decision making uncertain environment a queuing theory approach

International Journal of Advanced Engineering, Management and Science (IJAEMS) [Vol-2, Issue-6, June- 2016]
Infogain Publication (Infogainpublication.com) ISSN: 2454-1311
www.ijaems.com Page | 808
Decision Making Uncertain Environment - A
Queuing Theory Approach
Bhupender Kumar Som
Associate Professor, JIMS, Rohini, New Delhi, India
Abstract— Consumer behavior is one of the most uncertain
phenomenons. Customer impatience is one of those
uncertain phenomenon’s which is a threat to any business.
Customer impatience results in loss of customers and
business. Stochastic modeling provides numerical
measurement of necessary measures of performance in any
business up-to a certain extent. In this paper a multi-server
Markovian queuing system is developed with reverse
balking and impatient customers. Reverse balking is a very
new concept introduced in stochastic queuing models.
While reneging is one of the most known phenomenon in
queuing theory. Steady-state solution of the newly
developed model is derived. Necessary measures of
performance are obtained and numerical results are
presented. Sensitivity analysis of the model is also
performed. MATLAB and MS Excel are used as and when
needed.
Keywords— Reverse balking, customer impatience,
retention of customers, stochastic modeling, queuing
theory.
I. INTRODUCTION AND LITERATURE
REVIEW
In this era of globalization and liberalization managing
business has become a challenging task. Consumer behavior
is one of the most uncertain characteristics of business
environment. Customers have become more selective.
Brand switching is more frequent. Due to higher level of
expectations, customers get more impatient with a particular
firm. Customer impatience has also become a burning
problem in the corporate world. Queuing theory offers
various stochastic models that can be used in various
service systems facing customer impatience. By adopting
and applying these stochastic models strategy making
becomes highly effective. The premier work on customer
impatience in queuing theory appeared in [Haight, 1957,
1959], [Anker & Gafarian, 1963a, 1963b], [Bareer, 1957]
etc. Since then a number of papers have appeared on this
concept (reneging and balking). In these models, reneging
and balking is a function of system size/ queue length.
Larger is the system size more is the reneging and similar is
the case of balking. But, when it comes to sensitive
businesses like investment, selection of a food court,
selection of a service station etc. more number of customers
with a particular firm become the attracting (investing)
factor for more investing customers. Thus, the probability
of joining in such a firm is high. Modeling such a system as
a queuing system indicates that the probability of balking
will be low when the system size is more and vice-versa,
which is balking in the reverse sense (we call it Reverse
Balking).
The concept of reverse balking is introduced by [Jain, et.
al., 2014], they studied a single server Markovain queuing
system with reverse balking. [Kumar et. al., 2014] further
introduce notion of reverse reneging and applied it with
reverse balking. [Kumar et. al., 2013, 2014] designed
queuing systems for various environments and further
optimized them for various parameters.
Finding impatience a threat to business firms employ
various strategies to retain a reneging customer and they
manage to do it with some probability. [Kumar, et. al.,
2011] introduced the concept of retention of reneged
customer in their work. They study a single –server queuing
system with retention of reneged customers and balking.
[Kumar, et. al., 2012] also study a multi-server queue with
discouraged arrivals and retention of reneged customers.
[Kumar, et. al., 2012, 2012a] further extend their work on
single and multi-server feedback queues. Literature survey
unfolds the need of the study due to following reasons.
Once a customer moves in to the system by looking at the
large number of customers already present in the system, he
may find the service unsatisfactory, as it is difficult for the
firms to handle a huge chunk of customers at times. The
customer starts experiencing delay and dissatisfaction in
service. The customer becomes impatient due to this and
considers leaving the system without completion of his
service. This customer impatience can be termed as
Reneging in queuing literature. [Som, 2014] developed a

single-server queuing model by incorporating customer
impatience and reverse balking. He also performed
economic analysis of the model. Extending the work of the
paper.
Owing to the practically valid aspects of above mentioned
concepts, sensitive businesses with customer impatience are
formulated as queuing system in this paper. Consider any
life insurance company, where the purchase of policy refers
to the arrival of a customer in the queuing system
(insurance firms), the processed claim refers to as the
departure from the queuing system, where the claim
processing department is a multi-server and finite system
capacity (the number of policies it can accommodate). The
claims are processed in order of their arrival (i.e. the queue
discipline is FCFS). We incorporate the reverse balking and
reneging into this model. The model is based in Markovian
assumptions.
We present steady-state analysis of the stochastic models as
described above and derive important measures which help
in the management of sensitive businesses like investment
business. Numerical examples are provided for more clarity.
Rest of the paper is structured as follows: in section 2
assumptions under which the model is developed are
presented; section 3 deals with the mathematical
formulation; in section 4 steady state solution is derived;
section 5 deals with measures of performance; Numerical
illustrations and sensitivity analysis of the model is
performed in section 6; conclusions and future work are
provided in section7.
II. MODEL ASSUMPTIONS
1. The arrival to a queuing system (insurance firm) occur,
one by one in accordance with a Poisson process with
mean rateλ. The inter-arrival times are independently,
identically and exponentially distributed with
parameterλ.
2. There is a multi-server and the policy claims are
processed in parallel. The service times are
independently, identically and exponentially
distributed with parameter µ such as = for <
. = for ≥ .
3. The capacity of the system is finite, say N.
4. The policy claims are processed in order of their
arrival, i.e. the queue discipline is First-come, First-
served.
5. (a) When the system is empty, the customers balk (do
not purchase policy) with probability and may
purchase with probability p’ (= 1 – q’).
(b) When there is at-least one customer in the system,
the customers balk with a probability 1 − and join
the system with probability . Such kind of balking
is referred to as reverse balking.
6. The policy holders keeping their policies in force after
some time, say T may get impatient due to certain
reasons and decide to surrender their services before
completion (the customer wait up-to certain time T and
may leave the system before getting service due to
impatience). The reneging times (T) are independently,
identically and exponentially distributed with
parameterξ.
III. STOCHASTIC MODEL FORMULATION
Differential difference equations of the model is given by:
= −λ ′
+ µ
; n =0 (1)
= λ ′ −
1
− 1
λ + µ + 2µ
; n =1 (2)
=
− 1
− 1
λ −
− 1
λ + µ + ! + 1 µ " #
2 ≤ < (3)
=
− 1
− 1
λ −
− 1
λ + µ + − ξ + % µ + ! + 1 − "ξ & #
≥ (4)

= λ − % µ + − ξ &
; n = N (5)
IV. STEADY- STATE SOLUTION
In steady state lim*→∞ = , lim*→∞ ′ = 0. Therefore the equations (1) to (5) become:
0 = −λ ′
+ µ
; n =0 (6)
0 = λ ′ −
1
− 1
λ + µ + 2µ
; n =1 (7)
0 =
− 1
− 1
λ −
− 1
λ + µ + ! + 1 µ " #
2 ≤ < (8)
0 =
− 1
− 1
λ −
− 1
λ + µ + − ξ + % µ + ! + 1 − " ξ& #
≥ (9)
= λ − % µ + − ξ &
; n = N (10)
Steady-state solution of the model is obtained by solving (6) – (10) iteratively. Probability of n customers in the system can be
given by:
=
.
/
/
/
0
/
/
/
12
− 1 !
− 1
4
5
6
78
9 ′ , <
2
− 1 !
− 1
4
5
+ : − ξ
4
5
6
;
78<8;
9 ′ , ≥
2
− 2 !
− 1
4
5
+ : − ξ
4
5
6
;
78<8;
9 ′ , =
=
Using the normalization condition
1
1
N
n
n
P
=
=∑ , we get
+ > + >
8;
;
8
+ = 1
= ?1 + 2
− 1 !
− 1
4
5
6
78
9 ′
+ 2
− 1 !
− 1
4
5
+ : − ξ
4
5
6
;
78<8;
9 ′
+ 2
− 2 !
− 1
4
5
+ : − ξ
4
5
6
;
78<8;
9 ′@

V. MEASURES OF PERFORMANCE
5.1 Expected System Size
A< = >
8
A< = > + >
8;
;
8
+
A< = > 2
− 1 !
− 1
4
5
6
78
9 ′
;
8
+ > 2
− 1 !
− 1
4
5
+ : − ξ
4
5
6
;
78<8;
9 ′
8;
+ 2
− 2 !
− 1
4
5
+ : − ξ
4
5
6
;
78<8;
9 ′
5.2 Average rate of reneging
B7 = > − ξ
8;
B7 = > − 2
− 1 !
− 1
4
5
+ : − ξ
4
5
6
;
78<8;
9 ξ ′CD +
8;
− E 2
− 2 !
− 1
4
5
+ : − ξ
4
5
6
;
78<8;
9 ξ ′
5.3 Average rate of reverse balking
BF′ = G′
λ + > 1 −
− 1
λ
8
BF′ = G′λ + > 1 −
− 1
2
− 1 !
− 1
4
5
6
78
9 ′
+ > 1 −
− 1
λ 2
− 1 !
− 1
4
5
+ : − ξ
4
5
6
;
78<8;
9 ′
8;
;
8
VI. NUMERICAL ILLUSTRATION
In table -1, numerical results of all measures of performance are presented. Numerical results are obtained for various rates of
service.
Table.1:
λ =10, ξ =0.1, q′ =0.8, c=3, N =15
Rate of Service
(µ)
Expected System Size
(Ls)
Average Rate of Reneging
(Rr)
Average Rate of Reverse Balking
(Rb')
3.0 0.58730 0.00074 8.25796
3.1 0.56672 0.00060 8.25999
3.2 0.54789 0.00049 8.26106
3.3 0.53057 0.00040 8.26137
3.4 0.51455 0.00033 8.26110
3.5 0.49966 0.00028 8.26035
3.6 0.48576 0.00024 8.25923

3.7 0.47274 0.00020 8.25781
3.8 0.46051 0.00017 8.25616
3.9 0.44899 0.00015 8.25431
4.0 0.43811 0.00013 8.25232
4.1 0.42781 0.00011 8.25021
4.2 0.41804 0.00009 8.24801
4.3 0.40875 0.00008 8.24575
4.4 0.39991 0.00007 8.24343
4.5 0.39148 0.00006 8.24108
4.6 0.38343 0.00006 8.23870
4.7 0.37573 0.00005 8.23631
4.8 0.36836 0.00004 8.23392
4.9 0.36129 0.00004 8.23152
5.0 0.35451 0.00003 8.22914
An increasing rate of service ensures a large number of serviced customers leaving the system that leaves a negative impact on
system size. This can be observed from table -1. Following figure shows change in system size with increasing rate of service.
Fig.1: Ls Vs
6.1 Sensitivity Analysis
In this section sensitivity analysis of the model is presented. Variations in required measures of performance are observed with
respective variable. Results are presented through graphs for better insight
Table.2:
µ =3, ξ =0.1, q′ =0.8, c=3, N =15
Mean Arrival Rate
(λ)
(Ls)
Average Rate of Reneging
(Rr)
(Rb')
5 0.29905 0.00001 4.10322
6 0.35450 0.00003 4.93749
7 0.41003 0.00008 5.77227
8 0.46650 0.00018 6.60563
0.00000
0.10000
0.20000
0.30000
0.40000
0.50000
0.60000
0.70000
3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 5.0

9 0.52505 0.00038 7.43524
10 0.58730 0.00074 8.25796
11 0.65557 0.00138 9.06939
12 0.73319 0.00250 9.86309
13 0.82493 0.00436 10.62960
14 0.93730 0.00738 11.35519
15 1.07892 0.01209 12.02055
16 1.26051 0.01922 12.59966
17 1.49435 0.02960 13.05966
18 1.79298 0.04416 13.36238
19 2.16673 0.06371 13.46872
20 2.62057 0.08878 13.34601
21 3.15088 0.11936 12.97729
22 3.74367 0.15476 12.36948
23 4.37553 0.19362 11.55642
24 5.01751 0.23411 10.59444
25 5.64072 0.27431 9.55130
From table -2 it is clearly visible that, with increase in average arrival rate, expected system size increases. An increasing
expected system size leads to high confidence of customers with the firm and rate of reverse balking decreases therefore. Due to
this more and more arriving customers join the particular firm. The insight can be observed from graph below. On other hand rate
of reneging increases gradually as increasing number creates a dense network due to high system size that leads to high level of
impatience.
Fig.2: Rb vs λ
Figure -1 clearly states that more and more arrivals cause an increase in system size due to which rate of reverse balking
decreases.
0.00000
2.00000
4.00000
6.00000
8.00000
10.00000
12.00000
14.00000
16.00000
5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Rb vs λ

Table.3:
µ =3, λ =2, q′ =0.2, c=3, N =15
Rate of Reneging
(ξ)
(Ls)
Rate of Reneging
(Rr)
0.05 0.354518 0.000018
0.06 0.354516 0.000021
0.07 0.354515 0.000024
0.08 0.354513 0.000028
0.09 0.354512 0.000031
0.1 0.354510 0.000035
0.11 0.354509 0.000038
0.12 0.354507 0.000041
0.13 0.354505 0.000045
0.14 0.354504 0.000048
0.15 0.354502 0.000052
From table -3, it can be observe that increasing rate of reneging causes decrease in expected system size and increase in average
rate of reneging. This is because increasing rate of reneging states that more and more customers are moving out of the system
without completing their service.
Fig.3: E vs Rr
Figure -3 represents increase in average rate of reneging with increase in reneging rate that is obvious.
Table.4:
ξ =0.2, µ =3, λ =10, c=3, N =15
Probability of Reverse Balking
(q')
(Ls)
(Rb')
0.1 1.01918 0.00128
0.000000
0.000010
0.000020
0.000030
0.000040
0.000050
0.000060
0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15
ξ and Rr

0.2 0.99310 0.00125
0.3 0.96146 0.00121
0.4 0.92229 0.00116
0.5 0.87253 0.00109
0.6 0.80719 0.00101
0.7 0.71763 0.00090
0.8 0.58730 0.00074
0.9 0.38018 0.00048
1.0 0.00000 0.00000
It can be observed from table -5, that with increase in probability of reverse balking when there were no customers in the system
expected system size reduces and at q’ =1 (probability that an arriving customer does not join the system) expected system size
drops to zero. And Rb′ = 10, states that all arriving customers reverse balked.
Fig.4: q’ vs Rb’
Figure -4, represents increasing rate of reverse balking w.r.t. increase in probability of reverse balking.
VII. CONCLUSION
In this paper a multi-server Mrkovian queuing system with
reverse balking and reneging of customers is developed.
Steady-state solution of the model is derived. Necessary
measures of performance are obtained. Numerical results
are obtained by writing and algorithm in MS Excel and
MATLAB. Sensitivity analysis of the model is also
performed. Measures of performance with relevant
variables are studied.
The results are of immense use for making growth
strategies. The model mentioned above can be tailor-made
as per need and want for the firms operating in uncertain
business environment. In future cost-profit analysis of the
model can be presented with optimization.
REFERENCES
[1] Ancker. Jr., C.J. and Gafarian. A.V., Some queuing
problems with balking and reneging I, Operations
Research, 11, 1963, 88–100.
[2] Ancker. Jr., C.J. and Gafarian. A.V., Some queuing
problems with balking and reneging II, Operations
Research, 11, 1963b, 928–937.
[3] Haight, F. A., Queuing with balking I, Biometriika, 44,
1957, 360-369.
0.00000
2.00000
4.00000
6.00000
8.00000
10.00000
12.00000
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
q' and Rb'

[4] Haight, F. A., Queuing with reneging, Metrila 2, 1959,
186-197.
[5] Kumar R, Som B K, Design of Service System for
Insurance Busniess Facing Customer Impatience using
Queuing Theory, OJAS- An International Journal of
Research in Management, 2013, 2(1).
[6] Kumar R, Som B K , Optimization and Performance
Analysis of Insurance Business – A Queuing Modeling
Approach, International Journal of Mathematical
Sciences and Engineering Applications, 2013, 7(4).
[7] Kumar R, Som B K, Research paper; Profit
Optimization in Insurance Business Facing Customer
Impatience, Global Journal of Pure and Applied
Mathematics, 9 (1), 2013.
[8] Kumar R, Som B K, Modeling Insurance Business
facing Customer Impatience using Queuing Theory,
American Journal of Operational Research 3 (2A),
2013
[9] Jain N K, Kumar R, Som B K, An M/M/1/N Queuing
System with Reverse Balking, American Journal of
Operational Research, 2014, 4(2), 17-20.
[10]Kumar R, Som B K, An M/M/1/N queuing system
with reverse balking and reverse reneging, Advance
Modeling and Optimization, 2014, 16(2), 339-353.
[11]Kumar R, Som B K, Optimization of M/M/1/N
Feedback Queue with Retention of Reneged
Customers, Operations Research and Decisions, 2014,
24(3)
[12]Kumar R, Som B K, Optimization of M/M/1/N
queuing system with retention of reneged customers
and discouraged arrivals, ISST Journal of Mathematics
and Computing Systems, 2013 4(2).
[13]Kumar R, Som B K, Optimizing Service Rate and the
Capacity of an M/M/1/N Queuing System with
Retention of Reneged Customers, Indian Journal of
Industrial and Applied Mathematics, 2014 5(1).
[14]Kumar R, Som B K, Optimization of a Service System
Facing Customer Impatience, Presented in ICSSR
Sponsored International Conference Titled, Shifting
Paradigms in Applied Economics and Management:
Course Correction organized by faculty of
Management, Shri Mata Vaishno Devi, University,
August, 2014.
[15]Kumar R, and Sahrma K S, An M/M/1/N queuing
model with retention of reneged customers and
balking, American Journal of Operational Research,
2011, 1(1), 1-5
[16]Kumar R, and Sahrma K S, 2012, A Multi-server
Markovian queuing system with discouraged arrivals
and retention of reneged customers, International
Journal of Operations Research, 2012, 9(4), 173 –
184.
[17]Kumar R, and Sahrma K S, A Markovian Feedback
queue with retention of reneged customers and
balking, AMO – Advance modeling and optimization,
14(3), 681 – 668.
[18]Kumar R, and Sahrma K S, M/M/c/N queuing system
with renetion of reneged customers, International
Journal of Operations Research, 2012a 17(3), 333 –
344.
[19]Som B K, System design and economic analysis of a
Markovian queuing system with customer impatience,
presented in an international conference at JIMS, New
Delhi on Feb 6, 2015. Published in conference
compendium.

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