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Journal for Studies in Management and Planning Markov Chain Models in
Discrete Time Space and Application to Personnel Management
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Journal for Studies in Management and Planning
Available at http://internationaljournalofresearch.org/index.php/JSMaP
e-ISSN: 2395-0463
Volume 01 Issue 09
October 2015
Available online: http://internationaljournalofresearch.org/ P a g e | 351
Markov Chain Models in Discrete Time Space and
Application to Personnel Management
1.C. E. Okorie
Department of Mathematics and Statistics, Federal University Wukari, Taraba State, Nigeria.
E-mail of corresponding author: Chyokanmelu@yahoo.com
ABSTRACT
A Markov chain probability model is found
to fit personnel data of recruitment and
promotion pattern in El-Amin
International School, Minna. Manpower
planning is a useful tool for human
resource management in large
organizations. Classical Manpower
Planning models are analytical time –
discrete push and pull models. A mixed
push-pull model is developed for in this
study. This model allows taking into
account push and pull transitions of
employees through an organization at the
same time. In fact, in any organization, the
present number of staff in each level is
known and at any particular time each
member of staff is in a particular grade
either by promotion or recruitment into
that grade, We consider a Markov model
formulated to assist in making promotion,
recruitment policies for the next time
period given that existing staff structure is
known. Data from El-Amin International
School, Minna is used on the formulated
model. The data is collected for a period of
ten years,from 2000-2010 The result
shows that the probability of those on
promotion is 0.21 of the entire personnel
and that of the teachers retained but no
promotion is 0.52 while new recruitment is
0.27.
Keywords: Stochastic, Transition, Markov
chain, recruitment, promotion, pull, push.
INTRODUCTION
Manpower systems are hierarchical in
nature and consists of a finite number
ordered grades for which internal
movement or promotion of staff is possible
from one grade to another though there is
no promotion beyond the highest grade.
Members of staff in the same grade have
certain common characteristics and
attributes (such as rank, trade, age, or
experience) and the grades are mutually
exclusive and exhaustive so that any staff
must belong to one but only one grade at
any time (Georgiou and Tsantas, 2002)
Markov chain theory is one of the
Mathematical tools used to investigate
dynamic behaviours of a system (e.g.
workforce system, financial system, health
service system) in a special type of
discrete-time stochastic process in which
the time evolution of the system is
described by a set of random variables. It
is worth mentioning that variables are
called random if their values cannot be
predicted with certain and discrete-time
means that the state of the system can be
viewed only at discrete instant rather than
at any time (Howard, 1971).
Stochastic model are influential and have
been used widely in health care
management. Markov chain models have
been applied to many areas of health
related problems (Parker and Caine, 1996).
Some mathematical models of diseases in
populations (Epidemometric models) have
also been employed to study leprosy

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Volume 01 Issue 09
October 2015
disease. McClean (1991) has studied, the
incidences of new cases as a result of
prevalence of overt cases, using different
equation and McClean and Montgomery
(2000) have adopted the kinship
coefficient to determine correlation
between leprosy rates in village of
different distances apart. The mixed push –
pull model is capable of incorporating this
additional constraint. This model is based
on the assumption of the classical pull
models, in which vacancies arise in case
that the number of employees in a specific
group is less than the desired one. It allows
the organization to choose a policy to fulfil
those vacancies. According to the pull
strategy, the vacancies are filled by
promotions or by external recruitment.
Besides, in the mixed push-pull model,
push promotions are possible in case not
enough people had the opportunity to
promote after all vacancies at higher levels
were filled.
Homogeneous Markov chain models
having time independent (or stationary)
transition probability have been applied to
manpower planning in Winston (1994),
Bartholomew et al (1991) and Ekoko
(2006). Alem (1985) and krishnamurty
(1988) asserted manpower mobility from
one organization to another results in
policies of promotion and recruitment that
are within a systematic and qualitative
framework in some sectors of the
economy. The bivariate model in
Raghavendra (1991) is a non –
homogeneous Markov model by which
promotion and recruitment policies are
derived given the required future structure.
The model in Raghavendra (1991) uses
two fundamental equations: one is the
probability equation and the other is for
determining the number of staff in each
grade in the next time period.
Aim and Objectives
The ultimate aim of this study is to apply
Markov model for recruitment and
promotion systems and the objectives
include the following:
(1) Develop analytical time-discrete
push and pull models.
(2) Consider constant promotion
probabilities over time.
(3) Estimate transition and the future
number of employees in an
organization using push and pull
models.
MATERIALS AND METHODS
Mixed push and pull mode is developed
for this study. The model uses the
assumptions that push as well as pull
promotions are possible to occur in the
same system at the same time. An example
of a personnel system requiring a model in
which both push and pull transitions occur,
is an organization in which vacancies are
filled by promotions from groups of
employees that succeeded in an
examination. A transition between the
group of people that not yet passed an
examination and the group of people that
succeeded in the examination happens
with a certain probability. This is a typical
push movement. Meanwhile, the actual
promotion (only if there is a vacancy at
another level) has to be considered as a
pull transition. A mixed push-pull model
has an advantage from the practitioner’s
point of view. Often, organizations
promote employees because of several
reasons: Obviously, vacancies at higher
levels can be filled by promotions from
lower levels. The mixed push-pull model
allows considering several reasons for
promotion at the same time. Under
Markovian assumptions, the equation for
determining the number of staff in each
grade in the next time period is
         
t
t
t
t
t
N R
P
N
p d
d
i
k
j ij



  1
1
(1.1)

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October 2015
𝑓𝑜𝑟 𝑗 = 1, 2, … . . 𝑘
Where 𝑡 is the current time period and
(𝑡 + 1) is the next time period.
Members of staff could stay in the
same grade, move to another grade or
leave the system. There is therefore a
probability equation that governs the
way promotion is carried out in each
level. The probability equation is given
as;
    1
1


 
t
t P
P d
k
j ij
(1.2)
For all 𝑖 = 1, 2, … . . 𝑘
The promotion and new
recruitment to any grade in an
organization follows a prescribed
policy as expressed in Krishnamurthy,
(1988). These proportion and
recruitment are specified in the policy
to be translated into estimates of the
probability𝑃𝑖𝑗(𝑡) of moving from state
𝑖 to state 𝑗 in a time period 𝑡. Let
𝑒𝑗(< 1) represent the proportion of
staff to be promoted from grade level
𝑗 − 1 𝑡𝑜 𝑗 .The (1 − 𝑒𝑗) represent the
proportion of newly recruited staff to
grade level 𝑗.
As stated earlier 𝑃𝑑(𝑡) 𝑎𝑛𝑑 𝑃𝑑(𝑡)𝑁𝑑(𝑡)
are the probabilities of double promotion
respectively.
From period 𝑡 𝑡𝑜 𝑝𝑒𝑟𝑖𝑜𝑑 𝑡 + 1 and
starting from the highest grade level 𝑘.
                 
t
t
t
t
t
t
t
t P
N
P
N
P
N
P
N k
d
d
k
kk
k
k
k
k




 
 1
1
1
(1.3)
But , from (1.2)𝑎𝑛𝑑 (1.3).
   
t
t P
P d
kk

1
(1.4)
Where in the highest grade 𝑘, 𝑃𝑑(𝑡) is the
probability of double promotion into grade
𝑘 in period 𝑡 i.e. 𝑃𝑑(𝑡) = 𝑃(𝑘−2)𝑘(𝑡).
     
t
t P
P k
k
d 2


And substituting for 𝑃𝑘𝑘 in (1.3) and
simplifying we obtain:
                   
 
1
1
1
'
1
1










t
t
t
t
t
t
t
t
t N
N
P
N
P
N
R
N
P k
d
d
k
d
k
k
k
k
k
(1.5)
𝑁′𝑘(𝑡 + 1) can be easily determined since
𝑃𝑑(𝑡) is assumed known and given.
Since the number of promotions and
recruitments to grade 𝑘 should follow the
ratio
𝑒𝑘: (1 − 𝑒𝑘) respectively, it follows that
       
1
'
1
1




t
t
t N
e
N
P k
k
k
k
k
(1.6)
And      
1
1
'


 t
t N
e
R k
k
k
(1.7)
Equations (1.6) and (1.7) would give the
number of promotions from grade (k-1) to
k and the number of new recruitments to
grade k respectively. From (1.6),
 
 
  






 



t
t
t
N
N
e
P
k
k
k
k
k
1
'
)
1
(
1
(1.8)
Equation (1.6) and (1.7) would give the
number of promotions from grade (𝑘 − 1)
to 𝑘 and the number of new recruitments
to grade 𝑘 respectively.
      
t
t
i
t p
P
P d
j
ij



 1
1
(1.9)
𝑗 = 1, 2, … . . 𝑘 − 1
For example, at 𝑗 = 𝑘 − 1, we have;

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Volume 01 Issue 09
October 2015
         
t
t
t P
P
P d
k
k
k
k


 

 1
1
1
1
(1.10)
Using equations (1.7) and (1.8) and
(1.9), the number of promotions, the
number of recruitments and the
transition probabilities can be
estimated successfully for all other
state of the organization.
Development of the model and model
assumptions-The Mixed Push-pull
Model
We consider an organization in which the
total population of employees is divided
into 𝑘 homogeneous groups. The
homogeneous groups form a partition of
the total population. The number of people
in group 𝑖 at time 𝑡 is denoted by 𝑛𝑖(𝑡).
We use a discrete time scale. The length
of one time interval is chosen in such a
way that it can be assumed that one
employee can make at most one transition
during the time interval. This implicates
the assumption that vacancies are not filled
instantaneous. This is a realistic hypothesis
since it takes time to perform a promotion
or recruitment decision. This assumption
also implicates that a vacancy does not
disappear in the company when it is filled
by a promotion. When an employee is
promoted from group 𝑖 to group 𝑗, the
initial vacancy in group 𝑗 created a new
vacancy in group 𝑖.
In fact, the initial vacancy moves in the
opposite direction of the employee. This
means that 𝑛𝑖(𝑡) is possibly smaller than
the desired number of employees ni
*
(𝑡) at
time 𝑡.
Classical pull models most often assume
that vacancies which need to be filled in
the next time interval are determined at the
end of the period in which they turned up.
This way, the model is given by;
        
     
     
1
1
1
2
2
1
*
*
*
*
*
















t
t
t
t
t
W
t
V
t
t
i
t
V
t
V
n
n
n
n
ni
(1.11)
∆𝑛∗(𝑡) = ∆𝑛∗(𝑡)(𝑡) − 𝑛∗(𝑡 − 1)
        
     
     
1
1
1
2
2
1
*
*
*
*
*
















t
t
t
t
t
W
t
V
t
t
i
t
V
t
V
n
n
n
n
ni
Where
𝑉(𝑡) being a(1 + 𝑘) row vector
formed by the vacancies in every group
to be filled in time interval (𝑡 − 1, 𝑡);
𝑆(𝑡) 𝑖𝑠 𝑎 (𝑘 × 𝑘) matrix with elements
𝑠𝑖𝑗(𝑡);
𝑊(𝑡) 𝑏𝑒𝑖𝑛𝑔 (𝑘 × 𝑘) diagonal matrix
formed by the voluntary wastage
probabilities 𝑤𝑖(𝑡);
𝑤𝑖(𝑡) is the probability that an
employee in group 𝑖 will leave the
organization in time period (𝑡 − 1, 𝑡);
𝑛∗(𝑡) denoting the (1 × 𝑘) row vector
{𝑛𝑖
∗
(𝑡)} and 𝑛(𝑡) denoting the (1 × 𝑘)
row stock vector 𝑛𝑖(𝑡).
𝑛(𝑡) = 𝑛(𝑡 − 1)[1 − 𝑊(𝑡)]𝑄(𝑡) +
𝑅(𝑡)𝑟(𝑡) = 𝑛(𝑡 − 1)𝑃(𝑡) + 𝑅(𝑡)𝑟(𝑡)
     
               
t
r
t
R
t
P
t
n
t
r
t
R
t
Q
t
W
t
n
t
n 





 1
1
1
(1.12)
       
 
  0
1
1
;
0
*




 t
W
t
n
t
Max
t
V n
(1.13)
𝑄(𝑡) becomes the identity matrix because
there will be no push promotions. The
push recruitments 𝑅(𝑡) also becomes zero.
The model (1.12) becomes:
       
     
t
S
t
V
t
W
t
n
t
V
t
n 



 1
1

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Volume 01 Issue 09
October 2015
     
     
     
t
S
t
V
t
W
t
n
o
Max t
W
t
n
t
n 













 

 1
1
; 1
1
*
     
     
     
t
S
t
V
t
W
t
n
t
W
t
n
t
n 





 1
1
1
1
*
     
t
S
t
V
t
n 

*
(1.14) For computational reasons, we put
SQ
A 
1 and    
1
1
1 


 Q
S
W
M
(1.12) becomes:
    Rr
A
M
t
n
t
n n 



*
1
(1.15)
We use the Jordan normal form theorem
(Gantmacher,1964), which allows us to
rewrite 𝑀 as;
DA
M A


(1.16)
With 𝐴 𝑎(𝑘 × 𝑘) non-singular matrix and
𝐷 a block diagonal matrix with m the
number of eigenvalues of 𝑀 and
   
 
 


 m
m
j
j
j
diag
D (
,.....
( 2
2
1
1

(1.17)
With  
i
i
j a  
P
P i
i
 matrix with
eigenvaluei
(𝑤𝑖𝑡ℎ 𝑎𝑙𝑔𝑒𝑏𝑟𝑎𝑖𝑐 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑖𝑐𝑖𝑡𝑦 Pi
)
 






















i
i
i
i
i
i
j
1
0
..
..
0
0
0
0
0
0
..
..
..
..
..
..
0
..
..
0
0
0
..
..
0
1
To incorporate the vacancies arising out
of all pull promotions in the coming time
period, (1.11) in the mixed push-pull
model needs to be replaced by:
       
   ]
1
*
1
[
1
1
;
0 t
n S
t
W
t
n
t
Max
t
V

















(1.18)
Since the vacancies within one time period
evolve as a chain satisfying the Markov
properties with 𝑆 acting as a transition
matrix, the initial vacancies as estimated
by (1.11) need to be multiplied by the
fundamental matrix. Indeed, it is very well
known that the fundamental matrix gives
the expected number of visits to each state
before absorption occurs (Bartholomew et
al, 1991).
Results and Discussion
The implication of models in this study on
a common data for the purpose of
comparison is the concern in this chapter.
A teacher is seen to be recruited and also
promoted when the conditions specified
are met. The states considered are
promotion, retained and recruited status of
the teacher available over the period.
Discrete State time Markov model. A
follow-up summary statistics for
2009/2010 academic session on 130
teachers in El-Amin International school,
Minna provides the following transition
matrices for the second term. The
transition count matrix for the number of
teachers in first term (𝑀1)

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Volume 01 Issue 09
October 2015
Is given as:











14
20
6
11
24
7
9
16
23
1
M
The transition count matrix for those who
returned back to the school in second tern
and those who were recruited to make up
the required 130 teachers needed is ; 𝑀2











15
10
6
14
21
11
10
17
16
2
M
The following estimates of transition
probabilities are obtained











35
.
0
500
.
0
15
.
0
429
.
0
571
,
0
167
.
0
188
.
0
333
.
0
479
.
0
1
P
In this study of Attitudinal changes of
teachers moving from this school and
coming into this school based on the
grade/ rank, it is now pertinent to ask
whether these three sets of transition
probabilities reflect the same behaviour on
the part of teachers from one term to
another. If so, the data can be pooled to
give a single transition count matrix and
hence a single set of estimates.
The pooled transition count matrix
obtained is;











29
40
12
25
45
18
19
33
39
M
The pooled estimates of elements of the
transition probability matrix obtained are;











358
.
0
494
.
0
148
.
0
284
.
0
511
.
0
205
.
0
208
.
0
363
.
0
429
.
0
P
The model can be represented by a single
transition count matrix.
Thus, the maximum likelihood estimate of
the transition probability
matrix is given by;











`
358
.
0
494
.
0
148
.
0
284
.
0
511
.
0
295
.
0
208
.
0
363
.
0
429
.
0
P
Calculating 𝑃𝑛
, we have,











2952
.
0
426
.
0
2094
.
0
2791
.
0
4236
.
0
2593
.
0
2688
.
0
4455
.
0
3752
.
0
2
P











2487
.
0
5150
.
0
3630
.
0
2093
.
0
5261
.
0
3024
.
0
2812
.
0
5454
.
0
3556
.
0
3
P











2710
.
0
5241
.
0
2132
.
0
2714
.
0
5233
.
0
2104
.
0
2713
.
0
5243
.
0
2110
.
0
16
P











2710
.
0
5241
.
0
2132
.
0
2714
.
0
5233
.
0
2104
.
0
2713
.
0
5243
.
0
2110
.
0
17
P











27
.
0
52
.
0
21
.
0
27
.
0
52
.
0
21
.
0
27
.
0
52
.
0
21
.
0











366
.
0
488
.
0
146
.
0
304
.
0
457
.
0
239
.
0
233
.
0
395
.
0
375
.
0
2
P

e-ISSN: 2395-0463
Volume 01 Issue 09
October 2015
Corrected to 2 decimal places and for 𝑛 >
17, we find that 𝑃𝑛
gets
closer to exactly 𝑃17
that is
as 𝑛 increases,
P
P
P
n
n 0












27
.
0
52
.
0
21
.
0
27
.
0
52
.
0
21
.
0
27
.
0
52
.
0
21
.
0
0
3
0
2
0
1 P
P
P
= (0.21 0.52 0.27)
The limit state probability vector is given
by
𝜋 = 𝜋𝑝 = (0.21 0.52 0.27)
This shows that 21% of the teachers get
promoted,52% are retained in the school
but without promotion and 27% of the
teachers are recruited in the school at the
beginning of the term.
Based on the plan of this school the
desired personnel distribution is fixed
over time and it is given by
𝑛∞
= (50 100 20)
The current personnel size in every group
is smaller than the desired ones.
Consequently, both aspects of the mixed
push and pull model have an influence on
the changes or transition.
𝑛(0) = (43 46 41)
Under this recruitment policy
𝑅(𝑡)𝑟(𝑡) = 0
This school never reaches the desired
personnel structure 𝑛∞
The decision maker
might consider changing its policy. So the
optimal recruitment policy is
𝑅(1)𝑟(1) = (2 8 28)
It is clear that there exist a structural
problems in this school. The promotion
system is not compatible with the desired
personnel structure. So the school should
consider trying to influence and change its
promotion system.
Conclusion
According to the optimal recruitment
policy, it is clear that there exists a
structural problem in the organization
studies (El-Amin International school,
Minna). The promotion system is not
compatible with the desired personnel
structure. The school should reconsider
its (push) promotion and or recruitment
policy to increase its personnel size to the
desired personnel size.Also, the
organization should consider trying to
influence and change its promotion
system.
Aknowledgement and References
[1] Abubakar,U.Y.(2004) ’’A Three-State-
Semi Markov Model in Continuous Time
to study the Relapse Cases of Leprosy
Disease After the Treatment Using
Dapsone and Multi-Drug Therapy (M D
T)’’; Proceeding of the 41st Annual
National Conference of Mathematical
Association Nigeria. Page 15-23
[2] Alam, M.A.(1985), Steady State Career
Structure Model
For Indian Army Officers, Defence Mgmt.
Vol.12, 34-42
[3] Bartholomew, D. J., Forbes A.
F.,&McClean, S. I. (1991). Statistical
techniques for manpower planning,
Chichester; Wiley.

e-ISSN: 2395-0463
Volume 01 Issue 09
October 2015
[4] De Feyter, T. (2006),Modelling
heterogeneity in Manpoer Planning;
dividing the
personnel system in more homogeneous
subgroups. Applied Stochastic Models in
Business and Industry, 22(4), 321-334.
[5] Georgious, A. C. &Tsantas, N. (2002).
Modelling recruitment training in
mathematical human resource planning.
Applied Stocastic Models in Business and
Industry,18,53-74.
[6] Ekoro,P.O., (2006).’’Analyzing
Academic Staff Ptomotion Criteria in
Nigeria Universities Using a Nonergordic
Markov chain Model, The Nigerian
Academic Forum, Vol. 11,No. 3 130-137.
[7] Krishnamurthy,G.S.(1988),’’Out of
Turn Promotions: Government in a
Fix,’’Deccan Herald
[8 ]McClean, S. (1991). Manpower
planning models and their estimation.
European Journal of Operation Research,
51, 179-187.
[9] McClean, S.I. &Montgomery, E. J.
(2000). Estimation for semi-Markov
manpower models in a stocastic
environment.In.J.Janssen & N.Limnios
(Eds.), Semi-Markov models and
applicationa (pp. 219-227). Dordrecht:
Kluwer Academic.
[10] Parker, B.,& Caine, D. (1996).
Holonic modeling: human resource
planning and the two faces of Janus,
International Journal of Manpower, 17(8),
30-45.
[11] Raghavendra, B. G.(1991), “ A
Bivariate Model for Markov manpower
planning system, Journal
Opl.Res.Soc.Vol. 42, No. 7, 565-570.
[12] Staff Condition of Service of El-Amin
International School, Minna. Revised
Edition (2008), page 3and 7.
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