Title

在預算限制下分配隨機數位網路最佳頻寬之研究

Translated Titles

Analysis of bandwidth allocation on End-to-End QoS networks under budget control

Authors

王嘉宏

Key Words

資源分配 ; 等候理論 ; 壅塞管理 ; 利潤最佳化模型 ; 頻寬分配 ; 隨機數位網路 ; Resource Allocation ; Queueing Theory ; Congestion Control ; Revenue Management ; Bandwidth Allocation ; End-to-End QoS Networks

PublicationName

政治大學應用數學研究所學位論文

Volume or Term/Year and Month of Publication

2010年

Academic Degree Category

博士

Advisor

陸行

Content Language

英文

Chinese Abstract

本論文針對隨機數位網路提出一套可行的計算機制,以提供網路管理者進行資源分配與壅塞管理的分析工具。我們研究兩種利潤最佳化模型,探討在預算控制下的頻寬分配方式。因為資源有限,網路管理者無法隨時提供足夠頻寬以滿足隨機的網路需求,而量測網路連結成功與否的阻塞機率(Blocking Probability)為評估此風險之一種指標。我們利用頻寬分配、網路需求量和虛擬端對端路徑的數量等變數,推導阻塞機率函數,並證明阻塞機率的單調性(Monotonicity)和凸性(Convexity)等數學性質。在不失一般性之假設下,我們驗證阻塞機率是(1)隨頻寬增加而變小;(2)在特定的頻寬分配區間內呈凸性;(3)隨網路需求量增加而變大;(4)隨虛擬路徑的數量增加而變小。 本研究探討頻寬分配與阻塞機率之關係,藉由推導單調性和凸性等性質,提供此兩種利潤模型解的最適條件與求解演算法。同時,我們引用經濟學的彈性概念,提出三種模型參數對阻塞機率變化量的彈性定義,並分別進行頻寬分配、網路需求量和虛擬路徑數量對邊際利潤函數的敏感度分析。當網路上的虛擬路徑數量非常大時,阻塞機率的計算將變得複雜難解,因此我們利用高負荷極限理論(Heavy-Traffic Limit Theorem)提供阻塞機率的估計式,並分析其漸近行為(Asymptotic Behavior)。本論文的主要貢獻是分析頻寬分配與阻塞機率之間的關係及其數學性質。網路管理者可應用本研究提出的分析工具,在總預算限制下規劃寬頻網路的資源分配,並根據阻塞機率進行網路參數的調控。

English Abstract

This thesis considers the problem of bandwidth allocation on communication networks with multiple traffic classes, where bandwidth is determined under the budget constraint. Due to the limited budget, there exists a risk that the network service providers can not assert a 100% guaranteed availability for the stochastic traffic demand at all times. We derive the blocking probabilities of connections as a function of bandwidth, traffic demand and the available number of virtual end-to-end paths for all service classes. Under general assumptions, we prove that the blocking probability is directionally (i) decreasing in bandwidth, (ii) convex in bandwidth for specific regions, (iii) increasing in traffic demand, and (iv) decreasing in the number of virtual paths. We also demonstrate the monotone and convex relations among those model parameters and the expected path occupancy. As the number of virtual paths is huge, we derive a heavy-traffic queueing model, and provide a diffusion approximation and its asymptotic analysis for the blocking probability, where the traffic intensity increases to one from below. Taking the blocking probability into account, two revenue management schemes are introduced to allocate bandwidth under budget control. The revenue/profit functions are studied in this thesis through the monotonicity and convexity of the blocking probability and expected path occupancy. Optimality conditions are derived to obtain an optimal bandwidth allocation for two revenue management schemes, and a solution algorithm is developed to allocate limited budget among competing traffic classes. In addition, we present three elasticities of the blocking probability to study the effect of changing model parameters on the average revenue in analysis of economic models. The sensitivity analysis and economic elasticity notions are proposed to investigate the marginal revenue for a given traffic class by changing bandwidth, traffic demand and the number of virtual paths, respectively. The main contribution of the present work is to prove the relationship between the blocking probability and allocated bandwidth under the budget constraint. Those results are also verified with numerical examples interpreting the blocking probability, utilization level, average revenue, etc. The relationship between blocking probability and bandwidth allocation can be applied in the design and provision of broadband communication networks by optimally choosing model parameters under budget control for sharing bandwidth in terms of blocking/congestion costs.

Topic Category 基礎與應用科學 > 數學
理學院 > 應用數學研究所
Reference
  1. [1] Abramov, V.M., Asymptotic analysis of loss probabilities in GI/M/m/n queueing systems as n increases to infinity, Quality Technology and Quantitative Management, Vol. 4, pp. 379--393, 2007.
    連結:
  2. [2] Aktaran-Kalayci, T., Ayhan, H., Sensitivity of optimal prices to system parameters in a steady-state service facility, European Journal of Operational Research, Vol. 193, pp. 120--128, 2009.
    連結:
  3. [3] Alkahtani, A.M.S., Woodward, M.E., Al-Begain, K., Prioritised best effort routing with four quality of service metrics applying the concept of the analytic hierarchy process,
    連結:
  4. [4] Al-Manthari, B., Ali, N.A. Nasser, N., Hassanein, H., Dynamic multiple-frame bandwidth provisioning with fairness and revenue considerations for broadband wireless access systems, Performance Evaluation, 2011, doi:10.1016/j.peva.2010.11.001
    連結:
  5. [6] Anderson, E., Kelly, F.P., Steinberg, R., A contract and balancing mechanism for sharing capacity in a communication network, Management Science, Vol. 52, No. 1, pp. 39--53, 2006.
    連結:
  6. [7] Andrade, R., Lisser, A., Maculan, N., and Plateau, G., B&B frameworks for the capacity expansion of high speed telecommunication networks under uncertainty, Annals of Operations Research, Vol. 140, pp. 49--65, 2005.
    連結:
  7. [8] Antunes, N., Fricker, C., Robert, P., Tibi, D., Analysis of loss networks with routing, Annals of Applied Probability, Vol. 16, No. 4, pp. 2007--2026, 2006.
    連結:
  8. [9] Apostolopoulos, G., Tripathi, S.K., On the effectiveness of path pre-computation in reducing the processing cost of on-demand QoS path computation, Proc. 3rd IEEE Symposium Computers and Communications, pp. 42--46, 1998.
    連結:
  9. [10] Atar, R., A diffusion model of scheduling control in queueing systems with many servers, Annals of Applied Probability, Vol. 15, pp. 820--852, 2005.
    連結:
  10. [11] Atkinson, J.B., Two new heuristics for the GI/G/n/0 queueing loss system with examples based on the two-phase Coxian distribution, Journal of the Operational Research Society, Vol. 60, No. 6, pp. 818--830, 2009.
    連結:
  11. [12] Atov, I., Tran, H.T., Harris, R.J., OPQR-G: Algorithm for efficient QoS partition and routing in multiservice IP Networks, ph{Computer Communications}, Vol. 28, pp. 1987--1996, 2005.
    連結:
  12. [13] Ayesta, U., Mandjes, M., Bandwidth-sharing networks under a diffusion scaling, Annals of Operations Research, Vol. 170, pp. 41--58, 2009.
    連結:
  13. [14] Bai, Y., Ito, M.R., Class-based packet scheduling to improve QoS for IP video, Telecommunication Systems, Vol. 29, No. 1, pp. 47--60, 2005.
    連結:
  14. [16] Berger, A.W., Kogan, Y., Dimensioning bandwidth for elastic traffic in high-speed data networks, IEEE/ACM Transactions on Networking, Vol. 8, No. 5, pp. 643--654, 2000.
    連結:
  15. [20] Bonald, T., Proutiere, A., Insensitive bandwidth sharing in data networks, Queueing Systems, Vol. 44, No. 1, pp. 69--100, 2003.
    連結:
  16. [21] Bonald., T., Proutiere, A., On performance bounds for balanced fairness, Performance Evaluation, Vol. 55, pp. 25--50, 2004.
    連結:
  17. [22] Bonald, T., Massoulie, L., Proutiere, A., Virtamo, J., A queueing analysis of max-min fairness, proportional fairness and balanced fairness, Queueing Systems, Vol. 53, pp. 65--84, 2006.
    連結:
  18. [23] Bonnans, J.F., Haddou, M., Asymptotic analysis of congested communication networks, Mathematics of Operations Research, Vol. 25, No. 3, pp. 409--426, 2000.
    連結:
  19. [26] Bruni, C., Priscoli, F.D., Koch, G., Marchetti, I., Resource management in network dynamics: An optimal approach to the admission control problem, Computers and Mathematics with Applications, Vol. 59, pp. 305--318, 2010.
    連結:
  20. [27] Bruno, R., Conti, M., Pinizzotto, A., Routing Internet traffic in heterogeneous mesh networks: Analysis and algorithms, Performance Evaluation, 2011, doi:10.1016/j.peva.2011.01.006
    連結:
  21. [28] Cancela, H., Rodriguez-Bocca, P., Tuffin, B., End-to-end availability-dependent pricing of network services, Annals of Operations Research, Vol. 157, pp. 61--71, 2008.
    連結:
  22. [29] Catanzaro, D., Gourdin, E., Labbe, M., Ozsoy, F.A., A branch-and-cut algorithm for the partitioning-hub location-routing problem, Computers and Operations Research, Vol. 38, No. 2, pp. 539--549, 2011.
    連結:
  23. [30] Chen, H., Zhang, H., Diffusion approximations for some multiclass queueing networks with FIFO service disciplines, Mathematics of Operations Research, Vol. 25, No. 4, pp. 679--707, 2000.
    連結:
  24. [31] Cheng, Y.H., Luh, H., Wang, C.H., Modeling on weighted utilizations of network dimensioning problems, International Journal of Operations Research, Vol. 7, No. 1, pp. 41--52, 2010.
    連結:
  25. [32] Chiu, D.M., Tam, A.S.W., Fairness of traffic controls for inelastic flows in the Internet, Computer Networks, Vol. 51, pp. 2938--2957, 2007.
    連結:
  26. [33] Cho, H., Girard, A., Rosenberg, C., On the advantages of optimal end-to-end QoS budget partitioning, Telecommunication Systems, Vol. 34, pp. 91--106, 2007.
    連結:
  27. [34] Choi, B.D., Kim, B., Kim, J., Wee, I.S., Exact convergence rate for the distributions of GI/M/c/K queue as K tends to infinity, Queueing Systems,Vol. 44, pp. 125--136, 2003.
    連結:
  28. [35] Choi, D.W., Kim, N.K., Chae, K.C., A two-moment approximation for the GI/G/c queue with finite capacity, Informs Journal on Computing, Vol. 17, No. 1, pp. 75--81, 2005.
    連結:
  29. [36] Correa, J.R., Schulz, A.S., Stier-Moses, N.E., Fast, fair, and efficient flows in networks, Operations Research, Vol. 55, No. 2, pp. 215--225, 2007.
    連結:
  30. [37] Dutta, M.Kr., Chaubey, V.K., Performance analysis of all-optical WDM network with wavelength converter using Erlang C traffic model, Communications in Computer and Information Science, Vol. 70, pp. 238--244, 2010.
    連結:
  31. [38] Erera, A.L., Daganzo, C.F., Lovell, D.J., The access-control problem on capacitated FIFO networks with unique O-D paths is hard, Operations Research, Vol. 50, No. 4, pp. 736--743, 2002.
    連結:
  32. [39] Esteves, J.S., Craveirinha, J., Cardoso, D., Computing Erlang-B function derivatives in the number of servers, Communications in Statistics -- Stochastic Models, Vol. 11, No. 2, pp. 311--331, 1995.
    連結:
  33. [40] Farago, A., Efficient blocking probability computation of complex traffic flows for network dimensioning, Computers and Operations Research, Vol. 35, No. 12, pp. 3834--3847, 2008.
    連結:
  34. [41] Fashandi, S., Gharan, S.O., Khandani, A.K., Path diversity over packet switched networks: Performance analysis and rate allocation, IEEE/ACM Transactions on Networking, Vol. 18, No. 5, pp. 1373--1386, 2010.
    連結:
  35. [42] Gerstel, O., Sasaki, G., A general framework for service availability for bandwidth-efficient connection-oriented networks, IEEE/ACM Transactions on Networking, Vol. 18, No. 3, pp. 985--995, 2010.
    連結:
  36. [43] Gozdecki, J., Jajszczyk, A., Stankiewicz, R., Quality of service terminology in IP networks, IEEE Communications Magazine, Vol. 41, No. 3, pp. 153--159, 2003.
    連結:
  37. [44] Guan, Y., Yang, W., Owen, H., Blough, D.M., A pricing approach for bandwidth allocation in differentiated service networks, Computers and Operations Research, Vol. 35, pp. 3769--3786, 2008.
    連結:
  38. [45] Guerin, R.A., Orda, A., QoS routing in networks with inaccurate information: Theory and algorithms, IEEE/ACM Transactions on Networking, Vol. 7, No. 3, pp. 350--364, 1999.
    連結:
  39. [46] Guven, T., La, R.J., Shayman, M.A., Bhattacharjee, B., A unified framework for multipath routing for unicast and multicast traffic, IEEE/ACM Transactions on Networking, Vol. 16, No. 5, pp. 1038--1051, 2008.
    連結:
  40. [47] Halachmi, B., Franta, W.R., A diffusion approximation to multi-server queue, Management Science, Vol. 24, No. 5, pp. 522--529, 1978.
    連結:
  41. [48] Halfin, S., Whitt, W., Heavy-traffic limits for queues with many exponential servers, Operations Research, Vol. 29, No. 3, pp. 567--588, 1981.
    連結:
  42. [49] Harel, A., Convexity properties of the Erlang loss formula, Operations Research, Vol. 38, No. 3, pp. 499--505, 1990.
    連結:
  43. [50] Harrison, J.M., Zeevi, A., Dynamic scheduling of a multiclass queue in the Halfin-Whitt heavy traffic regime, Operations Research, Vol. 52, No. 2, pp. 243--257, 2004.
    連結:
  44. [51] Helber, S., Schimmelpfeng, K., Stolletz, R., Lagershausen, S., Using linear programming to analyze and optimize stochastic flow lines, Annals of Operations Research, Vol. 182, No. 1, pp. 193--211, 2011.
    連結:
  45. [52] Hernandez-Orallo, E., Vila-Carbo, J., Efficient QoS routing for differentiated services EF flows, Proc. 10th IEEE Symposium Computers and Communications, pp. 91--96, 2005.
    連結:
  46. [53] van Hoesel, S., Optimization in telecommunication networks, Statistica Neerlandica, Vol. 59, No. 2, pp. 180--205, 2005.
    連結:
  47. [54] Huang, Q., Ko, K.T., Iversen, V.B., A new convolution algorithm for loss probability analysis in multiservice networks, Performance Evaluation, Vol. 68, No. 2, pp. 76--87, 2011.
    連結:
  48. [55] Izady, N., Worthington, D., Approximate analysis of non-stationary loss queues and networks of loss queues with general service time distributions, European Journal of Operational Research, 2011, doi:10.1016/j.ejor.2011.03.029
    連結:
  49. [57] Jarvis, J.P., Approximating the equilibrium behavior of multi-server loss systems, Management Science, Vol. 31, No. 2, pp. 235--239, 1985.
    連結:
  50. [58] Jeong, S.B., Kim, S., Lee, H., Data traffic scheduling algorithm for multiuser OFDM system with adaptive modulation considering fairness among users, Computers and Operations Research, Vol. 32, No. 7, pp. 1723--1737, 2005.
    連結:
  51. [59] Jin, N., Jordan, S., The effect of bandwidth and buffer pricing on resource allocation and QoS, Computer Networks, Vol. 46, pp. 53--71, 2004.
    連結:
  52. [60] Johari, R., Tan, D.K.H., End-to-end congestion control for the Internet: Delays and stability, IEEE/ACM Transactions on Networking, Vol. 9, No. 6, pp. 818--832, 2001.
    連結:
  53. [61] Jordan, S., A recursive algorithm for bandwidth partitioning, IEEE Transactions on Communications, Vol. 58, No. 4, pp. 1026--1030, 2010.
    連結:
  54. [62] Kim, H.S., Shroff, N.B., Loss probability calculations and asymptotic analysis for finite buffer multiplexers, IEEE/ACM Transactions on Networking, Vol. 9, No. 6, pp. 755--768, 2001.
    連結:
  55. [63] Kim, B., Choi, B.D., Asymptotic analysis and simple approximation of the loss probability of the GI^X/M/c/K queue, Performance Evaluation, Vol. 54, pp. 331--356, 2003.
    連結:
  56. [64] Kimura, T., A consistent diffusion approximation for finite-capacity multiserver queues, Mathematical and Computer Modelling, Vol. 38, pp. 1313--1324, 2003.
    連結:
  57. [65] Kelly, F.P., Charging and rate control for elastic traffic, European Transactions on Telecommunications, Vol. 8, pp. 33--37, 1997.
    連結:
  58. [66] Kelly, F.P., Maulloo, A.K., Tan, D.K.H., Rate control for communication networks: Shadow prices, proportional fairness and stability, Journal of the Operational Research Society, Vol. 49, pp. 237--252, 1998.
    連結:
  59. [68] Kelly, F.P., Fairness and stability of end-to-end congestion control, European Journal of Control, Vol. 9, pp. 159--176, 2003.
    連結:
  60. [69] Kelly, F.P., Williams, R.J., Fluid model for a network operating under a fair bandwidth-sharing policy, Annals of Applied Probability, Vol. 14, No. 3, pp. 1055--1083, 2004.
    連結:
  61. [70] Kimura, T., A consistent diffusion approximation for finite-capacity multiserver queues, Mathematical and Computer Modelling, Vol. 38, pp. 1313--1324, 2003.
    連結:
  62. [71] Krile, S., Perakovic, D., Load control for overloaded MPLS/DiffServ networks during SLA negotiation, International Journal of Communications, Network and System Sciences,
    連結:
  63. Vol. 2, No. 5, pp. 422--432, 2009.
    連結:
  64. [72] Kumar, N., Saraph, G., End-to-end QoS in interdomain routing, Proc. International Conference on Networking and Services, pp. 82-82, 2006.
    連結:
  65. [74] Lin, Y.K., Yeh, C.T., Using minimal cuts to optimize network reliability for a stochastic computer network subject to assignment budget, Computers and Operations Research, Vol. 38, No. 8, pp. 1175--1187, 2011.
    連結:
  66. [75] Lee, C.Y., Cho, H.K., Discrete bandwidth allocation considering fairness and transmission load in multicast networks, Computers and Operations Research, Vol. 34, No. 3, pp. 884--899, 2007.
    連結:
  67. [76] Low, S.H., Lapsley, D.E., Optimization flow control--Part I: Basic algorithm and convergence, IEEE/ACM Transactions on Networking, Vol. 7, No. 6, pp. 861--874, 1999.
    連結:
  68. [77] Low, S.H., A duality model of TCP and queue management algorithms, IEEE/ACM Transactions on Networking, Vol. 11, No. 4, pp. 525--536, 2003.
    連結:
  69. [79] Maglaras, C., Zeevi, A., Pricing and capacity sizing for systems with shared resources: Approximate solutions and scaling relations, Management Science, Vol. 49, No. 8, pp. 1018--1038, 2003.
    連結:
  70. [80] Maglaras, C., Zeevi, A., Pricing and design of differentiated services: Approximate analysis and structural insights, Operations Research, Vol. 53, No. 2, pp. 242--262, 2005.
    連結:
  71. [82] Massoulie, L., Roberts, J., Bandwidth sharing: objectives and algorithms, IEEE/ACM Transactions on Networking, Vol. 10, No. 3, pp. 320--328, 2002.
    連結:
  72. [83] Messerli, E., Proof of a convexity property of the Erlang B formula, Bell System Technical Journal, Vol. 51, pp. 951--953, 1972.
    連結:
  73. Tarasiuk, H., Dugeon, O., Diaz, M., Baresse, L., Monteiro, E., EuQoS: End-to-end quality of service over heterogeneous networks, Computer Communications, Vol. 32, pp. 1355--1370, 2009.
    連結:
  74. [85] Minoux, M., Mathematical Programming: Theory and Algorithms, Wiley, Chichester, 1986.
    連結:
  75. [86] Mo, J., Walrand, J., Fair end-to-end window-based congestion control, IEEE/ACM Transactions on Networking, Vol. 8, No. 5, pp. 556--567, 2000.
    連結:
  76. [87] Nain, P., Qualitative properties of the Erlang blocking model with heterogeneous user requirements, Queueing Systems, Vol. 6, No. 2, pp. 189--206, 1990.
    連結:
  77. [88] Nilsson, P., Pioro, M., Solving dimensioning tasks for proportionally fair networks carrying elastic traffic, Performance Evaluation, Vol. 49, pp. 371--386, 2002.
    連結:
  78. [90] Orda, A., Routing with end-to-end QoS guarantees in broadband networks, IEEE/ACM Transactions on Networking, Vol. 7, No. 3, pp. 365--374, 1999.
    連結:
  79. [92] Paschalidis, I.Ch., Liu, Y., Pricing in multiservice loss networks: Static pricing, asymptotic optimality, and demand substitution effects, IEEE/ACM Transactions on Networking, Vol. 10, No. 3, pp. 425--438, 2002.
    連結:
  80. [93] Pioro, M., Malicsko, G., Fodor, G., Optimal link capacity dimensioning in proportionally fair networks, Lecture Notes in Computer Science, Vol. 2345, pp.277-288, 2002.
    連結:
  81. [94] Rardin, R.L., Optimization in Operations Research, Prentice-Hall, 1998.
    連結:
  82. [95] Roberts, J.W., A survey on statistical bandwidth sharing, Computer Networks, Vol. 45, pp. 319--332, 2004.
    連結:
  83. [96] Rodrigo, M.V., Latouche, G. Remiche, M.A., Blocking probability computation in reversible Markovian bufferless multi-server systems, Performance Evaluation, Vol. 67, No. 3, pp. 121--140, 2010.
    連結:
  84. [97] Ross, S.M., Stochastic Processes, New York: Wiley, 1983.
    連結:
  85. [99] Shakkottai, S., Srikant, R., Economics of network pricing with multiple ISPs, IEEE/ACM Transactions on Networking, Vol. 14, No. 6, pp. 1233--1245, 2006.
    連結:
  86. [100] Shenker, S., Fundamental design issues for the future Internet, IEEE Journal of Selected Areas in Communications, Vol. 13, pp. 1176--1188, 1995.
    連結:
  87. [102] Simonot, F., A comparison of the stationary distributions of GI/M/c/n and GI/M/c, Journal of Applied Probability, Vol. 35, No. 2, pp. 510--515, 1998.
    連結:
  88. [103] Smith, J.M., M/G/c/K blocking probability models and system performance, Performance Evaluation, Vol. 52, pp. 237--267, 2003.
    連結:
  89. [105] Takacs, L., Introduction to the Theory of Queues, Oxford University Press, New York/London (1962)
    連結:
  90. [109] Thomas, P., Teneketzis, D., Mackie-Mason, J.K., A market-based approach to optimal resource allocation in integrated-services connection-oriented networks, Operations Research, Vol. 50, No. 4, pp. 603--616, 2002.
    連結:
  91. [110] Vijaya Laxmi, P., Gupta, U.C., Analysis of finite-buffer multi-server queues with group arrivals: GI^{X}/M/c/N, Queueing Systems, Vol. 36, pp. 125--140, 2000.
    連結:
  92. [111] Wang, C.H., Luh, H., A precomputation-based scheme for QoS routing and fair bandwidth allocation, Lecture Notes in Computer Science, Vol. 4297, pp. 595--606, 2006.
    連結:
  93. [117] Wang, C.H., Luh, H., A utility maximization scheme for end-to-end QoS routing in communication networks with heterogeneous service classes, Proc. IEEE International Conference on Service Operations and Logistics, and Informatics (IEEE/SOLI'2008), pp. 488--493, 2008.
    連結:
  94. [122] Wang, C.H., Luh, H., Blocking probability on end-to-end QoS networks under budget control, working paper, submitted to Performance Evaluation, 2010.
    連結:
  95. [123] Wang, C.H., Luh, H., Analysis of bandwidth allocation on end-to-end QoS networks under budget control, Computers and Mathematics with Applications, 2011, doi:10.1016/j.camwa.2011.05.024
    連結:
  96. [124] Whitt, W., Heavy-traffic approximations for service systems with blocking, AT&T Bell Laboratories Technical Journal, Vol. 63, No. 5, pp. 689--708, 1984.
    連結:
  97. [125] Whitt, W., How multiserver queues scale with growing congestion-dependent demand, Operations Research, Vol. 51, No. 4, pp. 531--542, 2003.
    連結:
  98. [126] Whitt, W., A diffusion approximation for the G/Gl/n/m queue, Operations Research, Vol. 52, No. 6, pp. 922--941, 2004.
    連結:
  99. [127] Whitt, W., Heavy-traffic limits for the G/H_2*/n/m queue, Mathematics of Operations Research, Vol. 30, No. 1, pp. 1--27, 2005.
    連結:
  100. [129] Xiao, X., Ni, L.M., Internet QoS: A big picture, IEEE Network, Vol. 13, No. 2, pp. 8--18, 1999.
    連結:
  101. [130] Yacoubi, M., Emelianenko, M., Gautam, N., Pricing in next generation networks: A queuing model to guarantee QoS, Performance Evaluation, Vol. 52, pp. 59--84, 2003.
    連結:
  102. [131] Ye, H.Q., Qu, J., Stability of data networks: Stationary and bursty models. Operations Research,Vol. 53, No. 1, pp. 107--125, 2005.
    連結:
  103. [132] Yuksel, M., Kalyanaraman, S., Elasticity considerations for optimal pricing of networks, Proc. IEEE Symposium on Computer Communications, Vol. I, pp. 163--168, 2003.
    連結:
  104. [133] Zachariadis, G., Barria, J.A., Dynamic pricing and resource allocation using revenue management for multiservice networks, IEEE/ACM Transactions on Networking, Vol. 5, No. 4, pp. 215--226, 2008.
    連結:
  105. Computers and Operations Research, Vol. 33, pp. 559--580, 2006.
  106. [5] Amiri, A., Barkhi, R., The combinatorial bandwidth packing problem, European Journal of Operational Research, Vol. 208, No. 1, pp. 37--45, 2011.
  107. [15] Bakry, S.H., A new method for computing Erlang-B formula, Computers and Mathematics with Applications, Vol. 19, No. 2, pp. 73--74, 1990.
  108. [17] Bertsekas, D.P., Gallager, R., Data Networks, 2nd ed., Prentice Hall, New Jersey, 1992.
  109. [18] Bertsekas, D.P., Network Optimization, Athena Scientific, 1998.
  110. [19] Bogomolnaia, A., Holzman, R., Moulin, H., Sharing the cost of a capacity network, Mathematics of Operations Research, Vol. 35, No. 1, pp. 173--192, 2010.
  111. [24] Borovkov, A.A., Stochastic Processes in Queueing Theory, New York: Springer-Verlag, 1976.
  112. [25] Boutaba, R., Szeto, W., Iraqi, Y., DORA: Efficient routing for MPLS traffic engineering, Journal of Network and Systems Management, Vol. 10, No. 3, pp. 309--325, 2002.
  113. [56] Jagers, A.A., Van Doorn, E.A., On the continued Erlang loss function, Operations Research Letters, Vol. 5, No. 1, pp. 43--46, 1986.
  114. [67] Kelly, F.P., Mathematical modelling of the Internet, Mathematics Unlimited - 2001 and Beyond, Springer-Verlag, Berlin, pp. 685--702, 2001.
  115. [73] Lien, Y.N., Jang, H.C., Tsai, T.C., Luh, H., BBQ: A QoS management infrastructure for All-IP networks, Communications of Institute of Information and Computing Machinery: Mobile Communications and Wireless Networks, Vol. 7, No. 1, pp. 89--115, 2004.
  116. [78] Luh, H., Wang, C.H., Proportional bandwidth allocation for unicasting in All-IP networks, Proc. 2nd Sino-International Symposium on Probability, Statistics, and Quantitative Management, pp. 111--130, 2005.
  117. [81] Massoulie, L., Roberts, J., Bandwidth sharing and admission control for elastic traffic, Telecommunication Systems, Vol. 15, pp. 185--201, 2000.
  118. [84] Mingozzi, E., Stea, G., Callejo-Rodriguez, M.A., Enriquez-Gabeiras, J., Garcia-de-Blas, G., Ramon-Salquero, F.J., Burakowski, W., Beben, A., Sliwinski, J.,
  119. [89] Ogryczak, W., Sliwinski, T., Wierzbicki, A., Fair resource allocation schemes and network dimensioning problems, Journal of Telecommunications and Information Technology, Vol. 3, pp. 34--42, 2003.
  120. [91] Orda, A., Sprintson, A., Precomputation schemes for QoS routing, IEEE/ACM Transactions on Networking, Vol. 11, No. 4, pp. 578--591, 2003.
  121. [98] Rudin, W., Principles of Mathematical Analysis, McGraw-Hill Science/Engineering/Math, 3rd edition, 1976.
  122. [101] Shetty, N., Schwartz, G., Walrand, J., Internet QoS and regulations, IEEE/ACM Transactions on Networking, Vol. 18, No. 6, pp. 1725--1737, 2010.
  123. [104] Stockman, A.C., Introduction to Economics, 2nd ed., Dryden Press, 1999.
  124. [106] The Cooperative Association for Internet Data Analysis (CAIDA). [Online]. Available: http://www.caida.org/home
  125. [107] The MathWorks Company, MATLAB The Language of Technical Computing: Using MALTAB, Version 6, 2002.
  126. [108] The UMTS Forun: Enabling UMTS/Third Generation Services and Applications. UMTS Forun Report, Vol. 11, 2000.
  127. [112] Wang, C.H., Luh, H., Network dimensioning problems of applying achievement functions. In: Zhang, X.S., Liu, D.G., Wu, L.Y. (Eds.), Lecture Notes in Operations Research--Operations Research and Its Applications, Vol. 6, pp. 35--59, World Publishing Corporation, 2006.
  128. [113] Wang, C.H., Yue, W., Luh, H., Performance evaluation of predetermined bandwidth allocation for heterogeneous networks, IEICE Technical Report, Vol. 107, No. 6, pp. 37--42, 2007.
  129. [114] Wang, C.H., Luh, H., Two-phase modeling of QoS routing in communication networks, Proc. 16th International Conference on Computer Communications and Networks (ICCCN'07), pp. 1210--1216, 2007.
  130. [115] Wang, C.H., Luh, H., Bandwidth allocation and QoS routing for heterogeneous networks, Applied Mathematical Sciences, Vol. 1, No. 43, pp. 2139--2151, 2007.
  131. [116] Wang, C.H., Luh, H., A fair QoS scheme for bandwidth allocation by precomputation-based approach, International Journal of Information and Management Sciences, Vol. 19, No. 3, pp. 391--412, 2008.
  132. [118] Wang, C.H., Luh, H., Mathematical models of QoS management for communication networks. In: Alvarez, M.P. (Eds.), Leading-Edge Applied Mathematical Modeling Research, Chapter 8, pp. 295--317, Nova Science Publishers Inc., 2008.
  133. [119] Wang, C.H., Luh, H., Blocking probabilities of multiple classes in IP networks with QoS routing. In: Yue, W., Takahashi, Y., Takagi, H. (Eds.), Advances in Queueing Theory and Network Applications, Chapter 16, pp. 281--290, Springer, 2009.
  134. [120] Wang, C.H., Luh, H., A fast branch and bound based algorithm for bandwidth allocation and QoS routing on class-based IP networks, Proc. 1st World Congress on Global Optimization in Engineering and Science (WCGO2009), Hunan, China, June 1--5, 2009. Full paper is available in CD-ROM and ISTP database.
  135. [121] Wang, C.H., Luh, H., Measuring average revenue with bandwidth and traffic demand on communication networks, Proc. International Conference on Computational Intelligence and Software Engineering (CiSE 2009), Wuhan, China, December 11--13, 2009. Full paper is available in CD-ROM, IEEE Xplore and ISTP databases.
  136. [128] Wolfram, S., The Mathematica Book, Wolfram Media, Inc., 5th edition, 2004.
  137. [134] Zukerman, M., Mammadov, M., Tan, L., Ouveysi, I., Andrew, L.L.H., To be fair or efficient or a bit of both, Computers and Operations Research, Vol. 35, No. 12, pp. 3787--3806, 2008.
Times Cited
  1. 劉文龍(2006)。都市交通管理系統現場設備之無線通訊網路方案評選研究。淡江大學運輸管理學系碩士班學位論文。2006。1-153。