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A Study on Queuing Models in Banking Services
S.Dhivya Priya
1Department of Mathematics1, Dhanalakshmi Srinivasan Engineering College,Perambalur1 [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)
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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
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In general , by using the inductive principle , we get
n
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. between 0 and t , all the customers must have been serviced by
time t .Let s1 ,s2,....,sm denote service times of m customers
The distribution function of waiting time w for a customer who has to wait
Probability density function of service time T = t is given by s(t) = t > 0
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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 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
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.
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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
