Chapter 4 Availability Analysis by Simulation and Markov Chain

Transcription

1 Chapter 4 Availability Analysis by Simulation and Markov Chain

2 Chapter 4 Availability Analysis by Simulation and Markov Chain 4.1 Introduction: For a perfect design, an engineering systems, component and device should remain operational and attain system's objective without failure for a specific period of time. When we perform reliability analysis, it is important to distinguish between repairable and nonrepairable systems. To examine repairable systems and establish methods used to determine the failure characteristics of these systems, as well as the methods for predicting their reliability and availability. The term dependability or reliability, availability, and maintainability are used in a generic sense to cover reliability, availability, and maintainability, or in a particular sense that it may mean the probability of success. Several analytical techniques can be used to calculate either reliability or availability, depending upon the requirements, specific to the systems, and their objectives, input parameters and specific constraints. The reliability of the system can be described as the probability that the system does not fail in time interval <, t> and still functions at time t. The system is made up of a number of components of which some cab be repaired on-line while some are repaired offline. The term 'availability' is considered as the probability that the system will be functional at time t. The unavailability is complementary to availability, the evaluation of a system's reliability requires relevant methods to comply with the specific structural and functional definitions of the system in question. In general, the reliability of a system is the collection of the reliabilities of the components which make up the system. It is said that a complex system reliability is under consideration when physical interruptions caused by failure of the components or subsystems in an interaction of many components. The system fits into this definition involving a number of components interacting with each other in a complex way. For such a complex system's reliability, solutions are offered by various 85

3 nethods such as failure modes effect analysis (FMEA), reliability block diagrams (RBDs), TA and Markov analysis (MA). This chapter deals with the criticality analysis of the various units of a 'Reliability- Based' System using the various basic causes of failure or failure modes and then processing hem through Monte Carlo simulation, before evaluating the Risk Priority number. Operational Availability(A ), which is mainly dependent on the Mean Time To Repair MTTR) and Administrative Logistic Time (ALT), needs to be assessed by considering (i) ion-scheduled or breakdown maintenance (ii) scheduled maintenance eg. periodic preventive naintenance, and (iii) predictive preventive maintenance. These maintenances may be done >y departmental men or by external human resources and may require uses of spares. Thus, it s necessary to consider the probability of all of them in order to arrive at the extended but nore appropriate estimation of MTTR. Continuous Time Markov Chain analysis (CTMC), lave used for evaluating probability of each constituent state of transition, to know and ationalize the method of assessing the total "mean time to repair" and the 'non-availability' )f the system and suggesting rule for quantitative decision-making. 1.2 Monte Carlo Simulation: The Monte Carlo technique is a numerical method based on a probabilistic nterpretation of quantities obtained from algorithmically generated random variables. It was ntroduced in 1949 by N. Metropolis and S. Ulman. Basically, it can be used to estimate a 'alue (an unknown probability) or simulate (reproduce) the stochastic process describing the >ehavior of a complex system and one of the most important aspects of digital simulation is he ability to handle stochastic events, that is, events which occur randomly (48). The Monte Carlo method is used for reliability prediction when an exact nathematical model cannot be developed economically or when it becomes too complex to lermit timely evaluation. This method involves the determination of distributions of the larameters of the various elements in a system, selection of a random sample of each element nd its parameters, and combining of these samples to obtain a measure of the system erformance or reliability (49). 86

4 .3 Systematic Methodology for Analysis: Systematic methodology, involved for such analysis, requires initiation of the allowing stages: Step 1: Identify the system and its elements or sub-systems following the methods of system dynamics. Step 2: Classify the basic causes of failure of the system (sub-system) and put the causes, their effects and protections in the form of a Causes and Effects Diagram, known as Ishikawa Fishbone Diagram. Step 3: For each subsystem (SA) (i.e. basic mode) write down all the causes of failure, VIZ, OAi, SAM, SAiii, Step 4: Collect actual industrial data from maintenance history records to know the probability density function (pdf) of each mode and plot the data to get the cumulative probability distribution (cdf). Step 5: Use 'Monte-Carlo' Simulation to get an overall probability distribution of failure of each basic mode, using the following steps systematically. Step 6: Integrate probability distribution of all basic modes of failure into one probability distribution of failure of the total system sub-system. Step 7: Using Ishikawa 'Fish-Bone' Diagram and subsequently, based on data collected and a set of observations etc. analyse the reliability. Step 8: Subsequently analyse the criticality of each subsystem from the FMECA that Could have suffered performing its functions satisfactorily While doing this, it is necessary to use 'Di-graph' or Analytical Hierarchical Process (AHP) through global data matrix, in order to obtain the relative worth of each failure mode. The entire observation data and the output function should have been stored in the amputer memory for the purpose of using them as and when needed, while estimating the tal "unavailability" of the system and accordingly the total time lost in maintenance & jrvicing - thereby increasing the duration of MTTR (Mean Time To Repair). Computerized program can be developed using the flow diagram, as in Fig

5 System Analysis Identification of Units/Subsystems IDENTIFY Basic Modes of failures of each unit Collect failure of units over a Period Collect Further set of observation Use random number (Monte-Carlo) simulation using CDF of each elements 88

6 Get Probability Distribution (Normal) of total system Cause, effects & precautions analysis using Fishbone diagram. Ranking of critical modes by AHP & Digraph diagram No Modelling system Performance & Reliability Evaluation Establish the system Fig. 4.1: Objectives: Ranking of Failure Modes for evaluating criticality of units. 89

7 4.4 Typical cases undertaken: In an actual case study of a hydraulic transmission system (, the Pressure Line Filter, the "Off-line Filter" was a main source of adequate supply and proper functioning as per specification limit. The above system has been studied to find the failure mode (basic causes of failure / malfunctioning) effects, it could produce on the output requirement and a system of precautions to be attended in order to avoid future blockade arising of such malfunctioning. The causes, effect and suggestions for protection are found out and an Ishikawa Fishbone diagram is constructed as a preliminary to step to decide upon the reliable design of the unit, as shown in Fig 4.2. Non-return valve not working Motor Speed Inadequate Variation of Pump Speed Change of Filter - Valve Checked Occasionally Voltage Stabilizer Speed regulation Automation Off-line Filter Clogging of Filter Flow of oil improper Vibration and Noise Clogging of filter Inadequate pressure Reservoir Temperature rise Fig. 4.2: Ishikawa Fishbone Diagram for "Off-line" filtering unit of a hydraulic system. Considering the case of clogging of filter, following are the basic causes of failure: a) Filter not Working b) Non-return valve malfunctioning 9

8 c) Motor & pump speed inadequate d) DCV spool malfunctioning. All these basic causes have different type of probability distribution of malfunctioning over 4 hours of working. From the data collected as shown in Table 4.1 Cumulative density function (cdf) is plotted against time as shown in Fig 4.3. Monte-Carlo method of random number simulation is used for combining all the four probability distribution functions (pdf) of failures shown in Fig 4.4 into one overall probability distribution functions, so that the mean rate of failure could be evaluated. The Random number table and corresponding pdf values are given in Table 4.2. The overall pdf against time is plotted and is shown in Fig. 4.5, giving average operating hours in between failures and hence failure rate. Table 4.1 Time intervals with pdf and cdf Time Interval (Hours) Pdf of failure modes in % of various modes a b c d CDF of each mode a b c d

9 a) b) c) d) Filter not working Non return valve S malfunctioning jf>\-' Motor & pump speed,fc' s inadequate -'As* DCV spool malfunctioning*^.'' <5-._ -j:^ * cdf (%) <* / ' '' J 1/ 7 ''l / ' ' / r/.'f -- -a - - b --A-- C d 2 1 'S S Time (Hrs) Fig. 4.3: Cumulative distribution function for four basic causes of clogging of filter a) Filter not working b) Non return valve malfunctioning c) Motor & pump speed inadequate d) DCV spool malfunctioning. a -b c d Time Interval in Hours Fig. 4.4: Probability distribution function for four basic causes of clogging of filter 92

10 Table 4.2 Random numbers and hours for various failure modes SI. Random Failure Modes and Hours Minimum No. 1 Number 33 a 15 b 9 c 1 d

11 *-.2 o Time (Hrs) 2 pdfi Fig. 4.5: Combined Probability Distribution for clogging of filter 4.5 Markov Chain Model for MTTR evaluation Markov Analysis (MA) is used to study the dynamic behaviour of systems. Markov models are useful for finding the behaviour of a system when there are common cause failures, imperfect coverage, complex repair policies, degradation, shock effects, induced failures, dependent failures, and other sequence-dependent events (2) (5). The Markov model considers states and transitions where states represent the possible conditions of the system that can be described as failed or good by the failure of each component. The basic assumption in a Markov model is that the state transitions are memoryless, which means that the transition rates are determined only by the present state and not by history. Although this method is very powerful and accurate, it is not practical to apply if the number of the components involved in the analysis is large. This yields many states and transitions, which would make the analysis almost impossible to handle since the transition diagram has to be set up manually, presenting every single relationship between states (51). MA gives the opportunity to find out the probability of occurrence of the possible failure states for the cases which are randomly lost functionality failure of the internal components. Individual state probabilities are calculated using the information given in the state transition diagram (5)(5I). 94

12 To find out the maximum time spread of Mean Time To Repair (MTTR), used for determining the extent of unavailability of a machine unit mathematical model is being presented using Continuous Time Markov Chain Analysis. This is needed since Operational Availability of a machine A is given as: A = MTBF MTBF +MTTR + ALT, where ALT - Administrative Logistic Time. Unavailability which is (1-A ) requires accurate determination of MTTR. Considering a system as shown in Fig. 4.6 having five states as follows: State 1: Machine in working condition State2: Machine failed State3: Machine under scheduled maintenance condition State4: Machine under breakdown or spares part maintenance condition State5: Machine repaired and being set to make it ready for working condition The system has failure rate ( A ), planned maintenance rate ( a ), unplanned maintenance rate (/?), spare parts management rate (8), repair rate (y) and setting rate after repair ( /J. ). P Fig. 4.6: Transition state diagram

13 The transitional probability matrix is given as follows: Probability of each states States Ii 1 -X X 7T 2 2 -(a+5) a 5 TC3 3 -P P 7I4 K n -Y y -^ The equations obtained from steady state diagram shown in Fig Xn x - fj.n 5 Xn x = (a + 5)7i 2 - an 2 = /?7r 3 S7T 2 +/37r 3 =yn 4 (4.1) (4.2) (4.3) (4.4) r^4=m^$ From equation (4.1) we have 7T + 7T 2 + 7t i + 7T A + 7T 5 = 1 L = a 7T 5 From equation (4.1) & (4.2) we get X ^i. = M K 5 (a + S) From equation (4.3) & (4.8) we have ;r 3 _ an 7T 5 P(a + S) (4.5) (4.6) (4.7) (4.8) (4.9) 96

14 From equation (4.5) we get?± = JL (4.1) Dividing both sides of equation (4.6) by n s we get 71, 71-, 7T-, i A t *\ L ±- + 1 = (4.11) ff 5 7t $ 7C S 7T 5 7l 5 Using equations (4.7) (4.8) (4.9) and (4.1) in equation (4.11) we get = (4.12) A (a+ 8) P(a + S) y TT 5 1 _/J/3 y(a + S) + jua/3 y + {icta y+/ua fi (a + S) n s Afiy{a + 8) Hence steady state probability for State 5 can be given as * s - v>*.«+s) (413) HP y(a + S) + /ua/3 y + juaa y + piap(a + 8) Once the value of steady state probability for State 5 (7r 5 ) is calculated from the data available for the system, all remaining steady state probabilities for State 1, State 2, State 3, State 4 can be obtained as follows. From equation 4.9 and 4.13 we get x a fj, Ap y(a + 8) 3 =. -x- P(a + 8) /up y(a + 8) + fiap y + /jaa y + ^AP(a + 8) aay * 3 = (4.14) Py(a + 8) + Apy + aay + Ap(a + 8) From equation 4.1 and 4.13 we get 7T 4 = X M x Apy(a + 8) y npy(a + 8) + juapy + juaay+ ^AP(a + 8) Ap(a+8) it. = (4 15) py(a+8) + APy + aay+ap(a + 8) Total Sojourn Time involved in MTTR MTTR = 7r,a + 7t,(p + 8)+ n 5 y (4.16) It can be now calculated from the analysis, after evaluating for m, n 4> n$. 97

15 This shows how extended value of MTTR, could be apprehended, and hence evaluated to give an upper bound solution, including the contingent situation arising from inhouse repair/or repair by external agency and or uses of spare parts. Thus spare parts management and storage also become important in the total analysis of MTTR. The system availability is given by equation 4.17: MT where MT- stands for total mission time. 4.6 Discussion: From the foregoing analysis the some vital point can be drawn as under:- 1) While evaluating Mean Down Time used for evaluating the operational availability (A ) or unavailability which is (1-A ), it is necessary to assess properly the extent of actual time spent for MTTR. 2) This again consists of a unavailability due to (i) undetected failure (U r ), (ii) Unavailability due to scheduled testing or planned maintenance (U s ), (iii)unavailability due to contribution, associated with non-planned or unscheduled maintenance task (U n ) and also (iv)unavailability due to delay in procuring spare parts wherever needed, (U sp ). 3) Item (ii) and (iii) again may be done either by internal manpower or by external/hired manpower, requiring additional time delays, thereby there is a possibility of increased time content of total MTTR. 4) This MTTR, again, when added to Administrative logistic time, then we get a total extent of Mean Down Time (MDT). 5) A quantitative decision making equation for MTTR may be written as in equation