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Optimal redundant systems for works with random processing time
Author’s Accepted Manuscript
Optimal redundant systems for works with random processing time M. Chen, T. Nakagawa
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S0951-8320(13)00044-6 http://dx.doi.org/10.1016/j.ress.2013.02.009 RESS4785
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Reliability Engineering and System Safety
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17 August 2011 10 January 2013 9 February 2013
Cite this article as: M. Chen and T. Nakagawa, Optimal redundant systems for works with random processing time, Reliability Engineering and System Safety, http://dx.doi.or g/10.1016/j.ress.2013.02.009 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Optimal Redundant Systems for Works with Random Processing Time M. Chena,∗, T. Nakagawab,∗ a
Graduate Institute of Business Administration, Fu-Jen Catholic University, Taipei, 24205, R.O.C. b Department of Business Administration, Aichi Institute of Technology, 1247 Yachigusa, Yakusa-cho, Toyota, 470-0392, Japan
Abstract This paper studies the optimal redundant policies for a manufacturing system processing jobs with random working times. The redundant units of the parallel systems and standby systems are subject to stochastic failures during the continuous production process. First, a job consisting of only one work is considered for both redundant systems and the expected cost functions are obtained. Next, each redundant system with a random number of units is assumed for a single work. The expected cost functions and the optimal expected numbers of units are derived for redundant systems. Subsequently, the production processes of N tandem works are introduced for parallel and standby systems, and the expected cost functions are also summarized. Finally, the number of works is estimated by a Poisson distribution for the parallel and standby systems. Numerical examples are given to demonstrate the optimization problems of redundant systems. Keywords: Random processing time, Parallel system, Standby system, Random number of units, Random number of works, Optimal redundant policy
Correspondence to: Mingchih Chen, Graduate Institute of Business Administration, Fu-Jen Catholic University, Taipei, 24205, R.O.C., Tel.: +886-2-29053895; Fax: +886-229052015; E-mail: [email protected] ∗
Preprint submitted to Reliability Engineering & System Safety
February 21, 2013
1. Introduction Manufacturing systems in the real world are usually subject to much uncertainty or randomness. Such uncertainty might result from the machine failures or the variable process time required by each job [1]. Redundancy is commonly used in practical applications to ensure a high degree of reliability when the components of operating systems are subject to stochastic failures. There are two basic ways to provide component redundancy, namely, parallel redundancy and standby redundancy. For a system with several spare units [2], an optimal replacement policy which maximized mean time-to-failure (MTTF) was discussed. Recently, a parallel system with n identical units was studied and a simple asymptotic method to obtain MTTF is proposed in [3]. Many researchers have developed vast number of studies on this reliability field such as [4], [5] and so on. There are various optimization approaches and techniques such as dynamic programming, integer programming, heuristics and meta-heuristics adopted in these studies; see, for example, [6], [7] for an extensive overview. Another important issue of redundancy is how many units and what kinds of redundant systems should be provided to optimize the expected costs [8]. Nakagawa and Yasui [9] summarized some research results for a parallel system with n units and obtained the optimal number of units which minimized the expected costs. However, there are still relatively few papers that deal with the optimal redundant number problems based on cost considerations as in [10] and [11]. Our main purpose in this paper is to survey some redundant models in [11] and [12], and to add some new interesting results for a job with random working times by using reliability theory. A job with a single work having random processing time is considered in Section 2, with the expected cost functions being derived for two different redundant systems, namely, parallel and standby. Next, redundant systems with random numbers of units are assumed when processing a single work. The expected cost functions and the optimal expected numbers of units are derived in Section 3. Subsequently, a job with N tandem works is introduced that has a working time Y as its total operating time. A job consisting of N random working times would mean that the completion time of this job is also a random variable. Such a problem involves comparing variables Y and X, which is well-known as the stress-strength model in [13]. By introducing the redundant, excess and shortage costs, the total expected cost functions for standby and parallel systems under a linear cost structure are summarized in Section 4. Then, the 2
optimal redundant numbers which minimize the total expected costs for each system are derived. Two numerical examples are given to compare the effects of the redundant systems in which the optimal number n∗ of redundant units for the given N works, and conversely, the optimal number N ∗ of works for the given n units are obtained. In Section 5, the number of works is estimated by a Poisson distribution for the parallel and standby systems. Numerical examples are given to demonstrate the optimization problems of redundant units with a random number of works. Finally, the conclusions are provided in Section 5. 2. Redundant Systems for a Work Suppose that redundant systems with n units are operating for a job with one single work: It is assumed that each unit is independent and has an identical failure distribution F (t) with finite mean 1/λ. An operating cost c0 n occurs to the redundant systems with n units. The job has a working time Y , which R ∞is assumed to have a general distribution G(t) with a mean time 1/w ≡ 0 G(t)dt, where Φ(t) ≡ 1 − Φ(t) for any function Φ(t). In the following sections, two kinds of redundant systems are considered: parallel systems and standby systems. 2.1. Parallel System for a Work The following linear costs are set up for the expected cost functions: If a parallel system with n units fails at time t and the work of a job finishes at time u (< t), the excess cost c2 (t − u) is required, and conversely, if the system fails at time u and the work would finish at time t (> u), the shortage cost c1 (t − u) is required (see Fig. 1). Adding the operating cost c0 n, the total expected cost is Z ∞ Z t n C1 (n) = c1 (t − u)dF (u) dG(t) 0 0 Z ∞ Z t + c2 (t − u)dG(u) dF (t)n + c0 n 0 0 Z ∞ Z ∞ n = c1 F (t) G(t)dt + c2 [1 − F (t)n ] G(t)dt + c0 n 0
0
(n = 0, 1, 2, · · · ).
3
(1)
Clearly,
c1 , C1 (∞) = ∞. w Thus, there exists a finite number n∗1 (0 ≤ n∗1 < ∞) which minimizes C1 (n). In, particular, when F (t) = 1 − e−λt and G(t) = 1 − e−wt , the expected cost in (1) is Z ∞ n n −wt c1 c2 X 1 C1 (n) = + − (c1 + c2 ) 1 − 1 − e−λt e dt + c0 n w λ j=1 j 0 n n X 1 c1 c2 X 1 n j = + + (c1 + c2 ) (−1) + c0 n j w λ j=1 j jλ + w j=1 C1 (0) =
(n = 0, 1, 2, · · · ).
(2)
From the inequality C1 (n + 1) − C1 (n) ≥ 0, the following inequality holds: n X 1 c2 1 n j (−1) ≤ + c0 . (3) j (j + 1)λ + w (n + 1)λ c1 + c 2 j=0 Letting L1 (n) be the leftt-hand side of (3), n X λ n j L1 (n) − L1 (n + 1) = (−1) > 0. j (j + 2)λ + w j=0 Thus, L1 (n) decreases strictly to 0, and the left-hand side of (3) decreases strictly to c0 /(c1 + c2 ). Therefore, there exists a finite and unique n∗1 (0 ≤ n∗1 < ∞) which satisfies (3). Clearly, if c1 ≤ c2 (w/λ) + c0(λ + w) then n∗1 = 0. 2.2. Standby System for a Work We consider a standby system with n units under the same cost structure in Section 2.1. Then, the total expected cost is Z ∞ Z t (n) C2 (n) = c1 (t − u)dF (u) dG(t) 0 0 Z ∞ Z t + c2 (t − u)dG(u) dF (n) (t) + c0 n 0 0 Z ∞ c1 n = + c2 − (c1 + c2 ) 1 − F (n) (t) G(t)dt + c0 n w λ 0 (n = 0, 1, 2, · · · ). (4) 4
In particular, when F (t) = 1 − e−λt and G(t) = 1 − e−wt , the expected cost in (4) is n c1 n c1 + c 2 λ C2 (n) = + c2 − 1− + c0 n (n = 0, 1, 2, · · · ). (5) w λ w w+λ From the inequality C2 (n+1)−C2 (n) ≥ 0, the following inequality is derived: n w+λ λ c2 ≤ + c0 , (6) w+λ λ c1 + c 2 whose left-hand side decreases from 1 to zero. Therefore, there exists a finite and unique minimum n∗2 (0 ≤ n∗2 < ∞) which satisfies (6). Clearly, if c1 ≤ c2 (w/λ) + c0 (w + λ) then n∗2 = 0, i.e., we would provide no redundant system for such a job. 3. Redundant Systems with Random Number of Units It has been assumed that the number n of units is constant and is previously given. However, because objective systems would be complex or each redundant unit could experience stochastic failures, the number of remaining working units might not be a fixed number all the time. In this case, the number of units for redundant systems could be estimated statistically by a probability distribution. The processing time of each job is assumed to have a general distribution G(t) with a mean time 1/w as in Section 2. For such systems, the optimal expected numbers of units for processing a single work are derived to minimize the total expected costs. 3.1. Parallel System with Random Number of Units We consider a parallel system with N units in which N is a random variable following a Poisson distribution with mean θ. Then, the failure distribution of the system is F (t|θ) =
∞ X
F (t)n
n=0
θn −θ e = e−θF (T ) . n!
Thus, the expected cost is, from (1) Z ∞h i c1 −θF (t) C3 (θ) = + c2 1−e dt w 0 Z ∞h i − (c1 + c2 ) 1 − e−θF (t) G(t)dt + c0 θ. 0
5
(7)
(8)
Clearly,
c1 , C3 (∞) ≡ lim C3 (θ) = ∞. θ→∞ w Differentiating C3 (θ) with respect to θ and setting it equal to zero, the result is Z ∞ F (t)e−θF (t) c1 G(t) − c2 G(t) dt = c0 . (9) C3 (0) =
0
In particular, when F (t) = 1 − e−λt and G(t) = 1 − e−wt , the expected cost in (8) is Z ∞ c1 C3 (θ) = + c2 1 − exp(−θe−λt dt + c0 θ w 0 Z ∞ − (c1 + c2 ) e−wt 1 − exp(−θe−λt ) dt 0
∞ ∞ X (−θ)j 1 c1 c2 X (−θ)j − + c0 θ + (c1 + c2 ) . = w λ j=1 j!j j! w + jλ j=1
(10)
From (9) and (10), an optimal θ1∗ satisfies (c1 + c2 )
X (−θ)j 1 1 − c2 = c0 , w + (j + 1)λ j! (j + 1)λ j=0
∞ X (−θ)j
1 1 − c2 1 − e−θ = c0 . w + (j + 1)λ λθ
j=0
i.e., (c1 + c2 )
∞
∞ X (−θ)j
j=0
j!
j!
(11)
Table 1 presents the optimal n∗1 which satisfies (3) and θ1∗ for different λ/w when c2 /c1 = 0.25, 0.5, 0.75 and (λc0 )/c1 = 0.1. Both n∗1 and θ∗ increase with λ/w and decrease with c2 /c1 . It is noted that n∗1 ≤ θ1∗ for λ/w ≥ 2; however, the differences between them are small. 3.2. Standby System with Random Number of Units We consider a standby system with N units of which N have a Poisson distribution with mean θ. Then the failure distribution of the system is F (t|θ) =
∞ X
F (n) (t)
n=0
6
θn −θ e . n!
(12)
In particular, when F (t) = 1 − e−λt , (12) becomes F (t|θ) =
∞ X θn n=0
n!
e−θ
∞ X (λt)j j=n
j!
e−λt .
(13)
Thus, when G(t) = 1 − e−wt , the expected cost in (4) is ∞ n−1 Z X c1 θn −θ X ∞ (λt)j −λt C4 (θ) = + c2 e e dt + c0 θ w n! j! 0 n=0 j=0 ∞ n−1 Z ∞ X θn X (λt)j −λt −wt −θ − (c1 + c2 ) e e e dt n! j! n=0 j=0 0 n ∞ c1 c2 θ c1 + c2 X θn −θ λ = + − e 1− + c0 θ w λ w n=0 n! λ+w c1 c2 θ c1 + c 2 θw = + − 1 − exp − + c0 θ. w λ w λ+w
(14)
Clearly,
c1 , C4 (∞) ≡ lim C2 (θ) = ∞. θ→∞ w Differentiating C4 (θ) with respect to θ and setting it equal to zero, the resulting equation is λ θw c2 + c 0 λ exp − , (15) = λ+w λ+w c1 + c 2 C4 (0) =
whose left-hand side decreases strictly from λ/(λ + w) to zero. Thus, if λ c2 + c 0 λ > λ+w c1 + c 2 then, there exists a positive and unique θ2∗ (0 < θ2∗ < ∞) which satisfies (15). Table 2 presents the optimal n∗2 and θ2∗ for different λ/w when c2 /c1 = 0.25, 0.5, 0.75 and (λc0 )/c1 = 0.1. The optimal n∗2 and θ2∗ increase with λ/w and decrease with c2 /c1 just as the tendency in Table 1. It is also noted that both n∗2 and θ2∗ are much larger than n∗1 and θ1∗ when λ/w ≥ 10.
7
4. Redundant Systems for N Works Suppose that redundant systems with n units operate for a job consisting of N works (N = 1, 2, . . . ) (Fig. 2): It is assumed that Yj (j = 1, 2, . . . , N) is the processing time for each work and has an independent, identical distribution Pr{Yj ≤ t} ≡ G(t) with a finite mean 1/w, which is called N tandem works. Then, the probability that N works finish before time t is Pr{Y = Y1 + Y2 + · · · + YN ≤ t} = G(N ) (t), where Φ(j) (t) (j = 1, 2, . . . ) denotes the j-fold Stieltjes convolution of any function Φ(t) and Φ(0) (t) ≡ 1 for t ≥ 0. The assumptions of the redundant systems and the cost structures are the same as in previous sections. 4.1. Parallel System for N Works We consider a parallel system with n units and a linear cost structure as follows: If a parallel system with n units fails at time t and N works finish at time u(< t), the excess cost c2 (t − u) is required, and conversely, if the system fails at time u and the N works would finish at time t(> u), the shortage cost c1 (t − u) is required. Then, the total expected cost is Z ∞ Z ∞ n (N ) C5 (n, N) = c2 [1 − F (t) ] G (t)dt + c1 F (t)n 1 − G(N ) (t) dt 0
+ c0 n
0
(n, N = 1, 2, . . . ).
(16)
In particular, when G(t) = 1 − e−wt and F (t) = 1 − e−λt , the expected cost is C5 (n, N) Z ∞ n −1 X N c2 X 1 N (wt)i −wt −λt n = + c1 − (c1 + c2 ) 1 − (1 − e ) e dt + c0 n λ j=1 j w i! 0 i=0 " N # n n X X c2 1 N c1 + c 2 w n 1 + c1 + (−1)j 1− + c0 n = j λ j=1 j w λ j jλ + w j=1 (n, N = 1, 2, . . . ).
8
(17)
From the inequality C5 (n + 1, N) − C5 (n, N) ≥ 0, the following inequality holds: ( N ) n X 1 w c2 1 λ n j (−1) 1− ≤ + c0 . j j+1 (j + 1)λ + w λ n+1 c1 + c 2 j=0
(18)
Letting L5 (n) be the left-hand side of (18), L5 (n) − L5 (n + 1) =
n X
(−1)j
j=0
n j
1 j+2
(
w 1− (j + 2)λ + w
N )
> 0.
Thus, L5 (n) decreases strictly to 0, and the left-hand side of (18) decreases to c0 λ/(c1 + c2 ). Therefore, there exists a finite n∗5 (1 ≤ n∗5 < ∞) which satisfies (18) and increases with N. Next, from the inequality C5 (n, N + 1) − C5 (n, N) ≥ 0, the following inequality holds: n X j=1
(−1)
j+1
n j
w jλ + w
N +1
≤
c1 . c1 + c 2
(19)
It is easily proved that the left-hand side of (19) decreases strictly to 0. Therefore, there exists a finite and unique minimum N5∗ which satisfies (19) and increases with n. Table 3 presents the optimal n∗5 and N5∗ for different n and λ/w when c2 /c1 = 0.25 and (λc0 )/c1 = 0.1. The optimal n∗5 increases with λ/w and N, but the optimal N5∗ decreases with λ/w and increases with n. It is of interest that when λ/w = 0.6, n∗5 = N. 4.2. Standby System for N Works In this section, we consider a standby system with n units for the N consecutive processing works. Then, replacing F (t)n in (16) by F (t)(n) formally,
9
the total expected cost is Z ∞ Z t (N ) C6 (n, N) = c2 (t − u)dG (u) dF (n) (t) 0 0 Z ∞ Z t (n) + c1 (t − u)dF (u) dG(N ) (t) + c0 n Z0 ∞ 0 = c2 1 − F (n) (t) G(N ) (t)dt Z 0∞ + c1 F (n) (t) 1 − G(N ) (t) dt + c0 n 0 Z ∞ n N = c2 + c1 − (c1 + c2 ) 1 − F (n) (t) 1 − G(N ) (t) dt + c0 n λ w 0 (n, N = 1, 2, . . . ). (20) In particular, when G(t) = 1 − e−wt and F (t) = 1 − e−λt , the expected cost is C6 (n, N) = j i n−1 N −1 n N c1 + c 2 X X i + j λ w c2 + c 1 − + c0 n i λ w w + λ j=0 i=0 w+λ w+λ (n, N = 1, 2, . . . ).
(21)
We find an optimal n∗6 given N ≥ 1 and N6∗ given n ≥ 1, respectively. From the inequality C6 (n + 1, N) − C6 (n, N) ≥ 0, the following inequality holds: n N i −1 w+λ X λ w c2 n+i ≤ + c0 . (22) i w+λ w + λ λ c + c 1 2 i=0
Letting L6 (n) be the left-hand side of (22), N n λ w n+N L6 (n) − L6 (n + 1) = > 0. N −1 w+λ w+λ
Thus, L6 (n) decreases strictly to 0. Therefore, there exists a finite and unique minimum n∗6 (1 ≤ n∗6 < ∞) which satisfies (22) and increases with N. In particular, when N = 1, the optimal n∗6 is given by an integer which satisfies c2 + c 0 λ λ c2 + c 0 λ λ ∗ log / log − 1 ≤ n4 < log / log . c1 + c 2 w+λ c1 + c 2 w+λ 10
Next, from the inequality C6 (n, N + 1) − C6 (n, N) ≥ 0, the resulting inequality is
w w+λ
N +1 X n−1 j=0
N +j j
λ w+λ
j
≤
c1 , c1 + c 2
(23)
whose left-hand side decreases strictly to 0. Therefore, there exists a finite and unique minimum N6∗ (1 ≤ N6∗ < ∞) which satisfies (23) and increases with n. In particular, when n = 1, the optimal N6∗ is given by an integer which satisfies c1 w c1 w ∗ log / log − 1 ≤ N4 < log / log . c1 + c 2 w+λ c1 + c 2 w+λ Table 4 presents the optimal n∗6 and N6∗ for different n and λ/w when c2 /c1 = 0.25 and (λc0 )/c1 = 0.1 and exhibits a similar tendency to Table 3. Notice that when λ/w = 0.7, 0.8, n∗6 = N. Comparing Tables 3 and 4, it is easily noted that n∗5 ≥ n∗6 and N5∗ ≤ N6∗ , because a parallel system needs more units than a standby system for working the same number of works. 5. Redundant Systems with Random Number of Works It has been assumed that the number N of works is constant and is previously given. Suppose that the system operates for a job with works which arrive at a counter per day, per week, or per month, and so on. In this case, the number of works is not constant; however, it may be estimated statistically by a probability distribution. According to the assumption in Section 4, the processing time of each work has an independent and identical distribution G(t). 5.1. Parallel System for Random Number of Works In this section, a parallel system with n units operates for a random number of works. The random number can be estimated by a Poisson distribution with mean β. Then the probability that random works will be finished before time t is ∞ X β j −β (j) e G (t). j! j=0 11
Thus, the total expected cost is, from (16), ∞ X βj
C7 (n|β) = c2
j!
j=0 ∞ X
+ c1
j=0
−β
e
∞
Z
[1 − F (t)n ] G(j) (t)dt
0
β j −β e j!
Z
∞
F (t)n 1 − G(j) (t) dt + c0 n.
0
(24)
In particular, when G(t) = 1 − e−wt and F (t) = 1 − e−λt , the expected cost is, from (24) n β c2 X 1 C7 (n|β) = + c1 + c0 n λ j=1 j w
− (c1 + c2 )
Z
∞
0
=
c2 λ
n X
1− 1−e
−λt n
∞ X βj j=0
j!
−β
e
j−1 X (wt)i i=0
i!
e−wt dt
1 β + c1 + c0 n j w j=1 n c1 + c 2 X jλβ n 1 j (−1) 1 −exp − (n = 0, 1, 2, · · · ). + j λ j jλ + w j=0 (25)
From the inequality C7 (n+1|β)−C7 (n|β) ≥ 0, the following inequality holds: n X j=0
(−1)
j
n j
1 j+1
(j + 1)λβ c2 1 λ 1 − exp − ≤ + c0 . (j + 1)λ + w λ n+1 c1 + c 2 (26)
Letting L7 (n) be the left-hand side of (26), L7 (n)−L7 (n+1) =
n X j=0
(−1)
j
n j
1 j+2
(j + 2)λβ 1 − exp − (j + 2)λ + w
> 0.
Therefore, by using similar arguments in Section 4.1, there exists a finite n∗7 (1 ≤ n∗7 < ∞) which satisfies (26) and increases with β.
12
5.2. Standby System for Random Number of Works We consider a standby system with n units which operates for a job with a random number of works according to a Poisson distribution with mean β. Then, the total expected cost is, from (20), n β + c1 + c0 n λ w Z ∞ X β j −β ∞ − (c1 + c2 ) e 1 − F (n) (t) 1 − G(j) (t) dt. j! 0 j=0
C8 (n|β) = c2
(27)
In particular, when G(t) = 1 − e−wt and F (t) = 1 − e−λt , the expected cost is, from (27) n β + c1 + c0 n λ w j i X n−1 ∞ ∞ c1 + c 2 X X i + j λ w β k −β − e i w + λ j=0 i=0 w+λ w+λ k!
C8 (n|β) = c2
k=i+1
(n = 1, 2, · · · ).
(28)
From the inequality C8 (n+1|β)−C8 (n|β) ≥ 0, the following inequality holds:
λ w+λ
n X ∞ i=0
n+i i
w w+λ
i X ∞ w+λ β k −β c2 e ≤ + c0 . (29) k! λ c 1 + c2 k=i+1
Letting L8 (n) be the left-hand side of (29), n i i n X λ w β −β n+i L8 (n) − L8 (n + 1) = e >0 i−1 w + λ w + λ i! i=1 Therefore, by using similar arguments to those in Section 4.2, there exists a finite n∗8 (1 ≤ n∗8 < ∞) which satisfies (29). Table 5 presents the optimal n∗7 and n∗8 for different β and λ/w when c2 /c1 = 0.25 and (λc0 )/c1 = 0.1. It exhibits a similar tendency to Table 3 and Table 4 in that n∗7 ≥ n∗8 because a parallel system needs more units than a standby system for working the same number of works. Another interesting point is that when λ/w = 0.4, n∗7 = β and λ/w = 0.7, 0.8, n∗8 = β.
13
6. Conclusion This paper has summarized the optimal redundant policies when the job processing time is a random variable. For a single work, we have derived the optimal number of redundant units as well as the optimal expected number of units when the redundant units can only be estimated by a known probability distribution. The optimal numbers of units for both standby and parallel systems for N tandem works have been reviewed and, subsequently, the extended models when processing random numbers of works are studied. These results would be useful to set up the system configurations and to estimate how many units are needed in order to achieve N works in the most economic way. However, there are still many different kinds of works in the real world and a variety of redundant systems have been utilized for these works. Nevertheless, there are relatively few papers that have studied the redundancy optimization problems for continuously processing jobs while considering cost efficiency. Thus there is still much room for the further development and modification of our research. Different structures of redundancy can be modified to fit more practical and complicated applications as further studies. By modifying and extending the techniques in this research, these applications offer interesting topics for researchers and are useful in solving the redundant number problems in practical fields. Acknowledgment This research is supported by the National Science Council of the Republic of China, under the Grant No. NSC 100-2628-E030-002. [1] Pinedo M. Scheduling Theory, Algorithms and Systems. New Jersey: Prentice Hall; 2008. [2] Nakagawa T. A replacement policy maximizing MTTF of a system with several spare units. IEEE Trans Reliab, 1989;38(2):210-211. [3] Nakagawa T, Yun WY. Note on MRRF of a parallel system. Int J Reliab Qual Saf Eng 2011;18:1-8. [4] Coit DW, Smith AE. (1996). Reliability optimization of series-parallel systems using a genetic algorithm. IEEE Trans Reliab 1996;45:254-260.
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[5] Ushakov IA. Handbook of Reliability Engineering. New York: Wiley; 1994. [6] Kuo W, Prasad VR. An annotated overview of system reliability optimization. IEEE Trans Reliab, 2000;49(2):176-187. [7] Kuo W, Wan R. Recent advances in optimal reliability allocation. IEEE Trans Syst Man and Cyber - Part A: Syst and Hum, 2008;38:143-156. [8] Chen M, Nakagawa T. Optimal scheduling of random works with reliability application. To appear in Asia Pac J Oper Res. [9] Nakagawa T, Yasui K. Note on optimal redundant policies for reliability models. J Qual in Maint Eng, 2005;11:82-96. [10] Nakagawa T. Optimal number of units for a parallel system. J Appl Prob, 1984;21:431-436. [11] Nakagawa T. Maintenance Theory of Reliability. London: SpringerVerlag; 2005. [12] Nakagawa T. (2008). Advanced Reliability Models and Maintenance Policies. London: Springer-Verlag; 2008. [13] Durham SD, and Padgett WJ. Estimation for a probabilistic stressstrength model. IEEE Trans Reliab, 1990;39:199-203.
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• Figure 1: Excess cost and shortage cost of a job with one single work. • Figure 2: Process time of N works.
16
Table 1: Optimal n∗1 and θ1∗ for different λ/w when c2 /c1 = 0.25, 0.5, 0.75 and (λc0 )/c1 = 0.1.
λ/w 0.5 1 2 3 4 5 10 20 50 100
c2 /c1 =0.25 c2 /c1 =0.5 n∗1 θ1∗ n∗1 θ1∗ 1 0.427 0 0 1 1.687 1 0.831 3 3.210 2 2.419 4 4.240 3 3.489 4 5.004 4 4.297 5 5.595 4 4.933 7 7.248 6 6.774 8 8.436 8 8.146 9 9.320 9 9.188 9 9.650 9 9.581
c2 /c1 =0.75 n∗1 θ1∗ 0 0 1 0.127 1 1.761 2 2.846 3 3.676 4 4.341 6 6.326 7 7.863 9 9.058 9 9.512
Table 2: Optimal n∗2 and θ2∗ for different λ/w when c2 /c1 = 0.25, 0.5, 0.75 and (λc0 )/c1 = 0.1.
λ/w 0.5 1 2 3 4 5 10 20 50 100
c2 /c1 =0.25 n∗2 θ2∗ 1 0.262 1 1.159 3 2.603 4 3.941 5 5.249 6 6.543 13 12.954 26 25.708 64 63.911 127 127.565
c2 /c1 =0.5 c2 /c1 =0.75 n∗2 θ2∗ n∗2 θ2∗ 0 0.273 0 0 1 0.446 1 0.058 2 1.532 1 0.950 3 2.514 2 1.738 4 3.466 3 2.495 5 4.404 3 3.239 9 9.031 7 6.895 18 18.218 14 14.140 46 45.721 36 25.819 92 91.540 72 71.931
17
Table 3: Optimal n∗5 and N5∗ when
λ w
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1 1 1 1 1 1 1 1 1 1
2 1 1 1 1 2 2 2 2 3
N 3 1 1 1 2 2 3 3 4 4
4 1 1 2 3 3 4 5 5 6
5 1 2 2 3 4 5 6 6 7
1 2 1 1 1 1 1 1 1 1
Table 4: Optimal n∗6 and N6∗ when
λ w
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1 1 1 1 1 1 1 1 1 1
2 1 1 1 1 1 2 2 2 2
N 3 1 1 1 2 2 2 3 3 4
4 1 1 2 2 3 3 4 4 5
5 1 1 2 3 3 4 5 5 6
18
1 2 1 1 1 1 1 1 1 1
c2 c1
2 5 2 1 1 1 1 1 1 1
c2 c1
2 8 1 2 2 1 1 1 1 1
= 0.25 and
n 3 8 4 2 2 1 1 1 1 1
= 0.1
4 5 10 12 5 6 3 4 2 2 2 2 1 1 1 1 1 1 1 1
= 0.25 and
n 3 15 7 4 3 2 2 2 1 1
λc0 c1
4 22 11 7 5 4 3 3 2 2
λc0 c1
5 30 15 9 7 5 4 4 3 3
= 0.1
Table 5: Optimal n∗7 and n∗8 for different β and λ/w when c2 /c1 = 0.25 and (λc0 )/c1 = 0.1
1 λ w
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
n∗7 0 1 1 1 2 2 2 3 3
β 3
2 n∗8 0 0 0 0 1 1 1 1 1
n∗7 0 1 1 2 3 3 4 4 5
n∗8 0 1 1 1 1 2 2 2 2
n∗7 1 1 2 3 4 4 5 6 7
4 n∗8 0 1 1 2 2 3 3 3 4
n∗7 1 2 3 4 5 6 7 7 8
5 n∗8 1 1 2 2 3 3 4 4 5
n∗7 1 2 3 5 6 7 8 8 9
n∗8 1 1 2 3 3 4 5 5 6
Figure 1: Excess cost and shortage cost of a job with one single work.
Figure 2: Process time of N works.
19
Optimal redundant systems for works with random processing time M. Chen, T. Nakagawa
www.elsevier.com/locate/ress
PII: DOI: Reference:
S0951-8320(13)00044-6 http://dx.doi.org/10.1016/j.ress.2013.02.009 RESS4785
To appear in:
Reliability Engineering and System Safety
Received date: Revised date: Accepted date:
17 August 2011 10 January 2013 9 February 2013
Cite this article as: M. Chen and T. Nakagawa, Optimal redundant systems for works with random processing time, Reliability Engineering and System Safety, http://dx.doi.or g/10.1016/j.ress.2013.02.009 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Optimal Redundant Systems for Works with Random Processing Time M. Chena,∗, T. Nakagawab,∗ a
Graduate Institute of Business Administration, Fu-Jen Catholic University, Taipei, 24205, R.O.C. b Department of Business Administration, Aichi Institute of Technology, 1247 Yachigusa, Yakusa-cho, Toyota, 470-0392, Japan
Abstract This paper studies the optimal redundant policies for a manufacturing system processing jobs with random working times. The redundant units of the parallel systems and standby systems are subject to stochastic failures during the continuous production process. First, a job consisting of only one work is considered for both redundant systems and the expected cost functions are obtained. Next, each redundant system with a random number of units is assumed for a single work. The expected cost functions and the optimal expected numbers of units are derived for redundant systems. Subsequently, the production processes of N tandem works are introduced for parallel and standby systems, and the expected cost functions are also summarized. Finally, the number of works is estimated by a Poisson distribution for the parallel and standby systems. Numerical examples are given to demonstrate the optimization problems of redundant systems. Keywords: Random processing time, Parallel system, Standby system, Random number of units, Random number of works, Optimal redundant policy
Correspondence to: Mingchih Chen, Graduate Institute of Business Administration, Fu-Jen Catholic University, Taipei, 24205, R.O.C., Tel.: +886-2-29053895; Fax: +886-229052015; E-mail: [email protected] ∗
Preprint submitted to Reliability Engineering & System Safety
February 21, 2013
1. Introduction Manufacturing systems in the real world are usually subject to much uncertainty or randomness. Such uncertainty might result from the machine failures or the variable process time required by each job [1]. Redundancy is commonly used in practical applications to ensure a high degree of reliability when the components of operating systems are subject to stochastic failures. There are two basic ways to provide component redundancy, namely, parallel redundancy and standby redundancy. For a system with several spare units [2], an optimal replacement policy which maximized mean time-to-failure (MTTF) was discussed. Recently, a parallel system with n identical units was studied and a simple asymptotic method to obtain MTTF is proposed in [3]. Many researchers have developed vast number of studies on this reliability field such as [4], [5] and so on. There are various optimization approaches and techniques such as dynamic programming, integer programming, heuristics and meta-heuristics adopted in these studies; see, for example, [6], [7] for an extensive overview. Another important issue of redundancy is how many units and what kinds of redundant systems should be provided to optimize the expected costs [8]. Nakagawa and Yasui [9] summarized some research results for a parallel system with n units and obtained the optimal number of units which minimized the expected costs. However, there are still relatively few papers that deal with the optimal redundant number problems based on cost considerations as in [10] and [11]. Our main purpose in this paper is to survey some redundant models in [11] and [12], and to add some new interesting results for a job with random working times by using reliability theory. A job with a single work having random processing time is considered in Section 2, with the expected cost functions being derived for two different redundant systems, namely, parallel and standby. Next, redundant systems with random numbers of units are assumed when processing a single work. The expected cost functions and the optimal expected numbers of units are derived in Section 3. Subsequently, a job with N tandem works is introduced that has a working time Y as its total operating time. A job consisting of N random working times would mean that the completion time of this job is also a random variable. Such a problem involves comparing variables Y and X, which is well-known as the stress-strength model in [13]. By introducing the redundant, excess and shortage costs, the total expected cost functions for standby and parallel systems under a linear cost structure are summarized in Section 4. Then, the 2
optimal redundant numbers which minimize the total expected costs for each system are derived. Two numerical examples are given to compare the effects of the redundant systems in which the optimal number n∗ of redundant units for the given N works, and conversely, the optimal number N ∗ of works for the given n units are obtained. In Section 5, the number of works is estimated by a Poisson distribution for the parallel and standby systems. Numerical examples are given to demonstrate the optimization problems of redundant units with a random number of works. Finally, the conclusions are provided in Section 5. 2. Redundant Systems for a Work Suppose that redundant systems with n units are operating for a job with one single work: It is assumed that each unit is independent and has an identical failure distribution F (t) with finite mean 1/λ. An operating cost c0 n occurs to the redundant systems with n units. The job has a working time Y , which R ∞is assumed to have a general distribution G(t) with a mean time 1/w ≡ 0 G(t)dt, where Φ(t) ≡ 1 − Φ(t) for any function Φ(t). In the following sections, two kinds of redundant systems are considered: parallel systems and standby systems. 2.1. Parallel System for a Work The following linear costs are set up for the expected cost functions: If a parallel system with n units fails at time t and the work of a job finishes at time u (< t), the excess cost c2 (t − u) is required, and conversely, if the system fails at time u and the work would finish at time t (> u), the shortage cost c1 (t − u) is required (see Fig. 1). Adding the operating cost c0 n, the total expected cost is Z ∞ Z t n C1 (n) = c1 (t − u)dF (u) dG(t) 0 0 Z ∞ Z t + c2 (t − u)dG(u) dF (t)n + c0 n 0 0 Z ∞ Z ∞ n = c1 F (t) G(t)dt + c2 [1 − F (t)n ] G(t)dt + c0 n 0
0
(n = 0, 1, 2, · · · ).
3
(1)
Clearly,
c1 , C1 (∞) = ∞. w Thus, there exists a finite number n∗1 (0 ≤ n∗1 < ∞) which minimizes C1 (n). In, particular, when F (t) = 1 − e−λt and G(t) = 1 − e−wt , the expected cost in (1) is Z ∞ n n −wt c1 c2 X 1 C1 (n) = + − (c1 + c2 ) 1 − 1 − e−λt e dt + c0 n w λ j=1 j 0 n n X 1 c1 c2 X 1 n j = + + (c1 + c2 ) (−1) + c0 n j w λ j=1 j jλ + w j=1 C1 (0) =
(n = 0, 1, 2, · · · ).
(2)
From the inequality C1 (n + 1) − C1 (n) ≥ 0, the following inequality holds: n X 1 c2 1 n j (−1) ≤ + c0 . (3) j (j + 1)λ + w (n + 1)λ c1 + c 2 j=0 Letting L1 (n) be the leftt-hand side of (3), n X λ n j L1 (n) − L1 (n + 1) = (−1) > 0. j (j + 2)λ + w j=0 Thus, L1 (n) decreases strictly to 0, and the left-hand side of (3) decreases strictly to c0 /(c1 + c2 ). Therefore, there exists a finite and unique n∗1 (0 ≤ n∗1 < ∞) which satisfies (3). Clearly, if c1 ≤ c2 (w/λ) + c0(λ + w) then n∗1 = 0. 2.2. Standby System for a Work We consider a standby system with n units under the same cost structure in Section 2.1. Then, the total expected cost is Z ∞ Z t (n) C2 (n) = c1 (t − u)dF (u) dG(t) 0 0 Z ∞ Z t + c2 (t − u)dG(u) dF (n) (t) + c0 n 0 0 Z ∞ c1 n = + c2 − (c1 + c2 ) 1 − F (n) (t) G(t)dt + c0 n w λ 0 (n = 0, 1, 2, · · · ). (4) 4
In particular, when F (t) = 1 − e−λt and G(t) = 1 − e−wt , the expected cost in (4) is n c1 n c1 + c 2 λ C2 (n) = + c2 − 1− + c0 n (n = 0, 1, 2, · · · ). (5) w λ w w+λ From the inequality C2 (n+1)−C2 (n) ≥ 0, the following inequality is derived: n w+λ λ c2 ≤ + c0 , (6) w+λ λ c1 + c 2 whose left-hand side decreases from 1 to zero. Therefore, there exists a finite and unique minimum n∗2 (0 ≤ n∗2 < ∞) which satisfies (6). Clearly, if c1 ≤ c2 (w/λ) + c0 (w + λ) then n∗2 = 0, i.e., we would provide no redundant system for such a job. 3. Redundant Systems with Random Number of Units It has been assumed that the number n of units is constant and is previously given. However, because objective systems would be complex or each redundant unit could experience stochastic failures, the number of remaining working units might not be a fixed number all the time. In this case, the number of units for redundant systems could be estimated statistically by a probability distribution. The processing time of each job is assumed to have a general distribution G(t) with a mean time 1/w as in Section 2. For such systems, the optimal expected numbers of units for processing a single work are derived to minimize the total expected costs. 3.1. Parallel System with Random Number of Units We consider a parallel system with N units in which N is a random variable following a Poisson distribution with mean θ. Then, the failure distribution of the system is F (t|θ) =
∞ X
F (t)n
n=0
θn −θ e = e−θF (T ) . n!
Thus, the expected cost is, from (1) Z ∞h i c1 −θF (t) C3 (θ) = + c2 1−e dt w 0 Z ∞h i − (c1 + c2 ) 1 − e−θF (t) G(t)dt + c0 θ. 0
5
(7)
(8)
Clearly,
c1 , C3 (∞) ≡ lim C3 (θ) = ∞. θ→∞ w Differentiating C3 (θ) with respect to θ and setting it equal to zero, the result is Z ∞ F (t)e−θF (t) c1 G(t) − c2 G(t) dt = c0 . (9) C3 (0) =
0
In particular, when F (t) = 1 − e−λt and G(t) = 1 − e−wt , the expected cost in (8) is Z ∞ c1 C3 (θ) = + c2 1 − exp(−θe−λt dt + c0 θ w 0 Z ∞ − (c1 + c2 ) e−wt 1 − exp(−θe−λt ) dt 0
∞ ∞ X (−θ)j 1 c1 c2 X (−θ)j − + c0 θ + (c1 + c2 ) . = w λ j=1 j!j j! w + jλ j=1
(10)
From (9) and (10), an optimal θ1∗ satisfies (c1 + c2 )
X (−θ)j 1 1 − c2 = c0 , w + (j + 1)λ j! (j + 1)λ j=0
∞ X (−θ)j
1 1 − c2 1 − e−θ = c0 . w + (j + 1)λ λθ
j=0
i.e., (c1 + c2 )
∞
∞ X (−θ)j
j=0
j!
j!
(11)
Table 1 presents the optimal n∗1 which satisfies (3) and θ1∗ for different λ/w when c2 /c1 = 0.25, 0.5, 0.75 and (λc0 )/c1 = 0.1. Both n∗1 and θ∗ increase with λ/w and decrease with c2 /c1 . It is noted that n∗1 ≤ θ1∗ for λ/w ≥ 2; however, the differences between them are small. 3.2. Standby System with Random Number of Units We consider a standby system with N units of which N have a Poisson distribution with mean θ. Then the failure distribution of the system is F (t|θ) =
∞ X
F (n) (t)
n=0
6
θn −θ e . n!
(12)
In particular, when F (t) = 1 − e−λt , (12) becomes F (t|θ) =
∞ X θn n=0
n!
e−θ
∞ X (λt)j j=n
j!
e−λt .
(13)
Thus, when G(t) = 1 − e−wt , the expected cost in (4) is ∞ n−1 Z X c1 θn −θ X ∞ (λt)j −λt C4 (θ) = + c2 e e dt + c0 θ w n! j! 0 n=0 j=0 ∞ n−1 Z ∞ X θn X (λt)j −λt −wt −θ − (c1 + c2 ) e e e dt n! j! n=0 j=0 0 n ∞ c1 c2 θ c1 + c2 X θn −θ λ = + − e 1− + c0 θ w λ w n=0 n! λ+w c1 c2 θ c1 + c 2 θw = + − 1 − exp − + c0 θ. w λ w λ+w
(14)
Clearly,
c1 , C4 (∞) ≡ lim C2 (θ) = ∞. θ→∞ w Differentiating C4 (θ) with respect to θ and setting it equal to zero, the resulting equation is λ θw c2 + c 0 λ exp − , (15) = λ+w λ+w c1 + c 2 C4 (0) =
whose left-hand side decreases strictly from λ/(λ + w) to zero. Thus, if λ c2 + c 0 λ > λ+w c1 + c 2 then, there exists a positive and unique θ2∗ (0 < θ2∗ < ∞) which satisfies (15). Table 2 presents the optimal n∗2 and θ2∗ for different λ/w when c2 /c1 = 0.25, 0.5, 0.75 and (λc0 )/c1 = 0.1. The optimal n∗2 and θ2∗ increase with λ/w and decrease with c2 /c1 just as the tendency in Table 1. It is also noted that both n∗2 and θ2∗ are much larger than n∗1 and θ1∗ when λ/w ≥ 10.
7
4. Redundant Systems for N Works Suppose that redundant systems with n units operate for a job consisting of N works (N = 1, 2, . . . ) (Fig. 2): It is assumed that Yj (j = 1, 2, . . . , N) is the processing time for each work and has an independent, identical distribution Pr{Yj ≤ t} ≡ G(t) with a finite mean 1/w, which is called N tandem works. Then, the probability that N works finish before time t is Pr{Y = Y1 + Y2 + · · · + YN ≤ t} = G(N ) (t), where Φ(j) (t) (j = 1, 2, . . . ) denotes the j-fold Stieltjes convolution of any function Φ(t) and Φ(0) (t) ≡ 1 for t ≥ 0. The assumptions of the redundant systems and the cost structures are the same as in previous sections. 4.1. Parallel System for N Works We consider a parallel system with n units and a linear cost structure as follows: If a parallel system with n units fails at time t and N works finish at time u(< t), the excess cost c2 (t − u) is required, and conversely, if the system fails at time u and the N works would finish at time t(> u), the shortage cost c1 (t − u) is required. Then, the total expected cost is Z ∞ Z ∞ n (N ) C5 (n, N) = c2 [1 − F (t) ] G (t)dt + c1 F (t)n 1 − G(N ) (t) dt 0
+ c0 n
0
(n, N = 1, 2, . . . ).
(16)
In particular, when G(t) = 1 − e−wt and F (t) = 1 − e−λt , the expected cost is C5 (n, N) Z ∞ n −1 X N c2 X 1 N (wt)i −wt −λt n = + c1 − (c1 + c2 ) 1 − (1 − e ) e dt + c0 n λ j=1 j w i! 0 i=0 " N # n n X X c2 1 N c1 + c 2 w n 1 + c1 + (−1)j 1− + c0 n = j λ j=1 j w λ j jλ + w j=1 (n, N = 1, 2, . . . ).
8
(17)
From the inequality C5 (n + 1, N) − C5 (n, N) ≥ 0, the following inequality holds: ( N ) n X 1 w c2 1 λ n j (−1) 1− ≤ + c0 . j j+1 (j + 1)λ + w λ n+1 c1 + c 2 j=0
(18)
Letting L5 (n) be the left-hand side of (18), L5 (n) − L5 (n + 1) =
n X
(−1)j
j=0
n j
1 j+2
(
w 1− (j + 2)λ + w
N )
> 0.
Thus, L5 (n) decreases strictly to 0, and the left-hand side of (18) decreases to c0 λ/(c1 + c2 ). Therefore, there exists a finite n∗5 (1 ≤ n∗5 < ∞) which satisfies (18) and increases with N. Next, from the inequality C5 (n, N + 1) − C5 (n, N) ≥ 0, the following inequality holds: n X j=1
(−1)
j+1
n j
w jλ + w
N +1
≤
c1 . c1 + c 2
(19)
It is easily proved that the left-hand side of (19) decreases strictly to 0. Therefore, there exists a finite and unique minimum N5∗ which satisfies (19) and increases with n. Table 3 presents the optimal n∗5 and N5∗ for different n and λ/w when c2 /c1 = 0.25 and (λc0 )/c1 = 0.1. The optimal n∗5 increases with λ/w and N, but the optimal N5∗ decreases with λ/w and increases with n. It is of interest that when λ/w = 0.6, n∗5 = N. 4.2. Standby System for N Works In this section, we consider a standby system with n units for the N consecutive processing works. Then, replacing F (t)n in (16) by F (t)(n) formally,
9
the total expected cost is Z ∞ Z t (N ) C6 (n, N) = c2 (t − u)dG (u) dF (n) (t) 0 0 Z ∞ Z t (n) + c1 (t − u)dF (u) dG(N ) (t) + c0 n Z0 ∞ 0 = c2 1 − F (n) (t) G(N ) (t)dt Z 0∞ + c1 F (n) (t) 1 − G(N ) (t) dt + c0 n 0 Z ∞ n N = c2 + c1 − (c1 + c2 ) 1 − F (n) (t) 1 − G(N ) (t) dt + c0 n λ w 0 (n, N = 1, 2, . . . ). (20) In particular, when G(t) = 1 − e−wt and F (t) = 1 − e−λt , the expected cost is C6 (n, N) = j i n−1 N −1 n N c1 + c 2 X X i + j λ w c2 + c 1 − + c0 n i λ w w + λ j=0 i=0 w+λ w+λ (n, N = 1, 2, . . . ).
(21)
We find an optimal n∗6 given N ≥ 1 and N6∗ given n ≥ 1, respectively. From the inequality C6 (n + 1, N) − C6 (n, N) ≥ 0, the following inequality holds: n N i −1 w+λ X λ w c2 n+i ≤ + c0 . (22) i w+λ w + λ λ c + c 1 2 i=0
Letting L6 (n) be the left-hand side of (22), N n λ w n+N L6 (n) − L6 (n + 1) = > 0. N −1 w+λ w+λ
Thus, L6 (n) decreases strictly to 0. Therefore, there exists a finite and unique minimum n∗6 (1 ≤ n∗6 < ∞) which satisfies (22) and increases with N. In particular, when N = 1, the optimal n∗6 is given by an integer which satisfies c2 + c 0 λ λ c2 + c 0 λ λ ∗ log / log − 1 ≤ n4 < log / log . c1 + c 2 w+λ c1 + c 2 w+λ 10
Next, from the inequality C6 (n, N + 1) − C6 (n, N) ≥ 0, the resulting inequality is
w w+λ
N +1 X n−1 j=0
N +j j
λ w+λ
j
≤
c1 , c1 + c 2
(23)
whose left-hand side decreases strictly to 0. Therefore, there exists a finite and unique minimum N6∗ (1 ≤ N6∗ < ∞) which satisfies (23) and increases with n. In particular, when n = 1, the optimal N6∗ is given by an integer which satisfies c1 w c1 w ∗ log / log − 1 ≤ N4 < log / log . c1 + c 2 w+λ c1 + c 2 w+λ Table 4 presents the optimal n∗6 and N6∗ for different n and λ/w when c2 /c1 = 0.25 and (λc0 )/c1 = 0.1 and exhibits a similar tendency to Table 3. Notice that when λ/w = 0.7, 0.8, n∗6 = N. Comparing Tables 3 and 4, it is easily noted that n∗5 ≥ n∗6 and N5∗ ≤ N6∗ , because a parallel system needs more units than a standby system for working the same number of works. 5. Redundant Systems with Random Number of Works It has been assumed that the number N of works is constant and is previously given. Suppose that the system operates for a job with works which arrive at a counter per day, per week, or per month, and so on. In this case, the number of works is not constant; however, it may be estimated statistically by a probability distribution. According to the assumption in Section 4, the processing time of each work has an independent and identical distribution G(t). 5.1. Parallel System for Random Number of Works In this section, a parallel system with n units operates for a random number of works. The random number can be estimated by a Poisson distribution with mean β. Then the probability that random works will be finished before time t is ∞ X β j −β (j) e G (t). j! j=0 11
Thus, the total expected cost is, from (16), ∞ X βj
C7 (n|β) = c2
j!
j=0 ∞ X
+ c1
j=0
−β
e
∞
Z
[1 − F (t)n ] G(j) (t)dt
0
β j −β e j!
Z
∞
F (t)n 1 − G(j) (t) dt + c0 n.
0
(24)
In particular, when G(t) = 1 − e−wt and F (t) = 1 − e−λt , the expected cost is, from (24) n β c2 X 1 C7 (n|β) = + c1 + c0 n λ j=1 j w
− (c1 + c2 )
Z
∞
0
=
c2 λ
n X
1− 1−e
−λt n
∞ X βj j=0
j!
−β
e
j−1 X (wt)i i=0
i!
e−wt dt
1 β + c1 + c0 n j w j=1 n c1 + c 2 X jλβ n 1 j (−1) 1 −exp − (n = 0, 1, 2, · · · ). + j λ j jλ + w j=0 (25)
From the inequality C7 (n+1|β)−C7 (n|β) ≥ 0, the following inequality holds: n X j=0
(−1)
j
n j
1 j+1
(j + 1)λβ c2 1 λ 1 − exp − ≤ + c0 . (j + 1)λ + w λ n+1 c1 + c 2 (26)
Letting L7 (n) be the left-hand side of (26), L7 (n)−L7 (n+1) =
n X j=0
(−1)
j
n j
1 j+2
(j + 2)λβ 1 − exp − (j + 2)λ + w
> 0.
Therefore, by using similar arguments in Section 4.1, there exists a finite n∗7 (1 ≤ n∗7 < ∞) which satisfies (26) and increases with β.
12
5.2. Standby System for Random Number of Works We consider a standby system with n units which operates for a job with a random number of works according to a Poisson distribution with mean β. Then, the total expected cost is, from (20), n β + c1 + c0 n λ w Z ∞ X β j −β ∞ − (c1 + c2 ) e 1 − F (n) (t) 1 − G(j) (t) dt. j! 0 j=0
C8 (n|β) = c2
(27)
In particular, when G(t) = 1 − e−wt and F (t) = 1 − e−λt , the expected cost is, from (27) n β + c1 + c0 n λ w j i X n−1 ∞ ∞ c1 + c 2 X X i + j λ w β k −β − e i w + λ j=0 i=0 w+λ w+λ k!
C8 (n|β) = c2
k=i+1
(n = 1, 2, · · · ).
(28)
From the inequality C8 (n+1|β)−C8 (n|β) ≥ 0, the following inequality holds:
λ w+λ
n X ∞ i=0
n+i i
w w+λ
i X ∞ w+λ β k −β c2 e ≤ + c0 . (29) k! λ c 1 + c2 k=i+1
Letting L8 (n) be the left-hand side of (29), n i i n X λ w β −β n+i L8 (n) − L8 (n + 1) = e >0 i−1 w + λ w + λ i! i=1 Therefore, by using similar arguments to those in Section 4.2, there exists a finite n∗8 (1 ≤ n∗8 < ∞) which satisfies (29). Table 5 presents the optimal n∗7 and n∗8 for different β and λ/w when c2 /c1 = 0.25 and (λc0 )/c1 = 0.1. It exhibits a similar tendency to Table 3 and Table 4 in that n∗7 ≥ n∗8 because a parallel system needs more units than a standby system for working the same number of works. Another interesting point is that when λ/w = 0.4, n∗7 = β and λ/w = 0.7, 0.8, n∗8 = β.
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6. Conclusion This paper has summarized the optimal redundant policies when the job processing time is a random variable. For a single work, we have derived the optimal number of redundant units as well as the optimal expected number of units when the redundant units can only be estimated by a known probability distribution. The optimal numbers of units for both standby and parallel systems for N tandem works have been reviewed and, subsequently, the extended models when processing random numbers of works are studied. These results would be useful to set up the system configurations and to estimate how many units are needed in order to achieve N works in the most economic way. However, there are still many different kinds of works in the real world and a variety of redundant systems have been utilized for these works. Nevertheless, there are relatively few papers that have studied the redundancy optimization problems for continuously processing jobs while considering cost efficiency. Thus there is still much room for the further development and modification of our research. Different structures of redundancy can be modified to fit more practical and complicated applications as further studies. By modifying and extending the techniques in this research, these applications offer interesting topics for researchers and are useful in solving the redundant number problems in practical fields. Acknowledgment This research is supported by the National Science Council of the Republic of China, under the Grant No. NSC 100-2628-E030-002. [1] Pinedo M. Scheduling Theory, Algorithms and Systems. New Jersey: Prentice Hall; 2008. [2] Nakagawa T. A replacement policy maximizing MTTF of a system with several spare units. IEEE Trans Reliab, 1989;38(2):210-211. [3] Nakagawa T, Yun WY. Note on MRRF of a parallel system. Int J Reliab Qual Saf Eng 2011;18:1-8. [4] Coit DW, Smith AE. (1996). Reliability optimization of series-parallel systems using a genetic algorithm. IEEE Trans Reliab 1996;45:254-260.
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[5] Ushakov IA. Handbook of Reliability Engineering. New York: Wiley; 1994. [6] Kuo W, Prasad VR. An annotated overview of system reliability optimization. IEEE Trans Reliab, 2000;49(2):176-187. [7] Kuo W, Wan R. Recent advances in optimal reliability allocation. IEEE Trans Syst Man and Cyber - Part A: Syst and Hum, 2008;38:143-156. [8] Chen M, Nakagawa T. Optimal scheduling of random works with reliability application. To appear in Asia Pac J Oper Res. [9] Nakagawa T, Yasui K. Note on optimal redundant policies for reliability models. J Qual in Maint Eng, 2005;11:82-96. [10] Nakagawa T. Optimal number of units for a parallel system. J Appl Prob, 1984;21:431-436. [11] Nakagawa T. Maintenance Theory of Reliability. London: SpringerVerlag; 2005. [12] Nakagawa T. (2008). Advanced Reliability Models and Maintenance Policies. London: Springer-Verlag; 2008. [13] Durham SD, and Padgett WJ. Estimation for a probabilistic stressstrength model. IEEE Trans Reliab, 1990;39:199-203.
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• Figure 1: Excess cost and shortage cost of a job with one single work. • Figure 2: Process time of N works.
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Table 1: Optimal n∗1 and θ1∗ for different λ/w when c2 /c1 = 0.25, 0.5, 0.75 and (λc0 )/c1 = 0.1.
λ/w 0.5 1 2 3 4 5 10 20 50 100
c2 /c1 =0.25 c2 /c1 =0.5 n∗1 θ1∗ n∗1 θ1∗ 1 0.427 0 0 1 1.687 1 0.831 3 3.210 2 2.419 4 4.240 3 3.489 4 5.004 4 4.297 5 5.595 4 4.933 7 7.248 6 6.774 8 8.436 8 8.146 9 9.320 9 9.188 9 9.650 9 9.581
c2 /c1 =0.75 n∗1 θ1∗ 0 0 1 0.127 1 1.761 2 2.846 3 3.676 4 4.341 6 6.326 7 7.863 9 9.058 9 9.512
Table 2: Optimal n∗2 and θ2∗ for different λ/w when c2 /c1 = 0.25, 0.5, 0.75 and (λc0 )/c1 = 0.1.
λ/w 0.5 1 2 3 4 5 10 20 50 100
c2 /c1 =0.25 n∗2 θ2∗ 1 0.262 1 1.159 3 2.603 4 3.941 5 5.249 6 6.543 13 12.954 26 25.708 64 63.911 127 127.565
c2 /c1 =0.5 c2 /c1 =0.75 n∗2 θ2∗ n∗2 θ2∗ 0 0.273 0 0 1 0.446 1 0.058 2 1.532 1 0.950 3 2.514 2 1.738 4 3.466 3 2.495 5 4.404 3 3.239 9 9.031 7 6.895 18 18.218 14 14.140 46 45.721 36 25.819 92 91.540 72 71.931
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Table 3: Optimal n∗5 and N5∗ when
λ w
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1 1 1 1 1 1 1 1 1 1
2 1 1 1 1 2 2 2 2 3
N 3 1 1 1 2 2 3 3 4 4
4 1 1 2 3 3 4 5 5 6
5 1 2 2 3 4 5 6 6 7
1 2 1 1 1 1 1 1 1 1
Table 4: Optimal n∗6 and N6∗ when
λ w
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1 1 1 1 1 1 1 1 1 1
2 1 1 1 1 1 2 2 2 2
N 3 1 1 1 2 2 2 3 3 4
4 1 1 2 2 3 3 4 4 5
5 1 1 2 3 3 4 5 5 6
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1 2 1 1 1 1 1 1 1 1
c2 c1
2 5 2 1 1 1 1 1 1 1
c2 c1
2 8 1 2 2 1 1 1 1 1
= 0.25 and
n 3 8 4 2 2 1 1 1 1 1
= 0.1
4 5 10 12 5 6 3 4 2 2 2 2 1 1 1 1 1 1 1 1
= 0.25 and
n 3 15 7 4 3 2 2 2 1 1
λc0 c1
4 22 11 7 5 4 3 3 2 2
λc0 c1
5 30 15 9 7 5 4 4 3 3
= 0.1
Table 5: Optimal n∗7 and n∗8 for different β and λ/w when c2 /c1 = 0.25 and (λc0 )/c1 = 0.1
1 λ w
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
n∗7 0 1 1 1 2 2 2 3 3
β 3
2 n∗8 0 0 0 0 1 1 1 1 1
n∗7 0 1 1 2 3 3 4 4 5
n∗8 0 1 1 1 1 2 2 2 2
n∗7 1 1 2 3 4 4 5 6 7
4 n∗8 0 1 1 2 2 3 3 3 4
n∗7 1 2 3 4 5 6 7 7 8
5 n∗8 1 1 2 2 3 3 4 4 5
n∗7 1 2 3 5 6 7 8 8 9
n∗8 1 1 2 3 3 4 5 5 6
Figure 1: Excess cost and shortage cost of a job with one single work.
Figure 2: Process time of N works.
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