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International Journal of Research in Advent Technology (IJRAT) Special Issue, January 2019 E-ISSN: 2321-9637
Available online at www.ijrat.org International Conference on Applied Mathematics and Bio-Inspired Computations
10th & 11th January 2019
50
A Study on Queuing Models in Banking Services
S.Dhivya Priya 1
Department of Mathematics1, Dhanalakshmi Srinivasan Engineering College,Perambalur
1
Abstract- This paper deals with the Queuing theory and some mathematical models of queuing systems. The ultimate goal is to achieve
an economic balance between the cost of service and the cost associated with the waiting for that service. Then the basic laws and
formulas are introduced it highlights several recent advances and developments of the theory and new applications in the banking
services. It ends with the References of the most important sources.
Index Terms- Queues; Server; Customer
1. INTRODUCTION Queues (waiting line) are a part of everyday life. Providing
too much service involves excessive costs. And not providing
enough service capacity causes the waiting line to become
excessively long. The ultimate goal is to achieve an economic
balance between the cost of service and the cost associated with
the waiting for that service. Queuing theory is the study of
waiting in all these various guises.
Queuing theory was originated from the work of
A.K.Erlang, an engineer in Copenhagen Telephone Exchange
who studied the reason for the delay operators and published his
findings as a paper in 1909 under the title” The Theory of
Probabilities and Telephone Conversation” are his most
important work. Solutions of Some problems in the Theory of
Probabilities of Significance in Automatic Telephone Exchanges
was published in 1917, which contained formulas for loss and
waiting probabilities which are now known as Erlang’s loss
formula (or ErlangB-formula) and delay formula (or Erlang C-
formula), respectively
2. MODELS ON QUEUING THEORY
Definition: 2.1
Queuing theory is nothing but a waiting line.
Definition 2.2
This is the rate at which the customers arrive to be serviced .
This arrival rate may not be constant. Hence it is treated as
random variable for which a certain probability distribution is to
be assumed. In general in queuing theory arrival rate is
randomly distributed according to the Poisson distribution .
The mean value of the arrival rate is denoted by λ.
Definition: 2.3
This is the rate at which the service is offered to the
customers. This can be done by a single server or sometimes by
a multiple servers, but this service rate refers to service offered
by single service channel. This rate is also a random variable as
the service to one customer may be different from the other .
The mean value of service rate is μ .
Definition: 2.4
If the customer who arrive and form the queue are from a
large population then the queue is referred to as infinite queuing
model.
Definition: 2.4
If the customers arrive from a small number of
population then this is treated as a finite queue.
3. KENDALL’S NOTATION
Kendall’s notation expression is of the form
M/G/1 - LCFS preemptive resume (PR)
describes an elementary queuing system with exponentially
distributed inter arrival times arbitrarily distributed service
times, and a single server. The queuing discipline is LCFS
where a newly arriving job interrupts the job currently being
processed and replaces it in the server.
International Journal of Research in Advent Technology (IJRAT) Special Issue, January 2019 E-ISSN: 2321-9637
Available online at www.ijrat.org International Conference on Applied Mathematics and Bio-Inspired Computations
10th & 11th January 2019
51
3.1 MODEL (SINGLE SERVER)
{(M/M/1):(∞/FCFS)};Birth and Death model
This model deals with a queuing situation having Poisson
arrivals(exponential inter arrival times) and Poisson services
(exponential service times) , single server, infinite capacity of
the system and first come first served queue discipline .
The solution procedure of this queuing model can be
summarized into following steps
Step1
If is the probability of n customers at time t in the
system ,then the probability that the system will contain n
customers at time (t+∆t) can be expressed as the sum of the joint
probabilities of the three mutually exclusive and collectively
exhaustive cases. That is For n≥1 and t≥0
= { } {
} {
} = { }{ } { } { }{ } = { } Since ∆t is very small ,therefore terms involving (∆t)
2 can be
neglected .
Then [1] becomes
{ }
[or]
=λ
n≥1
Taking limit on both sides as ∆t→0, then above equation
reduces to
׳
=λ -(λ+μ) [2]
Similarly, if there is no customer in the system at time (t+∆t),
then there will be no service completion during ∆t. Thus for
n=0 and t≥0, We have only two probabilities instead of three.
The resulting equation is
{ }
Or
Taking limit on both sides as ∆t→0, we get
׳
=
Step 2
Obtain system of steady state equation
In the steady state, is independent of time t, and
the number of customers in the system initially,that is
and
{ }
Consequently, equation [2] and [3] may be written in the form
μ thus these equations constitute the system of steady state
difference equations. The solution of these equations can be
obtained by using iterative method .we shall find the values of
P1 ,P2,.... in terms of P0,λ,μ.
Step 3
Solve the system of difference equations
From equation[5] we get
(
)
International Journal of Research in Advent Technology (IJRAT) Special Issue, January 2019 E-ISSN: 2321-9637
Available online at www.ijrat.org International Conference on Applied Mathematics and Bio-Inspired Computations
10th & 11th January 2019
52
If we put n=1 in equation [4], we get
0= - (λ+μ)
or (
)
= (
= (
-
=
)(
– (
)
= (
n=2
In general , by using the inductive principle , we get
n
To obtain the value , we make use of the fact that
= 1
∑
∑
∑
Since
, therefore sum of infinite GP series
∑
(
)
=
(
)
And hence
=
And (
)
= This expression gives the required probability distribution of
exactly n customers in the system.
Step 4
Obtain probability density function of waiting time
excluding service time distribution
The waiting time distribution of each customer in the steady
state is same, and it is a continuous random variable except that
there is a non zero probability that the delay will be zero, that is
waiting time is zero. Let w be the time required by the server to
serve all the customers present in the system at a particular time
in the steady state.
Let be the probability distribution function of w, that
is
P{w ≤ t} ,0 ≤ t ≤ ∞
If an arriving customer finds m(≥ 1) customers already in the
system, then in order for a customer to get service at a time
between 0 and t , all the customers must have been serviced by
time t .Let s1 ,s2,....,sm denote service times of m customers
respectively . Thus
{
∑
The distribution function of waiting time w for a customer who
has to wait
{
{∑
We know that
Probability density function of service time T = t is given by
s(t) = t > 0
Thus the expression for may be written as
= P(w ≤ t) ={
∑ ∫
= {
∑ ∫
= {
∫ ∑
{
∫
This shows that waiting time distribution is discontinuous at t=0
and continuous in the range 0 < t < ∞. Thus expression for
may also be written as
(μ λ) ׳
= λ
Step 5
Calculate busy period distribution
For the busy period distribution,let the random variable w
denote the total time that customer had to spend in the system .
Then the probability density function for the distribution is
given by
.
Where
{ }.
=
∫
=
∫
from [6]
=
= (μ-λ) t > 0
International Journal of Research in Advent Technology (IJRAT) Special Issue, January 2019 E-ISSN: 2321-9637
Available online at www.ijrat.org International Conference on Applied Mathematics and Bio-Inspired Computations
10th & 11th January 2019
53
Now∫ ∫
Which is the required distribution of busy period.
4.1 Application
Queuing theory in banking service
By means of the queuing theory, the bank queuing
problem is studied as the following aspects: In reality we have
waiting lines in the bank, there are several service stations. Each
service station has a queue or a waiting line. If each service
station has a queue according to their schedule, the arrival
customers join in each queue as the probability 1/2,known as the
two scheduled queue. For example, when there are two lines, the
system can be considered as two isolated M/M/1 systems, and
the arrival rate of each service station λ=λ/2. If there is a line,
the system will be M/M/2, L, Lq, W and Wq are calculated
respectively and compared to know which one is more efficient,
we will analysis it from a technical point as following:
When there is a line, z=2, λ=50, μ=40, ρ=5/4
∑
]
-1
= 0.23
(
)
= 0.801
= 2.051
=0.041
=0.016
When there are two lines, λ=
=25, μ = 40, ρ =
=1.667
=1.041
=0.067
= 0.041
Similarly, the calculation is same for, when the lines are three,
four and five.
When there are n lines: it is means that there are n service
stations, each service station has a queue based on their
schedule, each arrival customer joins in each queue at the
probability 1/n. It is called as scheduled n queues. The mean
arrival rate is λ/n, the mean service rate is μ.
Expected number of customers in the system:
=
Expected number of the customers waiting on the queue:
=
Average time a customer spends in the system:
=
Expected waiting time of customers in the queue:
=
we can see that, in the case of two lines, the waiting time in
the system is 0.067 and at a line, it is 0.041. The staying time is
decreasing clearly, and the length of line is also less. This shows
that in banking services, in terms if “first come, first serve” the
principle of fairness or technically, a line is better than more
lines, so bank managers should have the attention on this
problem.
Optimal Service Station:
In order to guarantee the quality of service, we set up the
number of service stations. If we need customers need to line up
no more than 10% how many service stations should be setup.
When z=2, λ=50, μ=40, ρ=5/4
∑
(
)
]
-1
= 0.230.
C(z ,ρ) =∑
=
Optimal Service Rate: Here we only studied the condition of one service station that is;
consider the model M/M/1/∞. To determine the particular level
of service, which minimizes the total cost of providing service
and waiting for that service.
Let Cw = expected waiting cost/unit/unit time.
Ls = expected (average) number of units in the system
Cs = cost of servicing one unit.
Expected waiting cost per unit time,
Expected service cost per unit time is
Total cost, C= (
)
This will be minimum if
International Journal of Research in Advent Technology (IJRAT) Special Issue, January 2019 E-ISSN: 2321-9637
Available online at www.ijrat.org International Conference on Applied Mathematics and Bio-Inspired Computations
10th & 11th January 2019
54
The optimal service rate is
With which we can find out the optimal services, to improve the
efficiency of our service.
5. CONCLUSION The efficiency of commercial banks is improved by the
following three measures: the queuing number, the service
stations number and the optimal service rate are investigated by
means of queuing theory. By the example, the results are
effective and practical. The time of customer queuing is
reduced. The customer satisfaction is increased. It was proved
that this optimal model of the queuing is feasible.
REFERENCES
[1] Toshiba Sheikh, Sanjay Kumar Singh, Anil Kumar
Kashyap, Application Of Queuing Theory For The
Improvement Of Bank Service.
[2] Kantiswarup, Operations Research, Sulthan chand and
sons publications, New Delhi.
[3] Qun Zhang & Zhonghui Dong Zhian, “Dynamic
Optimization of Commercial Bank Major Channels,”
Fourth International Conference on Business
Intelligence and Financial Engineering,pp. 627-630,
2011.
[4] Yu-Bo WANG, Cheng QIAN and Jin-De CAO
“Optimized M/M/c Model and Simulation for Bank
Queuing System”, IEEE,2010.