Application of Queuing Systems with Many Classes of Customers for Structural Optimisation of Banks

Copyright © IFAC Management and Control of Production and Logistics, Carnpinas, SP, Brazil, 1997

APPLICATION OF QUEUING SYSTEMS

WITH MANY CLASSES OF CUSTOMERS FOR STRUCTURAL OPTIMISA nON OF BANKS

Boguslaw Filipowicz*, Boguslaw Bieda**,

*AGH-University ofMining and Metallurgy Institute ofAutomatics. AI. Mickiewicza 30. 30-059 KrakOw. Poland. faxltf048 34 15 68. e-mail:[email protected]. **AGH-University ofMining and Metallurgy Management Faculty. Al. Mickiewicza 30. 30-059 Krak6w. Poland

Abstract: A significant increase of demand for various services is one of the characteristics of the country economic development. In all the service processes service quality has become a predominant factor. Lengthening of time of a customer service due to firms ' and companies' mismanagement may be the reason for their smaller profits. The aim of this paper is to show the usability of a model based on the queuing system theory in the improvement of operations within a given bank. A model of queuing networks with many classes of customers has been worked out on the basis of a selected bank. Copyright © 1998lFAC Keywords: queuing systems, queuing network with many classes of customers

1. INTRODUCTION

investigation without the application of a model system seems to be impossible. Model researches are easier to carry out than experimental ones. In case of the former a model is not at risk.

A growing demand for various types of customer services in such areas as money turnover, passenger and freight transport, co-operation between companies, banking system development. is one of the critical features of economic reform in each country.

Model application makes it practically possible to compare decision ' s variants, choose the most appropriate solution according to the quality criteria, and to introduce a decision game which enables to make predictions of system behaviour after some decisions have been taken. The main objective of the paper is to show usefulness of a modeL based on a queuing network theory, in the improvement of interbank operations.

A predominant problem of service quality i.e. meeting customers' needs arises during such processes. Inefficient organisation has a critical impact on service realisation time which, in turn. can lead to significant reduction of companies' time profitability. Practice shows that some technical and economic problems are so complex. that their

149

2.1. An open queuing network with a Single class of customers

Queuing networks are tools which allow to observe interactions between components of a system, which can be used to mark and predict such parameters of a system efficiency as: a level of resources usage, capacity, queuing mean length, queuing mean time etc.

An open queuing network consists of service stations marked by symbols 1,2,,,.,M and two fictitious stations: 0 - a source of customers and exits (M+ I). An exit (M+ I) is fictitiously con~ected with an entrance 0 , which is sho\\'n by the conservation law of a customers flow. A queuing network is represented by a determined graph, in which nodes are service elementary systems, called stations, and edges show customer possible routes from one station to another when service is in progress. Numbers above the edges give probability values for a given route (Gelenbe and Pujolle, 1986: Walrand, 1989).

A queuing network model with many classes of customers has been designed for a chosen bank. Customers categorisation allows to obtain information which is necessary to evaluate the quality of customer service. Due to the model application the organisational improvements were introduced what will lead to the betterment of the service.

2. BASIC NOTIONS OF AN OPEN QUEUING NETWORKS THEORY

If Ym represents a number of customers in the m-th station, then

Nowadays a rapid development of a queuing networks theory and its practical application can be broadly observed. Many monographs and articles on queuing networks with a single or many classes of customers have been published. Among many authors who work on a queuing networks theory the following are worth mentioning: lackson J.R. , Gordon WJ., Newell G.F. , Chang A , Lavenberg S.S., Basket F., Chandy K M, Muntz P,. Palacios F.G., Kelly F.P, Fdida S., Pujolle G.

It is assumed that the matrix P is independent of a network state and, that:

Let us introduce the basic elements of a queuing network theory. Each queuing system [Fig. I] consists in three parts: - arriving process (customers' arrival) - queuing -up, service process at one or more channels, where customer' needs are realised.

---+--

queue

P i ,O

=0

for

i=O, .. . ,M

PM+l ,i

=0

for

j = 0, ... ,(M +1)

PO,M+l

=0

and

PM+l ,O

(2)

=1

A quantity e=(n p n 2 ,,,. ,n M ), where ni ~O , i = 1, ... , M is a number of customers in the ,/' -th station which belongs to the network, is called a state of a queuing network with single class of customers. Notions for the network with single class of customers are also used for networks with many classes of customers.

Only open queuing networks with a single or many classes of customers are able to map service processes properly.

arri val

defines Markov chain with a

matrix of routes

Naturally, the following question arises: why should a bank be analysed according to queuing network models? An answer to this question results from the following factors: customers' random arrival at a bank, heterogenous service demand and a complex bank structure.

customers'

{Ym } m

2.2. Open queuing networks with many classes of customers

Let us assume that R classes of customers can simultaneously stay in a queuing network. The customers' classes can change when the customers move among service stations within the network according to the routes matrix Pc, which for an open network with many classes of customers can be shown as

sc",icc channel (customers exit)

_-.L...LJII0

pcP =[

Fig. 1. Queuing system diagram

. ,]

' ,C;; ,C

In practice, queuing systems are connected with one another and make a network, in which customers' complex services are realised. A bank is a case when only open networks can be considered.

where

for

{O

i , j $, M + 1 $, c, C'$, R

$,

1

(3)

P . ' . is a probability of the event in which l,C,j .C

a customer from the class c, leaves the station " i" immediately after the service has been completed

150

and arrives at the station j . from the class c.

still as the customer

A set of variants of existing pairs E

= {(i, c)}has

for type-3 , where a notion "type" means classification of stations according to categorisation of BCMP see (Filipowicz, 1996).

to be defined for each application. It can be proved that the matrix Pc is a stochastic one because

(4)

"" . , =1 £..J Pl,C;j,C

If the probabilities of occurrence of every state of a network are known, then such parameters as mean quantity of customers who belong to a certain class in each station of network can be calculated.

(i .e')

If the matrix P c is known, we can calculate particular customers' arrival flows and categorise them by solving a matrix equation:

3. A DESCRIPTION OF A NETWORK MODEL

(5)

where

X

= [xi •e }

O~i,j~M+1; l~c~R

A branch of a national bank was the subject of a synthesis of an open queuing network model with many classes of customers, This baTIk operates in a building with several floors with transaction halls and pay-desks (cashier-offices). A transaction hall on the ground floor is for those customers whose transactions are below 500 PLN (new polish zloty) , Transactions above 500 PLN require supervision and are conducted on the first floor. All the pay-desks function according to a post office system in which a customer approaches a given pay-desk. where a cashier confirms bank operations. Service organisation on the first floor is different. From a check in point each customer is marshalled to a vacant pay -desk. Only exchange office works independently. Identification of customer routes resulted in the specification of pay-desks their allocation to particular queuing systems. [Table I] .

is a vector of an arrival flow in the i-th station of the class c. Now let us quote BCMP theorem (Baskett, Chandy, Muntz, Palacios) for open queuing networks with many classes of customers.

2.3. BCh.1P theorem (R classes of customers)

Assumption: An open queuing network with R classes of customers is given. A condition of network is defined as e = (n1 ,n 2, .. .n M) and all the conditions are feasible , because the network is open:

{eln i ~ 0 optional for i = 1,2, .. , M} Thesis: A probability of the event, that the network will be in the state e equals

Analysis of customers flow out of stations is based mainly on the data collected at pay-desks, and revealed by cashiers. Typical routes of customers were indicated by the bank staff . The routes visited rarely, were omitted in this model. It presents only those routes, which were occupied by at least 10% of customers moving from one station to another.

M

p(e)=ITpi(nJ

(6)

1=1

but

for type-I , m1

An open queuing network, which represents a customer flow between particular pay-desks, is shown in Fig. 2.

=1

One pay-desk or a group of them, in which a given set of operations is realised, is presented in Fig. 2 by service stations in which the number of channels equals the number of pay-desks. Each channel works according to FIFO queuing discipline to minimise a queue. Since the data obtained from the bank documents were insufficient to make structural and parametric identification of the bank, complementary observations were made and then a statistical analysis of the results was performed. Five classes of customers were specified during observation, but it has to be said that customers can change their attachment to a class when they move from one station to another. for type-2 and type-4, 151

Operation~=~===-_~===:JIC~2~ 3 4pay desks __IJcNumrer f ==-~~5~~

Station [%]

No

Typ

e

operations

2 3

90%

4 5 6 7

10%

8 9

rwming personal account savings bank books located bank books bonus flat savings bank books current bank accounts foreign exchange others others

clail

0

Numrer

87 8 15

ope [%)raltons

163

12

+

+

9,1 13 01 14 1 16 17 18 + +

476

35

+

+

+

+

+

+

+

+

+

+

°

3 4: 5, 7

6

12

+

14 245

18

55

4

14

1

40 140

3

Ciround·Floor

+ +

°

152

+

+

+ - +

not considered - - - - - - not considered - - - - - - -

Fir.:t-Flooc

Fig 2 DOlagram of network with many classes of custo mers °

of 1 2

ope
A set of eXIstmg pairs, which is necessalJ to define matrix of routes Pc, is as follows :

The model of network consists of eight stations and a customer flux is decomposed into five classes. All the stations in the network are of an exponential character and contain one or more service channels (with identical service intensity f1.) which work according to FIFO queuing discipline.

E = {(I,l), (1,5), (2,1,), (2,5), (3,1), (3,2), (3,4), (3,51

(4,11 (4,2), (4,4), (4,5), (5,1), (5,21 (5,3), (5,4), (5,5~ (6,1 ~ (6,2), (6,3), (6,5), (7,2), (7,5), (8,1)}

Let us assume that the customer service time does not depend on a class number. Table 2. Matrix of transitions for network of manv classes of customers

.......

....

4 I 2 1, 1 1,5 2.1 2, 5 3,1 3,2 3, 4 3,5 4,1 4, 2 4,4 4, 5 S, I (0,1) liS 0 0 0 0 0 3110 0 0 1110 0 0 III (0,2) 0 1n0 0 0 9110 0 0 0 0 0 0 (0,4) 0 0 0 0 0 4/S 0 0 lIS 0 0 (0, S) 0 1110 0 0 0 315 1120 0 I" 0 0 I (1,1) 0 lIS 0 0 0 0 0 0 (D) 0 0 !f10 0 0 0 0 0 In In 0 2 (2 , I) 0 0 0 0 0 0 !f10 (2,S) 0 0 3 (3. J) I,. III 0 0 0 III 0 (3,2) 0 0 0 0 0 (f.4) 0 0 0 0 (3, S) 0 0 7110 0 115 0 1110 0 4 (4, 1) 0 0 0 114 0 0 (4, 2) 0 0 0 0 0 0 (4, 4) 0 0 0 0 0 0 (4, S) 0 0 0 0 0 0 0 0 s (5, 1) 0 0 0 (5.2) 0 0 0 0 (5,3) 0 0 0 0 0 0 (5,4) 0 0 0 0 0 (5, S) 0 0 0 0 (6, 1) 0 0 6 0 0 (6,2) 0 0 0 0 0 0 (6, 3) 0 0 0 0 0 0 '(6.5) 0 0 0 0 7 (7,2) 0 0 0 0 0 (7, S) 0 0 0 0 0 0 8 (I, I) 0 0 0 0 0 0

6

S,2 S,3 0 0

0 114 0

5,4 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0

0

0 0

0 0 0

0 0

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 114

2n 0

0 0 0 0

0

...

A choice of the best organisation variants, using a network model and different modifications of a system structure (a number of class) and parameters concerning both the time of service and flux of customers who belong to the certain class, can be done To compare all the variants and to choose the best one, parameters which describe a network operation can be used:

5, S 6, 1 6,2 6, 3 6,5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3/4 3,. 0 0 0 2n sn

'0

c:

0

0 0

0

7 7, 2 7, 5 1, 1 9,1 0 0 1120 0 0 0 0 0 0 41S 0 0 0 5n 0 0 0 111 0

9,2 9,4 9,5 0 0 0 0 0 0

0 0 0 0

0 0 415 0 0 0

415 0 0

0 0 0

0 0

......

E :>

z

0

0 0

0 0

0 115

lIS

3,. 0 0

m

0

0

700

• class 5

...

.!:I

9/10

0 0 0 0

'" 600 E '0 ..... 0

9/10

500

• class 4

400

200

o class 3 o class 2

100

~ class

300

I

0

S.6 S.7 S.8 Number of station

As a result of calculations, the following parameters describing a network were determined: mean customer flows in each station with customers and customers categorisation parameters of usage of a particular station mean times of customers' stay in a given class, in a particular station, and in the network, mean numbers of customers in a given class in a particular station. Matrix of routes Pc corresponding to the network presented in Fig. 2, is shown in Table 2. Results of the analysis of a possible variant of network configuration is presented in Table 3 or graphically in Fig 3+6.

Fig. 3. Total number of demands 4,5 ..c:

ti. c:

~

., (ij

E bO

• class 5

4

3,5 3 2,5 2

• class 4

o class 3

.::

1,5 I

o class 2

::: CY

0,5 0

~

...

;j

class I

S.l S.2 S.3 S.4 S.5 S.6 S.7 S.8 Number of station

Fig. 4. Customer queuing mean length

153

Table 3. Simulation results of a selected organisational variant of network with manv classes of customers

Station

(1,1) (1 ,5) 1 (2 ,1) (2,5) 2 (3,1) (3,2) (3 ,4) (3.5) 3 (4,1) (4.2) (4.4) (4.5) 4 (5.l) (5 .2) (5 .3) (5.4) (5 .5) 5 (6.1 ) (6.2) (6.3) (6.5) 6 (7,2) (7,5) 7 8

Number of seIViced demands 149.75 16.125 165.875 363.75 34.375 398.125 66 16.625 2.875 8.875 94.375 109.375 261.75 17.5 144.875 533.5 25.125 62.25 88.625 17.375 41.25 234.625 78.875 187.875 232.875 97.625 597.25 200.625 136.75 337.375 39

Total number of demands 159.875 17.375 177.25 394.125 37.75 431.875 71.25 17.75 3 9.75 101.75 109.625 262.75 17.5 145.25 535.125 25.25 63 .25 90.5 17.5 41.5 238 84.125 198.5 246.875 103.375 632.875 203.25 138.875 342 .125 44

Number of refused demands 7.75 0.75 8.5 24.375 2.5 26.875 4.75 0.75 0 0.5 6 0.25 0.625 0 0.25 1.125 0 0.5 0.625 0 0.25 1.375 4 7.25 10 4.875 26.125 1.75 1.5 3.25 4

7

·r..,

6

5

• class 4

0

4

o class 3 o class 2

'" E

2

r1

~class

S.l S.2 S.3 S.4 S.5 S.6 S.7 S.8 Number of station

4

"'"

.,,

• class 4

2

o class 3 o class 2

.LJ

;:;

Z

~c1ass

o

5. REFERENCES

• class 5

~

I:

1.37 0.153 1.523 3.322 0.312 3.634 0.444 0.11 0.023 0.068 0.645 0.09 0.226 0.014 0.119 0.448 0.202 0.473 0.675 0.133 0.29 1.772 0.581 1.381 1.774 0.675 4.412 0.311 0.223 0.534 0.53

Network models today are applied fairly often. They have become useful tools for making and verifying decisions which are conducive to improvement in organisational structure of administration, health service, army etc. The paper presents a new area of application of the theory of queuing network with many classes of customers: structural optirnisation of a bank.

I

"'§'""

'0

System occupancy

4. CONCLUSIONS

Fig. 5. Queuing mean time

..,E

Waiting mean time 4.581 4.326 4.573 6.332 6.8 6.374 5.849 5.56 8.475 4.468 5.738 0.312 0.345 0.443 0.474 0.376 1.121 1.231 1.273 l.l27 1.509 1.278 4.092 3.941 3.89 4.191 3.977 0.848 0.911 0.874 5.43

• class 5

!=

0

Queuing mean length 1.263 0.131 1.394 4.184 0.417 4.601 0.723 0.164 0.041 0.077 1.006 0.058 0.153 0.013 0.114 0.338 0.049 0.134 0.197 0.033 0.104 0.518 0.592 1.309 1.606 0.738 4.245 0.294 0.214 0.509 0.457

The results were confirmed in a simulation using a file designed for simulation of a queuing network with many classes of customers.

9

8

~ '-

Number of non-accepted demands 2.375 0.5 2.875 6 0.875 6.875 0.5 0.375 0.125 0.375 1.375 0 0.375 0 0.125 0.5 0.125 0.5 1.25 0.125 0 2 1.25 3.375 4 0.875 9.5 0.875 0.625 1.5

Filipowicz B. (1996). Stochastic Models in Operation Research. Analysis and Synthesis of the QueUing Systems and Networks. WNT, Warszawa (in Polish). Gelenbe E. , Pujolle. G. (1986). introduction to queuing networks. John Wiley, New York. Walrand J (1989). A introduction to of the Queuing networks. Prentice-Hall, Englewood Cliffs, New York.

I

S. l S.2 S.3 S.4 S.5 S.6 S.7 S.8 Number of station

Fig. 6. System occupancy

154