Power Allocation Analysis for Dynamic Power Utility in Cognitive Radio Systems

Transcription

1 Power Allocation Analysis for Dynamic Power Utility in Cognitive Radio Systems Mahdi Ben Ghorbel 1, Bassem Khalfi 2#, Bechir Hamdaoui 3#, Mohsen Guizani 4 Qatar University, Doha, Qatar, # Oregon State University, Oregon, USA 1 mahdibenghorbel@queduqa, 2,3 {khalfib,hamdaoui@eecsorstedu, 4 mguizani@ieeeorg Abstract The focus of is paper is to investigate e fundamental limits of power allocation when taking into account a dynamic power pricing scheme This paper proposes an optimal power allocation analysis for wireless systems when real time power pricing is available We propose to minimize e total power consumption cost while ensuring minimum individual and total roughput limits We consider different models for e power pricing function Analytic solutions for e power allocation are derived for each model The derived solutions are shown to be modified versions of e water-filling solution Lowcomplexity algorims are proposed for e resource allocation wi each pricing model Performance comparison and pricing effect are shown rough simulations Index Terms Energy consumption awareness, spectrum access efficiency, power pricing I INTRODUCTION Optimal resource allocation is a crucial task in wireless communication systems to respond to e continuously increasing demand in terms of target data rates and network coverage which require higher and higher resources Thus, an efficient use of e available resources, mainly e power and spectrum, became one of e principal challenges for future communications The power consumption concerns were driven by e growing worries about e effect of is explosive demand in terms of power which were recognized as a maor reat from environmental and economic perspectives In fact, e huge demand in terms of power not only leads to higher cost but also contributes to e global warming phenomena rough e increase of e carbon footprint This tradeoff led to e emergence of e concept of green communication 1], which encourages developing energy-efficient communication systems The spectrum awareness was driven by e problem of spectrum scarcity due to increasing number of wireless devices as observed by e Federal Communications Commission FCC 2] Cognitive radio systems have attracted a great interest recently as a means to enhance e spectrum efficiency and overcome e problem of spectrum scarcity 3 5] It enables opportunistic access to e spectrum, and has en been seen as a key candidate for enabling dynamic spectrum access by taking advantage of e awareness about e surrounding environment The challenges related to cognitive networks have been addressed in e literature, ranging from spectrum sensing and protocol design6, 7] to spectrum access and analysis 8 1] The resource allocation task in cognitive radio systems is of great importance This task should allow to maximize spectrum utilization efficiency while taking into consideration e impacts of power consumption expressed in terms of cost and carbon footprint Resource allocation has also been intensively investigated in e literature 11 13] In 12], e auors proposed e optimal power allocation in an OFDM based underlay spectrum sharing by targeting e maximization of e total rate The merits of e proposed scheme compared to e classical water-filling is at e latter does not account for e interference constraint In 13], e auors formulated e problem of secondary users capacity maximization while considering a given quality of service of e primary user as well as e SU They used a geometric program to solve e problem Oer works considered e power allocation for oint underlay and overlay in multiband cognitive systems by considering an auction based approach to deal wi e competition between e secondary users 14] In 15], e auors used a game eoretical approach to allocate e power among e secondary users while accounting for eir quality of service as well as protecting e primary system Inspired by e emergence of dynamic pricing in future power grids smart grids as well as e need for providing green communication, we consider to propose an optimal power allocation for a cognitive radio system at minimizes a generic cost function of e allocated power while ensuring e performance requirements in terms of minimum individual and total roughput and at e same time protecting e licensed users by setting a reshold interference temperature Even ough e targeted power gains are not important to generate profit in e case of classic wireless systems due to small power consumption, is work can be of paramount importance for large wireless systems wi high power consumption and high targeted roughput 6 Ghz communication, free space optical communication which are expected to replace backhaul connections based on wired links We are targeting analytical solutions to allow analysis of e system s performance and obtained gains Obtained results can be employed later in decision algorims for multiple service providers The rest of is paper is organized as follows In Section II, we describe e system model and present e dynamic power cost models used in is work In Section III, we formulate e resource allocation problem In Section IV, we analyze and propose e power allocation solutions for different cost functions Then, simulation results are presented in Section V Finally, conclusion is drawn in Section VI

2 II SYSTEM MODEL AND DYNAMIC PRICING A System Model We consider a dynamic spectrum access system where a cognitive system, called secondary system, is sharing e spectrum wi a licensed system, called e primary system The cognitive system is composed of a secondary transmitter ST communicating wi its secondary receiversr while e primary system is composed of a primary transmitter PT and a primary receiver PR We assume at e communication pattern follows an underlay spectrum sharing scheme; e primary system is operating wiout paying attention to e presence of e secondary system while e secondary system is using power control mechanism to control e interference caused to e primary receiver PR Hence, e caused interference should be kept below an interference temperature I We assume at N channels are available for e secondary system Note at e diversity scheme is not specified in our work Thus, it could be applied for various access schemes time, frequency, space, etc For example, is could be applied for a multi-antenna system where different channel gains arepresentatesecondarysystemitcanalsobeappliedfora multi-carrier spectrum access where e secondary could send its data over more an one band We denote e instantaneous channel gains between e secondary transmitter ST and e secondary receiver SR by {h c 1,,hc,,h c N while e instantaneous channel gains of e interference channels between e secondary transmitter ST and e primary receiver PR are denoted by {h p 1,,hp,,h p N The received roughput in a channel when employing a power P is written as wi γ c r P = log 2 1P γ c, 1 = hc 2 w N where N is e noise power density at is assumed to be constant for all e channels The cognitive user tries to find e optimal power allocation among e different channels wi regards to e given cost function B Dynamic Power Cost In conventional power allocation problems, e cognitive transmitter tries to minimize its used power to meet certain requirements in terms of roughput This problem was solved in different previous works using mainly e well-known water-filling algorim 12] In a more sophisticated set-up, we consider a more general case where e obective is to minimize a cost function C at takes into account various aspects related to economic and/or environment impact of e used power For instance, is obective function could model e cost of e power procured from a utility company wi a dynamic pricing variation depending on demand level In anoer context taking into consideration green communications obectives, e cost function could model e carbon emission of e used power For generality purposes, we consider a generic cost function at could include different parts modeling bo energy consumption, cost, and environment impact Our study will not focus on modeling is cost but raer on proposing an optimal power allocation scheme for a generic cost function Thus, Let c P 1,,P N be e cost of e power P consumed by e channel Hence, e total cost in is case could be modeled as CP 1,,P N = c P 1,,P N 2 =1 where c P 1,,P N is continuously differentiable function, increasing and convex as of P III PROBLEM FORMULATION The focus of is work is to investigate e optimal power allocation when taking into account oer aspects an e channel gains Hence, we are seeking e optimal power level s selection at allows to minimize our cost function C wi regards to e different system requirements Maematically, we formulate our optimization problem as follows min P 1,,P N CP 1,,P N 3a Sub to r P rtot 3b =1 r P r {1N, 3c P g p I {1N, 3d where rtot denotes e required total roughput while r denotes e required individual roughput per channel Equation 3b constrains e system to achieve a total desired roughput using e N channels while Eequation 3c requires at each channel to achieve a minimum roughput at corresponds to a minimum quality of service However, e constraint 3d is e requirement of e underlay paradigm used to respect e interference level at e primary system wi g p = h p 2 From practical considerations, is system is recurrent in a wide number of applications An example of application is when e transmitter have different pas wi his correspondent receiver different time slots, bands, antennas rtot is e total roughput at e whole system should reach while r is e minimum quality of service at each pa In e optimization problem3, e values for e minimum total roughputrtot and e minimum roughputper channel r along wi e interference temperature I may lead to some conflicting constraints If N =1 r r Tot, e constraint 3b will automatically be guaranteed On e oer hand, e problem could be unfeasible in a number of cases; if r I < r g p or if r I < r g p Tot In ese cases, e interference constraint imposed by e primary system will limit e maximum transmit power at e cognitive transmitter side and fails to reach eier e minimum roughput per channel or e minimum total roughput

3 IV PROBLEM ANALYSIS WITH DIFFERENT PRICING MODELS Given at e cost function of e power c P 1,,P N is continuously derivable and convex, e problem 3 is a convex optimization problem Thus, we propose to alternatively solve its dual problem using e Karush-Kuhn-Tucker KKT conditions as duality gap is zero under e Slater condition 16] For our case, e Slater condition is satisfied when e problem is feasible Maematically, e feasibility conditions are written as r I g p I r =1 g p r, {1,,N 4 r Tot 5 Thedualproblemofecategoryproblem3canbewritten as follows L {P N =1,λ,{λ N =1,{ν N =1 = N c P 1,,P N λ rtot r P =1 =1 λ r r P =1 =1 ν P g p I, 6 where λ, {λ N =1, and {ν N =1 are e KKT multipliers After simplifications, e KKT constraints are written as follows c ip 1,,P N γ c λ = {1N 7 P i=1 1P γ c log2 λ rtot r P = 8 P 2r 1 γ c =1 {1N 9 P I {1N 1 g p λ 11 Hence, e optimal power allocation for every channel is deduced such at it satisfies e following Equation: 1Pγ c N c ip 1,,P N = γ c P i=1 Sub to P P P, 12 where is a constant proportional to λ determined such at e total roughput constraint is saturated ie, N =1 r P = rt, while P and P are defined as P = 2r 1 γ c 13 P = I g p Note at uniqueness of P could be verified in e interval P,P ] if it exists in is interval Proof Let us denote g P = 1P γ c N c i P 1,,P N λ P γ c 14 i=1 Solving Equation 12 is equivalent to finding e zero of e function g in e interval P,P ] Thus, we compute its derivative as follows g P = 1P γ c N 2 c i P 1,,P N P 2 15 i=1 γ c i=1 c i P 1,,P N P, which is positive since c i P 1,,P N are by definition increasing and convex function of P, i, Thus, we prove e uniqueness of e solution The existence of is solution could be checked by testing e interval bounds ie, it exists if and only if g P g P Depending on e cost function expression, e solution could be simplified furer Thus, in e following we will consider some families of cost functions and express e power allocation for each case starting from e most simple models to generic expressions A Constant Unit Price In is section, we consider a cost function wi a variable unitary power price across e channel wiout depending on e consumed power In fact, as specified in e system model, e channels in our case represent generic diversity of e pas Thus, for instance, e variable price could be applicable when channels represent different time slots or different power providers In is case, e power cost function is expressed as follows c P 1,,P N = µ P, 16 where µ is e unitary power cost for e - channel Given, is model, e allocated power expression is deduced from 12 as P = µ γ c ] x x if x > x, where x = x x if x < x, x oerwise The water-level,, is expressed as = 2 r S c S p 2 r 2 r Tot {S c S p P γc, 17 µ wi S c and S p defind as { S c = {1,,N such at λ { S p = {1,,N such at λ 1 N Sc Sp, 18 µ γ c µ γ c < P > P 19

4 The obtained roughput per channel is deduced en as a function of e signal-to-noise ratio and unit price per channel as r = log 2 γ c ] r 2 µ r We obtain a water-filling expression used in resource allocation algorims over multichannel systems 17] wi e modification at e channel unit price power will affect e power allocated in each channel The allocated power is obtained as e difference between e water-level and e ratio of e unit price by e channel gain instead of e inverse of e channel gain in ordinary water-filling B Power Consumption Dependent Unit Price In is section, we assume at e unitary power cost depends not only on e channel but also on e allocated power in at channel In fact, in practical scenarios, power providers impose higher unitary power prices when e consumption increases Similarly, to penalize high power consumers, higher factors are associated when e allocated power increases in e carbon impact computation Thus, in is section, we study e following model for e cost function c P 1,,P N = µ P P, 21 where µ P is e unitary cost function 1 Linear Unit Price function: We consider e unitary cost as a linear function of e consumed power, ie, µ P = a b P, 22 where a and b are power pricing coefficients fixed by e power provider and can be obtained in real-time rough e back-haul network Inserting 22 and 21 in 12, we obtain e following equation to solve for e allocated power per channel P = a 2b P γ c P 23 Alough is is a quadratic equation, non-negativity of e allocated power per channel results in obtaining a unique solution which is written as P = aγ c 2b aγ c 2b 2 8b γ c 2 4b γ c P 24 The expression of can not be derived analytically in is case but it can be obtained by solving e total roughput constraint which is transformed into finding e zero of e function f for wi f = {S c S p 2 r Tot 2 r 2 r S c S p 2b a γ c a γ c 2b 2 8b 4b c 2 γ =, 25 wi S c and S p defined as { S c = {1,,N such at { {1,,N such at S p = a 2b P < P γ c > P a 2b P γ c 26 It is easy to check at f is continuous and decreasing wi f > and lim = us it has a unique zero which can be obtained fλ numerically Alough in is case we do not obtain a strictly speaking water-filling expression, a similar algorim can be developed where will represent an "imaginary" water-level as it remains constant for all channels The pseudo water-filling expression can be deduced from 23 as follows P = λ ˆµ N g c P, 27 wi ˆµ = a 2b P is e effective power cost in e - channel Note at is is not a water-filling equation as ˆµ depends on e allocated power P but it only allows to analyze e allocated power function to e channel gains and e price coefficients Thus, we obtain a system of nonlinear coupled Equations 24 and 25 An iterative approach allows us to determine is water-level and us obtain e optimal power allocation per channel by solving at each step consecutively 24 and 25 until convergence This algorim has e same convergence speed as e regular water-filling algorim The only difference is at e water-level is determined analytically in regular water-filling while it is obtained numerically by solving 25 in is case 2 Polynomial Unit Price function: We consider e unitary cost as a general polynomial function of e consumed power as follows p µ P = a,i P i, 28 i= where p is e polynomial degree and a,i are power pricing coefficients fixed by e power provider Inserting 28 and 21 in 12, we obtain e following Equation to solve for e allocated power per channel P = p 1 i= a,i1i1p i γ c P 29 Since e solution of is equation is unique if it exists as shown earlier, we transform e problem into root finding problem of e following polynomial wi p1 α,i P, i 3 i= a, λ γ c, if i = a α,i =,i i1a γ c,i 1 i, if 1 i p a,p p1, if i = p1 31

5 Theobtainedpowerwilldependonewater-level is obtained by solving e total roughput constraint which which is transformed into finding e zero of e function f λ for wi ] f = rtot r r r P S c S p {S c S p =, 32 wi S c and S p defined as { S c = {1,,N such at { S p = {1,,N such at p 1 i= a,i1i1p i p 1 i= a,i1i1p i < P γ c > P γ c 33 Alough in is case we do not obtain a strictly speaking water-filling expression, a similar algorim can be developed where will represent an "imaginary" water-level as it remains constant for all channels The pseudo water-filling expression can be deduced from 29 as follows P = λ ˆµ N g c P, 34 wi ˆµ = p 1 i= a,i1i1p i is e effective power cost in e - channel Note at is is not a water-filling equation as ˆµ depends on e allocated power P but it only allows to analyze e allocated power function to e channel gains and e price coefficients Thus, we obtain a system of nonlinear coupled Equations 3 and 32 An iterative approach allows us to determine is water-level and us obtain e optimal power allocation per channel by solving at each step consecutively 3 and 32 until convergence V SIMULATION RESULTS We consideracognitiveuserrandomlylocatedin a cell wi a radius d = 1 Km We assume at e CU is equipped wi a smart meter at could provide it wi instantaneous unit pricing in real-time Unless notified for a different usage of e parameters, we consider N = 2 channels The total required = 5 Mbps while e individual required = 1 Mbps, We consider a Rayleigh fading channel model The interference reshold is fixed to be equal to e noise floor I = N = 12 dbm Using e different pricing cost models presented in section IV, we compute e optimal cost of e power needed to reach e required roughput, C P opt, en compare it to e cost of e power if dynamic pricing is not available This reference power allocation P ref is obtained by minimizing e total power consumed instead of cost of e power we use algorim proposed in 17] for is reference allocation Then we compute e relative power cost gain as follows roughput rtot roughput per channel is r Cost gain = C P ref C P opt C Pref 35 In Fig 1, we plot e cost gain wi reference to e case where pricing is not considered minimization of e total power cost for e channel dependent unitary cost 16 as a function of e standard deviation of is unitary cost for different numbers of channels using e channel dependent pricing model 16 We observe at e gain increases as e variance of e unitary price increases This is due to e increase of e variability between channels which allows a better exploitation of e channels On e oer hand, e gain is more important when e number of channels is lower This can be explained by e fact at increasing e number of channels limits users freedom to allocate e power due to additional individual constraints for e new channels Cost gain percentage N=1 N=2 N=3 N= Standard deviation of e unitary price coefficient Fig 1 Percentage of cost gain function to e unitary cost variance wi Eγ c ] = 2 db In Fig 2, we plot e obtained cost gain compared to absence of pricing for different values of e pricing coefficients to observe eir effect on e total cost We use e pricing model 21 wi linear unit pricing as in 22 wi uniform pricing coefficients for all channels We fix a = 1 and vary b as shown in e legend The cost gain is increasing wi e increase of e pricing coefficient since power savings became more valuable wi e increase of e unitary cost Cost gain percentage Eγ c ] in db Fig 2 Total power consumption cost wi different pricing parameters function to average channels SNR b =1 b =2 b =5 b =1

6 In Fig 3, we plot e power cost gain compared to e case where pricing is not taken in consideration as a function of e average channel gains Eγ c ] We use power dependent pricing model 21 wi polynomial unit pricing as in 28 wi different degrees The different polynomials are generated using Chebychev polynomial approximation 18] from e same cost function but wi different degrees of approximation p In is figure, we observe two behaviors of e cost gain as a function of e SNR In e first part corresponding to low SNRs, e cost gain is a decreasing function of e SNR but it is higher wi higher values of e polynomial approximation degree In fact, for low SNRs, high power levels are needed to meet roughput constraints which results in higher cost savings when e polynomial approximation is more accurate higher degree of e polynomial In e second part corresponding to high SNRs, e cost gain is a slowly increasing function of e SNR and also function of e polynomial degree In fact, for high SNRs, lower power levels are needed to reach roughput requirements Thus, in is case, e effect of e channel SNRs on e total cost gain becomes dominant over e power effect In addition, even ough we still observe at higher degrees of polynomial approximations result in higher cost savings, e difference between e gains become negligible which ustifies e use of a lower polynomial degree for is case since resource allocation is easier wi low polynomial degrees Cost gain percentage Eγ c ] in db Fig 3 Cost gain wi different polynomial approximations function to average channels SNR VI CONCLUSION This paper proposes a resource allocation scheme for dynamic cost of e consumed power for a cognitive radio system while ensuring total and individual roughput requirements The proposed power allocation allows to profit from available information about e cost and e channels diversity to better employ e power to meet e roughput requirements and minimize e power cost Analytic expressions of e allocated power are developed for different cost functions and low-cost algorims are presented for e power allocation Simulation results show e gain at e cognitive system achieved p=2 p=3 p=4 p=6 p=8 by profiting from e dynamic power pricing rough e proposed power allocation scheme VII ACKNOWLEDGMENT This work was made possible by NPRP grant # NPRP from e Qatar National Research Fund a member of Qatar Foundation The statements made herein are solely e responsibility of e auors REFERENCES 1] Y Chen, S Zhang, S Xu, and G Li, Fundamental trade-offs on green wireless networks, IEEE Communications Magazine, vol 49, no 6, pp 3 37, June 211 2] Spectrum efficiency working group, FCC Spectrum Policy Task Force, Tech Rep, Nov 22 3] B Hamdaoui, Adaptive spectrum assessment for opportunistic access in cognitive radio networks, IEEE Transactions on Wireless Communications, vol 8, no 2, pp , Feb 29 4] J Mitola and J Maguire, GQ, Cognitive radio: making software radios more personal, IEEE Personal Communications, vol 6, no 4, pp 13 18, Aug Venkatraman, B Hamdaoui, and M Guizani, Opportunistic bandwid sharing rough reinforcement learning, IEEE Tran on Vehicular Technology, vol 59, no 6, pp , July 21 6] O Altrad, S Muhaidat, A Al-Dweik, A Shami, and P Yoo, Opportunistic spectrum access in cognitive radio networks under imperfect spectrum sensing, Vehicular Technology, IEEE Transactions on, vol 63, no 2, pp , 214 7] B Hamdaoui and K G Shin, OS-MAC: An efficient MAC protocol for spectrum-agile wireless networks, IEEE Transactions on Mobile Computing, vol 7, no 8, pp , August 28 8] M NoroozOliaee, B Hamdaoui, and K Tumer, Efficient obective functions for coordinated learning in large-scale distributed osa systems, Mobile Computing, IEEE Transactions on, vol 12, no 5, pp , 213 9] L-C Wang, C-W Wang, and C-J Chang, Modeling and analysis for spectrum handoffs in cognitive radio networks, Mobile Computing, IEEE Transactions on, vol 11, no 9, pp , 212 1] M NoroozOliaee, B Hamdaoui, X Cheng, T Znati, and M Guizani, Analyzing cognitive network access efficiency under limited spectrum handoff agility, Vehicular Technology, IEEE Transactions on, vol 63, no 3, pp , ] M Ben Ghorbel, B Hamdaoui, R Hamdi, M Guizani, and M NoroozOliaee, Distributed dynamic spectrum access wi adaptive power allocation: Energy efficiency and cross-layer awareness, in Computer Communications Workshops INFOCOM WKSHPS, 214 IEEE Conference on IEEE, 214, pp ] G Bansal, J Hossain, and V Bhargava, Optimal and suboptimal power allocation schemes for ofdm-based cognitive radio systems, IEEE Transactions on Wireless Communications, vol 7, no 11, pp , November 28 13] S Singh, P Teal, P Dmochowski, and A Coulson, Power allocation in underlay cognitive radio systems wi feasibility detection, in Communications Theory Workshop AusCTW, 212 Australian, Jan 212, pp ] J Zou, H Xiong, D Wang, and C W Chen, Optimal power allocation for hybrid overlay/underlay spectrum sharing in multiband cognitive radio networks, IEEE Transactions on Vehicular Technology, vol 62, no 4, pp , May ] C gang Yang, J dong Li, and Z Tian, Optimal power control for cognitive radio networks under coupled interference constraints: A cooperative game-eoretic perspective, IEEE Transactions on Vehicular Technology, vol 59, no 4, pp , May 21 16] S Boyd and L Vandenberghe, Convex Optimization New York, NY, USA: Cambridge University Press, 24 17] N Papandreou and T Antonakopoulos, Bit and power allocation in constrained multicarrier systems: The single-user case, EURASIP Journal on Advances in Signal Processing, vol 28, pp 1 14, 28 18] A Gil, J Segura, and N Temme, Numerical Meods for Special Functions Society for Industrial and Applied Maematics, 27 Online] Available: