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MINISTRY OF
EDUCATION AND TRAINING

MINISTRY OF
NATIONAL DEFENSE

ACADEMY OF MILITARY SCIENCE AND TECHNOLOGY
---------------------------

NGUYEN TRUNG DUNG

SOME TYPES OF QUEUE AND HANDLING PRINCIPLES
Speciality: Mathematical Foundation for Informatics
Code:
9460110.

SUMMARY OF MATHEMATICAL DOCTORAL THESIS

HA NOI - 2018


THE THESIS WAS COMPLETED AT ACADEMY OF MILITARY SCIENCE
AND TECHNOLOGY

Supperviors:
1. Senior Researcher.Dr Nguyen Hong Hai.
2. Dr Tran Quang Vinh.

Reviewer 1:
Assoc Prof. Dr Phan Viet Thu.
VNU University of Science.

Reviewer 2:
Assoc Prof. Dr Tran Nguyen Ngoc.
Military Technical Academy.

Reviewer 3:
Assoc Prof. Dr Ngo Quynh Thu.
Hanoi university of science and technology.

The thesis will be defended in front of the Doctoral Evaluating Council at
Academy of Military Science and Technology at
……/….../….../2018.

Thesis can be found at:
- Library of Academy of Military Science and Technology.
- Vietnam National Library.


1

INTRODUCTION
1. Rationale
1.1. Alongside with the development of science and technology, multiple
queueing network were created and practically applied in daily social life such
as telecommunication networks, computer networks, production system ... The
research, performance evaluation of these schemes is one of the most important
and complex issues. To study and evaluate the scheme, we can apply various
mathematical tools and one of the most important ones that can be used is
queueing theory and theoretical queueing networks.
1.2. Research findings which are aimed to determine the probability

distribution of queueing network state and performance parameters of the
network ... from authors all over the world had solely reached few results
corresponding to a queueing network conditions such as Poisson arrival flow,
service time of the network nodes according to random variables with
exponential distribution, queueing network operating at equilibrium. To
queueing network under general arrival flow assumption, service time of
network nodes are of randomly distributed variables, the researchers had just
restricted to determine the approximate probability distribution of the queueing
network state under certain conditions.
1.3. There are many problems from practical to theoretical issues which
require examination of queueing network model with more generous
assumptions of the queueing network structure, such as the assumption of
external job stream to service network; assumption of service duration, the
hypothesis of priority scheme; assumption of mechanism which build up
transitioned probability matrix within queueing network.
2. Research target: The thesis is aimed at general queueing network.
3. Research content: The author would like to study two layers of the
problems: the issue of determining job’s rotation process in queueing network
and related problem of state process at the nodes and queueing network.
4. Pratical and scientific meanings: Objectives and research targets have
been of interest and studied by many authors all over the world. The research
contents are proved to be practical.
5. Research Methods: Using the method of queueing theory and queueing
network, combining with several methodologies of probability theory and
mathematical statistics to research and solve certain important issues in the
general queueing network model.
6. Thesis structure: Besides the introduction; conclusion; published
scientific works; references; the thesis’ content is presented in three chapters as
follows:

2

Chapter 1. Some basic issues about queueing theories and queueing networks.
Chapter 2. The general multi-class network - Decomposing and
Synthesising algorithm.
Chapter 3. Evaluation on state process of general queueing network.
CHAPTER 1
SOME BASIC ISSUES ABOUT QUEUEING THEORIES AND
QUEUEING NETWORKS
Chapter 1 presents some of the knowledge that will be used for further
research on queueing network in chapter 2 and chapter 3 of this thesis. At the
same time, chapter 1 presents the research worlwide so far about the queueing
network, and then identifies the contents to be studied in the thesis.
1.1. RELATED PROBABILITY DEFINITIONS
In this part, the thesis shall present several basic and related probability
definitions such as random variables; distribution function of random variables;
characteristics of random variables ([5],[10],[44]).
1.2. MARKOV PROCESS
Within this section, the thesis shall present several definitions on Markov
process in relation to the thesis such as definition of stochastic process;
transition matrix of Markov chain; probability distribution of Markov chain;
steady state distribution and limit state distribution of discretetime Markov
chains ([9],[10],[34],[36]).
1.3. QUEUEING THEORY AND QUEUEING NETWORK
Herein this part, the thesis shall present basic definitions relating to
queueing
theory
and
queueing

network
theory
([19],[20],[29],[32],[33],[34],[36],[47]).
1.3.1. Queueing
Mathematically, queueing is described by:
A / B / m / K − mechanism of service piority
with
Period times between two continuous coming jobs are random variables
which have the same distribution, and this distribution is symbolized by A;
times for serving jobs are random variables which have the same distribution,
and this distribution is symbolized by B; m is the number of server ( m  1 ); K is
the size of queueing (the maximum jobs that the queueing can have).
The following symbos are often used with A và B : M (exponential
distribution); Ek , (Erlang distribution with parameters k ,  ); D (degenerate
distribution (time for serving job is constant)); G (general distribution).


3

Mechanism of service piority: There are some mechanisms of service
piority wich are often used, such as: FIFO(First in first out); FCFS (First come
first served); LCFS (Last come first served); SIRO (service in random order)
and so on.
Some performance parameters of the queue:
Probability distribution of queueing state; throughput of the queue; average
time that a job maintain in the queue; average jobs that are inside the queue and
so on.
Definition 1.13. A queue is working in balance if the total rate of coming
jobs is equal with the total rate of leaving jobs.
1.3.2. Queueing network

Definition 1.14. A queueing network is working in balance when all nodes
are working in balance.
a. Single-class queueing network
A one layer queueing network (all jobs are belong to one class) is featured
by the following components:
- N (or J ): number of nodes; ki (or xi ): number of jobs inside a node

(

)

i i = 1, N and is called state of node i ; ( k1 ,..., k N ) : state of queueing network.

- mi : number of servers which can work parallel inside node i ; i : serving
rate of node i ; 1  : average serving time of node i .
i

- pij : transition probability that a job moves from node i to node j

(i, j = 0, N )

with: p0j is the probability that a job comes from outside the

queueing network to node j và pi 0 is the probability that a job leaves the
N

network after being served at node i ( pi 0 = 1 −  pij ).
j =1

- 0i : coming rate of jobs from outside network that move to node i .

N

i =   ji : coming rate of jobs from outside node i to node i, with: 0i coming
j =0

rate of jobs from outside network to node i ,  ji is the coming rate of jobs
from node j ( j = 1, N , j  i ) to node i and ii is the coming rate of jobs from
node i return to node i .
b. Multi-class queueing network
Multilayer queueing network is featured by the following components:
- N (or J ): number of nodes in the network; R : number of job-class.


4

- kir ( r = 1, R ) : number of jobs in class r that is in node i at the reviewing
N

time; K r =  kir : number of jobs in class r that is in the network.
i =1

- Si = ( ki1 ,..., kiR ) : state of node i . S = ( S1 ,..., S N ) : state of the queueing
network.
- ir : serving rate for job class r in node i .
- pir,js : transition probability that a job in class r moves from node i to
node j and becomes job in class s ; p0,js is transition probability that a job
from outside network moves to node j and becomes job in class s; pir,0 is the
probability that a job class r from node i leaves the network.
-  : total coming rate of jobs from outside network that moves to nodes;
0,ir =  p0,ir : coming rate of jobs from outside network that move to node i and

becomes job in class r ; ir : coming rate of jobs that move to node i and
becomes job in class r .
Some working performance parameters of queueing network
With the definition: The state of node i at time t is the number of jobs that
inside node i at time t , we have some working performance parameters such as:
Probability distribution of node state and of queueing network; probability of
blocking traffic; Throughtput of node and throughput of queueing network;
Average number of jobs inside a node, a network and so on.
d. Applications of Queuing theory in telecommunications networks,
computer networks
Upon review, assessing the operation of telecommunication networks and
computer networks, we are particularly interested in elements of traffic data in
the network transmission and is based on the mathematical tools including
mathematical tools probability theory and stochastic process theory are two
important mathematical tools to review and evaluate the operation of
telecommunication networks and computer networks.
1.4. The national and international research on queueing networks
Research findings which are aimed to determine performance parameters of
the network such as: probability distribution of node state and of queueing
network; probability of blocking traffic; throughtput of node and throughput of
queueing network... from authors all over the world had solely reached few
results corresponding to a queueing network conditions such as Poisson arrival
flow, service time of the network nodes according to random variables with
exponential distribution, queueing network operating at equilibrium. To


5

queueing network under general arrival flow assumption, service time of
network nodes are of randomly distributed variables, the researchers had just

restricted to determine the approximate probability distribution of the queueing
network state under certain conditions.
Practicality and theory requires reviewing the queue network models with
broader assumptions. Thus, the thesis focuses on two classes of problems: the
problem class examining the state of the nodes, the state process of the
queueing network, and the problem class determining the process of job flow in
the network with the research object being the general queueing network.
Conclusion of chapter 1
In this chapter, the thesis presents several concepts of probability and the
theory of Markov processes as well as queueing theory, queueing network. At
the same time, chapter 1 presents the research worlwide so far about the
queueing network, the open problems in the queueing network models which
have been published. Based on that the thesis identifies two classes of problems
to be studied which are the problem class examining the state of the nodes, the
state process of the queueing network, and the problem class determining the
process of job flow in the network with the research object being the general
queueing network. The content which was discussed in chapter 1 shall be
applied for further research on queuing network in chapter 2 and chapter 3 of
this thesis.
CHAPTER 2
THE GENERAL MULTI-CLASS NETWORK- DECOMPOSING AND
SYNTHESISING ALGORITHM
Chapter 2 uses the results presented in articles [2], [3] on the list of
published works. The movement of jobs in the queueing network is the number
1 concern in the research on the queues and queueing networks. With any
queueing network, in theory, the external job flow can enter any node in the
network, the job flow after leaving a node can enter another node in the
netnwork or can go outside the network and at the same time between two
nodes i and j may occur: job a moves from node i to node j, job b moves from
node j to node i. Thus, the study of job flow in the queueing network is very

complicated. In this thesis, we propose decomposing and synthesising
techniques to study the job flow in the queueing network.
2.1. Decomposing general queueing network into component networks
For any queueing networks, the job flows between the nodes of the network
interweave each other in different directions.


6

With the queueing network denoted as ( i, j ) ( i, j = 1, J ; J 

+

) , the job flow in

the queueing network ( i, j ) is described by routing probability matrix
P( i , j ) (t ) =  pk( i,l, j ) (t )  k ,l =0, J in which pk( i,l, j ) (t ) is the routing probability of the job

moving from node k to node l in the queueing network ( i, j ) at time t
( k , l = 1, J ) , p0,(ik, j ) (t ) is the routing probability of the job outside the queueing
network ( i, j ) into node k in the queueing network ( i, j ) at time t and pl(,0i , j ) (t ) is
the routing probability of job moving from node l in the queueing network
( i, j ) out of the queueing network ( i, j ) at time t .
Definition 2.1. Assumably the network has the nodes denoted as
0,1, 2,..., J , J  + (in which 0 is formal node added to the network as
mentioned above) and i, j 1, 2,..., J  , the component network ( i, j ) is the
queueing network satisfying the following conditions:
 p (ji,0, j ) (t )  0
 p (t ) = 1


(i )  ( i , j )
; (ii )  pk( i,0, j ) (t ) = 0 k  j , k = 1, J
 pk ,i (t ) = 0 k  i, k = 1, J
 (i , j )
 p j ,l (t ) = 0 l  j , l = 1, J .
 pk( i,l, j ) (t ) pl(,ik, j ) (t ) = 0 t , k  l
(iii ) 
k , l = 1, J
(i , j )
0,i

(2.1)

With definition 2.1 of directional component network, we always
decompose a queueing network with J nodes into directional component
networks ( i, j ) ( i, j = 1, J ) and the total of directional component networks is a
unconformity repetition convolution 2 of J (and equal to J 2 directional
component network). With arguments and proofs above, we have the following
theorem:
Theorem 2.1. A network with J nodes ( J  + ) is always decomposed into
J 2 directional component networks (according to definition 2.1).
2.2. Synthesising the general queueing network from the component
networks
This section presents the technique “convolution” the directional component
networks. Based on that result, we can direct the study of complex general
network into the study of simpler directional component networks.


7

2.2.1. Moving jobs in the general queueing network G/G/J in the context
of job flow among component networks
Hypothesis 2.1. Knowing the job flow in the component network. In
convolution network of the component networks, the component networks
operate not independently of each other and knowking the mechanism of job
flow among the component networks (this mechanism is also known as phase
change mechanism in the queueing network queue).
Definition 2.2. The process of job flow in convolution network is devided
into the steps and defined as below:
Considering the process of job flow in a queueing. The symbol ij ( t ) is the
number of jobs moving from node i to node j at time t ( i, j = 0, J ), in which 0j ( t )
is the number of jobs moving from outside the network into node j ,  j0 ( t ) is the
number of jobs moving from node j out of the network and ij ( t ) is the number
of jobs moving from node i to node j ( i, j = 1, J ) . As assigned  0 is the time to
observe the original starting point and inductively defined as follows:

 1 = min t   0 |

...


J

J

 ( t ) +    ( t )  0 
j =1

J

0j

i =1 j =0, j i

J

 n = min t   n−1 |   0j ( t ) + 




J

j =1

ij




J

  ( t )  0  , n = 2,3,...

i =1 j =0, j i

ij



Then each point  n ( n = 0,1,...) is regarded as the nth moving step of the job
flow in the queueing network.
To describe the job flow from the component network at the nodes of
convolution network out of convolution network and the job flow outside
convolution network into component networks at the nodes of convolution
network, we add the component network  = ( 0,0) | i  i 1,..., J  (formal
component network). This formal network does not contain any nodes of the
convolution network. The job flow outside the convolution network into the
component networks at the nodes of the convolution network is the job flow
from the network  = ( 0,0 ) into the component networks at the nodes of the
convolution network. The job flow leaving the component networks at the
nodes of the convolution network is the job flow from the component networks
at the nodes of the convolution network into the network  = ( 0,0 ) .


8

Symbols:
- L = (i, j ) | i, j 1, 2,..., J  is the set of all the component networks of the queueing
network; Li is the set of the component networks containing node i ( i = 1, J ) . At
step n ( n = 1,2,...) :

+ Pc (n) =  pic, j (n)  i , j =0, J is the routing probability matrix of the component
network c ( c  L ) in which pic, j (n) is the routing probability of the job flow
from node i to node j in the component network c .
 0

0 

 si (n) Si (n) 

+ Si ( n) = 

in which

Si (n) =  Sic,d (n) 

c , dLi

is the routing probability matrix

of job flow in node i between component networks with Sic,d (n) being the
probability of job flow in node i from component network c to component
T
network d and si (n) = ( sic (n) )cL is the probability vector of job flow from node
i

i out of the queueing network with sic (n) being the probability of job flow
from node i in component network c out of the queueing network.
+ ai ( n ) = ( aic ( n ) )c  L is the vector showing traffic of the job flow to node i in
 

i

the queueing network. In which aic ( n ) is traffic of the job flow to node i of
component network c ( c  Li ) and ai ( n ) = 0 .
+ bi ( n ) = ( bic ( n ) )cL is the vector showing traffic of the job flow at node i in the
i

queueing network. In which bic ( n ) is traffic of the job flow at node i in

component network c .
+ vi ( n ) = ( vic ( n ) )c L is the vector showing traffic of the external job flow into
i

node i in the queueing network. In which vic ( n ) is traffic of the external job
flow into component network c ( c  Li ) at node i and vi ( n ) = 0 .
+ di ( n ) is traffic of the job flow from node i going out of the queueing
network.
From the definition 2.1 of the component network, then:
- With all nodes i being used by component network c  L at step n we have:
(2.2)
sic (n) +  Sic ,d (n) = 1 .
d Li

- With all nodes i being used by component network ( h, l )  L at step n we
have:


9
( h ,l )
( h ,l )
 p0,i
(n) = 1 if i = h; pi,h
(n) = 0 if i  h
J
 p ( h ,l ) ( n ) = 1
 i,j
.
 j =0
 p ( h ,l ) (n) p ( h ,l ) (n) = 0 with j = 1, J and j  i

j,i
 i,j
(
h
,
l
)
 p (n) = 0 if i  l ; p ( h ,l ) (n) = 0, s ( h ,l ) (n) = 0 if i  l
i ,0
i
 l ,i

(2.3)

 J ( h ,l )
( h ,l )
 pi,j (n) + p j,i (n) = 0
.
 j =0
 s ( h ,l ) ( n ) = 0
 i

(2.4)

- With all nodes i not being used by component network ( h, l )  L at step n we
have:

2.2.1.1. Identifying the job flow in the queueing network Γ is convoluted
by two component networks (1):=(i1,j1) and (2):=(i2,j2)
From the characteristics of the job flow in the component network, then:

(1),(2) if i1  i2 or j1  j2

- If i1  j1 and i2  j2 : Li = 
- If i1  j1 and
- If i1 = j1 and

- If i1 = j1 and

.

(1) if i1 = i2 and j1 = j2
 Li = (1),(2)
.
i2 = j2 then  2
L
=
(1)
with

i

i


 i
2
 Li = (1),(2)
i2  j2 then  1
.
 Li = (2) with i  i1

 Li1 = (1)

 Li = (2)
i2 = j2 then  2
 Li =  with i  i1 , i  i2

i1  i2

or

 Li1 = Li2 = (1)

 Li =  with i  i1 .
i = i
 1 2

- Moving jobs in the queueing network Γ at step n (n≥1)
Traffic of the external job flow into node i ( i = 1, J ) in the queueing network
vi (n) = ( vic (n) )

Γ at step n is:

c Li

.

(2.5)

Traffic of the job flow into node i in the queueing network Γ at step n is:

(

ai (n) = aic (n)

)

c Li

with

aic (n) = vic (n) +

J



j =1, j  i ,cL j

bcj (n − 1) p cji (n − 1) .

(2.6)

Traffic of the job flow bewteen component networks at node i at step n is:


10

(
(

)

 ai(1) (n) si(1) (n), ai(1) (n) Si(1),(1) (n) if Li = (1)

 ai(2) (n) si(2) (n), ai(2) (n) Si(2),(2) (n) if Li = (2)

.
ri (n) =  a (1) (n) s (1) (n) + a (2) (n) s (2) (n), a (1) (n) S (1),(1) (n) + a (2) (n) S (2),(1) (n), 
i
i
i
i
i
i
i
 i
 if Li = (1), (2)
 ai(1) (n) Si(1),(2) (n) + ai(2) (n) Si(2),(2) (n)




)

(2.7)

Traffic of the job flow at node i of the queueing network Γ at step n is:

(
(

)

 ai(1) (n) Si(1),(1) (n) + bi(1) (n − 1) pi(1)
,i ( n − 1) if Li = (1)

 ai(2) (n) Si(2),(2) (n) + bi(2) (n − 1) pi(2)
,i ( n − 1) if Li = (2)

bi (n) =  a (1) (n) S (1),(1) (n) + a (2) (n) S (2),(1) (n) + b (1) (n − 1) p (1) (n − 1), 
.
i
i
i
i
i
i ,i

if
L
=
(1),
(2)



i

 ai(1) (n) Si(1),(2) (n) + ai(2 ) (n) Si(2),(2) (n) + bi(2) (n − 1) pi(2)
,i ( n − 1) 




)

(2.8)

And traffic of the job flow from node i going out of the queueing network
Γ at step n is:
ai(1) (n) si(1) (n) + bi(1) (n − 1) pi(1),0 (n − 1) if Li = (1)

.(2.9)
di (n) = ai(2) (n) si(2) (n) + bi(2) (n − 1) pi(2)
,0 ( n − 1) if Li = (2)
 (1)
(1)
(2)
(2)
(1)
(1)
(2)
(2)
ai (n) si (n) + ai (n) si (n) + bi (n − 1) pi ,0 (n − 1) + bi (n − 1) pi ,0 (n − 1) if Li = (1),(2)

Thus, this section of the thesis presents the process of job flow of the
queueing network convoluted by two component networks. Formula (2.8) shows
the change in traffic of the job flow in the network node at the step through
which we can see the job flow in the queueing network. Formula (2.9) shows
the serving capacity of the queueing network.
2.2.1.2. Identifying the job flow in the queueing network Γ convoluted

by J2 component networks.
- Moving jobs in the queueing network Γ at step n (n≥1):
With traffic of the external job flow into node i ( i = 1, J ) of the queueing
network Γ at step n is:

(

vi ( n ) = vic ( n )

)

c Li

.

(2.10)

Then traffic of the job flow to node i of the queueing network Γ at step n is:

(

ai (n) = aic (n)

)

c Li

with

aic (n) = vic (n) +

J



j =1; j  i :cL j

bcj (n − 1) p cji (n − 1) .

(2.11)

The traffic of job flow between the component networks at node i at step n is:
ri (n) := ai (n)Si (n) .
(2.12)
The traffic of job flow at node i of the queueing network Γ at step n is:
with bic (n) = ric (n) + bic (n −1) piic (n −1) .
(2.13)
bi (n) = ( bic (n) )
cL
i


11

The traffic of job flow from node i going out of the queueing network Γ at
step n is:
(2.14)
di (n) =  aic (n)sic (n) + bi(c ) (n − 1) pi(,0c ) (n − 1) .
cLi

Thus, this section of the thesis presents the job flow of the queueing network
convoluted by J 2 component networks. Formula (2.13) shows the change in
traffic of the job flow in the network node at the steps through which the job
change in the queueing network can be seen. Formula (2.14) shows the serving
capacity of the queueing network at the steps.
With the above arguments and proofs, we have the following theorem:
Theorem 2.2. Any queueing networks with J nodes can be expressed as the
convolution of J 2 directional component networks (according to definition 2.1)
and with traffic of the job flows (components vi ( n ) , ai ( n ) , bi ( n ) , di ( n ) ) at step n
are calculated according to the formulas (2.10),(2.11),(2.13),(2.14).
2.2.2. Considering the particular case – in convolution network without
the job flows between component networks
Symbols:
- L = (i, j ) | i, j 1, 2,..., J  is the set of all the component networks of the
general queueing networks with J nodes.
- P( h,l ) =  pi,j( h,l ) i , j =0, J is the routing probability matrix of component network

( h, l ) ( (h, l )  L ). In which pi,j( h,l ) is the routing probability of job flow from node i
to node j in component network ( h, l ) at time t , p0,i( h,l ) is the routing probability
of jobs from outside the component network ( h, l ) to node i in component
network ( h, l ) at time t and p (j,0h,l ) is the routing probability of jobs from node j
in component network ( h, l ) going out of component network ( h, l ) at time t ;
- P =  pi,j i , j =0, J is the routing probability matrix of convolution network. In
which pi,j is the routing probability of jobs from node i to node j in
convolution network at time t , p0,i is the routing probability of jobs from
outside the convolution network to node i in convolution network at time t and
p j ,0 is the routing probability of jobs from node j going out of convolution
network at time t .
- Ai(,hj,l ) is event job moving from node i to node j in component network
( h, l ) ( (h, l )  L ) at time t , A0,( hi,l ) is event job moving from outside the component

network ( h, l ) to node i in component network ( h, l ) at time t and A(j h,0,l ) is event
job from node j in component network ( h, l ) out of the component network


12

( h, l ) at time t ; Ai , j is event job moving from node i to node j in convolution
network at time t , A0,i is event job moving from outside convolution network to
node i in convolution network at time t and Aj ,0 is event job from node j in
convolution network out of convolution network at time t .
Hypothesis 2.2. It is assumed that the job flow in component networks is
known. In convolution network of the component networks, it is assumed that
the job flows in component networks are independent of each other and there is
no job flow from one component network to another one.
From the definition 2.1 of the component network:
- With all nodes i being used by component network ( h, l )  L at time t we have:
 J ( h ,l )
 pi,j ( t ) = 1
 j =0
 ( h ,l )
( h ,l )
 pi,0 ( t ) = 0 if i  l ; p0,i ( t ) = 0 if i  h .
 ( h ,l )
( h ,l )
 pi,j ( t ) p j,i ( t ) = 0 j = 1, J and j  i


(2.15)

- With all nodes i not being used by component network ( h, l )  L at time t

J

 p (t ) = 0 .

we have:

j =0

( h ,l )
i, j

(2.16)

2.2.2.1. Identifying the job flow in the queueing network Γ convoluted
by two component networks.
Considering queueing network Γ is convoluted by two component networks
( i1 , j1 ) and ( i2 , j2 ) . Since Ai(,ij , j ) ( t ) ( k = 1, 2 ) is event job moving from node i to
node j in component network ( ik , jk ) at time t and Ai , j is event job moving
from node i to node j in queueing network Γ at time t . Then we have:
Ai , j ( t ) = Ai(,ij , j ) ( t ) Ai(,ij , j ) ( t ) .
(2.17)
From hypothesis 2.2, then two component networks ( i1 , j1 ) and ( i2 , j2 ) operate
independently of each other. Thus we have:
 Ai , j ( t ) = pi(,ij, j ) ( t ) + pi(,ij , j ) ( t ) − pi(,ij, j ) ( t ) pi(,ij , j ) ( t ) .
(2.18)
From hypothesis 2.2 and formula (2.18) if queueing network Γ is convoluted
by two component networks and the job flow between nodes in two component
networks is known, then the job flow between nodes in queueing network Γ will
be identified.
2.2.2.2. Identifying the job flow in queueing network Γ convoluted by J2

component networks.
Since the queueing network has J nodes, queueing network Γ is convoluted
by J 2 component networks. Thus we have:
k

k

1

1

1

1

2

2

2

2

1

1

2

2

13
Ai , j ( t ) =

Ai(,kj,l ) ( t ) .

(2.19)

( k ,l )L

From hypothesis 2.2, then the component networks operate independently, it
means events Ai(,kj,l ) ( t ) ( ( k , l )  L ) are independent of each other. Thus we have:
 Ai , j ( t )  = 1 −  (1 − pi(,kj,l ) ( t ) ) .
(2.20)
( k ,l )L

From the hyothesis of problem 2.2 and formula (2.20), if the probabilty of
job flow between nodes in all component networks is known, then the
probability of job flow between nodes in the queueing network will be
identified.
2.3. Regarding a specific queueing network model
This section presents the 5-node queueing network. Applying the results of
the section 2.2.1 in chapter 2 to calculate traffic of the job flow in the queueing
network at different steps.
2.3.1. Set of component networks
For ease of presentation, we implement the indexation of the component
networks from 1 to 25 as in the following table:
Table 2.1. Indexing component networks
Index Corresponding Index Corresponding Index Corresponding

component
component
component
network
network
network
1
(1,1)
10
(2,5)
19
(4,4)
2
(1,2)
11
(3,1)
20
(4,5)
3
(1,3)
12
(3,2)
21
(5,1)
4
(1,4)
13
(3,3)
22
(5,2)

5
(1,5)
14
(3,4)
23
(5,3)
6
(2,1)
15
(3,5)
24
(5,4)
7
(2,2)
16
(4,1)
25
(5,5)
8
(2,3)
17
(4,2)
9
(2,4)
18
(4,3)
Then we have:
- The set of 1-node component networks is:
L1 = 1, 2,3, 4,5,6,8,9,10,11,12,14,15,16,17,18, 20, 21, 22, 23, 24
- The set of 2-node component networks is:

L2 = 2,3, 4,5,6,7,8,9,10,11,12,14,15,16,17,18, 20, 21, 22, 23, 24
- The set of 3-node component networks is:
L3 = 2,3, 4,5,6,8,9,10,11,12,13,14,15,16,17,18, 20, 21, 22, 23, 24
- The set of 4-node component networks is:


14
L4 = 2,3, 4,5,6,8,9,10,11,12,14,15,16,17,18,19, 20, 21, 22, 23, 24

- The set of 5-node component networks is:
L5 = 2,3, 4,5,6,8,9,10,11,12,14,15,16,17,18, 20, 21, 22, 23, 24, 25
2.3.2. The job flow in the queueing network at step n (n≥1)
a. Traffic of the external job flow to the nodes of the queueing network at step n
v1 (n) = ( 0, v11 (n), v12 (n), v13 (n), v14 (n), v15 (n),0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0 )

v2 (n) = ( 0,0,0,0,0, v26 (n), v27 (n), v28 (n), v29 (n), v10
2 ( n),0,0,0,0,0,0,0,0,0,0,0,0 )

v3 (n) = ( 0,0,0,0,0,0,0,0,0, v311 ( n), v312 ( n), v313 ( n), v314 ( n), v315 ( n),0,0,0,0,0,0,0,0 )

17
18
19
20
v4 (n) = ( 0,0,0,0,0,0,0,0,0,0,0,0,0, v16
4 ( n), v4 ( n), v4 ( n), v4 ( n), v4 (n),0,0,0,0 )

v5 (n) = ( 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, v521( n), v522 ( n), v523 (n), v524 (n), v525 (n) )

b. Traffic of the job flow to the nodes of the queueing network at step n

aic ( n ) = vic ( n ) +

J



j =1, j i:cL j

bcj ( n − 1) p cji ( n − 1) víi c  Li , i = 1,5 .

(2.21)

c. Traffic of the job flow between component networks at step n
ri d (n) =  aic (n)Sic ,d (n) d  Li , i = 1,5 .
(2.22)
cLi

d. Traffic of the job flow at a node of the queueing network at step n
bic ( n ) = ric ( n ) + bic ( n − 1) pii (n − 1) c  Li , i = 1,5 .
(2.23)
e. Traffic of the job flow going out of the queueing network at step n
- Traffic of the job flow going out of the queueing network at node 1:
c
(2.24)
d1 (n) =  a1c (n) s1c (n) + b1c (n − 1) p1,0
(n − 1) .
c1,6,11,16,21

- Traffic of the job flow going out of the queueing network at node 2:
c

(2.25)
d2 (n) =  a2c (n) s2c (n) + b2c (n − 1) p2,0
(n − 1) .
c2,7,12,17,22

- Traffic of the job flow going out of the queueing network at node 3:
c
(2.26)
d3 (n) =  a3c (n) s3c (n) + b3c (n − 1) p3,0
(n − 1) .
c3,8,13,18,23

- Traffic of the job flow going out of the queueing network at node 4:
c
(2.27)
d4 (n) =  a4c (n) s4c (n) + b4c (n − 1) p4,0
(n − 1) .
c4,9,14,19,24

- Traffic of the job flow going out of the queueing network at node 5:
(2.28)
d 5 ( n) =
 a5c (n)s5c (n) + b5c (n − 1) p5,0c (n − 1) .
c5,10,15,20,25

Thus, the thesis has done traffic of the job flow calculations switching
between component networks at each node and traffic of the existing job flow at


15

each node in the steps through which the process of job flow move in the 5-node
queueing network can be seen.
2.4. Building the program to calculate the job flow in the queueing network
The program is designed and built to allow a queueing network with
arbitrary network nodes. The network parameters can be customized and saved
into configuration files to allow reuse in program running times instead of
reseting parameters when running the program. Some calculation results of
traffic of the job flow in the queueing network is shown visually in charts.
Conclusion of chapter 2
Chapter 2 studies, proposes the technique to decompose and synthesise the
queueing network in which the decomposing technique aims at decomposing a
network into simpler directional component networks and synthesising
technique to allow “convolution” directional component networks into a given
general queueing network. This decomposing and synthesising technique
allows us to study any queueing networks with multi-directional job flow
viewed as “convolution” (“superposition”) network of the directional
component networks and from thereby lead the research problem of arbitrary
queueing network to the problem of simpler directional component networks.
CHAPTER 3
EVALUATION THE STATE PROCESS OF GENERAL QUEUEING
NETWORK
Chapter 3 uses the results presented in articles [1] on the list of published
works. For queueing networks, the problem class on researching the states of
the nodes and the queueing networks is both scientifically and practically
significant and is of great concern to many authors worldwide. Chapter 3
studies this problem class with assumptions the job flow enter the queueing
network is a general distribution and service time at the network node is a
random variable with general distribution. Specifically, the queueing network
satisfies following assumptions:

Hypothesis 3.1. The queueing network has J nodes ( J  N + ), with each node
having a service station. The service time at each network node has general
probability distribution. It is a random variable with an arbitrary distribution
independent from other nodes. A job after being served at node i (i 1, J ) moves
either to node j ( j 1, J ) or outside the network in case it is fully served. The
assumptions about the network are the arrival flows are independent of the
network state and the internal flows are independent of each other and of the
state of the arrival nodes.


16

3.1. State and equation of state transition in the network node
3.1.1. Definitions and symbols
Let  n is the nth time ( n = 0,1, 2,... ) the external job event appears in the
queueing network or the job is served at a certain network node (See Definition
2.2 in Chapter 2). In which  0 is the time of the original starting observed.
a. Definition 3.1. Quasi-binomial distribution
Let n ( n  + ) independent random variables i | i = 1, nwith A(qi )
distribution and they are denoted by i  A(qi )

(i = 1, n)

(Here A(qi ) is the
n

distribution of the Bernoulli random variable with parameter qi ). Set  = i ,
i =1

then  is called a random variable with quasi-binomial distribution and

denoted by   B(n; q1 ,..., qn ) .
Properties: After some calculations we get:

1−
(3.2)
[ =k] = 
 ( qi ) (1 − qi ) .
i

i

1 +...+ n = k 0i  n
1 ,..., n 0,1

n

E ( ) =  qi

n

D ( ) =  qi (1 − qi ) .

and

(3.3)

i =1

i =1

In the special case when qi = q i = 1, n then  is the binomially distributed
random variable B(n; q) .
b. Some symbols:
- X j ( n ) is the number of jobs at node j ( j = 1, J ) at the time  n and called the
state of node j at the time  n . X ( n ) = ( X1 ( n ),..., X J ( n )) and is called the state of
network at the time  n .
- pij is the probability of jobs transferring from node i to node j
(i = 1, J , j = 0, J ) , where pi 0 is the probability that jobs after being served at node i
and leaving the network and pii is the probability of jobs that continues to serve
at node i (we consider the transition probability (routing probability) is not
changed over time in this chapter).
- N j is size of the queue in node j of the queueing network ( j = 1, J , N j  );
E j = 0,1,..., N j  .
3.1.2. The equation of state transition network node
Because ij is the number of jobs moving from node i ( i = 1, J ) to node j at
time  n so the total number of jobs leaving node i at time  n is:
d ( n ) =

J



j = 0, j  i

ij ( n ) .

(3.4)


17

Since assuming the queueing network and from the definition of a time  n
when we have:
ij ( n )  A( pij ) i = 1, J

d ( n )  A(1 − pii ) i = 1, J

.
 J
  ij ( n )  B( J − 1, p1 j ,..., p j −1, j , p j +1, j ,..., pJj )
i =1,i  j
n = 0,1, 2,...


(3.5)

Symbol Aj ( n ) is the number of jobs from outside to Node j within the time
period  n =[ n−1 , n ] . Then the number of jobs in Node j at the time  n is defined
as follows:
X j ( n ) = X j ( n −1 ) + Aj ( n ) −

J



i =0,i  j

 ji ( n ) +

J



i =1,i  j

ij

( n ) .

(3.6)

Formula (3.6) is the equation of state transition in node j of the queueing
network.
3.1.3. State transition probability distribution of the network node
Q j ( n ) = q j ( n−1 , xn−1 , n , xn )  x , x E ,
Symbol
in
which
n−1

n

j

q j ( n−1 , xn−1 , n , xn ) =  X j ( n ) = xn | X j ( n−1 ) = xn−1  is the state transition probability
of the state process at node j from state xn−1 at the time  n−1 to state xn at the

time  n . We have:
- If xn  xn−1 − 1 then q j ( n−1 , xn−1 , n , xn ) = 0 .

(3.7)

- If xn = xn−1 − 1 then q j ( n−1, xn−1, n , xn ) = (1 − p jj )  (1 − pij )  Aj ( n ) = 0 .
J

- If xn = xn−1 + k ( k  ) then
q j ( n−1 , xn−1 , n , xn ) = p jj
+ (1 − p jj )

mink , J −1



mink +1, J −1


y =0

y =0

(3.8)

i =1,i  j

  ( p ) (1 − p )
J

i

1− i

ij

1 +...+ n = y i =1;i  j
1 ,..., n 0,1

ij

  ( p ) (1 − p )
J

1 +...+ n = y i =1;i  j
1 ,..., n 0,1

i

ij

1− i

ij

 Aj ( n ) = k − y 

 Aj ( n ) = k + 1 − y 

. (3.9)

From (3.7), (3.8) and (3.9) we have the state shifting diagram of
( X j ( n ) ) ; n = 0,1,... after one step:

18

0

1

2

…

m-1

m

m+1

m+2 … m+k

…

Figure 3.1. the state shifting diagram of the state process at the network node

3.2. Distribution and nature of the state process
3.2.1. The probability distribution of state at the network node after one step
Assuming that at the time  n−1 we knew the probability distribution of
X j ( n−1 ) . From the equation of state transition (3.5) and with m  E j . So we
have:
m+1

(3.10)
 X j ( n ) = m  =   X j ( n −1 ) = l H j (m, l , n) .
l =0

with

1 if l  m
g (m, l ) = 
0 if l > m

and

 J
 minm−l , J −1  J

H j (m, l , n) = g (m, l )    ji ( n ) = 0  
  ij ( n ) = y   Aj ( n ) = m − l − y 
i =0,i  j
 y =0
 i =1,i  j

minm +1−l , J −1
J
J




+    ji ( n ) = 1
  ij ( n ) = y   Aj ( n ) = m + 1 − l − y 


y =0
i =0,i  j

 i =1,i  j


Reviews 3.1. If the state probability distribution of node j at the present
time and the probability distribution of job flow moving from the outside
network to node j are known, then (3.10) identifies the state probability
distribution of node j at the next step.
3.2.2. The probability distribution of State at the network node after k steps
From (3.6), the probability distribution of State at the node j after k steps is:
l +1
l +1
m+1
(3.11)
[ X j ( n+k ) = m] =  H (m, ln+k , n+k )  H (ln+k , ln+k −1, n+k −1 )...  H (ln+2 , ln+1, n+1 ) [ X j ( n ) = ln+1] .
n+k

ln +k =0

ln + k −1 =0

n+2

ln +1 =0

Reviews 3.2. If the state probability distribution of node j at the present
time and the probability distribution of job flow moving from the outside

network to node j are known, then (3.11) identifies the state probability
distribution of node j at the next k steps ( j = 1, J ).
3.2.3. Conditions for the state process of the network node and the state
process of network is Markov
a. Conditions for the state process of the network node is Markov process
Definitions 3.2. Random matrix
Matrix P =  pij i , jE is called random matrix if the following conditions are
met:
i)

pij  0 i, j  E .

(3.12)


19

ii)

p
jE

ij

= 1 i  E .

(3.13)

Lemma 3.1.
(i) If X =  X ( n )n=0,1,... is the Markov chain in the state space E , then transition

matrix
is
random
matrix
(In
which
P( n ) =  p( n−1 , in−1 , n , in )i ,i E
n−1 n

p( n−1 , in−1 , n , in ) = P  X ( n ) = in | X ( n−1 ) = in−1  ).

(ii) It is assumed that

Q( n ) =  q( n−1 , in−1 , n , in )i

n−1 ,in E

is random matrix. Then

there exists a Markov chain with the state space E and Q( n ) is its transition
matrix.
Symbol Q j ( n ) = q j ( n−1, in−1, n , in ) i ,i E with q j ( n−1, in−1, n , in ) = P  X j ( n ) = in | X j ( n−1) = in−1  . Then
n−1 n

j

we have the following theorem:
Theorem 3.1. With the assumption (3.1) about the queueing network,
X j =  X j ( n )
is the Markov chain identified in the state space E j if and only if

n =0,1,...
Nj



k =−1

J
J


A
(

)
−

(

)
+
ij ( n ) = k  = 1 .


 j n
ji n
i =0,i  j
i =1,i  j



(3.14)

b. Conditions for the state process of the network is Markov process
Lemma 3.2. Assumably X =  X n n=0,1,... is the Markov chain identified in the
state space E X with the transition matrix P X . And Y = Yn n=0,1,... is the Markov
chain identified in the state space E Y with the transition matrix PY . Building the
process Z = ( X n , Yn )n=0,1,... with the state space E = E X  EY = ( x, y) | x  E X , y  EY  and
the transition probability of Z defined as PZ = P X  PY that means:
( AX  E X ; AY  EY )
and
A  E X  EY  A = AX  AY = ( x, y ) | x  A X , y  AY 
P Z ( A) := P X ( AX ).PY ( AY ) . Then Z is the Markov chain identified in the state space
E and P Z = P X .PY is its transition matrix.
Symbol:





-

E j := 0,1,..., N with N = max  N j 

-

Q j ( n ) := q j ( n −1 , in −1 , n , in ) 

in−1 ,in E j

j =1, J





; E := Jj =1 E j = ( x1 ,..., xJ ) : x j  E j , j = 1, J .

with q j ( n−1 , in−1 , n , in ) := P  X j ( n ) = in | X j ( n−1 ) = in−1  .

Theorem 3.2. With the assumption (3.1) about the queueing network, the
state process of the queueing network X ( n ) = ( X1 ( n ),...,. X J ( n ) )n=0,1,2,... is the
J

Markov chain identified in the state space

E

and P X ( n ) =  Q j ( n ) is its
j =1

transition matrix if
Nj



k =−1

J
J



A
(

)
−

(

)
+
ij ( n ) = k  = 1 .


 j n
ji n
i =0,i  j
i =1,i  j



(3.15)


20

with n = 1, 2,...; j = 1, J .
Along with the process X ( n ) = ( X1 ( n ),...,. X J ( n ) )n=0,1,2,... the state of the queueing
network is also related to a one-way stochastic process that is:

J

.
L( n ) =  X j ( n ) 
 j =1
n=0,1,...

(3.16)

Lemma 3.3. Suppose E X and E Y are set in R d and X =  X n n=0,1,... is the
Markov chain identified in the state space E X with the transition matrix P X and
Y = Yn n=0,1,... is the Markov chain identified in the state space E Y with the
transition matrix PY . It is assumed that X and Y are two independent Markov
chains. Then Z = X + Y is the Markov chain identified in the state space
E Z = E X  EY = i := (iX + iY ) | iX  E X , iY  E Y  .
Symbol: E := E1  E2  ...  EJ = i := (i1 + i2 + ... + iJ ) | i1  E1, i2  E2 ,..., iJ  EJ  . Then we
have the following theorem:
Theorem 3.3.
With the assumption (3.1) about the queueing network,


J



the process L( n ) =  X1 ( n ) 
 j =1

space

E

is the Markov chain identified in the state

n =0,1,...

if
Nj



k =−1

J
J


A
(

)
−

(

)
+
ij ( n ) = k  = 1



 j n
ji n
i =0,i  j
i =1,i  j



với

n = 1, 2,...; j = 1, J .

(3.17)

3.3. Applying to calculate the characteristics of the network queue
3.3.1. Average number of jobs at a node
Assumaby we know the distribution of X j ( n−1 ) , then the average number
of jobs at node j ( j = 1, J ) at the time  n is:
E ( X j ( n ) ) = E ( X j ( n −1 ) ) + mAj ( n ) +  pij − 1
J

.

(3.18)

i =1

with mA ( n ) = E ( Aj ( n ) ) .
3.3.2. Throughput of each node
Ther are a lof of definitions about throughput. Here we use the term
“throughput” defined as follows: Througput of node j is the average number of

jobs leaving j in a unit of time. Therefore, if symbol TH j ( n ) is the throughput of
node j ( j = 1, J ) to the time  n , then after some calculations we have:
j

  E (
n

TH j ( n ) =

J

l =1 i = 0;i  j

n

ji

( l ) )

=

n (1 − p jj )

n

.

(3.19)

21

3.3.3. Excessive probability at each node
The excessive probability at node j ( j = 1, J ) at the time  n is identified by the
following formula:
M
(3.20)
[ X j ( n )  M ] = 1 −  [ X j ( n ) = m] .
m =0

with P[ X j ( n ) = m] identified by (3.10).
3.3.4. The average number of jobs in the queueing network
The sum of jobs in the queueing network at the time  n is:
J

L( n ) =  X j ( n ) .

(3.21)

j =1

Then, the average number of jobs in the queueing network at the time  n is:
J
J
 J

E ( L( n ) ) =  E ( X j ( n −1 ) ) + mAj ( n ) +   pij − 1 J .
j =1
j =1
 i =1



(3.22)

3.3.5. Throughput of the queueing network
Symbol TH ( n ) is the throughput of the queueing network to the time  n .
According to the definition: throughput of the network is the average number of
jobs leaving the network in a unit of time. So we have:

 E (
n

TH ( n ) =

J

l =1 j =1

n

j0

( l ) )

=

n

n

J

p
j =1

j0

.

(3.23)

3.3.6. A mechanism to divide the job flow from the outside of the network
Suppose Jobs from outside of the network are categorized into two types,
the first is the job that can move to any nodes in the network, the second is the
job that has to move to some certain nodes to be served before moving to other
nodes. In this section the thesis present a plan to split optimize the job flow
from outside the network to the network so that the average length of queues at
the network nodes and the entire network is minimal.
a. The job flow from the outside of the network
(2)
Symbols A(1)
j ( n ) , Aj ( n ) are respectively the numbers of jobs type 1 and
type 2 from outside to node j within the time period [ n−1 , n ] .
It is assumed that in the time period [ n−1 , n ] there are k jobs type 1 arriving
J


at the network. Symbol  k = a = (a1 , a2 ,..., aJ ) :  ai = k , ai  N  is the set of



i =1



possibilities to divide k jobs from outside to the nodes of the network with
a j being the number of jobs assigned to node j . Symbol pa(1) is the probability to
select the dividing solution a (a   k ) with the assumption:


22

p

a k

(1)
a

=1.

(3.24)

Symbol ph(1) ( n ) is the probability over the time period [ n−1 , n ] with h type 1
jobs moving into the network and p(2)
j , h ( n ) is the probability over the time period
[ n−1 , n ] with h type 2 jobs moving into node j . Then we have:
 Aj ( n ) = m  = [ A

(1)

j

m +

( n ) + A ( n ) = m] =  pl(1) ( n )
(2)
j

k =0 l = k

pa(1) p(2)

j ,m −k ( ) .
a : a =k
n

l

(3.25)

j

b. A mechanism to divide the job flow from the outside of the network
Symbol a(2)
j ( j = 1, J )
j ( n ) is the number of jobs type 2 moving to node
within the time period  n and k (k  N + ) is the number of jobs type 1 from the
outside of the network in the time period  n .
Because the service capacity of each node is different we use quantity  j ( n )
with the following main characteristics to select dividing solutions:

- In direct proportion to the service capacity of node j . In inverse proportion
to X j ( n−1 ) + a(2)
j ( n ) (in which X j ( n −1 ) is the number of jobs in node j at the time
 n −1 ) .
J
 i ( n ) = 1
- 
.
i =1
  i ( n )  0
Symbol  j ( n )  f ( j , X j ( n−1 ) + a(2)
j ( n )) ; x j

(3.26)

is the number of jobs assigned to
node j in k jobs from the outside of the network.
The necessary to select solutions to divide k jobs into J nodes of the network
must be the optimal root of the following integral linear programming:
J

z ( x) =   j ( n ) x j → max .

(3.27)

j =1

Subject to:
 J
 x j = k

 j =1
 x ,..., x 
J
 1

.

(3.28)

Therefore, the integral linear programming (3.27) always has optimal roots.
The symbol of the optimal root set of the integral linear programming (3.27) is
 k . The sufficient to select solutions to divide k jobs into J nodes of the
network must be the optimal root of the following integral linear programming:
(3.29)
max{ f (i , X i ( n−1 ) + ai(2) ( n ) + xi )}i =1, J → min .
Subject to:
(3.30)
x  k .


23

The integral linear programming (3.29) always has optimal roots. Symbol
 is the set of optimal roots of the integral linear programming (3.29) 
*k   k . The probability to select the solutions to divide k jobs into J nodes of
the network is identified as follows:
*
k

1

 (1)
*
 Pa = | * | : a   k
k

 P (1) = 0 : a  *
k
 a

.

(3.31)

In which | *k | is the number of items in set *k .
Thus, with the solution to divide k jobs into J nodes of the queueing
network identified by (3.31), from (3.25) we have:
m  +
 (2)
(2)
(1)
(1)
[ A(1)

+
A

=
m
]
=

(
)
(
)
  pl ( n )  pa  p j ,m −k ( n ) .

j
n
j
n
k =0  l = k
a*l :a j =k



(3.32)

Thus, assuming jobs from outside of the network are categorized into two
types as above when *k is the optimal plan divides k jobs from outside the
network to the network so that the average number of jobs in the network and in
network nodes are minimal.
Conclusion of chapter 3
Chapter 3 of the thesis studies on the queueing network to satisfy condition
3.1. The thesis presents some main results focusing on the features of the
process X j (t ) and the multi-dimensional process X (t ) under gerenal job flow.
Specifically, the thesis studies and determines the equation of state transition of
network node; the state probability distribution of network node; state
transitions probability distribution in the network node. Find conditions for the
state process at the network nodes and the network to be the Markov process.
Calculate the characteristics of the queueing network. Propose an optimal plan

to divide the external job from so that the average number of jobs in the
network node is minimal.
CONCLUSION
I. Key findings of the thesis
This thesis is presented on 163 pages, divided into 03 major chapters
containing core content; introduction; conclusion; all published scientific
works; references; appendix.
The research focused on queueing network G/G/J. With such general
research target, the study is about to focus on two main issues as follows:
Analysis and assessment of job rotation within queueing network; Researching
process state of the network node and state of the entire network. Theoretical
results in Chapter 2 and Chapter 3 are published in 03 articles in national