CATV by yurtgc548

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									    Network Planning Algorithms
        in CATV Networks


          博士論文計劃

            Kuo-Wei Peng
            PhD. Student
Department of Information Management
      National Taiwan University
              6/20/2006
Outline
   Introduction
   Problem Formulation
   Single-Layered Solution Procedure and
    Computational Experiments
   Multi-Layered Solution Procedure and Computational
    Experiments
   Conclusion and Future Work




                                                         2
Outline
   Introduction
   Problem Formulation
   Single-Layered Solution Procedure and
    Computational Experiments
   Multi-Layered Solution Procedure and Computational
    Experiments
   Conclusion and Future Work




                                                         3
      Introduction
         Overview
         Research Scope
         Research Background




Introduction of CATV Communication Networks   4
Overview
   有線電視網路已經廣泛使用在各個地區。
   在有線電視網路上,提供雙向數位服務是可行的。
   有線電視網路的優點:
       高頻寬
       高覆蓋率
       易於擴充
   有線電視網路適於作為資訊基礎建設中的一部份。




                             5
Overview
   建構一個服務品質符合要求的有線電視網路是不容易
    的。
   政府法規再加上各類新式服務的興起,這個工作變得
    更複雜而不易預測。
       雙向服務的通訊品質如何滿足。
   再加上網路成本的考量,這個問題變成了一個網路最
    佳化問題。




                              6
Overview
   傳統的網路規劃方法,建構的網路品質有賴於網路規
    劃者的經驗
       必須滿足所有通訊品質的限制
       如何降低所需的成本
   本論文的目標,在以最低的成本,建構符合服務品質
    要求的有線電視網路。




                              7
Research Scope
   有線電視網路規劃問題的數學模型的建立
       數學模型的建立
       數學方程式的調整
       對偶問題的轉換
   單層網路解題程序
       解題程序
       相關參數的影響
       解題過程中參數的設定與調整
   多層網路解題程序
       分群演算法
       次層網路的頭端(下節點, drop points)的選擇演算法



                                          8
Research Background
   CATV Communication Network Technology
       Network Architecture
       Noise-funneling effect
       Traditional Network Planning Methods
   Research Methods
       Mathematical Programming
       Geometric Programming




                                               9
CATV Communication Network
Technology

                             TRUNK
                            NETW ORK
   Satellite dish
                                                                       Trunk
                                                                      Amplifier


                 Head End



   Radio tower


                             Bridger
                            Amplifier                                     Tap




                                 Splitter     Distribution
                                Direction      Network
                                Couplers



                                            Line Extender


                                                             Television




Figure 1-1. The Network Structure of CATV Networks
                                                                                  10
Noise Funneling Effect



                             C able Modem




                                C able Modem



                                            C able MOdem



     H ead End



                                            C able Modem



                                C able Modem




                             C able Modem




     Figure1-7. Noise-funnelling effect
                                                           11
CATV Network Planning --- Traditional
Approaches
   製圖
   幹線系統設計
   餽線系統設計
   反向系統設計




                                        12
幹線系統設計
Figure 1-8. 頭端幹線系統




                     13
餽線系統設計
   Figure 1-9




                 14
Concluding Remark
   It is difficult to design an CATV network systems
       Intensive computational work.
       Number of possible solutions is very large.
   CAD tools for CATV network design
       To help designer to reduce the overhead of computational
        work.
       To track the signal quality and to make sure the end-to-end
        signal quality is feasible.
       Unable to suggest or create a good design of CATV system
       The quality of design is still relied on the experience and
        expertise of the designers.




                                                                      15
Research Methods
   Mathematical Programming
   Geometric Programming Method
   Steepest Descent Method
   Enhanced Steepest Descent Method
   Surrogate Functions
   Projection Method
   Integer Programming
   Linear Relaxation




                                       16
Geometric Programming Method
     Formulation of the Primal Problem
        min g 0 (t )                                                                       (IP)
        s.t. : t1  0 , t 2  0 , ... , t m  0                                            (1)

              g1 ( t )  1 , g 2 ( t )  1 , ... , g p (t )  1
                                                                                           (2)


              g k (t )   ci t 2 i 1 t 2 i 2 ... t mim , k  0, 1, ..., p,
                                     a  a           a
                         iJ [ k ]


                J [k ]  { mk , mk  1, m k  2 , ..., n k } ,         k  0, 1, ..., p,



               m0  1 , m1  n0  1 , m 2  n1  1 , ... , m p  n p 1  1 ,




                np  n                                                                     (3)




                                                                                                  17
Geometric Programming Method
    Formulation of the Dual Problem
                                              p
                           n       ci
       max v ( )  [ (                ) i ]  k ( ) k ( )                       (IP)
                         i 1      i        k 1


    s.t. :  1  0,  2  0, ... n  0                                                (1)

             i  1                                                                   (2)
          jJ [ 0 ]


             n
             a ij 1  0                j  1,2,..., p                                (3)
            i 1



            k ( )    i ,                k  1,2,..., p
                       iJ [ k ]


            J [k ]  {mk , mk  1, mk  2,..., nk },               k  0.1., , ,. p,



m0  1, m1  n0  1, m2  n1  1,..., m p  n p 1  1, n p  n.

                                                                                              18
Outline
   Introduction
   Problem Formulation
   Single-Layered Solution Procedure and
    Computational Experiments
   Multi-Layered Solution Procedure and Computational
    Experiments
   Conclusion and Future Work




                                                     19
Problem Formulation
   Mathematical Formulation of the CATV Network
    Planning Problem
   Reformulation of the original problem
   The Dual Problem




                                                   20
       Mathematical Formulation and Network
       Optimization
          Basic ideas: formulate the network and try to optimize
           it.


Head End                Al     Gv
   CNR                  Fl     Mv
  X-MOD
                        l     Ov
                                                  CNR
                               Bv                X-MOD
                                                  CSO
                               Fv                 CTB
                                                           User
                               
                               Gv
                               
                               Fv
                               
                               Mv
                                Av
                        (link ) ( equipment )


                                                                  21
Performance Requirements
   Performance requirements in downstream
       CNR (Carrier to Noise Ratio) ≧43dB
       X-MOD (Cross Modulation ) ≦-46dB
       CSO (Composite Second Order) ≦-53dB
       CTB (Composite Triple Beat) ≦-53dB




                                              22
Problem Formulation
    Problem description
        Given :
             downstream performance objectives
             upstream performance objectives
             specifications of network components
             cost structure of network components
             number and position of endusers
             terrain which networks will pass through and the associated cost
        Determine:
             routing
             allocation of network components
             operational parameters (e.g., gain of each amplifier)




                                                                                 23
Problem Formulation
   Features
       Nonlinear problems
       Hard to solve directly by standard methods
       Some technique needed
            Problem Decomposition
                  Stiner Tree Problem
                  Network Optimization
            Geometric Programming
                  Posynomial form
            Gradient-based Optimization




                                                     24
Reformulation of the CATV Network
Design Problem
   Surrogate Function
       Surrogate function of the objective function
       Surrogate functions of the constraints




                                                       25
Surrogate function of the objective
function
     Original objective function
min         [ y l C l  y l  l ( Al , Fl )] 
           lL
                                                      
                                                        
 [ z v   ( Fv , Gv , M v , Bv , Ov )  z v   ( Fv , Gv , M v )  z v   ( Av )]
                                                
vV



     Surrogate function of the objective function
                                         
min  [d1 ( Fl )1  d 2 ( Al )1 ]   {zv [d3 ( Fv )1  d 4 (Gv )1  d5 ( M v )1  d6 ( Bv )1  d7 (Ov )1 ] 
    lL                              vV
      
              1           1            1        
                                                      
     zv [d8 ( Fv )  d9 (Gv )  d10 ( M v ) ]  zv [d11 ( Av )1 ]}




                                                                                                             26
 Surrogate functions of the constraints
      Original Constraints for X-Mod
                                                                                      M sys
                            M pn
           Hpc                       n                       n                         20
                                                                             10
   x p  [ z 10      pn
                             20
                                     Gpi Api   Apj pj ] 
 pPw       n 1                    i 1                    j 1                     S

      Surrogate function for X-Mod
                                                                                             M sys
         Hpc                                                                                   20
                
                                                      n             n                10
 x p  [z      pn   (1.25 * 10   13
                                         )M   8.94
                                              pn     Gpi Api   Apj pj ] 
pPw     n 1                                        i 1          j 1                     S




                                                                                                      27
Surrogate functions of the constraints
   Figure 2-2. SURROGATE FUNCTIONS OF X-MOD,
    CTB, AND CSO
 Name Original Function     Value Set (dB)            Surrogate Function     Avg. Error (%)
X-MOD    10^(M/20)         (61, 88, 96, 97)    (1.250386E-13)*M^(8.935579)    0.009329198
 CTB     10^(B/20)        (61, 90, 102, 110)   (9.290291E-15)*B^(9.567963)    0.022317717
 CSO     10^(O/10)      (66.5, 79.5, 88, 90.5)   (1.1106E-11)*O^(10.22383)    0.062867807



   Figure 2-3. Comparison of functions for X-MOD
                    80000


                    70000


                    60000


                    50000
            X-MOD




                                                                 Original
                    40000
                                                                 Surrogate


                    30000


                    20000


                    10000


                       0
                            61   88                    96   97
                                      Value Set (dB)




                                                                                              28
Outline
   Introduction
   Problem Formulation
   Single-Layered Solution Procedure and
    Computational Experiments
   Multi-Layered Solution Procedure and Computational
    Experiments
   Conclusion and Future Work




                                                     29
Single-Layered Solution Procedure and
Computational Experiments
   Solution Procedure
   Analysis of Starting Points
   Analysis of Initial Step Size
   Analysis of Computing Time




                                        30
Solution Procedure

                                              i
                   n c                           p
         v      i                             k   k
                                                                
max                                      
                                                                                (IP4)
                   i 1   i
                                                  k 1
                                                   


s.t. :   1  0,  2  0,...,  n  0                                            (4.1)

          
         jJ 0 
                     i       1
                                                                                 (4.2)

          n

         a 
         i 1
                    ij       i       0   j  1,2,..., m                         (4.3)

                       , k  0,1,...,
                                     i         p,
                         i J k 

          J k   m k , m k  1, m k  2 ,..., n k , k  0 ,1,..., p ,        (4.4)
         m 0  1, m1  n 0  1, m 2  n1  1,..., m p  n p  1  1, n p  n .




                                                                                         31
The Penalty Function

            min  ln W  JX 2

Where
             n  c  i  p
        W    i   k   k
                                    
                    
             i 1   i   k 1
                          

                n                       n
        X 2  (       i  1) 2  ( aij  i ) 2
               i 1   jJ 0          i 1




                                                     32
Comparison of Gradient Methods
   Figure 3-2. Comparison of Solution Quality
                                Enhanced Steepest
              Net#   Steepest
              c00    2.83746
                     -                    7.979516
                                          -
              c01    2.27522
                     -                    7.771648
                                          -
              c02    6.11063
                     -                    8.268778
                                          -
              c03    1.69108
                     -                    7.446525
                                          -
              c04      0.5784
                       -                  8.155407
                                          -
              c05    3.86666
                     -                    7.412583
                                          -
              c06    3.42424
                     -                    7.351371
                                          -
              c07    3.94931
                     -                    7.379032
                                          -
              c08    2.37665
                     -                    7.159171
                                          -
              c09    2.33859
                     -                    7.523604
                                          -


                                                     33
Analysis of Starting Points
   Figure 3-7. Comparison of starting point: network
    example 3.

           8
                         User 4
           7
           6
                                  User 5
           5
           4
                    HE      User 1
           3
           2
                   User 3
                                                   User 2
           1
           0
               0     1        2      3     4   5      6     7   8   9




                                                                        34
Analysis of Starting Points
   Figure3-8. Comparison of starting point: data for
    network example 3
           Starting Point   DualMin Computing Time (Iterations)
                      0.1    17.43283
                             -                          698232
                      0.2    17.44020
                             -                          585561
                      0.4    17.44225
                             -                         1252148
                      0.8    17.31683
                             -                         4317955
                        1    17.27297
                             -                         7069196
                        2   5.651758*
                            -                         10000000
                        4    58.02029*                10000000
                        8    621.0294*                10000000
                      16     2955.969*                10000000




                                                                  35
Analysis of Initial Step Size
   Figure 3-7. Comparison of starting point: network
    example 3.

           8
                         User 4
           7
           6
                                  User 5
           5
           4
                    HE      User 1
           3
           2
                   User 3
                                                   User 2
           1
           0
               0     1        2      3     4   5      6     7   8   9




                                                                        36
Analysis of Initial Step Size
   Figure 3-11. Comparison of initial step size: data for
    network example3
       Initial Step
       Size(10^-n)    DualMin       Computing Time(Iterations)
                  1
                  -   3777702                           <100000
                  2
                  -   45382.62                          <100000
                  3
                  -     1517.157*                       <100000
                  4
                  -     17.46237
                        -                               <100000
                  5
                  -     17.45711
                        -                               <100000
                  6
                  -     17.44020
                        -                                698232
                  7
                  -     16.06451
                        -                               3868412
                  8
                  -     12.43347
                        -                               7224569
                  9
                  -     2.344709
                        -                              10000000


                                                                  37
Analysis of Initial Step Size
       Initial Step Size vs. Number of Nodes on Steiner Tree
        Constructed
    Initial Step Size(10^-N)



                               5
                               4
                               3
                               2
                               1
                               0
                                   0   5    10        15     20   25
                                           Number of Nodes




                                                                       38
Analysis of Initial Step Size
   Initial Step Size vs. Penalty Parameter J
                                        Pe         .
                                          naltyJ vs Effe                 p
                                                        ctive Initial Ste Size

                               14

                               12
     Initial Step Size=10^-k




                               10

                               8

                               6

                               4

                               2

                               0
                                    0       2           4            6           8   10
                                                            J=10^m




                                                                                          39
       Adjustment Procedure for Initial Step Size
       and Penalty parameter J
          Initial Step Size vs. Number of Nodes on Steiner Tree
           Constructed

Set initial step size ss=10^-k:         Set J=10*J,
        If #(tree)<2, k=2                 k=k+1,
        Else If #(tree) < 7, k=3
        Else if #(tree) < 25, k=4
        Else k=6;                   Compute the optimal
Set J=1;



                    End                   X^2 == 0


                                                               40
Analysis of Computing Time
   Figure 3-12. Number of Network Users versus
    Computing Time
                      100
                      90
                      80
                      70
     Computing Time




                      60
                      50
                      40
                      30
                      20
                      10
                       0
                            User Number



                                                  41
Analysis of Computing Time
   Figure 3-13. Network Size versus Computing Time
                         350

                         300

                         250
        Computing Time




                         200

                         150

                         100

                         50

                          0
                          2

                               4

                                   6

                                       8
                                           10

                                                14

                                                     16

                                                          20

                                                                24

                                                                     28

                                                                          35

                                                                               42
                                                 Network Size



                                                                                    42
Outline
   Introduction
   Problem Formulation
   Single-Layered Solution Procedure and
    Computational Experiments
   Multi-Layered Solution Procedure and Computational
    Experiments
   Conclusion and Future Work




                                                     43
Multi-Layered Solution Procedure and
Computational Experiments
   Multi-layered Solution Procedure
   Adaptive Placement Algorithms for Drop Points
   Conclusion




                                                    44
Multi-layered Solution Procedure:
Concept
   Figure 4-1. 階層式規劃:第一層
       91                                                                 100
                                                City1                    90
       81

       71                   City2                                        80

       61                                               City3             70

       51          City4                                                 60

       41
                                        City5                             50

       31   Head End                                            City6    40

       21                   City7                                        30

       11                               City8           City9             20

       1           City10
               2       3     4      5    6        7      8       9      10 (10Km)


                                                                                    45
Multi-layered Solution Procedure:
Concept (Cont.)
   Figure 4-2. 階層式規劃:第二層
       91                                          100

                                        3         90
       81   1               2


       71                                         80

       61       City1               4             70

       51                                   5
                                                  60

       41        6                      7         50

       31                       8
                                                  40

       21               9                         30

       11                                   10     20

        1
            2    3      4   5   6   7   8   9    10 (1Km)

                                                            46
Modified Agglomerative Hierarchical分群
演算法
    給定:網路使用者座標,最大容忍半徑R
    求解:將網路使用者分群,每個使用群的半徑皆不得
     大於R
1.   將所有網路使用者各自為一群,此時所有使用群的半
     徑為0。
2.   建立一距離矩陣,記錄所有使用群間的距離。
3.   找到距離矩陣中,距離最近的二個使用群i與j。
4.   計算i與j合併後的使用群半徑為R’,比較半徑R’與R。
     若R’>R,則程式結束。
5.   若R’<R,則合併使用群i與j為使用群i’,並更新距離矩
     陣。
6.   回到步驟3.

                                        47
     Network Example for Clustering
        Figure 4-4. Network Example for Clustering
35



30



25



20



15



10



5



0
     0       5      10     15     20     25     30    35



                                                           48
         Network Example after Clustering
            Figure 4-5. Network Example after Clustering
35



30



25



20



15



10



5



0
     0          5      10     15     20     25     30       35



                                                                 49
Adaptive Placement Algorithms for Drop
Points
   Figure 4-7. Different placement for drop points




           DP

                                      DP



HE                               HE
       Centroid                        Near-HE



                                                      50
Comparison of
                                Cluster Cost   centroid   Near-HE
Network cost                        c00            2855      3173

                                    c01            3300      3560
   Centroid Placement vs.
                                    c02            3432      2396
    Near-HE
                                    c03            1450      1126
   No one is good for all          c04            3535      2300
    clusters                                       1443      1509
                                    c05
   Adaptive placement              c06            1331      1324
    algorithm for drop points       c07            1690      1353
       Globally adaptive                          1241      1245
                                    c08
        placement algorithm
                                    c09            1662      1755
       Partially adaptive
                                                    707       707
        placement algorithm         c10

                                    c11             456       456

                                    Layer 1       13554     17028

                                    Total         36656     37934


                                                                    51
Global Adaptive placement algorithm for
drop points
   步驟一:採用分群演算法,將所有節點分成多個使用
    群。
   步驟二:以重心為下一層的下節點,計算網路的總體
    成本。
   步驟三:考慮各個使用群,以不同選取策略計算網路
    總體成本,選擇最低成本的選取策略為該使用群的下
    節點。
   步驟四:重複步驟三,直到所有使用群都被考慮過為
    止。




                                          52
         Layer 1 network topology
            Figure 4-11. Layer 1 network topology for different
             placement of drop points
35



30
               c04
                                       c06
25
                                                                                c10
                     c05                                       c01
20                               c02                                      c09


15           c08
                                                        c07

10
                           c03
                                                                          c00
5
                                                  c11

0 HE
     0               5           10          15           20         25               30   35




                                                                                           53
Partially Adaptive Placement algorithm
Partially Adaptive placement algorithm for drop points
步驟一:採用分群演算法,將所有節點分成多個使用群。
步驟二:以重心為下一層的下節點,計算網路的總體成本。
步驟三:選擇任一終端節點所代表的使用群,以不同選取
策略計算網路總體成本,選擇最低成本的選取策略為該使
用群的下節點。
步驟四:重複步驟三,直到所有第一層網路的終端節點都
被考慮過為止。




                                                         54
                             Cluster    Centroid   NearHE Partial AP    Global AP

Comparison                     c00         2855      3173        2855         2855

                               c01         3300      3560        3300         3300
   Adaptive placement         c02         3432      2396        3432         2396
    is better than both        c03         1450      1126        1450         1126
    Centroid-based and
                               c04         3535      2300        2300         2300
    NearHE-based
                               c05         1443      1509        1443         1443
    placement.
                               c06         1331      1324        1324         1324
   Global Adaptive            c07         1690      1353        1690         1353
    algorithm is better
                               c08         1241      1245        1241         1241
    than Partial Adaptive
                               c09         1662      1755        1662         1662
    algorithm.
                               c10          707       707         707          707
       However, it spends
                               c11          456       456         456          456
        more time.
                               Layer1     13554     17028       13005        12909

                               Total      36656     37934       34865        33073



                                                                                55
Conclusion
   Multi-layered solution procedure
       Clustering
            To separate the large network into several small networks that can be
             solved by single-layered solution procedure.
       The placement of drop points
            Different placement algorithms for different total costs.
            Globally adaptive and Partially adaptive




                                                                                 56
Outline
   Introduction
   Problem Formulation
   Single-Layered Solution Procedure and
    Computational Experiments
   Multi-Layered Solution Procedure and Computational
    Experiments
   Conclusion and Future Work




                                                     57
Conclusion and Future Work
   Conclusion
       Mathematical Model
            Mathematical formulation for the CATV network planning problem is
             constructed.
            The mathematical formulation is re-formulated to conform the posynomial
             form.
            By applying the Geometric Programming Method, the dual problem is
             formulated.
       Single-layered solution procedure
            Gradient-based methods
                 Steepest descent method

                 Enhanced steepest descent method

            Initial value for dual variables
            Initial step size
            Adjustment procedure for Initial step size and penalty parameter J.
            Analysis of computing time


                                                                                       58
Conclusion and Future Work
   Conclusion
       Multi-layered solution procedure
            Clustering
                Modified agglomerative hierarchical clustering algorithm

            Placement of drop points
                Different placement strategy

                Adaptive placement algorithm

                     Global adaptive

                     Partial adaptive




                                                                            59
Future Work
   Locally adjustment of network parameters may
    improve the total cost of networks.
   How to apply our work to CATV networks with more
    different constraints and cost structure
       Multimedia applications
       Network expansion problems
   Hybrid network elements
       Hybrid Fiber-Optical(HFC) network model
       Other possibility.




                                                       60
Q&A

								
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