Maximization of reliability of a k-out-of-n system with repair by a facility attending external customers in a retrial queue Текст научной статьи по специальности «Математика»

MAXIMIZATION OF RELIABILITY OF A K-OUT-OF-N SYSTEM WITH REPAIR BY A FACILITY ATTENDING EXTERNAL CUSTOMERS IN A RETRIAL QUEUE

A. Krishnamorthy, Vishwanath C. Narayanan, T. G. Deepak

e-mail: ak@cusat.ac.in. krishna.ak@gmail.com

Abstract. In this paper, we study a A-out-of-o system with single server who provides service to external customers also. The system consists of two parts: (i) a main queue consisting of customers (failed components of the A>out-of-n system) and (ii) a pool (of finite capacity M) of external customers together with an orbit for external customers who find the pool full. An external customer who finds the pool full on arrival, joins the orbit with probability 7 and with probability 1 — 7 leave the system forever. An orbital customer, who finds the pool full, at an epoch of repeated attempt, returns to orbit with probability 5 (< 1) and with probability 1 —6 leaves the system forever. We compute hte steady starts system size probability. Several performance measures are computed, numerical illustrations are provided.

1. Introduction

We study a A>out-of-n system with single server who provides service to external customers also as described in the following paragraphs.

The system consists of two parts: (i) a main queue consisting of customers (failed components of the /;-out-of-?i system) and (ii) a pool (of finite capacity M) of external customers together with an orbit for external customers who find the pool full. An external customer who finds the pool full 011 arrival, joins the orbit with probability 7 and with probability 1 — 7 leave tin? system forever. An orbital customer, who finds the pool full, at an epoch of repeated attempt, returns to orbit with probability 8 (< 1) and with probability 1 — 6 leaves the system forever.

The arrival process : Arrival of main customers have interoccurence time exponentially distributed with parameter A, when the number of operational components of the A>out-of-n system is i. By taking A, = — we notice that the cumulative failure rate is a constant A. We assume that the A>out-of-n system is COLD (components fail only when system is operational). The case of WARM and HOT system can be studied 011 the same lines (see Krishnamoorthy and Ushakumari [4]). External customers arrive according to a Markovian Arrival Process (MAP) with representation /.',•. / 'i • where Do and D\ are assumed to be matrices of order m. Fundamental arrival rate Xg = —irDoe

The service process : Service to the failed components of the main system is governed by the iV-policy. That is each epoch the system starts with all components operational (mi., all n components are in operation), the server starts attending one

Research supported by nbhm (DAE, Govt, of India): nbhm 48/5/2003/R&DII/3269.

by one the customers from the pool (if there is any). The moment the number of failed components of the main system reaches N, no more customer from the pool is taken for service until there is no components of the main system waiting for repair. However service of the external customer, if there is any, will not be disrupted even when N components accumulate in the main queue (that is the external customer in service will not get pre-empted on realization of the event that N components of the main system failed and got accumulated; instead the moment the service of the present external customer is completed, the server is switched to the service of main customers).

Service time of main customers follow PH distribution or order r?i and representation (a. Si) and that of external customers have PH distribution of order «2 with representation (3, So)'.

$1 and Si? are such that 5',-e + s® = 0, i — 1,2 where e is column vector of ones. The two service times are independent of each other and also independent of tin? failure of components of the main system as well as the arrival of external customers.

Objective : To utilize server idle time without affecting the system reliability.

Krishnamoorthy and Ushakumari [4] deals with the study of the reliability of a fc-out-of-n system with repairs by server in a retrial queue. They do not give any priority to the failed components of the main system nor do they investigate any control policy. Krishnamoorthy, Ushakumari and Lakshmi [5] introduced the repair of failed components of a fc-out-of-n system under tin? iV-policy. For further details one may refer to the paper and references therein as well as Ushakumari and Krishnamoorthy [7 Bocharov et til [1 examine an M/G/l/r retrial queue with priority of primary customers. They obtain the stationary distribution of the primary queue size, an algorithm for the factorial moments of the number of retrial customers and an expression for the expected number of customers in the system. Nevertheless, we wish to emphasise that their paper does not distinguish between the priority and ordinary customers. This is distinctly done in this paper (our priority customers are the failed components of the k-out-oi-n system):

We also consider an intermediate pool of finite capacity to which external customers join after seeing a busy server on arrival or after a successful retrial from the orbit. We expect that this intermediate pool from which an external customer can be selected for service, whenever the server becomes idle, will help us to decrease the server idle time.

The steady state distribution is derived. Note that the non-persistence of orbital customers together with the fact that an external customer, finding the pool full, may not join the pool ensures that even under very heavy traffic the system can attain stability. Several performance measures are obtained.

One can refer Deepak, Joshua, and Krishnamoorthy [3] for a detailed analysis of queues with pooled customers (postponed work).

2. Modelling and analysis

The following notations are used in the equal: Ni (t) — # orbital customers at time t

iV2(t) = # customers in the pool (including the one getting service, if any,) at time t.

.Y;: ; ) = # failed components (including the one under repair, if any) at time t

of a failed component of the main system 2 if the server is attending an external customer at time t

Phase of service of the customer, if any, in service at t

is a continuous time Markov chain on the state space

S = f(ii,0,j3,0,j5,0)|ji > 0; 0 < ¿3 < N 1; 1 < j5 < m \ - IC?i, J2,h, 1, J5. J6)b'i > o, 0 < ¿2 < M; 1 < J3 < « - fc + 1 1 < J5 < m\ 1 < j6 < ni |

{C?'i, J2, J3,2, j5, je)|ji >0: 1 < j2 < M;

0 < j3 < n -k+ 1; 1 < J5 < m; 1 < je < «2}

Arranging the states lexicographically, and then partitioning the state space into levels i, where each level i correspond to the collection of states with i customers in the orbit, we get the infinitesimal generator of the above chain as

0 if the server is idle f if the server is busy with repair

N5(t) = Phase of the arrival process,

I 0, if no service is going on at time t. It follows that (X(i) : t > 0} where

X(t) = (JVi(t), N2{t),Nz{t),NA{t), N5(t), JVe(i))

M i,. A0 0 0 ... .421 hi -4o 0... Q = 0 Â22 AI-2 AQ . .

where

Wo W. TT 3 II I

Wt We W4 Wt We

'5

w4 Wt We W4 Mo

where

and where

Where

I!'-.,, = Im ® St W31

0 0 0

m(ni+na) X m(ni H-na)

w32 =

WA =

0

In, s (SZa) . ^ ,

■ • * 'J m(ni+no) xmjii

~E0 0 0

0 I.\ 1 .. /:, 0

0 0 In-k-n+2®E2

/',. = im s (s°ff), /•:, =

r. _ 0 o"

2 Im S (5|a) 0

W5 -

0 0

0 im a, (S023)

m(ni -\-7i2 ) X m(ni H-ris)

Fo 0 0

0 Fi 0

0 0 f2

/;. />, ® a /', = /v_ 1 ® /•;. /;

0 D

Pi

F2 = /„_ fc+2_ JV 8 K. K = [£>1 ® 0] rHo 0

1 ® 8]

0 J

We

Ho = Z?i ® 7na, iii =

/''i ® Jni 0 0 Di ®

ii. = • im - Au for i > 1

An =

MLa 0 0 ¿9(1 -S)ILl

Li — (n - k + 2)rnn2 + (n — k + l)nini L2 - Nm + (n - k + l)i?ii?i + (M - 1 )Li

fo z, 0

A2l

1 0 WI(M-i)Li

0 0 id(l - 5)Il1

i > 1

Z, -

'Zu 0

0 /.v I /j:

0

Z'n —

0

0 0

T(n-k-N+2)

® Zu

, = Im C3 (¡08)

0 In, € (¿00)

16L

mni

0

Ao =

— \idlmni 0]

0 0 0 Ao

An =

(iDt)®^

0 L-

n-k+l

A,

(i)

4a) -' o —

0

(7D i)®In

3. System stability

Theorem 1. The assumption that after each retrial a eustcnner may leave the system uritli probability 1 —5 makes the system stable irrespective of the parameter values.

Proof. To prove the theorem we use a result due to Tweedie [6 . For the model under consideration we consider the following Lvapunov function:

4>(s) = i if s is a state belonging to level i

The mean drift ys for an .s belonging to level i > 1 is given by

Vs = V qsp(<p(p) ~ <KS))

p = s

= - Ms))+w - )

s' s"

+ {№'")-m)

s'"

where s', s", $"' varies over the states belonging to levels i — 1, i. i + 1 respectively. Then by definition of <f>, <j>{s) = i, <f>(s') — i — 1, <f>(s") — i, c/>(s'") — i + 1 So that

Vs = -

if s e h if s e I,

where I, denotes the collection of states in level i which corresponds to N<i(t) < M, and Ij denotes the collection of states in level i which correspond to jV2(t) = M.

We note that £ ,„ qss», is bounded by some fixed constant for any s in any level i > f. So, let Yls'" Qes'" < some real number a > 0, for all states s belonging

to level i > 1. Also since 1 — 6 > 0, for any e > 0, we can find N' large enough that ys < e for any s belonging to level i > N'.

Hence by Tweedie's result, the theorem follows. □

4. Steady state distribution

Since the process under consideration is an LDQBD, to calculate the steady state distribution, we use the methods described in Bright and Taylor [2 .

By partitioning the steady state vector xasx= (a;o, X2,.. .) we can write

k-1

x-h — xo yj R{ for k > 1 i=0

where the family of matrices { h'i. k > 0} are minimal non-negative solutions to the system of equations:

i 1) + RkAi + ii. !i:,. |. i^+2] = 0, k > 0

xo is calculated by solving

(2) x0[4io + iM2i] = 0 such that

.X A"—1

(3) ■'..(• • /„V \[ /.'; <■ < x

fc=l i=0

The calculation of the above infinite sums does not seem to be practical, so we approximate x^s by Xk{K")a where (x/,:(K~)) , 0 < k < is defined as the stationary probability that X(t) is ill the jth state of level k, conditional on X(t) being in level i, 0 < i < K*.

Then xk(K*), 0 < k < K* is given by

k — 1

(4) xk(K*) =xo(K<) [] R,

1=0

where xq(K*) satisfies (2) and

e = 1

K* k-1

(5) ,„ K )e • ,,, K . V \[i':

k=1 ;=o

Here we have that for all i > 1, and for all k, there exists j such that [-42?.] a, j > 0. So we can construct a dominating process X(t) of X(t) and can use it to find the truncation level K~ in the same way as in [2], as follows. The dominating process X'(f) has generator

~AW .4 0 0 0 0 0 An Aq 0 0 0 i22 A12 Ao 0 0 0 .42:3 A13 AQ

0 =

where

(A0)ij = ¿[(^oe)max], (A2k)i,j = jj ((A2!fe-i)e) for k > 2, (Ait)^- = (Alk)i:j,

\ / mill

j ± i, k > 1; and C = Nm + (M + l)(n - k + l)m»i + M(n - k + 2)mn2 is the dimension of a level i > 1.

5. Performance measures

We partition the steady state vector x as x = (a'o, , x2,. . .) where the sub-vectors 3'^s are again partitioned as Xjt = x(j\, j2, J3, .74) which correspond to

•V,;;': ./.. I < ' 1

(1) Fraction of time the system is down is given by

K* M 2

'Pdown = V V x(jl, J2, n - k + 1, J4)e.

Ji = 0 j-2=0 ji = l

(2) System reliability, defined as the probability that atleast k components are operational, 'Prei is given by

rel f down ■

(3) Average no. of external units waiting in the pool is given by

M K* n-k+1

ApooL = 53 j2( X] x(3i,h,h^ l)e

¿2 = 1 J 1=0 .73 = 1

M K" n-k+1

+ 1' zL zL - /1 •.!-• ./'••• 2)e

¿2=2 ¿1=0 ¿3=0

(4) Average no. of external units in the orbit is given by

a"*

Aorbit - 53 ■■M;ru'0e ¿1=1

(5) Average no. of failed components is given by

n-k+1 K* M

A/faic = V j\i ( 53 53 -r (31, h, r., 2)e

¿3=1 ¿1=0 ¿2 = 1

K * M N-l a"*

+ 53 ¿3, f)e) + 53 •<'• 5Z n-"-./--..":«'

¿I=0j2 = 0 ¿3=1 ¿1=0

(6) The probability that an external unit, on its arrival joins the queue in the pool is given by

K" M-l n-k+1 2

V,

queue

J1=UJ2 = 1 J3 = i 3 4 =

^ XI in X lb n-yi

— { V V ¿3 V ., !>1 /. . e

a"* n-fc+1

V V i.',/■>,. I ) ( / 'i •: )e}

¿1=0 ¿3=1

(7) The probability that an external unit, on its arrival gets service directly is given by

K* N-1

r'.. —{ v V .•• i•'''•./'•'''1 'ij

9 ¿1=0 ¿3=0

(8) The probability that an external unit, on its arrival enters orbit is given by

1

Vorbit = — | 53 r:}

/ =0

(9) Fraction of time the server is busy with external customers is given by

K* M n-k+1

v V V z3 J2, J3,2)e

¿1=0 ¿2=1 ¿3=0

(10) Probability that the server is found idle is given by

K* N-l

■Pidle = •'!•"•

ji =0 j-2 =0

(11) Probability that the server is found busy is given by

>n s y f ^idle

(12) Expected loss rate of external customers is given by

K* ji-fc+l

Aioss V V x(j1Mj2,m~J)(D1 ®/ni)e

ji —0 J2 = 1

K* n. k | I

V V :r(j1.M.j2)2)( 1-7)(D1®/,J.i)e

Jl=0 J 2 — 0

A"* ra-fc+l

V V 1 <5 n»r !:<•

jl = l ¿2 = 1

a" ra-fc+l

V V (1 -6)jx6x{juM,h,2)e

¿1 = 1 ¿2 = 0

(13) We construct a cost function as where C\ is the holding cost per unit time per customer waiting in the pool, C2 is the loss per unit time due to the system becoming down, is the loss per unit time due to a customer leaves the system without taking service, C4 is the holding cost per unit time per failed component in the system, C5 is the loss per unit time due to the server becoming idle and Cg is the profit per unit time due to the server becoming busy with an external customer.

6. Numerical illustration Set e = 15.0, A = 1.0, 7 - 0.7, 5 = 0.7, n = 11, k = 4. M = 5, N = 4

Si =

-6.5 4.0 1.5 -4.5

S2 =

S? =

2.5 3.0

s?2 =

3.0

2.5

-5.06 2.06 4.0 -6.5 /? = (0.5, 0.5)

Cx = 10.0. C2 = 1500.0, C:i = 100.0, C4 = 20.0. C5 = 50.0. C6 = 200.0.

a = (0.5,0.5)

Effect of correlation : The additional parameters for table 1 are the following

(Al)

Do =

-5.5 3.5 1.0 -3.5

D, =

1.0 1.0 1.0 1.5

average arrival rate = 2.34615, correlation =-0.00029

(A2)

Do =

-4.05 1.55 3.5 -5.5

Dt -

average arrival rate = 2.34615, correlation =0.00029

2.05 0.45 1.0 1.0

(Bl)

Do -

-6.5 4.0 " 1.5 -4.5

D, -

1.5 1.0

1.0 2.0

average arrival rate = 2.83333, correlation =-0.00042

(B2)

Da =

-5.06 2.06 4.0 -6.5

D, -

2.56 1.0

average arrival rate — 2.83333, correlation =0.00042

(CI)

Ai =

-6.6 4.05 1.55 -4.6

D! -

1.55 1.0

average arrival rate = 2.88224, correlation =-0.00041

0.44

1.5

1.0

2.05

(C2)

Ml =

-5.15 2.1

4.05 -6.6

A =

2.6 1.0

0.45 1.55

average arrival rate = 2.88224, correlation =0.00041

In the above correlation is between two inter-arrival times.

Table 1

^down ■V' - v pool л Г - * orbit faic ^exbusy Pidle Cost

Al .2805 x 10" -2 3.262 0.1204 2.2281 0.5620 0.0842 37.8228

A 2 .2803 x 10" -2 3.2572 0.1207 2.2278 0.5612 0.0850 38.1696

Bl .2923 x 10" -2 3.6689 0.1822 2.2431 0.5940 0.0522 68.2556

B2 .2922 x 10" -2 3.6647 0.1824 2.2429 0.5935 0.0526 68.4537

CI .2932 x 10" -■J 3.7031 0.1888 2.2442 0.5964 0.0497 71.6377

С 2 .2931 x 10" -■J 3.6992 0.1890 2.2440 0.5960 0.0502 71.8214

The table 1 shows that as the external arrival rate increases the system down probability increases; but this increase is narrow as compared to the decrease in server idle probability. Also as expected, the expected number in the pool, in the orbit and the expected number of failed components and the fraction of time the server is found busy with an external customer increases as the external arrival rate increases. The table also shows that as the correlation changes from negative to positive, there is a slight increase in cost and in the server idle probability. Also when correlation changes from negative to positive, the expected number of pooled customers and failed components decrease while the expected number in the orbit increases. The increase in probability Pexbusy being small compared to the increase in other parameters can be thought of as the reason behind increase in cost. But all these changes are narrow as the difference between negative and positive correlation is small.

Effect of component failure rate : Take в — 20.0. 7 = 0.7, 5 = 0.7, n — 11,

к = 4, M = 5, N = 4.

Arrival process is according to (Al).

Table 2 shows that when the component failure rate A increases, the system down probability as well as expected number of failed components increase and the idle time probability of the server decreases, as expected. But note that as A increases, the fraction of time the server is found busy with an external customer, decreases and as a result the expected pool size increases. Also note that the expected orbit

TABLE 2. Effect of component failure rate

A ^down A/ pool -'Vor bit A faic ^exbusy ''Adle Cost

0.05 .196 x 10-" 2.1163 0.0285 1.5266 0.7513 0.2310 -67.3177

0.1 .5933 x 10"7 2.1765 0.0311 1.5538 0.7432 0.2213 -63.3658

1.0 .2801 x 10~'2 3.2399 0.0907 2.2276 0.5607 0.0855 38.4979

2.0 0.04702 4.2095 0.1748 3.5505 0.3029 0.0208 261.502

3.0 0.17207 4.7390 0.2362 5.1091 0.1149 0.0038 580.397

size is small, which shows that the orbital customers are either transferee! to the pool (when A is small) or leaves the system forever (when A is large). Since the probability jPdown increases and the probability Pexbusy decreases, as A increases, tin? cost also increases.

Effect of N policy level : 6 = 20.0, A = 2.0, n - 13, k = 4, M - o The other parameters are same as for table 2. Table 3 shows that the system performance measure which is most affected by

Table 3. Effect of iV-policy level

N ^down A/ pool A orbit A/ faic Pexbusy ^idle Cost

4 0.02245 4.2521 0.1802 3.8666 0.2866 0.01969 203.559

5 0.02795 4.2249 0.1801 4.2456 0.2869 0.02325 219.258

6 0.03528 4.1968 0.1796 4.6087 0.2882 0.02717 237.002

7 0.04509 4.1658 0.1787 4.9473 0.2910 0.03135 257.358

8 0.05830 4.1300 0.1771 5.2518 0.2959 0.03577 281.200

the A—policy level is the expected number of failed components; which is expected because as N increases, time for the service of failed components to be started, once the system started with all components operational, increases so that during this time more components may fail. For the same reason a pooled customer lias a better chance of getting service and as a result Pexbusy increases, Arpool and Aorbit decreases. Also note that the server idle probability is small. The increase in A fak. might be the reason behind the increase in cost.

Effect of retrial rate 0 : Take A - 1.0, a - 11, k - 4, M - 5, N = 4

The other parameters are the same as in table 2.

Table 4 shows that as 0 increases, expected number in the orbit decreases but the expected pool size also decreases which tells that retrying customers may be

Table 4. Effect of retrial rate

e ^down A pool r orbit r - v faic ^exbusy Pidle cost

5.0 .2832 x 10" -■2 3.3908 0.3501 2.2315 0.5704 0.07579 33.688

10.0 .2813 x10" -■1 3.3008 0.1790 2.2290 0.5644 0.08176 36.612

15.0 .2805 x 10- -2 3.2620 0.1204 2.2281 0.5620 0.08415 37.823

20.0 .2801 x 10 -1 3.2399 0.0907 2.2276 0.5607 0.08546 38.498

25.0 .2798 x 10 -1 3.2255 0.0728 2.2272 0.5598 0.08630 38.932

leaving the system. Note that the idle probability of the server is very small and the expected pool size is also close to the maximum pool capacity so that retrying customers may choose to leave the system after a failed retrial. Also this can be thought of as the reason behind the decrease in the fraction of time the server is found busy with an external customer and the increase in cost as 0 increases.

Effect of pool size Mi 0 = 10.0, A = 1.0 The other parameters are same as for table 2.

Table 5. Effect of pool size

M vdown A i ' v pool A r » V orbit A'fak. ^exbusy ^idle cost

3 .2655 x 10"' 1.9658 0.2155 2.2090 0.5084 0.1377 65.402

4 .2743 x 10"2 2.6238 0.1942 2.2201 0.5410 0.1051 55.047

5 .2813 x 10 - 3.3008 0.1790 2.2290 0.5644 0.0818 36.612

Table 5 shows that as M, the pool size, increases, expected number of pooled customers increases and as a result the expected number of failed components, the system down probability and the fraction of time the server is found busy with and external customer increases. But the expected number in the orbit decreases, which is expected because as M increases more customers can join the pool. As expected, the idle probability of the server decreases as M increases.

Comparison with the case where no external customers are allowed :

Below we compare the fc-out-of-n-system with a fc-out-of-n system where no external customers are allowed.

Case 1: fc-out-of-r.' system where no external customers are allowed, Case 2: &-out-of-n system

e = 10.0. A = 1.0, T = 0.7. 5 = 0.7. n = 11, k = 4, N = 4

Dn =

=

-5.5 3.5 1.0 -3.5

-7.5 2.0 2.1 -7.7

5.5

5.6

a = [0.5 0.5]

S? =

D! =

So =

5° =

1.0 1.0

-5.06 4.0

3.0" 2.5

1.0

1.5

2.06 -6.5

¡3 = [0.5 0.5]

Table 6. Comparison with the fc-out-of-« system where no external customers are allowed

A = 0.1 A = 0.5 A = 1.0 'O II A = 2.0 A = 2.5

'' ^ down Case 1 < io-1:1 .3956 x 10s .9124 x № -0 .2081 x 10- -4 .1822 x 10" -3 .9335 x 10 -3

M = 1 Case 2 .129 x 10-7 .2379 x .u 1 .4329 x Kr -3 .2039 x 10- -2 .5728 x 10" -■j .01237

Pbusy Case 1 0.0180 0.0901 0.1802 0.2703 0.3603 0.4501

Case 2 0.5347 0.5836 0.6415 0.6958 0.7458 0.7914

'' ^ down Case 1 < 10"" .3956 x 10s .9124 x 10" -ti .2081 x 10" -4 .1822 x 10" -3 .9335 x 10" -3

CI II Case 2 .1801 lu .3289 x 10-4 .5952 x 10- -3 .2782 x 10- -3 .7689 x 10- -2 .1616 x 10 -i

- i:. . Case 1 Case 2 0.0180 0.7500 0.0901 0.7941 0.1802 0.8434 0.2703 0.8848 0.3603 0.9179 0.4501 0.9433

Table 7. Variation in IE).,

ID,.ost A = 0.1 A = 0.5 A = 1.0 A = 1.5 A = 2.0 A = 2.5

Cu = 100 Case 1 -0.1800 -0.9010 -1.8019 -2.7009 -3.5848 -4.4077

C12 10 Case 2 -5.3470 -5.8336 —6.3717 -6.7541 -6.8852 -6.6770

M = 1 ' = 1000 Case 1 -0.1800 -0.9010 -1.8011 -2.6822 -3.4208 -3.5675

C12 = 10 Case 2 -5.3470 -5.8122 -5.9821 -4.9190 -1.7300 4.4560

Cu = 10000 Case 1 -0.1800 -0.9010 -1.7929 -2.4949 -1.7810 4.8340

C12 = 10 Case 2 —26.7349 -28.9421 -27.7460 -14.4000 19.9900 84.1300

Cu = 100 Case 1 -0.1800 -0.9010 -1.8019 -2.7009 -3.5848 -4.4077

Ci2 = 10 Case 2 -7.5000 -7.9377 -8.3745 -8.5698 -8.4101 -7.8170

M 4 Cu = 1000 Case 1 -0.1800 -0.9010 -1.8011 -2.6822 -3.4208 -3.5675

C12 = 10 Case 2 -7.5000 -7.9081 -7.8388 -6.0660 -1.4900 6.7270

Cu = 10000 Case 1 -0.1800 -0.9010 -1.7929 -2.4949 -1.7810 4.8340

C12 = 10 Case 2 -7.4998 -7.6121 -2.4820 18.9720 67.7110 152.167

'Table 6 shows that compared to the increase in the fraction of time the server is found busy, the increase in the system down probability is not high, if we provide service to external customers in a fc-out-of-n system 'To make these statements more clear we consider the cost function

id cost = cu • 'pdown _ cl2 ' Pbusy

where Cu is the loss per unit time the system being down and is the profit per-unit time due to the server being busy.

Table 7 shows that when M = 1 and A < 1.5, IDcost is smaller in case 2 than case 1, even when Cu is 1000 times bigger than C12. But when A = 2.0 and 2.5, IDcost is larger in case 2 than case 1, when Cu is 100 times larger than C12. When M - 4 and A < 1.0, the table shows that IDcost is smaller in case 2 than in case 1, even when Cu is 1000 times bigger than C-'ia- But when A = 2.0 and 2.5, IDcost is larger in case 2 than case 1, when Cu is 100 times larger than C10.

Table 7 proves atleast numerically that we are able to utilize server idle time without much effecting system reliability.

References

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[4] A, Krishnamoorthy and P. V. Ushakumari. Reliability of fe-out-of-n system with repair and retrial of failed units, TOP, 7(2): 293-304, 1999.

[5] A. Krishnamoorthy, P. V. Ushakumari, and B, Lakshmi, fr-out-of-n system with the repair: the A'-policy, Asia Pacific Journal of Operations Research, 19:47-61, 2002.

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[7] P. V, Ushakumari and A. Krishnamoorthy. A'-out-of-n system with repair: the max(Ar,T) policy. Performance Evaluation, 57:221-234, 2004.

A. Krishnamoorthy, Vishwanath C. Narayanan, Department of Mathematics, CUSAT. Kochi, India

T. G. Deepak, M. G. University Regional Centre, Kochi, India