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Reliability Of A fc-out-of-n System With A Single Server Extending Non-Preemptive Service To External Customers-
Part I
A. Krishnamoorthy •
Dept. of Mathematics, Cochin University of Science & Technology, Kochi-682022
e-mail: achyuthacusat@gmail.com
M. K. Sathian •
Dept. of Mathematics, Panampilly Memorial Govt. College, Chalakudy, Thrissur.
e-mail: sathianmkk@yahoo.com
Narayanan C Viswanath •
Dept. of Mathematics Govt. Engg. College, Thrissur Thrissur 680 009. e-mail: narayanan_viswanath@yahoo.com
Abstract
We study repairable k-out-of-n system with single server who provides service to external customers also. N-policy is employed for the service of main customers. Once started, the repair of failed components is continued until all components become operational. When not repairing main customers, the server attends external customers (if there is any) who arrive according to a Poisson process. Once selected, the external customers receive a service of non-preemptive nature. When at least N main customers accumulate in the system and/or when the server is busy with such customers, external customers are not allowed to join the system. Otherwise, they join an infinite capacity queue of external customers. Life time distribution of components, service time distribution of main and external customers are all assumed to follow independent exponential distributions. Steady state analysis has been carried out and several important system performance measures based on the steady state distribution derived. A numerical study comparing the current model with those in which no external customers are provided service, is carried out. This study suggests that rendering service to external customers helps to utilize the server idle time profitably, without affecting the system reliability.
Keywords: k-out-of-n system; non-preemptive service.
1 Introduction
A fc-out-of-n system can be defined as an n-component system which works if and only if at least k of its components operational. Application of such systems can be seen in many real-world phenomena. For instance almost all machines, of different complexity, are subjected to failure. One would expect a machine to work, even if some of its components have failed. A hospital providing
emergency service is a typical example.We would expect the hospital to run even if some of its doctors/nurses/other staffs are on leave since it is supposed to have these personal in excess of the actual requirement. However, keeping these extra resources could be costly and not even feasible in some cases. A probabilistic study of a real world system such as a fc-out-of-n system, often helps to develop an optimal strategy for maintaining high system reliability. Literature on such studies is vast (for example,see Chakravarthy et al.[1]).
Dudin et al.[2], Krishnamoorthy et al.[3, 4, 5] are among the studies on the reliability of a fc-out-of-n system, where the server provides service to external customers in addition to repairing failed components of the main system. Such models are suitable for many real world situations. For example, a big telecom company may decide to share its resources like optical cables, mobile towers etc., for additional revenue. In doing so there is the risk that it may lead to dissatisfaction of the companies own customers. Therefore, the company would like to develop an optimal strategy for sharing its resources. Krishnamoorthy et al.[5] studied an N-policy for rendering service to external customers. They gave priority to the main customers through N-policy: the moment N failed components of the main system get accumulated, the ongoing service of an external customer (if there is any) is preempted and service to failed components is started.
In the present study, we consider a variant of the model in [5]. We assume N-policy for starting repair of failed components. However, the priority of the main customers is a bit reduced by assuming that an ongoing service of an external customer is not preempted when the number of failed components reaches N. This can be a serious compromise on the reliability of the fc-out-of-n system. As in [5] it is assumed here also that an external customer, not allowed to join the system when the server is busy with service of main customers and/or when there are at least N failed components in the system. The external customer joins a queue of infinite capacity.
This paper is arranged as follows. In section 2 , we define the queuing model; section 3 conducts the steady state analysis, where we have obtained the stability condition explicitly and we also present an efficient method for computing the steady state probability vector. In section 4, we derive some important system performance measures and in section 5 the effect of N-policy and rendering service to external customers on the system reliability is examined. A cost function has also been studied in section 5 .
2 The queueing model
Here we consider a fc-out-of-n system with a single server, offering service to external customers also. Commencement of service to failed components of the main system is governed by N-policy. That is at the epoch the system starts with all components operational, the server starts attending one by one the external customers (if there is any).When the number of failed components in the system is > N, the server in service of external customer (if there is any) is switched on to the service of the main customers after completing the ongoing service of the external customer. We assume that the failure rate of a component is y, when i components are operational so that the inter-failure time of components of the fc-out-of-n system remains exponentially distributed with parameter A. Arrival of external customers follows a Poisson process with parameter A. External customers are not allowed to join the system when the server is busy with main customers or when there is > N failed components. An external customer, who on arrival finds an idle server is directly taken for service. Service times of main and external customers follow exponential distribution with parameters ^ and p respectively.
2.1 The Markov Chain
Let X1(t) = number of external customers in the system including the one getting service (if any) at time t,
X2 (t) = number of main customers in the system including the one getting service (if any)
at time t,
0, if the server is idle or is busy with external customers
if the server is idle or is busy with main customers. Let X(t) = (X1(t):5(t):X2(t)) then X = (X(t), t > 0} is a continuous time Markov chain on the state space
5 = {(0,0:72)/0 < 72 < N - 1} U {(/i, 0,72)//I > 1,0 < 72 < n - ft + 1} U {(/1:1:72)/7i > 0:1 < 72 < n - ft + 1}.
Arranging the states lexicographically and partitioning the state space into levels i, where each level i corresponds to the collection of the states with number of external customers in the system at any time t equal to i, we get an infinitesimal generator of the above chain as
•^10 -^00 A,
Q =
A,
A,
^0 ¿i
¿2
¿0 ¿1
¿0
In order to describe the entries in the above matrix we introduce some notations below. [(i)]
1. /m denotes an identity matrix of order m and / denotes an identity matrix of appropriate order.
2. em denotes a mx1 column matrix of 1s and e denotes a column matrix of 1s of appropriate order.
3. £m denotes a square matrix of order m defined as
/-1 if j = i: 1 < i < m £m(i:7) = (1 if j = i+1:1 < i < m-1 0 otherwise
4. £"m = Transpose (£m).
5. rm(i) denotes a 1xn row matrix whose ith entry is 1 and all other entries are zeros.
6. cm(i) = Transpose (rm(i)).
7. 0 denotes Kronecker product of matrices.
The transition within level 0 is represented by the matrix
. TBi B21 , ^10 = D D : where
= - Ah,.
B2 is a Wx(n-ft + 1) matrix whose (N:W)th entry is A and all other entries are zeroes. B3 is a Wx(n-ft + 1) matrix whose (1:1)th entry is ^ and all other entries are zeroes. £4 = A£„-fc+i + Ac„-fc+i(n - ft + 1) 0 r„-fc+i(n - ft + 1) + ^£„-fc+i'. The transition from level 0 to level 1 is represented by the matrix
-^nn —
0
WX(2n-2fc + 3-W)
^(n-fc+i)XW ^(n-fc+i)X(2n-2fc+3-W)
Transition from level 1 to 0 is represented by the matrix
0
=
0
H
wherefl = [0(„-fc+2-w)x(w-i) ^(n-fc+2-w)]. Transition within level 1 is represented by the matrix
0(n-fc+i)xw 0
flu 0
A = 0 0 where
fl3i 0 «4-
Hii = Bi - Hi2 = (N)®rn_fc + 2_„(1):
H22 = ^E-n-k+2-N
+ Àcn-k+2-N(n - k + 2 -N)0 rn-k+2-N(n - k + 2 - N) - ^In-k+2-A
H31 is an (n — k + 1)xN matrix whose (l,l)th entry is /J..
An =
AIn
0,
NX(2n-2k+3-N)
0(2n-2k+3-N)XN 0(2n-2k + 3-N)X(2n-2k + 3-N)
A, =
0
0(n-k+2-N)X(n-k + 2-N) H
0(n-k + 1)XN 0
where H = [0(n-k+2-N)x(N-i) n-k+2-N)]•
3 Steady state analysis
3.1 Stability condition
Consider the generator matrix A = A0 + A1 + A2
aen H12 0
A = 0 H 22 F23 with
F31 0 B4
F23 = [0(n-k + 2-N)X(N-1) V-In -k+2-N], F31 = № n-k + 1
(1)®rN(1).
Let ( = ((o,^,^) be the steady state vector of the generator matrix A, where
(0 = (((0,0),((0,1), ■■■,((0,N-1)), <1 = (((0 ,N), i(0,N+1)' ■■■ , ((0,n-k + 1))• (2 = (((1,1)'((1,2)' ■■■ '((1,n-k + 1)).
The Markov chain {X(t), t > 0} is stable if and only if ^A0e < $A2e (please refer Neuts [6]).
It follows that (A0e = A(0e and (A2e = ~p((0e + ^e). Therefore the stability condition
becomes
It follows from the relation (A = 0 that
X Ç0e
< 1.
From (4), it follows that Substituting this in (2) we get
M itoe + tie)
(0AEn + Ï2F31 = 0, (0^12 + Ç1H22 = 0, (1F23 + (2B4 = 0. (2 = (1F23 B4 1.
(O^EN - $1F23B41F31 =
A(oe = (—(1F23B4-1F31)(—E-1e).
Notice that the first column of the matrix F31 is —B4e and all other columns of it are zero columns. This implies that the first column of the matrix B-1F31 is —e and its all other columns are zero columns. Hence the first column of the matrix —F23B-1F31 is and all other columns are zero columns. The first entry of the row matrix —(1F23B4-1F31 is thus and its all other entries are
(1) (2)
(3)
(4)
(5)
(6) (7)
0
zeros. It can be seen that the first entry of the column matrix -£,iVie is N. These two facts together tell us that (-iF3i)(-£,i-ie) is N^u^e. Thus, equation (7) becomes
= N^e.
Adding N^e on both sides of the above equation, we get
(A + N^e = W^e + Cie):
which implies
(foe + fie) (A+W^).
Hence the stability condition (1) becomes
3.2 Computation of steady state vector
Let n = (^(0):^(1):^(2): ...) the steady state vector of the Markov chain X, where ^(0) =
(^(0i0): ^(0i1)) with ^(0.0) = (^(0i0i0): ^(0i0i1): ."■ : —1)) and (^^(0i1i1): ."' : fc + 1)). For ¿>1, = (^(i,0)::'Í(i,0):^(i,1)) with ^(1:0) = A0): ^(¿Al): : —1)), = ^(¿AW): : ^(¿An— fc+1)),
^(¿:1) = (^(¿jiji): ^(¿ji^): .■: ^(i^n—fc+1)). Now from rcQ = 0, we can write
^(0:0)^1 + ^(0:1)^3 + = 0: (8)
^(0:0)^2 + ^(0:1)^4 + = 0: (9)
and for i > 1,
^(i — lA^V + ^(1:0)^11 + ^(1:1)^31 + ^(¿+1:0)^^« = 0: (10)
^(1:0)^12 + ^¿A^ = 0: (11)
^(1:1)^4 + ^(¿ + ^0)^ = 0. (12)
^(1:0) = -^(1:0)^12 (fl221). (13) ^1) = -Vl^^1). (14)
*(U) = ^+1Atfl2№2!)tf(54—1). (15)
From (11), we get, for i > 1 From (12), we get Substituting (13) in (14), we get Substituting (15) in (10), we get
^(¿—lA^V + ^(1:0)^11 + ^(i + 1Afl12(fl2~21)fl№~1)fl31 + ^(i+1A^V = (16)
We notice that the first column of the matrix fl31 is -B4e and all other columns of fl31 are zero columns. Hence the first column of the matrix (B4"l)H31 is -e and its all other columns are zero columns. This tells us that the first column of the matrix //(B4"l)H31 is -jue and all other columns are zeros. But -^e is fl22e and hence the first column of the matrix (fl2—1)H(#4~1)H31 is e and all other columns are zeros. This fact leads us to conclude that the first column of the matrix fl12(fl2—21)//(B4"1)fl31 is fl12e = Acw(N) and all other columns are zeros. In other words
tfi2№2z1)^1)^! = Ac„(N) 0 r„(1).
Now equation (16) becomes
^ — 1:0)1/, + + ^(i + 1:0)^CW(W) 0 rW(1) + ^¿+1:0)^ = 0.
That is
^¿ — 1:0)^ + ^(1:0)^11 + ^(¿ + 1:0) (Acw(W)0rw(1)+^/w) = 0. (17)
Now from equation (9), we can write
^(0:1) = -^(0:0)^2 №~!) - ^(lA^^i^ (18) However, from equation (13), we have
^(1:0) = ^(1:0)^12 (19)
Hence equation (18) becomes
(01)
= n
(0,0)B2(B4 x) + n(lfi)Hl2(H22)H(B4 x).
Substituting (20) in (8), we get
ft(o,o)B1 + (—Tt(o,o)B2(B-1) + n(1,o)H12(H22)H(B-1))B3 + n(1fi)pIN =
Since the first column of the matrix B3 is —B4 e, a similar reasoning as for equation (16) leads us to write:
—B2(B-1)Bз=AcN(N)®rN(1)' H12(H-21)H(B-1)B3 = Acn(N) ® rN(1). Hence equation (21) becomes
n{o,o)(B1 + Acn(N) ® rw(1)) + ni1fi)(AcN(N) ® rN(1) + ~pIN) = 0. Equations (17) and (22) shows that the vector n = (^^(0,0)'^(1,0)'^(2,0)' ■■■) satisfies the relation nQ = 0, where Q isa generator matrix defined as
Aw Ao A2 A1 Ao
Q =
In the above, A10 = B1+ Acn(N) ® rN(1), A0 = MN,A1 = H11 and A2 = Acn(N) ® rw(1) + pIN. Hence the vector n is a constant multiple of the steady state vector x= (t(0),t(1), ■..) of the generator matrix Q. The vector t can be obtained by applying the matrix analytic methods (see Neuts [6]) as
T(i) = T(0)Ri, i > 0,
where the matrix R is the minimal non-negative solution of the matrix quadratic equation:
A0 + RA1 + R2A2 = 0.
Equation (23) implies
n(o,o) = xt(0),
n(i,0) = n(0,0)Rl' i >
Now the vector ft is obtained up to a constant K as ft = Kt, the other component vectors ft^fi), i > 1, f(i,1), i > 0 of n can be obtained from the equations (13), (14) and (20), up to the constant K, which is finally obtained from the normalizing condition ne = 1.
(20) (21)
(22)
(23)
(24)
4 Performance measures 4.1 Busy period of the server with the failed components of the main system
Let Ti denote the server busy period with failed components which starts with i failed components and with j external customers in the system. Consider the absorbing Markov chain Y = [Y(t), t > 0}, where Y(t) is the number of failed components of the main system, with the state space {0,1,2,...,N,N + 1,...,n — к + 1} and having infinitesimal matrix given by
HBF = \0 0 ,
\—HBFe HBF\
where
HBF = AEn-k+1 + Acn-k+1(n — k + i.)0 rn-k+1 + у.Е'п-к+г. Notice that Y(t) = 0 is an absorbing state. T is the time until absorption in the Markov chain {Y(t}} assuming that it starts in the state i. The expected value ET of Ti is therefore the ith entry of the column matrix — H—e as given by (please see Krishnamoorthy et al. [5]):
n-k + 1-i „n-к s 1 . „ -4 {^Л
ET, =±(iZi-r-1 gy + Zj-Lk+2-i (n-k + 1-j)gy). We notice that once the service of failed components starts, the external customers has no effect on it and hence ET is independent of j the number of external customers. Define
Krishnamoorthy A., Sathian M., Narayanan C Viswanath RT&A, No3 (42) RELIABILITY of k-out-of-n SYSTEM. PART I_Volume 11, September 2016
= ^AW—I) + £J=i n^'AW) and P/(i) = £°°=l rcaAi) forW < i < n - ft + 1 Py.(i) will then denote the system steady state probability just before starting service to failed components with i number of failed components. The expected length of the busy period of the server with failed components is then given by
2F=-vft+1P/(i) .
4.2 Other performance measures
1. Fraction of time the system is down,
^down = £A = 0 ^(AAn—fc + 1) + £A=0 ^"(AA^—k + 1V
2. System reliability, Prei = 1 - Pdow„.
3. Average number of external customers waiting in the queue,
Wq = 7i(£"3—=0+1 ^(AA^O + £A = 1 O'l - 1)(£?3—=0+1 ^(AAA^
4. Average number of failed components of the main system,
N/aa = £™3—=0+1 73(£H=0 ^OiAAO + £™3—=1+1 73(£A=0 ^(AAA^
5. Average number of failed components waiting when server is busy with external
customers
N£/aa = £™3—=0+1y3(£H = 1 ^(AAA)).
6. Expected number of external customers joining the system,
e3 = ^{£A=i (£^3—=!0 ^OiAAO + £A—1 ^(0i0j3)}.
7. Expected number of external customers on its arrival gets service directly
^^direct = £A=0 ^(0:0:73).
8. Fraction of time the server is busy with external customers,
^ext^usy = £A = 1 (£"3=fc0+! ^(AAA^
9. Probability that server is found idle,
^idie = £A=0 ^(0:0:73) = ^(0A0).
10. Probability that the server is found busy,
^fcusy = 1 - £7W3=!0 ^(0:0:73) = 1 - ^(0A0).
11. Expected loss rate of external customers,
e4 = ^{£H=0 (£53—ii"1 ^(AAAO + £A=i (£A~=+1 ^(AAAO).
12. Expected service completion rate of external customers,
05 = A£H=0 (£"3—=fc0+! ^(AAA^
13. Expected number of external customers when server is busy with external customers,
06 = £H=0 7i(£"3—=fc0+! ^(AAA)^
5 Numerical Study of the Performance of the System
5.1 The Effect of N Policy on the Server Busy Probability
The main purpose of introducing N-policy while studying a fc-out-of-n system with a single server offering service to external customers, in a non pre-emptive nature, was optimization of the system revenue, by utilizing the server idle time, without compromising the reliability of the system much. Tables 1 and 2 reports the variation in the server busy probability when external customers are allowed and not allowed respectively. A comparison of the two tables suggest that there is an increase in the server busy probability, when external customers are allowed. Table 3, which report the effect of the N-policy level on the fraction of time the server remains busy with external customers, tells that there is an increase in the reported measure with an increase in N. Hence, it can be concluded that the N-policy has helped in improving the attention towards external customers slightly. Now, we want to check whether the introduction of the N-policy has badly affected the system reliability.
5.2 The effect of N policy on system reliability
We study two cases X < n and X > n . We expected a decrease in Prel with an increase in N. This is because as N increases, the server spends more time for external customers, which we thought might cause a decrease in the system reliability. This was verified from Table 4, where we assumed X < /j.. However, Table 4 shows very high system reliability over 95 %. The magnitude of decrease in reliability was found lesser when the total number of components n was high. In short Table 4 shows that reliability of the system is not much affected by increasing N-policy level. In Table 5 where it was assumed that the component failure rate X is greater than their service rate ^, it was again found that Prel decreases with increase in N and that the magnitude of decrease is not high. More importantly, the reliability of the system was found less than 91.5 %. To check whether this was actually due to the introduction of external customers, we compared the system reliability of the current model with that of a fc-out-of-n system where no external customers are entertained. Table 6 shows that allowing external customers in the system has only a narrow effect on the system reliability and the decrease in reliability is actually due to the assumption X > n .
5.3 Analysis of a Cost function
Table 1 shows that as N increases, even though the server busy probability increases first, it decreases as N crosses some value. Note that the overall server busy probability is the sum of the server busy probability with external customers and the server busy probability with main customers. Table 3 shows that the fraction of time server remaining busy with external customers is ever increasing with N. Now as N increases, there is a decrease in the server busy probability with main customers. Hence, the above said behavior of the overall server busy probability can be concluded to be due to the conflicting nature of the two entities constituting it. This behavior of the server busy probability lead us to construct a cost function in the hope of finding an optimal value for the N-policy level defined as follows:
Expected cost per unit time
c3
= C1 ■ Pdown + C2 ■ Nq + C4 ■ 64 + C5 ■ Nfaii + -—+ C6- pidie
EH
In the above, C1 denote the cost per unit time incurred if the system is down, C2 denote the holding cost per unit time per external customer in the queue, C3 denote the cost incurred for starting failed components service, C4 denote the cost due to loss of 1 external customer, C5 denote the holding cost per unit time of one failed component, C6 denote the cost per unit time if the server is idle. The values of the cost function presented in Table 7, for various failure rates of the
Krishnamoorthy A., Sathian M., Narayanan C Viswanath RT&A, No3 (42) RELIABILITY of k-out-of-n SYSTEM. PART I_Volume 11, September 2016
components, shows an optimal value for N in each case.
Table 1: Variation in the server busy probability when external customers are allowed k = 20,X =
4,1 = 3.2,\i = 5.5, p = 8
N n=45 n=50 n=60 n=65
1 0.823494 0.823522 0.823529 0.823529
3 0.829935 0.829973 0.829983 0.831354
5 0.832187 0.832243 0.832256 0.832891
7 0.833255 0.833338 0.833358 0.833717
9 0.833839 0.833968 0.834 0.83423
11 0.834162 0.834367 0.834417 0.834577
13 0.834295 0.834627 0.834708 0.834827
15 0.834239 0.834789 0.834923 0.835093
17 0.833936 0.834861 0.835085 0.835224
19 0.833252 0.834829 0.835211 0.835329
21 0.831922 0.834652 0.835306 0.835413
23 0.829445 0.834239 0.835375 0.83548
25 0.824871 0.833426 0.835412 0.83553
Table 2: Variation in the server busy probability when external customers are not allowed k = 20, A = 4,n = 5.5
N n=45 n=50 n=60 n=65
1 0.72722 0.72726 0.72727 0.72727
3 0.7272 0.72726 0.72727 0.72727
5 0.72717 0.72725 0.72727 0.72727
7 0.72711 0.72724 0.72727 0.72727
9 0.72703 0.72722 0.72727 0.72727
11 0.72688 0.72719 0.72727 0.72727
13 0.72663 0.72714 0.72727 0.72727
15 0.72622 0.72706 0.72726 0.72727
17 0.7255 0.72691 0.72726 0.72727
19 0.72425 0.72666 0.72725 0.72727
21 0.72206 0.72623 0.72723 0.72726
23 0.71814 0.72546 0.7272 0.72726
0.728 0.726 0.724 0.722 0.72 0.718 0.716
0 10 20 30
Table 3: Effect of the N-policy level on the fraction of time server is busy with external customers
with ft = 20: A = 4:A= 3.2:^ = 5.5:^= 8
N n=40 n=45 n=50 n=55 n=60
1 0.096351 0.096276 0.096261 0.096257 0.096257
2 0.100557 0.100464 0.100445 0.100441 0.10044
3 0.102853 0.10274 0.102717 0.102712 0.102711
4 0.104255 0.104117 0.104089 0.104083 0.104082
5 0.105198 0.105028 0.104993 0.104986 0.104985
6 0.105882 0.105672 0.105629 0.105621 0.105619
7 0.106413 0.106153 0.1061 0.106089 0.106087
8 0.106853 0.106528 0.106462 0.106449 0.106446
9 0.107241 0.106832 0.106749 0.106733 0.106729
10 0.107605 0.107088 0.106984 0.106963 0.106958
11 0.107968 0.107313 0.10718 0.107153 0.107148
12 0.108354 0.107517 0.107348 0.107314 0.107307
13 0.108786 0.107711 0.107495 0.107451 0.107442
14 0.109291 0.107904 0.107626 0.10757 0.107559
15 0.109905 0.108106 0.107747 0.107675 0.10766
17 0.111651 0.108581 0.107976 0.107854 0.107829
19 0.114606 0.109249 0.108092 0.108008 0.107966
21 0.110301 0.108216 0.108153 0.10808
23 0.112079 0.10851 0.108308 0.108182
25 0.115216 0.108928 0.1085 0.108281
27 0.110699 0.108771 0.108387
29 0.112652 0.109196 0.108516
31 0.116153 0.10991 0.108697
33 0.111158 0.108978
35 0.113399 0.109446
Table 4: Variation in the system reliability with increase in N (A < ^ case) k = 20, A = 4, A =
3.2,^ = 5.5, p = 8
N n=40 n=45 n=50 n=55 n=60 n=65
1 0.99963 0.99993 0.99998 1 1 1
3 0.99948 0.99989 0.99998 1 1 1
5 0.99924 0.99985 0.99997 0.99999 1 1
7 0.99885 0.99977 0.99995 0.99999 1 1
9 0.9982 0.99964 0.99993 0.99998 1 1
11 0.99712 0.99942 0.99988 0.99998 1 1
13 0.9953 0.99905 0.99981 0.99996 0.99999 1
15 0.99217 0.99843 0.99968 0.99994 0.99999 1
17 0.98668 0.99736 0.99947 0.99989 0.99998 1
19 0.97689 0.9955 0.99909 0.99982 0.99996 0.99999
21 0.95915 0.99223 0.99844 0.99968 0.99994 0.99999
23 0.98638 0.9973 0.99945 0.99989 0.99998
25 0.97578 0.99528 0.99905 0.99981 0.99996
27 0.99165 0.99833 0.99966 0.99993
29 0.98509 0.99705 0.9994 0.99988
31 0.97315 0.99475 0.99894 0.99979
1.005 1
0.995 0.99 0.985 0.98 0.975 0.97 0.965 0.96 0.955
Table 5: Variation in the system reliability with increase in N (A > ^ case) A = 6:^ = 5.5: A =
3.2:^ = 8
N n=40 n=50 n=55 n=60
1 0.90191 0.91106 0.91312 0.91441
2 0.90118 0.91081 0.91297 0.91431
3 0.90041 0.91055 0.91281 0.91421
4 0.89961 0.91028 0.91264 0.91411
5 0.89876 0.91 0.91247 0.914
6 0.89758 0.90971 0.91229 0.91389
7 0.89696 0.90941 0.91211 0.91377
8 0.896 0.9091 0.91192 0.91366
9 0.895 0.90878 0.91173 0.91354
10 0.89396 0.90845 0.91153 0.91341
11 0.89287 0.90812 0.91133 0.91329
12 0.89174 0.90777 0.91112 0.91316
13 0.89055 0.90741 0.9109 0.91303
14 0.88932 0.90705 0.91068 0.91289
15 0.88804 0.90667 0.91046 0.91275
16 0.8867 0.90628 0.91 0.91261
17 0.88531 0.90589 0.90951 0.91247
18 0.88386 0.90548 0.90901 0.91232
19 0.88235 0.90507 0.90848 0.91217
21 0.88079 0.90464 0.90794 0.91186
23 0.87916 0.90421 0.90738 0.91155
25 0.90331 0.90679 0.91122
27 0.90237 0.9062 0.91088
29 0.90139 0.90558 0.91053
31 0.90036 0.90494 0.91018
33 0.8993 0.90462 0.90981
35 0.90944
37 0.90905
39 0.90866
41 0.90827
Table 6: Variation in the system reliability with increase in N (case when no external customers
are allowed) ft = 20, A = 6, ^ = 5.5
N n=40 n=45 n=50 n=55 n=60 n=65
1 0.902225 0.907874 0.911180 0.913196 0.914453 0.915247
3 0.900740 0.907001 0.910662 0.912877 0.914252 0.915120
5 0.899093 0.906080 0.910108 0.912537 0.914040 0.914985
7 0.897301 0.905082 0.909519 0.912176 0.913815 0.914843
9 0.895355 0.904014 0.908894 0.911796 0.913578 0.914693
11 0.893242 0.902873 0.908232 0.911395 0.913329 0.914537
13 0.890948 0.901655 0.907531 0.910974 0.913069 0.914373
15 0.888461 0.900358 0.906793 0.910533 0.912797 0.914202
17 0.885763 0.898979 0.906016 0.910071 0.912514 0.914025
19 0.882837 0.897514 0.905200 0.909589 0.912219 0.913841
21 0.879662 0.895960 0.904345 0.909087 0.911913 0.913651
23 0.894313 0.903450 0.908566 0.911597 0.913454
25 0.892570 0.902514 0.908025 0.911271 0.913252
27 0.901539 0.907465 0.910934 0.913044
29 0.900523 0.906886 0.910588 0.912831
31 0.899465 0.906289 0.910233 0.912613
33 0.905674 0.909868 0.912390
35 0.905041 0.909495 0.912162
37 0.909114 0.911930
39 0.908724 0.911693
41 0.908327 0.911453
43 0.911209
45 0.910961
Table 7: Analysis of a cost function for finding optimal N value, n = 50, k = 20,^ = 5.5, A = 3.2,p = 8, C1 = 2000, C2 = 20, C3 = 800, C4 = 1000, C5 = 10, C6 = 200
N A = 4 A = 4.5 A = 5 A = 5.5
1 4925.877 4937.695 5079.029 5226.181
3 4710.059 4856.852 5057.425 5221.212
5 4630.354 4825.835 5050.332 5218.775
7 4591.702 4812.151 5048.243 5216.965
9 4571.3 4806.745 5048.411 5215.313
11 4561.086 4806.248 5049.849 5213.713
13 4558.217 4809.556 5052.345 5212.268
15 4563.915 4817.604 5056.578 5211.373
17 4588.216 4835.444 5064.896 5211.922
18 4605.19 4846.938 5070.21 5212.65
19 4624.185 4859.68 5076.196 5213.701
21 4670.646 4890.628 5091.4 5217.34
23 4735.585 4934.206 5114.597 5224.719
25 4837.829 5004.721 5155.522 5240.069
27 5032.125 5144.138 5241.815 5274.736
29 5546.901 5525.659 5482.957 5371.341
31 8780.95 7911.995 6932.789 5918.758
Acknowlegment: Krishnamoorthy acknowledges financial support from Kerala State Council for Science, Technology and Environment, Grant
No.001/KESS/2013/CSTE, under the Emeritus Scientist scheme.
Authors would like to thank Prof.V.Rykov for some suggestion which improved the presentation of the paper.
References
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