a study on queuing models in banking services id-10.pdf · describes an elementary queuing system...

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

[email protected]

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.




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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

(

)




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




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




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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.

a study on queuing models in banking services id-10.pdf · describes an elementary queuing system...

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