RELIABILITY AND AVAILABILITY ANALYSIS OF SOME SYSTEMS WITH COMMON-CAUSE FAILURES USING SPICE CIRCUIT SIMULATION PROGRAM

The effectiveness of SPICE circuit simulation program in calculating probabilities, reliability, steady-state availability and mean-time to failure of repairable systems described by Markov models is demonstrated. Two examples are presented. The first example is a warm standby system with common-cause failures and human errors. The second example is a non-identical unit parallel system with common-cause failures. In both cases recourse to numerical solution is inevitable to obtain the Laplace transforms of the probabilities. Results obtained using SPICE are compared with previously published results obtained using the Laplace transform method. Full SPICE listings are included.


INTRODUCTION
In the design, development, tuning and upgrading of engineering sys- tems, performance and reliability are two ofthe main factors which must be taken into consideration.Whether it is for predicting the behavior of new designs or studying possible changes to existing ones, models of such systems are essential tools for such investigations.Several *Corresponding author.mathematical evaluation techniques can be used to analyze these models.Of particular interest here is the Markov analysis technique which is widely used for calculating the reliability and availability of repairable systems.
According to the state transition diagram, the state transition equation of a repairable electrical system, with N state probabilities, can be represented by N equations of the form dP (t) alnPl(t) + annPn(t) + aNnPN(t), n 1,2,...,N, (In)   where (dPn(t)/dt) represents the differentiation of the nth-state prob- ability Pn(t), all a12 + a13 + + alN ann an "4-an2 + -+-anN and aNN aN1 + aN2 + + aN(N-l) where N is the number of states.Eq. (In), n 1, 2,...,N, can be solved using the Laplace transform method [1-5] yielding expressions for the probability, P,(t), that the overall system under consideration is in state n at time for n 1, 2,..., N.Alternatively, the steady state probabilities, P,, n 1, 2, N can be obtained by setting the deri- vatives with respect to time, of equations (la)-(1N), equal to zero and using the relationship N E Pn (t)   (2) n=l to solve the resulting N algebraic equations [6, 7].While these ap- proaches are attractive for relatively simple systems, it is virtually im- possible to obtain a general time dependent expressions for the probabilities of more complicated systems.Moreover, in some cases recourse to numerical methods may be inevitable in order to solve high order algebraic equations before obtaining the Laplace transforms of the different probabilities [1].
On the other hand, the SPICE circuit simulation program is a general- purpose program which can be used for d.c., transient and a.c.analysis.Although it was initially developed for integrated circuit analysis, it is now widely used for many non-integrated circuit applications.Recently, the use of SPICE capabilities in studying the behavior of multi-state systems described by Markov models was investigated [8].The equiva- lent circuits used and the results obtained for the steady-state probabil- ities of four-state and five-state models prove that SPICE circuit simulation program is a useful tool for studying the behavior of multi- state systems described by the Markov models.
The major intention of this paper is to present equivalent circuits for studying the behavior of two multi-state systems described by the Markov models.The first system is a non-identical unit parallel system with common-cause failures [5].The second system is warm standby system with common-cause failures and human errors [1].The first sys- tem requires the algebraic solution of a fourth-order polynomial before obtaining the Laplace transforms of the different probabilities and the second system requires numerical solution of a fifth-order polynomial before obtaining the Laplace transforms of the different probabilities.Using the proposed equivalent circuits, reliability and availability analysis of these systems can be easily performed using SPICE.From the SPICE output file the following can be obtained: 1.The probability, Pn(t), that the system is in state n at time t. 2. The system reliability, Rs(t), that is the probability that the system is in an up state.3. The system steady state availability, A Vs, that is the long term probability that the system is in an up state.4. The system mean time to failure, MTTF.

PROPOSED MODEL
According to Ref. [7], an appropriate source for the model must be identified and this source-state is recommended to be the equation having the larger number of terms.The source equation is replaced by Eq. (2).It is worth mentioning that, the choice of a particular equation as the source is not critical and any of the state Eq.(ln) can be assigned as the source-state.After replacing the selected source equation by Eq. (2), performing the necessary modifications and rearrange- ments, Eq. (In), n 1,2,..., N can be rewritten as Ps(t) 1 The electrical equivalent circuit of Eq. ( 3) is shown in Figure 1.In the circuit, the current is the analog of Pn(t), the voltage across the in- ductor is the analog of the derivative with respect to time, (dPn(t)/dt), the voltage across the capacitor is the analog of the integration with respect to time, f P(t)dt, and the voltage-controlled current sources (VCCSs), G1, G2,..., Gv, represent the right-hand sides of Eq. ( 3).The resistors, R, in parallel with the capacitors prevent floating nodes to satisfy SPICE rules.These resistors are too large to avoid any significant effect on the circuit dynamics.The integrations f P(t)dt, n 1, 2,...,N, n s are required to calculate the VCCSs, Gn, n 1, 2,..., N, n =/: s, when Eq. (3n) is in the form dPn(t) N %rePro(t), n=l,...,N, ns (an)   dt m=l,mn In such case (or cases), the probability Pn(t) can be expressed as f0t The polynomial feature of SPICE will be used to represent these VCCSs [8].In the SPICE input file of the circuit of Figure 1, the initial conditions of P(t), n 1,2,..., N are realized by the IC option, the transient analysis option TRAN will be used to perform the transient Ps(t) P-n(t) /.. analysis of the circuit of Figure and the SPICE output file will contain the probabilities P,(t), n 1, 2,..., N as a function of time.
Using the probabilities P,(t), n 1,2,... ,N, the system reliability can be calculated using Eq. ( 6).M Rs(t) P.(t) where M < N is the number of states with the system up.The electrical equivalent circuit of Eq. ( 6) is shown in Figure 2. In the circuit the current is the analog of the system reliability R(t) and the VCCS, G, represents the right hand-side of Eq. ( 6).
The steady-state availability, defined as the long term probability that the system is in the up state, can be expressed as where M < N is the number of states with the system up.Thus, the steady-state system availability can be obtained directly from the SPICE output file.
The system mean-time to failure can be expressed as MTTF Rs(t)dt (6b) The electrical equivalent circuit of Eq. ( 6b) is shown in Figure 3.In the circuit the voltage across the capacitor is the analog of the system MTTF.

EXAMPLES
The circuits of Figures 1-3 can be used for obtaining the probabilities P(t), n 1,2,..., N, the reliability R(t), the steady-state availability A Vss and the MTTF of any system described by a set of state transi- tion equations.Ofparticular interest here are systems for which recourse to numerical solution of high order polynomials is inevitable.
A non-identical unit parallel system with common-cause Figure 4 shows the state transition diagram along with the corre- sponding failure rates and repair rates of the non-identical unit parallel system with common-cause failures [5]. of this system can be written as: T AaPo(t)-(Za "F-/b "a t-Ac3)Pl(t) (7b) dt Solving Eqs.(7a-7e) with the aid of Laplace transforms yields the state probabilities in terms of the real roots of a fourth-order polynomial.
However, obtaining the real roots ofa fourth-order polynomial is deeply involved in a sea of algebra [5].
Example 2 Warm standby systems with common-cause failures and human errors [1].

CONCLUSION
It is evident, from the analysis presented and the examples considered in this paper, that SPICE circuit simulation program is an effective tool for calculating the time dependent probabilities, the system reliability, the system steady-state availability and the mean-time to failure of repairable multi-state systems described by Markov models.Especially for complicated systems where recourse to numerical solu- tion of higher-order polynomial is inevitable, the use of SPICE is highly recommended as it saves time and avoids involvement in a sea of algebra.

FIGURE 6
FIGURE 6 SPICE input file for the circuit of Figure 5 corresponding to the state transition diagram of Example 1.

FIGURE 10
FIGURE 10 SPICE input file of the circuit of Figure 9 corresponding to the state transition diagram of Example 2.
The state transition equations State transition diagram of Example [5] State 0: Both units operating, State 1: Unit A failed, unit B working, State 2: Unit B failed, unit A working, State 3: Both units failed, State 4: System failed due to common cause failures.

TABLE III Steady
[1]ate system availability ofModel III ofExample with Aa 0.0001, Ab 0.0002, #, #, #c #3 At2 Ac3 0.0 State transition diagram of Example 2[1]State 0: One unit operating and standby operating, State 1: Operating unit failed and standby operating, State 2: Stand by unit failed in its standby mode, operating unit working, State 3: both units failed due to other than common-cause failure and human errors, State 4: both units failed due to common-cause failure, State 5: both units failed due to a human error.
time.Tables IV and V show the