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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: Senior Researcher.Dr Nguyen Hong Hai 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 INTRODUCTION 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 Research target: The thesis is aimed at general queueing network 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 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 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 Thesis structure: Besides the introduction; conclusion; published scientific works; references; the thesis’ content is presented in three chapters as follows: Chapter Some basic issues about queueing theories and queueing networks Chapter The general multi-class network - Decomposing and Synthesising algorithm Chapter Evaluation on state process of general queueing network CHAPTER SOME BASIC ISSUES ABOUT QUEUEING THEORIES AND QUEUEING NETWORKS Chapter presents some of the knowledge that will be used for further research on queueing network in chapter and chapter of this thesis At the same time, chapter 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  ); K is the size of queueing (the maximum jobs that the queueing can have) The following symbos are often used with A B : M (exponential distribution); Ek , (Erlang distribution with parameters k ,  ); D (degenerate distribution (time for serving job is constant)); G (general distribution) 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 ;  : 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 pi is the probability that a job leaves the N network after being served at node i ( pi = −  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 - 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 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 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 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 shall be applied for further research on queuing network in chapter and chapter of this thesis CHAPTER THE GENERAL MULTI-CLASS NETWORK- DECOMPOSING AND SYNTHESISING ALGORITHM Chapter 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 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 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 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 )   p (t ) =  (i )  ( i , j ) ; (ii )  pk( i,0, j ) (t ) = k  j , k = 1, J  pk ,i (t ) = k  i, k = 1, J  (i , j )  p j ,l (t ) = l  j , l = 1, J  pk( i,l, j ) (t ) pl(,ik, j ) (t ) = 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 of J (and equal to J 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 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 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  is the time to observe the original starting point and inductively defined as follows:   = t   |   J J  ( t ) +    ( t )   j =1 J 0j i =1 j =0, j i J  n = t   n−1 |   0j ( t ) +    J j =1 ij   J   ( t )   , 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 ) 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    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 + ( 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 ) = + 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 ) = + 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) = d Li - With all nodes i being used by component network ( h, l )  L at step n we have: 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 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 directional component networks (according to definition 2.1) and with traffic of the job flows (components vi ( n ) , ( 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 ) =  j =0  ( h ,l ) ( h ,l )  pi,0 ( t ) = if i  l ; p0,i ( t ) = if i  h  ( h ,l ) ( h ,l )  pi,j ( t ) p j,i ( t ) = 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 ) = 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, ) 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 component networks Thus we have: k k 1 1 2 2 1 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 − 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 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 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) 10 (2,5) 19 (4,4) (1,2) 11 (3,1) 20 (4,5) (1,3) 12 (3,2) 21 (5,1) (1,4) 13 (3,3) 22 (5,2) (1,5) 14 (3,4) 23 (5,3) (2,1) 15 (3,5) 24 (5,4) (2,2) 16 (4,1) 25 (5,5) (2,3) 17 (4,2) (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 ( 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 ( 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 ( 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 Chapter 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 EVALUATION THE STATE PROCESS OF GENERAL QUEUEING NETWORK Chapter 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 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  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 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 − then q j ( n−1 , xn−1 , n , xn ) = (3.7) - If xn = xn−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 + − 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 … 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 ) =     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 + − 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  i, j  E (3.12) 19 ii) p jE ij = 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  =    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  =    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  =    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 − 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 ] = −  [ 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 and type 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 arriving J   at the network Symbol  k = a = (a1 , a2 , , aJ ) :  = k ,  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 jobs moving into the network and p(2) j , h ( n ) is the probability over the time period [ n−1 , n ] with h type 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 moving to node within the time period  n and k (k  N + ) is the number of jobs type 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 ) = -  i =1   i ( n )  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  (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 → 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) *  Pa = | * | : a   k k   P (1) = : 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 Chapter 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 and Chapter are published in 03 articles in national 24 magazines Some findings were published in the yearbook of the th Vietnam Mathematical Congress and presented in Probability and Statistic Board of the 8th Vietnam Mathematical Congress took place from 10-14/08/2013 in Nha Trang city, Khanh Hoa Province The mentioned thesis content as well as published findings are relevant and meet the stated objectives II Latest contribution of the thesis Propose technique to decompose general multidimensional network into directed component networks and technique to synthesize (convolution superposition - integrated) component networks to build up respective general network This technique enables to change the study of job rotation in general complex network into the study of component networks with simpler tendency and this techniques also allows the capability of applying usual mathematical tools for research (such as: random graph theory; transportation case ) Provided with necessary and sufficient conditions for the state process of network nodes, state process of the entire network is Markov process These results, besides their new findings, also lay the foundation for analyzing and forecasting future state process when observing network status at the moment, create basis for statistical and forecast analysis of state process The thesis calculates a number of network parameters such as throughput of each node and throughput of the entire network; probability of congestion; Average number of jobs available in the network nodes and queueing network The thesis proposed a methodology of dividing optimal job line in terms of average jobs within network nodes is minimal III Recommendations, further researches Stability of a network is related to the problem that the steady state distribution, limit state distribution (incase the state process satisfy Markov conditions) exsits or not; After this thesis, we are going to research this problem LIST OF PUBLISHED SCIENTIFIC WORKS Nguyen Hai Nam, Nguyen Trung Dung.(10-2013) Some studies on the state of general queueing network type G/G/J (J∈N+) Journal of Science and Technology - Le Quy Don Technical University, No 157, pp.16-31 Nguyen Trung Dung, Tran Quang Vinh.(04-2015) The mechanism of routing the job flows in the general queueing network G/G/J Journal of Militarial Sciences and Technology Researches, No 36, pp.62-70 Nguyen Trung Dung, Ha Manh Tien.(10-2015) An analytical method of general muticlass queueing network G/G/J Journal of Science and Technology - Le Quy Don Technical University, No 172, pp.15-31