The SIJ Transactions on Industrial, Financial & Business Management (IFBM), Vol. 1, No. 5, November-December 2013
Variance of Time to Recruitment in a
Two Graded Manpower System using
Order Statistics for Attrition
J. Sridharan*, P. Saranya** & A. Srinivasan***
*Assistant Professor in Mathematics, Government Arts College (Autonomous), Kumbakonam, Tamilnadu, INDIA.
E-Mail: jayabala_dharan@yahoo.in
**Assistant Professor (Sr.Gr) in Mathematics, T.R.P. Engineering College (SRM Group), Trichirappalli, Tamilnadu, INDIA.
E-Mail: saranya.panchu@yahoo.in
***Associate Professor in Mathematics, Bishop Heber College (Autonomous), Trichirappalli, Tamilnadu, INDIA.
E-Mail: mathsrinivas@yahoo.com
Abstract—An organization with two grades subjected to loss of man power due to the policy decisions taken
by the organization is considered in this paper As the exit of personal is unpredictable, a new recruitment
policy involving two thresholds for each grade one is optional and the other is 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 is obtained for the models when (i) the loss of man-hours form an
order statistics (ii) the inter-decision time forms a sequence of independent and identically distributed
exponential random variables (iii) the optional and the 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—Man Power Planning; Mean and Variance of the Time for Recruitment; Order Statistics; Shock
Model; Univariate Recruitment Policy.
2000 MSC Subject Classification: Primary 90B70, Secondary: 91B40, 91D35
I.
R
INTRODUCTION
ANDOM depletion of manpower occurs in any
marketing organization due to the attrition of
personnel when the management takes policy
decisions regarding pay, perquisites and targets. This attrition
will adversely affect the smooth functioning of the
organization in due course of time when the loss of man
power is not compensated by recruitment. Frequent
recruitment is not advisable as it involves more cost. In view
of this situation organization should frame a suitable
recruitment policy to plan for recruitment. In this context, for
a two -graded organization three mathematical models are
constructed in this paper using a univariate recruitment policy
based on shock model approach.
As the loss of man-hours is unpredictable, a suitable
recruitment policy has to be designed to overcome this loss.
Esary et al., (1973) have stated a replacement policy for a
device, which is exposed to shocks. A number of models can
be seen from Grinold & Marshall (1977), Bartholomew &
Forbes (1979). The problem of finding the time to
ISSN: 2321 – 242X
recruitment is studied for a single grade and multi grade
system by several authors under different conditions.
Recently Muthaiyan et al., (2009) have obtained system
characteristic for a single grade man-power system when the
inter-decision times form an order statistics.
For a single graded system, Esther Clara (2012), has
considered a recruitment policy involving two thresholds for
the loss of manpower 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 the thresholds according as the inter-decision time are
independent and identically distributed random variables or
the inter-decision time are exchangeable and constantly
correlated exponential random variables. Srinivasan and
Vasudevan (2011A-D) have extended the results of Esther
Clara (2012) 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.
Sridharan et al., (2012A-C & 2013A-D) have extended
the results of Muthaiyan et al., (2009) for a two-grade system
© 2013 | Published by The Standard International Journals (The SIJ)
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The SIJ Transactions on Industrial, Financial & Business Management (IFBM), Vol. 1, No. 5, November-December 2013
sample 𝑋1 , 𝑋2 , 𝑋3 , … , 𝑋𝑘 with respective density functions
𝑔𝑥 1 . , 𝑔𝑥 2 . , … , 𝑔𝑥 𝑘 . . Let the inter-decision times are
independent and identically distributed with cumulative
distribution function F(.), probability density function f(.).
Let Y1, Y2 (Z1, Z2) be the random variables denoting optional
(mandatory) thresholds for the loss of man-hours in grades 1
and 2, with cumulative distribution function H(.), probability
density function is h(.). 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 the
mandatory thresholds are 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 Fk(.) be the k fold
convolution of F(.). Let l*(.) and f*(.), be the Laplace
transform of l(.) and f(.), respectively. Let Vk(t) be the
probability that there are exactly k decision epochs in (0, t].
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 manhours exceeds the mandatory threshold Z, the recruitment is
necessary.
involving two thresholds by assuming different distributions
for thresholds. In the above cited research works of
Muthaiyan et al., (2009) and Sridharan et al., (2012A-C &
2013A-D) it is assumed that the inter-decision time form an
order statistics and loss of man-hours forms a sequence of
independent and identically distributed exponential random
variables.
The objective of the present paper is to obtain the mean
and variance of the time to recruitment for a two grade
system using a univariate recruitment policy assuming that (i)
the inter–decision time forms an sequence of independent and
identically distributed exponential random variables (ii) the
loss of man-hours form an order statistics and (iii) the
thresholds for the loss of man-hours in each grade 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 𝑋𝑖 be the loss of
man-hours due to the ith decision epoch, i=1,2,3 Let
𝑋 1 , 𝑋 2 , 𝑋 3 , … , 𝑋 𝑘 be the order statistics selected from the
Main Results
We note that


 k

 k
  k

P(T  t )   V k (t ) P  X i  Y   p  V k (t ) P  X i  Y   P  X i  Z 
k 0
k 0
 i 1

 i 1
  i 1

For r=1, 2, 3…k the probability density function of X(r) is given by
g x( r ) (t )  r kcr [G (t )]
r 1
g (t )[1  G (t )]
k r
, r  1,2,3..k
(1)
(2)
If 𝑔 𝑡 = 𝑔𝑥 1 𝑡 ,
In this case it is known that
k 1
g x(1) (t )  k g (t ) 1  G(t ) 
By hypothesis 𝑓 𝑡 = 𝜆e−𝜆𝑡 and 𝑔 𝑡 = ce−c𝑡
Therefore from (3) and (4) we get,

kc
g x (1) ( ) 
kc  
(3)
(4)
(5)
If 𝑔 𝑡 = 𝑔𝑥 𝑘 𝑡 ,
In this case it is known that
k 1
g x ( k ) (t )  k G (t ) 
g (t )
(6)
Therefore from (4) and (6) we get
*
g x ( k ) ( ) 
E (T )  
*
d (l ( s ))
ds
ISSN: 2321 – 242X
k! c
k
(  c )(  2c )...(  kc )
2 *
d (l ( s )
2
2
2
, E (T ) 
and V (T )  E (T )  ( E (T ))
2
ds
s 0
s 0
© 2013 | Published by The Standard International Journals (The SIJ)
(7)
(8)
160
The SIJ Transactions on Industrial, Financial & Business Management (IFBM), Vol. 1, No. 5, November-December 2013
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
If 𝑔 𝑡 = 𝑔𝑥 1 𝑡 ,
E (T )  C I 1  C I 2  C I 3  p (C I 4  C I 5  C  H




I6
I 1,4 H I 1,5 H I 1,6 H I 2, 4 H I 2,5

(9)

(10)
 H I 2,6  H I 3,4  H I 3,5  H I 3,6
2
2
2
2
2
2
2
2
2
2
2
2
E (T )  2 C I 1  C I 2  C I 3  p C I 4  C I 5  C I 6  H I 1, 4  H I 1,5  H I 1,6  H I 2, 4  H I 2,5
2
2
2
2
 H I 2,6  H I 3, 4  H I 3,5  H I 3,6


where for a = 1, 2…6. b=1, 2, 3 and d=4, 5, 6.
CIa

1
1
and HIb,d 
(1  DIa )
(1  DIb DId )
(11)
*
*
*
DI 1  g x (1) (1), DI 2  g x (1) ( 2), DI 3  g x (1) (1   2)
*
*
*
D I 4  g x(1) (1), D I 5  g x(1) ( 2), D I 6  g x(1) (1   2)
If 𝑔 𝑡 = 𝑔𝑥
are given by (5)
(12)
𝑡 ,
𝑘
E (T )  PK1  PK 2  PK 3  p ( PK 4  PK 5  P




K 6  Q K1, 4 Q K1,5 Q K1,6 Q K 2,4 Q K 2,5
Q

K 2,6
Q
K 3, 4
Q
K 3,5

Q
K 3,6

(13)
2
2
2
2
2
2
2
2
2
2
2
2
E (T )  2 PK1  PK 2  PK 3  p PK 4  PK 5  PK 6  Q




K1,4 Q K1,5 Q K1,6 Q K 2,4 Q K 2,5
(14)
2
2
2
2

Q




K 2,6 Q K 3,4 Q K 3,5 Q K 3,6 

where for a = 1, 2…6. b=1, 2, 3 and d=4, 5, 6.
PKa 
1
 (1  D Ka)
and Q
Kb,d

1
 (1  D Kb D Kd )
*
*
*
D K1  g x ( k ) ( 1), D K 2  g x ( k ) ( 2), D K 3  g x ( k ) ( 1   2)
*
*
*
DK 4  g x( k ) (1), DK 5  g x( k ) ( 2), DK 6  g x( k ) (1   2)
are given by (7)
(15)
(16)
The variance of time to recruitment can be calculated from (9), (10), (13) and (14)
Case (ii): The distributions of optional and mandatory thresholds follow extended exponential distribution with shape
parameter 2.
If 𝑔 𝑡 = 𝑔𝑥 1 𝑡 ,

E (T )  2C I 1  2C I 2  2C I 7  2C I 8  C I 9  C I 10  C I 11  4C I 3  p 2C I 4  2C I 5  2C I 12  2C I 13  C I 14
 C I 15  C I 16  4C I 6  4 H I 1, 4  4 H I 1,5  4 H I 1,12  4 H I 1,13  2 H I 1,14  2 H I 1,15  2 H I 1,16  8 H I 1,6
 4 H I 2, 4  4 H I 2,5  4 H I 2,12  4 H I 2,13  2 H I 2,14  2 H I 2,15  2 H I 2,16  8 H I 2,6  4 H I 7, 4  4 H I 7,5
 4 H I 7,12  4 H I 7,13  2 H I 7,14  2 H I 7,15  2 H I 7,16  8 H I 7,6  4 H I 8, 4  4 H I 8,5  4 H I 8,12  4 H I 8,13
 2 H I 8,14  2 H I 8,15  2 H I 8,16  8 H I 8,6  2 H I 9, 4  2 H I 9,5  2 H I 9,12  2 H I 9,13  H I 9,14  H I 9,15
(17)
 H I 9,16  4 H I 9,6  2 H I 10, 4  2 H I 10,5  2 H I 10,12  2 H I 10,13  H1I 0,14  H I 10,15  H I 10,16  4 H I 10,6
 2 H I 11, 4  2 H I 11,5  2 H I 11,12  2 H I 11,13  H I 11,14  H I 11,15  H I 11,16  4 H I 11,6  8 H I 3, 4  8 H I 3,5
 8 H I 3,12  8 H I 3,13  4 H I 3,14  4 H I 3,15  4 H I 3,16  16 H I 3,6



2
2
2
2
2
2
2
2
2
2
2
2
2
E (T )  2 2C I 1  2C I 2  2C I 7  2C I 8  C I 9  C I 10  C I 11  4C I 3  p 2C I 4  2C I 5  2C I 12  2C I 13  C I 14
2
2
2
2
2
2
2
2
2
2
2
 C I 15  C I 16  4C I 6  4 H I 1, 4  4 H I 1,5  4 H I 1,12  4 H I 1,13  2 H I 1,14  2 H I 1,15  2 H I 1,16  8 H I 1,6
2
2
2
2
2
2
2
2
2
2
2
 4 H I 2, 4  4 H I 2,5  4 H I 2,12  4 H I 2,13  2 H I 2,14  2 H I 2,15  2 H I 2,16  8 H I 2,6  4 H I 7, 4  4 H I 7,5
2
2
2
2
2
2
2
2
2
2
 4 H I 7,12  4 H I 7,13  2 H I 7,14  2 H I 7,15  2 H I 7,16  8 H I 7,6  4 H I 8, 4  4 H I 8,5  4 H I 8,12  4 H I 8,13
2
2
2
2
2
2
2
2
2
2
 2 H I 8,14  2 H I 8,15  2 H I 8,16  8 H I 8,6  2 H I 9, 4  2 H I 9,5  2 H I 9,12  2 H I 9,13  H I 9,14  H I 9,15
(18)
2
2
2
2
2
2
2
2
2
2
 H I 9,16  4 H I 9,6  2 H I 10, 4  2 H I 10,5  2 H I 10,12  2 H I 10,13  H I 10,14  H I 10,15  H I 10,16  4 H I 10,6
2
2
2
2
2
2
2
2
2
2
 2 H I 11, 4  2 H I 11,5  2 H I 11,12  2 H I 11,13  H I 11,14  H I 11,15  H I 11,16  4 H I 11,6  8 H I 3, 4  8 H I 3,5
2
2
2
2
2
2
 8 H I 3,12  8 H I 3,13  4 H I 3,14  4 H I 3,15  4 H I 3,16  16 H I 3,6

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.
𝐶𝐼𝑎 , 𝐻𝐼𝑏,𝑑 are given by (5) and (11)
D I7
 g *x (1) ( 21   2), D I8  g *x (1) (1 2 2), D I9  g *x (1) ( 21), D I10  g *x (1) ( 2 2)
D I11  g *
x (1) ( 21  2 2), D I12
 g *x (1) ( 21 2), D I13  g *x (1) ( 2 21), D I14  g *x (1) ( 21),
(19)
*
D I15  g *
x (1) ( 2 2), D I16  g x (1) ( 21 2 2)
ISSN: 2321 – 242X
© 2013 | Published by The Standard International Journals (The SIJ)
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The SIJ Transactions on Industrial, Financial & Business Management (IFBM), Vol. 1, No. 5, November-December 2013
If 𝑔 𝑡 = 𝑔𝑥
𝑘 𝑡 ,
E (T )  2P K1  2P K 2  2P K 7  2P K8  P K 9  P K10  P K11  4P K 3  p2P K 4  2P K 5  2P K12  2P K13  P K14
 P K15  P K16  4P K 6  4Q K1, 4  4Q K1,5  4Q K1,12  4Q K1,13  2Q K1,14  2Q K1,15  2Q K1,16  8Q K1,6
 4Q K 2, 4  4Q K 2,5  4Q K 2,12  4Q K 2,13  2Q K 2,14  2Q K 2,15  2Q K 2,16  8Q K 2,6  4Q K 7, 4  4Q K 7,5
 4Q K 7,12  4Q K 7,13  2Q K 7,14  2Q K 7,15  2Q K 7,16  8Q K 7,6  4Q K8, 4  4Q K8,5  4Q K8,12  4Q K8,13
 2Q K8,14  2Q K8,15  2Q K8,16  8Q K8,6  2Q K 9, 4  2Q K 9,5  2Q K 9,12  2Q K 9,13  Q K 9,14  Q K 9,15
(20)
 Q K 9,16  4Q K 9,6  2Q K10, 4  2Q K10,5  2Q K10,12  2Q K10,13  Q K10,14  Q K10,15  Q K10,16  4Q K10,6
 2Q K11, 4  2Q K11,5  2Q K11,12  2Q K11,13  Q K11,14  Q K11,15  Q K11,16  4Q K11,6  8Q K 3, 4  8Q K 3,5
 8Q K 3,12  8Q K 3,13  4Q K 3,14  4Q K 3,15  4Q K 3,16  16Q K 3,6


2
2
2
2
2
2
2
2
2
2
2
2
E ( T 2)  2 2 P 2
K1  2 P K 2  2 P K 7  2 P K 8  P K 9  P K10  P K11  4 P K 3  p2 P K 4  2 P K 5  2 P K12  2 P K13  P K14
2
2
2
2
2
2
2
2
2
2
 P2
K15  P K16  4 P K 6  4Q K1, 4  4Q K1,5  4Q K1,12  4Q K1,13  2Q K1,14  2Q K1,15  2Q K1,16  8Q K1,6
2
2
2
2
2
2
2
2
2
 4Q 2
K 2, 4  4Q K 2,5  4Q K 2,12  4Q K 2,13  2Q K 2,14  2Q K 2,15  2Q K 2,16  8Q K 2,6  4Q K 7, 4  4Q K 7,5
2
2
2
2
2
2
2
2
2
 4Q 2
K 7,12  4Q K 7,13  2Q K 7,14  2Q K 7,15  2Q K 7,16  8Q K 7,6  4Q K 8, 4  4Q K8,5  4Q K8,12  4Q K 8,13
2
2
2
2
2
2
2
2
2
 2Q 2
K8,14  2Q K 8,15  2Q K8,16  8Q K8,6  2Q K 9, 4  2Q K 9,5  2Q K 9,12  2Q K 9,13  Q K 9,14  Q K 9,15
(21)
2
2
2
2
2
2
2
2
2
 Q2
K 9,16  4Q K 9,6  2Q K10, 4  2Q K10,5  2Q K10,12  2Q K10,13  Q K10,14  Q K10,15  Q K10,16  4Q K10,6
2
2
2
2
2
2
2
2
2
 2Q 2
K11, 4  2Q K11,5  2Q K11,12  2Q K11,13  Q K11,14  Q K11,15  Q K11,16  4Q K11,6  8Q K 3, 4  8Q K 3,5
2
2
2
2
2
 8Q 2
K 3,12  8Q K 3,13  4Q K 3,14  4Q K 3,15  4Q K 3,16  16Q K 3,6

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.
𝑃𝐾𝑎 , 𝑄𝐾𝑏,𝑑 are given by (7) and (15)
D K 7  g *x ( k ) (21   2), D K8  g *x ( k ) (1 2 2), D K9  g *x ( k ) (21), D K10  g *x ( k ) (2 2), D K11  g *x ( k ) (21  2 2),
D K12  g *x ( k ) (21 2), D K13  g *x ( k ) (221), D K14  g *x ( k ) (21), D K15  g *x ( k ) (2 2), D K16  g *x ( k ) (21 2 2)
(22)
The variance of time to recruitment can be calculated from (17),(18), (20) and (21).
Case (iii): The distributions of optional thresholds follow exponential distribution and mandatory thresholds follow
extended exponential distribution with shape parameter 2.
If 𝑔 𝑡 = 𝑔𝑥 1 𝑡 ,
E (T )  C I1  C I 2  C I3  p2C I 4  2C I5  2C I12  2C I13 C I14  C I15  C I16  4C I6
 2 H I1, 4  2 H I1,5  2 H I1,12  2 H I1,13 
H I1,14
 2 H I 2, 4  2 H I 2,5  2 H I 2,12  2 H I 2,13 

2 H I3, 4


2 H I3,5

2 H I3,12



H I1,15
H I 2,14
2 H I3,13  H I3,14


H I 2,15
H I1,16


4 H I1,6
H I 2,16

4 H I 2,6
(23)
 H I3,15  H I3,16  4 H I3,6 
2
2
2
2
2
2
2
2
2
2
E ( T 2)  2 C 2
I1  C I 2  C I3  p 2C I 4  2C I5  2C I12  2C I13  C I14  C I15  C I16  4C I 6
2
2
2
2
2
2
2
 2H 2
I1, 4  2 H I1,5  2 H I1,12  2 H I1,13  H I1,14  H I1,15  H I1,16  4 H I1,6
(24)
2
2
2
2
2
2
2
 2H 2
I 2, 4  2 H I 2,5  2 H I 2,12  2 H I 2,13  H I 2,14  H I 2,15  H I 2,16  4 H I 2,6
2
2
2
2
2
2
2
 2H 2
I3, 4  2 H I3,5  2 H I3,12  2 H I3,13  H I3,14  H I3,15  H I3,16  4 H I3,6

where for a=1,2,3,4,5,6,12,13,14,15,16,b=1,2,3 and d=4,5,6,12,13,14,15,16 𝐶𝐼𝑎 , 𝐻𝐼𝑏,𝑑 are given by (5) and (11)
If 𝑔 𝑡 = 𝑔𝑥 𝑘 𝑡 ,
E (T ) 
P K1  P K 2
 P K 3  p2 P K 4  2 P K 5 
2 P K12

2 P K13  P K14
 P K15  P K16  4 P K 6
 2Q K1, 4  2Q K1,5  2Q K1,12  2Q K1,13  Q K1,14  Q K1,15  Q K1,16 
4Q K1,6
 2Q K 2, 4  2Q K 2,5  2Q K 2,12  2Q K 2,13  Q K 2,14  Q K 2,15  Q K 2,16 

2Q K 34

E (T 2)  2P 2
K1 
2Q K 3,5
P2
K2

2Q K 3,12

2Q K 3,13  Q K 3,14
2
2
 P2
K 3  p2 P K 4  2 P K 5 
2P 2
K12


Q K 3,15

2
2P 2
K13  P K14

2Q 2
K 3,5

2Q 2
K 3,12

2
2Q 2
K 3,13  Q K 3,14

4Q 2
K1,6
2
2
2
2
2
2
 2Q 2
K 2, 4  2Q K 2,5  2Q K 2,12  2Q K 2,13  Q K 2,14  Q K 2,15  Q K 2,16 
2Q 2
K 3, 4
(25)
2
2
 P2
K15  P K16  4 P K 6
2
2
2
2
2
2
 2Q 2
K1, 4  2Q K1,5  2Q K1,12  2Q K1,13  Q K1,14  Q K1,15  Q K1,16 

4Q K 2,6
Q K 3,16  4Q K 3,6
4Q 2
K 2,6
2
2
 Q2
K 3,15  Q K 3,16  4Q K 3,6
(26)

where for a=1,2,3,4,5,6,12,13,14,15,16,b=1,2,3 and d=4,5,6,12,13,14,15,16 𝑃𝐾𝑎 , 𝑄𝐾𝑏 ,𝑑 are given by (7) and (15)
The variance of time to recruitment can be calculated from (23), (24), (25) and (26).
ISSN: 2321 – 242X
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The SIJ Transactions on Industrial, Financial & Business Management (IFBM), Vol. 1, No. 5, November-December 2013
Case (iv): The distributions of optional and mandatory thresholds possess SCBZ property.
If 𝑔 𝑡 = 𝑔𝑥 1 𝑡 ,
E(T)  p 2 C I1  q 2 CI 2  p1 CI3  p1p 2 C I 4  p1q 2 CI5  q1 CI6  p 2q1 C I7  q1q 2 C I8  p( p 4 C I9  q 4 C I10  p3 C I11  p3p 4 C I12
 p3q 4 CI13  q3 CI14  p 4q3 C I15  q 3q 4 CI16  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 q3 H I1,14  p 2 p 4q3 H I1,15  p 2 q 3q 4 H I1,16  q 2 p 4 H I 2,9  q 2q 4 H I 2,10  q 2 p3 H I 2,11  q 2 p3p 4 H I 2,12  q 2 p3q 4 H I 2,13
 q 2 q3 H I 2,14  q 2 p 4q3 H I 2,15  q 2 q 3q 4 H I 2,16  p1p 4 H I3,9  p1q 4 H I3,10  p1p3 H I3,11  p1p3p 4 H I3,12  p1 p3q 4 H I3,13 
pp p p
p1 q 3 H I3,14  p1 p 4q 3 H I3,15  p1 q 3q 4 H I3,16  p1p 2 p 4 H
p p q
p p p
I 4,9  1 2 4 H I 4,10  1 2 3 H I 4,11  1 2 3 4 H I 4,12
 p1 p 2 p3q 4 H I 4,13  p1 p 2 q3 H I 4,14  p1 p 2 p 4q3 H I 4,15  p1 p 2 q 3q 4 H I 4,16  p1q 2 p 4 H I5,9  p1 q 2q 4 H I5,10  p1 q 2 p3 H I5,11
(27)
 p1q 2 p3p 4 H I5,12  p1 q 2 p3q 4 H I5,13  p1 q 2 q3 H I5,14  p1 q 2 p 4q3 H I5,15  p1 q 2 q 3q 4 H I5,16  q1p 4 H I6,9  q1q 4 H I6,10
 q1p3 H I6,11  q1p3p 4 H I6,12  q1 p3q 4 H I6,13  q1 q3 H I6,14  q1 p 4q3 H I6,15  q1 q 3q 4 H I6,16  q1p 2 p 4 H I7,9  q1 p 2q 4 H I7,10
 q1 p 2 p3 H I7,11  q1p 2 p3p 4 H I7,12  q1 p 2 p3q 4 H I7,13  q1 p 2 q3 H I7,14  q1 p 2 p 4q3 H I7,15  q1 p 2 q 3q 4 H I7,16  q1q 2 p 4 H I8,9
 q1 q 2q 4 H I8,10  q1 q 2 p3 H I8,11  q1q 2 p3p 4 H I8,12  q1 q 2 p3q 4 H I8,13  q1 q 2 q3 H I8,14  q1 q 2 p 4q3 H I8,15  q1 q 2 q 3q 4 H I8,16 
E (T 2)  2p 2 C2I1  q 2 C2I 2  p1 C2I3  p1p 2 C2I 4  p1q 2 C2I5  q1 C2I6  p 2q1 C2I7  q1q 2 C2I8  pp 4 C2I9  q 4 C2I10  p3 C2I11
 p3p 4 C2I12  p3q 4 C2I13  q3 C2I14  p 4q3 C2I15  q3q 4 C2I16  p 2p 4 H 2I1,9  p 2q 4 H 2I1,10  p 2 p3 H 2I1,11
 p 2p3p 4 H 2I1,12  p 2 p3q 4 H 2I1,13  p 2 q3 H 2I1,14  p 2 p 4q3 H 2I1,15  p 2 q3q 4 H 2I1,16  q 2p 4 H 2I 2,9
 q 2q 4 H 2I 2,10  q 2p3 H 2I 2,11  q 2p3p 4 H 2I 2,12  q 2 p3q 4 H 2I 2,13  q 2 q3 H 2I 2,14  q 2 p 4q3 H 2I 2,15
 q 2 q3q 4 H 2I 2,16  p1p 4 H 2I3,9  p1q 4 H 2I3,10  p1p3 H 2I3,11  p1p3p 4 H 2I3,12  p1 p3q 4 H 2I3,13
 p1 q3 H 2I3,14  p1 p 4q3 H 2I3,15  p1 q3q 4 H 2I3,16  p1 p 2 p 4 H 2I 4,9  p1 p 2q 4 H 2I 4,10  p1 p 2p3 H 2I 4,11
 p1p 2p3p 4 H 2I 4,12  p1 p 2 p3q 4 H 2I 4,13  p1 p 2 q3 H 2I 4,14  p1 p 2 p 4q3 H 2I 4,15  p1 p 2 q3q 4 H 2I 4,16
(28)
 p1 q 2p 4 H 2I5,9  p1 q 2q 4 H 2I5,10  p1 q 2 p3 H 2I5,11  p1q 2p3p 4 H 2I5,12  p1 q 2 p3q 4 H 2I5,13
 p1 q 2 q3 H 2I5,14  p1 q 2 p 4q3 H 2I5,15  p1 q 2 q 3q 4 H 2I5,16  q1p 4 H 2I6,9  q1q 4 H 2I6,10  q1p3 H 2I6,11
 q1p3p 4 H 2I6,12  q1 p3q 4 H 2I6,13  q1 q3 H 2I6,14  q1 p 4q3 H 2I6,15  q1 q3q 4 H 2I6,16  q1p 2p 4 H 2I7,9
 q1 p 2q 4 H 2I7,10  q1 p 2p3 H 2I7,11  q1p 2p3p 4 H 2I7,12  q1 p 2 p3q 4 H 2I7,13  q1 p 2 q3 H 2I7,14
 q1 p 2 p 4q3 H 2I7,15  q1 p 2 q3q 4 H 2I7,16  q1q 2 p 4 H 2I8,9  q1 q 2q 4 H 2I8,10  q1 q 2p3 H 2I8,11
 q1q 2p3p 4 H 2I8,12  q1 q 2 p3q 4 H 2I8,13  q1 q 2 q3 H 2I8,14  q1 q 2 p 4q3 H 2I8,15  q1 q 2 q3q 4 H 2I8,16

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.
C Ia 
p1 
1  1
1  1  1
,p 
2
 2   2
 2   2   2
,p 
3
1
 (1  BIa )
 3  3
3   3  3
and H Ib ,d 
,p 
4
1
 (1  BIb BId )
(29)
 4   4
 4   4   4
q1  1  p1, q2  1  p2 , q3  1  p3 , q4  1  p4
where 𝑔𝑥∗
1
(. ) are given by (5)
B I1
 g *x (1) ( 2   2), B I 2  g *x (1) ( 2), B I3  g *x (1) (1  1), B14  g *x (1) (11 2   2),
B I5
 g *x (1) (1   2   2), B I6  g *x (1) (1), B I7  g *x (1) (1   2   2), B I8  g *x (1) (1   2),
B I9
 g *x (1) ( 4   4), B I10  g *x (1) ( 4), B I11  g *x (1) ( 3   3), B I12  g *x (1) (33 4   4),
B I13
ISSN: 2321 – 242X
(30)
 g *x (1) (3 4   3), B I14  g *x (1) ( 3), B I15  g *x (1) (3 4   4), B I16  g *x (1) (3 4)
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The SIJ Transactions on Industrial, Financial & Business Management (IFBM), Vol. 1, No. 5, November-December 2013
If 𝑔 𝑡 = 𝑔𝑥
𝑘
𝑡 ,
E(T)  p 2 P K1  q 2 P K 2  p1 P K 3  p1p 2 P K 4  p1q 2 P K 5  q1 P K 6  p2q1 P K 7  q1q 2 P K8  p(p 4 P K 9  q 4 P K10  p 3 P K11  p3p 4 P K12
 p3q 4 P K13  q 3 P K14 p4q 3 P K15  q3q 4 P K16  p2p4Q K1,9  p2q 4 Q K1,10  p2p 3 Q K1,11  p2p3p 4 Q K1,12  p 2 p3q 4 Q K1,13
 p 2 q 3 Q K1,14  p 2 p4q 3 Q K1,15  p 2 q3q 4 Q K1,16  q 2p4Q K 2,9  q 2q 4 Q K 2,10  q 2p 3 Q K 2,11  q 2p3p 4 Q K 2,12  q 2 p3q 4 Q K 2,13
 q 2 q 3 Q K 2,14  q 2 p4q 3 Q K 2,15  q 2 q3q 4 Q K 2,16  p1p4Q K 3,9  p1q 4 Q K 3,10  p1p 3 Q K 3,11  p1p3p 4 Q K 3,12  p1 p3q 4 Q K 3,13
 p1 q 3 Q K 3,14  p1 p4q 3 Q K 3,15  p1 q3q 4 Q K 3,16  p1p2p4Q K 4,9  p1 p2q 4 Q K 4,10  p1 p2p 3 Q K 4,11  p1p2p3p 4 Q K 4,12
 p1 p 2 p3q 4 Q K 4,13  p1 p 2 q 3 Q K 4,14  p1 p 2 p4q 3 Q K 4,15  p1 p 2 q3q 4 Q K 4,16  p1q 2p4Q K 5,9  p1 q 2q 4 Q K 5,10
(31)
 p1 q 2p 3 Q K 5,11  p1q 2p3p 4 Q K 5,12  p1 q 2 p3q 4 Q K 5,13  p1 q 2 q 3 Q K 5,14  p1 q 2 p4q 3 Q K 5,15  p1 q 2 q3q 4 Q K 5,16
 q1p4Q K 6,9  q1q 4 Q K 6,10  q1p 3 Q K 6,11  q1p3p 4 Q K 6,12  q1 p3q 4 Q K 6,13  q1 q 3 Q K 6,14  q1 p4q 3 Q K 6,15
 q1 q3q 4 Q K 6,16  q1p2p4Q K 7,9  q1 p2q 4 Q K 7,10  q1 p2p 3 Q K 7,11  q1p2p3p 4 Q K 7,12  q1 p 2 p3q 4 Q K 7,13
 q1 p 2 q 3 Q K 7,14  q1 p 2 p4q 3 Q K 7,15  q1 p 2 q3q 4 Q K 7,16  q1q 2p4Q K8,9  q1 q 2q 4 Q K8,10  q1 q 2p 3 Q K8,11
 q1q 2p3p 4 Q K8,12  q1 q 2 p3q 4 Q K8,13  q1 q 2 q 3 Q K8,14  q1 q 2 p4q 3 Q K8,15  q1 q 2 q3q 4 Q K8,16 
E (T 2)  2p 2 P 2K1  q 2 P 2K 2  p1 P 2K 3  p1p 2 P 2K 4  p1q 2 P 2K 5  q1 P 2K 6  p 2q1 P 2K 7  q1q 2 P 2K8  pp 4 P 2K 9  q 4 P 2K10  p3 P 2K11  p3p 4 P 2K12
 p3q 4 P 2K13  q 3 P 2K14  p 4q 3 P 2K15  q 3q 4 P 2K16  p 2 p 4 Q 2K1,9  p 2q 4 Q 2K1,10  p 2 p3 Q 2K1,11  p 2 p3p 4 Q 2K1,12
 p 2 p3q 4 Q 2K1,13  p 2 q 3 Q 2K1,14  p 2 p 4q 3 Q 2K1,15  p 2 q 3q 4 Q 2K1,16  q 2 p 4 Q 2K 2,9  q 2q 4 Q 2K 2,10  q 2 p3 Q 2K 2,11
 q 2 p3p 4 Q 2K 2,12  q 2 p3q 4 Q 2K 2,13  q 2 q 3 Q 2K 2,14  q 2 p 4q 3 Q 2K 2,15  q 2 q 3q 4 Q 2K 2,16  p1p 4 Q 2K 3,9  p1q 4 Q 2K 3,10
 p1p3 Q 2K 3,11  p1p3p 4 Q 2K 3,12  p1 p3q 4 Q 2K 3,13  p1 q 3 Q 2K 3,14  p1 p 4q 3 Q 2K 3,15  p1 q 3q 4 Q 2K 3,16  p1 p 2 p 4 Q 2K 4,9
 p1 p 2q 4 Q 2K 4,10  p1 p 2 p3 Q 2K 4,11  p1p 2 p3p 4 Q 2K 4,12  p1 p 2 p3q 4 Q 2K 4,13  p1 p 2 q 3 Q 2K 4,14  p1 p 2 p 4q3 Q 2K 4,15
(32)
 p1 p 2 q 3q 4 Q 2K 4,16  p1 q 2 p 4 Q 2K 5,9  p1 q 2q 4 Q 2K 5,10  p1 q 2 p3 Q 2K 5,11  p1q 2 p3p 4 Q 2K 5,12  p1 q 2 p3q 4 Q 2K 5,13
 p1 q 2 q 3 Q 2K 5,14  p1 q 2 p 4q 3 Q 2K 5,15  p1 q 2 q 3q 4 Q 2K 5,16  q1p 4 Q 2K 6,9  q1q 4 Q 2K 6,10  q1p3 Q 2K 6,11  q1p3p 4 Q 2K 6,12
 q1 p3q 4 Q 2K 6,13  q1 q 3 Q 2K 6,14  q1 p 4q3 Q 2K 6,15  q1 q 3q 4 Q 2K 6,16  q1p 2 p 4 Q 2K 7,9  q1 p 2q 4 Q 2K 7,10  q1 p 2 p3 Q 2K 7,11
 q1p 2 p3p 4 Q 2K 7,12  q1 p 2 p3q 4 Q 2K 7,13  q1 p 2 q 3 Q 2K 7,14  q1 p 2 p 4q 3 Q 2K 7,15  q1 p 2 q 3q 4 Q 2K 7,16  q1q 2 p 4 Q 2K8,9
 q1 q 2q 4 Q 2K8,10  q1 q 2 p3 Q 2K8,11  q1q 2 p3p 4 Q 2K8,12  q1 q 2 p3q 4 Q 2K8,13  q1 q 2 q 3 Q 2K8,14  q1 q 2 p 4q 3 Q 2K8,15  q1 q 2 q 3q 4 Q 2K8,16

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.
P Ka

1
1
and QKb,d 
(1  BKa )
(1  BKb BKd )
(33)
B K1
 g *x ( k ) ( 2   2), B K 2  g *x ( k ) ( 2), B K 3  g *x ( k ) (1  1), B K 4  g *x ( k ) (11 2   2),
B K5
 g *x ( k ) (1   2   2), B K 6  g *x ( k ) (1), B K 7  g *x ( k ) (1   2   2), B K8  g *x ( k ) (1   2),
B K9
 g *x ( k ) ( 4   4), B K10  g *x ( k ) ( 4), B K11  g *x ( k ) ( 3   3), B K12  g *x ( k ) (33 4   4),
B K13
(34)
 g *x ( k ) (3 4   3), B K14  g *x ( k ) ( 3), B K15  g *x ( k ) (3  4   4), B K16  g *x ( k ) (3 4)
The variance of time to recruitment can be calculated from (27), (28), (31) and (32).
Case (v): The distributions of optional thresholds follow exponential distribution and the distribution of mandatory
thresholds possess SCBZ property.
If 𝑔 𝑡 = 𝑔𝑥 1 𝑡 ,
E(T)  C 'I1  C 'I 2  C 'I3  pp 4 C I9  q 4 C I10  p 3 C I11  p3p 4 C I12  p3q 4 C I13  q 3 C I14  p 4 q 3 C I15  q 4 q 3 C I16 
p 4 H 'I1,9  q 4 H 'I1,10  p 3 H 'I1,11  p3p 4 H 'I1,12  p3q 4 H 'I1,13  q 3 H 'I1,14  p 4 q 3 H 'I1,15  q 4 q 3 H 'I1,16 
p 4 H 'I 2,9  q 4 H 'I 2,10  p 3 H 'I 2,11  p3p 4 H 'I 2,12  p3q 4 H 'I 2,13  q 3 H 'I 2,14  p 4 q 3 H 'I 2,15  q 4 q 3 H 'I 2,16 
p 4 H 'I3,9  q 4 H 'I3,10  p 3 H 'I3,11  p3p 4 H 'I3,12  p3q 4 H 'I3,13  q 3 H 'I3,14  p 4 q 3 H 'I3,15  q 4 q 3 H 'I3,16


E(T 2)  2 C 'I21  C 'I22  C 'I23  pp 4 CI9 2  q 4 CI10 2  p 3 CI11 2  p3p 4 CI12 2  p3q 4 CI13 2  q 3 CI14 2  p 4 q 3 CI15 2  q 4 q 3 CI16 2 
'2
'2
'2
'2
'2
'2
p p
p q
p 4 H '2  q 4 H '2
p
q
p q
q q
I1,9
I1,10  3 H I1,11  3 4 H I1,12  3 4 H I1,13  3 H I1,14  4 3 H I1,15  4 3 H I1,16 
'2
'2
'2
'2
'2
'2
p p
p q
p 4 H '2  q 4 H '2
I 2,9
I 2,10  p 3 H I 2,11  3 4 H I 2,12  3 4 H I 2,13  q 3 H I 2,14  p 4 q 3 H I 2,15  q 4 q 3 H I 2,16 
'2
'2
'2
'2
'2
'2
p p
p q
p 4 H '2  q 4 H '2
p
q
p q
q q
I3,9
I3,10  3 H I3,11  3 4 H I3,12  3 4 H I3,13  3 H I3,14  4 3 H I3,15  4 3 H I3,16
(35)
(36)

where for b =1, 2,3 and d=9,10,11,12,13,14,15,16.
C 'Ib
ISSN: 2321 – 242X

1
1
1
, C Id 
and H 'Ib, d 
(1  D Ib)
(1  B Id )
(1  D Ib B Id )
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If 𝑔 𝑡 = 𝑔𝑥
𝑘
𝑡 ,
E(T)  P 'K1  P 'K 2  P 'K 3  pp 4 P K 9  q 4 P K10  p 3 P K11  p3p 4 P K12  p3q 4 P K13  q 3 P K14  p 4 q 3 P K15  q 4 q 3 P K16 
p 4 Q 'K1,9  q 4 Q 'K1,10  p 3 Q 'K1,11  p3p 4 Q 'K1,12  p3q 4 Q 'K1,13  q 3 Q 'K1,14  p 4 q 3 Q 'K1,15  q 4 q 3 Q 'K1,16 
(38)
p 4 Q 'K 2,9  q 4 Q 'K 2,10  p 3 Q 'K 2,11  p3p 4 Q 'K 2,12  p3q 4 Q 'K 2,13  q 3 Q 'K 2,14  p 4 q 3 Q 'K 2,15  q 4 q 3 Q 'K 2,16 
p 4 Q 'K 3,9  q 4 Q 'K 3,10  p 3 Q 'K 3,11  p3p 4 Q 'K 3,12  p3q 4 Q 'K 3,13  q 3 Q 'K 3,14  p 4 q 3 Q 'K 3,15  q 4 q 3 Q 'K 3,16


E(T 2)  2 P 'K21  P 'K2 2  P 'K2 3  pp 4 PK 9 2  q 4 PK10 2  p 3 PK11 2  p3p 4 PK12 2  p3q 4 PK13 2  q 3 PK14 2  p 4 q 3 PK15 2  q 4 q 3 PK16 2 
'2
'2
'2
'2
'2
'2
p p
p q
p 4 Q '2  q 4 Q '2
K1,9
K1,10  p 3 Q K1,11  3 4 Q K1,12  3 4 Q K1,13  q 3 Q K1,14  p 4 q 3 Q K1,15  q 4 q 3 Q K1,16 
(39)
'2
'2
'2
'2
'2
'2
'2
p p
p q
p 4 Q '2
K 2,9  q 4 Q K 2,10  p 3 Q K 2,11  3 4 Q K 2,12  3 4 Q K 2,13  q 3 Q K 2,14  p 4 q 3 Q K 2,15  q 4 q 3 Q K 2,16 
'2
'2
'2
'2
'2
'2
'2
p p
p q
p 4 Q '2
K 3,9  q 4 Q K 3,10  p 3 Q K 3,11  3 4 Q K 3,12  3 4 Q K 3,13  q 3 Q K 3,14  p 4 q 3 Q K 3,15  q 4 q 3 Q K 3,16

where for b =1, 2,3 and d=9,10,11,12,13,14,15,16
'
P Kb 
1
 (1  D
Kb
)
,P
Kd

'
1
and Q

Kb,d  (1 
 (1  B )
D Kb B Kd )
Kd
1
(40)
The variance of time to recruitment can be calculated from (35), (36), (38), (39).
Case (vi): The distributions of optional thresholds follow extended exponential distribution with shape parameter 2 and the
distribution of mandatory thresholds possess SCBZ property.
If 𝑔 𝑡 = 𝑔𝑥 1 𝑡 ,
E(T)  2C 'I1  2C 'I 2  2C 'I7  2C 'I8  C 'I9  C 'I10  C 'I11  4C 'I3  pp 4 C I9  q 4 C I10  p 3 C I11  p3p 4 C I12  p3q 4 C I13
 q 3 C I14  p4q 3 C I15  q3q 4 C I16  2p 4 H 'I1,9  2q 4 H 'I1,10  2p 3 H 'I1,11  2p3p 4 H 'I1,12  2p3q 4 H 'I1,13
 2q 3 H 'I1,14  2p4q 3 H 'I1,15  2q3q 4 H 'I1,16  2p 4 H 'I 2,9  2q 4 H 'I 2,10  2p 3 H 'I 2,11  2p3p 4 H 'I 2,12  2p3q 4 H 'I 2,13
 2q 3 H 'I 2,14  2p4q 3 H 'I 2,15  2q3q 4 H 'I 2,16  2p 4 H 'I7,9  2q 4 H 'I7,10  2p 3 H 'I7,11  2p3p 4 H 'I7,12  2p3q 4 H 'I7,13
 2q 3 H 'I7,14  2p4q 3 H 'I7,15  2q3q 4 H 'I7,16  2p 4 H 'I8,9  2q 4 H 'I8,10  2p 3 H 'I8,11  2p3p 4 H 'I8,12  2p3q 4 H 'I8,13
(41)
 2q 3 H 'I8,14  2p4q 3 H 'I8,15  2q3q 4 H 'I8,16  p 4 H 'I9,9  q 4 H 'I9,10  p 3 H 'I9,11  p3p 4 H 'I9,12  p3q 4 H 'I9,13  q 3 H 'I9,14
 p4q 3 H 'I9,15  q3q 4 H 'I9,16  p 4 H 'I10,9  q 4 H 'I10,10  p 3 H 'I10,11  p3p 4 H 'I10,12  p3q 4 H 'I10,13  q 3 H 'I10,14  p4q 3 H 'I10,15
 q3q 4 H 'I10,16  p 4 H 'I11,9  q 4 H 'I11,10  p 3 H 'I11,11  p3p 4 H 'I11,12  p3q 4 H 'I11,13  q 3 H 'I11,14  p4q 3 H 'I11,15  q3q 4 H 'I11,16  4p 4 H 'I13,9

 4q 4 H 'I13,10  4p 3 H 'I13,11  4p3p 4 H 'I13,12  4p3q 4 H 'I13,13  4q 3 H 'I13,14  4p4q 3 H 'I13,15  4q3q 4 H 'I13,16
2
E(T )  2 2C 'I21  2C 'I22  2C 'I27  2C 'I28  C 'I29  C 'I210  C 'I211  4C 'I23  pp 4 CI9 2  q 4 CI10 2  p 3 CI11 2  p3p 4 CI12 2  p3q 4 CI13 2
 q 3 CI14 2  p4q 3 CI15 2  q3q 4 CI16 2  2p 4 H 'I21,9  2q 4 H 'I21,10  2p 3 H 'I21,11  2p3p 4 H 'I21,12  2p3q 4 H 'I21,13

 2q 3 H 'I21,14  2p4q 3 H 'I21,15  2q3q 4 H 'I21,16  2p 4 H 'I22,9  2q 4 H 'I22,10  2p 3 H 'I22,11  2p3p 4 H 'I22,12  2p3q 4 H 'I22,13
 2q 3 H 'I22,14  2p4q 3 H 'I22,15  2q3q 4 H 'I22,16  2p 4 H 'I27,9  2q 4 H 'I27,10  2p 3 H 'I27,11  2p3p 4 H 'I27,12  2p3q 4 H 'I27,13
 2q 3 H 'I27,14  2p4q 3 H 'I27,15  2q3q 4 H 'I27,16  2p 4 H 'I28,9  2q 4 H 'I28,10  2p 3 H 'I28,11  2p3p 4 H 'I28,12  2p3q 4 H 'I28,13
(42)
 2q 3 H 'I28,14  2p4q 3 H 'I28,15  2q3q 4 H 'I28,16  p 4 H 'I29,9  q 4 H 'I29,10  p 3 H 'I29,11  p3p 4 H 'I29,12  p3q 4 H 'I29,13  q 3 H 'I29,14
 p4q 3 H 'I29,15  q3q 4 H 'I29,16  p 4 H 'I210,9  q 4 H 'I210,10  p 3 H 'I210,11  p3p 4 H 'I210,12  p3q 4 H 'I210,13  q 3 H 'I210,14
 p4q 3 H 'I210,15  q3q 4 H 'I210,16  p 4 H 'I211,9  q 4 H 'I211,10  p 3 H 'I211,11  p3p 4 H 'I211,12  p3q 4 H 'I211,13  q 3 H 'I211,14
 p4q 3 H 'I211,15  q3q 4 H 'I211,16  4p 4 H 'I213,9  4q 4 H 'I213,10  4p 3 H 'I213,11  4p3p 4 H 'I213,12  4p3q 4 H 'I213,13  4q 3 H 'I213,14  4p4q 3 H 'I213,15  4q3q 4 H 'I213,16

where for b =1,2,3,7,8,9,10,11 and d=9,10,11,12,13,14,15,16.
C 'Ib
ISSN: 2321 – 242X

1
1
1
, C Id 
and H 'Ib, d 
(1  D Ib)
(1  B Id )
(1  D Ib B Id )
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If 𝑔 𝑡 = 𝑔𝑥
𝑘 𝑡 ,
'
E(T)  2P K1  2P 'K 2  2P 'K 7  2P 'K8  P 'K 9  P 'K10  P 'K11  4P 'K 3  pp 4 P K 9  q 4 P K10  p 3 P K11  p3p 4 P K12  p3q 4 P K13
 q 3 P K14  p4q 3 P K15  q3q 4 P K16  2p 4 Q 'K1,9  2q 4 Q 'K1,10  2p 3 Q 'K1,11  2p3p 4 Q 'K1,12  2p3q 4 Q 'K1,13
 2q 3 Q 'K1,14  2p4q 3 Q 'K1,15  2q3q 4 Q 'K1,16  2p 4 Q 'K 2,9  2q 4 Q 'K 2,10  2p 3 Q 'K 2,11  2p3p 4 Q 'K 2,12  2p3q 4 Q 'K 2,13
 2q 3 Q 'K 2,14  2p4q 3 Q 'K 2,15  2q3q 4 Q 'K 2,16  2p 4 Q 'K 7,9  2q 4 Q 'K 7,10  2p 3 Q 'K 7,11  2p3p 4 Q 'K 7,12  2p3q 4 Q 'K 7,13
 2q 3 Q 'K 7,14  2p4q 3 Q 'K 7,15  2q3q 4 Q 'K 7,16  2p 4 Q 'K8,9  2q 4 Q 'K8,10  2p 3 Q 'K8,11  2p3p 4 Q 'K8,12  2p3q 4 Q 'K8,13
(44)
 2q 3 Q 'K8,14  2p4q 3 Q 'K8,15  2q3q 4 Q 'K8,16  p 4 Q 'K 9,9  q 4 Q 'K 9,10  p 3 Q 'K9,11  p3p 4 Q 'K 9,12  p3q 4 Q 'K 9,13  q 3 Q 'K 9,14
 p4q 3 Q 'K9,15  q3q 4 Q 'K 9,16  p 4 Q 'K10,9  q 4 Q 'K10,10  p 3 Q 'K10,11  p3p 4 Q 'K10,12  p3q 4 Q 'K10,13  q 3 Q 'K10,14  p4q 3 Q 'K10,15
 q3q 4 Q 'K10,16  p 4 Q 'K11,9  q 4 Q 'K11,10  p 3 Q 'K11,11  p3p 4 Q 'K11,12  p3q 4 Q 'K11,13  q 3 Q 'K11,14  p4q 3 Q 'K11,15  q3q 4 Q 'K11,16  4p 4 Q 'K13,9
 4q 4 Q 'K3,10  4p 3 Q 'K 3,11  4p3p 4 Q 'K 3,12  4p3q 4 Q 'K 3,13  4q 3 Q 'K 3,14  4p4q 3 Q 'K 3,15  4q3q 4 Q 'K 3,16


E(T 2)  2 2P 'K21  2P 'K2 2  2P 'K2 7  2P 'K2 8  P 'K2 9  P 'K210  P 'K211  4P 'K2 3  pp 4 PK 9 2  q 4 PK10 2  p 3 PK11 2  p3p 4 PK12 2  p3q 4 PK13 2
 q 3 PK14 2  p4q 3 PK15 2  q3q 4 PK16 2  2p 4 Q 'K21,9  2q 4 Q 'K21,10  2p 3 Q 'K21,11  2p3p 4 Q 'K21,12  2p3q 4 Q 'K21,13
 2q 3 Q 'K21,14  2p4q 3 Q 'K21,15  2q3q 4 Q 'K21,16  2p 4 Q 'K2 2,9  2q 4 Q 'K2 2,10  2p 3 Q 'K2 2,11  2p3p 4 Q 'K2 2,12  2p3q 4 Q 'K2 2,13
 2q 3 Q 'K2 2,14  2p4q 3 Q 'K2 2,15  2q3q 4 Q 'K2 2,16  2p 4 Q 'K2 7,9  2q 4 Q 'K2 7,10  2p 3 Q 'K2 7,11  2p3p 4 Q 'K2 7,12  2p3q 4 Q 'K2 7,13
 2q 3 Q 'K2 7,14  2p4q 3 Q 'K2 7,15  2q3q 4 Q 'K2 7,16  2p 4 Q 'K2 8,9  2q 4 Q 'K2 8,10  2p 3 Q 'K2 8,11  2p3p 4 Q 'K2 8,12  2p3q 4 Q 'K2 8,13
(45)
 2q 3 Q 'K2 8,14  2p4q 3 Q 'K2 8,15  2q3q 4 Q 'K2 8,16  p 4 Q 'K2 9,9  q 4 Q 'K2 9,10  p 3 Q 'K2 9,11  p3p 4 Q 'K2 9,12  p3q 4 Q 'K2 9,13  q 3 Q 'K2 9,14
 p4q 3 Q 'K2 9,15  q3q 4 Q 'K2 9,16  p 4 Q 'K210,9  q 4 Q 'K210,10  p 3 Q 'K210,11  p3p 4 Q 'K210,12  p3q 4 Q 'K210,13  q 3 Q 'K210,14
 p4q 3 Q 'K210,15  q3q 4 Q 'K210,16 p 4 Q 'K211,9  q 4 Q 'K211,10  p 3 Q 'K211,11  p3p 4 Q 'K211,12  p3q 4 Q 'K211,13  q 3 Q 'K211,14
 p4q 3 Q 'K211,15  q3q 4 Q 'K211,16 4p 4 Q 'K213,9  4q 4 Q 'K213,10  4p 3 Q 'K213,11  4p3p 4 Q 'K213,12  4p3q 4 Q 'K213,13  4q 3 Q 'K213,14  4p4q 3 Q 'K213,15  4q3q 4 Q 'K213,16

where for b =1, 2,3,7,8,9,10,11 and d=9,10,11,12,13,14,15,16.
'
P Kb 
1
 (1  D
Kb
)
,P
Kd

1
 (1  B
'
1
and Q

Kb,d  (1 
D Kb B Kd )
Kd
)
(46)
The variance of time to recruitment can be calculated from (41), (42), (44) and (45).
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 = 𝑔𝑥 1 𝑡 ,
E (T )  C
 p (C

)
I3
I 6 H I 3,6

2
2
2
E (T )  2 C
 p (C

I3
I 6 H I 3,6 \
2

(47)
(48)
Where 𝐶𝐼𝑎 , 𝐻𝐼𝑏,𝑑 are given by equation(5), (11)and (12) for a=3,6, b=3and d=6.
If 𝑔 𝑡 = 𝑔𝑥 𝑘 𝑡 ,
E (T )  P K 3  p ( P K 6  Q
)
K 3,6
(49)
2
2
2
2

E (T )  2
 P K 3  p ( P K 6  Q K 3,6 \ 


(50)
where 𝑃𝐾𝑎 , 𝑄𝐾𝑏,𝑑 are given by (7), (15) and (16) for a=3,6, b=3 and d=6.
The variance of time to recruitment can be calculated from (47), (48), (49), (50).
Case (ii): The distribution of optional and mandatory thresholds follow extended exponential distribution
For this case the first two moments of time to recruitment are found to be
If 𝑔 𝑡 = 𝑔𝑥 1 𝑡 ,
E(T)  4C I3  2C I7  2C I8  C I11  p4C I6  2C I12  2C I13  C I16  16H I3,6  8H I3,12
 8H I3,13  4H I3,16  8H I7,6  4H I7,12  4H I7,13  2H I7,6  8H I8,6  4H I8,12  4H I8,13
 2H I8,16  4H I11,6  2H I11,12  2H I11,13  H I11,16 
ISSN: 2321 – 242X
© 2013 | Published by The Standard International Journals (The SIJ)
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

2
2
2
2
2
2
2
2
2
E(T 2)  2 4C 2
I3  2C I7  2C I8  C I11  p 4C I6  2C I12  2C I13  C1I6  16H I3,6  8H I3,12
2
2
2
2
2
2
2
 8H 2
I3,13  4H I3,16  8H I7,6 4H I7,12  4 H I7,13  2 H I7,16  8H I8,6  4H I8,12
2
2
2
2
2
 4H 2
I8,13  2 H I8,16  4 H I11,6  2H I11,12  2H I11,13  H I11,16
(52)

where 𝐶𝐼𝑎 , 𝐻𝐼𝑏,𝑑 are given by equation (5), (11), (12) and (19) for a=3, 6,7,8,11,12, 13,16, b=3,7,8,11, and d=6,12,13,16
If 𝑔 𝑡 = 𝑔𝑥 𝑘 𝑡 ,
E(T)  4P K3  2P K 7  2P K8  P K11  p4P K 6  2P K12  2P K13  P K16  16QK3,6  8QK 3,12
 8QK3,13  4QK3,16  8QK 7,6  4QK 7,12  4QK 7,13  2QK 7,6  8QK8,6  4QK8,12  4QK8,13
 2QK8,16  4QK11,6  2QK11,12  2QK11,13  QK11,16 

(53)

E(T 2)  2 4P 2K 3  2P 2K 7  2P 2K8  P 2K11  p 4P 2K 6  2P 2K12  2P 2K13  P 2K16  16Q2K 3,6  8Q2K 3,12
 8Q2K 3,13  4Q2K 3,16  8Q2K 7,6 4Q2K 7,12  4Q2K 7,13  2Q2K 7,16  8Q2K8,6  4Q2K8,12
 4Q2K8,13  2Q2K8,16  4Q2K11,6  2Q2K11,12  2Q2K11,13  Q2K11,16
(54)

where 𝑃𝐾𝑎 , 𝑄𝐾𝑏,𝑑 are given by (7), (15), (16) and (22) for a=3, 6,7,8,11,12, 13,16, b=3,7,8,11, and d=6,12,13,16.
The variance of time to recruitment can be calculated from (51), (52), (53),(54).
Case (iii): The distributions of optional thresholds follow exponential distribution and mandatory thresholds follow
extended exponential distribution with shape parameter 2.
For this case the first two moments of time to recruitment are found to be
If 𝑔 𝑡 = 𝑔𝑥 1 𝑡 ,
E(T)  C I3  p4C I6  2C I12  2C I3  C I16  4H I3,6  2H I3,12  2H I3,13  H I3,6
E(T 2)
 2
C2
I3 
p
2
2
2
2
2
2
2
4C 2
I6  2C I12  2C I13  C I16  4H I3,6  2H I3,12  2H I3,13  H I3,16

(55)
(56)
where 𝐶𝐼𝑎 , 𝐻𝐼𝑏,𝑑 are given by (5), (11), (12) and (19) for a=3, 6,12, 13,16, b=3 and d=6,12,13,16.
If 𝑔 𝑡 = 𝑔𝑥 𝑘 𝑡 ,
E(T)  P K3  p4P K6  2Q K12  2Q K13  Q K16  4Q K3,6  2Q K3,12  2Q K3,13  Q K3,6


E(T 2)  2 P 2K3  p 4P 2K6  2P 2K12  2P 2K13  P 2K16  4Q 2K3,6  2Q 2K3,12  2Q 2K3,13  Q 2K3,16

(57)
(58)
where 𝑃𝐾𝑎 , 𝑄𝐾𝑏,𝑑 are given by (7), (15), (16) and (22) for a=3, 6,12, 13,16, b=3 and d=6,12,13,16.
The variance of time to recruitment can be calculated from (55), (56), (57), (58).
Case (iv): The distributions of optional and mandatory thresholds possess SCBZ property.
If 𝑔 𝑡 = 𝑔𝑥 1 𝑡 ,
E(T)  p1p 2 CI 4  p1q 2 CI5  p 2q1 CI7  q1q 2 CI8  p(p3p 4 CI12  p3q 4 CI13  p 4q3 CI15  q3q 4 CI16
 p1p 2p3p 4 H I 4,12  p1 p 2 p3q 4 H I 4,13  p1 p 2 q3p 4 H I 4,15  p1 p 2 q3q 4 H I 4,16  p1q 2p3p 4 H I5,12
 p1 q 2 p3q 4 H I5,13  p1 q 2 p 4q3 H I5,15  p1 q 2 q3q 4 H I5,16  q1p 2p3p 4 H I7,12  q1 p 2 p3q 4 H I7,13
(59)
 q1 p 2 p 4q3 H I7,15  q1 p 2 q3q 4 H I7,16  q1q 2p3p 4 H I8,12  q1 q 2 p3q 4 H I8,13  q1 q 2 p 4q3 H I8,15  q1 q 2 q3q 4 H I8,16 

E ( T 2)  2

p1p 2 C2  p1q 2 C2  p 2q1 C2  q1q 2 C 2  p p3p 4 C 2  p3q 4 C 2  p 4q3 C2  q 3q 4 C2
I5
I7
I4
I8
I12
I13
I15
I16
2
2
2
2
2
p
p q p p
p p p q
p p p q
p p q q
p q p q
 1p 2 p3p 4 H 2
I 4,12  1 2 3 4 H I 4,13  1 2 4 3 H I 4,15  1 2 3 4 H I 4,16  1 2 3 4 H I5,12  1 2 3 4 H I5,13
2
2
2
2
2
q p p p
p q q q
q p p q
q p p q
q p q q
 p1 q 2 p 4q3 H 2
I5,15  1 2 3 4 H I5,16  1 2 3 4 H I7,12  1 2 3 4 H I7,13  1 2 4 3 H I7,15  1 2 3 4 H I7,16
2
2
2
2
q q p q
q q q
q q p q
q q q q
 q1q 2 p3p 4 H 2
I8,12  1 2 3 4 H I8,13  1 2 3 H I8,14  1 2 4 3 H I8,15  1 2 3 4 H I8,16
(60)

where 𝐶𝐼𝑎 , 𝐻𝐼𝑏,𝑑 are given by (5), (29) and (30) for a=4,5,7,8, 12, 13,15,16, b=4,5,7,8 and d=12, 13,15,16.
If 𝑔 𝑡 = 𝑔𝑥 𝑘 𝑡 ,
E(T)  p1p2 P K 4  p1q 2 P K5  p2q1 P K 7  q1q 2 P K8  p(p3p4 P K12  p3q 4 P K13  p 4q3 P K15  q3q 4 P K16
 p1p 2p3p4 QK 4,12  p1 p2 p3q 4 QK 4,13  p1 p2 q3p4 QK 4,15  p1 p2 q3q 4 QK 4,16  p1q 2p3p4 QK5,12
 p1 q 2 p3q 4 QK5,13  p1 q 2 p 4q3 QK5,15  p1 q 2 q3q 4 QK5,16  q1p 2p3p4 QK 7,12  q1 p2 p3q 4 QK 7,13
(61)
 q1 p2 p 4q3 QK 7,15  q1 p2 q3q 4 QK 7,16  q1q 2p3p4 QK8,12  q1 q 2 p3q 4 QK8,13  q1 q 2 p4q3 QK8,15  q1 q 2 q3q 4 QK8,16 
E(T 2)  2p1p 2 P 2K 4  p1q 2 P 2K5  p 2q1 P 2K 7  q1q 2 P 2K8  pp3p 4 P 2K12  p3q 4 P 2K13  p 4q3 P 2K15  q 3q 4 P 2K16
 p1p 2 p3p 4 Q 2K 4,12  p1 p 2 p3q 4 Q 2K 4,13  p1 p 2 p 4q3 Q 2K 4,15  p1 p 2 q 3q 4 Q 2K 4,16  p1q 2 p3p 4 Q 2K5,12  p1 q 2 p3q 4 Q 2K5,13
 p1 q 2 p 4q3 Q 2K5,15  p1 q 2 q 3q 4 Q 2K5,16  q1p 2 p3p 4 Q 2K 7,12  q1 p 2 p3q 4 Q 2K 7,13  q1 p 2 p 4q3 Q 2K 7,15  q1 p 2 q 3q 4 Q 2K 7,16
 q1q 2 p3p 4 Q 2K8,12  q1 q 2 p3q 4 Q 2K8,13  q1 q 2 q3 Q 2K8,14  q1 q 2 p 4q3 Q 2K8,15  q1 q 2 q 3q 4 Q 2K8,16
(62)

where 𝑃𝐾𝑎 , 𝑄𝐾𝑏,𝑑 are given by (7), (33) and (34) for a=4,5,7,8, 12, 13,15,16, b=4,5,7,8 and d=12, 13,15,16.
The variance of time to recruitment can be calculated from (59), (60), (61), (62).
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Case (v): If the distributions of optional thresholds follow exponential distribution and the distribution of mandatory
thresholds possess SCBZ property
If 𝑔 𝑡 = 𝑔𝑥 1 𝑡 ,
E(T)  C 'I3  pp 3 p 4 C I12  p 3 q 4 C I13  p 4 q 3 C I15
 q 3 q 4 C I16  p 3 p 4 H 'I3,12  p 3 q 4 H 'I3,13  p 4 q 3 H 'I3,15  q 3 q 4 H 'I3,16 
(63)
C 'I23  pp 3 p 4 CI12 2  p 3 q 4 CI13 2  p 4 q 3 CI15 2  q 3 q 4 CI16 2  p 3 p 4 H 'I23,12  p 3 q 4 H 'I23,13
E(T 2)  2
 p 4 q 3 H 'I23,15  q 3 q 4 H 'I23,16

(64)
′
′
Where 𝐶𝐼𝑑 , 𝐶𝐼𝑏
, 𝐻𝐼𝑏,𝑑
are given by equation (5), (11), (12), (29) and (37), b=3 and d=12,13,15,16.
If 𝑔 𝑡 = 𝑔𝑥 𝑘 𝑡 ,
E(T)  P 'K3  pp 3 p 4 P K12  p 3 q 4 P K13  p 4 q 3 P K15

 q 3 q 4 P K16  p 3 p 4 Q 'K3,12  p 3 q 4 Q 'K3,13  p 4 q 3 Q 'K3,15  q 3 q 4 Q 'K3,16

2
2
2
'2
'2
2  pp p P K12 2  p q Q
E(T 2)  2 P 'K
3 4
3 4 K13  p 4 q 3 QK15  q 3 q 4 QK16  p 3 p 4 Q K 3,12  p 3 q 4 Q K 3,13
3
2
'2
q q
 p 4 q 3 Q 'K
3,15  3 4 Q K 3,16

(65)
(66)
′
′
Where 𝑃𝐾𝑑 , 𝑃𝐾𝑏
, 𝑄𝐾𝑏,𝑑
are given by (7),(15),(16),(33), (34) and (40) for b=3 and d=12,13,15,16.
The variance of time to recruitment can be calculated from (63), (64), (65), and (66).
Case (vi): If the distributions of optional thresholds follow extended exponential distribution and the distribution of
mandatory thresholds possess SCBZ property.
If 𝑔 𝑡 = 𝑔𝑥 1 𝑡 ,
E(T)  4C'I32C'I72C'I8C'I11 pp 3 p 4 C I12  4 p 3 p 4 4H 'I3,12  2 p 3 p 4 H 'I7,12  2 p 3 p 4 H 'I8,12  p 3 p 4 H 'I11,12
q3p 4 C I15  4 q3p 4 H 'I3,15  2 q3p 4 H 'I7,15  2 q3p 4 H 'I8,15  q3p 4 H 'I11,15  q 4p 3 C I13  4 q 4p 3 H 'I3,13  2 q 4p 3 H 'I7,13
 2 q 4p 3 H 'I8,13  q 4p 3 H 'I11,13  q3q 4 C I16  4 q3q 4 H 'I3,16  2 q3q 4 H 'I7,16  2 q3q 4 H 'I8,16  q3q 4 H'I11,16 '


(67)
E(T 2)  2 4C'I232C'I272C'I28C'I211 pp 3 p 4 CI12 2  4 p 3 p 4 4H 'I23,12  2 p 3 p 4 H 'I27,12  2 p 3 p 4 H 'I28,12  p 3 p 4 H 'I211,12
2
'2
'2
'2
'2
'2
q3p 4 CI15 2  4 q3p 4 H '2
q p
q p
q p
q p
q p
q p
I3,15  2 3 4 H I7,15  2 3 4 H I8,15  3 4 H I11,15  4 3 CI13  4 4 3 H I3,13  2 4 3 H I7,13

(68)
 2 q 4p 3 H 'I28,13  q 4p 3 H 'I211,13  q3q 4 CI16 2  4 q3q 4 H 'I23,16  2 q3q 4 H 'I217,16  2 q3q 4 H 'I218,16  q3q 4 H 'I211,16
′
′
𝐶𝐼𝑑 , 𝐶𝐼𝑏
, 𝐻𝐼𝑏,𝑑
are given by equation (5),(11),(12),(19),(29) and (30), for b=3,7,8,11 and d=12,13,15,16.
where
If 𝑔 𝑡 = 𝑔𝑥
𝑡 ,
𝑘
E(T)  4P'K 32P'K 72P'K8P'K11pp 3 p 4 P K12  4 p 3 p 4 4Q 'K3,12  2 p 3 p 4 Q 'K 7,12  2 p 3 p 4 Q 'K8,12  p 3 p 4 Q 'K11,12
q3p 4 P K15  4 q3p 4 Q 'K 3,15  2 q3p 4 Q 'K 7,15  2 q3p 4 Q 'K8,15  q3p 4 Q 'K11,15  q 4p 3 P K13  4 q 4p 3 Q 'K 3,13  2 q 4p 3 Q 'K 7,13
(69)
'
 2 q4p 3 Q 'K8,13  q4p 3 Q 'K11,13  q3q 4 P K16  4 q3q 4 Q 'K3,16  2 q3q 4 Q 'K 7,16  2 q3q 4 Q 'K8,16  q3q 4 Q'K11,16 


E(T 2)  2 4P'K2 32P'K2 72P'K2 8P'K211pp 3 p 4 PK12 2  4 p 3 p 4 4Q 'K2 3,12  2 p 3 p 4 Q 'K2 7,12  2 p 3 p 4 Q 'K2 8,12  p 3 p 4 Q 'K211,12
2
q3p 4 P K15 2  4 q3p 4 Q '2
q p '2
q p '2
q p '2
q p
q p '2
q p '2
K 3,15  2 3 4 Q K 7,15  2 3 4 Q K8,15  3 4 Q K11,15  4 3 P K13  4 4 3 Q K 3,13  2 4 3 Q K 7,13

(70)
 2 q4p 3 Q 'K2 8,13  q 4p 3 Q 'K211,13  q3q 4 PK16 2  4 q3q 4 Q 'K2 3,16  2 q3q 4 Q 'K2 7,16  2 q3q 4 Q 'K2 8,16  q3q 4 Q 'K211,16
′
′
𝑃𝐾𝑑 , 𝑃𝐾𝑏
, 𝑄𝐾𝑏,𝑑
are given by (7),(15),(16),(22),(34) and(40) for b=3,7,8,11 d=12,13,15,16
where
The variance of time to recruitment can be calculated from (67), (68), (69), (70).
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 = 𝑔𝑥 1 𝑡 ,

E (T )  A C
 A C
 p A C
 A C
 A A H
2 I2
1 I1
5 I5
4 I4
1 4
I 1, 4
 A A H
 A A H
 A A H
2 4
I 1, 4
1 5
I 1,5
2 5
I 2, 5


2
E(T 2)  2A 2 C I 2 2  A1 C I12  p A 5 C 2
I5  A 4 C I 4  A1 A 4 H I1, 4
2
2
 A 2 A 4 H I1, 4  A1 A 5 H I1,5  A 2 A 5 H I 2,5
2
(71)
2

(72)
Where 𝐶𝐼𝑎 , 𝐻𝐼𝑏,𝑑 are given by equation (5), (11) and (12) for a=1, 2, 4, 5, b=1,2 and d=4, 5
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A1 
If 𝑔 𝑡 = 𝑔𝑥
𝑘
2
𝑡 ,
E (T ) 
A 2 P K2

,A 
2
1   2
1
1   2
,A 
4
2
1   2
1
,A 
5
(73)
1   2
 A1 P K1  pA 5 P K 5  A 4 P K 4  A1 A 4 Q K1, 4
A 2 A 4 Q K1, 4
 A1 A 5 Q K1,5  A 2 A 5 Q K 2,5
2
2
Q
E(T 2)  2A 2 PK 2 2  A1 PK12  pA 5 P 2
K 5  A 4 PK 4  A1 A 4 K1, 4
 A 2 A 4 QK1, 4 2  A1 A 5 QK1,5 2  A 2 A 5 QK 2,5 2

(74)
(75)
where 𝑃𝐾𝑎 , 𝑄𝐾𝑏,𝑑 are given by (7), (15) and (16) for a=1,2,4,5, b=1,2 and d=4,5
The variance of time to recruitment can be calculated from (71), (72), (74), (75).
Case (ii): If the distributions of optional and mandatory thresholds follow extended exponential distribution with shape
parameter 2.
If 𝑔 𝑡 = 𝑔𝑥 𝑙 𝑡 ,
E (T )  S1 C I1  S 2 C I9  S 3 C I 2  S 4 C I10  pS 5 C I 4  S 6 C I14  S 7 C I5  S8 C I5
 S1 S 5 H I1, 4  S1 S6H I1,14  S1 S7 H I1,5  S1 S8 H I1,15
 S 2 S 5 H I9, 4  S 2 S6H I9,14  S 2 S7 H I9,5  S 2 S8 H I9,15
(76)
 S 3 S 5 H I 2, 4  S 3 S6H I 2,14  S 3 S7 H I 2,5  S 3 S8 H I 2,15
 S 4 S 5 H I10, 4  S 4 S6H I10,14  S 4 S7 H I10,5  S 4 S8 H I10,15 

E (T 2)  2S1 CI12  S 2 CI9 2  S 3 CI 2 2  S 4 CI10 2  p S 5 CI 4 2  S 6 CI14 2  S 7 CI5 2  S8 CI5 2
 S1 S 5 H I1, 4 
2
2
S1 S6H I1,14
 S1 S7 H I1,5 2
 S1 S8H I1,15 2
 S 2 S5 H I9, 4 2  S 2 S6H I9,14 2 S 2 S7 H I9,5 2  S 2 S8H I9,15 2
(77)
 S3 S5 H I 2, 4 2  S 3 S6H I 2,14 2  S 3 S7 H I 2,5 2  S 3 S8H I 2,15 2
 S 4 S5 H I10 , 4 2  S 4 S6H I10 ,14 2  S 4 S7 H I10 ,5 2  S 4 S8H I10 ,15 2

where 𝐶𝐼𝑎 , 𝐻𝐼𝑏,𝑑 are given by equation (5), (11), (12) and (19) for a=1, 2,4,5, 9,10,14,15 b=1,2,9,10 and d=4,5,14,15

S1
If 𝑔 𝑡 = 𝑔𝑥
𝑘
S4

S7

4 2
2
1   2 1  2 2 
2
1
, S2 
1   2 1  2 2 
4 12
, S5 
1   2 21   2 
2
2
1   2 21   2 
4 2
2
, S3 
1   2 1  2 2 
, S8 
4 12
1   2 21   2 
, S6 
2
2
1   2 21   2 
(78)
2
1
1   2 1  2 2 
𝑡 ,
E(T)  S1 P K1  S 2 P K 9  S3 P K 2  S 4 P K10  pS5 P K 4  S 6 P K14  S 7 P K 5  S8 P K 5
 S1 S5 Q K1,4  S1 S6Q K1,14  S1 S7Q K1,5  S1 S8 Q K1,15  S 2 S5 Q K 9,4  S 2 S6Q K 9,14
 S 2 S7Q K 9,5  S 2 S8 Q K 9,15  S3 S5 Q K 2,4  S3 S6Q K 2,14  S3 S7Q K 2,5  S3 S8 Q K 2,15
(79)
 S 4 S5 Q K10,4  S 4 S6Q K10,14  S 4 S7Q K10,5  S 4 S8 Q K10,15

E(T 2)  2S1 PK12  S 2 PK 9 2  S3 PK 2 2  S 4 PK10 2  p S5 PK 4 2  S 6 PK14 2  S 7 PK 5 2  S8 PK15 2
 S1S5 QK1, 4 2  S1 S6QK1,14 2  S1 S7QK1,5 2  S1S8QK1,15 2  S 2 S5 QK 9, 4 2  S 2 S6QK 9,14 2
S 2 S7QK 9,5
2
 S 2 S8QK 9,15 2  S3 S5 QK 2, 4 2  S3 S6QK 2,14 2  S3 S7QK 2,5 2  S3 S8QK 2,15 2
 S 4 S5 QK10, 4 2  S 4 S6QK10,14 2  S 4 S7QK10,5 2  S 4 S8QK10,15 2
(80)

where 𝑃𝐾𝑎 , 𝑄𝐾𝑏,𝑑 are given by (7), (15) and (16) for a=1, 2,4,5, 9,10,14,15 b=1,2,9,10 and d=4,5,14,15.
The variance of time to recruitment can be calculated from (76), (77), (79),(80).
Case (iii): If the distributions of optional thresholds follow exponential distribution and mandatory thresholds follow
extended exponential distribution.
If 𝑔 𝑡 = 𝑔𝑥 1 𝑡 ,
E (T )  A 2 C I 2  A1 C I1  pS5 C I 4  S6 C I14  S7 C I5  S8 C I15  A2S5 H I 2, 4
 A2S6 H I 2,14  A2S7 H I 2,5  A2S8 H I 2,15  A1S5 H I1, 4  A1S6 H I1,14
 A1S7 H I1,5  A1S8 H I1,15
ISSN: 2321 – 242X
© 2013 | Published by The Standard International Journals (The SIJ)
(81)
169
The SIJ Transactions on Industrial, Financial & Business Management (IFBM), Vol. 1, No. 5, November-December 2013

E(T 2)  2 A2 CI 22  A1 CI12  pS5 CI 42  S6 CI142  S7 CI52  S8 CI152  A 2 S5 H I 2, 4
2
2
2
2
2
2
 A 2 S6 H I 2,14  A 2 S7 H I 2,5  A 2 S8 H I 2,15  A1S5 H I1, 4  A1S6 H I1,14
2
2
 A1S7 H I1,5  A1S8 H I1,15
(82)

where 𝐶𝐼𝑎 , 𝐻𝐼𝑏,𝑑 are given by equation (5),(11), (12) and (19) for a=1, 2,4,14,15,5, b=1,2 and d=4,5,14,15
If 𝑔 𝑡 = 𝑔𝑥 𝑘 𝑡 ,
E (T )  A 2 P K 2  A1 P K1  pS5 P K 4  S6 P K14  S7 P K 5  S8 P K15  A2S5 Q K 2, 4
 A2S6 Q K 2,14  A2S7 Q K 2,5  A2S8 Q K 2,15  A1S5 Q K1, 4  A1S6 Q K1,14
 A1S7 Q K1,5  A1S8 Q K1,15


E(T 2)  2 A 2 P K 22  A1 P K12  pS5 P K 42  S6 P K142  S7 P K 52  S8 P K152  A 2 S5 QK 2, 4
2
2
2
2
2
2
 A 2 S6 QK 2,14  A 2 S7 QK 2,5  A 2 S8 QK 2,15  A1S5 QK1, 4  A1S6 QK1,14
2
(83)
(84)
2
 A1S7 QK1,5  A1S8 QK1,15  

where 𝑃𝐾𝑎 , 𝑄𝐾𝑏,𝑑 are given by (7),(15), (16) and (22) for a=1,2,4,5, b=1,2and d=4,5
The variance of time to recruitment can be calculated from (81), (82), (83),(84).
Case (iv): The distributions of optional and the mandatory thresholds possess SCBZ property.
If 𝑔 𝑡 = 𝑔𝑥 1 𝑡 ,
E(T)  R 1  R 2  C I1  R 3  R 4  C I3  R 5  R 6  C I6  R 7  R 8 C I 2  pR 9  R 10  C I9  R 13  R 14  C I10
 R 11  R 12  C I11  R 15  R 16  C I14  R 1  R 2 R 9  R 10  H I1,9  R 13  R 14  H I1,10
 R 11  R 12  H I1,11  R 15  R 16 H I1,14   R 7  R 8R 9  R 10  H I 2,9  R 13  R 14  H I 2,10
 R 11  R 12  H I 2,11  R 15  R 16 H I 2,14   R 3  R 4 R 9  R 10  H I3,9  R 13  R 14  H I3,10
(85)
 R 11  R 12  H I3,11  R 15  R 16 H I3,14   R 5  R 6 R 9  R 10  H I6,9  R 13  R 14  H I6,10
 R 11  R 12  H I6,11  R 15  R 16 H I6,14 
E ( T 2)
 2R 1  R 2  CI12  R 3  R 4  CI3 2  R 5  R 6  CI 6 2  R 7  R 8 CI 2 2  pR 9  R 10  CI9 2  R 13  R 14  CI10 2
 R 11  R 12  CI11 2  R 15  R 16  CI14 2  R 1  R 2 R 9  R 10  H I1,9 2  R 13  R 14  H I1,10 2


2 
R 5  R 6 R 9  R 10  H
2 
 R 11  R 12  H I1,11 2  R 15  R 16 H I1,14 2  R 7  R 8R 9  R 10  H I 2,9 2  R 13  R 14  H I 2,10 2
 R 11  R 12  H I 2,11 2  R 15  R 16 H I 2,14 2  R 3  R 4 R 9  R 10  H I3,9 2  R 13  R 14  H I3,10 2
 R 11  R 12  H I3,11 2  R 15  R 16 H I3,14
 R 11  R 12  H I 6,11 2  R 15  R 16 H I 6,14
I 6,9
(86)
2
2
R 13  R 14  H I 6,10
where 𝐶𝐼𝑎 , 𝐻𝐼𝑏,𝑑 are given by equation (5), (11), (29) and (30) for a=1,3,6,9,10,11,12,14, b=1,2,3,6 and d=9,10,11,14
1  1p1 p2 ,  1q1 p2 ,  2  2p1 p2 ,  2 p1q 2
  2q1 p2 ,
 2
1  2  1  2 R 2 1  2  2 R 3 1  2  1  2 R 4 1  2  1 R 5 1  2  2
3  3p3 p4 ,  3 q3 p4 ,
   p q
2 q1 q 2
q q
,
 1 1 1 2,
 1 1 2

R6 
1  2 R 7 1  1  2 R 8 1  2 R 9 3  4  3  4 R10 3  4  4
4  4p3 p4
  3q 4 p3 ,  3 q3 q 4 ,  4  4p4 q3 ,  4 q3 q 4
4 p3 q 4
,

 3
R11 
3  4  3  4 R12 3  4  3 R13 3  4  3 R14 3  4 R15 3  4  4 R16 3  4
R1 
𝑘 𝑡 ,
R 1  R 2  P K1  R 3  R 4  P K 3  R 5  R 6  P K 6  R 7  R 8 P K 2  pR 9  R 10  P K9  R 13  R 14  P K10
 R 11  R 12  P K11  R 15  R 16  P K14  R 1  R 2 R 9  R 10  Q K1,9  R 13  R 14  Q K1,10
 R 11  R 12  Q K1,11  R 15  R 16 Q K1,14   R 7  R 8R 9  R 10  Q K 2,9  R 13  R 14  Q K 2,10
 R 11  R 12  Q K 2,11  R 15  R 16 Q K 2,14   R 3  R 4 R 9  R 10  Q K 3,9  R 13  R 14  Q K 3,10
 R 11  R 12  Q K 3,11  R 15  R 16 Q K 3,14   R 5  R 6 R 9  R 10  Q K 6,9  R 13  R 14  Q K 6,10
 R 11  R 12  Q K 6,11  R 15  R 16 Q K 6,14 
2
E(T )  2R 1  R 2  P K12  R 3  R 4  P K 3 2  R 5  R 6  P K 6 2  R 7  R 8 P K 2 2  pR 9  R 10  P K 9 2  R 13  R 14  P K10 2
 R 11  R 12  P K11 2  R 15  R 16  P K14 2  R 1  R 2 R 9  R 10  QK1,9 2  R 13  R 14  QK1,10 2
 R 11  R 12  QK1,11 2  R 15  R 16 QK1,14 2   R 7  R 8R 9  R 10  QK 2,9 2  R 13  R 14  QK 2,10 2
 R 11  R 12  QK 2,11 2  R 15  R 16 QK 2,14 2   R 3  R 4 R 9  R 10  QK 3,9 2  R 13  R 14  QK 3,10 2
 R 11  R 12  QK 3,11 2  R 15  R 16 QK 3,14 2   R 5  R 6 R 9  R 10  QK 6,9 2  R 13  R 14  QK 6,10 2
 R 11  R 12  QK 6,11 2  R 15  R 16 QK 6,14 2 
where 𝑃𝐾𝑎 , 𝑄𝐾𝑏,𝑑 are given by (7), (15), (33)and (34) for a=1,3,6,9,10,11,12,14, b=1,2,3,6 and d=9,10,11,14
The variance of time to recruitment can be calculated from (85), (86), (88), (89).
(87)
If 𝑔 𝑡 = 𝑔𝑥
E (T ) 
ISSN: 2321 – 242X
© 2013 | Published by The Standard International Journals (The SIJ)
(88)
(89)
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The SIJ Transactions on Industrial, Financial & Business Management (IFBM), Vol. 1, No. 5, November-December 2013
Case (v): The distributions of optional thresholds follow exponential distribution and the distribution of mandatory
thresholds possess SCBZ property.
If 𝑔 𝑡 = 𝑔𝑥 1 𝑡 ,
E(T)  A 2 C 'I2  A1 C 'I1  pR 9  R10  C I9  R13  R14  C I10  R11  R12  C I11  R15  R16  C I14  A 2 R 9  R10  H 'I2,9 








'
'
'
A1 R 13  R 14  H I1,10  A1 R 11  R 12  H I1,11  A1 R 15  R 16  H I1,14
2
E(T )  2A 2 C 'I22  A1 C 'I21  pR 9  R10  CI9 2  R13  R14  CI10 2  R11  R12  CI11 2  R15  R16  CI14 2  A 2 R 9  R10  H 'I22,9
A 2 R 13  R 14 H 'I 2,10  A 2 R 11  R 12 H 'I 2,11  A 2 R 15  R 16 H 'I 2,14  A1 R 9  R 10 H 'I1,9 
(90)
 A 2 R13  R14  H 'I22,10  A 2 R11  R12  H 'I22,11  A 2 R15  R16  H 'I22,14  A1 R 9  R10  H 'I21,9 
(91)

A1 R 13  R 14  H 'I21,10  A1 R 11  R 12  H 'I21,11  A1 R 15  R 16  H 'I21,14
′
𝐶𝐼𝑑 , 𝐶𝐼𝑏
, 𝐻𝐼𝑏,𝑑 are given by equation (5), (11), (12), (29) and
where
If 𝑔 𝑡 = 𝑔𝑥
𝑘
(30), for b=1,2 and d=9,10,11,14.
𝑡 ,
E(T)  A 2 P 'K 2  A1 P 'K1  pR 9  R10  P K9  R13  R14  P K10  R11  R12  P K11  R 15  R16  P K14  A 2 R 9  R10  Q 'K 2,9 








A1 R 13  R 14  Q 'K1,10  A1 R 11  R 12  Q 'K1,11  A1 R 15  R 16  Q 'K1,14
E(T 2)  2A 2 P 'K2 2  A1 P 'K21  pR 9  R10  PK 9 2  R13  R14  PK10 2  R11  R12  PK11 2  R15  R16  PK14 2
 A 2 R 9  R10  Q 'K2 2,9  A 2 R13  R14  Q 'K2 2,10  A 2 R11  R 12  Q 'K2 2,11  A 2 R15  R16  Q 'K2 2,14  A1 R 9  R10  Q 'K21,9 
2
'2
'2
A1 R 13  R 14  Q 'K
1,10  A1 R 11  R 12  Q K1,11  A1 R 15  R 16  Q K1,14
A 2 R 13  R 14 Q 'K 2,10  A 2 R 11  R 12 Q 'K 2,11  A 2 R 15  R 16 Q 'K 2,14  A1 R 9  R 10 Q 'K1,9 
(92)
(92)
′
where 𝑃𝐼𝑑 , 𝑃𝐼𝑏
, 𝑄𝐼𝑏,𝑑 are given by (7), (15), (16), (33) and (34) for b=1,2, a=9,10,11,14 and d=12,13,15,16.
The variance of time to recruitment can be calculated from (90), (91), (92) and (93).
Case (vi): If the distributions of optional thresholds follow extended exponential distribution and the distribution of
mandatory thresholds possess SCBZ property.
If 𝑔 𝑡 = 𝑔𝑥 1 𝑡 ,
E(T)  S1 C 'I1  S3 C 'I2  S 4 C 'I10  S 2 C 'I9  pR 9  R10  C I9  R13  R14  C I10  R11  R12  C I11  R15  R16  C I14
 S1 R 9  R10  H 'I1,9  S1 R13  R14  H 'I1,10  S1 R11  R12  H 'I1,11  S1 R15  R16  H 'I1,14  S3 R 9  R10 H 'I2,9
(94)
 S3 R13  R14  H 'I2,10  S3 R11  R12  H 'I2,11  S3 R15  R16  H 'I2,14  S 4 R 9  R10  H 'I10,9  S 4 R13  R14  H 'I10,10
 S 4 R11  R12  H 'I10,11  S 4 R15  R16  H 'I10,14  S 2 R 9  R10  H 'I9,9  S 2 R13  R14  H 'I9,10  S 2 R11  R12  H 'I9,11  S 2 R15  R16  H 'I9,14 
E(T 2)  2S1 C 'I21  S3 C 'I22  S 4 C 'I210  S 2 C 'I29  pR 9  R10  C 2I9  R13  R14  C 2I10  R11  R12  C 2I11  R15  R16 C 2I14
 S1 R 9  R10  H 'I21,9  S1 R13  R14  H 'I21,10  S1 R11  R12 H 'I21,11  S1 R15  R16  H 'I21,14  S3 R 9  R10  H 'I22,9
(95)
 S3 R13  R14  H 'I22,10  S3 R11  R12 H 'I22,11  S3 R15  R16  H 'I22,14  S 4 R 9  R10  H 'I210,9  S 4 R13  R14  H 'I210,10
 S 4 R11  R12  H 'I210,11  S 4 R15  R16  H 'I210,14  S 2 R 9  R10  H 'I29,9  S 2 R13  R14  H 'I29,10  S 2 R11  R12  H 'I29,11  S 2 R15  R16  H 'I29,14

where
S1 
4 22
1  21  22
, S1 
22
1  221  2
, S3 
4 12
1  221  2
, and S4 
12
(96)
1  21  22
′
where 𝐶𝐼𝑑 , 𝐶𝐼𝑏
, 𝐻𝐼𝑏,𝑑 are given by equation (5), (11), (12), (19), (29) and (30), for b=1,2,9,10, d=9,10,11,14
If 𝑔 𝑡 = 𝑔𝑥 𝑘 𝑡 ,
E(T)  S1 P 'K1  S3 P 'K 2  S 4 P 'K10  S 2 P 'K9  pR 9  R10 P K9  R13  R14  P K10  R11  R12  P K11  R15  R16 P K14
 S1 R 9  R10 Q 'K1,9  S1 R13  R14 Q 'K1,10  S1 R11  R12  Q 'K1,11  S1 R15  R16  Q 'K1,14  S3 R 9  R10  Q 'K 2,9
(97)
 S3 R13  R14  Q 'K 2,10  S3 R11  R12  Q 'K 2,11  S3 R15  R16  Q 'K 2,14  S 4 R 9  R10  Q 'K10,9  S 4 R13  R14  Q 'K10,10
 S 4 R11  R12 Q 'K10,11  S 4 R15  R16  Q 'K10,14  S 2 R 9  R10 Q 'K9,9  S 2 R13  R14  Q 'K9,10  S 2 R11  R12  Q 'K9,11  S 2 R15  R16  Q 'K9,14
E(T 2)  2S1 P 'K21  S3 P 'K2 2  S 4 P 'K210  S 2 P 'K2 9  pR 9  R10  P 'K2 9  R13  R14  P 'K210  R11  R12  P 'K211  R15  R16  P 'K214

 S1 R 9  R10  Q 'K21,9  S1 R13  R14  Q 'K21,10  S1 R11  R12  Q 'K21,11  S1 R15  R16  Q 'K21,14  S3 R 9  R10  Q 'K2 2,9
 S3 R13  R14  Q 'K2 2,10  S3 R11  R12  Q 'K2 2,11  S3 R15  R16  Q 'K2 2,14  S 4 R 9  R10  Q 'K210,9  S 4 R13  R14  Q 'K210,10
 S 4 R11  R12  Q 'K210,11  S 4 R15  R16  Q 'K210,14  S 2 R 9  R10  Q 'K2 9,9  S 2 R13  R14  Q 'K2 9,10  S 2 R11  R12  Q 'K2 9,11  S 2 R15  R16  Q 'K2 9,14
(98)

′
𝑃𝐾𝑑 , 𝑃𝐾𝑏
, 𝑄𝐾𝑏,𝑑
where
are given by (7), (15), (16), (22), (34), (40) for b=1,2,9,10, d=9,10,11,14.
The variance of time to recruitment can be calculated from (94), (95), (97), (98).
ISSN: 2321 – 242X
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The SIJ Transactions on Industrial, Financial & Business Management (IFBM), Vol. 1, No. 5, November-December 2013
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),(iv),(v),(vi)
respectively by keeping 𝜃1 = 0.4, 𝜃2 = 0.6, 𝛼1 = 0.5, 𝛼2 = 0.8, 𝑃 = 0.8. 𝛿1 = 0.6, 𝜂1 = 0.3, 𝜇1 = 0.7, 𝛿2 = 0.4, 𝜂2 = 0.7, 𝜇2 = 0.4, 𝛿3 =
0.5, 𝜂3 = 0.2, 𝜇3 = 0.5, 𝛿4 = 0.8, 𝜂4 = 0.4, 𝜇4 = 0.2 fixed and varying c, k, λ one at a time and the results are tabulated below.
Table 1: Effect of c, k, λ on Performance Measures
K
C
λ
E(T) (n=1)
V(T) (n=1)
E(T) (n=k)
V(T) (n=k)
E(T) (n=1)
V(T) (n=1)
E(T) (n=k)
V(T) (n=k)
E(T) (n=1)
V(T) (n=1)
E(T) (n=k)
V(T) (n=k)
E(T) (n=1)
V(T) (n=1)
E(T) (n=k)
V(T) (n=k)
E(T) (n=1)
V(T) (n=1)
E(T) (n=k)
V(T) (n=k)
E(T) (n=1)
V(T) (n=1)
E(T) (n=k)
V(T) (n=k)
1
1.5
1
6.9679
25.6665
6.9679
25.6665
7.7324
43.6104
7.7324
43.6104
7.6598
27.3582
7.6598
27.3582
6.7761
29.2323
6.7761
29.2323
6.3514
23.1721
6.3514
23.1721
7.9834
28.7849
7.9834
28.7849
2
1.5
1
12.7114
77.4020
4.730
12.6869
14.1669
144.2556
5.3442
19.7909
14.0574
81.5901
5.1876
13.5404
12.3669
92.6577
3.9552
9.8094
11.5099
69.8898
4.3220
11.4928
14.7965
86.5084
7.8489
28.7427
3
1.5
1
18.4451
156.5814
3.9241
9.1335
20.6183
302.2485
4.4931
13.4981
20.4440
164.0955
4.297
9.7527
17.9494
191.6421
3.1877
6.5992
16.6596
141.5116
3.6021
8.2743
21.6066
174.4565
4.4903
10.2917
2
0.5
1
5.0451
14.5131
2.1596
3.5011
5.5940
22.9082
2.4912
4.3699
5.5180
15.5572
2.3247
3.7578
4.9055
15.9851
1.8868
2.8943
4.6242
13.1266
2.013
3.2180
5.7088
16.2611
4.4390
10.1313
MODEL-I
2
1
1
8.8847
39.8637
3.4497
7.4237
9.8742
70.7631
3.9306
10.5594
9.7949
42.2954
3.7619
7.9558
8.6416
46.4246
2.9243
5.8699
8.0730
35.9789
3.1717
6.7497
10.2552
44.6674
4.4913
10.1873
2
1.5
1
12.7114
77.4020
4.73
12.6869
14.1669
144.2556
5.3442
19.7909
14.0574
81.5901
5.1876
13.5404
12.3669
92.6577
3.9552
9.8094
11.5099
69.8898
4.3220
11.4928
14.7965
86.5084
4.5322
10.4479
2
1.5
0.6
24.2139
153.1010
9.1033
27.8966
29.7872
257.3404
11.0848
40.9110
27.1538
144.2872
10.0894
28.1981
23.0643
200.5393
7.5765
22.3054
21.6086
144.1815
8.2299
25.7557
27.4189
159.6305
5.8079
11.9987
2
1.5
0.7
20.1059
127.7187
7.5414
22.5103
24.2085
227.0730
9.0346
34.3086
22.4765
126.6968
8.3388
23.3401
19.2438
160.3362
6.2832
17.7392
18.0019
117.6021
6.8342
20.5339
22.9109
135.3133
5.3523
10.9724
2
1.5
0.8
17.0248
107.4767
6.37
18.4372
20.0245
196.9932
7.4970
28.6710
18.9686
110.0029
7.0258
19.4218
16.3784
131.4045
5.3132
14.3899
15.2969
97.7016
5.7874
16.7108
19.5299
115.7645
5.0106
10.5471
2
1.5
0.6
9.3989
42.1363
3.6202
7.6222
14.7677
52.1370
5.9568
13.0870
11.4176
46.9688
4.8561
11.1229
8.9574
43.1655
4.0552
9.6770
8.6575
38.4553
3.9560
9.0636
12.4654
55.6542
5.1947
12.3144
2
1.5
0.7
7.8165
32.8947
2.9750
5.8456
12.2392
43.9696
4.9193
10.4268
9.4759
37.5616
4.0065
8.6697
7.4628
33.4669
3.3449
7.4454
7.2073
29.8989
3.2603
6.9736
10.4286
44.2466
4.3112
9.6421
2
1.5
0.8
6.6296
26.3603
2.4912
4.5759
10.3429
37.1718
4.1411
8.4267
8.0196
30.6033
3.3692
6.8903
6.3498
26.7180
2.8121
5.8653
6.1198
23.9082
2.7386
5.4889
8.9009
36.1055
3.6485
7.7311
Table 2: Effect of c, k, λ on Performance Measures
K
C
λ
E(T) (n=1)
V(T) (n=1)
E(T) (n=k)
V(T) (n=k)
E(T) (n=1)
V(T) (n=1)
E(T) (n=k)
V(T) (n=k)
E(T) (n=1)
V(T) (n=1)
E(T) (n=k)
V(T) (n=k)
E(T) (n=1)
V(T) (n=1)
E(T) (n=k)
V(T) (n=k)
E(T) (n=1)
V(T) (n=1)
E(T) (n=k)
V(T) (n=k)
E(T) (n=1)
V(T) (n=1)
E(T) (n=k)
V(T) (n=k)
ISSN: 2321 – 242X
1
1.5
1
3.0441
7.1810
3.0441
7.1810
4.4497
10.6673
4.4497
10.6673
3.5702
8.4758
3.5702
8.4758
2.9380
7.1321
2.9380
7.1321
2.8512
6.5790
2.8512
6.5790
3.9369
9.8711
3.9369
9.8711
2
1.5
1
4.9681
17.7836
2.1339
3.8254
7.6880
26.6871
3.0517
5.6193
5.9807
21.0017
2.4771
4.486
4.7725
18.0254
2.0662
3.7882
4.5971
16.1355
2.0081
3.5292
6.7623
25.3047
2.7209
5.1899
3
1.5
1
6.8787
33.0360
1.8138
2.8885
10.9106
49.4034
2.5507
4.2102
8.3777
38.8392
2.0880
3.3744
6.5941
33.8971
1.7357
2.8182
6.3298
29.8839
1.6877
2.6356
9.5806
41.959
2.2689
3.8730
2
0.5
1
2.3932
4.6687
1.1612
1.3288
3.3580
6.8065
1.5918
2.1447
2.7560
5.4729
1.4025
1.8087
2.3177
4.5951
1.2719
1.5791
2.2599
4.3129
1.2535
1.5321
2.9886
6.2277
1.4672
1.9539
MODEL-II
2
1
1
3.6884
10.2009
1.4750
2.0241
5.5327
15.2649
2.3263
3.7298
4.3768
12.0623
1.9395
3.0247
3.5523
10.2126
1.6632
2.5610
3.4360
9.3013
1.6248
2.4273
4.8804
14.2632
2.0916
3.4137
2
1.5
1
4.9681
17.7836
1.8138
2.8885
7.6880
26.6871
3.0517
5.6193
5.9807
21.0017
2.4771
4.486
4.7725
18.0254
2.0662
3.7882
4.5971
16.1355
2.0081
3.5292
6.7623
25.3047
2.7209
5.1899
© 2013 | Published by The Standard International Journals (The SIJ)
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The SIJ Transactions on Industrial, Financial & Business Management (IFBM), Vol. 1, No. 5, November-December 2013
K
C
λ
E(T) (n=1)
V(T) (n=1)
E(T) (n=k)
V(T) (n=k)
E(T) (n=1)
V(T) (n=1)
E(T) (n=k)
V(T) (n=k)
E(T) (n=1)
V(T) (n=1)
E(T) (n=k)
V(T) (n=k)
E(T) (n=1)
V(T) (n=1)
E(T) (n=k)
V(T) (n=k)
E(T) (n=1)
V(T) (n=1)
E(T) (n=k)
V(T) (n=k)
E(T) (n=1)
V(T) (n=1)
E(T) (n=k)
V(T) (n=k)
1
1.5
1
8.7025
34.8211
8.7025
34.8211
11.9437
44.2200
11.9437
44.2200
9.8360
37.8434
9.836
37.8434
8.4755
40.082
8.4755
40.0820
7.9598
32.0076
7.9598
32.0076
10.6990
43.2219
10.6990
43.2219
2
1.5
1
16.1601
107.5901
5.8844
17.0293
22.5534
133.7518
8.0357
21.7335
18.3833
115.3035
6.6357
18.5379
15.7514
129.9102
4.3769
11.6592
14.7112
99.3772
5.3930
15.6775
20.2346
133.8285
7.2243
21.0947
VI.
Table 3: Effect of c, k, λ on Performance Measures
MODEL-III
3
2
2
2
1.5
0.5
1
1.5
1
1
1
1
23.6093
6.2092
11.1903
16.1601
219.6924
19.3
54.7082
107.5901
4.8678
2.5533
4.2237
5.8844
12.1710
4.3730
9.7346
17.0293
33.1521
8.3979
15.4828
22.5534
270.1036
24.7717
68.8635
133.7518
6.6236
3.3
5.6738
8.0357
15.5694
5.6084
12.5201
21.7335
26.9212
6.9787
12.6873
18.3833
233.7771
21.1329
59.1073
115.3035
5.4806
2.8196
4.7327
6.6357
13.2601
4.8008
10.66
18.5379
23.0206
6.0443
10.9024
15.7514
270.8655
21.4945
64.3452
129.9102
3.3948
2.0278
3.2051
4.3769
7.4287
3.2325
6.8214
11.6592
21.4554
5.7029
10.2120
14.7112
203.4804
17.7082
50.3838
99.3772
4.4673
2.3770
3.8894
5.3930
11.2170
4.0428
8.9588
15.6775
29.7701
7.5192
13.8777
20.2346
273.0144
23.8208
68.0251
133.8285
5.9683
2.9798
5.1042
7.2243
15.0517
5.2428
11.9675
21.0947
2
1.5
0.8
21.6851
145.0958
7.9273
24.1856
30.6159
154.9058
10.9233
28.3197
24.8833
148.5502
9.0029
25.7287
20.8823
180.2652
5.8879
17.0686
19.5708
135.5157
7.2146
22.3241
26.6568
170.7740
9.5663
28.5052
REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
[7]
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.
ISSN: 2321 – 242X
2
1.5
0.7
25.6315
168.0582
9.3865
28.9995
36.3748
151.6207
12.9858
31.1588
29.5262
163.3494
10.6938
30.0184
24.5472
216.4212
6.9672
20.9829
23.0420
159.9947
8.5158
27.0374
31.2440
192.6595
11.2391
33.4406
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 if k increases, the mean and
variance of the time to recruitment of all the models
increase when the probability density function of
loss of man-hours is probability density function of
first order statistics and decreases when it is
probability density function of k-th order statistics.
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.
iii.
As λ increases, the average inter-decision time
decreases, which, in turn, shows that frequent
decisions are made on the average and hence mean
and variance of the time to recruitment decrease for
all the models.
VII.
2
1.5
0.6
30.8934
193.7135
11.3321
35.0322
44.0534
123.3951
15.7359
32.5263
35.7166
171.4821
12.9483
34.6404
29.4338
264.8733
8.4063
26.2674
27.6702
190.9037
10.2507
33.2772
37.3603
216.0017
13.4696
39.5573
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J. Sridharan, P. Saranya & A. Srinivasan (2013B), “Variance of
Time to Recruitment in a Two-Grade Man-Power System with
Extended Exponential Thresholds using Order Statistics for
Inter-Decision Times”, Archimedes Journal of Mathematics,
Vol. 3, No. 1, Pp.19–30.
ISSN: 2321 – 242X
J. Sridharan, P. Saranya & A. Srinivasan (2013C), “Variance of
Time to Recruitment in a Two- Grade Manpower System with
Exponential and Extended Exponential Thresholds using Order
Statistics for Inter-Decision Times”, Bessel Journal of
Mathematics, Vol. 3, No. 1, Pp. 29–38.
[16] P. Saranya, J. Sridharan & A. Srinivasan (2013D), “Expected
Time to Recruitment in an Organization with Two-Grades
Involving Two Thresholds following SCBZ Property”,
Antarctica Journal of Mathematics, Vol. 10, No. 4, Pp. 379–
394.
J. Sridharan is an Assistant Professor in Department of
Mathematics, Government Arts college, Kumbakonam. He has
published more than 25 papers in international journals and
presented more than 5 papers in international conference, and he has
guided 15 M.Phil scholars. At present, he is guiding 4 research
scholars for their research programme in Mathematics.
P. Saranya is an Assistant Professor (Sr.Gr) in Department of
Mathematics, TRP Engineering College Tiruchirappalli. She has
published 9 papers in international journals and presented two
papers in International /National conference.
A. Srinivasan is a Professor Emeritus in Department of
Mathematics, Bishop Heber College, Tiruchirappalli. He has
published more than 131 papers in International /National Journals
and presented 30 papers in Conference. He has guided 8 Ph.D
scholars and 25 M.phil Scholars. At present, he is guiding 3 Ph.D
scholars and 8 M.Phil scholars.
[15]
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