7Yadavalli.qxd


In many organizations, it is quite commonly found that

promotion of staff from one grade to the next higher grade is

linked to product ivit y improvement. But, one of the

important criteria for promotion is the employee’s up-to-date

knowledge and technical skills for which training is required.

Staff training is very important to developing organizations.

For example, a nurse assisting in surgical operations may

need to assist surgeons performing rare operations such as

removal of brain tumor, heart transplantation etc., to gain

additional experience which gives her an opportunit y for

further promotion. As another example, a computer soft ware

professional may come across several opportunities to be

exposed to recent advances in the industry, and if these are

utilised properly, he has an increased chance of being

promoted to a higher cadre or of leaving the job for better

opportunities. These examples indicate it is reasonable to

assume that an employee may get one or more in-service

training opportunities and thus becomes qualified to be

promoted to the next higher grade; but, if promotion is not

given to a trained person in due course, there is a chance that

the person may leave the organisation either because of

frustration or leaving the organization for better benefits and

recognition of the acquired skills. Accordingly, depending of

the amount of training undergone in the present grade, the

employee may choose to remain in the organization, and

possibly be promoted, or leave the organization. The impact

of training on the movement of staff has not been given due

importance in literat ure. The organization can expect more

efficiency from their employee by giving him/her more

training opportunities. This can positively impact on his/her

career development. A manpower system with a single grade

allowing wastages and recruitment was studied by Yadavalli

and Natarajan (2001). Models considering the training aspect

have been studied by Guardabassi et al. (1969), Purkiss

(1969), Grinold & Marshall (1977), Vajda (1978), Nakamura

and Shingu (1984) and Goh et al. (1987). However, their

objective is to minimize the reference cost or maximize the

expected ret urn for the planning period. Promot ional

policies to reduce wastages have been dealt with very

extensively by Bartholomew (1967), Young and Vassiliou

(1974) and Agrafiotis (1983, 1984, 1991). Not much work has

been done regarding training-linked promot ions and

wastages. A stochast ic model of a manpower system

incorporating these aspects is presented in this paper. The

model assumptions and notation are given in Section 2. The

probabilit y of remaining in the organizat ion without

promotion up to time t and getting promotion in the interval

(t, t + ∆) is derived in Section 3. The probabilit y of remaining
in the organization without getting promotion upto t is

obtained in Section 4. The probabilit y distribution of the

number of promotions received by an employee in (0,t) is

derived in Section 5. A numerical example is provided in

Section 6 to illustrate the results obtained in the model.

ASSUMPTIONS AND NOTATION

We consider a manpower system which is graded

hierarchically. An employee who enters into the system in a

grade undergoes some kind of training before getting

promoted to the next higher grade. The number of n training

opportunities/events that the employee may encounter while

in service in a grade is unlimited and the rate of promotion

depends on n. The employee with n training already

completed but not having got promotion yet may leave the

organization either out of frustration or due to better

monetary opport unit y in a different organization. The

employee may also leave the system due to other rare reasons

arising out of medical grounds. Specifically, we made the

following assumptions:

1. The epochs of trainings in one grade form a Poisson process

with parameter �1.

2. The probability that a person gets promotion from a grade to

the next higher grade in the interval (t, t + ∆) given that the
employee had exactly n training opportunities/events in that

grade on hand at time t, is n��∆.
3. The probability that an employee leaves the organization

due to dissatisfaction in the interval (t, t + ∆) given that the
employee has exactly n promotions on hand at time t, is

n��∆.
4. The probability that an employee leaves the organization in

the interval (t, t + ∆) due to medical reasons is v∆.
5. The probability of more than one event occurring in (t, t + ∆)

is 0(∆).

V S S YADAVALLI
Department of Industrial &Systems Engineering

University of Pretoria

R NATARAJAN 

S UDAYABHASKARAN
Department of Mathematics

Presidency College

University of Madras

Chennai, India

ABSTRACT
An attempt is made in this paper to propose and analyse a stochastic model of a labour potential system

incorporating training-linked promotions and wastages.  A numerical example illustrates and highlights the findings

of the model.

OPSOMMING
Hierdie artikel poog om ‘n stogastiese model van ‘n arbeidspotensiaalstelsel voor te stel en te ontleed waarin

opleidingsverwante promosies en verlies van werkkragte geïnkorporeer is.  ‘n Numeriese voorbeeld illustreer en lig

die bevindinge van die model toe.

TRAINING DEPENDENT PROMOTIONS AND WASTAGES

Requests for copies should be addressed to:VSS Yadavalli, Department of

Industrial & Systems Engineering, University of Pretoria, Pretoria, 0002

46

SA Journal of Industrial Psychology, 2002, 28(2), 46-48

SA Tydskrif vir Bedryfsielkunde, 2002, 28 (2), 46-48



NOTATION:

N1(t) : Number of training opportunities undergone by the employee

in (0, t].

N2(t) : Number of promotions to the employee in (0, t].

e : The event that the employee is present in the system at time t.

A : The event that the employee leaves the system in (0, t] out of

dissatisfaction.

B : The event that the employee leaves the system in (0, t], due to

medical reasons.

f(t) : lim P[e, N2(0) = n, N2(t) = n, N2(t + ∆) = n + 1] / ∆

Sn(t) : P[{N2(t) = n}, e]; n = 0, 1, …

Lan(t) : P[{N2(t) = n}, a]; n = 0, 1, …

Lbn(t) : P[{N2(t) = n}, b]; n = 0, 1, …

Pn(t) : P[N2(t) = n]; n = 0, 1, …

�*(s) : Laplace transform of �(t).

� : Convolution symbol.

EXPRESSION FOR f(t)

f(t) is the pdf of the interval between two successive promotions.

In probabilistic terms, the quantity of f(t)∆ represents the
probability that an employee waits in a grade n without

promotion upto time t and gets promotion to (n + 1)th grade in

(t, t + ∆). Since at least one training should take place in (0, t)
and the employee should not leave the system for any reason

whatsoever before a promotion can occur in (t, t + ∆), we have,
by using probabilistic arguments, 

Taking L a place transforms both sides of (1), we obtain 

Inverting (2), we get

(3)

Simplif ying (3), we get

(4)

EXPRESSION FOR S0(t):

S0 (t) is the survivor function, which is the probability that an

employee remains without promotion in the organization upto

time t. Noting the fact that the employee might have received

either no or one or more training (0, t), we have

Taking the Laplace transforms on both sides and solve for S*0(s),

we get

(6)

Inverting (6), we get

(7)

EXPRESSION FOR Pn(t):

By definition, Pn(t)=P[N2(t)=n]; n=0, 1, …; where Pn(t) is the

probability of getting n promotions. But the event [N2(t)=n] can

be expressed as the union of three mutually exclusive and

exhaustive events [N2(t)=n;  e], [N2(t)=n;  a], [N2(t)=n;  b].

Therefore 

Pn(t)=Sn(t)+L
a
n(t)+L

b
n(t) (8)

For finding Sn(t), L
a
n(t), L

b
n(t), we need the following auxiliary

functions

Sm,n(t)=P[N1(t)=m, N2(t)=n; e]

Since �m,n(t)∆ represents the probability that an employee
remains in the system upto t, has undergone exactly m training

in (0, t) and gets nth promotion in (t, t+∆). Therefore

where q(r)n(t) is r-fold convolution of �1e . In the same

way,since �m,n(t)∆ represents the probability that an employee
remains in the system upto t, has obtained n promotions and

goes for mth training in (t, t+∆), we get

Now, we can readily obtain Sm,n(t),

using Sm,n(t), we get

NUMBERICAL ILLUSTRATION

We present in Table 1, the values of S0(t) for the values of the

parameters given in Agrafiotis (1984) with �1 = �2 = �. From the

table we find that the probability of remaining in the system

without promotion decreases as time progresses.

CONCLUSIONS

This paper is a study of the link between promotion and the

amount of training, and obtains, analytically, the chance for an

n m n
m

t
a
n m n

m

t
b
n m n

m

S t S t

L t S u n du

L t S u vdu

,
0

,
0 0

,
0 0

( ) ( )

( ) ( )

( ) ( )

∞

=

∞

=

∞

=

= ∑

= ∑ ∫ µ

= ∑ ∫

v n t
m n m n m nS t t t e

–( )1
, , ,( ) { ( ) ( )}

λ + + µ= φ + ψ 

v n t
m n m n nt t e

–( )1
, – , 1( ) ( )

λ + + µΨ = Ψ λ +

v n t–( )
1

λ + + µ

m n
r

m n m r n n
r

t t q t m e
– 1

( )
, – , –1 2

1
( ) ( ) ( )

+

=
φ = ∑ φ   λ

m n

m n

t P N t m N t n N t n e

t P N t m N t m N t m e

, 1 2 2

, 2 1 2

lim 1
( ) [ ( ) , ( ) – 1, ( ) ; ]

0

lim 1
( ) [ ( ) , ( ) – 1, ( ) ; ]

0

φ = = = + ∆ =
∆ → ∆

Ψ = = = + ∆ =
∆ → ∆

t
e

v t
S t e e

–1 2(1– )
–( ) 21

0( )

λ λ
λ + λ= +

n

n
S s

s v s v s v n

* 1
0

0 1 1 2 2

( )
( ) ( ) ... ( )

∞

=

λ
= ∑

+ λ + + λ + + λ + λ + + λ

t
e

v t t
f t e e e

–1 2(1– )
–( ) – 21 2

1( ) (1 – )

λ λ
λ + λ λ= λ

n

v j tj
c j

n j
f t n e

n

–1

1

–( )1 2
1

1 0

2
( ) (–1)

– 1)

∞ ∞
λ + + λ

= =

λ 
 λ = λ ∑ ∑

n

n

n
f s

s v s v s v n
1 2

1 1 1 2 1 2

*( )
( ) ( ) ... ( )

∞

=

λ λ
= ∑

+ λ + + λ + + λ + λ + + λ

TRAINING DEPENDENT PROMOTIONS 47

v t v t

v t v t

n

f t e e

e e e

–( ) – ( )1 1 2
2

–( ) – ( )1 2 1 2
1 2 1

2

( ) [

{ [ ...

λ + λ + + λ

∞
λ + + λ λ + + λ

=

=  λ

+ ∑ λ  λ   λ v n t v n te– ( ( –1) ) – ( )1 2 1 22 }]
λ + + λ λ + + λ λ

(2)

(1)

–( ) – ( ) –( )1 1 1 2
0 1 1

–( ) – ( )1 1 2
1 2

S ( )
v t v t v t

v t v t

t e e e e

e e

λ + λ + λ + + λ

λ + λ + + λ

= + λ  + λ

+ λ λ 

– ( ) – ( ) – ( 2 )1 1 2 1 2
2

v t v t v t
e

λ + λ + + λ λ + + λλ 

(5)

m v n t–( )1 2λ + λ + + µ

v n t
m n t e

–( )1 1
–1, 1( )

λ + + λφ λ



employee to remain in the organization without being promoted

up to time t. A numerical illustration for S0(t), answers the

important question raised in the motivation of the paper. If the

empirical data is available, the results given in the paper can be

used to obtain important measures such as mean time to

promotion, for a practical human resource management problem.

TABLE 1

PROBABILITY S
0
(T)

Time Probability S0(t)

T

� = 0,839, v = � = 0,805, v = � = 0,382, v = � = 0,576, v =

0,378 0,383 0,116 0,332

0 1,00000 1,00000 1,00000 1,00000

1 0,52249 0,52989 0,83486 0,62491

2 0,19775 0,20682 0,63015 0,32239

3 0,06511 0,07041 0,44401 0,14932

4 0,02018 0,02255 0,29850 0,06510

5 0,00610 0,00703 0,14435 0,02743

10 0,00001 0,00002 0,01828 0,00031

20 0,0000 0,00000 0,00013 0,00000

30 – – 0,00001 –

40 – – 0,00000 –

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YADAVALLI, NATARAJAN, UDAYABHASKARAN48