Reliability Engineering and System Safety 25 (1989) 365-369

Technical Note

Optimal Number of Redundant Units for a Standby System

ABSTRACT

This note considers the problem of determining the optimal number of redundant units in a standby system. The expected cost rate is obtained The number of redundant units minimizing the expected cost rate is shown to be finite and unique. The case in which the system is periodically inspected is also considered.

NOTATION

C1

C2 C3 C(n) n

st s~

Acquisition cost of a unit and a switching device Monitor cost Inspection cost Expected cost rate of the system with n units Number of redundant units in a system Expected cost during a renewal period with n units Expected duration of a renewal period with n units

'~'2 '~'3

Failure rate of unit Failure rate of the switching device Failure rate of the monitor

1 INTRODUCTION

Redundancy of components is used to achieve a required reliability for a system with unreliable parts. Nakagawa 1"2 studied the problem of

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366 Won Young Yun

determining the optimal numbers of redundant units which minimize the expected cost rate in parallel and k out of n systems. In this paper, the optimal redundant units of a standby system are determined. If there are n redundant units for successful operation [(n - 1) standby units], and if each unit requires a switching device, the reliability block diagram of a general model can be shown as Fig. 1 (see Ref. 3). In this model, the monitor represents the failure-detection and switching-control functions (the failure of the monitor may be considered as common mode failures). The expected cost rate is obtained. It is shown that the optimal value of redundant units is finite and unique. The case in which the system is periodically inspected is also studied.

Assumptions

1. Failure times of all units are independent and identically distributed. 2. Inspection and replacement times are negligible. 3. Switching is in one direction only. 4. Standby units and switching devices cannot fail if not energized. 5. Switching devices should respond only when directed to switch by

the monitor; false switching operation is detected by the monitor as a unit failure, and switching is initiated.

2 OPTIMAL NUMBERS OF UNITS WITHOUT INSPECTION

When the system fails, the system is replaced by a new one. Then, the reliability function and the expected duration of a renewal period is given by

n - 1

R.(t) = exp [-- t(21 + 22 + 23)] ) ' , [(21 + 22)t]i/i!

i = 0

n - 1

Sn= f o R n ( t ) d t = ~ ( ) ~ l +)~2)i/(,~l +,~2+,~3)i+l i = 0

Therefore, the expected cost rate is

C(n) = (cln + Cz)/S .

A necessary condition for n to minimize C(n) is

C(n + 1) > C(n) and C(n) < C(n - 1) (1)

If we let

L ( n ) = S , / ( S , + , - S , ) - n n = 1, 2 , . . .

Optimal number of redundant units for a standby system 367

I n p u t

(

(

Unit I

U n i t 2

Unit n

Fig. 1.

Monitor I--

Structure of standby system.

then cond i t i on (1) is equ iva len t to

L ( n ) > c 2 / c l and L ( n - 1 ) < c 2 / q

Since

& + l - s . = 0 1 + &)"/E,L + 22 + 23] "+1

decreases in n a n d

l im (S.+ 1 - S.) = 0 n--~ O0

we have

L(n + 1) - L(n) = 8 . [1 / (3 .+2 - S.+ x) - I r iS .+ x - S.)] > 0

Hence , L(n) is increas ing in n. F u r t h e r m o r e L ( 0 ) = 0 and There fo re , the va lue o f n min imiz ing C(n) is finite and unique.

TABLE 1 Optimal n of Standby System without Inspection

C2/C1 /'1 + 22

0"1 0"2 0"4 0"6 0"8

2 1 1 2 2 2 5 1 2 2 2 3 8 1 2 2 3 4

10 1 2 2 3 4 20 2 2 3 4 5

O u t p u t

(2)

L ( ~ ) = oo.

368 Won Young Yun

Example 1 Suppose that 21 + 2 2 q-23 = 1. Table 1 gives the optimal values of n for cz/c I = 2, 5, 8, 10, 20 and indicates that the optimal n is increasing in c2/c 1 and 2~ + 22.

3 OPTIMAL NUMBERS OF UNITS WITH INSPECTION

In this section, it is assumed that a switching device and a unit consist of a subsystem. Therefore, if either switching device or unit fails, the subsystem (a switching device and a unit) fails. When the system fails before age T, it is replaced. Otherwise, it is inspected at age T and all failed subsystems are replaced by new ones.

Let X be the number of failed subsystems when the system is inspected. Then, the expected cost during a renewal period is

n - - I

Sc = exp [ - 2 3 T - ] I S (ClXWC3)er{X=x}+( l ' lC l - t -c2)Pr{X=n} l

x = O

+ (1 -- exp [ - 2 3 T ] X n c 1 + c2)

= can + c 2 + exp [ - (21 + 22 + 23)T][-c1(21 + 2z)T]

n-2 n-1

x )' [(2x + 22)T]'/i! + (c a- c,n--c2) )' C(2x 22)T]i/i~ +

i = 0 i = 0

n - 1

S. = )', [(21 + 2z)'/i!] [~ exp C-t(21 + 22 + 2 3 ] / i dt

i = 0

Therefore the expected cost rate is

C(n, T) = So~S,,

It is difficult to show that the optimal n is unique. However, numerical experience shows that the expected cost rate function is unimodal. Hence, the optimal n may be investigated using numerical search techniques with a computer.

Example 2 Suppose that 21 = 1022 = 1023 and c3 = 0"1cl. Table 2 gives the optimal n for the selected values of the various cost and distribution parameters. Table 2 indicates that the optimal n is increasing in c2 but decreasing in 21.

Optimal number of redundant units for a standby system 369

TABLE 2 Optimal n of Standy System with Inspection

T 0.2 0.5 1 2

2c2 ~ 0-1 0.5 0.1 0.5 0-1 0.5 0.1 0.5

2 2 1 2 1 2 1 2 2 5 2 2 2 2 2 2 2 2 8 3 2 3 2 3 3 3 3

10 3 3 3 3 4 3 4 3

4 C O N C L U D I N G R E M A R K S

The problem of determining the opt imal number of redundant units in redundant systems is discussed. A standby structure is considered. The expected cost rate is obtained and the opt imal solution is shown to be finite and unique. Numerical examples are also included. The opt imal number of redundant units of a periodically inspected system is also obtained. In this note, the single type of r andom failure is assumed. In many cases, however, multi-types of failure may exist. Hence, redundancy optimizat ion of complex systems with mult i- type failures is promising problems. Further, other structures (series-parallel, bridge structures, etc.) may be studied and the inspection interval may be considered as a decision variable. .

R E F E R E N C E S

1. Nakagawa, T., Optimal number of units for a parallel system, J. Applied Probability, 21 (1984) 431-36.

2. Nakagawa, T., Optimization problems in k-out-of-n systems, IEEE Trans. Reliability, R-34 (1985) 248-50.

3. ARINC Research Corp., Reliability Engineering, Prentice-Hall, Inc., Englewood Cliffs, 1964.

Won Young Yun Department o f Industrial Engineering, Pusan National University, Pusan, 609-735, Republic o f Korea

(Received 10 November 1988; accepted 18 November 1988)