The Jackson Networks As a Quite Good Solution For Analysing The Financial
Analytical Service Facilities
Vladimir Simovic, Savo Vojnovic
University of Zagreb, Police College,
Avenija Gojka Suska 1, HR

10000 Zagreb
The Republic of Croatia
Tel: +385 1 23
9 1341; Fax: +385 1 239 1415
E

mail:
vladimir.simovic@zg.tel.hr
Branko Kovacevic
University of Zagreb, Faculty of Economics
Kennedyev trg 6, HR

10000 Zagreb
The Republic of Croatia
Tel: +385 1 239 1341
; Fax: +385 1 239 1415
E

mail: bkovac@oliver.efzg.hr
ABSTRACT:
Objective of this work is to explain the modelling concept of the financial analytical
function in the financial knowledge discovery model. We discuss the queuing networks with infinite
queue
s in series and especially Jackson networks with m financial analytical service facilities as a
quite good solution for analysing each service facility independently of other financial analytical
service facilities. Also, in this work we discuss a conceptu
al solution for reducing the sum of costs
(as a function of financial analytical service cost and cost of analytical waiting). To solve the large
number of simulations we used the queuing M/M/s model with priorities that is based on a Poisson
input (for ex
ponential financial information inter

arrival time) and exponential output (for financial
analytical service time), and which is mainly based on the birth

and

death process as a special type
of continuous time Markov chain. An application of this model inc
reases the group effectiveness,
efficiency, and quality of the operational and strategic financial market investigative operations
that are in usage during the whole financial knowledge discovery process.
Keywords:
applied Jackson networks, financial ana
lytical function
1
INTRODUCTION
This work is a short explanation of the main Operations Research (OR) concepts and results, which
are accomplished during the computing process of the modern financial analytical function (based
on queuing networks), which
is also a solid base for all simulation and modelling works in the
future. Here are some remarks about “application of queuing theory and queuing networks”, with
comments about financial analytical systems, based on “infinite queues in series and Jackson
N
etworks". An illustrative example is presented at the end.
2
THE APPLICATION OF QUEUING THEORY AND QUEUING NETWORKS
2.1 Analytical queuing theory
How does analytical queuing theory contribute important information required for achieving an
economic (or ot
her) balance between the cost of some financial analytical service and the cost
associated with waiting for that financial analytical service? Indirectly, by predicting various
characteristics of the analytical waiting line (such as the average waiting tim
e, and similar).
Providing too much financial analytical service involves excessive analytical costs. In the opposite
situation we have almost the same problem. Not providing enough financial analytical service
capacity causes the waiting line to become ex
cessively long at times. This is costly in a sense,
whether it is a financial or a social cost, the cost of lost financial analytical information, or the
analytical cost of idle analytical employees and analytical servers. Because of that, analytical
queui
ng theory is a specific concept, which involves the mathematical study of analytical queues
and analytical waiting lines. The basic process assumed by this queuing model is the following
(Figure 1). Customers are various data and information (in special an
alytical input forms) requiring
(mainly) financial analytical service and generated over time, by an input source (financial events of
the real world). These special analytical input forms (data and information about the financial
world) enter the queuing
system (analytical information queuing system of the whole financial
analytical process) and join an analytical queue. At certain times, an analytical queue member is
selected for analytical service by a rule known as the analytical queue discipline. The r
equired
analytical service is then performed for entered data and information (concerned financial business
spheres) by the analytical service mechanism (of the whole financial analytical process), after which
entered data and information leaves the analyt
ical information queuing system.
Analytical information
Queuing systems
Input
Analytical data
Queue
Service
Served analytical
Sources
and information
System
Mechanism
d
ata and information
Channels
Facilities
Figure 1. The basic analytical information queuing systems
Financial events of the real world as the input source are usually assumed as an infinite (unlimited)
rather then a finite (limited) popu
lation, because the input source size is the total number of distinct
potential data and information that might require analytical service from time to time. The
population, from which arrivals come, is referred to as the calling population. The number of
data
and information in the financial analytical information queuing system significantly affects the rate
at which the input source generates new potential data and information. The assumption is that the
statistical pattern by which the number of data an
d information are generated until any specific time
(or over time) has a Poisson distribution. This case is the one where arrivals to the financial
analytical information queuing system occur randomly, but at a certain fixed mean rate, regardless
of how ma
ny data and information already are there (the size of the input source is infinite). The
probability distribution of the time between consecutive arrivals is an exponential distribution, and
is referred as interarrival time.
For this queuing model the qu
eue is an infinite queue, and the analytical priority

discipline is based
on analytical priority system. The analytical order in which members of the queue are selected for
analytical service is based on their assigned priorities. Their assigned priorities
are influenced with
the financial analysis phase and with the mark of the financial analytical "data contents &
information source evaluation system" (so called «4x4x2» evaluation system). The financial
analytical "data contents & information source evalu
ation system" can be viewed like a conceptual
tool for reducing the entropy of the modern financial analytical function. Modern financial
analytical "data contents & information source evaluation system" («4x4x2» evaluation system) has
very specific criter
ion, which in reality deals with minimally (4
4
2=) 32 linguistic variables for data
contents and information source evaluation purposes. Assigned priorities of the analytical queue
members are in fact something like: “1
st
class priority”, “2
nd
class prior
ity”, “…”, and “
N
class
priority”. The “1
st
class priority” has the highest priority and “
N
class priority” has the lowest. In
other words, data and information are selected to begin analytical service in the order of their
priority classes, but on a first

come

first

served basis within each priority class. A Poisson input
process and exponential analytical service times are assumed for each priority class, with restrictive
assumption that the analytical expected service time is the same for all priority cl
asses, and the
mean arrival rate differs among priority classes. Also, the rule is: once an analytical server has
begun serving a potential analytical data (or information), the analytical service must be completed
without interruption. This is referred to
as a non pre

emptive analytical queuing model, where a
potential analytical data (or information) being analytically served cannot be ejected back into the
queue (pre

empted) if higher

priority class data (or information) enters the analytical queuing
sys
tem. In fact the order in which a potential analytical data (or information) are analytically served
is different from the “normal” or FIFO (First

come

In

First

served

Out) basis queuing order. The
analytical service mechanism consists of one or more analy
tical service facilities, each of which has
one or more parallel analytical service channels, so called analytical servers. For more than one
analytical service facility, a potential analytical data (or information) may receive analytical service
from a se
quence of these analytical servers (service channels in series). A specific analytical
queuing model specifies the arrangement of the analytical service facilities and the number of
servers (parallel analytical service channels) at each one. Usually basic
models assume one
analytical service facility, with only one or a finite number of analytical servers.
2.2 Terminology and notation used by queuing theory
The following extension of standard terminology and notation was used:
state of analytical system
=
number of a analytical data (or information) in queuing system
analytical queue length
=
number of a analytical data (or information) waiting for service
(state of analytical system minus number of a analytical data
being analytically served)
N(t)
=
num
ber of a analytical data (or information) in queuing system
at time
t (t)
P
n
(t)
=
probability of exactly n analytical data (or information) in
queuing system at time t, given number at time 0
s
=
number of analytical servers (parallel analytical service
channels) in queuing
n
=
mean arrival rate (expected number of arrivals per unit time) of
new analytical data (or information) when n data (or
information) are in the queuing system
n
=
mean analytical service rate for overall financial analytical
sys
tem (expected number of data (or information) completing
analytical service per unit time) when n data (or information)
are in the queuing system (Note: that is combined rate at which
all busy analytical servers (serving data) achieve analytical
service c
ompletions)
constant
=
when the mean arrival rate
n
is a constant for all
n
constant
=
when the mean analytical service rate per busy analytical server
is a constant, for all
n
1
(Note: in this case,
n
= s
when
n
s
, or when all analytical ser
vers s are busy)
1/
=
expected interarrival time
1/
=
expected analytical service time
=
the utilization factor (
=
/s
) for the analytical service
facility, or the expected fraction of time the individual
analytical servers are busy
stead
y

state condition
of analytical system
=
specific state of analytical system that is reached after
sufficient time has elapsed (after transient condition of
analytical system is finished), and when analytical system is
essentially independent of the initi
al system state and elapsed
time (where the probability distribution of the state of the
analytical system remains the same)
P
n
=
probability of exactly n potential analytical data (or
information) in the analytical queuing system
L
=
expected number of
analytical data (or information) in the
analytical queuing system
L
n
=
expected analytical queue length (excludes analytical data or
information being served)
W
=
waiting time in analytical queuing system (includes analytical
service time) for each indi
vidual analytical data or information
W
k
=
steady

state or total expected waiting time in the whole
analytical system (including analytical service time, or
analytical supplying time), where
W = E(
W
)
and
W
k
is for a
member of priority class
k
, which is
k
= 1, 2, ... , N
W
q
=
waiting time in queue (excludes analytical service time) for
each individual analytical data or information, where
W
q
= E(
W
q
)
2.3 Mathematically based queuing model formulation
In relation to the proposed classification of the s
imulation models (see p. 14 in [12]) here we are
dealing with abstract, dynamic, discrete, and stochastic models with numerical and analytical end
solutions, which are primarily accomplished with a lot of discrete simulations
and with recording
data on a d
eveloped model (for more details see pp. 16

21 in [12]). We are using a simplified
explanation of the analytical function and basically complex multi

channel supplying system (for
analytical information). For better clarity, suppose that in all financial a
nalytical investigations we
are analytically dealing mainly with financial based information and data. Also, we are dealing with
a lot of “analytical information”, which are coming from legal sources (financial analytical process)
and in some order that is
proposed from well known “supplying theory” or “queuing theory” (see
[12]). To solve the large number of the stochastic simulations of the modern financial analytical
function we have used the queuing
M/M/s
model with non pre

emptive analytical priorities
, which is
based on a Poisson input (for exponential financial information interarrival time), and exponential
output (for financial analytical service time) that is mainly based on the birth

and

death process (as
a special type of continuous time Markovia
n chain). The queuing
M/M/s
model with non
pre

emptive analytical priorities assumes that both expected interarrival times (
1/
) and expected
analytical service times (
1/
) have an exponential distribution, and that number of analytical servers
is any pos
itive integer (
s
). The benefit of this new simulation model of the financial analytical
function is in the simple method (based mainly on statistical simulations) of measuring analytical
capacity and capability of analysis, which is now in usage in the fin
ancial field (and partly in field of
financial law). The financial analytical function of the financial analytical was prepared for
investigations of various financial events, financial markets, subjects or entities, and for financial
business operations c
ontrol methods, etc. The model of modern financial analytical function
(MFAF) has two basic analytical sub

systems: analytical service receivers (analytical function
clients) and service suppliers (analytical function servers). Simplified, analytical servi
ce receivers
(data or information) are coming in channels (with and without queues) with all kind of “analytical
data or information” (maybe) interesting for financial analytical investigations (and especially for
the financial analysis function). If there
are free analytical service suppliers, then analytical service
will be done in that moment. But, if there is no free analytical service supplier, service receivers are
waiting for the service in queues. After the analytical service request was accomplishe
d, analytical
service receivers (fully analytical prepared information’s) are leaving the analytical system, or
coming to the end of it (or to the specific destination point). The analytical supplying system can be
simple or complex, open or closed, one or
multi

channel, and also with and without priorities (see
pp. 108

175 in [12]). Simplified, MFAF is a complex multi

channel analytical supplying system
with priorities. It has minimally two or “n” analytical servers more then the classical financial
analyt
ical function, and it have a backward feedback sub

system (well known as «4x4x2» evaluation
sub

system) [2]. MFAF is functionally complex, has multi

channel structure, backward feedbacks,
tails with queuing and analytical information supplying sub

system w
ith non pre

emptive priorities.
That is reason why we are usually researching rather simplified (without backward feedbacks sub

system), but still complex multi

channel analytical supplying system with priorities.
Financial analytical service has added va
lue as the result of OR based analysing and interpretation.
It must be clear that the serious financial analytical practice is basically done with various experts,
analysts and scientific financial analytical departments and that financial analytical resou
rces are
always finite. The operational financial analytical practice with finite number of financial analytical
specialists and usually a lot of financial analytical cases during the same time period, produce a need
for parallel and network working. The w
hole financial analytical process has almost clear heuristics,
because intelligence is the resulting product from various systematically connected OR based
processes, like: estimation, collection, evaluation, collation, integration, analysing and interpret
ation
of data and information, development of hypotheses, dissemination of information, intelligence
acting, co

ordination and
automation (see [9

10]). During the research preserved we are not using
the elementary model
M/M/1
of the “queuing theory”, which
is in fact one

channel (one server)
analytical supplying system model with exponential distribution of inter

arrival times (of analytical
information) and of (analytical supplying) service times. We are using the specific
M/M/s
model (for
more details see
pp. 628

755 in [6]), which assumes: that all inter

arrival times are independently
and identically distributed, according to an exponential distribution (our input process is Poisson);
that all analytical service times are independently and identically di
stributed according to another
exponential distribution (our analytical service process is Poisson); and that the number of servers is
s
(any positive integer), but in the Croatian financial analytical practice and related analytical
function they vary fro
m a minimum
1
to maximum
7
. With the equal distribution of analytical
supplying time, with expected analytical service time about
1/
(
n
is mean analytical service rate
for overall system, or expected number of clients (data or information) completing ana
lytical service
per unit time), and with exponentially distributed inter

arrival time of analytical information at
expected average rate of
1/
(
n
is mean arrival rate, or expected number of arrivals per unit time),
that is the most simplified type of Mar
kovian analytical system with supposed infinite analytical
capacity (
Y =
), and with priorities in queue discipline (or without supposed FIFO queue
discipline). We are researching analytical financial analytical cases in which there are no
possibilities f
or any analytical closeness of multi

channels model of the analytical supplying
function, or when the utilisation factor for the analytical service facility is
s
< 1
< s
(because
s
=
/s
). For better clarity of the priorities concept, firstly we ar
e talking about an elementary
model of analytical supplying system
M/M/1
, which has a stationary state represented with these
relations (see pp. 122

123 in [12]):
P
n
=
n
(1

) ;
=
/
< 1 ;
P
o
= (1

) ;
L =
/ (1

) =
/ (

) ;
L
q
=
2
/ (1

)
=
2
/
(

)
;
0
1
0
1
)
(
1
t
for
e
t
for
t
W
t
q
W
q
=
/
(

)
;
W = 1 / (

) ; T
occupied
= 1 / (

) .
With introducing only two relative priority classes of the analytical supplying function in the same
M/M/1
model, we have analytical inform
ation with higher priority of relative analytical supplying
order, which have a mean arrival rate
1
(
i
priority class is equal
1
), and analytical information with
a lower priority of relative analytical supplying order, which have a mean arrival rate
2
(
i
priority
class is equal
2
). The parameter of their corporate (coupled) input exponential distribution is
, and
it is their arithmetic sum (
=
1
+
2
). The analytical supplying function is the same for both types
of analytical information, and has a me
an service rate
. Basic results for average measures of
success (in stationary state condition) can be represented with these relations (see pp. 126

128 in
[12]):
L
(1)
= (
1
/
) (1 +

1
/
) / (1

1
/
) ;
L
q
(1)
= (
1
/
) / (1

1
/
) ;
W
q
(1)
=
/
(

1
)
;
L
(2)
=
(
2
/
) (1

1
/
+
1
/
)
/
(1

) (1

1
/
)
;
L
q
(2)
= (
2
/
) /
(1

) (1

1
/
)
;
W
q
(2)
=
/
(

) (

1
)
;
P
n
=
n
(1

) for n > 0 ;
P
o
= (1

) ;
=
/
< 1 ;
L = L
(1)
+ L
(2)
;
L
q
= L
q
(1)
+ L
q
(2)
;
W
q
= (
1
/
) W
q
(1)
+ (
2
/
) W
q
(2)
;
W
q
(2)
/ W
q
(1)
=
/ (

) .
But in multi

channels model
M/M/s
we have priority sub

system with
N
(where
N = 1, 2, ... , k
)
relative priorities classes, and where
W
k
is the steady

state or total expec
ted waiting time in the
whole analytical system (including service time, or analytical supplying time). In stationary model
situations
W
k
can be explained in relation (see pp. 706

707 in [6]):
1
1
1
k
k
k
B
AB
W
, for a member of priority class
k
, whi
ch is
k = 1, 2, ... , N
,
where:
A = s!
s
j
r
r
s
s
j
j
s
1
0
!
,
B
0
= 1,
B
k
= 1
s
k
i
i
1
, for k = 1, 2, ... , N ,
s
= number of analytical servers;
= mean service rate per busy server;
1
= mean arrival rate for
priority class
i
, for
i
=1, 2, ... , N
;
1
=
N
i
i
1
;
r =
/
(what assume that:
s
k
i
i
1
).
The steady state expected number of members of priority class
k
in the queuing system (including
those being analytically served) is
L
k
, and it can be
explained in relation:
L
k
=
k
W
k
, for k = 1, 2, ... , N .
The expected waiting time in the queue (excluding service time) for priority class
k
is
W
q
(k)
, and it
can be explained in relation:
W
q
(k) = W
k
–
1 /
.
The corresponding expected queue length (“
tail length”
) is
L
q
(k)
, and it can be explained in relation:
L
q
(k)=
k
W
q
(k)
.
2.4 Queuing Networks
When customers must receive not a single service facility with one or more servers, but service at
some of or all these facilities we have “
networks of ser
vice facilities
” or “
queuing networks
”. It is
therefore necessary to study the entire financial analytical network to obtain such information as the
expected total waiting time, expected number of customers in the entire system, expected waiting
time, and
so similar. New term and property is “
equivalence property
” for the input process of
arriving customers and the output process of departing customers for certain financial analytical
queuing systems. Here we must assume that a service facility with
s
serve
rs and an infinite queue
has a Poisson input with parameter
,
and the same exponential service

time distribution with
parameter
for each server (the queuing
M/M/s
model), where
s
>
. Then the steady

state output
of this service facility is also a Poisso
n process, with parameter
. This property makes no
assumption about the type of queue discipline used (it can be of any sort), but the served customers
will leave the service facility according to a Poisson process. The crucial implication of this fact fo
r
analytical queuing network is that, if these customers must then go to another service facility for
further financial analytical service, this second facility also will have a Poisson input. With an
exponential service

time distribution, the equivalence
property will hold for this facility as well,
which can then provide a Poisson input for a third facility, etc. Here we have the consequences for
two basic kinds of networks: “
System of infinite queues in series
” and “
Jackson Networks
”.
2.5 System of infi
nite queues in series and Jackson Networks
When the customers must all receive service at a series of
m
service facilities in one fixed sequence,
and each facility has an infinite queue, the allowed situation is: no limitations on the number of
customers
in the queue, and the series of facilities form a system of “
infinite queues in series
”. Also,
in further situations the customers arrive at the first facility according to a Poisson process with
parameter
,
and each facility
i
(
j = 1, 2, 3, …, m
) has an
exponential service

time distribution with
parameter
i
for its
s
i
servers, where
s
i
i
>
. From the specific property named “
equivalence
property
”, it follows that each service facility has a Poisson input with parameter
, under the
steady

state conditions
. Therefore, the elementary
M/M/s
queuing model and its priority

discipline
counterparts can be used in the process of the whole financial analysis, to analyse each service
facility independently of the others. This is significant and tremendous simplifica
tion. Rather then
analysing interactions between facilities, now we are able to use the elementary
M/M/s
queuing
model to obtain all measures of performance for each facility independently. Consequently, the
probability of having
n
customers at a given fac
ility is given by the formula for
P
n
for the
elementary
M/M/s
queuing model. The “
joint probability
” of
n
1
customers at facility
1
,
n
2
customers
at facility
2
, … ,
n
m
customers at facility
m
, then, is the simple product of individual probabilities
obtained
in the simple way. It can be expressed like “
a product form solution
”:
P
(N
1
, N
2
, …, N
m
) = (n
1
, n
2
, … , n
m
)
=
1 2
m
n n n
P P P
.
Similarly, the expected total waiting time and the expected number of customers in the entire
financial analytical
system can be obtained merely summing the corresponding quantities obtained
as the respective facilities. This simplification doesn’t hold for the case of finite queues, which is
quite important in practice, because there is often a definite limitation on
the queue length in front
of classical service facilities in real and classical analytical networks (for example: classical
analytical networks without computers, without Intranet and Internet solutions). For such systems
the facilities must be analysed j
ointly instead, and only limited results can be obtained.
Another prominent kind of queuing network with “
a product form solution
” property is the Jackson
network, where the elementary
M/M/s
queuing model can be used to analyse each service facility
indep
endently of the others. This is a realistic solution and a quite good solution for analysing the
financial analytical service facilities, as a unique system. The characteristics of a Jackson network
are the same as assumed for the system of infinite queues
in series, except now the customers visit
the facilities in different orders, or even they may not visit them all. For each facility, its arriving
customers come from both, outside the financial analytical system (according to a Poisson input
process) and
the other facilities. It is expressed like the following construction, a Jackson network is
a system of
m
service facilities where facility
i
(
i = 1, 2, …, m
) has

an infinité queue,

customers arriving from outside the system,
according to a Poisson input
process with
parameter
a
i
,

s
i
servers with an exponential service

time distribution with parameter
i
.
A customer leaving facility
i
is routed to facility
j
(
j = 1, 2, … , m
) with probability
p
ij
or departs the
system with probability:
q
i
= 1

m
j
ij
1
p
.
Any such network has the following key property:

each facility
j
(
j = 1, 2, … , m
) in a Jackson network behaves as if it were an independent
M/M/s
queuing system with arrival rate:
j
= a
j
+
m
i
ij
i
1
p
, where
s
j
j
>
j
.
For
each facility
i
, its input processes from the various sources (outside and other facilities) are
independent Poisson processes, so the aggregate input process is a Poisson with parameter
i
. The
equivalence property than says that the aggregate output pro
cess for facility
i
must be a Poisson with
parameter
i
. Consequently by disaggregating this output process, the process for customers going
from facility
i
to facility
j
must be a Poisson with parameter
i
p
ij
, and this process becomes one of
the Poisson i
nput process for facility
j
, thereby helping to maintain the series of Poisson input
processes in the overall financial analytical system. The equation for obtaining
j
is based on the
fact that
i
is the departure rate, as well as the arrival rate for all
customers using facility
i
. Because
p
ij
is the proportion of customers departing from facility
i
who go next to facility
j
, the rate at which
customer from facility
i
arrives at facility
j
is
i
p
ij
. Summing this product over all
i
, and then adding
this su
m to
a
j
, gives the total arrival rate to facility
j
from all sources. To calculate
j
from this
equation requires knowing the
i
for
i
j
, but these
i
are also unknowns, given by the
corresponding equations. The procedure is to solve s
imultaneously for
1
,
2
, … ,
m
by obtaining
the simultaneous solution of the entire system of linear equations for
j = 1, 2, … , m
. This way
solving
j
and the whole computer

based simulation modelling design was prepared with the
program «ProbMod» and p
artly with the program «MathProg», which are McGraw

Hill modular
developed programs for various simulation solutions (see pp. 947

950 and accompanied disks in
[6]). That design can be controlled with modular simulation programs solutions (designed in
Fortr
an’77, Fortran’90 and Lahey language, and supported with C++ source code for: DOS,
QuickWin, Windows’95, Windows’98, Xwindow’X11 and NT Windows) taken from Springer

Verlag Compact Disk (see pp. 595

608 and accompanied CD in [4]).
3
ILLUSTRATIVE EXAMPLE
3.1
Analytical system like a Jackson network with five facilities (
m
) and ten servers (
s
i
)
Let us consider the financial analytical system which looks like a Jackson network with five
facilities (
m=5
) and ten servers (
s=10
), and that have parameters shown in
Table 1.
Table 1. Data for the illustrative example of the financial analytical system
structurally similar to a Jackson network with five facilities (
m=5
) and ten servers (
s=10
)
p
ij
Facility
j
s
j
j
a
j
i = 1
i = 2
i = 3
i = 4
i = 5
j = 1
1
10
1
0
0.1
0.4
0
0
j = 2
2
10
4
0.6
0
0.4
0
0
j = 3
1
10
3
0.3
0.3
0
0
0
j = 4
3
10
2
0
0
0
0.2
0.4
j = 5
3
10
3
0
0
0.6
0
0.4
3.2 Analytical results
Plugging the parameters into formula for
j
for
j = 1, 2, 3, 4, 5
(because
m = 5
), we must obtain
the
total arrival rate:
1
= 1
+ 0.1
2
+ 0.4
3
2
= 4 + 0.6
1
+ 0.4
3
3
= 3 + 0.3
1
+ 0.3
2
4
= 2 + 0.2
4
+ 0.4
5
5
= 3 + 0.6
3
+ 0.4
5
.
The simultaneous solution for the analytical system (given by the program «ProbMod») is:
1
= 5,
2
= 10,
3
= 7.5,
4
= 8.75,
5
= 12.5.
Each of the five service facilities now can be analysed independently, by using the formulas for the
elementary
M/M/s
queuing model given in section 2.3 (see “Mathematically based queuing model
formulation”). For example, to obtain the distribution of the number of customers
N
i
= n
i
at facility
i
, we must note that:
i
i
i
i
s
q
0.5
0.5
0.75
0.2917
0.4167
f or
f or
f or
f or
f or
5
4
3
2
1
i
i
i
i
i
.
Plugging these values and the parameters into formula for
P
n
we obtain:
1
n
P
= 0.5 (0.5)
1
n
for facility 1,
2
n
P
=
2
1
0.333
0.333
0.333 0.5
n
for
for
for
2
2
2
0,
1
2,
n
n
n
for facility 2,
3
n
P
= 0.25 (0.75)
3
n
for facility 3,
4
n
P
=
4
1
0.065
0.065
0.065 0.2917
n
for
for
for
4
4
4
0,
1
2,
n
n
n
for facility 4,
5
n
P
=
5
1
0.155
0.155
0.155 0.4167
n
for
for
for
5
5
5
0,
1
2,
n
n
n
for facility 5.
The joint probability of (
n
1
, n
2
, n
3
, n
4
, n
5
) then is given simply by the product form solution:
1 2 3 4 5
1 2 3 4 5 1 2 3 4 5
,,,,,,,,
n n n n n
P N N N N N n n n n n P P P P P
.
Similarly, the
expected number of customers
L
i
at facility
i
was calculated, and the results were:
L
1
= 1,
L
2
= 1.333333,
L
3
= 3,
L
4
= 0.901871,
L
5
= 1.361052.
Consequently, the expected total number of customers in the entire financial analytical system then
was:
L
= L
1
+ L
2
+ L
3
+ L
4
+ L
5
= 1 + 1.333333 + 3 + 0.901871 + 1.361052 = 7.596256
Obtaining
W
, as the expected total waiting time in the entire financial analytical system (including
service times) for a customer, was more complex, because we cannot simply add
the expected
waiting times at the respective facilities, because a customer does not necessarily visit each facility
exactly once. But, we used Little’s formula, where the system arrival rate
is the sum of the arrival
rates from outside to the facilitie
s, or:
= a
1
+ a
2
+ a
3
+ a
4
+ a
5
= 13
.
Thus, finally:
1 2 3 4 5
7.596256
a + a + a + a + a 13
L
W
0.584327
.
Also, there do exist other and more complicated kinds of queuing networks, where the individual
service facilities can be analysed independently from the others,
but the Jackson network represents
a quite good solution for analysing the financial analytical service facilities in the Croatian case.
4
CONCLUSION
In this work we describe how analytical queuing theory can be adopted and used to help design
effective
queuing financial analytical systems. Because queuing systems are prevalent throughout
financial analytical services and throughout financial analytical environment, we discuss the
queuing networks with infinite queues in series and especially Jackson net
works with
m
financial
analytical service facilities as a quite good solution for analysing of each service facility
independently of the others financial analytical service facilities. The adequacy of these systems can
have an important effect on the qual
ity of the whole financial analytical work and on the quality of
analysis of the financial analytical service facilities. Also, in this work, we discuss a conceptual
solution that can be used (in the near future) for reducing the sum of costs (as function
of financial
analytical service cost and cost of analytical waiting time). We use queuing systems by formulating
mathematical models of their operation, and then we use these models in practice to derive some
interesting measures of financial analytical sy
stem performance. To solve the large number of
simulations we used the queuing M/M/s model with priorities that is based on a Poisson input (for
exponential financial information inter

arrival time) and exponential output (for financial analytical
service
time), and which is mainly based on the birth

and

death process as a special type of
continuous time Markov chain. This analysis provides vital information for effectively designing
queuing systems that can achieve an appropriate balance between the cost o
f providing a financial
analytical service and the cost associated with waiting for that service. An application of this model
increases the group effectiveness, efficiency, and especially quality of the operational financial
market investigative operation
s that are in usage during the whole financial knowledge discovery
process. This conclusion and this work are a solid base for future financial analytical modelling and
other simulation processes that are connected with specific customers needs. Specially,
in similar
situations, when customers must receive financial analytical service at several different service
facilities.
5
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