RELIABILITY AND CAPABILITY MODELING OF TECHNOLOGICAL SYSTEMS WITH BUFFER STORAGE

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1 RT&A # 0 (7 (Vol. 00, June RELIABILITY AND CAPABILITY MODELING OF TECHNOLOGICAL SYSTEMS WITH BUFFER STORAGE Aren S. Stepanyants, Valentina S. Victorova Institute of Control Science, Russian Acadey of Sciences 7997 Moscow, Profsouznay 65 E-ail: ray@ipu.ru, viktorova_v@ail.ru ABSTRACT The paper is devoted to reliability and capability investigation of technological systes, inclusive of developent of dynaic reliability odel for two-phase product line with buffer storages and ultiphase line decoposition Key words: Reliability analysis, arkov process, ultiphase systes, ulti-flow structure decoposition INTRODUCTION Multiphase systes are the systes where technological process and supporting euipent are divided into sections referred as phases. One of the approaches to iproving reliability and capability is to include into ultiphase syste tie redundancy using buffer storages. When failure of input section euipent occurs buffer storage ensures uninterrupted technological process in output sections. Valid choice of placeent location and capacity of buffer storages is ipossible without reliability odeling and analysis of syste projects alternatives. Coon prediction odels of ultiphase systes describe only single-flow structures and suppose absolute reliability of buffer storage (Cherkesov 974. In this paper we suggest analytical ethod for calculation reliability and capability of ultiphase systes based on two-paraeter arkov process. The prediction odel takes into account different ratio of input and output devices capability and unreliable buffers. The odel decoposition techniue is developed. This akes it possible to analyze ulti-flow systes with tree-type structures. Procedure of construction state space and transition graph of the two-paraeter arkov process is created. The procedure is founded on selection of state subsets, corresponding to interediate and arginal (axiu or iniu level of resource (inventory in buffer, and generation of boundary and liiting transition. Process of generation of difference euation and boundary condition are described. TWO-PHASE SYSTEM DESCRIPTION Schea of single-flow two-phase syste with input ( and output ( processing devices and transient buffer (3 is shown in Figure. Figure. Single-flow two-phase syste with buffer storage. Each processing devices is characterized by capability i, failure rate λ i, recovery rate i ; buffer is characterized by capacity z (0 z z M, failure rate λ, recovery rate. Let us denote the state of arkov graph for two-phase syste by three-digit binary code. The first two digits indicate 40

2 RT&A # 0 (7 (Vol. 00, June the states of the devices and the third digit indicates the buffer state. Digit indicates that the state of device (buffer is good, 0 is failed. Reliability behavior of the syste depends on inventory level in the buffer: zero level = 0; we will designate zero level subset of arkov reliability odel state set as G axiu level = z M ; we will designate axiu level subset of arkov reliability odel state set as V interediate level (0 < z < z M ; we will designate interediate level subset of arkov reliability odel state set as W 3 METHODOLOGY OF TWO-PARAMETER MARKOV MODEL CONSTRUCTION Let us define the arkov odel construction seuence:. Definition of all possible states for subsets G, V, W. Analysis of the states in copliance with characteristics of perforance and failures, reoving the states which can not stand in given subset and which have not transition fro another states 3. Deterination of the states which have arginal (liiting transitions fro another states (these are transitions fro subset W into V and G, assignable with buffer inventory level axiization (iniization. Marginal transitions are indicated as dotted line. 4. Deterination of boundary transitions fro subsets V and G into subset W. These transitions exist for the states in subsets V and G, for which failure or recovery of the syste devices result in buffer arginal inventory level decrease (increase. Boundary transitions are also indicated as dotted arc, waited with appropriate failure (recovery rate. After arkov graph construction we can define atheatical odel of the syste. Let us denote state probability for subset W as P, and for subsets V and G as F M, and F(0, respectively. Now we can set up difference euation for characteristic states of the syste. Characteristic states are the following:. The states which have input and output transitions in the range of one subset. The states which have input liiting transitions 3. The states which have output boundary transitions (euations for these states deterine boundary conditions Figure shows graphs with characteristic state α i and input (outpu transition. Graph I shows transitions in the range of one subset. Graph II shows boundary transition. Difference euation for case I (transitions in the range of one subse is of the for P i, t Δ P i ± Δz, Δt ψi ± Δz, + Δt ϕ ± Δz, ( α + α i Partial differential euation is: n Pα α α i, P + i, i ψi, + ϕi i, z ( P, Let us consider stationary area and take into account the fact that = 0 when t. Then n Pα α i i ψi α i + ϕi i z i (3 n i i i 4

3 RT&A # 0 (7 (Vol. 00, June Under (3 we can forulate the following rule for setting up differential euation for any state with transitions in the range of one subset. Figure. Graphs for case I (transitions in area of one state subse and case II (liiting transition. Rule. Derivative of state probability with respect to buffer inventory level ultiplied by rate of level change ( is eual to product of state probability by su of output transition rates, signed with inus, plus su of product of input transition rate by probability of state fro which transition is done. Siilarly we get differential euation for the case II. Here state α i in the range of one subset has input transitions with rate ϕ i, output transitions with rate ψ I and liiting transition fro subset W =0 or z=z. n Fα i(0 z, = ϕif i(0 z, ( ψi α i(0 z, + α i(0 z, (4 For stationary area (t we have algebraic euation n ( ψi α i(0 z = ϕif i(0 z + α i(0 z (5 Then it is possible to forulate the rule for states with input liiting transition. Rule. Probability of considering state ultiplied by su of output transition rate is eual to su of transition probabilities fro other states to given state and probability of liiting transition. Probability of liiting transition is probability of state fro which transition is done ultiplied by absolute value of rate of level change. Boundary condition occurs when transition exists fro states of subsets V and G into states of subset W: ψ ϕ Stationary boundary condition is:, = P (0, = P (0,, (6 4

4 RT&A # 0 (7 (Vol. 00, June ψ ϕ = P (0 = P (0 (7 4 MARKOV RELIABILITY MODEL FOR TWO-PHASE SYSTEM Proceeding fro rules and euations of previous section one can construct reliability odels for two-phase single-flow syste. Models were constructed for three alternatives of relationship of processing devices capability ( = =; > ; <. 4. Model for euality of input and output capability Markov graph for euality of capability of input and output processing devices ( = = is shown on Figure 3. Syste of partial differential euation is: P, P, z,, + P, P, ,, + z P, λ, + 0, + 0, + P00, + 00, 0, 0, P00, + 00, 0, P00, + 00, 0, P0, = 0, + 00, + 00,, F(0, λ (0, + 0(0, + 0 (0, F0(0, = 0(0, (0, + 0(0, F0 (0, = 0 (0, (0, F, λ, + 0, + 0, F0, = 0,, + 0, F0, = 0,, Boundary condition: , +, +, 00 00,, (8 43

5 RT&A # 0 (7 (Vol. 00, June 0 0, = λ (0, = λ (0,, (9 Figure 3. Markov graph for two-phase syste ( = =. At stationary area (t syste (8 turns into the syste of differential-algebraic euation: λ = λ (0 + 0(0 + 0 (0 0 = 0(0 (0 + 0(0 0 = 0 (0 (0 λ = = 0 We use the following boundary and noralizing condition when solving syste (0: (0 44

6 RT&A # 0 (7 (Vol. 00, June = λ (0 = λ Pijk z + Fijk + Fijk (0 = ijk 0 0 z (0 0 ijk ijk ( 4. Model for uneual input and output capability ( > Markov graph for uneual capability of input and output processing devices ( > is shown on Figure 4. Figure 4. Markov graph for two-phase syste ( >. Let us directly consider stationary area (t and syste of differential-algebraic euation: ( 0 λ = = 0(0 + 0(0 λ = = Boundary and noralizing condition: ( ( 45

7 RT&A # 0 (7 (Vol. 00, June ( = λ (0 = Pijk z + Fijk + Fijk (0 = ijk z 0 * 0 0 ijk ijk (0 (3 4.3 Model for uneual input and output capability ( < Markov graph for uneual capability of input and output processing devices ( < is shown on Figure 5. Figure 5. Markov graph for two-phase syste ( < Syste of differential-algebraic euation (t : ( 0 0 λ = * λ (0 + 0(0 + 0 (0 + ( 0 = 0(0 (0 + 0(0 0 = 0 (0 (0 0 = ( (4 46

8 RT&A # 0 (7 (Vol. 00, June Boundary and noralizing condition: ( (0 = λ = Pijk z + Fijk + Fijk (0 = ijk z 0 * (0 0 0 ijk ijk Coputer-oriented procedure was developed for analytical solving systes (0, (, (4. In accordance with this procedure at first one have to obtain probability density function P 0 : where a λ + a z 0 P0(0 e P (5 =, (6 ( λ b + + λ + + Further probability F 0 is defined via density function z a z 0 P0dz = 0(0 e = C H a 0 F =. (7 Then each i th unknown F ijk is represented as product of invariable and variable actors C H i, where H i recursively calculated fro H i-, and С is calculated fro noralizing condition (,3,5. Stationary availability K г and atheatical expectation of capability С of two-phase syste are K г =F +F 0 +F (0+F ; С=К г. (8 5 RELIABILITY AND CAPABILITY ANALYSIS OF MULTIPHASE SYSTEM Multiphase systes are aggregate of two-phase systes. Exaples of ultiphase ultiflow systes, specified in graphical editor of software ipleented described above odels, are shown in Figure 6. The procedure of calculation estiate of availability of ultiphase syste includes the following steps:. Pick out the triplet (buffer, input device, output device with iniu buffer capacity. Calculate availability and average capability indexes (8 for evolved triplet via appropriate odels (0,, 4 3. Replace the triplet by one processing device with euivalent availability and capability calculated on previous step 4. Repeat steps -3 until all ultiphase structure will be represented by one euivalent device It was shown in (Victorova 009 that above procedure ensures derivation of availability low estiate. 47

9 RT&A # 0 (7 (Vol. 00, June Figure.6. Screen shot of software for reliability and capability analysis of ultiphase systes. 6 CONCLUSION For adeuate reliability and capability odeling of technological systes it is necessary to take into account unreliability of buffer storages. Statistical analysis of failure data of buffer storages shows failure rate growth with increasing capacity. On assuption of absolute buffer reliability one can ake pitfall about continuous capability growth with increasing capacity (see upper curves on Figure 7. Analysis based on the odels suggested in this paper shows that inflection point exist on the curve of capability as function of capacity. After this point one can observe decrease of reliability and capability of ultiphase systes as it is shown on lower curves of Figure 7. 48

10 RT&A # 0 (7 (Vol. 00, June Figure 7. Multiphase syste capability dependence on buffer storage capacity. 7 REFERENCES Cherkesov, G.N. (974. Reliability of Technical Systes with Tie Redundancy, Sovetskoe Radio: Moscow. Victorova V.C. (009. Aggregation of reliability and safety odels of coplex technical systes, Abstract of Dr.Sci thesis, Institute of Control Sciences RAS: Moscow 49

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