Resource Allocation for QoS Provisioning in Wireless Ad Hoc Networks

Transcription

1 Resource Allocation for QoS Provisioning in Wireless Ad Hoc Networks Mung Chiang, Daniel ONeill, David Julian andstehenboyd Electrical Engineering Deartment Stanford University, CA 94305, USA Abstract- For wireless ad hoc networks with multiho transmissions and Rayleigh fading, this aer maximizes the overall system throughut subect to QoS constraints on ower, robability of outage, and data rates. Formulations are also given which minimize delay and otimize network resources in a wireless ad hoc network, where each link is shared by multile streams of traffic from different QoS classes, and each traffic traverses many links. Although these otimal resource allocation roblems are non-linear, they can be osed as geometric rograms, which are transformed into convex otimizations, and can be solved globally and efficiently through interior-oint methods. I. Introduction Quality of service (QoS) has become amortant issue in various kinds of data networks as some users are no longer satisfied with resource allocation based on service rovisioning. Three maor considerations of QoS suort are bandwidth, delay and delivery guarantee. Voice, data, image, and video have different bandwidth requirements. Some classes of traffic, such as voice, are also much more sensitive to delay than background classes, such as data. QoS rovisioning in a wireless network is a articularly difficult issue because hysical layer roblems; such as ath loss, fading, and multiath; can make the communication links unreliable. This makes delivery guarantee a necessary feature in wireless ad hoc network QoS rovisioning. The challenge is to first rescribe a feasible QoS scheme for different classes of traffic, and then to otimize the use of network resources, mainly link caacities and transmitter owers, to satisfy QoS requirements for all classes while maximizing either the total network erformance, or the QoS for the remier class. Within the wireless arena, ad hoc wireless networks ose additional technical challenges for QoS suort. Unlike cellular wireless networks, ad hoc networks have no fixed infrastructure, and long range communications require multiho transmissions where a acket is routed through the network by other transceivers that act as relay nodes. In sections 3 and 4, the following resource allocation roblems for QoS rovisioning in wireless ad hoc networks are solved: This work was suorted by Hertz Foundation Fellowshi and Stanford Graduate Fellowshi. This work was suorted by NSF Grant CCR and the Stanford Network Research Center. P As a secial case of resource allocation, the ower control of user nodes are otimized to maximize the overall system throughut. P Turning to the general cases, feasibility of service level agreement (SLA) terms are determined under network resource constraints. P 3 Taking delay into consideration, the total delay for the most time sensitive class of traffic is minimized by otimizing over owers, caacities, and SLA terms. P 4 Otimizing over owers, caacities, and SLA terms; the unused caacity of the network is maximized. Because the mobile radio channel is fast varying and the number of user nodes is large, a fast and robust decision making algorithm is needed that accommodates a large number of variables for dynamic resource allocation to be feasible. Several ad hoc heuristics have been roosed to tackle the above roblem, but they cannot meet all of the following criteria: otimality, seed, and the ability to accommodate a variety of constraints and a large number of variables. Solutions for wireless cellular networks have been roosed in []. This aer tackles the more difficult roblems of resource allocation for wireless ad hoc networks. A global solution to non-linear roblems P top4is found by transforming the roblems into convex otimization roblems. Solution methods for these roblems not only roduce globally otimal solutions as efficiently as for linear rograms, but also unambiguously determine feasibility. This second roerty is used to determine the feasibility and ricing scheme of admitting a new user with a defined QoS requirement. II. Convex otimization and geometric rogramming An efficient algorithm is needed in order to find the otimal solution to the above nonlinear roblems in a high seed dynamic network with a large number of links and nodes. Formulations for P top4 are rovided that can be turned into convex otimizations, which have fast algorithms, such as the interior oint method and the rimal dual method, that make them as easy to solve as linear rograms. Convex otimization refers to minimizing a convex obective function over convex constraint sets. The articular tye of convex otimization used in this aer is in the form of geometric rogramming []. First consider the following definition Definition A monomial is a function h : R n R,where the domain contains all real vectors with ositive comonents: /0/$ IEEE 9

2 h(x) =cx a xa xan n, c 0 and a i R () A sum of monomials f(x) = c k kx a k x a k x a nk s called a osynomial. Geometric rogramming is an otimization roblem with the following form: minimize f 0(x) subect to f i(x) h (x) = where f 0 and f i are osynomials and h are monomials. Geometric rogramming in the above form is not a convex otimization roblem. However, with a change of variables: y i =logx i and b ik =logc ik, the geometric rogramming form is ut into convex form: minimize 0(y) = log k ex(at 0ky + b 0k ) subect to i(y) = log k ex(at iky + b ik ) 0 q (y) = a T y+b = 0 (3) It can be verified that the log of sum of exonentials is a convex function []. Therefore i are convex functions and q are affine functions, and the roblem is a convex otimization roblem. Note that if all osynomials are in fact monomials, geometric rogramming becomes linear rogramming. Convex otimization roblems can be solved globally and efficiently through interior oint and rimal dual methods [3], with running times that usually scale to the square root of the roblem size. These methods also offer duality interretations, stability analyses and accommodate a variety of constraints. This aer shows how geometric rogramming can solve many versions of QoS rovisioning roblems in wireless ad hoc networks. III. Power control for throughut otimization First a secial case of resource allocation otimizatios formulated and solved in this section. The variables are user node owers and the obective is to maximize the overall system throughut in bs. The method in this aer exlicitly takes into account the statistical variation of the received signal and the interference ower. A. Multi-ho network model and Rayleigh fading Consider a wireless ad hoc network with n transmitter/receiver airs, labeled,...,n, which transmit at owers P...,P n. The ower received from transmitter, at receiver i is given by () G if ip (4) The nonnegative number G i reresents the ath gain the absence of fading from the th transmitter to the i th receiver. G i can encomass ath loss, shadowing, antenna gain, coding gain, and other factors. The Rayleigh fading between each transmitter and receiver i is given by F i. TheF i s are assumed to be indeendent and have unit mean. The G i s are aroriately scaled to reflect variations from this assumtion. The distribution of the received ower between any air of transmitter and receiver i is exonential with mean value, E [G if ip ]=G ip (5) The signal to interference ratio (SIR) for user i determines the quality of the received signal and is defined as SIR i = P ig ii N i PGi + ni (6) Unlike SNR, SIR cannot be increased by simly increasing all users transmitting owers since that would raise both the signal level and the interference level. This introduces a bit error floor and a QoS bottleneck. B. Outage robability and system throughut An outage is declared when the received SIR falls below a given threshold defined as SIR th, often comuted from a BER requirement. The outage robability associated with the i th ho is given by O i = Pr(SIR i SIR th ) = Pr(G iif iip i SIR th k i G ikf ik P k ) The outage robability can be exressed as [4] O i = k i (7) + SIR th G ik P k (8) Outage robability over a ho induces an outage robability over a ath S O aths = Prob(outage along the ath S) = ( Oi) s S =. (9) s S k s (+ SIR th G ik P k ) The constellation size M used by a ho can be closely aroximated for MQAM modulation as follows M =+.5 SIR (0) ln(5ber) where BER is the bit error rate. Defining K =.5 ln(5ber) leads to a monotonic exression for the data rate of the i th ho as a function of the received SIR: R i =(/T )log ( + KSIR i) () The aggregate data rate for the system can then be written simly as the sum of terms of this form. R system = i Ri =(/T )log ( + KSIRi) () Overall system throughut is defined as the maximum aggregate data rate suortable by the system given a set of users with defined QoS. i 9

3 C. Throughut otimization B Theorem (Otimize ower for throughut maximization) The following roblem of otimizing user node owers to maximize total network throughut is a convex otimization roblem. A D 0m maximize subect to R system 3 4 C R i R i,lb, i Pr k i + SIR th G ik P outi i k Pr s S k s out ath s S (+ SIR th G ik P k ) P i P max (3) The obective functios the overall system throughut. It is otimized over the set of all feasible owers P i. The first set of constraints are the data rates demanded by existing system users. The second set of constraints are the outage robability limitations demanded by users using single hos. The third set of constraints are the outage robability limitations for users using a multi-ho ath. Lastly, the forth set of constraints are regulatory or system limitations on transmitter owers. D. Simulation A simle four node multi-ho network is considered in the following simulation. As shown figure, the network consists of 4 nodes A, B, C, and D, and 4 links,, 3, and 4. On link node A is the transmitter and node B is the receiver; similarly, the transmitter and receiver nodes for each link are shown the figure. Note that node A is also the transmitter on link 3, illustrating that a node can be a transmitter and/or receiver on many links. Nodes A and D as well as B and C are searated by a distance of 0m. By geometry the distance of each transmit ath is 0 m. For this simulation each link has a maximum transmit ower of W. Alternatively, the ower constraint could be laced on each node instead of each link by adding a constraint that P + P 3 W. All nodes are using MQAM modulation. The baseband bandwidth for each link is 0kHz, the minimum data rate for each link is 00bs for maintenance data, and the target BER is 0 3. For the Rayleigh fading a robability of outage of P out =0. is required for an SIR threshold of 0dB. [ The gains for each link are comuted as G i = ] 4 00 d for i, andg ii = [ 4, d] with the excetion of G and G 34 which are set equal to 0 since it is assumed that a node does not transmit and receive at the same time. The factor of 00 can be viewed as the sreading gain a CDMA system, or the ower falloff with frequency in a FDMA system. This gives the following gain matrix: G = (4) 0m Fig. : Network Toology for Simulation The maximum aggregate data rate, found using the geometric rogramming otimization method, is R = 6.8kbs, with M =4.8QAM modulation for each link, R i =54.kbs for each link, and P = P 3 =0.709W and P = P 4 =W link transmit owers. The resulting SIR =.7dB on each link. The symmetry in modulation levels and SIR is due to the symmetries in the network toology, and not due to any exlicit otimization constraint. E. Admission control and ricing The methodology used in this section can also be extended for other resource management uroses in a wireless ad hoc network. For examle, a new user is admissible if his QoS requirements can be suorted by the system without disturbing the existing QoS requirements of current users. In this model a user is admissible if a feasible solution of the roblem in Theorem exists after the new user s QoS constraints have been added. Anfeasible solutios a definitive statement that this new user may not be added to the system without a change in required QoS. Our model also suorts several service-ricing aroaches. The decrease in the overall system throughut associated with a new user can be used to estimate the incremental cost of suorting that new user, and should be roortional to the rice that might be charged. A second aroach is to calculate the effect of different levels of QoS for a given data rate. The effect on overall system throughut for each of the chosen different levels of QoS can then be used to set relative ricing for these levels. One examle is to set ricing as a function of robability of outage for a given data rate. IV. Resource allocation for delay and efficiency otimization The more general cases of resource allocation are formulated and solved in this section. Otimization variables include owers, the number of ackets in each traffic, bandwidth, delay and delivery guarantee required for each QoS class, and caacity for each link. Potential obective criteria include delay, unused caacity and SLA feasibility, in addition to the overall system throughut in the last section. A. Problem formulations 93

4 Consider a network with J links with caacity of C ackets er second for each link. There are K classes of traffic with different QoS requirements to be transorted over the network. For each QoS class k, the bandwidth required is b k Hz, and the delay guarantee in the service level agreement (SLA) is d k,ub seconds. Also, a minimum robability of delivering the acket across the unreliable network is required in the SLA, denoted by k,lb. In this roblem formulation the delay is the delay due to transmission time; roagation delay is ignored because it is constant for the otimization arameters. The more refined model with queueing delay is treated in the extension. Similar to the last section, each stream of traffic from source s to destination d will traverse certain secific links as dictated by the articular routing rotocol used for the network. Denote by K the set of traffic using link and by J k the set of links traversed by QoS class k. Denote by n k the number of ackets dynamically admitted in the k th class of traffic. In an ad hoc network each link may fail due to either ower shut down of a user or dee fading that causes an outage. Therefore, a real number between 0 and, is attached to each link as the robability that this link will be maintained during the transmission. By increasing transmitter ower over a link while keeing other arameters of the network constant, SIR of link and therefore can be increased. As will be shown the following roblem formulations, with the above constraints on link caacity, bandwidth requirement, delay, and delivery robability guarantees, the roblem is not a linear rogramming roblem. However, these nonlinear otimizations can be turned into geometric rogramming roblems and solved as efficiently as linear rograms. The first formulatios the following. constraint, the second one is the delay guarantee constraint and the third one the delivery robability constraint. The fourth constraint delivers a guaranteed data rate to each class of traffic. The fifth constraint makes room for SLA terms that give a class of traffic the sole right to traverse a link.this could be for bandwidth requirements or for security reasons. The sixth constraint allows for SLA terms that secify not ust an end to end total delay guarantee, but also an exact delay requirement for a articular traffic class k on a link. The other constraints are ositivity constraints on the variables, and uer bound constraints on. The following arameters are all otential otimization variables: b k,n k,,c,d k,ub and k,lb. Variables b k,d k,ub and k,lb are terms in the SLA. The link caacities C and robability of maintaining a link are network resources to be otimized over. Admission control is reflected in n k. The formulatios a non-linear otimization roblem because otimization variables are multilied together, such as b k n k in the first constraint or the roduct of in the third constraint, and aear in the denominator, such as C in the second and sixth constraints. However, all the inequality constraints are in osynomial form and all the equality constraints are in monomial form. In the second formulation, the unused caacity of a articular link 0 is maximized. The link may be a bottleneck link, or the most often traversed link in the network. [5]. Theorem 3 (Unused caacity maximization) The following roblem of maximizing the unused caacity under SLA and network constraints is a convex otimization roblem. Theorem (SLA feasibility under network constraints) The following roblem of testing SLA feasibility is a convex otimization roblem. minimize subect to No Obective Function b, ( ) i K J k C J k b k = C C = d k,,ub b k,c,,d k,ub, k,lb 0 (5) No obective functios necessary since in this formulation only the feasibility of the SLA terms,d k,ub and k,lb is being tested. Alternatively, a cost function as the obective function could be used for relative ricing during admission control. Note that the first constraint is the link caacity maximize subect to C 0 k K 0 b k n k b, ( ) i K J k C J k b k = C C = d k,,ub b k,c,,d k,ub, k,lb 0 (6) The obective functios to maximize unused caacity of a link 0 by keeing the used caacity to the minimum under all network and QoS constraints. The constraints are the same as in Theorem. In the third formulation, the total delay for a articular class of traffic is minimized. Theorem 4 (Delay Minimization) Delay minimization under SLA and network constraints is a convex otimization roblem. 94

5 minimize subect to ( i K ) J k0 C + α C b, ( ) i K J k C J k b k = C C = d k,,ub b k,c,,d k,ub, k,lb 0 (7) Where α is the marginal tradeoff of caacity for delay. By increasing caacities available on each link at the relative cost α through bandwidth allocation or bandwidth leasing, delay of the most time sensitive QoS class can be decreased. Therefore, a weighted sum, arametrized by α, of the remier QoS class s delay and the cost of caacity rovisioning is maximized. B. Simulation The following is a simulation for Theorem 4, which investigates the tradeoff between delay and cost of caacity. For the network in Fig. there are three classes of traffic. The first class is audio data traffic sent along ath ABCD requiring a rate of 50 ackets/second and a maximum delay of 0. seconds. The second class is also audio data traffic sent along ath DFEA with the same rate and delay requirements. The third class of traffic is video data sent along ath ABFD with a rate requirement of 50 ackets/second. The goal is to minimize both the delay of the video data and the cost of caacity that must be rovisioned or leased. This is accomlished by minimizing a weighted sum of the video data delay and the total caacity used subect to meeting the rate constraints on all traffic classes, and the delay constraints on the audio data traffic. For each α, the marginal tradeoff value between delay and caacity, Fig. 3 shows the minimum delay achievable for the video traffic given. The x-axis uses a log scale. The tradeoff curve shows that the minimum delay increases raidly with increasing cost of caacity until it reaches the delay associated with the minimum caacity required to suort the video signal; from that oint onwards the tradeoff curve is flat. C. Extensions Some extensions to Theorems to 4 include minimizing the maximum delay and accounting for queueing delay at the nodes. Minimizing the maximum delay can be accomlished by minimizing a dummy variable subect to all delays less than that dummy variable. Queueing delay in a store and forward network can also be accounted for in the geometric rogramming construction [5]. A 4 B E Fig. : Network Toology for Simulation Minimized Delay for Video Traffic α, seconds of delay er acket of caacity Fig. 3: Trade off between video traffic delay and caacity cost Imortant erformance metrics are otimized, such as throughut and delay, subect to a variety of realistic constraints on both SLA terms and network resources. This is done for wireless ad hoc networks with multiho transmissions, mutual interference, and intrinsic unreliability of links and nodes. Although these resource allocation otimizations are non-linear roblems, they can be solved efficiently using fast convex otimization algorithms. Acknowledgments We would like to thank Nicholas Bambos and Andrea Goldsmith for discussions of wireless network roblems. References [] D. Julian, M. Chiang and D. O Neill, Constrained ower control otimization for wireless cellular networks, to aear in IEEE Vehicular Technology Conference, Oct. 00. [] S. Boyd and L. Vandenberghe, Convex Otimization and Its Engineering Alications, Stanford University EE 364 Course Reader 999. [3] Yu. Nesterov and A. Nemirovsky, Interior Point Polynomial Methods in Convex Programming, SIAM 994. [4] S. Kandukuri and S. Boyd, Otimal ower control interference limited fading wireless channels with outage robability secifications. To aear IEEE JSAC, 00. [5] D. ONeill, D. Julian, M. Chiang and S. Boyd, Resource allocation otimization for QoS rovisioning in wireless cellular and ad hoc networks, Submitted to IEEE Transaction on Communication, 00. C F 3 6 D V. Summary 95