Barria, J A

Mayer, D

IEEE Communications Letters

English

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634 IEEE COMMUNICATIONS LETTERS, VOL. 11, NO. 7, JULY 2007Bandwidth Allocation for aRevenue-Aware Network Utility MaximizationDavid Mayer and Javier A. Barria, Member, IEEEAbstract— In this letter, we consider the Network UtilityMaximization (NUM) problem in which a loss-related revenueindicator is set up by an operator as a constraint. We proposean algorithm which artificially modifies capacity constraints of astandard NUM algorithm. This results in a price-based mecha-nism that drives users into an operating point which maximizestotal user utility for a given revenue indicator constraint.Index Terms— Bandwidth allocation, network utility maxi-mization, revenue optimization, loss network.I. INTRODUCTIONBANDWIDTH allocation has been extensively studiedwith various objectives in view, especially objectivesconcerning the users (e.g. utility maximization and end-to-end delay minimization) and objectives of the operator’sinterest (e.g. revenue maximization and cost minimization).The question of combining the users’ and operators’ concernsremains an open issue. The motivation for this work stemsfrom the fact that selfish user behavior comes at high costfor the operator and is conflicting with Traffic Engineeringefforts [1].In this letter, we attempt to solve the problem of NetworkUtility Maximization (NUM) with a loss-related constraintset up by the operator. Traditional NUM algorithms [2][3]result in fully utilized links. Assuming that users only pay forsuccessfully delivered traffic, the loss associated with fullyutilized links reduces operator’s revenue. Hence we choose asa loss-related metric an operator’s revenue indicator and solvethe problem of NUM with this revenue indicator used as aconstraint. To our knowledge, there is not a published workon trade-offs between utility and revenue in loss networks.However, several papers exist on optimization involving bothrevenue and some aspect of utility, typically fairness (e.g. [4]).Our approach draws from the work of Mitra [5] on lossnetworks and uses a price-based mechanism to control userdemand [2].II. MODEL AND PROBLEM FORMULATIONConsider a set L of L links and a set F of F flows.Associated with each link l ∈ L there is a loss function Ll(xl),where xl is the flow on link l. Link l capacity is denoted byCl. Flow configuration is given and represented by routingmatrix A = (ai,l; i ∈ F , l ∈ L), where ai,l = 1 if flow iManuscript received February 23, 2007. The associate editor coordinatingthe review of this letter and approving it for publication was Dr. AleksandraSmiljanic. The work of D. Mayer was supported by the UK Engineering andPhysical Sciences Research Council under Grant GR/S52360/01.The authors are with the Department of Electrical and Electronic Engineer-ing, Imperial College London, London SW7 2AZ, UK (e-mail: {david.mayer,j.barria}@imperial.ac.uk).Digital Object Identifier 10.1109/LCOMM.2007.070280.uses link l and ai,l = 0 otherwise. Each flow i uses one pathdenoted by Ri = {l ∈ L|Ai,l = 1}. Flow i ∈ F is associatedwith a user i and a utility function Ui(fi), where fi is thebandwidth demand of user i. Let f = (fi, i ∈ F). We assumethat utility functions are concave increasing.Our goal is to optimize the capacity constraints providedto the standard NUM algorithm ([2][3]) in order to maximisetotal user utility U achieved by the NUM algorithm subjectto the operator’s revenue indicator π being equal to or greaterthan a positive constant πmin. The optimization problem ofthe operator isπ-NUM:maxk≥0U(k) (1)s.t. πmin − π(k) ≤ 0,where k = (k1, . . . , kL) is the control variable and U(k) isthe maximum of the following utility maximising problem.NUM(k):U(k) = maxf≥0∑i∈FUi(fi) (2)s.t.∑i∈F(fiai,l − klCl) ≤ 0, ∀l ∈ L, kl ∈ k.Denote f∗(k) the maximiser of NUM(k). Vector k modifiesthe capacity constraints. Once found, the standard distributedNUM algorithm can be run, realizing thus (1). Associated withthe solution of (2) (and obtained via the distributed algorithm),is a vector of Lagrange multipliers λ(k) = (λl(k), l ∈ L).Using a price-based control of user demand framework, weassume that users maximise their surplus. Hence if for flowi the price per unit of bandwidth is pi, the user will choosedemand fi, such thatfi = arg maxf{Ui(f) − pif}. (3)In this setting, and for concave utility functions, surplus ismaximized at a point where marginal utility equals price.When the price seen by flow i is given by the sum of Lagrangemultipliers λ(k) over all links on path Ri,pi(k) =∑l∈Riλl(k), (4)then the solution fi found by each user corresponds to thesolution f∗i (k) of NUM(k) [3]. The value of the revenueindicator π (referred to as revenue from now on) is calculatedas if the operator charged prices pi(k) for successfully carriedtraffic,π(k) =∑i∈Fpi(k)f∗i (k)(1 − L(i)), (5)where L(i) denotes the loss encountered by flow i along itspath.1089-7798/07$25.00 c© 2007 IEEEMAYER and BARRIA: BANDWIDTH ALLOCATION FOR A REVENUE-AWARE NETWORK UTILITY MAXIMIZATION 635We use a single-server queue with a buffer of size Kto model link loss in the network. Note that our algorithmis not dependent on the specific loss function. We givenumerical results for both small and large buffer, accountingfor different loss behavior. We assume that packets arrive tolink l according to a Poisson process with mean rate xl and thepacket length is exponentially distributed with mean s. Servicerate is then exponentially distributed with mean µl = Cl/s.This is an M/M/1/K queue, for which the loss probability isdefined as Ll(xl) = ρKl1−ρl1−ρK+1l, where ρl = xlµl =xlCls isthe offered load. Furthermore, assuming link independence,the loss encountered by flow i can be calculated as L(i) ≈1−∏l∈Ri (1 − Ll), where Ll, l ∈ L is obtained from a systemof fixed-point equations as in [6].III. PROPOSED SOLUTIONProblem (1) has a concave objective function, but noncon-vex constraints (revenue (5) is a non-monotonic function ofdemand) for which a suitable optimization method is thatof Augmented Lagrangians [7]. This is essentially a penaltymethod, which converts constrained problems into a sequenceof unconstrained problems Qγt(·), t = 0, 1, . . . by adding apenalty function Pγ(·) and an increasing penalty factor γ. Thesolution of Qγt(·) is used as the starting point for solvingQγt+1(·). This method does not require a feasible initial pointand is well-suited for nonlinear constraints.1 The augmentedLagrangian function isQγ(k, µ) = U(k) − Pγ(k, µ). (6)The inclusion of Lagrange multiplier µ within the penaltyfunction enables convergence without the penalty factor γ →∞.2 We use the following penalty (adopted from [7])Pγ(k, µ) =12γ([max{0, µ + γg(k)}]2 − µ2), (7)where g(k) = πmin − π(k). The maximization of theunconstrained Lagrangian (6) requires the sensitivity of totalutility and revenue generated by the solution of (2) withrespect to capacity multipliers k. From the theory of Lagrangemultipliers it is known that sensitivity of the objective functionwith respect to constraints equals the corresponding Lagrangemultipliers, here ∂U∂Cj = λj , hence∂U∂kj= Cjλj . Note thatthe sensitivity of revenue ∂π∂kj has to be replaced by a finitedifference ∆π∆kj due to discontinuities of Lagrangians causedby the change of the set of active constraints in (2). Sensitivityof revenue is derived below by extending the results in [5] forinterdependent demand and price.The proposed optimization consists of repeated increasesof the augmented Lagrangian (6) by shifting vector k in thesteepest ascent direction,kt+1 = kt + α · ∇Qγ(kt, µt), (8)1Convergence to global maximum requires that in each iteration a globalmaximum of Q(·) is found. We rely on the initial point lying in proximityof the global maximum. This is in practice, given by common packet lossfunctions, k = 1, i.e. the point corresponding to the original link capacities.2The idea is to increase the penalty factor to approach feasible range andthen to adjust only the Lagrangian multiplier in the near-optimum range,preventing γ → ∞.where α is a decreasing step size. The gradient of (6) atiteration t is∇Qγt(kt, µt) = C · λ(kt) +max{0, µt + γtg(kt)} · ∆π(kt)∆k . (9)Penalty factor γ and multiplier µ are updated as in [7, p.123]:µt+1 = max{0, µt + γt(πmin − π(kt))} (10)γt+1 ={βγt if|πmin−π(kt)||πmin−π(kt−1)| ≥ δγt otherwise.(11)Scalar β is typically chosen β ∈ [1.1, 2] and δ = 0.75. Thealgorithm terminates when the following condition holds:|πmin − π| ≤ επ ∧ ‖∇Qγt(kt, µt)‖ ≤ εP , (12)where typically επ = 0.01 and εP = 1. The pseudocode ofthe proposed algorithm for solving (1) is shown in Table I.Sensitivity of revenue to capacity constraintsIn order to update the capacity multipliers vector k in (8)we need to obtain the sensitivity of revenue with respect tok, ∆π∆k = (∆π∆kj, j = 1, 2, . . . , L). Let us write revenue (5) asa function of demand and price, π = y(f ,p). First, we derivethe total derivative with respect to demand fi,dπdfi=∂π∂fi+∂π∂pi∂pi∂fi. (13)Revenue sensitivity is known [5] and can be calculated as∂π∂fi= (1 − L(i))(pi −∑l∈Ri cil), where cil is the impliedcost of flow i on link l calculated using the expected bufferoccupancy of an M/M/1/K queue. From revenue definition (5)we have ∂π∂pi = fi(1 − L(i)). From (3) we have pi = Ui′,hence ∂pi∂fi = Ui′′(fi). Then (13) can be expressed asdπdfi= (1 − L(i))(pi −∑l∈Ricil + fiU′′i (fi)). (14)The term fiU′′i (fi) is negative and accounts for the depen-dency of price and demand. We use (14) in the calculation ofthe revenue sensitivity, which is given by the following totalfinite difference∆π∆kj= Cj∆π∆Cj= Cj∑i∈Fdπdfi∆fi∆Cj. (15)Sensitivity of demand to capacity constraintsThe remaining unknown in (15) is the change of demand ∆fiupon adding a small ∆C to link j in (2), ∆fi∆Cj , which weapproximate as follows. Total utility U is given by a sum ofindividual utilities, hence it can be approximated by a sum ofTaylor series approximations, here of second order:U(f∗ + ∆f) ≈ U(f∗) +∑i∈FU ′i∆fi +∑i∈F12U ′′i (∆fi)2, (16)where f∗ is the current utility maximising demand vector and∆f is the vector of demand changes. Vector ∆jf∗ of optimalchanges upon adding small ∆C to link j is approximated bythe solution of the following optimization:max∆jf∑i∈F[U ′i(f∗i )∆jfi +12U ′′i (f∗i )(∆jfi)2](17)s.t. xl +∑i∈F∆fi · ai,l ≤{Cl, l ∈ L\jCl + ∆C, l = j636 IEEE COMMUNICATIONS LETTERS, VOL. 11, NO. 7, JULY 2007TABLE IProposed algorithm First iteration fornetwork in Fig. 1a1. Choose minimum revenue πmin. πmin = 5.11t ← 0; kt ← 1; k0 = (1, 1)2. Obtain demands f∗(kt) and Lagrangians f∗ = (0.00, 3.00, 2.00)λ(kt) from the solution of NUM(kt) (2). λ= (0.91, 1.05)3. Calculate revenue π from (5). π = 4.82Evaluate ∇U = Cλ(kt) and ∆π (15). ∇U = (1.83, 3.15)If the termination condition (12) holds, ∆π = (0.01,−4.75)then k∗ ← kt and TERMINATE.4. Update multiplier µt (10) and then µ0 = 0.00 µ1 = 1.38penalty factor γt (11).4 γ0 = 4.33 γ1 = 4.765. t ← t + 1. Update vector kt (8). k1 = (1.00, 0.98)GO TO STEP 2.Fig. 1. Networks used in the numerical examples. Circles in network b)specify link capacities.The rationale for this optimization is that the sum of util-ity changes∑i∈F ∆Ui approximated by the last two termsin (16) must be maximal and that link flow must not exceedany link capacity except for the augmented link j. Thisoptimization has the standard form of a quadratic objectivefunction ∆jf y + 12∆jf H (∆jf)T where, in our case,y = (U ′1(f∗1 ), . . . , U′F (f∗F ))T , H is a diagonal matrix ofsecond derivations with H(i, i) = U ′′i (f∗i ) and so (17) canbe easily solved using standard algorithms (e.g. [7]). Solution∆jf∗ of (17) is used to approximate ∆fi∆Cj in (15).IV. NUMERICAL RESULTSFirst we show the proposed procedure on a 2-link networkaccommodating 3 users (see Fig. 1a). Utility functions of theusers are in the form Ui(fi) = ai(1− 2−bifi) and parametersof the problem are: a1 = 2, a2 = 5, a3 = 10, b1 = b2 = b3 =0.8, C1 = 2, C2 = 3. Solutions for various revenue constraintsare shown in Fig. 2, where axes represent elements of vectork. Contours of revenue and total utility were obtained byexhaustive search. All solutions were reached starting fromk0 = (1, 1), although in practice previous solutions couldserve as a starting point. Table I shows the data for the firstiteration of the algorithm for πmin = 5.11. Fig. 3a shows therevenue-utility trade-off for various buffer size to demonstratethe dependence of the trade-off on loss characteristics of thesystem.3 Results for a larger 8-link network (see Fig. 1b) areshown in Fig. 3b.V. FINAL REMARKSThis letter addresses network utility maximization with aconstraint on an operator’s revenue indicator and proposes an3Large buffer size corresponds to effective bandwidth close to the averagerate (small loss), whereas small buffer size models effective bandwidth closeto the peak rate (high loss).4Multipliers are initialized as µ0 = 0 and γ0 =∣∣∣ ‖∇U‖‖∇π‖·(πmin−π)∣∣∣.0.4 0.6 0.8 1 1.20.40.60.811.2k(2)123454.824.865.115.315.43k(1)11.4611.5011.1210.6110.15Σ Uπm1 2345Fig. 2. Contours of revenue (full line) and utility (dashed line) for networkin Fig. 1a. Circles show solutions for particular revenue constraints πminindexed by m. Solutions were reached individually from starting point k0 =(1, 1) using the proposed algorithm and for buffer size K = 24.Fig. 3. Total utility as a function of minimum revenue constraint for differentbuffer sizes K as obtained by the proposed algorithm for network in Fig. 1a(a) and Fig. 1b (b).algorithm for controlling this trade-off. Although some param-eters of the proposed algorithm are computed centrally (therevenue sensitivity vector), once they are communicated tonetwork nodes, the NUM algorithm runs in a fully distributedway. At present we are developing the distributed version ofthis framework which will be reported in a forthcoming paper.Regarding the assumption of concave utility functions,recent work on maximization of non-concave utility functionsoffers new methods based on the standard NUM algorithm.We are therefore also extending our work in this direction.REFERENCES[1] L. Qiu, Y. R. Yang, Y. Zhang, and S. Shenker, “On selfish routing inInternet-like environments,” in Proc. SIGCOMM03, Karlsruhe, Germany,2003.[2] F. Kelly, A. Maulloo, and D. Tan, “Rate control in communication net-works: shadow prices, proportional fairness and stability,” J. OperationalResearch Society, vol. 49, 1998.[3] S. Low and D. Lapsely, “Optimization flow control. I. Basic algorithmand convergence,” IEEE/ACM Trans. Networking, vol. 7, no. 6, pp. 861–874, 1999.[4] M. Zukerman, L. Tan, H. Wang, and I. Ouveysi, “Efficiency-fairnesstradeoff in telecommunications networks,” IEEE Commun. Lett., vol. 9,no. 7, pp. 643–645, 2005.[5] D. Mitra, J. Morrison, and K. Ramakrishnan, “ATM network design andoptimization: a multirate loss network framework,” IEEE/ACM Trans.Networking, vol. 4, no. 4, pp. 531–543, 1996.[6] F. Kelly, “Loss networks,” The Annals of Applied Probability, vol. 1,no. 3, pp. 319–378, 1991.[7] D. P. Bertsekas, Constrained Optimization and Lagrange MultiplierMethods. Academic Press, INC. (LONDON) LTD., 1982.

Bandwidth Allocation for a Revenue-Aware Network Utility Maximisation

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Bandwidth Allocation for a Revenue-Aware Network Utility Maximisation

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