time State Probability availability from day P t P t P 3 t P 4 t MSD approach * * Steady tate availability

8 0 M. S. Rao and Vallayil N. A. Naikan By conidering thi type of failure and repair rate, the enitivity analyi for the ytem availability can be eaily performed by uing thi MSD approach. It i worth mentioning here that the MSD modeling will be much eaier compared to the traditional approache to tudy the availability of complex ytem. Switchover from conventional method to MSD imulation eem to be the mot promiing availability modeling trategy 5. Concluion In thi paper, a hybrid approach called a Markov Sytem Dynamic MSD approach ha been propoed for availability modeling. The propoed framework i illutrated for a two component active redundant ytem with repair. The imulation reult obtained when compared with that by the traditional Markov analyi for contant failure and repair rate clearly validate the MSD approach a an alternative approach for availability modeling. The procedure for the development of the MSD approach for thi ytem i explained and the model i run to oberve all of it tate. It i alo hown that the MSD approach clearly indicate the time at which the ytem reache it teady tate. Moreover, tranient availability modeling with time varying failure and repair rate i performed in thi paper. It i worth mentioning here that finding the teady tate condition and tranient availability modeling are extremely difficult or impoible uing traditional approache. However, the propoed MSD approach can calculate the point, interval availabilitie with time varying failure and repair rate or for any uer defined ditribution. Reference [] Liu, L. M., and D. H. Shi. Availability Analyi of a Two-Unit Serie Sytem with a Priority Shut off Rule. Naval Reearch Logitic, 996; 437: [] Mathew, A. G., S. M. Rizwan, M. C. Majumder, K. P. Ramachandran, and G. Taneja. Reliability Analyi of an Identical Two-Unit Parallel CC Plant Sytem Operative with Full Intalled Capacity. International Journal of Performability Engineering, 0; 7: [3] Birolini, A. Quality and Reliability of Technical Sytem, Theory, Practice, Management. Springer Verlag, Berlin, 997. [4] Gupta, P., A.K. Lal, R.K. Sharma, and J. Singh. Numerical Analyi of Reliability and Availability of the Serial Procee in Butter Oil Proceing Plant. International Journal of Quality & Reliability Management, 005; 3: [5] Menni, E., and P. Agapio. Availability Aement of Dieel Generator Sytem of a Ship: A Cae Study. International Journal of Performability Engineering, 03; 95: [6] Vaneeta, J., S. Dharmaraja, and S.K. Trivedi. Markov Modeling Approach for Survivability Analyi of Cellular Network. International Journal of Performability Engineering, 0; 75: [7] Upadhya, K. S., and N. K. Srinivaan. A Simulation Model for Availability under Battlefield Condition. Simulation, 000; 74 6: [8] Liandro, M., and E. S. Quiroga. A Simulation Approach to the Optimization of Railway Infratructure Maintenance Strategie. International Journal of Performability Engineering, 0; 76: [9] Srinivaa Rao, M., and V.N.A. Naikan. A Managerial Tool for Reliability Analyi Uing a Novel Markov Sytem Dynamic MSD Approach. International Journal of Management Science and Engineering Management, 009; 43: [0] Srinivaa Rao, M., and V. N. A. Naikan. A Novel Markov Sytem Dynamic Framework for Reliability Analyi of Sytem. International Journal of Economic Quality and Control, 009; 4: 0-6.

9 A Sytem Dynamic Model for Tranient Availability Modeling of Repairable Redundant Sytem [] Forreter, J. W. Indutrial Dynamic. MIT Pre, Cambridge MA, 96. [] Dumolo, R. N. Application of Sytem Engineering to Railway Project. International Journal of Performability Engineering, 007; 3: [3] Burge, T.F. Modeling Quality Cot Dynamic. International Journal of Quality & Reliability Management, 996; 33: 8-6. M. Srinivaa Rao i Profeor in Mechanical Engineering department at GMR Intitute of Technology, Rajam, India. He took hi Doctorate degree in Reliability engineering from Reliability engineering centre at Indian Intitute of Technology, Kharagpur, India. He received hi B.E in Mechanical Engineering and M.E in Indutrial Engineering from Andhra Univerity, Viakhapatnam, India. He ha publihed hi reearch work in many international journal and preented paper in conference. Hi current reearch interet include reliability and availability analyi and modeling of ytem uing imulation methodologie. V.N.A. Naikan, i currently Profeor and Head of the Reliability Engineering at the Indian Intitute of Technology Kharagpur, India. He did hi M.Tech. and Ph.D. degree from thi centre. He graduated in mechanical engineering with econd rank from the Univerity of Kerala. Before joining the Reliability Engineering Centre, he alo worked with Indian Space Reearch Organization, Chinee Univerity of Hong Kong, Indian Intitute of Management, Ahmedabad and Union Carbide India Limited in variou capacitie. He ha publihed more than 00 reearch paper in international journal and conference, a book on Reliability Engineering and Life Teting and a chapter on SPC in the Handbook on Performability Engineering, Ed. Krihna B. Mira publihed by Springer 008. He i the aitant editor-in-chief of the International Journal of Performability Engineering, adviory and editorial board member of everal other journal. He had been the technical chair of everal international conference. He ha guided about 00 tudent for their B.Tech., M.Tech., and Ph.D. thee. He i a FIE and member of profeional ocietie including, IEEE, IEI, SREI, Sytem ociety of India, etc. He ha been doing conultancy and reearch project for organization like ISRO, BARC, BHEL, Defence Force, and Minitry of Textile. He ha been alo a viiting profeor at CALCE, Univerity of Maryland, USA.