隨機數位網路上最佳路徑選擇之研究(II)
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(128) 行政院國家科學委員會補助國內專家學者出席國際學術會議報告 報告人姓名. 時間 會議. 陸. 行. August 8-12, 2006 Xinjiang, China. 地點. 服務機構 及職稱. 年 8 月 20 日. 政大應數系教授. 本會核定 補助文號. 會議. (中文) 國際作業研究第六屆學術會議. 名稱. (英文) the 6th International Symposium on Operations Research and Its Applications. 發表 論文 題目. (中文). 利用成就函數描述網路設計與評估問題. (英文). Network Dimensioning Problems of Applying Achievement Functions. 報告內容應包括下列各項： 一、參加會議經過 因為筆者是台灣作業研究學會的理事長，接受大會邀請，擔任論文發表與評審。. 二、與會心得 這是亞太地區作業研究重要的會議，每二年舉辦一次。與會人員包括來自日本、 韓國、科威特、台灣和中國的學者專家，都是有重要影響力的意見領袖。 以後值得作推廣和舉辦是項會議。 三、考察參觀活動(無是項活動者省略) 參觀吐魯番盆地。. 四、建議 亞太地區發展作業研究突飛猛進。值得我們學習。特別是做跨領域的合作。. 五、攜回資料名稱及內容 大會手冊. 六、其他. 表 Y04. 附件三. 2006.
(129) ISORA'2006 Tentative Programs*. The 6th International Symposium on Operations Research and Its Applications August 8-12, 2006, Xinjiang, China August 7 (Monday): Participants arrive Urumqi, check in Yilite Hotel, and Registration package pick up. August 8 (Tuesday): Technical sessions. (Xinjiang time = Beijing time +2 hours.) 10:00-10:20 Opening Session Welcome address from ISORA2006 co-chairs: Prof. Xiangsun Zhang, Prof. Tatsuo Oyama. 10:20-12:05 Plenary Session I (Session Chair: Xiangsun Zhang) 10:20-11:05 “ Newsvendor Bounds and Heuristics for Optimal Policy of Serial Supply Chains with and without Expedited Shippings” , Xiuli Chao, North Carolina State University, USA. 11:05-11:50 “Applying Network Flow Optimization Techniques for Measuring the Robustness of Water Supply Network System in Tokyo” , Hiroshi Ashida (Tokyo Metropolitan Government), Hozumi Morohoshi, and Tatsuo Oyama (National Graduate Institute for Policy Studies). 11:50-12:10 Coffee Break 12:10-13:40 Plenary Session II (Session Chair: Tatsuo Oyama) 12:10-12:55 “Ne t wor kDi me ns i oni ngPr o b l e msb yAppl y i ngAc hi e v e me ntFunc t i ons ”, Hsing Luh, Taiwan Zhengchi University, Taiwan 12:55-13:40 “ILOG Optimization Software and Industrial Solutions”Kiat Shi Tan and and Lily Deng, ILOG Beijing. (ILOG Software Demo is available during the symposium) 14:00-15:30 Lunch 16:00-17:40 Parallel Session A1 (Session Chair: Prof. Wuyi Yue) 16:00-16:25 “Al gor i t hmi cS ol ut i onsf ort heS t at i ona r yDi s t r i but i onofM/ M/ c / K Re t r i alQue ue ” Yang Woo Shin, Dug Hee Moon, Changwon National University, Korea 16:25-16:50 “An al y t i c alNe t wor kPr o c e s sofGr o upCh oi c ef oraFor e i gnMar k e tEnt r yMode ”, Su-Chuan Shih, Taiwan Providence University, Taiwan 16:50-17:15 “Gl oball ogi s t i c sr oadpl anni ng:age ne t i cal gor i t hmapp r oac h”Wu Yue, University of Southampton, UK 17:15-17:40 ”S t oc h as t i cOp t i malCont r olPr obl e mswi t haBound e dMe mor y ”, Tao Pang, North Carolina State University, USA 16:00-17:40 Parallel Session B1 (Session Chair: Prof. Masanori Fushimi) 16:00-16:25 “DualSc al i ngUs i ngMat he mat i c alPr o gr ammi ngandI t sAppl i c at i on”, Tohru Ueda,. 表 Y04.
(130) 行政院國家科學委員會補助國內專家學者出席國際學術會議報告 報告人姓名. 時間 會議. 陸. 行. August 8-12, 2006 Xinjiang, China. 地點. 服務機構 及職稱. 年 8 月 20 日. 政大應數系教授. 本會核定 補助文號. 會議. (中文) 國際作業研究第六屆學術會議. 名稱. (英文) the 6th International Symposium on Operations Research and Its Applications. 發表 論文 題目. (中文). 利用成就函數描述網路設計與評估問題. (英文). Network Dimensioning Problems of Applying Achievement Functions. 報告內容應包括下列各項： 一、參加會議經過 因為筆者是台灣作業研究學會的理事長，接受大會邀請，擔任論文發表與評審。. 二、與會心得 這是亞太地區作業研究重要的會議，每二年舉辦一次。與會人員包括來自日本、 韓國、科威特、台灣和中國的學者專家，都是有重要影響力的意見領袖。 以後值得作推廣和舉辦是項會議。 三、考察參觀活動(無是項活動者省略) 參觀吐魯番盆地。. 四、建議 亞太地區發展作業研究突飛猛進。值得我們學習。特別是做跨領域的合作。. 五、攜回資料名稱及內容 大會手冊. 六、其他. 表 Y04. 附件三. 2006.
(131) ISORA'2006 Tentative Programs*. The 6th International Symposium on Operations Research and Its Applications August 8-12, 2006, Xinjiang, China August 7 (Monday): Participants arrive Urumqi, check in Yilite Hotel, and Registration package pick up. August 8 (Tuesday): Technical sessions. (Xinjiang time = Beijing time +2 hours.) 10:00-10:20 Opening Session Welcome address from ISORA2006 co-chairs: Prof. Xiangsun Zhang, Prof. Tatsuo Oyama. 10:20-12:05 Plenary Session I (Session Chair: Xiangsun Zhang) 10:20-11:05 “ Newsvendor Bounds and Heuristics for Optimal Policy of Serial Supply Chains with and without Expedited Shippings” , Xiuli Chao, North Carolina State University, USA. 11:05-11:50 “Applying Network Flow Optimization Techniques for Measuring the Robustness of Water Supply Network System in Tokyo” , Hiroshi Ashida (Tokyo Metropolitan Government), Hozumi Morohoshi, and Tatsuo Oyama (National Graduate Institute for Policy Studies). 11:50-12:10 Coffee Break 12:10-13:40 Plenary Session II (Session Chair: Tatsuo Oyama) 12:10-12:55 “Ne t wor kDi me ns i oni ngPr o b l e msb yAppl y i ngAc hi e v e me ntFunc t i ons ”, Hsing Luh, Taiwan Zhengchi University, Taiwan 12:55-13:40 “ILOG Optimization Software and Industrial Solutions”Kiat Shi Tan and and Lily Deng, ILOG Beijing. (ILOG Software Demo is available during the symposium) 14:00-15:30 Lunch 16:00-17:40 Parallel Session A1 (Session Chair: Prof. Wuyi Yue) 16:00-16:25 “Al gor i t hmi cS ol ut i onsf ort heS t at i ona r yDi s t r i but i onofM/ M/ c / K Re t r i alQue ue ” Yang Woo Shin, Dug Hee Moon, Changwon National University, Korea 16:25-16:50 “An al y t i c alNe t wor kPr o c e s sofGr o upCh oi c ef oraFor e i gnMar k e tEnt r yMode ”, Su-Chuan Shih, Taiwan Providence University, Taiwan 16:50-17:15 “Gl oball ogi s t i c sr oadpl anni ng:age ne t i cal gor i t hmapp r oac h”Wu Yue, University of Southampton, UK 17:15-17:40 ”S t oc h as t i cOp t i malCont r olPr obl e mswi t haBound e dMe mor y ”, Tao Pang, North Carolina State University, USA 16:00-17:40 Parallel Session B1 (Session Chair: Prof. Masanori Fushimi) 16:00-16:25 “DualSc al i ngUs i ngMat he mat i c alPr o gr ammi ngandI t sAppl i c at i on”, Tohru Ueda,. 表 Y04.
(132) Network Dimensioning Problems by Applying Achievement Functions Hsing Luh Department of Mathematical Sciences National Chengchi University August 8, 2006. – OR Lab, NCCU –.
(133) 1. Outline. • Introduction • Achievement functions with proportional fairness • Fair bandwidth allocation on network dimensioning problems • Numerical examples • Conclusions. – OR Lab, NCCU –.
(134) 2. Introduction. e6 node 2. e1. e5. node 5. e9. e4. e13. e11 e3. node 1. node 4. e2. e10. e7. node 3. node 7. e8. e12. node 6. Fig. 1. A Sample Network Topology – OR Lab, NCCU –. e14.
(135) 3. Introduction UMTS network services have different QoS classes for four types of traffic: • Conversational class (voice, video telephony, video gaming) • Streaming class (multimedia, video on demand, webcast) • Interactive class (web browsing, network gaming, database access) • Background class (email, SMS, downloading) Table 1. The Characteristics of UMTS Service Classes Traffic Classes Conversational Streaming Interactive class Background class. – OR Lab, NCCU –. Sensitivity to Jitter high high low very low. Sensitivity to Delay high high low low. Sensitivity to Packet Loss low low high high.
(136) 4. Proportional Fairness Kelly et al. (1998, 2001, 2003, 2004) advocated proportional fairness characterized by log(θi). This log utility function is strictly concave. The proportional fair bandwidth allocation is determined by the following objective function: max. X. Ki log(Kiθi).. i∈I. Mo and Walrand (2000) characterized the class of (w, α)- proportionally fair bandwidth allocation, for any given number α (α > 0, α 6= 1), as the following objective function: max. X i∈I. where wi is a fixed parameter. – OR Lab, NCCU –. 1−α α (Ki θi ) , wiKi. 1−α.
(137) 5. Achievement functions with proportional fairness. Proportionally fair bandwidth allocation problems are considered by Pioro et al. (2002), Ogryczak et al. (2003), Park and Choi (2004), Sarkar and Tassiulas (2004), Ye and Qu (2005), Wang and Luh (2005), etc. Depending on the specified aspiration and reservation levels, ai and ri, respectively, we construct our achievement function of θi as follows: µi(θi) = logαi. θi ai , whereαi = . ri ri. (1). Proposition 1. The achievement function µi(θi) is continuous, increasing, and concave.. – OR Lab, NCCU –.
(138) 6. Achievement functions with proportional fairness. Fig. 2. The Graph of an Achievement Function µi(zi) – OR Lab, NCCU –.
(139) 7. Achievement functions with proportional fairness Lemma 2. Let κ be the cheapest cost per unit bandwidth given in an end-to-end path. Suppose the total budget is B. There exists a finite number Mi ≤ B/κKi such that θi ≤ Mi, ∀ i, where Ki is the number of connections in class i.. −M ρ0 · (θi − ki,0) ρ1 · (θi − ki,1) + µi(ki,1) ρ2 · (θi − ki,2) + µi(ki,2) µ ˆi(θi) = .. ρn−1 · (θi − ki,n−1) + µi(ki,n−1) ρn · (θi − ki,n) + 1 ρM · (θi − Mi) + µi(Mi). if if if if. 0 ≤ θi < bi bi ≤ θi < ri ri ≤ θi < ki,1 ki,1 ≤ θi < ki,2. (2). if ki,n−2 ≤ θi < ki,n−1 if ki,n−1 ≤ θi < ai if ai ≤ θi ≤ Mi.. Proposition 3. The achievement function µ ˆi(θi) is continuous, increasing, and concave. – OR Lab, NCCU –.
(140) 8. Achievement functions with proportional fairness. (n). Lemma 4. Let µ ˆi (θi) : [ri, ai] → [0, 1], where n means the number of break points, be defined as the achievement function (2) restricted on [ri, ai]. Then the (n) sequence of functions {ˆ µi (θi)}∞ n=1 converges uniformly to µi (θi ) = logαi (θi /ri ) on [ri, ai]. Theorem 5. If ri ≤ θi ≤ ai, then the ε-proportionally fair bandwidth allocation obtained by using (2) as objective function approaches to proportional fairness as n → ∞.. – OR Lab, NCCU –.
(141) 9. Fair bandwidth allocation on network dimensioning problems. Given a network topology G =< V, E >, where V and E denote the set of nodes and the set of links in the network respectively. There is given a set S of m classes, i.e., |S| = m. We denote by S i a set of sessions in class i. There is also given the maximal possible number Ki in each class i, that is |S i| = Ki. Denote xe and θi be the bandwidth allocated to the link e and the connection j of class i respectively. We also let χi,j (e) be a binary variable which determines whether the link e is chosen for connection j in class i. Given the total available budget B and the marginal cost κe of bandwidths for each link e ∈ E, we want to allocate the bandwidths in order to provide each class with maximal possible QoS.. – OR Lab, NCCU –.
(142) 10. Fair bandwidth allocation on network dimensioning problems. Let x denote the vector of decision variables and Q denote the feasible set. We consider a resource allocation problem defined as an optimization problem with m objective functions fi(x): max{ f (x) : x ∈ Q }, (3) where f (x) is a vector-function that maps the decision space Rn into the criterion space Rm.. – OR Lab, NCCU –.
(143) 11. Majorization. For the n-dimensional decision vector x=( x1, . . . , xn ) of reals, let x(1) ≤ . . . ≤ x(n) denote the components of x in increasing order. Pn Pn n Definition 6. For x and y in R , x ≤M y if i=1 x(i) = i=1 y(i) and Pk Pk i=1 x(i) ≥ i=1 y(i) , for k = 1, . . . , n − 1. When x ≤M y then x is said to be majorized by y. Definition 7. A function g : Rn → R is called Schur-concave, if x ≤M y implies g(x) ≥ g(y). Pn Theorem 8. Let h be an arbitrary real function and define g(x)= i=1 h(xi) for x ∈ Rn, then g is Schur-concave if and only if h is concave.. – OR Lab, NCCU –.
(144) 12. Majorization. Typical solution concepts for multiple criteria problems are defined by aggregation functions g : Rm → R to be maximized. Thus, (3) ⇒ max{g(f(x)) : x ∈ Q}.. (4). An aggregation (4) is fair if it is defined by a strictly increasing and strictly Schur-concave function g. Theorem P 9. For a strictly concave, increasing function µi : R → R, the function m g(f(x)) = i=1 wifi(x) is a strictly monotonic and strictly Schur-concave function. Theorem 10. For a strictly concave, increasing function µi : R → R, the optimal Pm solution of the problem max{ i=1 wifi(x) : x ∈ Q} is a fair solution for resource allocation problem (3).. – OR Lab, NCCU –.
(145) 13. Mathematical Model. max. m X. wifi(x). i=1. s.t.. X. κexe ≤ B. e∈E. XX i. X. χi,j (e)θi = xe, ∀e ∈ E. j. Ki · ci ≤ B. i. θi ·. X. κeχi,j (e) = ci, ∀j ∈ Si, ∀i = 1, . . . , m. e. xe ≤ Ue, ∀e ∈ E – OR Lab, NCCU –.
(146) 14. θi ≥ bi, ∀i = 1, . . . , m X χi,j (e) = 1, ∀j ∈ Si, ∀i = 1, . . . , m e∈Eo. X e∈Eν. X. χi,j (e) =. X. χi,j (e), ∀ν ∈ V \ {o, d}, ∀j ∈ Si, ∀i = 1, . . . , m. e∈Eν0. χi,j (e) = 1, ∀j ∈ Si, ∀i = 1, . . . , m. e∈Ed. xe ≥ 0, ∀e ∈ E θi ≥ 0, ∀i = 1, . . . , m χi,j (e) = 0 or 1, ∀e ∈ E, ∀j ∈ Si, ∀i = 1, . . . , m, where wi ∈ (0, 1) is given for each i and. – OR Lab, NCCU –. Pm. i=1 wi. = 1..
(147) 15. Numerical Example 1. e6. 2. 5. e1. e5. e13. e4. Class 1. Class 2. e9. 1. Class 1. e11. 4. e3. Class 3. e10. e7. e2 3. e12. e8. Fig. 3. Sample Network 1 – OR Lab, NCCU –. C lass 2. Class 3. e14 6. 7.
(148) 16. Numerical Example 1. Class 1 2 3. Table 2. The Characteristics of Each Class Bandwidth Requirement Aspiration Level Reservation Level 160 kbps 334 kbps 167 kbps 80 kbps 166 kbps 83 kbps 25 kbps 56 kbps 28 kbps. Suppose the number of connections in each class i is Ki for i = 1, 2, 3. Under the total available budget B = $1, 000, 000, we want to allocate the bandwidths in order to provide each class with maximal possible QoS defined via the achievement function (1).. – OR Lab, NCCU –.
(149) 17. Numerical Example 1. Given (K1, K2, K3) = (80, 120, 150) and B = 1, 000, 000. We change the weight assigned to each class, and the computational result is shown in Table 3. Table 3. Change in the Weight for Example 1 weight (w1 , w2 , w3 ) 1 1 (1 3, 3, 3) (0.4, 0.3, 0.3) (0.4, 0.4, 0.2) (0.5, 0.3, 0.2) (0.5, 0.4, 0.1) (0.6, 0.2, 0.2) (0.6, 0.3, 0.1) (0.7, 0.2, 0.1) (0.8, 0.1, 0.1). – OR Lab, NCCU –. selected path. e1 − e5 − e9 − e13 e2 − e8 − e14 e1 − e5 − e9 − e13 e2 − e8 − e14 e2 − e8 − e14 e1 − e5 − e9 − e13 e2 − e8 − e14 e2 − e8 − e14 e2 − e8 − e14. bandwidth (kbps). budget ($). optimal value. (θ1 , θ2 , θ3 ). (c1 , c2 , c3 ). (satisfaction). (334,159,28) (334,159,28) (334,162.75,25) (453.625,83,25) (334,162.75,25) (458.125,80,25) (453.625,83,25) (458.125,80,25) (458.125,80,25). (6680,3180,560) (6680,3180,560) (6680,3255,500) (9072.5,1660,500) (6680,3255,500) (9162.5,1600,500) (9072.5,1660,500) (9162.5,1600,500) (9162.5,1600,500). 0.442 0.482 0.514 0.561 0.589 0.628 0.646 0.713 0.781. ratio K1 c1 K2 c2 K3 c3 ( , , ) B B B (0.534,0.382,0.084) (0.534,0.382,0.084) (0.534,0.391,0.075) (0.726,0.199,0.075) (0.534,0.391,0.075) (0.733,0.192,0.075) (0.726,0.199,0.075) (0.733,0.192,0.075) (0.733,0.192,0.075).
(150) 18. Numerical Example 1. Given (w1, w2, w3) = (0.6, 0.3, 0.1) and B = 1, 000, 000. We change the numbers of connections in each class, and the computational results are shown in Table 4, Table 5, and Table 6. Table 4. Change in the Number of Connections in Class 1 for Example 1 number of connections (K1 , K2 , K3 ) (150, 100, 100) (140, 100, 100) (130, 100, 100) (120, 100, 100) (110, 100, 100) (100, 100, 100) (90, 100, 100) (80, 100, 100) (70, 100, 100) (60, 100, 100) (50, 100, 100). – OR Lab, NCCU –. selected path. bandwidth (kbps). budget ($). optimal value. e1 − e5 − e9 − e13 e1 − e5 − e9 − e13 e2 − e8 − e14 e1 − e5 − e9 − e13 e1 − e5 − e9 − e13 e2 − e8 − e14 e1 − e5 − e11 e1 − e5 − e11 e2 − e8 − e14 e2 − e8 − e14 e2 − e8 − e14. (θ1 , θ2 , θ3 ) (261.333,83,25) (280,83,25) (301.539,83,25) (326.667,83,25) (334,107.6,25) (334,141,25) (334,150.591,25) (460.238,83,25) (560,83,25) (653.333,83,25) (790,80,25). (c1 , c2 , c3 ) (5226.667,1660,500) (5600,1660,500) (6030.769,1660,500) (6533.333,1660,500) (6680,2152,500) (6680,2820,500) (7014,3162.4,525) (9665,1743,525) (11200,1660,500) (13066.67,1660,500) (15800,1600,500). (satisfaction) 0.510 0.530 0.5536 0.5807 0.6019 0.62 0.6251 0.6492 0.6971 0.7419 0.8046. ratio K1 c1 K2 c2 K3 c3 ( , , ) B B B (0.784,0.166,0.05) (0.784,0.166,0.05) (0.784,0.166,0.05) (0.784,0.166,0.05) (0.735,0.215,0.05) (0.668,0.282,0.05) (0.631,0.316,0.053) (0.773,0.174,0.053) (0.784,0.166,0.05) (0.784,0.166,0.05) (0.79,0.16,0.05).
(151) 19. Numerical Example 1. Table 5. Change in the Number of Connections in Class 2 for Example 1 number of connections (K1 , K2 , K3 ) (100, 150, 100) (100, 140, 100) (100, 130, 100) (100, 120, 100) (100, 110, 100) (100, 100, 100) (100, 90, 100) (100, 80, 100) (100, 70, 100) (100, 60, 100) (100, 50, 100). – OR Lab, NCCU –. selected path. bandwidth (kbps). budget ($). optimal value. e1 − e5 − e9 − e13 e2 − e8 − e14 e2 − e8 − e14 e1 − e5 − e9 − e13 e1 − e5 − e9 − e13 e2 − e8 − e14 e1 − e5 − e9 − e13 e1 − e5 − e9 − e13 e1 − e5 − e9 − e13 e1 − e5 − e9 − e13 e2 − e8 − e14. (θ1 , θ2 , θ3 ) (350.5,83,25) (358.8,83,25) (367.1,83,25) (375.4,83,25) (334,128.181,25) (334,141,25) (334,156.667,25) (342.2,166,25) (358.8,166,25) (375.4,166,25) (392,166,25). (c1 , c2 , c3 ) (7010,1660,500) (7176,1660,500) (7342,1660,500) (7508,1660,500) (6680,2563.636,500) (6680,2820,500) (6680,3133.333,500) (6844,3320,500) (7176,3320,500) (7508,3320,500) (7840,3320,500). (satisfaction) 0.597 0.601 0.605 0.609 0.613 0.620 0.628 0.637 0.645 0.653 0.661. ratio K1 c1 K2 c2 K3 c3 ( , , ) B B B (0.701,0.249,0.05) (0.718,0.232,0.05) (0.734,0.216,0.05) (0.751,0.199,0.05) (0.668,0.282,0.05) (0.668,0.282,0.05) (0.668,0.282,0.05) (0.684,0.266,0.05) (0.718,0.232,0.05) (0.751,0.199,0.05) (0.784,0.166,0.05).
(152) 20. Numerical Example 1. Table 6. Change in the Number of Connections in Class 3 for Example 1 number of connections (K1 , K2 , K3 ) (100, 100, 150) (100, 100, 140) (100, 100, 130) (100, 100, 120) (100, 100, 110) (100, 100, 100) (100, 100, 90) (100, 100, 80) (100, 100, 70) (100, 100, 60) (100, 100, 50). – OR Lab, NCCU –. selected path. bandwidth (kbps). budget ($). optimal value. e1 − e5 − e9 − e13 e1 − e5 − e9 − e13 e2 − e8 − e14 e1 − e5 − e9 − e13 e1 − e5 − e9 − e13 e2 − e8 − e14 e1 − e5 − e9 − e13 e1 − e5 − e9 − e13 e2 − e8 − e14 e1 − e5 − e9 − e13 e1 − e5 − e9 − e13. (θ1 , θ2 , θ3 ) (334,128.5,25) (334,131,25) (334,133.5,25) (334,136,25) (334,138.5,25) (334,141,25) (334,143.5,25) (334,146,25) (334,148.5,25) (334,151,25) (334,152,28). (c1 , c2 , c3 ) (6680,2570,500) (6680,2620,500) (6680,2670,500) (6680,2720,500) (6680,2770,500) (6680,2820,500) (6680,2870,500) (6680,2920,500) (6680,2970,500) (6680,3020,500) (6680,3040,560). (satisfaction) 0.613 0.615 0.616 0.617 0.619 0.620 0.621 0.623 0.624 0.625 0.627. ratio K1 c1 K2 c2 K3 c3 ( , , ) B B B (0.668,0.257,0.075) (0.668,0.262,0.07) (0.668,0.267,0.065) (0.668,0.272,0.06) (0.668,0.277,0.055) (0.668,0.282,0.05) (0.668,0.287,0.045) (0.668,0.292,0.04) (0.668,0.297,0.035) (0.668,0.302,0.03) (0.668,0.304,0.028).
(153) 21. Numerical Example 2. e4. B. e5. e1 Class 1. Class 2. A. e8. Class 3. e3. F. e12 e13. e10. G. Class 1. M. e20. L. e21. e22 e17. e25. K. e19. I e16. e23. e24 e18. e9. D. – OR Lab, NCCU –. e11 e7. C. H. e15. e6. e2. e14. E. J. e26. Class 2. Class 3.
(154) 22. Numerical Example 2. Table 7. Change in the Weight for Example 2 weight (w1 , w2 , w3 ) 1 1 (1 3, 3, 3) (0.4, 0.3, 0.3) (0.4, 0.4, 0.2) (0.5, 0.3, 0.2) (0.5, 0.4, 0.1) (0.6, 0.2, 0.2) (0.6, 0.3, 0.1) (0.7, 0.2, 0.1) (0.8, 0.1, 0.1). – OR Lab, NCCU –. selected path. (e2 , e7 , e11 , e14 , e23 ) (e2 , e7 , e11 , e14 , e23 ) (e2 , e7 , e11 , e14 , e23 ) (e2 , e7 , e11 , e14 , e23 ) (e2 , e7 , e11 , e14 , e23 ) (e2 , e7 , e13 , e16 , e19 , e25 ) (e2 , e7 , e11 , e14 , e23 ) (e2 , e7 , e13 , e16 , e19 , e25 ) (e2 , e7 , e13 , e16 , e19 , e25 ). bandwidth (kbps) (θ1 , θ2 , θ3 ) (334,166,39.94) (356.39,166,28) (356.39,166,28) (486.52,83,25) (362.02,166,25) (491.02,80,25) (486.52,83,25) (491.02,80,25) (491.02,80,25). budget ($) (c1 , c2 , c3 ) (12692,6308,1517.87) (13543,6308,1064) (13543,6308,1064) (18487.75,3154,950) (13756.75,6308,950) (18658.75,3040,950) (18487.75,3154,950) (18658.75,3040,950) (18658.75,3040,950). total flow (kbps) 52631.58 52631.58 52631.58 52631.58 52631.58 52631.58 52631.58 52631.58 52631.58. optimal value (satisfaction) 0.454 0.493 0.525 0.574 0.603 0.644 0.662 0.732 0.802.
(155) 23. Numerical Example 2. Table 8. Change in the Number of Connections in Class 1 for Example 2 number of connections (K1 , K2 , K3 ) (150, 100, 100) (140, 100, 100) (130, 100, 100) (120, 100, 100) (110, 100, 100) (100, 100, 100) (90, 100, 100) (80, 100, 100) (70, 100, 100) (60, 100, 100) (50, 100, 100). – OR Lab, NCCU –. selected path. (e2 , e7 , e13 , e16 , e20 , e26 ) (e2 , e7 , e13 , e16 , e19 , e25 ) (e2 , e7 , e13 , e16 , e19 , e25 ) (e2 , e6 , e14 , e23 ) (e2 , e7 , e13 , e16 , e19 , e25 ) (e2 , e7 , e13 , e16 , e20 , e26 ) (e2 , e7 , e11 , e14 , e23 ) (e2 , e7 , e13 , e16 , e20 , e26 ) (e2 , e7 , e13 , e16 , e19 , e25 ) (e2 , e7 , e11 , e14 , e23 ) (e2 , e7 , e13 , e16 , e19 , e25 ). bandwidth (kbps) (θ1 , θ2 , θ3 ) (278.878,83,25) (298.797,83,25) (321.781,83,25) (334,100.516,25) (334,133.916,25) (335.316,166,25) (372.573,166,25) (522.895,83,25) (597.594,83,25) (697.193,83,25) (842.632,80,25). budget ($) (c1 , c2 , c3 ) (10597.333,3154,950) (11354.286,3154,950) (12227.692,3154,950) (12692,3819.6,950) (12692,5088.8,950) (12742,6308,950) (14157.778,6308,950) (19870,3154,950) (22708.571,3154,950) (26493.333,3154,950) (32020,3040,950). total flow (kbps) 52631.58 52631.58 52631.58 52631.58 52631.58 52631.58 52631.58 52631.58 52631.58 52631.58 52631.58. optimal value (satisfaction) 0.529 0.551 0.575 0.598 0.616 0.634 0.652 0.679 0.715 0.763 0.830.
(156) 24. Numerical Example 2. Table 9. Change in the Number of Connections in Class 2 for Example 2 number of connections (K1 , K2 , K3 ) (100, 150, 100) (100, 140, 100) (100, 130, 100) (100, 120, 100) (100, 110, 100) (100, 100, 100) (100, 90, 100) (100, 80, 100) (100, 70, 100) (100, 60, 100) (100, 50, 100). – OR Lab, NCCU –. selected path. (e2 , e7 , e13 , e16 , e19 , e25 ) (e2 , e7 , e13 , e16 , e20 , e26 ) (e2 , e7 , e13 , e16 , e19 , e25 ) (e2 , e7 , e13 , e16 , e19 , e25 ) (e2 , e7 , e13 , e16 , e20 , e26 ) (e2 , e7 , e13 , e16 , e20 , e26 ) (e2 , e7 , e13 , e16 , e19 , e25 ) (e2 , e7 , e13 , e16 , e19 , e25 ) (e2 , e7 , e13 , e16 , e20 , e26 ) (e2 , e7 , e13 , e16 , e20 , e26 ) (e2 , e7 , e13 , e16 , e20 , e26 ). bandwidth (kbps) (θ1 , θ2 , θ3 ) (376.816,83,25) (385.116,83,25) (393.416,83,25) (401.716,83,25) (334,152.105,25) (335.316,166,25) (351.916,166,25) (368.516,166,25) (385.116,166,25) (401.716,166,25) (334,334.632,25). budget ($) (c1 , c2 , c3 ) (14319,3154,950) (14634.4,3154,950) (14949.8,3154,950) (15265.2,3154,950) (12692,5780,950) (12742,6308,950) (13372.8,6308,950) (14003.6,6308,950) (14634.4,6308,950) (15265.2,6308,950) (12692,12716,950). total flow (kbps) 52631.58 52631.58 52631.58 52631.58 52631.58 52631.58 52631.58 52631.58 52631.58 52631.58 52631.58. optimal value (satisfaction) 0.609 0.613 0.617 0.621 0.626 0.634 0.642 0.650 0.658 0.666 0.674.
(157) 25. Numerical Example 2. Table 10. Change in the Number of Connections in Class 3 for Example 2 number of connections (K1 , K2 , K3 ) (100, 100, 150) (100, 100, 140) (100, 100, 130) (100, 100, 120) (100, 100, 110) (100, 100, 100) (100, 100, 90) (100, 100, 80) (100, 100, 70) (100, 100, 60) (100, 100, 50). – OR Lab, NCCU –. selected path. (e2 , e7 , e13 , e16 , e20 , e26 ) (e2 , e7 , e13 , e16 , e20 , e26 ) (e2 , e7 , e13 , e16 , e20 , e26 ) (e2 , e7 , e13 , e16 , e19 , e25 ) (e2 , e7 , e13 , e16 , e19 , e25 ) (e2 , e7 , e13 , e16 , e20 , e26 ) (e2 , e7 , e13 , e16 , e20 , e26 ) (e2 , e7 , e13 , e16 , e20 , e26 ) (e2 , e7 , e13 , e16 , e20 , e26 ) (e2 , e7 , e13 , e16 , e19 , e25 ) (e2 , e7 , e13 , e16 , e19 , e25 ). bandwidth (kbps) (θ1 , θ2 , θ3 ) (334,154.816,25) (334,157.316,25) (334,159.816,25) (334,162.316,25) (334,164.816,25) (335.316,166,25) (337.816,166,25) (340.316,166,25) (342.816,166,25) (343.516,166,28) (346.316,166,28). budget ($) (c1 , c2 , c3 ) (12692,5883,950) (12692,5978,950) (12692,6073,950) (12692,6168,950) (12692,6263,950) (12742,6308,950) (12837,6308,950) (12932,6308,950) (13027,6308,950) (13053.6,6308,1064) (13160,6308,1064). total flow (kbps) 52631.58 52631.58 52631.58 52631.58 52631.58 52631.58 52631.58 52631.58 52631.58 52631.58 52631.58. optimal value (satisfaction) 0.627 0.629 0.630 0.632 0.633 0.634 0.635 0.637 0.638 0.639 0.640. (0.6.
(158) 26. Conclusions. • We present an approach for the fair resource allocation problem in All-IP networks that offer multiple services to users. • Users’ utility functions are summarized by means of achievement functions. We find that the achievement function can map different criteria onto a normalized scale. • The achievement function also can work in the Ordered Weighted Averaging method. Moreover, it may be interpreted as a measure of QoS on All-IP networks. • Using the bandwidth allocation model, we can find a Pareto optimal allocation of bandwidth on the network under a limited available budget. • Numerical results show that this scheme can provide each connection with its fair share of the bandwidth, which is proportional to the user’s preferences. – OR Lab, NCCU –.
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