Adaptive Resource Allocation for Electric Environmental Pollution through the Control Network
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1 Available olie at Eergy Procedia 6 (202) Iteratioal Coferece o Future Eergy, Eviromet, ad Materials Adaptive Resource Allocatio for Electric Evirometal Pollutio through the Cotrol etwork Yig Yag Zhehai Petrochemical Jia a Egieerig Co., Ltd. igbo, Zhejiag, P. R. Chia Abstract This paper focus o the problem of adaptive resource allocatio for the electric eviromet pollutio cotrol through the web-based GIS. The paper provides a optimizatio problem for a geeral utility fuctio such that it could be ococave ad odifferetiable. To solve this problem, the paper decompose the origial optimizatio problem ito the dual optimizatio problem, which ca be easily solved by a simple iterative bisectio algorithm to adaptively cotrol the electric evirometal pollutio through the web-based GIS. 20 Published by Elsevier B.V. Selectio ad/or peer-review uder resposibility of Iteratioal Materials Sciece Society. 20 Published by Elsevier Ltd. Selectio ad/or peer-review uder resposibility of [ame orgaizer] Keywords: adaptive resource allocatio; web-based GIS; dual optimizatio problem;. Itroductio These years, researches have a sigificat amout of iterest i the area of web-based GIS, which aims to trasmit the iformatio of the eviromet through the etwork to cotrol the eviromet pollutio. Curretly, most services i the web-based system are elastic to some degree, i.e., the etwork sources ca be adaptively assiged to adjust their trasmissio data rates accordig to cogestio levels withi the etwork. Hece, by appropriately exploitig the elasticity through rate cotrol, oe ca provide high etwork efficiecy while at the same time alleviatig etwork cogestio i web-based GIS. Usually, there are two classes of the utility fuctios i the GIS etwork: oe is the traditioal date service, whose utility fuctio is modeled as a cocave utility fuctio. This utility fuctio refers to the low data trasmissio such as file trasfer. The other correspods to high data rate trasmissio, such as video or audio services, which is modeled as a o-covex utility fuctio. I the past years, the first type of the utility fuctios is exteded studied [2]-[4]. Curretly, researches start pay attetio from the traditioal type of the utility fuctios to the secod type of the utility fuctios [5]-[7] Published by Elsevier B.V. Selectio ad/or peer-review uder resposibility of Iteratioal Materials Sciece Society. doi:0.06/j.egypro
2 Yig Yag / Eergy Procedia 6 (202) However, there is o article to solve the electric evirometal pollutio usig the web-based GIS. Furthermore, some urealistic assumptios sigificatly limit the applicatio of above approximate utility fuctios. Thus, it should ivestigate a more geeral type of utility fuctio for the adaptive electric evirometal pollutio etwork. I this paper, i order to exted the result of the recet articles, a more geeral utility fuctio, which ca be ococave ad odifferetiable, is ivestigated i the web-based etwork for the electric evirometal pollutio cotrol. A dual-decompositio-based optimizatio problem is derived, ad a simple iterative bisectio algorithm ca be used to deal with data trasmissio i the web-based etwork for the cotrol of electric evirometal pollutio. The remaider of this paper is orgaized as follows. The system model ad problem framework are give i sectio 2. We the preset the umerical results ad coclusios are give i Sectio 3 ad Sectio 4, respectively. 2. System Descriptio ad Problem Formulatio I the web-based GIS etwork for the electric eviromet pollutio cotrol, the goal of adaptive trasmissio allocatio i the etwork is to fid the optimal resource assigmet i order to maximize the total trasmissio utility i the web-based GIS, which ca be expressed as max x = = U ( x ) st.. x Ctotal () x 0, where Ctotal is the etwork resource costraits for users, ad the utility fuctio U has the followig properties. ) U is a cotiue fuctio of x. 2) U a icreasig fuctio of the allocated etwork resource x. 3) U ca be a ocovex ad odifferetiable fuctio. Sice the objective fuctio U( x ) is ocovex ad odifferetiable, it is difficult to deal with the optimizatio problem i (). Therefore, we wat to obtai the dual optimizatio of the optimizatio problem of (), ad the solve the optimiztio problem i a easier way. First, we ca get the Lagragia fuctio of () as follows. L( x, ) = U ( x ) x Ctotal (2) = = where x = { x, x2,, x m } (3) 0 Based o [], the optimal solutio of problem () ca be rewritted as x = arg mi L( x, ) (4) xi 0, i=,2,, m where x is the global optimal solutio. * Based o [], we has the optimal solutio ( x, ) as x Ctotal = 0 (5) =
3 62 Yig Yag / Eergy Procedia 6 (202) Sice x Ctotal 0 ad 0, we have = Based o (2), (5) ad (6), we get The, we have x Ctotal 0 (6) = L x (, ) = max { L( x, ) } L x (, ) L( x, ) { L( x ) x 0, =,2,, } = max mi, mi ( ) x 0, =,2,, = = { Ctotal U ( x ) x, x 0 } = max U x + x Ctotal = max + mi - + which meas that we ca trasfer the optimizatio problem to be a separable optimizatio problem. The, we ca formulate the dual optimizatio problem as g = max q (9) where Cosiderig the costrait = x 0 ( ) ( ) mi (, ) q = C + L x (0) C formulate the dual optimizatio problem as total x 0 = ( ) arg mi L ( x, ) β = () x 0 (, ) ( ) L x = U x + x (2) total i the origial optimizatio problem (), we ca further g st.. = max 0 = About the dual optimizatio problem of (3), we ca use the iterative bisectio algorithm to solve this problem. 3. umerical Results I the umerical results, we provide umerical results to illustrate the performace of the proposed adaptive resource allocatio. I the simulatio, we cosider a electric pollutio through the web-based GIS i Fig., which cotais two iformatio sources S ad S2, ad two cotrol users D ad D2. For simple, we choose the sigmoid utility fuctio to simulate the performace. β q ( ) ( ) C total (7) (8) (3)
4 Yig Yag / Eergy Procedia 6 (202) S D GIS S2 D2 Fig.. Electric Evirometal Pollutio Cotrol System. Fig. 2 compares the performace of the dual optimizatio usig the iterative bisectio algorithm with the global optimizatio. From Fig. 2, we ca see that the performace of the proposed algorithm is almost the same as the global optimizatio, which idicates that the duality gap of the dual optimizatio problem is very small. Thus, it is reasoable for us to use the proposed iterative bisectio algorithm to solve the adaptive resource allocatio for electric eviromet pollutio cotrol i the web-based GIS Utility Global Optimizatio Proposed Algorithm Average Costs Fig. 2. Compariso of proposed algorithm ad Equal power allocatio. 4. Summary I this paper, we studied a adaptive resource allocatio for the iformatio trasmissio for the electric pollutio cotrol i the web-based GIS. A dual optimizatio problem is derived ad a simple
5 64 Yig Yag / Eergy Procedia 6 (202) iterative bisectio algorithm is proposed to maximize the trasmissio data rates. We obtai that the performace of the proposed algorithm is almost the same as the Global optimizatio, which meas that the electric pollutio iformatio ca be effectively trasmitted through the GIS usig the proposed algorithm. Refereces [] Bertsekas, D. P.: oliear Programmig. Athea Scietific, 999. [2] Sheker, S.: Fudametal desig issues for the future Iteret. IEEE J. Sel. Areas Commu.995, p [3] Kelly F. P., Chargig ad rate cotrol for elastic traffic. Eur. Tras. Telecommu. 997, p [4] Low, S. H., Lapsley, D. E.: Optimizatio flow cotrol, I: Basic algorithm ad covergece. IEEE/ACM Tras. etw. 999, p [5] La, R. J., Aatharam, V.: Utility-based rate cotrol i the Iteret for elastic traffic. IEEE/ACM Tras. etw. 2002, p [6] J. W. Lee, R. R. Mazumdar, ad. B. Shroff, o-covexity Issues for Iteret Rate Cotrol with. Multi-class Services: Stability ad Optimality, IEEE Ifocom, 2004, p [7] Lee, J.-W., Mazumdar, R. R., Shroff,. B.: o-covex Optimizatio ad Rate Cotrol for Multi-Class Services i the Itere. IEEE/ACM Tras. etw. 2005, p
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