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1. P. Saranya et al Int. Journal of Engineering Research and Applications
ISSN : 2248-9622, Vol. 4, Issue 2( Version 1), February 2014, pp.315-330
RESEARCH ARTICLE
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OPEN ACCESS
Expected Time to Recruitment in A Two - Grade Manpower
System Using Order Statistics for Inter-Decision Times and
Attrition
J. Sridharan*, P. Saranya**, A. Srinivasan***
*
(Department of Mathematics, Government Arts College (Autonomous), Kumbakonam-1)
(Department of Mathematics, TRP Engineering College (SRM Group), Trichirappalli-105)
***
(Department of Mathematics, Bishop Heber College (Autonomous), Trichirappalli-17)
**
ABSTRACT
In this paper, a two-grade organization subjected to random exit of personal due to policy decisions taken by the
organization is considered. There is an associated loss of manpower if a person quits. As the exit of personnel is
unpredictable, a new recruitment policy involving two thresholds for each grade-one is optional and the other
mandatory is suggested to enable the organization to plan its decision on recruitment. Based on shock model
approach three mathematical models are constructed using an appropriate univariate policy of recruitment.
Performance measures namely mean and variance of the time to recruitment are obtained for all the models
when (i) the loss of man-hours and the inter decision time forms an order statistics (ii) the optional and
mandatory thresholds follow different distributions. The analytical results are substantiated by numerical
illustrations and the influence of nodal parameters on the performance measures is also analyzed.
Keywords-Manpower planning, Shock models, Univariate recruitment policy, Mean and variance of the time to
recruitment.
I. INTRODUCTION
Consider an organization having two grades in which depletion of manpower occurs at every decision
epoch. In the univariate policy of recruitment based on shock model approach, recruitment is made as and when
the cumulative loss of manpower crosses a threshold. Employing this recruitment policy, expected time to
recruitment is obtained under different conditions for several models in [1], [2] and [3].Recently in [4],for a
single grade man-power system with a mandatory exponential threshold for the loss of manpower ,the authors
have obtained the system performance measures namely mean and variance of the time to recruitment when the
inter-decision times form an order statistics .In [2] , for a single grade manpower system ,the author has
considered a new recruitment policy involving two thresholds for the loss of man-power in the organization in
which one is optional and the other is mandatory and obtained the mean time to recruitment under different
conditions on the nature of thresholds. In [5-8] the authors have extended the results in [2] for a two-grade
system according as the thresholds are exponential random variables or geometric random variables or SCBZ
property possessing random variables or extended exponential random variables. In [9-17], the authors have
extended the results in [4] for a two-grade system involving two thresholds by assuming different distributions
for thresholds under different condition of inter-decision time and wastage. The objective of the present paper is
to obtain the performance measures when (i) the loss of man-hours and the inter decision time forms an order
statistics (ii) the optional and mandatory thresholds follow different distributions.
II. MODEL DESCRIPTION AND ANALYSIS OF MODEL –I
Consider an organization taking decisions at random epoch in (0, ∞) and at every decision epoch a
random number of persons quit the organization. There is an associated loss of man-hours if a person quits. It is
assumed that the loss of man-hours are linear and cumulative. Let X i be the loss of man-hours due to the ith
decision epoch , i=1,2,3…n Let X 1, X 2, X 3,....X n be the order statistics selected from the sample
X1, X 2 ,....X n with respective density functions
g x (1) (.), g x ( 2) (.),.... g x ( n ) (.) . Let Ui be a continuous random
variable denoting inter-decision time between (i-1)th and ith decision, i=1,2, 3….k with cumulative distribution
1
function F(.), probability density function f(.) and mean
(λ>0). Let U(1) (U(k)) be the smallest (largest) order
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2. P. Saranya et al Int. Journal of Engineering Research and Applications
ISSN : 2248-9622, Vol. 4, Issue 2( Version 1), February 2014, pp.315-330
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statistic with probability density function. f u (1) (.), f u (k ) (.) .Let Y1, Y2 (Z1, Z2) denotes the optional (mandatory)
thresholds for the loss of man-hours in grades 1 and 2, with parameters 1, 2 , 1, 2 respectively, where
𝜃1 , 𝜃2 , 𝛼1 , 𝛼2 are positive. It is assumed that Y1 < Z1 and Y2 <Z2. Write Y=Max (Y1, Y2) and Z=Max (Z1, Z2),
where Y (Z) is the optional (mandatory) threshold for the loss of man-hours in the organization. The loss of
man- hours, optional and mandatory thresholds are assumed as statistically independent. Let T be the time to
recruitment in the organization with cumulative distribution function L (.), probability density function l (.),
mean E (T) and variance V (T). Let F k(.) be the k fold convolution of F(.). Let l *(.) and f*(.), be the Laplace
transform of l (.) and f (.), respectively. Let V k(t) be the probability that there are exactly k decision epochs in
(0, t]. It is known from Renewal theory [18] that Vk(t) = Fk(t) - Fk+1(t) with F0(t) = 1. Let p be the probability
that the organization is not going for recruitment whenever the total loss of man-hours crosses optional
threshold Y. The Univariate recruitment policy employed in this paper is as follows: If the total loss of manhours exceeds the optional threshold Y, the organization may or may not go for recruitment. But if the total loss
of man-hours exceeds the mandatory threshold Z, the recruitment is necessary.
MAIN RESULTS
P (T t )
k
i 1
k 0
k
k
i 1
i 1
V k ( t ) P Xi Y p V k ( t ) P Xi Y P Xi Z
k 0
(1)
For n=1, 2, 3…k the probability density function of X(n) is given by
g x(n) (t) n kc n [G(t)]n 1 g(t)[1 G(t)]k n , n 1,2,3..k
(2)
If g( t ) g x (1) ( t )
In this case it is known that
g x(1) (t) k g(t) 1 G(t)k 1
(3)
By hypothesis g( t ) ce ct
Therefore from (3) and (4) we get,
(4)
g x (1) ()
(5)
kc
kc
If g(t ) g x (k ) (t )
In this case it is known that
g x(k) (t ) kG(t )k 1g(t )
Therefore from (4) and (6) we get
g* ( k ) ()
x
(6)
k!c k
( c)( 2c)...( kc)
(7)
For r=1, 2, 3…k the probability density function of U(r) is given by
f u(r ) (t ) r kcr [F(t )]r 1f (t )[1 F(t )]k r , r 1,2,3..k
(8)
If f ( t ) f u (1) ( t )
In this case it is known that
f u(1) (t) k f (t) 1 F(t)k1
(9)
Therefore from (8) and (9) we get,
k
(10)
f u (1) (s) k s
If f (t ) f u(k) (t )
In this case it is known that
f u (k ) (t ) kF(t )k 1f (t )
*
f u ( k ) (s)
(11)
k!k
(s )(s 2)...(s k)
(12)
It is known that
E(T)
d(l* (s))
ds
, E(T 2 )
s 0
d 2 (l* (s)
ds 2
and V(T) E(T 2 ) (E(T))2
(13)
s 0
Case (i): The distribution of optional and mandatory thresholds follow exponential distribution
For this case the first two moments of time to recruitment are found to be
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316 | P a g e
3. P. Saranya et al Int. Journal of Engineering Research and Applications
ISSN : 2248-9622, Vol. 4, Issue 2( Version 1), February 2014, pp.315-330
If g(t ) g
x (1)
( t ), f ( t ) f u (1) ( t )
E(T) CI1 CI2 CI3 p(CI4 CI5 CI6 H
I1,4
2
2
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2
2
2
2
2
(14)
HI1,5 HI1,6 HI2,4 HI2,5 HI2,6 HI3,4 HI3,5 HI3,6
2
2
2
2
2
2
2
2
2
E(T ) 2 CI1 CI2 CI3 p CI4 CI5 CI6 HI1,4 HI1,5 HI1,6 HI2,4 HI2,5 HI2,6 HI3,4 HI3,5 HI3,6
(15)
where for a = 1, 2…6. b=1, 2, 3 and d=4, 5, 6.
1
(16)
1
CIa k(1
and HIb,d
k(1 DIb DId)
DIa)
*
*
*
*
*
*
DI1 gx(1) (1), DI2 gx (1) (2), DI3 gx (1) (1 2), DI4 gx (1) (1), DI5 gx(1) (2), DI6 gx (1) (1 2)
are given by (5)
If g(t ) g
(t ), f (t ) f
x (k )
(17)
(t )
u (1)
E(T) LK1 LK2 LK3 p(LK4 LK5 LK6 M
2
2
2
2
K1,4
2
2
2
MK1,5 MK1,6 MK2,4 MK2,5 MK2,6 MK3,4 MK3,5 MK3,6
2
2
2
2
2
2
2
2
2
E(T ) 2 LK1 LK2 LK3 p LK4 LK5 LK6 MK1,4 MK1,5 MK1,6 MK2,4 MK2,5 MK2,6 MK3,4 MK3,5 MK3,6
(18)
(19)
where for a = 1, 2…6. b=1, 2, 3 and d=4, 5, 6.
1
(20)
1
LKa k(1
and MKb,d
k(1 DKb DKd)
DKa)
*
*
*
*
*
*
DK1 gx(k) (1), DK 2 gx (k ) (2), DK3 gx(k ) (1 2), DK 4 gx (k ) (1), DK5 gx (k ) (2), DK6 gx (k ) (1 2)
are given by (7)
If g(t ) g
( t ), f ( t ) f
(t)
u(k)
x (k )
E (T ) PK1 PK 2 PK 3 p(PK 4 PK 5 P K 6 Q
2
E(T ) 2 P
2
K1
P
2
p
K3
N M 1 1
p
P
2
K2
P
2
2
DK 4
(21)
2
K4
P
2
K5
P
2
K6
K1, 4
Q
K1,5
Q
Q
K1,6
Q
K 2, 4
K 2, 5
Q
K 2, 6
Q
1
QK1,4 QK1,5 QK1,6 QK 2,4 QK 2,5 QK 2,6 QK3,4 QK3,5 QK3,6 2
2
2
2
2
2
2
2
2
2
K 3, 4
Q
K 3,5
Q
(22)
(23)
K 3,6
N
2
1
1
1
M
1
DK1 1 DK 2 1 DK3
1
1
1
1
1
1
1
1
1
1
1
1 DK5 1 DK 6 1 DK1 DK 4 1 D D
1 D K1D
1 D K2 D
1 D K2 D
1 D K2 D
1 D K3 D
1 D K3 D
1 D K3 D
K1 K5
K6
K4
K5
K6
K4
K5
K6
where for a = 1, 2…6. b=1, 2, 3 and d=4, 5, 6.
P Ka
If
g( t ) g
x (1)
( t ), f ( t ) f u ( k ) ( t )
E(T) R I1 R I2 R I3 p(R I4 R I5 R I6 S
I1,4
2
2
2
2
2
2
2
2
2
2
SI1,5 SI1,6 SI2,4 SI2,5 SI2,6 SI3,4 SI3,5 SI3,6
2
2
2
2
2
2
1 N M 1 1
E(T ) 2 R I1 R I 2 R I3 p R I 4 R I5 R I6 SI1,4 SI1,5 SI1,6 SI 2,4 SI 2,5 SI 2,6 SI3,4 SI3,5 SI3,6
p
2
(24)
k
k
N
N
1
1
,Q
,N
and M
2
Kb, d
(1 D Ka)
(1 D Kb D Kd)
n 1 n
n 1 n
N M 1 1
2
DI 4
2
2
DI1
(25)
(26)
1
1
1 DI 2 1 DI 3
1
1
1
1
1
1
1
1
1
1
1
1 DI5 1 DI6 1 DI1 DI 4 1 D D
I1 I5 1 D I1DI6 1 D I2 DI 4 1 D I2 DI5 1 D I2 DI6 1 D I3 DI 4 1 D I3 DI5 1 D I3 DI6
where for a = 1, 2…6, b=1, 2, 3 and d=4, 5
R Ia
N
,
(1 DIa) SIb,d
(1
N
,N
k
1
and M
k
1
(27)
Case (ii): The distributions of optional and mandatory thresholds follow extended exponential distribution with
shape parameter 2.
If g(t ) g (t ), f (t ) f
(t )
DIb DId)
n 1 n
n 1
n
2
u (1)
x (1)
E (T ) 2CI1 2CI 2 2CI7 2CI8 CI9 CI10 CI11 4CI3 p 2CI 4 2CI5 2CI12 2CI13 CI14 CI15 CI16 4CI6 4H I1,4 4HI1,5 4H I1,12 4H I1,13
(28)
2HI1,14 2H I1,15 2H I1,16 8HI1,6 4H I 2,4 4H I 2,5 4HI 2,12 4H I 2,13 2HI 2,14 2HI 2,15 2HI 2,16 8H I 2,6 4HI7,4 4HI7,5 4H I7,12 4H I7,13
2HI7,14 2H I7,15 2HI7,16 8H I7,6 4HI8,4 4H I8,5 4H I8,12 4HI8,13 2HI8,14 2HI8,15 2H I8,16 8HI8,6 2HI9,4 2HI9,5 2H I9,12 2HI9,13
H I9,14 H I9,15 H I9,16 4HI9,6 2HI10,4 2H I10,5 2H I10,12 2H I10,13 H1I0,14 H I10,15 HI10,16 4H I10,6 2H I11,4 2H I11,5 2HI11,12 2H I11,13
H I11,14 H I11,15 HI11,16 4HI11,6 8HI3,4 8H I3,5 8H I3,12 8H I3,13 4H I3,14 4HI3,15 4H I3,16 16H I3,6
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
I 2, 6
2
I 7, 4
2
I 7 ,5
E(T ) 2 2CI1 2CI 2 2CI7 2CI8 CI9 CI10 CI11 4CI3 p 2CI 4 2CI5 2CI12 2CI13 CI14 CI15 CI16 4CI6 4HI1,4 4HI1,5 4HI1,12
4H
2
I1,13
2
I1,14
2
I1,15
2
I1,16
2
I1,6
2
I 2, 4
2
I 2,5
2
I 2,12
2
I 2,13
2
I 2,14
2
I 2,15
2
I 2,16
(29)
8H 4H 4H 4H
8H 4H 4H
2H
2H
2H
4H
2H
2H
2H
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
4HI7,12 4HI7,13 2HI7,14 2HI7,15 2HI7,16 8HI7,6 4HI8,4 4HI8,5 4HI8,12 4HI8,13 2HI8,14 2HI8,15 2HI8,16 8HI8,6 2HI9,4
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2HI9,5 2HI9,12 2HI9,13 HI9,14 HI9,15 HI9,16 4HI9,6 2HI10,4 2HI10,5 2HI10,12 2HI10,13 HI10,14 HI10,15 HI10,16 4HI10,6 2HI11,4
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2HI11,5 2HI11,12 2HI11,13 HI11,14 HI11,15 HI11,16 4HI11,6 8HI3,4 8HI3,5 8HI3,12 8HI3,13 4HI3,14 4HI3,15 4HI3,16 16HI3,6
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4. P. Saranya et al Int. Journal of Engineering Research and Applications
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where for a=1, 2, 3…16, b=1, 2, 3,7, 8, 9, 10,11 and d=4, 5, 6, 12, 13,14,15,16.
given by (16)
C Ia , H Ib, d are
(30)
DI7 gx (1) (21 2), DI8 gx (1) (1 22), DI9 gx (1) (21), DI10 gx (1) (22) DI11 gx (1) (21 22),
*
*
*
*
*
DI12 gx(1) (21 2), DI13 gx(1) (2 2 1), DI14 gx (1) (21), DI15 gx (1) (22), DI16 gx (1) (21 22)
*
*
*
If g ( t ) g x ( k ) ( t ), f ( t ) f u (1) ( t )
*
*
E(T) 2LK1 2LK 2 2LK 7 2LK8 LKI9 LK10 LK11 4LK3 p 2LK 4 2LK5 2LK12 2LK13 LK14 LK15 LK16 4LK 6 4MK1,4 4MK1,5 4MK1,12 4MK1,13
2MK1,14 2MK1,15 2MK1,16 8MK1,6 4MK 2,4 4MK 2,5 4MK 2,12 4MK 2,13 2MK 2,14 2MK 2,15 2MK 2,16 8MK 2,6 4MK 7,4 4MK 7,5 4MK 7,12 4MK 7,13
2MK 7,14 2MK 7,15 2MK 7,16 8MK 7,6 4MK8,4 4MK8,5 4MK8,12 4MK8,13 2MK8,14 2MK8,15 2MK8,16 8MK8,6 2MK9,4 2MK9,5 2MK9,12 2MK9,13
MK9,14 MK9,15 MK9,16 4MK9,6 2MK10,4 2MK10,5 2MK10,12 2MK10,13 MKI0,14 MI10,15 MK10,16 4MK10,6 2MK11,4 2MK11,5 2MK11,12 2MK11,13
MK11,14 MK11,15 MK11,16 4MK11,6 8MK3,4 8MK3,5 8MK3,12 8MK3,13 4MK3,14 4MK3,15 4MK3,16 16MK3,6
2
2
2
2
2
2
2
2
(31)
2
2
2
2
2
2
2
2
2
E(T ) 2 2L K1 2L K 2 2L K 7 2L K8 L K 9 L K10 L K11 4L K 3 p 2L K 4 2L K 5 2L K12 2L K13 L K14 L K15 L K16 4L K 6
2
2
2
2
2
2
2
2
2
2
2
2
2
4M K1,4 4M K1,5 4M K1,12 4M K1,13 2M K1,14 2M K1,15 2M K1,16 8M K1,6 4M K 2,4 4M K 2,5 4M K 2,12 4M K 2,13 2M K 2,14
2
2
2
2
K8,12
2
K8,13
2
K8,14
2
2
2
2
2
2
2
2
2
2
2M K 2,15 2M K 2,16 8M K 2,6 4M K 7,4 4M K 7,5 4M K 7,12 4M K 7,13 2M K 7,14 2M K 7,15 2M K 7,16 8M K 7,6 4M K8,4 4M K8,5
4M
4M
2
2M
2
2M
2
2
K8,15
2M
2
2
K8,16
8M
2
2
K8,6
2M
2
2
K 9, 4
2M
2
2
K 9,5
2M
2
K 9,12
2
2M
2
K 9,13
2
M
2
K 9,14
2
M
2
K 9,15
2
M
2
K 9,16
2
2
4M K 9,6 2M K10,4 2M K10,5 2M K10,12 2M K10,13 M K10,14 M K10,15 M K10,16 4M K10,6 2M K11,4 2M K11,5 2M K11,12 2M K11,13
M
2
K11,14
M
2
K11,15
M
2
K11,16
4M
2
K11,6
8M
2
K 3, 4
8M
2
K 3,5
8M
2
K 3,12
8M
2
K 3,13
4M
2
K 3,14
4M
2
K 3,15
4M
2
K 3,16
16M
2
K 3,6
(32)
where for a=1,2,3,4,5,….16,b=1,2,…8 and d=9,10…..16. L Ka , M Kb, d are given by (20) .
If g(t ) g
(t ), f (t ) f u (1) (t )
x (k )
E (T ) 2PK1 2PK 2 2PK 7 2PK8 PK 9 PK10 PK11 4PK 3 p 2PK 4 2PK 5 2PK12 2PK13 PK14 PK15 PK16 4PK 6 4Q
4Q
K1,13
4Q
K 7,12
2Q
K 9 ,5
2Q
K11,5
2
2Q
K1,14
4Q
2Q
K 7,13
K 9,12
2Q
2
2Q
K1,15
2Q
2Q
K11,12
2
2Q
K 7,14
K 9,13
2Q
2
K1,16
2Q
Q
K11,13
2
K 7,15
K 9,14
Q
8Q
2
2Q
Q
K11,14
K1,6
K 7,16
K 9,15
Q
2
4Q
Q
K11,15
K 2, 4
8Q
K 9,16
Q
2
4Q
K 7,6
4Q
4Q
K11,16
2
K 2 ,5
K 9, 6
4Q
4Q
K 8, 4
4Q
2Q
K11,6
2
K 2,12
K10, 4
8Q
2
4Q
K 8,5
4Q
2Q
K 3, 4
K 2,13
K 8,12
K10,5
8Q
2
2Q
4Q
2Q
K 3,5
2
K 2,14
K 8,13
K10,12
8Q
2
2Q
2Q
2Q
K 3,12
K 8,14
K10,13
8Q
2
K 2,15
4Q
2
2Q
2
Q
2Q
K 7,14
2
Q
K1,14
2
K 9,14
K1,15
2Q
Q
K11,14
2
2
K 7,15
2
K 9,15
Q
2Q
K11,15
K1,16
2Q
Q
2
2
2
K 9,16
Q
8Q
2
K 7,16
K11,16
K1,6
8Q
4Q
2
2
2
K 9, 6
4Q
4Q
2
K 7,6
K11,6
K 2, 4
4Q
2Q
2
2
2
K10, 4
8Q
4Q
2
K 8, 4
K 3, 4
K 2, 5
4Q
2Q
2
2
2
K 8,5
2
K10,5
8Q
4Q
K 3,5
K 2,12
4Q
2Q
2
2
2
K 8,12
2
K10,12
8Q
4Q
2
K 3,12
2
K 2,13
4Q
2Q
8Q
2Q
2
K 8,13
2
K 3,13
K 2,14
2Q
Q
4Q
2Q
2
K 8,14
2
2
K10,13
2
2
K10,14
K 3,14
2
K 2,15
2Q
Q
2
K 8,15
2
K10,15
4Q
2Q
2
K 3,15
2
K 2,16
2Q
Q
2
K 8,16
2
K10,16
4Q
8Q
K 3,16
K 2, 6
8Q
4Q
2
2
2
K 8,6
2
K10,6
16Q
2
4Q
K1, 4
2
K 7, 4
2Q
2Q
2
K 9, 4
2
K11, 4
4Q
2
K1,5
K 7 ,5
K 9,5
K11,5
K1,5
4Q
K 7, 4
4Q
8Q
Q
K10,16
4Q
4Q
4Q
2
2
4Q
4Q
K 8,16
K 3,15
2
2Q
2Q
K 2, 6
K10,15
4Q
4Q
2
K1, 4
2Q
Q
K 3,14
2
8Q
K 8,15
K10,14
E(T ) 2 2PK1 2PK 2 2PK 7 2PK8 PK 9 PK10 PK11 4PK 3 p 2PK 4 2PK 5 2PK12 2PK13 PK14 PK15 PK16 4PK 6 4Q
2Q
K 2,16
2Q
Q
K 3,13
2
2Q
K1,12
2
K 7,12
2Q
2Q
2
K 9,12
2
K11,12
4Q
2Q
K11, 4
(33)
K 3,6
2
K1,13
2
K 7,13
2Q
2Q
K 9, 4
K10,6
16Q
4Q
2
2Q
K 8,6
4Q
K 3,16
K1,12
K 7 ,5
2
K 9,13
2
K11,13
2
2
1 ( 2 M ) 2
2 N
1
DK1 1 DK 2 1 DK 7
K 3,6
2
1
1
1
4 p
2
2
2
2
1
1
1
4
4
2
(
M )
2 N
1
1 DK8 1 DK 9 1 DK10 1 DK11 1 DK 3
DK 4 1 DK 5 1 DK12 1 DK13 1 DK14 1 DK15 1 DK16 1 DK 6 1 DK1 DK 4
4
4
4
2
2
2
8
4
4
4
4
1 DK1 DK 5 1 DK1 DK12 1 DK1 DK13 1 DK1 DK14 1 DK1 DK15 1 DK1 DK16 1 DK1 DK 6 1 DK 2 DK 4 1 DK 2 DK 5 1 DK 2 DK12 1 DK 2 DK13
2
2
2
8
4
4
4
4
2
2
2
1 DK 2 DK14 1 DK 2 DK15 1 DK 2 DK16 1 DK 2 DK 6 1 DK 7 DK 4 1 DK 7 DK 5 1 DK 7 DK12 1 DK 7 DK13 1 DK 7 DK14 1 DK 7 DK15 1 DK 7 DK16
8
4
4
4
4
2
2
2
8
2
2
1 DK 7 DK 6 1 DK8 DK 4 1 DK8 DK 5 1 DK8 DK12 1 DK8 DK13 1 DK8 DK14 1 DK8 DK15 1 DK8 DK16 1 DK8 DK 6 1 DK 9 DK 4 1 DK 9 DK 5
2
2
1
1
1
4
2
2
2
2
1
1 DK 9 DK12 1 DK 9 DK13 1 DK 9 DK14 1 DK 9 DK15 1 DK 9 DK16 1 DK 9 DK 6 1 DK10 DK 4 1 DK10 DK 5 1 DK10 DK12 1 DK10 DK13 1 DK10 DK14
1
1
4
2
2
2
2
1
1
1
4
1 DK10 DK15 1 DK10 DK16 1 DK10 DK 6 1 DK11 DK 4 1 DK11 DK 5 1 DK11 DK12 1 DK11 DK13 1 DK11 DK14 1 DK11 DK15 1 DK11 DK16 1 DK11 DK 6
8
8
8
8
4
4
4
16
1 DK 3 DK 4 1 DK 3 DK 5 1 DK 3 DK12 1 DK 3 DK13 1 DK 3 DK14 1 DK 3 DK15 1 DK 3 DK16 1 DK 3 DK 6
(34)
where for a=1,2,3,4,5,….16,b=1,2,…8 and d=9,10…..16. P Ka , Q Kb, d are given by (24) .
If g(t ) g
x (1)
(t ), f (t ) f u (k ) (t )
E (T ) 2R I1 2R I 2 2R I7 2R I8 R I9 R I10 R I11 4R I3 p 2R I 4 2R I5 2R I12 2R I13 R I14 R I15 R I16 4R I6 4SI1,4 4SI1,5 4SI1,12
(35)
4SI1,13 2SI1,14 2SI1,15 2SI1,16 8SI1,6 4SI 2,4 4SI 2,5 4SI 2,12 4SI 2,13 2SI 2,14 2SI 2,15 2SI 2,16 8SI 2,6 4SI7,4 4SI7,5 4SI7,12
4SI7,13 2SI7,14 2SI7,15 2SI7,16 8SI7,6 4SI8,4 4SI8,5 4SI8,12 4SI8,13 2SI8,14 2SI8,15 2SI8,16 8SI8,6 2SI9,4 2SI9,5 2SI9,12
2SI9,13 SI9,14 SI9,15 SI9,16 4SI9,6 2SI10,4 2SI10,5 2SI10,12 2SI10,13 SI10,14 SI10,15 SI10,16 4SI10,6 2SI11,4 2SI11,5 2SI11,12
2SI11,13 SI11,14 SI11,15 SI11,16 4SI11,6 8SI3,4 8SI3,5 8SI3,12 8SI3,13 4SI3,14 4SI3,15 4SI3,16 16SI3,6
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318 | P a g e
5. P. Saranya et al Int. Journal of Engineering Research and Applications
ISSN : 2248-9622, Vol. 4, Issue 2( Version 1), February 2014, pp.315-330
2
2
2
2
2
2
2
2
2
2
I 2, 5
2
I 2,12
2
2
2
2
2
2
2
2
2
2
I7, 4
2
I 7 ,5
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2
2
2
E(T ) 2 2R I1 2R I 2 2R I7 2R I8 R I9 R I10 R I11 4R I3 p 2R I 4 2R I5 2R I12 2R I13 R I14 R I15 R I16 4R I6 4S1,4 4SI1,5 4SI1,12 4SI1,13
2
I1,14
2
I1,15
2
I1,6
2
2S
2
I1,16
2
2
I 2, 4
2
2
I 2,13
2
I 2,14
2
I 2,15
2
I 2,16
2
I 2,6
2
I7,12
2
I7,13
2
I7,14
2
2S
2S
8S 4S
4S 4S
4S
2S
2S
2S
8S
4S
4S 4S
4S
2S
2SI7,15
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2S
8SI7,6 4SI8,4 4SI8,5 4SI8,12 4SI8,13 2SI8,14 2SI8,15 2SI8,16 8SI8,6 2SI9,4 2SI9,5 2SI9,12 2SI9,13 SI9,14 SI9,15 SI9,16 4SI9,6 2SI10,4
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2SI10,5 2SI10,12 2SI10,13 SI10,14 SI10,15 SI10,16 4SI10,6 2SI11,4 2SI11,5 2SI11,12 2SI11,13 SI11,14 SI11,15 SI11,16 4SI11,6 8SI3,4 8SI3,5
2
I7,16
2
2
2
8SI3,12 8SI3,13 4SI3,14 4SI3,15 4SI3,16 16SI3,6
1
2
2
2
2
2
1
1
1
4
2
( N M )
1
D I1 1 D I 2 1 D I7 1 D I8 1 D I9 1 D I10 1 D I11 1 D I3
2
2
2
2
1
1
1
4
4
4
4
4
2
( N M )
1
D I 4 1 D I5 1 D I12 1 D I13 1 D I14 1 D I15 1 D I16 1 D I6 1 D I1 D I 4 1 D I1 D I5 1 D I1 D I12 1 D I1 D I13
2
2
2
8
4
4
4
4
2
2
2
1 D I1 D I14 1 D I1 D I15 1 D I1 D I16 1 D I1 D I6 1 D I 2 D I 4 1 D I 2 D I5 1 D I 2 D I12 1 D I 2 D I13 1 D I 2 D I14 1 D I 2 D I15 1 D I 2 D I16
p
2
8
4
4
4
4
2
2
2
8
4
4
1 D I 2 D I6 1 D I7 D I 4 1 D I7 D I5 1 D I7 D I12 1 D I7 D I13 1 D I7 D I14 1 D I7 D I15 1 D I7 D I16 1 D I7 D I6 1 D I8 D I 4 1 D I8 D I5
4
4
2
2
2
8
2
2
2
2
1
1 D I8 D I12 1 D I8 D I13 1 D I8 D I14 1 D I8 D I15 1 D I8 D I16 1 D I8 D I6 1 D I9 D I 4 1 D I9 D I5 1 D I9 D I12 1 D I9 D I13 1 D I9 D I14
1
1
4
2
2
2
2
1
1
1
4
1 D I9 D I15 1 D I9 D I16 1 D I9 D I6 1 D I10 D I 4 1 D I10 D I5 1 D I10 D I12 1 D I10 D I13 1 D I10 D I14 1 D I10 D I15 1 D I10 D I16 1 D I10 D I6
2
2
2
2
1
1
1
4
8
8
8
1 D I11 D I 4 1 D I11 D I5 1 D I11 D I12 1 D I11 D I13 1 D I11 D I14 1 D I11 D I15 1 D I11 D I16 1 D I11 D I6 1 D I3 D I 4 1 D I3 D I5 1 D I3 D I12
8
4
4
4
16
1 D I3 D I13 1 D I3 D I14 1 D I3 D I15 1 D I3 D I16 1 D I3 D I6
(36)
where for a=1,2,3…16, b=1, 2, 3, 7, 8, 9, 10,11 and d=4, 5, 6, 12, 13,14,15,16.
R Ia , S Ib, d are given by (27).
Case (iii): The distributions of optional and mandatory thresholds possess SCBZ property.
If g(t ) g (t ), f (t ) f
(t )
u (1)
x (1)
E (T ) p
q CI 2 p CI3 p p CI 4 p q CI5 q CI6 p q CI7 q q CI8 p(p CI9 q CI10 p CI11 p p CI12 p q CI13 q CI14
3 4
3 4
1 2
1 2
2 1
1 2
2 CI1
2
1
1
4
4
3
3
p 4 q3 CI15 q 3q4 CI16 p 2 p 4 HI1,9 p 2 q4 HI1,10 p 2 p3 HI1,11 p 2 p 3 p4 HI1,12 p2 p 3q4 HI1,13 p2 q3 HI1,14 p2 p 4 q3 HI1,15 p2 q 3q4 HI1,16
q
2 p 4 H I 2 ,9
q
2 q 4 HI 2,10
q
2 p3 HI 2,11
q 2p
p
3 4 HI 2,12
q
2
(37)
p 3q4 HI2,13 q2 q3 HI2,14 q2 p 4 q3 HI2,15 q2 q 3q4 HI2,16 p1p 4 HI3,9
p q HI3,10 p p HI3,11 p1p p HI3,12 p p q HI3,13 p q HI3,14 p p q H I3,15 p q q H I3,16 p1p p H
1 4
1 3
2 4 I 4,9 p1 p 2 q 4 HI 4,10
1 3 4
1 3
1 4 3
1 3 4
3 4
p
p 2 p3 HI4,11 p1p 2 p 3 p4 HI4,12 p1 p2 p 3q4 HI4,13 p1 p2 q3 HI4,14 p1 p2 p 4 q3 HI4,15 p1 p2 q 3q4 HI4,16 p1q 2 p 4 HI5,9 p1 q 2 q4 HI5,10
p
q 2 p3 HI5,11 p1q 2 p 3 p4 HI5,12 p1 q2 p 3q4 HI5,13 p1 q2 q3 HI5,14 p1 q2 p 4 q3 HI5,15 p1 q2 q 3q4 HI5,16 q1p 4 HI6,9 q1q4 HI6,10 q1p3 HI6,11
1
1
q1p
3 4 H I6,12
q1p
2 3 p4 HI7,12
q
1
2
p
q
1
p
q 2 p3
p 3q4 HI6,13 q1 q3 HI6,14 q1 p 4 q3 HI6,15 q1 q 3q4 HI6,16 q1p 2 p 4 HI7,9 q1 p 2 q4 HI7,10 q1 p 2 p3 HI7,11
q
1
HI8,11 q1q
p2 p 3q4 HI7,13 q1 p2 q3 HI7,14 q1 p2 p 4 q3 HI7,15 q1 p2 q 3q4 HI7,16 q1q 2 p 4 HI8,9 q1 q 2 q4 HI8,10
p
2 3 p4 HI8,12 q1 q 2 p 3q 4 HI8,13 q1 q 2 q3 HI8,14 q1 q 2 p 4 q3 H I8,15 q1 q 2 q 3q 4 H I8,16
2
2
2
2
2
2
2
2
2
2
2
2
2
2
C I1 q 2 C I2 p1 C I3 p1p 2 C I4 p1q 2 C I5 q1 C I6 p 2q1 C I7 q1q 2 C I8 pp 4 C I9 q 4 C I10 p3 C I11 p3p 4 C I12 p3q 4 C I13 q 3 C I14
2
2
2
2
2
2
2
2
2
p 4q 3 C I15 q 3q 4 C I16 p 2 p 4 H I1,9 p 2q 4 H I1,10 p 2 p3 H I1,11 p 2 p3p 4 H I1,12 p 2 p3q 4 H I1,13 p 2 q 3 H I1,14 p 2 p 4q 3 H I1,15 p 2 q 3q 4 H 21,16
I
E(T ) 2 p
2
q
(38)
2
2 p 4 H I 2,9
q
2
2 q 4 H I 2,10
2
q
2
2 p 3 H I 2,11
2
q 2p
3
p 4 H 22,12 q 2 p3q 4 H 22,13 q 2 q 3 H 22,14 q 2 p 4q 3 H 22,15 q 2 q 3q 4 H 22,16 p1p 4 H 23,9 p1q 4 H 23,10
I
I
I
I
I
I
I
2
2
2
2
2
2
2
p p H I3,11 p1p p H I3,12 p p q H I3,13 p q H I3,14 p p q H I3,15 p q q H I3,16 p p p H I 4,9 p p q H I 4,10 p p p H I 4,11
1 3
1 3 4
1 3
1 4 3
1 3 4
1 2 4
1 2 4
1 2 3
3 4
p1p
p1q
p
2
I 4,12
p
2
I5,12
2 3p 4 H
2 3p 4 H
p1 p 2 p3q 4 H
2
I 4,13
p1 q 2 p3q 4 H
2
I5,13
p1 p 2 q 3 H
2
I 4,14
p1 q 2 q 3 H
2
I5,14
p1 p 2 p 4q 3 H
p1 q 2 p 4q 3 H
2
I 4,15
2
I5,15
p1 p 2 q 3q 4 H
p1 q 2 q 3q 4 H
2
I 4,16
2
I5,16
p1 q 2 p 4 H
q1p 4 H
2
I5,9
2
I6,9
p1 q 2q 4 H
q1q 4 H
2
I6,10
2
I5,10
p1 q 2 p3 H 25,11
I
q1p3 H 26,11 q1p3p 4 H 26,12
I
I
q
1
p3q 4 H 26,13 q1 q 3 H 26,14 q1 p 4q 3 H 26,15 q1 q 3q 4 H 26,16 q1p 2 p 4 H 27,9 q1 p 2q 4 H 27,10 q1 p 2 p3 H 27,11 q1p 2 p3p 4 H 27,12 q1 p 2 p3q 4 H 27,13
I
I
I
I
I
I
I
I
I
q
1
p 2 q 3 H 27,14 q1 p 2 p 4q 3 H 27,15 q1 p 2 q 3q 4 H 27,16 q1q 2 p 4 H 28,9 q1 q 2q 4 H 28,10 q1 q 2 p3 H 28,11 q1q 2 p3p 4 H 28,12 q1 q 2 p3q 4 H 28,13 q1 q 2 q 3 H 28,14
I
I
I
I
I
I
I
I
I
1
q 2 p 4q 3 H 28,15 q1 q 2 q 3q 4 H 28,16
I
I
q
where for a=1,2….16, b=1,2,3,4,5,6,7,8 and d=9,10,11,12,13,14,15,16.
CIa
p1
(39)
1
1
and HIb,d
k(1 BIa )
k(1 BIb BId)
3 3
2 2
4 4
1
,p
,p
,p
1 1 2 2 2 2 3 3 3 3 4 4 4 4
1
1
q1 1 p1 , q 2
BI1
*
g x (1)
1 p 2 , q 3 1 p3 , q 4 1 p 4
(2 ), BI 2
2
*
g x (1)
( ), BI3
2
*
g x (1)
(40)
(1 ), B
14
1
g x (1) (1 1 2 2),
*
(41)
BI5 g x (1) (1 2 2), BI6 g x (1) (1), BI7 g x (1) (1 2 2), BI8 g x (1) (1 2),
*
*
*
*
BI9 g x (1) (4 4), BI10 g x (1) (4), BI11 g x (1) (3 3), BI12 g x (1) (3 3 4 4),
*
*
*
*
BI13 g x (1) (3 4 3), BI14 g x (1) (3), BI15 g x (1) (3 4 4), BI16 g x (1) (3 4)
*
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*
*
*
319 | P a g e
6. P. Saranya et al Int. Journal of Engineering Research and Applications
ISSN : 2248-9622, Vol. 4, Issue 2( Version 1), February 2014, pp.315-330
www.ijera.com
If g ( t ) g x ( k ) ( t ), f ( t ) f u (1) ( t )
q
LK1 q2 LK 2 p1 LK3 p1p2 LK 4 p1q2 LK5 q1 LK6 p 2 q1 LK7 q1q2 LK8 p(p4 LK9 q4 LK10 p3 LK11 p 3 p4 LK12 p 3q4 LK13 q3 LK14 p 4 q3 LK15
p
p
p
3q 4 LK16 p 2 4 M K1,9 p 2 q4 M K1,10 p 2 p3 M KI1,11 p 2 3 p4 MK1,12 p2 p 3q4 M K1,13 p2 q3 M K1,14 p2 p 4 q3 M K1,15 p2 q 3q4 M K1,16 q 2 4 M K 2,9
q
2 q 4 M K 2,10
E (T ) p
2
p1p
q p M K 2,11 q 2 p p M K 2,12 q p q M K 2,13 q q M K 2,14 q p q M K 2,15 q q q MK 2,16 p p M
1 4 K 3,9 p1q 4 M K 3,10 p1p3 M K 3,11
2 3
2 3 4
2 3
2 4 3
2 3 4
3 4
p4 MK3,12 p1 p 3q4 MK3,13 p1 q3 MI3,14 p1 p 4 q3 MI3,15 p1 q 3q4 MI3,16 p1p 2 p 4 MI4,9 p1 p 2 q4 MK 4,10 p1 p 2 p3 MK 4,11 p1p 2 p 3 p4 MK 4,12
3
p2 p 3q4 MI4,13 p1 p2 q3 MI4,14 p1 p2 p 4 q3 MI4,15 p1 p2 q 3q4 MI4,16 p1q 2 p 4 MK5,9 p1 q 2 q4 MK5,10 p1 q 2 p3 MK5,11 p1q 2 p 3p4 MK5,12 p1 q2 p 3q4 MK5,13
p
1
p
q2 q3 MK5,14 p1 q2 p 4 q3 MK5,15 p1 q2 q 3q4 MK5,16 q1p 4 MK6,9 q1q4 MK6,10 q1p3 MK6,11 q1p 3p4 MK6,12 q1 p 3q4 MK6,13 q1 q3 MK6,14 q1 p 4 q3 MK6,15
q 3q4 MK6,16 q1p 2 p 4 MK7,9 q1 p 2 q4 MK7,10 q1 p 2 p3 MK7,11 q1p 2 p 3p4 MK7,12 q1 p2 p 3q4 MK7,13 q1 p2 q3 MK7,14 q1 p2 p 4 q3 MK7,15 q1 p2 q 3q4 MK7,16
1
1
q
q1q
2
p
2 4 M K8,9
q
1
q 2 q4 MK8,10 q1 q 2 p3 MK8,11 q1q 2 p 3p4 MK8,12 q1 q2 p 3q4 MK8,13 q1 q2 q3 MK8,14 q1 q2 p 4 q3 MK8,15 q1 q2 q 3q4 MK8,16
(42)
2
2
2
2
2
2
2
2
2
L q L p L p p L p q L q LK6 p 2 q1 LK7 q1q2 LK8 pp4 LK9 q4 LK10 p3 LK11 p3p4 LK12 p3q4 LK13 q3 LK14
p q L q q L p p M p q M p 2 p3 M2 1,11 p 2 p3p4 M2 1,12 p2 p 3q4 M2 1,13 p2 q3 M2 1,14 p2 p 4q3 M2 1,15 p2 q 3q4 M2 1,16
K
K
K
K
K
K
q p M
q p p M2 2,12 q2 p3q4 M2 2,13 q2 q3 M2 2,14 q2 p 4q3 M2 2,15 q2 q 3q4 M2 2,16 p1p4 M2 3,9 p1q4 M2 3,10
qqM
q pM
K
K
K
K
K
K
K
2 4
p p M
pppM
pp q M
p q M2 3,14 p1 p 4q3 M2 3,15 p1 q 3q4 M2 3,16 p1 p 2 p4 M2 4,9 p1 p 2q4 M2 4,10 p1 p 2 p3 M2 4,11
K
K
K
K
K
K
1 3
2
2 K1
2
2
4 3 K15
2
2
2
2
K2
1 2 K 4 1 2 K5 1
1 K3
2
2
2
3 4 K16 2 4 K1,9 2 4 K1,10
2
2
2
K 2 ,9
2 4 K 2,10 2 3 K 2,11 2 3 4
2
2
2
1
K 3,11
3 4 K3,12 1 3 4 K3,13 1 3
E(T ) 2 p
p1p
2
p
2 3 p4 M K 4,12
p
(43)
p p3q4 M2 4,13 p1 p2 q3 M2 4,14 p1 p2 p 4q3 M2 4,15 p1 p2 q 3q4 M2 4,16 p1 q 2 p4 M2 5,9 p1 q 2 q4 M2 5,10 p1 q 2 p3 M2 5,11 p1q 2 p3p4 M2 5,12
K
K
K
K
K
K
K
K
1 2
q p3q4 M2 5,13 p1 q2 q3 M2 5,14 p1 q2 p 4q3 M2 5,15 p1 q2 q 3q4 M2 5,16 q1p4 M2 6,9 q1q4 M2 6,10 q1p3 M2 6,11 q1p3p4 M2 6,12 q1 p3q4 M2 6,13 q1 q3 M2 6,14
K
K
K
K
K
K
K
K
K
K
2
2
2
2
2
2
2
2
2
q p q MK 6,15 q q q MK 6,16 q p p MK 7,9 q p q MK 7,10 q p p MK 7,11 q1p p p MK 7,12 q p p q MK 7,13 q p q MK 7,14 q p p q MK 7,15
1 2 4
2 3
1 4 3
1 3 4
1 2 4
1 2 3
1 2 3 4
1 2 3
1 2 4 3
p
1 2
4
q
p q 3q4 M2 7,16 q1q 2 p4 M2 8,9 q1 q 2q4 M2 8,10 q1 q 2 p3 M2 8,11 q1q 2 p3p4 M2 8,12 q1 q2 p3q4 M2 8,13 q1 q2 q3 M2 8,14 q1 q2 p 4q3 M2 8,15 q1 q2 q 3q4 M2 8,16
K
K
K
K
K
K
K
K
K
1 2
where for a=1,2….16, b=1,2,3,4,5,6,7,8 and d=9,10,11,12,13,14,15,16.
(44)
1
1
and M Kb, d
k (1 B Ka )
k (1 B Kb B Kd )
L Ka
BK1 g x ( k ) (2 2), BK 2 g x ( k ) (2), BK 3 g x ( k ) (1 1), BK 4 g x ( k ) (1 1 2 2),
*
*
*
*
BK 5 g x (k ) (1 2 2), BK 6 g x ( k ) (1), BK 7 g x (k ) (1 2 2), BK8 g x (k ) (1 2),
*
*
*
*
BK 9 g x ( k ) (4 4), BK10 g x ( k ) (4), BK11 g x ( k ) (3 3), BK12 g x ( k ) (3 3 4 4),
*
*
*
*
BK13 g x ( k ) (3 4 3), BK14 g x (k ) (3), BK15 g x (k ) (3 4 4), BK16 g x ( k ) (3 4)
*
*
*
*
(45)
If g ( t ) g x ( k ) ( t ), f ( t ) f u ( k ) ( t )
E (T )
p2 PK1 q 2 PK 2 p1 PK 3 p1p2 PK 4 p1q 2 PK 5 q1 PK 6 p 2 q1 PK 7 q1q 2 PK8 p(p4 PK 9 q 4 PK10 p3 PK11 p 3 p4 PK12
p
p
3q 4 P K13 q3 PK14 p 4 q3 P K15 q 3q 4 P K16 p 2 4 QK1,9 p 2 q 4 QK1,10 p 2 p3 QK1,11 p 2 3 p4 QK1,12 p2 p 3 q 4 QK1,13
p
p
q
2
q3 QK1,14 p2 p 4 q3 QK1,15 p2 q 3q 4 QK1,16 q 2 p 4 QK 2,9 q 2 q4 QK 2,10 q 2 p3 QK 2,11 q 2 p 3 p4 QK 2,12 q 2 p 3q 4 QK 2,13
q QK 2,14 q 2 p 4 q3 QK 2,15 q2 q 3q 4 QK 2,16 p1p 4 QK 3,9 p1q 4 QK 3,10 p1p3 QK 3,11 p1p 3 p4 QK 3,12 p1 p 3q4 QK 3,13
2 3
p
q3 QK 3,14 p1 p 4 q3 QK 3,15 p1 q 3q 4 QK 3,16 p1p 2 p 4 QK 4,9 p1 p 2 q4 QK 4,10 p1 p 2 p3 QK 4,11 p1p 2 p 3 p4 QK 4,12
p
p2 p 3q 4 QK 4,13 p1 p2 q3 QK 4,14 p1 p2 p 4 q3 QK 4,15 p1 p2 q 3q 4 QK 4,16 p1q 2 p 4 QK 5,9 p1 q 2 q4 QK 5,10 p1 q 2 p3 QK 5,11
1
1
p1q
q
2
p 3 p4 QK 5,12 p1 q 2 p 3q 4 QK 5,13 p1 q2 q3 QK 5,14 p1 q 2 p 4 q3 QK 5,15 p1 q2 q 3q 4 QK 5,16 q1p 4 QK 6,9 q1q 4 QK 6,10
q1p p Q
p
1p3 QK 6,11 q1 3 p4 QK 6,12 q1 p 3 q 4 QK 6,13 q1 q3 QK 6,14 q1 p 4 q3 QK 6,15 q1 q 3 q 4 QK 6,16
2 4 K 7,9 q1 p 2 q 4 QK 7,10
q
p 2 p3 QK 7,11 q1p 2 p 3 p4 QK 7,12 q1 p2 p 3q 4 QK 7,13 q1 p2 q3 QK 7,14 q1 p2 p 4 q3 QK 7,15 q1 p2 q 3q 4 QK 7,16 q1q 2 p 4 QK8,9
q
q 2 q 4 QK8,10 q1 q 2 p3 QK8,11 q1q 2 p 3 p4 QK8,12 q1 q2 p 3q 4 QK8,13 q1 q2 q3 QK8,14 q1 q 2 p 4 q3 QK8,15 q1 q2 q 3q 4 QK8,16
1
1
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(46)
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ISSN : 2248-9622, Vol. 4, Issue 2( Version 1), February 2014, pp.315-330
2
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2
2
2
2
2
2
2
2
2
2
2
2
P K1 q 2 P K 2 p1 P K3 p1p2 P K 4 p1q 2 P K5 q1 P K 6 p2 q1 P K 7 q1q 2 P K8 pp4 P K9 q 4 P K10 p3 P K11 p3 p4 P K12
2
2
2
2
2
2
2
2
p q P K13 q P K14 p q P K15 q q P K16 p p Q
p
3 4
3 4
4 3
2 4 K1,9 p 2 q 4 Q K1,10 p 2 p3 Q K1,11 p 2 3 p 4 Q K1,12
3
2
2
2
2
2
2
2
p p q Q
p q Q
p p q Q
p q q Q
q p Q
2 4 K 2,9 q 2 q 4 Q K 2,10 q 2 p3 Q K 2,11
K1,13
K1,14
K1,15
K1,16
2 3 4
2 3
2 4 3
2 3 4
E(T ) 2 p
2
q2 p
2
p
3 4 Q K 2,12
q
2
2
2
2
2
2
2
p3 q 4 QK 2,13 q 2 q3 QK 2,14 q 2 p4 q3 QK 2,15 q 2 q3 q 4 QK 2,16 p1p4 QK3,9 p1q 4 QK3,10
2
2
2
2
2
2
2
p p Q
p
1 3 K 3,11 p1 3 p 4 Q K 3,12 p1 p3 q 4 Q K 3,13 p1 q 3 Q K 3,14 p1 p 4 q 3 Q K 3,15 p1 q3 q 4 Q K 3,16 p1 p 2 p 4 Q K 4,9
2
p
1
2
2
2
2
2
p2 q 4 QK 4,10 p1 p2 p3 QK 4,11 p1p2 p3 p4 QK 4,12 p1 p2 p3 q 4 QK 4,13 p1 p2 q3 QK 4,14 p1 p2 p4 q3 QK 4,15
2
2
2
2
2
2
p
p2 q3 q 4 QK 4,16 p1q 2 p4 QK5,9 p1q 2 q 4 QK5,10 p1q 2 p3 QK5,11 p1q 2 p3 p4 QK5,12 p1q 2 p3 q 4 QK5,13
p
q 2 q3 QK5,14 p1q 2 p4 q3 QK5,15 p1q 2 q3 q 4 QK5,16 q1p4 QK 6,9 q1q 4 QK 6,10 q1p3 QK 6,11 q1p3 p4 QK 6,12
q
p3 q 4 QK 6,13 q1q3 QK 6,14 q1 p4 q3 QK 6,15 q1q3 q 4 QK 6,16 q1p2 p4 QK 7,9 q1 p2 q 4 QK 7,10 q1 p2 p3 QK 7,11
1
2
1
2
1
q1 p
1
2 3 p 4 Q K 7,12
2
2
2
1
2
2
2
2
2
2
2
2
2
2
p2 p3 q 4 QK 7,13 q1 p2 q3 QK 7,14 q1 p2 p4 q3 QK 7,15 q1 p2 q3 q 4 QK 7,16 q1q 2 p4 QK8,9
2
2
2
2
2
2
2
(N
2
p
p
q2
p1
p1 p2
p1q 2
q1
q1 p2
q1q 2 p
q4
2
2
4
M )
(
M )
2 N
1 BK1 1 BK 2 1 BK 3 1 BK 4 1 BK 5 1 BK 6 1 BK 7 1 BK8
1 BK 9 1 BK10
p3
q
2
q 2 q 4 QK8,10 q1q 2 p3 QK8,11 q1q 2 p3 p4 QK8,12 q1q 2 p3 q 4 QK8,13 q1q 2 q3 QK8,14 q1q 2 p4 q3 QK8,15 q1q 2 q3 q 4 QK8,16
1
2
p
2
2
2
q
2
1 BK11
p3 p 4
1 BK12
p3 q 4
1 BK13
q3
1 BK14
q3 p4
1 BK15
q3 q 4
1 BK16
p4 p2
1 BK1 BK 9
p2 q 4
1 BK1 BK10
p 2 p3
1 BK1 BK11
p 2 p3 p 4
p 2 p3 q 4
p2 q3 p4
p2 q3 q 4
q 2 p3
p2 q3
q 2 p4
q2 q4
1 BK1 BK12 1 BK1 BK13 1 BK1 BK14 1 BK1 BK15 1 BK1 BK16 1 BK 2 BK 9 1 BK 2 BK10 1 BK 2 BK11
q 2 p3 p 4
q 2 p3 q 4
q 2 q3 p4
q 2 q3 q 4
p1p3
q 2 q3
p4 p1
p1q 4
1 BK 2 BK12 1 BK 2 BK13 1 BK 2 BK14 1 BK 2 BK15 1 BK 2 BK16 1 BK 3 BK 9 1 BK 3 BK10 1 BK 3 BK11
p1p3 p4
p1p3 q 4
p1q3 p4
p1q3 q 4
p1 p2 p3
p1q3
p1 p4 p2
p1 p2 q 4
1 BK 3 BK12 1 BK 3 BK13 1 BK 3 BK14 1 BK 3 BK15 1 BK 3 BK16 1 BK 4 BK 9 1 BK 4 BK10 1 BK 4 BK11
p1 p2 p3 p4 p1 p2 p3 q 4
p1 p2 q3 p4 p1p2 q3 q 4
p1q 2 p3
p1 p2 q3
p1 p4 q 2
p1q 2 q 4
1 BK 4 BK12 1 BK 4 BK13 1 BK 4 BK14 1 BK 4 BK15 1 BK 4 BK16 1 BK 5 BK 9 1 BK 5 BK10 1 BK 5 BK11
q1p3
p1q 2 p3 p4 p1q 2 p3 q 4
p1q 2 q3 p4 p1q 2 q3 q 4
p1q 2 q3
p4 q1
q1q 4
1 BK 5 BK12 1 BK 5 BK13 1 BK 5 BK14 1 BK 5 BK15 1 BK 5 BK16 1 BK 6 BK 9 1 BK 6 BK10 1 BK 6 BK11
q1p3 p4
q1p3 q 4
q1q3 p4
q1q3 q 4
q1 p2 p3
q1q3
q1 p4 p2
q1 p2 q 4
1 BK 6 BK12 1 BK 6 BK13 1 BK 6 BK14 1 BK 6 BK15 1 BK 6 BK16 1 BK 7 BK 9 1 BK 7 BK10 1 BK 7 BK11
q1 p2 p3 p4 q1 p2 p3 q 4
q1 p2 q3 p4 q1 p2 q3 q 4
q1q 2 p3
q1 p2 q3
q1 p4 q 2
q1q 2 q 4
1 BK 7 BK12 1 BK 7 BK13 1 BK 7 BK14 1 BK 7 BK15 1 BK 7 BK16 1 BK8 BK 9 1 BK8 BK10 1 BK8 BK11
q1q 2 p3 p4 q1q 2 p3 q 4
q1q 2 q3 p4 q1q 2 q3 q 4
q1q 2 q3
1 BK8 BK12 1 BK8 BK13 1 BK8 BK14 1 BK8 BK15 1 BK8 BK16
(47)
where for a=1,2….16, b=1,2,3,4,5,6,7,8 and d=9,10,11,12,13,14,15,16.
N
PKa (1
If g ( t )
E (T ) p
2
BKa )
and Q
Kb,d
g x (1) ( t ), f ( t )
R I1 q
2
(48)
N
(1 BKb BKd)
f u (k ) (t )
R I2 p
p p R I 4 p q R I5 q R I6 p q R I7 q q R I8 p(p R I9 q R I10 p R I11 p p R I12 p q R I13
3 4
3 4
1 2
1 2
2 1
1 2
1 R I3
1
4
4
3
p4 q3 R I15 q3 q 4 R I16 p2 p4SI1,9 p2 q 4 SI1,10 p2 p3SI1,11 p2 p3 p4 SI1,12 p2 p3 q 4 SI1,13 p2 q3SI1,14 p2 p4 q3SI1,15
3 R I14
p q q SI1,16 q p S
2 4 I 2,9 q 2 q 4 SI 2,10 q 2 p3SI 2,11 q 2 p3 p4 SI 2,12 q 2 p3 q 4 SI 2,13 q 2 q3SI 2,14 q 2 p4 q3SI 2,15 q 2 q3 q 4 SI 2,16
2 3 4
q
p
p p S
1 4 I3,9 p1q 4 SI3,10 p1 p3SI3,11 p1 p3 p 4 SI3,12 p1 p3 q 4 SI3,13 p1 q3SI3,14 p1 p4 q3SI3,15 p1 q3 q 4 SI3,16 p1 2 p4 SI 4,9
p
p2 q 4 SI4,10 p1 p2 p3SI4,11 p1p2 p3 p4 SI4,12 p1 p2 p3 q 4 SI4,13 p1 p2 q3SI4,14 p1 p2 p4 q3SI4,15 p1 p2 q3 q 4 SI4,16 p1q 2 p4SI5,9
p
q 2 q 4 SI5,10 p1q 2 p3SI5,11 p1q 2 p3 p4 SI5,12 p1q 2 p3 q 4 SI5,13 p1q 2 q3SI5,14 p1q 2 p4 q3SI5,15 p1q 2 q3 q 4 SI5,16 q1p4SI6,9
1
1
q q SI6,10 q p SI6,11 q1 p p SI6,12 q p q SI6,13 q q SI6,14 q p q SI6,15 q q q SI6,16 q1 p p S
1 4
1 3
2 4 I7,9 q1 p2 q 4 SI7,10
1 3 4
1 3
1 4 3
1 3 4
3 4
q
1
p2 p3SI7,11 q1p2 p3 p4 SI7,12 q1 p2 p3 q 4 SI7,13 q1 p2 q3SI7,14 q1 p2 p4 q3SI7,15 q1 p2 q3 q 4 SI7,16 q1q 2 p4SI8,9
(49)
q q p SI8,11 q1q p p SI8,12 q q p q SI8,13 q q q SI8,14 q q p q SI8,15 q q q q SI8,16
q q
2 3 4
1 2 4 SI8,10
1 2 3
1 2 3 4
1 2 3
1 2 4 3
1 2 3 4
q
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ISSN : 2248-9622, Vol. 4, Issue 2( Version 1), February 2014, pp.315-330
2
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2
2
2
2
2
2
2
2
2
2
2
2
2
R I1 q 2 R I2 p1 R I3 p1 p2 R I4 p1q 2 R I5 q1 R I6 p2 q1 R I7 q1q 2 R I8 pp4 R I9 q 4 R I10 p3 R I11 p3 p4 R I12 p3 q 4 R I13
2
2
2
2
2
2
2
2
2
2
q R I14 p q R I15 q q R I16 p p SI1,9 p q SI1,10 p p SI1,11 p2 p p SI1,12 p p q SI1,13 p q SI1,14 p p q SI1,15
3 4
4 3
2 4
2 4
2 3
3
2 3 4
2 3
2 4 3
3 4
2
2
2
2
2
2
2
2
2
p q q SI1,16 q p SI 2,9 q q SI 2,10 q p SI 2,11 q 2 p p SI 2,12 q p q SI 2,13 q q SI 2,14 q p q SI 2,15 q q q SI 2,16
2 4
2 4
2 3
2 3 4
2 3 4
2 3
2 4 3
2 3 4
3 4
E(T ) 2 p
2
2
2
2
2
2
2
2
2
2
p p SI3,9 p q SI3,10 p p SI3,11 p1 p p SI3,12 p p q SI3,13 p q SI3,14 p p q SI3,15 p q q SI3,16 p p p SI 4,9
1 4
1 4
1 3
1 3 4
1 3
1 4 3
1 3 4
1 2 4
3 4
p
1
p
1
2
I 4,10
p2 q 4 S
2
I5,10
q2 q4S
2
I 4,11
p1 p2 p3S
2
I5,11
p1 q 2 p3S
2
2
I 4,12
p1 p2 p3 p4 S
2
I5,12
p1q 2 p3 p4 S
2
2
I 4,13
p1 p2 p3 q 4 S
2
I5,13
p1 q 2 p3 q 4 S
2
2
2
I 4,14
2
I 4,15
p1 p2 p4 q3S
p1 p2 q3S
2
I5,14
p1 q 2 q3S
2
2
I5,15
p1 q 2 p4 q3S
2
2
I 4,16
p1 p2 q3 q 4 S
2
I5,16
p1 q 2 q3 q 4 S
2
p1 q 2 p4 S25,9
I
q1 p4 S26,9
I
2
2
q q SI6,10 q p SI6,11 q1 p p SI6,12 q p q SI6,13 q q SI6,14 q p q SI6,15 q q q SI6,16 q p p SI7,9 q p q SI7,10
1 2 4
1 4
1 3
1 3 4
1 3
1 4 3
1 3 4
1 2 4
3 4
2
I7,11
2
I7,12
q1 p2 p3 p4 S
2
I 7,13
2
I7,14
2
I7,15
2
I7,16
q
p 2 p3 S
q
q 2 p3S28,11 q1q 2 p3 p4 S28,12 q1q 2 p3 q 4 S28,13 q1 q 2 q3S28,14 q1 q 2 p4 q3S28,15 q1 q 2 q3 q 4 S28,16
I
I
I
I
I
I
1
1
p1
1 BI3
q3 q 4
1 BI16
p1 p2
1 BI 4
p1q 2
1 BI5
p4 p2
1 BI1 BI9
q1 p2 p3 q 4 S
q1
1 BI6
p2 q 4
1 BI1 BI10
q1 p2
1 BI7
q1 p2 q3S
q1 p2 p4 q3S
q1 p2 q3 q 4 S
2
I8,9
q1q 2 p4 S
1
2
q
1
q 2 q 4 S28,10
I
p
q2
2
2
( N M )
1 BI1 1 BI 2
p
p3
p3 p 4 p3 q 4
q3
q3 p4
q1q 2 p 2
q4
4
(
M )
2 N
1 BI8
1 BI9 1 BI10 1 BI11 1 BI12 1 BI13 1 BI14 1 BI15
p 2 p3
p 2 p3 p 4
p 2 p3 q 4
p2 q3 p4
p2 q3 q 4
p2 q3
q 2 p4
q2 q4
1 BI1 BI11 1 BI1 BI12 1 BI1 BI13 1 BI1 BI14 1 BI1 BI15 1 BI1 BI16 1 BI 2 BI9 1 BI 2 BI10
(50)
q 2 p3
q 2 p3 p 4
q 2 p3 q 4
q 2 q3 p4
q 2 q3 q 4
p1 p3
p1 p3 p4
p1 p3 q 4
q 2 q3
p4 p1
p1q 4
1 BI 2 BI11 1 BI 2 BI12 1 BI 2 BI13 1 BI 2 BI14 1 BI 2 BI15 1 BI 2 BI16 1 BI3 BI9 1 BI3 BI10 1 BI3 BI11 1 BI3 BI12 1 BI3 BI13
p1 q3
1 BI3 BI14
p1 p4 q 2
1 BI5 BI9
p1q3 p4
p1q3 q 4
p1 p2 p3 p1 p2 p3 p4 p1 p2 p3 q 4 p1 p2 q3 p1 p2 q3 p4 p1 p2 q3 q 4
p1 p4 p2
p1 p2 q 4
1 BI3 BI15 1 BI3 BI16 1 BI 4 BI9 1 BI 4 BI10 1 BI 4 BI11 1 BI 4 BI12 1 BI 4 BI13 1 BI 4 BI14 1 BI 4 BI15 1 BI 4 BI16
p1q 2 q 4
1 BI5 BI10
q1 p3
p1 q 2 p3 p1 q 2 p3 p4 p1 q 2 p3 q 4 p1 q 2 q3 p1 q 2 q3 p4 p1q 2 q3 q 4
p4 q1
q1 q 4
1 BI5 BI11 1 BI5 BI12 1 BI5 BI13 1 BI5 BI14 1 BI5 BI15 1 BI5 BI16 1 BI6 BI9 1 BI6 BI10 1 BI6 BI11
q1 p3 p4
q1 p3 q 4
q1q3 p4
q1q3 q 4
q1 p2 p3 q1 p2 p3 p4 q1 p2 p3 q 4 q1 p2 q3
q1 q3
q1 p4 p2
q1 p2 q 4
1 BI6 BI12 1 BI6 BI13 1 BI6 BI14 1 BI6 BI15 1 BI6 BI16 1 BI7 BI9 1 BI7 BI10 1 BI7 BI11 1 BI7 BI12 1 BI7 BI13 1 BI7 BI14
q1 p2 q3 p4 q1 p2 q3 q 4 q1 p4 q 2
q1q 2 p3 q1q 2 p3 p4 q1q 2 p3 q 4 q1 q 2 q3 q1q 2 q3 p4 q1 q 2 q3 q 4
q1 q 2 q 4
1 BI7 BI15 1 BI7 BI16 1 BI8 BI9 1 BI8 BI10 1 BI8 BI11 1 BI8 BI12 1 BI8 BI13 1 BI8 BI14 1 BI8 BI15 1 BI8 BI16
where for a=1,2….16, b=1,2,3,4,5,6,7,8 and d=9,10,11,12,13,14,15,16.
R Ia
(51)
N
N
and S Ib, d
(1 B Ia)
(1 B Ib B Id)
III. MODEL DESCRIPTION AND ANALYSIS OF MODEL-II
For this model, the optional and mandatory thresholds for the loss of man-hours in the organization are
taken as Y=min (Y1, Y2) and Z=min (Z1, Z2). All the other assumptions and notations are as in model-I.
Case (i): The distribution of optional and mandatory thresholds follow exponential distribution
For this case the first two moments of time to recruitment are found to be
Proceeding as in model-I, it can be shown for the present model that
If g ( t ) g
( t ), f ( t ) f u (1) ( t )
x (1)
E (T )
C I3 p(C I6 H I3,6)
E (T 2 ) 2 C I 3 p(C I 6 H I 3, 6
2
2
2
(52)
(53)
where for a = 1, 2…6. b=1, 2, 3 and d=4, 5, 6 C Ia , H Ib, d are given by (16)
If g(t ) g
(t ), f (t ) f
(t )
u (1)
x (k )
E (T )
(54)
LK 3 p(LK 6 M K 3,6)
2
2
2
2
E(T ) 2(LK3 p(LK6 MK3,6
(55)
where for a = 1, 2…6. b=1, 2, 3 and d=4, 5, 6 L Ka , M Kb, d are given by (20).
If g(t ) g
(t ), f (t ) f u (k ) (t )
x(k)
E(T) PK3 p(PK6 Q
(56)
)
K3,6
2
2
E (T ) 2 PK 3 p
P
2
K6
2
Q
1
2
K 3,6
N
2
1
p
M
1
DK3 2
N
2
1
1
M
1
DK 6 1 D K3 DK 6
(57)
where for a = 1, 2…6. b=1, 2, 3 and d=4, 5, 6 P Ka , Q Kb, d are given by (24)
If g ( t ) g x (1) ( t ), f ( t ) f u ( k ) ( t )
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E (T )
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(58)
R I3 p(R I6 SI3,6)
E (T 2 ) 2
R
p
2
R
2
2
1
S I 3, 6
M
2
p
1
2
M
1
1
1 DI 3 D
I6
N
N
1
1
D
D
(59)
where for a = 1, 2…6. b=1, 2, 3 and d=4, 5, 6 R Ia , SIb, d are given by (26).
Case (ii): The distribution of optional and mandatory thresholds follow extended exponential distribution
If g(t ) g (t ), f (t ) f
(t )
I3
I6
2
2
I3
u (1)
x (1)
I6
E(T) 4CI3 2CI7 2CI8 CI11 p 4CI6 2CI12 2CI13 CI16 16HI3,6 8HI3,12 8HI3,13 4HI3,16 8HI7,6
(60)
4HI7,12 4HI7,13 2HI7,6 8HI8,6 4HI8,12 4HI8,13 2HI8,16 4HI11,6 2HI11,12 2HI11,13 HI11,16
(61)
E(T ) 2 4CI3 2CI7 2CI8 CI11 p 4CI6 2CI12 2CI13 C1I6 16HI3,6 8HI3,12 8HI3,13 4HI3,16 8HI7,6 4HI7,12
2
2
4H
2
2
I7,13
2
2H
2
I7,16
2
8H
2
I8,6
2
4H
2
2
I8,12
4H
2
2
I8,13
2
2H
2
I8,16
2
4H
2
I11,6
2
2H
2
I11,12
2
2H
2
I11,13
2
2
H
2
2
I11,16
where for a=3, 6,7,8,11,12, 13,16 ,b=3,7,8,11,and d=6,12,13,16 C Ia , H Ib, d are given by (16)
If g ( t ) g x ( k ) ( t ), f ( t ) f u (1) ( t )
E(T) 4LK 3 2LK 7 2LK8 LK11 p 4LK 6 2LK12 2LK13 LK16 16MK 3,6 8MK 3,12 8MK 3,13 4MK 3,16 8MK 7,6
(62)
4MK 7,12 4MK 7,13 2MK 7,6 8MK8,6 4MK8,12 4MK8,13 2MK8,16 4MK11,6 2MK11,12 2MK11,13 MK11,16
2
2
2
2
2
2
2
2
2
2
2
2
2
2
E(T ) 2 4LK3 2LK 7 2LK8 LK11 p 4LK 6 2LK12 2LK13 LK16 16LK3,6 8LK3,12 8LK3,13 4LK3,16 8LK 7,6
8L
4L
4L
2L
4L
4L
2L
4L
2L
2L
L
where for a=3, 6,7,8,11,12, 13,16 ,b=3,7,8,11,and d=6,12,13,16 L Ka , M Kb, d are given by (20).
If g(t ) g
(t ), f (t ) f u (k ) (t )
2
K 7,12
2
K 7,13
2
K8,6
2
K 7,16
2
K8,12
2
K8,13
2
K8,16
2
K11,6
2
K11,12
2
K11,13
2
K11,16
(63)
x (k )
E(T) 4PK 3 2PK 7 2PK8 PK11 p
4Q
K 7,12
2
E(T ) 2 4P
8Q
2
K3
2
K8,6
2P
4Q
K 7,13
2
K7
4Q
2P
2
K8,12
2
K8
4Q
P
2Q
4PK6 2PK12 2PK13 PK16 16QK3,6 8QK3,12 8QK3,13 4QK3,16 8QK7,6
K 7, 6
2
p
K11
2
K8,13
8Q
2
K6
2
K12
4P
2Q
2
K8,16
K8,6
2P
4Q
2
K11,6
2P
4Q
K8,12
2
K13
2Q
P
2
K11,12
4Q
K8,13
2Q
K8,16
4Q
K11,6
2Q
K11,12
2Q
K11,13
Q
K11,16
16QK3,6 8QK3,12 8QK3,13 4QK3,16 8QK7,6 4QK7,12 4QK7,13 2QK7,16
2
2
K16
2Q
2
K11,13
Q
2
2
2
2
2
2
(64)
2
(65)
1
1 ( 2 M) 4 2 2
2 N
K11,16
1
DK3 1 DK7 1 DK8 1 DK11
2
4
2
2
1
16
8
8
4
8
4
2
(
M)
2 N
1
DK 6 1 DK12 1 DK13 1 DK16 1 DK3 DK 6 1 DK3 DK12 1 DK3 DK13 1 DK3 DK16 1 DK 7 DK 6 1 DK 7 DK12
p
4
2
8
4
4
2
4
2
2
1
1 DK 7 DK13 1 DK 7 DK 6 1 DK8 DK 6 1 DK8 DK12 1 DK8 DK13 1 DK8 DK16 1 DK11 DK 6 1 DK11 DK12 1 DK11 DK13 1 DK11 DK16
where for a=3, 6,7,8,11,12, 13,16 ,b=3,7,8,11,and d=6,12,13,16 P Ka , Q Kb, d are given by (24) .
If g ( t ) g x (1) ( t ), f ( t ) f u ( k ) ( t )
E(T) 4R I3 2R I7 2R I8 R I11 p 4R I6 2R I12 2R I13 R I16 16SI3,6 8SI3,12 8SI3,13 4SI3,16 8SI7,6
(66)
4SI7,12 4SI7,13 2SI7,6 8SI8,6 4SI8,12 4SI8,13 2SI8,16 4SI11,6 2SI11,12 2SI11,13 SI11,16
E(T ) 2 4R I3 2R I7 2R I8 R I11 p 4R I6 2R I12 2R I13 R1I6 16SI3,6 8SI3,12 8SI3,13 4SI3,16 8SI7,6 4SI7,12 4SI7,13 2SI7,16 8SI8,6
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
4
4
2
2
1 p
2
2
2
( N M )
(
M )
2 N
1
1
DI3 1 DI7 1 DI8 1 DI11
DI6 1 DI12
2
1
16
8
8
4
8
4
4
2
8
1 DI13 1 DI16 1 DI3 DI6 1 DI3 DI12 1 DI3 DI13 1 DI3 DI16 1 DI7 DI6 1 DI7 DI12 1 DI7 DI13 1 DI7 DIK 6 1 DI8 DI6
2
2
2
2
2
2
2
4SI8,12 4SI8,13 2SI8,16 4SI11,6 2SI11,12 2SI11,13 SI11,16
1
2
4
4
2
4
2
2
1
1 DI8 DI12 1 DI8 DI13 1 DI8 DI16 1 DI11 DI6 1 DI11 DI12 1 DI11 DI13 1 DI11 DI16
(67)
where for a=3, 6,7,8,11,12, 13,16 ,b=3,7,8,11,and d=6,12,13,16 R Ia , SIb, d are given by (26).
Case (iii): The distributions of optional and mandatory thresholds possess SCBZ property.
If g ( t ) g
( t ), f ( t ) f u (1) ( t )
x (1)
(68)
p1p2 CI 4 p1q 2 CI5 p2 q1CI7 q1q 2 CI8 p(p3 p4 CI12 p3 q 4 CI13 p4 q3 CI15 q3 q 4 CI16
p1p p p H I 4,12 p p p q H I 4,13 p p q p H I 4,15 p p q q H I 4,16 p1q p p H I5,12
2 3 4
2 3 4
1 2 3 4
1 2 3 4
1 2 3 4
E (T )
p
q 2 p3 q 4 H I5,13 p1q 2 p4 q3 H I5,15 p1q 2 q3 q 4 H I5,16 q1p2 p3 p4 H I7,12 q1 p2 p3 q 4 H I7,13
q
p2 p4 q3 H I7,15 q1 p2 q3 q 4 H I7,16 q1q 2 p3 p4 H I8,12 q1q 2 p3 q 4 H I8,13 q1q 2 p4 q3 H I8,15 q1q 2 q3 q 4 H I8,16
1
1
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2
2
2
2
2
2
2
2
2
E(T ) 2 p p CI 4 p q CI5 p q CI7 q q CI8 p p p CI12 p q CI13 p q CI15 q q CI16
3 4
3 4
1 2
2 1
1 2
4 3
1 2
3 4
p2 p3q4 H24,13 p1p2 p4 q3 H24,15 p1p2 q3q4 H24,16 p1q2 p3 p4 H25,12
I
I
I
I
2
2
2
2
2
2
p q p q H I5,13 p q p q H I5,15 p q q q H I5,16 q1p p p H I7,12 q p p q H I7,13 q p p q H I7,15
2 3 4
1 2 3 4
1 2 4 3
1 2 3 4
1 2 3 4
1 2 4 3
p1p
q
1
2
p
2 3 p4 H I 4,12
p
1
p2 q3q4 H27,16 q1q2 p3 p4 H28,12 q1q2 p3q4 H28,13 q1q2 q3 H28,14 q1q2 p4 q3 H28,15 q1q2 q3q4 H28,16
I
I
I
I
I
I
(69)
where for a=4,5,7,8, 12, 13,15,16 ,b=4,5,7,8 and d=12, 13,15,16. C Ia , H Ib, d are given by (39) .
If g ( t ) g x ( k ) ( t ), f ( t ) f u (1) ( t )
E (T )
p1p2 LK 4 p1q 2 LK5 p2 q1 LK 7 q1q 2 LK8 p(p3 p4 LK12 p3 q 4 LK13 p4 q3 LK15 q3 q 4 LK16
p1q p p M K 5,12
p
2 3 p4 M K 4,12 p1 p2 p3 q 4 M K 4,13 p1 p2 q3 p4 M K 4,15 p1 p2 q3 q 4 M K 4,16
2 3 4
(70)
p1 p
p
q 2 p3 q 4 M K5,13 p1q 2 p4 q3 M K5,15 p1q 2 q3 q 4 M K5,16 q1p2 p3 p4 M K 7,12 q1 p2 p3 q 4 M K 7,13
q
p2 p4 q3 M K 7,15 q1 p2 q3 q 4 M K 7,16 q1q 2 p3 p4 M K8,12 q1q 2 p3 q 4 M K8,13 q1q 2 p4 q3 M K8,15 q1q 2 q3 q 4 M K8,16
1
1
2
2
2
2
2
2
2
2
2
2
E(T ) 2 p p LK 4 p q LK 5 p q LK 7 q q LK8 p p p LK12 p q LK13 p q LK15 q q LK16 p1p p p M K 4,12
3 4
3 4
2 3 4
1 2
2 1
1 2
4 3
1 2
3 4
p2 p3 q4 M2 4,13 p1 p2 p4 q3 M2 4,15 p1 p2 q3 q 4 M2 4,16 p1q2 p3 p4 M2 5,12 p1q 2 p3 q 4 M2 5,13 p1q 2 p4 q3 M2 5,15
K
K
K
K
K
K
2
2
2
2
2
2
p q q q M K 5,16 q1p p p M K 7,12 q p p q M K 7,13 q p p q M K 7,15 q p q q M K 7,16 q1q p p M K8,12
2 3
2 3
1 2 3 4
1 2 3 4
1 2 4 3
1 2 3 4
p
1
4
4
q p q 2 q1q 2 q3 M2 8,14 q1q 2 p4 q3 M 2 8,15 q1q 2 q3 q 4 M2 8,16
K
K
K
1 2 3 4 M K 8,13
q
(71)
where for a=4,5,7,8, 12, 13,15,16 ,b=4,5,7,8 and d=12, 13,15,16. L Ka , M Kb, d are given by (44).
If g ( t ) g x ( k ) ( t ), f ( t ) f u ( k ) ( t )
p1p2 PK 4 p1q 2 PK 5 p2 q1 PK 7 q1q 2 PK8 p(p3 p4 PK12 p3 q 4 PK13 p4 q3 PK15 q3 q 4 PK16
p1p p p Q
p1q p p Q
2 3 4 K 4,12 p1 p2 p3 q 4 QK 4,13 p1 p2 q3 p4 QK 4,15 p1 p2 q3 q 4 QK 4,16
2 3 4 K 5,12
E (T )
p
q 2 p3 q 4 QK 5,13 p1q 2 p4 q3 QK 5,15 p1q 2 q3 q 4 QK 5,16 q1p2 p3 p4 QK 7,12 q1 p2 p3 q 4 QK 7,13
q
p2 p4 q3 QK 7,15 q1 p2 q3 q 4 QK 7,16 q1q 2 p3 p4 QK8,12 q1q 2 p3 q 4 QK8,13 q1q 2 p4 q3 QK8,15 q1q 2 q3 q 4 QK8,16
1
1
(72)
E(T ) 2 p p P K 4 p q P K 5 p q P K 7 q q P K8 p p p P K12 p q P K13 p q P K15 q q P K16 p1 p p p Q
3 4
3 4
2 3 4 K 4,12
1 2
2 1
1 2
4 3
1 2
3 4
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
p
p2 p3 q 4 QK 4,13 p1 p2 p4 q3 QK 4,15 p1 p2 q3 q 4 QK 4,16 p1q 2 p3 p4 QK5,12 p1q 2 p3 q 4 QK5,13 p1q 2 p4 q3 QK5,15
p
q 2 q3 q 4 QK5,16 q1p2 p3 p4 QK 7,12 q1 p2 p3 q 4 QK 7,13 q1 p2 p4 q3 QK 7,15 q1 p2 q3 q 4 QK 7,16 q1q 2 p3 p4 QK8,12
q
q 2 p3 q 4 QK8,13 q1q 2 q3 QK8,14 q1q 2 p4 q3 QK8,15 q1q 2 q3 q 4 QK8,16
1
1
2
2
2
2
1
p p
p1q 2
q1 p2
2
1 2
( N M)
1
DK 4 1 DK 5 1 DK 7
p p
p1 p2 p3 p4
p1 p2 p3 q 4
p1 p2 q3 p4
p3 q 4
q3 p4
q3 q 4
q1q 2 p 2
3 4
(
M)
2 N
1 DK8
1 DK12 1 DK13 1 DK15 1 DK16 1 DK 4 DK12 1 DK 4 DK13 1 DK 4 DK15
p1p2 q3 q 4
p1q 2 q3 q 4
p1q 2 p3 p4
p1q 2 p3 q 4
p1q 2 q3 p4
q1 p2 p3 p4
q1 p2 p3 q 4
1 DK 4 DK16 1 DK 5 DK12 1 DK 5 DK13 1 DK 5 DK15 1 DK 5 DK16 1 DK 7 DK12 1 DK 7 DK13
q1 p2 q3 p4
q1 p2 q3 q 4
q1q 2 p3 p4
q1q 2 p3 q 4
q1q 2 q3 p4
q1q 2 q3 q 4
1 DK 7 DK15 1 DK 7 DK16 1 DK8 DK12 1 DK8 DK13 1 DK8 DK15 1 DK8 DK16
1
2
(73)
where for a=4,5,7,8, 12, 13,15,16 ,b=4,5,7,8 and d=12, 13,15,16. P Ka , Q Kb, d are given by (48).
If g ( t ) g x (1) ( t ), f ( t ) f u ( k ) ( t )
E (T ) p p R I 4 p q R I5 p q R I7 q q R I8 p(p p R I12 p q R I13 p q R I15 q q R I16
3 4
3 4
3 4
1 2
1 2
2 1
1 2
4 3
p1p p p SI 4,12 p p p q SI 4,13 p p q p SI 4,15 p p q q SI 4,16 p1q p p SI5,12
2 3 4
2 3 4
1 2 3 4
1 2 3 4
1 2 3 4
p q p q SI5,13 p q p q SI5,15 p q q q SI5,16 q1p p p SI7,12 q p p q SI7,13
2 3 4
1 2 3 4
1 2 4 3
1 2 3 4
1 2 3 4
q
1
p2 p4 q3 SI7,15 q1 p2 q3q 4 SI7,16 q1q2 p3 p4 SI8,12 q1 q 2 p3q 4 SI8,13 q1 q 2 p4 q3 SI8,15 q1 q 2 q3q 4 SI8,16
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(74)
324 | P a g e
11. P. Saranya et al Int. Journal of Engineering Research and Applications
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2
2
2
2
2
2
2
2
2
2
2
E(T ) 2 p p R I 4 p q R I5 p q R I7 q q R I8 p p p R I12 p q R I13 p q R I15 q q R I16 p1 p p p SI 4,12 p p p q SI 4,13
3 4
3 4
2 3 4
1 2
2 1
1 2
4 3
1 2 3 4
1 2
3 4
p2 p4 q3S24,15 p1 p2 q3 q 4 S24,16 p1q 2 p3 p4 S25,12 p1 q 2 p3 q 4 S25,13 p1 q 2 p4 q3S25,15 p1 q 2 q3 q 4 S25,16 q1 p2 p3 p4 S27,12
I
I
I
I
I
I
I
2
2
2
2
2
2
2
q p p q SI7,13 q p p q SI7,15 q p q q SI7,16 q1q p p SI8,12 q q p q SI8,13 q q q SI8,14 q q p q SI8,15
2 3 4
1 2 3 4
1 2 4 3
1 2 3 4
1 2 3 4
1 2 3
1 2 4 3
p
1
q
q 2 q3 q 4 S28,16
I
1
p p
p p
p3 q 4
q3 p4
q3 q 4
p1 q 2 q1 p2 q1 q 2 p
2
2
3 4
( N M ) 1 2
(
M )
2 N
1 D I 4 1 D I5 1 D I7 1 D I8
1 D I12 1 D I13 1 D I15 1 D I16
p1 p2 p3 p4 p1 p2 p3 q 4 p1 p2 q3 p4 p1 p2 q3 q 4 p1 q 2 p3 p4 p1 q 2 p3 q 4 p1 q 2 q3 p4 p1q 2 q3 q 4
1 D I 4 D I12 1 D I 4 D I13 1 DI 4 D15 1 D I 4 D I16 1 D I5 D I12 1 DI5 D I13 1 D I5 D I15 1 D I5 D I16
q p2 p3 p4 q1 p2 p3 q 4 q1 p2 q3 p4 q1 p2 q3 q 4 q1 q 2 p3 p4 q1 q 2 p3 q 4 q1 q 2 q3 p4 q1 q 2 q3 q 4
1
1 D I7 D I12 1 D I7 D I13 1 DI7 D I15 1 D I7 D I16 1 DI8 D I12 1 D I8 D I13 1 D I8 D I15 1 D I8 D I16
1
2
(75)
where for a=4,5,7,8, 12, 13,15,16 ,b=4,5,7,8 and d=12, 13,15,16. R Ia , SIb, d are given by (51).
IV. MODEL DESCRIPTION AND ANALYSIS OF MODEL-III
For this model, the optional and mandatory thresholds for the loss of man-hours in the organization are
taken as Y=Y1+Y2 and Z=Z1+ Z2 .All the other assumptions and notations are as in model-I. Proceeding as in
model-I, it can be shown for the present model that
Case (i): The distributions of optional and mandatory thresholds follow exponential distribution.
If g ( t ) g x (1) ( t ), f ( t ) f u (1) ( t )
E (T ) A2 CI 2 A1 CI1 p A5 CI5 A4 CI 4 A1 A4 HI1,4 A2 A4 HI1,4 A1 A5 HI1,5 A2 A5 HI 2,5
(76)
2
2
2
2
2
2
2
2
2
E(T ) 2 A2 C I2 A1 C I1 p A5 CI5 A4 C I4 A1 A4 H I1,4 A2 A4 H I1,4 A1 A5 H I1,5 A2 A5 H I2,5
where
(77)
2 ,
1 ,
2 ,
1
A2
A4
A5
1 2
1 2
1 2
1 2
A1
for a=1, 2, 4, 5, b=1,2and d=4, 5 C Ia , H Ib, d are given by equation (16).
If g ( t ) g
E (T )
x (k )
( t ), f ( t ) f u (1) ( t )
A 2 LK 2 A1 LK1 pA5 LK5 A 4 LK 4 A1 A 4 M K1,4 A 2 A 4 M K1,4 A1 A5 M K1,5 A 2 A5 M K 2,5
(78)
(79)
2
2
2
2
2
2
2
2
2
E(T ) 2 A2 LK 2 A1 LK1 p A5 LK5 A 4 LK 4 A1 A4 MK1,4 A 2 A4 MK1,4 A1 A5 MK1,5 A2 A5 MK 2,5
where for a=1, 2, 4, 5, b=1,2and d=4, 5 L Ka , M Kb, d are given by (20).
If g ( t ) g x ( k ) ( t ), f ( t ) f u ( k ) ( t )
E(T ) A 2 P K 2 A1 P K1 p A5 P K 5 A 4 P K 4 A1 A 4 Q
K1, 4
A2 A4 Q
K1, 4
A1 A5 Q
K1,5
A 2 A5 Q
(80)
K 2, 5
A2
A1
2
2
2
E(T ) 2 A2 P K 2 A1 P K1 p A5 PK 5 A4 P K 4 A1 A4 Q K1,4 A2 A4 Q K1,4 A1 A5 Q K1,5 A2 A5 Q K 2,5
( N M )
1
2
DK 2 1 DK1
2
p
2
2
2
2
2
2
2
1
A5
A4 A1A4 A 2 A4 A1A5 A 2 A5
2
( N M)
1
DK 5 1 DK 4 1 DK1 DK 4 1 DK1 DK 4 1 DK1 DK 5 1 DK 2 DK 5
(81)
where for a=1, 2, 4, 5, b=1,2and d=4, 5 P Ka , Q Kb, d are given by (24) .
If g ( t ) g x (1) ( t ), f ( t ) f u ( k ) ( t )
E(T) A2 R I 2 A1 R I1 p
2
E (T ) 2
A5 R I5 A4 R I4 A1 A4 SI1,4 A2 A4 SI1,4 A1 A5 SI1,5 A2 A5 SI2,5
(82)
A2 R I22 A1 R I12 pA5 R 25 A4 R I42 A1 A4 SI1,42 A2 A4 SI1,42 A1 A5 SI1,52 A2 A5 SI2,52
I
A2
1
A1 p ( 2 M) A5 A4 A1A4 A 2 A4 A1A5 A 2 A5
2
(
M )
N
2 N
1
1
DI2 1 DI1 2
DI5 1 DI 4 1 DI1 DI 4 1 DI1 DI 4 1 DI1 DI5 1 DI 2 DI5
(83)
where for a=1, 2, 4, 5, b=1,2and d=4, 5 R Ia , SIb, d are given by (27).
Case (ii): If the distributions of optional and mandatory thresholds follow extended exponential distribution
with shape parameter 2.
If g ( t )
( t ), f ( t )
(t)
g x (1)
f u (1)
E(T) S1 CI1 S2 CI9 S3 CI2 S4 CI10 p S5 CI4 S6 CI14 S7 CI5 S8 CI5 S1S5 HI1,4 S1S6 H
S2 S7 H
I9,5
S2 S8 HI9,15 S3S5 HI2,4 S3S6 H
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I2,14
S3S7 H
I2,5
I1,14
S1S7 H
S3S8 HI2,15 S4 S5 HI4,10 S4 S6 H
I1,5
I10,14
S1S8 HI1,15 S2 S5 HI9,4 S2 S6 H
S4 S7 H
I10,5
S4 S8 HI10,15
I9,14
(84)
325 | P a g e
12. P. Saranya et al Int. Journal of Engineering Research and Applications
ISSN : 2248-9622, Vol. 4, Issue 2( Version 1), February 2014, pp.315-330
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2
2
2
2
2
2
2
2
2
2
2
2
2
2
E(T ) 2 S1 CI1 S2 CI9 S3 CI2 S4 CI10 pS5 CI4 S6 CI14 S7 CI5 S8 CI5 S1S5 H I1,4 S1S6 H I1,14 S1S7 H I1,5 S1S8H I1,15 S2 S5 H I9,4
S2 S6 H I9,14
2
2
2
2
2
2
2
2
2
2
2
S2 S7HI9,5 S2 S8HI9,15 S3 S5 HI2,4 S3 S6HI2,14 S3 S7HI2,5 S3 S8HI2,15 S4 S5 HI4,10 S4 S6HI10,14 S4 S7HI10,5 S4 S8HI10,15
(85)
Where for a=1, 2, 4, 5, 9,10,14,15 b=1, 2,9,10 and d=4, 5, 14, 15. C Ia , H Ib, d are given by (15)
4
4 1
2
1
,S
,S
S1 2
S
1 2 1 22 2 1 221 2 3 1 221 2 4 1 21 22
2
2
2
2
4 2
4 1
2
1
S5
1 22 , S6 1 221 2 S7 1 221 2 , S8 1 21 22
1 2
If g(t ) g
(t ), f (t ) f u (1) (t )
2
2
x (k )
2
2
E (T ) S1 LK1 S2 LK 9 S3 LK 2 S4 LK10 p S5 LK 4 S6 LK14 S7 LK 5 S8 LK 5 S1S5 M K1, 4 S1S6 M
S1S7 M
K1,5
S3S7 M
E(T
2
S1S8 M K1,15 S2 S5 M K 9, 4 S2 S6 M
K 9,14
S3S8 M K 2,15 S4 S5 M K 4,10 S4 S6 M
K 2 ,5
S2 S7 M
K10,14
S4 S7 M
K1,14
S2 S8 M K 9,15 S3S5 M K 2, 4 S3S6 M
K 9 ,5
K10,5
(86)
K 2,14
S4 S8 M K10,15
(87)
2
2
2
2
2
2
) 2 S1 LK1 S2 LK9 S3 LK 2 S4 LK10 p S5 LK 4 S6 LK14 S7 LK5 S8 LK15 S1S5 M K1,4 S1S6 M K1,14
2
2
2
2
2
2
2
S1S7 M K1,5 S1S8 M K1,15 S2 S5 M K9,4 S2 S6 M K9,14
2
2
2
2
S2 S7 M K9,5 S2 S8 M K9,15 S3S5 M K 2,4 S3S6 M K 2,14
2
(88)
2
2
2
2
2
2
S3S7 M K 2,5 S3S8 M K 2,15 S4 S5 M K 4,10 S4 S6 M K10,14 S4 S7 M K10,5 S4 S8 M K10,15
where for a=1, 2,4,5 ,9,10,14,15 b=1,2,9,10 and d=4,5,14,15 L Ka , M Kb, d are given by (20).
If g ( t ) g
x (k )
( t ), f ( t ) f u ( k ) ( t )
E(T) S1 PK1 S2 PK 9 S3 PK 2 S4 PK10 p S5 PK 4 S6 PK14 S7 PK 5 S8 PK 5 S1 S5 Q
S1 S7 Q
S1 S8 Q
S3 S7 Q
S3 S8 Q
K1,5
K 2,5
K1,15
S2 S5 Q
K 9, 4
K 2,15
S4 S5 Q
K1, 4
S2 S6 Q
K 4,10
K 9,14
S4 S6 Q
S2 S7 Q
K 9,5
K10,14
S4 S7 Q
S2 S8 Q
K 9,15
K10,5
S4 S8 Q
S1 S6 Q
K1,14
S3 S5 Q
K 2, 4
S3 S6 Q
K 2,14
K10,15
(89)
2
E(T ) 2 S1 P K1 S2 P K9 S3 P K 2 S4 P K10 p S5 P K 4 S6 P K14 S7 P K5 S8 P K15 S1S5 QK1,4 S1S6 QK1,14
2
2
2
2
2
2
2
2
2
2
2
S1S7 QK1,5 S1S8 QK1,15 S2 S5 QK9,4 S2 S6 QK9,14
S2 S7 QK9,5
2
2
2
2
2
2
2
S2 S8 QK9,15 S3S5 QK 2,4 S3S6 QK 2,14
S
2
2
2
2
2
2
1
2
1
S3S7 QK 2,5 S3S8 QK 2,15 S4 S5 QK 4,10 S4 S6 QK10,14 S4 S7 QK10,5 S4 S8 QK10,15
( N M )
1
2
DK1
S2
1 DK9
S3
1 DK 2
S1S5
S4 p ( 2 M) S5 S6 S7 S8
2 N
1
1 DK10
DK 4 1 DK14 1 DK5 1 DK15 1 DK1D
K4
S1S6
1 DK1 D
K14
S3S5
S3S6
S1S7
S1S8
S2S5
S2S6
S2S7
S2S8
1 DK1 D
1 DK1 D
1 DK 2 D
1 DK 2 D
1 DK9 D
1 DK9 D
1 DK9 D
1 DK9 D
K5
K15
K4
K14
K4
K14
K5
K15
S3S7
S3S8
S4S5
S4S6
S4S7
S4S8
1 DK 2 D
1 DK 2 D
1 DK10 D
1 DK10 D
1 DK10 D
1 DK10 D
K5
K15
K4
K14
K5
K15
(90)
where for a=1, 2,4,5 ,9,10,14,15 b=1,2,9,10 and d=4,5,14,15 P Ka , Q Kb, d are given by (24) .
If g ( t ) g x (1) ( t ), f ( t ) f u ( k ) ( t )
E (T ) S1 R I1 S2 R I9 S3 R I 2 S4 R I10 p S5 R I 4 S6 R I14 S7 R I5 S8 R I5 S1S5 SI1, 4 S1S6S
S1S7S
I1,5
S3 S7S
I1,14
S1S8 SI1,15 S2 S5 SI9, 4 S2 S6S
I 2,5
I9,14
S3 S8 SI 2,15 S4 S5 SI 4,10 S4 S6S
S2 S7S
I10,14
I 9, 5
S4 S7S
S2 S8 SI9,15 S3 S5 SI 2, 4 S3 S6S
I10,5
I 2,14
S4 S8 SI10,15
(91)
2
2
2
2
2
2
2
2
2
2
2
E (T ) 2 S1 R I1 S2 R I9 S3 R I 2 S4 R I10 p S5 R I 4 S6 R I14 S7 R I5 S8 R I5 S1S5 SI1,4 S1S6 SI1,14
2
2
2
2
S1S7 SI1,5 S1S8SI1,15 S2 S5 SI9,4 S2 S6 SI9,14
S2 S7 SI9,5
2
2
2
2
S2 S8SI9,15 S3S5 SI 2,4 S3S6 SI 2,14
2
2
2
2
2
2
1
2
S3S7 SI 2,5 S3S8SI 2,15 S4 S5 SI 4,10 S4 S6 SI10,14 S4 S7 SI10,5 S4 S8SI10,15
( N M ) S1
1
2
DI1
S2
1 D I9
S3
1 DI 2
S4 p ( 2 M) S5 S6 S7 S8 S1S5
2 N
1
1 DI10
DI 4 1 DI14 1 DI5 1 DI15 1 DI1D
I4
S1S6
1 DI1D
I14
S3S5
S3S6
S1S7
S1S8
S2S5 S2S6
S2S7
S2S8
1 DI1D
1 DI1D
1 DI 2 D
1 DI 2 D
1 DI9 D
1 DI9 D
1 DI9 D
1 DI9 D
I5
I15
I4
I14
I4
I14
I5
I15
S3S7
S3S8
S4S5
S4S6
S4S7
S4S8
1 DI 2 D
1 DI 2 D
1 DI10 D
1 DI10 D
1 DI10 D
1 DI10 D
I5
I15
I4
I14
I5
I15
(92)
where for a=1, 2,4,5 ,9,10,14,15 b=1,2,9,10 and d=4,5,14,15 R Ia , SIb, d are given by (27).
Case (iii):The distributions of optional and the mandatory thresholds possess SCBZ property.
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ISSN : 2248-9622, Vol. 4, Issue 2( Version 1), February 2014, pp.315-330
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If g(t ) g (t ), f (t ) f
(t )
u (1)
x (1)
E (T ) R1 R 2 CI1 R 3 R 4 CI3 R 5 R 6 CI6 R 7 R 8 CI 2 pR 9 R10CI9 R13 R14CI10
R11 R12CI11 R15 R16CI14 R1 R 2 R 9 R10 H I1,9 R13 R14 H I1,10 R11 R12 H I1,11
R15 R16H I1,14 R 7 R 8 R 9 R10 H I 2,9 R13 R14 H I 2,10 R11 R12 H I 2,11 R15 R16H I 2,14
R 3 R 4 R 9 R10 H I3,9 R13 R14 H I3,10 R11 R12 H I3,11 R15 R16H I3,14 R 5 R 6 R 9 R10 H I6,9
R13 R14 H I6,10 R11 R12 H I6,11 R15 R16H I6,14
(93)
R1 R 2 C I12 R 3 R 4 C I32 R 5 R 6 C I62 R 7 R 8C I22 pR 9 R10C I92
2
2
2
2
R13 R14C I10 R11 R12C I11 R15 R16C I14 R1 R 2 R 9 R10 H I1,9
2
2
2
2
R13 R14 H I1,10 R11 R12 H I1,11 R15 R16H I1,14 R 7 R 8 R 9 R10 H I 2,9
2
2
2
2
R13 R14 H I 2,10 R11 R12 H I 2,11 R15 R16H I 2,14 R 3 R 4 R 9 R10 H I3,9
2
2
2
2
R13 R14 H I3,10 R11 R12 H I3,11 R15 R16H I3,14 R 5 R 6 R 9 R10 H I6,9
2
E (T ) 2
R13 R14H I6,102 R11 R12H I6,112 R15 R16H I6,142
(94)
where for a=1, 2, 3,6,9,10,11,14 b=1, 2,3,6and d=9,10,11,14. C Ia , H Ib, d are given by (39).
1 p p2
2 p p2
2 q1p
q p
pq
qq
R1 1 1 , R 2 1 1 2 , R 3 2 1 , R 4 21 2 R 5 2 2 , R 6 2 1 2
1 2 1 2
2 2
1 2 1 2
1 2 1
2 2
1
2
4 1 p p4
1 1p1q2 , 1q11 2 , 3 3p3 p4
q
q p4
p q4
,R7
R 8 R 9 R10 3 3 , R11 4 3 , R12 4 3
1 2
3 4
3 4 3 4
3 4 3
2
3 4
3 1q4 p3 13q3q4 , 4 4p34 q3 ,4 4 q3q4 4
3
R13 R14 R15 R16
3 4 3
3
4
3 4
4
3
4
If g ( t ) g
x (k )
( t ), f ( t )
(95)
f u (1) ( t )
R1 R 2 LK1 R 3 R 4LK3 R 5 R 6 LK 6 R 7 R 8LK 2 pR 9 R10LK9 R13 R14LK10
R11 R12LK11 R15 R16LK14 R1 R 2 R 9 R10M K1,9 R13 R14M K1,10 R11 R12M K1,11
R15 R16M K1,14 R 7 R 8R 9 R10M K 2,9 R13 R14M K 2,10 R11 R12M K 2,11 R15 R16M K 2,14
R 3 R 4R 9 R10M K3,9 R13 R14MK3,10 R11 R12M K3,11 R15 R16MK3,14
R 5 R 6 R 9 R10M K 6,9 R13 R14M K 6,10 R11 R12M K 6,11 R15 R16M K 6,14
2
2
2
2
2
2
2
2
E(T ) 2R1 R 2 LK1 R 3 R 4 LK3 R 5 R 6 LK 6 R 7 R 8LK 2 pR 9 R10LK9 R13 R14LK10 R11 R12LK11
2
2
2
2
2
R15 R16LK14 R1 R 2 R 9 R10M K1,9 R13 R14M K1,10 R11 R12M K1,11 R15 R16M K1,14
2
2
2
2
R 7 R 8R 9 R10M K 2,9 R13 R14M K 2,10 R11 R12M K 2,11 R15 R16M K 2,14
2
2
2
2
R 3 R 4 R 9 R10M K3,9 R13 R14M K3,10 R11 R12M K3,11 R15 R16M K3,14
2
2
2
2
R 5 R 6 R 9 R10M K 6,9 R13 R14M K 6,10 R11 R12M K 6,11 R15 R16M K 6,14
E (T )
(96)
(97)
where for a=1, 2, 3,6,9,10,11,14 b=1, 2,3,6and d=9,10,11,14 L Ka , M Kb, d are given by (44).
If g ( t ) g x ( k ) ( t ), f ( t ) f u ( k ) ( t )
R 9 R10PK9 R13 R14PK10
K1,9 R13 R14QK1,10 R11 R12QK1,11
R15 R16Q
R 7 R 8R 9 R10QK 2,9 R13 R14QK 2,10 R11 R12QK 2,11 R15 R16QK 2,14
K1,14
R 3 R 4 R 9 R10Q
R13 R14Q
R11 R12Q
R15 R16Q
K 3,9
K 3,10
K 3,11
K 3,14
R 5 R 6 R 9 R10Q
R13 R14Q
R11 R12Q
R15 R16Q
K 6,9
K 6,10
K 6,11
K 6,14
E (T ) R1 R 2 PK1 R 3 R 4 PK 3 R 5 R 6 PK 6 R 7 R 8 PK 2 p
R11 R12 PK11 R15 R16 PK14 R1 R 2 R 9 R10 Q
2
R1 R 2PK12 R 3 R 4PK32 R 5 R 6PK62 R 7 R 8PK 22 pR 9 R10PK92 R13 R14PK102
2
2
2
2
2
R11 R12P K11 R15 R16P K14 R1 R 2 R 9 R10QK1,9 R13 R14QK1,10 R11 R12QK1,11
2
2
2
2
2
R15 R16QK1,14 R 7 R 8R 9 R10QK 2,9 R13 R14QK 2,10 R11 R12QK 2,11 R15 R16QK 2,14
(98)
E (T ) 2
(99)
R 3 R 4R 9 R10QK3,92 R13 R14QK3,102 R11 R12QK3,112 R15 R16QK3,142 R 5 R 6R 9 R10QK6,92
R R R R R R R R
2
2
2
1
2
3
4
5
6
7
8
1
2
M
R13 R14 QK 6,10
R11 R12 QK 6,11
R15 R16 QK 6,14
N
1
1 BI3
1 BI6
1 BI 2
2
BI1
R 9 R10 R13 R14 R11 R12 R15 R16
R 9 R10 R13 R14 R11 R12 R15 R16
2 N M
R1 R 2 1
1
1 BI10
1 BI11
1 BI14
BI9
BI1 BI9 1 BI1 BI10 1 BI1 BI11 1 BI1 BI14
R 7 R 8 R 9 R10 R13 R14 R11 R12 R15 R16 R 3 R 4 R 9 R10 R13 R14 R11 R12 R15 R16
1
1
BI2 BI9 1 BI2 BI10 1 BI2 BI11 1 BI2 BI14
BI3 BI9 1 BI3 BI10 1 BI3 BI11 1 BI3 BI14
R R
9
10 R 13 R 14 R 11 R 12 R 15 R 16
R5 R6
1
BI6 BI9 1 BI6 BI10 1 BI6 BI11 1 BI6 BI14
p
2
where for a=1, 2, 3,6,9,10,11,14 b=1, 2,3,6and d=9,10,11,14 P Ka , Q Kb, d are given by (48).
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327 | P a g e
14. P. Saranya et al Int. Journal of Engineering Research and Applications
ISSN : 2248-9622, Vol. 4, Issue 2( Version 1), February 2014, pp.315-330
If g ( t ) g
x (1)
( t ), f ( t )
www.ijera.com
f u (k ) (t )
R1 R 2R I1 R 3 R 4R I3 R 5 R 6R I6 R 7 R8R I2 pR 9 R10R I9 R13 R14R I10 R11 R12R I11 R15 R16R I14
R1 R 2R 9 R10SI1,9 R13 R14SI1,10 R11 R12SI1,11 R15 R16SI1,14 R 7 R8R 9 R10SI2,9 R13 R14SI2,10
R11 R12SI2,11 R15 R16SI2,14 R 3 R 4R 9 R10SI3,9 R13 R14SI3,10 R11 R12SI3,11 R15 R16SI3,14
R 5 R 6R 9 R10SI6,9 R13 R14SI6,10 R11 R12SI6,11 R15 R16SI6,14
(100)
E (T )
2
R1 R 2R I12 R 3 R 4R I32 R 5 R 6R I62 R 7 R 8R I22 pR 9 R10R I92 R13 R14R I102 R11 R12R I112
R15 R16R I142 R1 R 2R 9 R10SI1,92 R13 R14SI1,102 R11 R12SI1,112 R15 R16SI1,142 R 7 R 8R 9 R10SI2,92
2
2
2
2
2
2
R13 R14SI2,10 R11 R12SI2,11 R15 R16SI2,14 R 3 R 4 R 9 R10SI3,9 R13 R14SI3,10 R11 R12SI3,11
R R
2
2
2
2
2
1
2 M 1 2
E(T ) 2
R15 R16 SI3,14
R 5 R 6 R 9 R10 SI6,9
R13 R14 SI6,10
R11 R12 SI6,11
R15 R16 SI6,14
(101)
2 N
1
BI1
R R
R13 R14
9
10
R1 R 2 1
2 N
1 BI3
1 BI6
1 BI 2
1 BI9
1 BI10
1 BI11
1 BI14
BI1 BI9 1 BI1 BI10
R R
R R
R15 R16
R R
R R
R R
R R
R R
R11 R12
R 7 R 8 1 9 10 1 13 14 1 11 12 1 15 16 R 3 R 4 1 9 10 1 13 14 1 11 12
1 BI1 BI11 1 BI1 BI14
BI2 BI9
BI2 BI10
BI2 BI11
BI2 BI14
BI3 BI9
BI3 BI10
BI3 BI11
R15 R16 R 9 R10 R13 R14 R11 R12 R15 R16
1 BI3 BI14 R 5 R 6 1 BI6 BI9 1 BI6 BI10 1 BI6 BI11 1 BI6 BI14
R 3 R 4 R 5 R 6 R 7 R 8
p
2
R 9 R10 R13 R14 R11 R12 R15 R16
M
where for a=1, 2, 3,6,9,10,11,14 b=1, 2,3,6and d=9,10,11,14 R Ia , SIb, d are given by (51).
V. NUMERICAL ILLUSTRATIONS
The mean and variance of the time to recruitment for the above models are given in the following
tables for the cases(i),(ii),(iii) respectively by keeping 1 0.4, 2 0.6, 1 0.5, 2 0.8, p 0.8.
1 0.6, 1 0.3, 1 0.7, 2 0.4, 2 0.7, 2 0.4, 3 0.8, 3 0.9, 3 0.5, 4 1, 4 1.5, 4 0.2, 1
fixed and varying c, k one at a time and the results are tabulated below .
MODEL-I
r=1
Case (i)
r=k
r=1
Case(ii)
r=k
r=1
Case(iii)
r=k
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Table – I (Effect of c, k, λ on performance measures)
c
1.5
1.5
1.5
0.5
k
1
2
3
2
λ
1
1
1
1
E
6.9679
6.3557
6.1484
2.5225
n=1
V
25.6665 19.3505
17.3979
3.6283
E
6.9679
2.3650
1.3080
1.0798
n=k
V
25.6665 3.1717
1.0148
0.8753
E
6.9679
19.0671
33.8154
7.5676
n=1
V
25.6665 161.4276 488.6549 28.1272
E
6.9679
7.0959
7.1940
3.2394
n=k
V
25.6665 24.4003
24.2223
6.6552
E
7.7324
7.0834
6.8728
2.7970
n=1
V
43.6104 36.0639
33.5832
5.7270
E
7.7324
3.5027
1.4977
1.2753
n=k
V
43.6104 3.1907
1.4998
1.0853
E
7.7324
21.2503
37.7996
8.3909
n=1
V
43.6104 319.2949 998.6626 50.7338
E
7.7324
8.0163
8.2373
3.7368
n=k
V
43.6104 38.4822
35.3702
7.2110
E
6.7761
6.1834
5.9831
2.4527
n=1
V
29.2323 23.1644
21.2936
3.9963
E
6.7761
2.2433
1.2782
1.0315
n=k
V
29.2323 3.6124
1.1285
0.8973
E
6.7761
18.5503
32.9067
7.3582
n=1
V
29.2323 196.1130 608.2081 31.0609
E
6.7761
6.7298
7.0299
3.0944
n=k
V
29.2323 28.0247
26.4670
6.0124
1
2
1
4.4423
9.9659
1.7248
1.8559
13.3270
81.0386
5.1745
13.9834
4.9371
17.6908
2.1966
2.3381
14.8114
156.1554
5.8959
19.4178
4.3208
11.6062
1.6384
2.0285
12.9625
87.1721
4.9151
14.9796
1.5
2
1
6.3557
19.3505
2.3650
3.1717
19.0671
161.4276
7.0959
24.4003
7.0834
36.0639
3.5027
3.1907
21.2503
319.2949
8.0163
38.4822
6.1834
23.1644
2.2433
3.6124
18.5503
196.1130
6.7298
28.0247
328 | P a g e
15. P. Saranya et al Int. Journal of Engineering Research and Applications
ISSN : 2248-9622, Vol. 4, Issue 2( Version 1), February 2014, pp.315-330
MODEL-II
n=1
r=1
n=k
Case (i)
r=k
n=1
n=k
n=1
r=1
n=k
Case(ii)
r=k
n=1
n=k
n=1
r=1
n=k
Case(iii)
r=k
n=1
n=k
MODEL-III
n=1
r=1
n=k
Case (i)
r=k
n=1
n=k
n=1
r=1
n=k
Case(ii)
r=k
n=1
n=k
n=1
r=1
n=k
Case(iii)
r=k
n=1
n=k
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c
k
λ
E
V
E
V
E
V
E
V
E
V
E
V
E
V
E
V
E
V
E
V
E
V
E
V
c
k
λ
E
V
E
V
E
V
E
V
E
V
E
V
E
V
E
V
E
V
E
V
E
V
E
1.5
1
1
3.0491
7.1810
3.0491
7.1810
3.0441
7.1810
3.0441
7.1810
4.4497
10.6673
4.4497
10.6673
4.4497
10.6673
4.4497
10.6673
2.9380
7.1321
2.9380
7.1321
2.9380
7.1321
2.9380
7.1321
1.5
1
1
8.7025
34.8211
8.7025
34.8211
8.7025
34.8211
8.7025
34.8211
11.9437
44.2200
11.9437
44.2200
11.9437
44.2200
11.9437
44.2200
8.4755
40.0820
8.4755
40.0820
8.4755
40.0820
8.4755
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1.5
2
1
2.4841
4.4459
1.0669
0.9564
7.4522
40.0132
3.2008
6.4733
3.8440
6.6718
1.5259
1.4048
11.5320
52.3579
4.5776
9.5917
2.3862
4.5063
1.0271
0.9467
7.1587
33.4547
1.0271
0.9467
1.5
3
1
2.2929
3.6707
0.6046
0.3209
12.6107
111.0337
3.3253
6.0806
3.6369
5.4893
0.8502
0.4678
20.0024
144.2232
4.6762
9.0489
2.1980
3.7663
0.5761
0.3126
12.0889
94.9477
0.5761
0.3126
0.5
2
1
1.1966
1.1672
0.6467
0.4026
3.5897
10.5046
1.9402
2.3297
1.6790
1.7016
0.7959
0.5362
5.0370
11.9566
2.3877
3.2339
1.1588
1.1488
0.6355
0.3945
3.4765
6.4008
0.6355
0.3945
1
2
1
1.8442
2.5502
0.8546
0.6514
5.5326
22.9520
2.5638
4.1530
2.7663
3.8162
1.1631
0.9324
8.2990
28.8134
3.4894
6.0657
1.7761
2.5532
0.8291
0.6393
5.3284
17.4633
0.8291
0.6393
1.5
2
1
2.4841
4.4459
1.0669
0.9564
7.4522
40.0132
3.2008
6.4733
3.8440
6.6718
1.5259
1.4048
11.5320
52.3579
4.5776
9.5917
2.3862
4.5063
1.0271
0.9467
7.1587
33.4547
1.0271
0.9467
1.5
2
1
8.0800
26.8975
2.9422
4.2573
24.2401
225.9177
8.8266
32.4314
11.2767
33.4379
4.0179
5.4334
33.8301
278.3881
12.0536
40.8646
7.8757
32.4776
2.8689
4.8388
23.6271
276.5466
8.6067
1.5
3
1
7.8698
24.4103
1.6226
1.3523
43.2829
691.1652
8.9241
31.1711
11.0507
30.0115
2.2079
1.7299
60.7777
841.5110
12.1431
39.0814
7.6735
30.0962
1.5833
1.5246
42.2037
864.3347
8.7081
0.5
2
1
3.1046
4.8250
1.2767
1.0932
9.3139
37.2158
3.83
7.2859
4.1989
6.1929
1.6500
1.4021
12.5968
47.3386
4.95
9.3189
3.0222
5.3736
1.2449
1.1397
9.0665
42.3183
3.7341
1
2
1
5.5952
13.6770
2.118
2.4337
16.7855
111.9030
6.3355
17.6793
7.7414
17.2159
2.8369
3.1300
23.2243
139.4601
8.5106
22.4964
5.4512
16.0863
2.0589
2.6751
16.3536
133.8743
6.1768
1.5
2
1
8.0800
26.8975
2.9422
4.2573
24.2401
225.9177
8.8266
32.4314
11.2767
33.4379
4.0179
5.4334
33.8301
278.3881
12.0536
40.8646
7.8757
32.4776
2.8689
4.8388
23.6271
276.5466
8.6067
329 | P a g e
16. P. Saranya et al Int. Journal of Engineering Research and Applications
ISSN : 2248-9622, Vol. 4, Issue 2( Version 1), February 2014, pp.315-330
V
40.0820
37.8112
36.6164
7.7674
www.ijera.com
19.9579
37.8112
VI. FINDINGS
The influence of nodal parameters on the performance measures namely mean and variance of the time
to recruitment for all the models are reported below.
i.
It is observed that the mean time to recruitment decreases, with increase in k for the cases r=1,n=1 and
r=1 ,n=k but increases for the cases r=k,n=1 and r=k,n=k.
ii.
If c increases, the average number of exits increases, which, in turn, implies that mean and variance of
the time to recruitment increase for all the models.
VII.
CONCLUSION
Note that while the time to recruitment is postponed in model-III, the time to recruitment is advanced
in model-I and II .Therefore from the organization point of view, model III is more preferable.
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