Optimal Link Capacity Dimensioning in Proportionally Fair Networks

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1 Optimal Link Capacity Dimensioning in Proportionally Fair Networks Micha l Pióro 1,Gábor Malicskó 2, an Gábor Foor 3 1 Department of Communication Systems, Lun Institute of Technology, Sween, Michal.Pioro@telecom.lth.se 2 Ericsson Traffic- an Performance Laboratory, Hungary, Gabor.Malicsko@eth.ericsson.se 3 Ericsson Research, Sween, Gabor.Foor@era-t.ericsson.se Abstract. We consier the problem of link capacity imensioning an banwith allocation in networks that support elastic flows an maintain proportional fairness among these flows. We assume that a certain allocate banwith to a user eman generates revenue for the network operator. On the other han, the operator is incurre a capacity epenent cost for each link in the network. The operator s profit is the ifference between the revenue an the total link cost. Uner this assumption the problem is to etermine the banwith of the flows an the link capacities such that the profit is maximize. We first show that uner fairly general assumptions, the optimum allocation of flows leas to selecting the lowest cost paths between O-D pairs. We also erive explicit formulae for the banwith allocate to these flows. We istinguish the case when the operator s capacity buget is fixe ("equality buget constraint", in which case the profit is maximize when the revenue is maximize) an the case when the buget is upper-boune ("inequality buget constraint", in which case the profit can - in general - be maximize by using some portion of the capacity buget). Finally, we show numerical examples to highlight some of the trae-offs between profit maximization, revenue maximization an fairness. Keywors: network imensioning, banwith allocation, routing, traffic engineering, linear programming, convex optimization 1 Introuction After years of research an stanarization efforts, there seems to be a growing consensus that some form of traffic engineering an in particular the separation of flows with ifferent quality of service (QoS) emans are necessary to avoi too costly over-imensioning of IP networks [1], [2]. To this en, MPLS provies a set of stanars that can be applie to explicitly allocate banwith resources between originator-estination (O-D) pairs [13]. In aition, traffic engineering algorithms can also be useful to provie some kin of a "reference engineere network" that can help operators to etermine the level of over-imensioning in a non-engineere network. Despite these obvious motivations, it is however still the topic of research an stanarization exactly which mechanisms an algorithms can be use in for instance MPLS networks for various aspects of traffic engineering. E. Gregori et al. (Es.): NETWORKING 2002, LNCS 2345, pp , c Springer-Verlag Berlin Heielberg 2002

2 278 M. Pióro, G. Malicskó, an G. Foor Despite numerous recent avances (see Section 2 for a more etaile iscussion), aopting the "traitional" traffic engineering methos (incluing link capacity imensioning, routing an banwith allocation) from circuit switche networks (such as PSTN s, ISDN s or even ATM networks) is non-trivial, because of the presence of elastic traffic classes. Therefore, in this paper we concentrate on eveloping a moel an algorithms that take into account the above three aspects of engineering for elastic services. This paper buils upon the results of the first author in [17] where the etaile proofs of the results are presente. Specifically, we assume that between each O-D pair, there may be only a single user flow realizing the eman, i.e. we exclue the case of the "eman split". Uner this assumption, the network operator faces the following problems: The capacity of each link must be etermine such that the network can accommoate the offere traffic (imensioning). The traffic eman between the O-D pair must be associate either with a single flow or - in the case when eman split is acceptable - with a set of flows, an an appropriate route must be foun for each flow (routing). Banwith must be allocate for each flow such that some notion of fairness among the user flows is maintaine (banwith allocation). In our moel, each user flow generates a banwith epenent revenue for the operator. On the other han, each link incurs a capacity epenent cost for the operator, who is therefore motivate in maximizing the resource utilization in the network (an thereby its profit which is efine as the ifference between revenue an cost). The network operator may be intereste in maximizing its profit while keeping the total link cost fixe ("equality link cost buget constraint") or he may want to maximize the profit while keeping the total link cost uner some boun ("inequality buget constraint"). Note that in the first case the profit is maximize when the revenue is maximize. While there is no "killer argument" or any clear technical or economical evience why a certain (if any) fairness between user flows shoul be maintaine, maximizing the network throughput may lea to extremely unfair allocations (incluing the situation where some flows are eeme to complete starvation), see for instance [16]. Therefore, we will formulate the traffic engineering problem as a series of optimization tasks, where one is intereste in maximizing the profit/revenue subject to capacity an fairness constraints. Specifically, we will consier the following two cases: 1. Dimensioning uner a fixe buget constraint, the routes are consiere fixe an pre-etermine to each flow. (Task 1) 2. Profit (revenue-cost) maximization, where the buget (i.e. the sum of all link costs) is not fixe but can be freely chosen up to a limit. (Task 2) Throughout the numerical evaluation of the techniques enumerate above we will use the proportional fair sharing with fixe link capacities an shortest path routing as a reference metho (Task 3). We organize the paper as follows. In the next section we take a look at recent research results in the area of banwith allocation in (max-min an proportional) fair networks. In Section 3 we formulate the problem of link capacity imensioning an routing as a

3 Optimal Link Capacity Dimensioning in Proportionally Fair Networks 279 (revenue) optimization problem uner buget an fairness constraints. We assume that the total link cost in the network shoul be equal to a pre-etermine buget ("equality buget constraint"). For this case, we present explicit formulae to etermine the allocate banwith for each flow. Next, we efine the profit optimization task, in which case the sum of all link costs is boune ("inequality buget constraint"). Here the task is to fin both the link cost buget an the banwith allocate to flows (an associate revenue) such that the profit is maximize. In Section 6 we iscuss numerical results. We conclue in Section 7. 2 Relate Works In the context of routing an resource allocation uner fairness constraints, most paper consier the popular max-min fairness mostly in Asynchronous Transfer Moe (ATM) Available Bit Rate (ABR) context, since the ATM Forum aopte the max-min fairness criterion to allocate network banwith for ABR connections [7], [19], [21]. However, these papers o not consier the issue of path optimization in the boune elastic environment. For instance, [21] stuies the spee of convergence of max-min fair allocation algorithms rather than focusing on path or link capacity optimization. From the point of view whether the flows are static (also calle "long live") or ynamic (where some arrival an eparture patterns are also consiere) these papers can be ivie into two groups. Representative papers for the static case inclue for instance the papers by Kleinberg et al., see [11] an [12]. Another important series of work on the static case is the papers that evelop on-line fair routing algorithms where the eman matrix is not a-priori known, see for instance [6], [18] an more recently [8]. The "ynamic case" uner both max-min an proportional fairness is analyze, mostly focusing on stability aspects, by Veciana et al. in [22]. Here, routes an link capacities are assume fixe. Max-min fair routing is the topic of the paper by Ma et al. [14], where the wiestshortest, shortest-wiest an the shortest-ist algorithms are stuie. These algorithms o not aim to explicitly maximize the carrie traffic an consequently the path allocation is not formulate as an optimization task. An important reference for both the static- an ynamic cases is the series of works by Massoulie an Roberts, see e.g. [15] an [16]. Here, a number of fairness notions are iscusse an associate optimization tasks are presente for the case of the unboune flows an assuming fixe routes an link capacities. Although the max-min allocation has been wiely accepte an stuie in the literature, its appropriateness can be questione because of the relatively low banwith utilization. One of the promising alternatives to the max-min fairness is the proportional-rate fairness propose by Kelly in [9], [10] an also summarize by Massoulie an Roberts in [16]. Because of its superior characteristics in terms of overall network utilization, we in this paper concentrate on the proportional fair allocation metho. Accoring to the proportional-rate fairness criterion, the rate allocations x are fair, if they maximize log x (or in the weighte case w log x, where w is the weight of eman ) uner the capacity constraints. This objective may be interprete as being

4 280 M. Pióro, G. Malicskó, an G. Foor to maximize the overall revenue of allocations assuming each route has a logarithmic revenue function. Banwith allocation algorithms can further be ivie into two main groups. The ERAQLES algorithm in ATM [3] or the algorithms of [16] provie examples on istribute algorithms, while the application of a banwith broker facilitates the use of centralize banwith allocation algorithms [20]. In an earlier work [5], we evelope centralize algorithms for networks with given (fixe) link capacities that optimize routing in orer to improve the available level of fairness (either max-min or proportional rate) between flows. In [4] we combine network imensioning an proportional fair banwith allocation into a single optimization task without the inequality buget constraint. Our contribution to this line of works is the evelopment of explicit analytical formulae for a set of optimization tasks that allow for the joint optimization of routing an link capacity imensioning. To the best of our knowlege, such a profit/revenue optimization formulation an associate formulae an algorithms have not yet been propose in the literature. 3 Fixe Buget Network Dimensioning for Non-boune Elastic Flows As a basic case, consier the following optimization task, where the objective is to fin the rate allocations x an link capacities y e such that the logarithmic revenue function corresponing to proportional fairness is maximize. Note that each link e is associate with a marginal cost c e an so the operator s total link cost is c e y e. In this basic setting we assume that the operator s total link cost buget (C) is kept fixe such that C = c e y e. Task 1. Dimensioning uner Fixe (Equality) Buget Constraint with Fixe Routes inices: =1, 2,..., D emans e =1, 2,..., E links constants: { 1 if link e belongs to the path realizing eman a e = 0 otherwise. w weight of eman c e marginal cost of link e C assume buget variables: x flow allocate to eman y e capacity of link e

5 Optimal Link Capacity Dimensioning in Proportionally Fair Networks 281 maximize: F (x) = w log(x ) (1) constraints: c e y e = C; (2) e a e x y e ; e =1, 2,..., E (3) x,y non-negative. The explicit solution of the optimization task is given by the following theorem. Theorem 1. Let x 0 an y 0 be the solution of the above task. Then: F (x) x=x 0 = log(c) w w log( e c e a e )+ + w log(w ) log( w ) w (4) x 0 = Cw /(( w )( e c e a e )) (5) y 0 e = a e x 0. (6) Proof: As shown in [17], the ual function for Task 1 is of the form (7) where σ is a non-negative ual variable (multiplier) corresponing to constraint (2). W (σ) = (w w log(w /σ e c e a e )) σc. (7) The maximum of the ual function is attaine at the stationary point of (7) with respect to the multiplier : w /σ C =0. (8) Hence, the optimal multiplier σ 0 is given by σ 0 = w /C (9) an this immeiately implies (4) an (5). Naturally, at the optimum the constraint of inequality (3) is bining. We note that the maximum value of the objective function (1) epens only on C; (1) implies that this maximum is of the form F (C) =αlog(c)+β γ. (10)

6 282 M. Pióro, G. Malicskó, an G. Foor where α = w, β = w log(w ) log( w ) w an γ = w log( e c ea e ). Formula (10) implies that when the paths for the flows realizing the emans an the link capacities are also subject to optimization then the optimal solution will assign each flow to its shortest (lowest cost with respect to the c e -s) path. This is because in (10) only γ epens on the path selection an it is minimize when the lowest cost paths are use for realizing the emans flows. 4 Profit Maximization During the imensioning of a network the buget constraint usually appears only as an upper limit on the isposable amount of money an the target is to achieve an investment that maximizes the ifference of the revenue an the total link costs ("profit"). The revenue associate with eman epens on the operator s charging moel an is not necessarily a linear function of the allocate banwith (x ). In fact, a logarithmic revenue function can be consiere appropriate [23]. This logarithmic function is also motivate by the observation that the revenue can become negative if the allocate banwith is smaller than a threshol value. The optimization task in accorance with this objective is the following: Task 2. Profit Maximization uner Flexible (Inequality) Buget Constraint with Fixe Routes maximize: Ψ = w log(x ) c e y e (11) e constraints: c e y e C 0 ; (12) e a e x y e ; e =1, 2,..., E (13) x,y non-negative. The maximal value of the objective function (11) is attaine at the maximum, with respect to variable C, of the function F (C) = log(c) w C (14) over 0 C C 0. (15) The optimum of (14) is attaine either at C = w (if w C 0 )oratc 0 (if w >C 0 ). Of course, the optimal C is the optimal total cost of links e c ey e. Furthermore, from (4) it follows that in the case of w C 0 the optimal value of (11) is equal to w log(w /ξ ) w (16)

7 Optimal Link Capacity Dimensioning in Proportionally Fair Networks 283 (where ξ is the length of the shortest path of eman ), an the optimal flows are given by x 0 = w /ξ ; =1, 2,..., D. (17) 5 The Fixe Link Capacity Metho We are intereste in analyzing the ifferences between the metho outline in Task 1 an Task 2 an the proportional fair allocation mechanism use for fixe link capacities. In orer to o this we nee to formulate explicitly the allocation task use for comparisons. The numerical results are presente in Section 6. Task 3. Proportional Fair Allocations for Links of Fixe Capacity an Lowest Cost Path Routing inices: =1, 2,..., D emans e =1, 2,..., E links constants: w c e { 1 if link e belongs to the path realizing eman a e = 0 otherwise. weight of eman marginal cost of link e C assume buget y e = e C c ce e capacity of link e variables: x flow allocate to eman maximize: F (x) = w log(x ) (18) constraints: a e x y e ; (19) for e =1, 2,..., E =1, 2,...,D In Task 3, between each O-D pair we choose the lowest cost path with respect to the links marginal cost c e (lowest cost path routing). Note that the formulation allows for links of ifferent capacities, but uring the numerical evaluation we will consier equal size links (where c e 1). There is no close formula available for the calculation of the allocations in this case, therefore we use the optimization tool "Solver" inclue in Microsoft Excel for numerical evaluations. We use the piece-wise approximation of the logarithmic revenue function as escribe in [5].

8 284 M. Pióro, G. Malicskó, an G. Foor Seattle Argonne Ann Arbor Princeton Cambrige Palo Alto Bouler Arlington Urbana San Diego Houston Atlanta Fig. 1. The 12-noe Example Network 6 Numerical Examples 6.1 Input Parameters We consier the network of Figure 1. This network consists of 12 noes an E =16 links. We assume that there is one eman between each network noe pair yieling in total D =66emans. We assume that each link has unit marginal cost (i.e. c e 1 e) an that each eman has equal weight (w 10 ). 6.2 Numerical Results Recall from the formulation of Task 1 an Task 2 that the revenue is associate with the allocate banwith only, while the profit takes into account the link costs as well. Figure 2 compares the optimum revenue an profit as the function of the link cost buget C of Task 1 ("Revenue" an "Profit") with those of Task 2 ("Max-Profit" an "Max-Revenue"). Recall that for a fixe buget C, Task 1 etermines the banwith of each flow (x ) an the capacity of each link (y e ) such that the revenue is maximize (see (1)). Since Task 1 uses the equality buget constraint, in this case the profit is maximize when the revenue is maximize an the ifference between these two quantities is exactly C. (Inee, see the "Revenue" an "Profit" curves.) For Task 2, x, y e an the actually use link buget are etermine such that the profit is maximize (see (11)). (That is, for Task 2, the horizontal axis correspons to the maximum allowable buget C 0 of (12).) Figure 2 plausibly highlights the ifference between the revenue an profit maximization tasks. Increasing the buget up to a certain limit leas to the increase of both the "Profit" an "Max-Profit". Increasing the buget beyon this point (in this example at C = 660) ecreases the "Profit" of Task 1 an leas to the saturation of "Max-Profit" of Task 2, since in the profit maximization case the actually use buget may be smaller than the allowable. Task 2 effectively "freezes" the revenue increase that woul lea to profit ecrease (see the curve "Max-Revenue").

9 Optimal Link Capacity Dimensioning in Proportionally Fair Networks Revenue Max_Profit Buget Profit Max_Revenue Fig. 2. Revenue an Profit for Task 1 an Task Buget PRF_link_opt PRF PRF_Link_opt_profit PRF_profit Fig. 3. Revenue an profit for fixe an optimize link PRF In Figure 3 we compare the revenue an the profit as the function of the link cost buget C of Task 1 ("PRF link opt" an"prf link opt profit") with those provie by the fixe link proportional fair metho of Task 3 ("PRF" an "PRF profit"). We can see again analogously to Figure 2 the logarithmic increase of the revenue in case of both methos an the saturation of the profit function aroun the same link cost buget value of C = 660. Note that both the revenue an consequently the profit values are significantly higher when the link capacities are optimize (Task 1). Moreover, the ifference between the two methos is approximately constant for both aspects. The ifferences are even more visible in Figure 4 that highlights the ifference between the revenue ("Revenue") an the profit ("Profit") values of Task 1 an Task

10 286 M. Pióro, G. Malicskó, an G. Foor 100% 90% 80% 70% 60% 50% 40% 30% 20% 10% 0% Revenue Buget Profit 1540 Fig. 4. Revenue an profit for fixe an optimize link PRF II. 3. The vertical axis is the "gain of Task 1 as compare to Task 3": it is calculate as the profit (an revenue) given by the optimize link proportional fair metho (Task 1) ivie by that of the fixe link case (Task 3), minus 1. In accorance with the approximately constant ifference between the revenues provie by Task 1 ("PRF link opt") an Task 3 ("PRF") outline in Figure 3, the relative gain provie by Task 1 is monotonously ecreasing. We can observe in Figure 3, that at low capacity buget values, the profit of both Task 1 ("PRF link opt profit") an Task 3 ("PRF profit") are low ue to the insufficient buget. Thus, the relative gain in Figure 4 when optimizing the capacity (Task 1) as compare to the fixe capacity case (Task 3) is high. As the capacity buget increases, the reachable optimum profit grows, an as the ifference between the Task 1 an Task 3 is more or less constant, the relative profit gain ecreases, arriving to its minimum at the optimal capacity buget. Note, that even in this case link capacity imensioning as (in our example approximately 30%) to the profit as compare to the fixe capacity case. When the buget goes beyon the optimum, the profit ecreases again an the relative gain of Task 1 as compare to Task 3 becomes higher again. 7 Conclusion In this paper we consiere link capacity imensioning an banwith allocation in proportionally fair networks. We first consiere the "basic task" (Task 1) where one is intereste in fining the flow banwiths an link capacities such that the revenue is maximize. For this case we provie explicit exact formulae. An important variant of this type of optimization problem arises when the cost associate with the total link capacities is not kept fixe but upper-boune. In this case (Task 2) the profit (efine as the ifference between revenue an cost) is maximize by using a portion of the total allowe link capacity buget. As a "reference case" for comparisons, we also consiere the proportional fair sharing metho with fixe link capacities (Task 3).

11 Optimal Link Capacity Dimensioning in Proportionally Fair Networks 287 We summarize our finings as follows: Uner the fixe buget constraint, when both the paths for the flows realizing the emans an the link capacities are subject to optimization, the optimal solution assigns each flow to its shortest (with respect to the links marginal cost) path. Uner the inequality buget constraint, i.e. when the actually use link capacity buget can be chosen up to a boun, the maximum profit can be reache at a significantly lower capacity buget value than the maximum allowe buget. Link capacity imensioning in proportionally fair networks may significantly increase the profit as compare to the case when the link capacities are fixe. We believe that our problem formulations in Task 1 an Task 2 can provie important insight into traffic engineering problems an can serve as a basis for practically useful engineering tools. In future work we plan to present results for the case when the user flows are both lower- an upper boune as the moel in [17]. References 1. G. R. Ash, "Traffic Engineering & QoS Methos for IP-, ATM-, & TDM-Base Multiservice Networks", raft-ietf-tewg-qos-routing-00.txt, Internet Engineering Task Force, work in progress, November D. O. Awuche, A. Chiu, A. Elwali, I. Wijaja, Xipeng Xiao, "A Framework for Internet Traffic Engineering" raft-ietf-tewg-framework-02.txt, Internet Engineering Task Force, work in progress, July A. Fichou, C. Galan, S. Fia, Y. Moret, "Evaluation of the ER Algorithm ERAQLES in Different ABR Environments", 5 th IFIP TC6 Workshop on Performance Moeling an Evaluation of ATM Networks, pp. 48/1-48/3, Ilkley, UK, July G. Foor, G. Malicsko, M. Pioro, "Link Capacity Dimensioning an Path Optimization for Networks Supporting elastic Services", submitte to ICC 2002, New York, USA, G. Foor, G. Malicsko, M. Pioro an T. Szymanski, "Path Optimization for Elastic Traffic uner Fairness Constraints", 17 th International Teletraffic Congress, Salvaor a Bahia, Brasil, A. Goel, A. Meyerson, S. Plotkin, "Combining Fairness with Throughput: Online Routing with Multiple Objectives", To appear in JCSS, special issue on Internet Algorithms (Invite Paper) Extene abstract in STOC Y. T. Hou, H. H-Y. Tzeng, S. S. Panwar, "A Generic Weight-Base Network Banwith Sharing Policy for ATM ABR Service", IEEE International Conference on Communications, ICC 98, pp , K. Kar, M. Koialam, T. V. Lakshman, "Minimum Interference Routing of Banwith Guarantee Tunnels with MPLS Traffic Engineering Applications", IEEE Journal on Selecte Areas in Communications, Vol. 18, No. 12, pp , Dec F. P. Kelly, A. K. Mauloo, D. K. H. Tan, "Rate Control for Communication Networks: Shaow Prices, Proportional Fairness an Stability", Journal of the Operational Research Society, (49), pp , August, F. P. Kelly, A. K. Maulloo, D. K. H. Tan, "Rate Control for Communication Networks: Shaow Prices, Proportional Fairness an Stability", Journal of the Operational Research Society, (49), pp , 1998.

12 288 M. Pióro, G. Malicskó, an G. Foor 11. J. Kleinberg, "Single-Source Unsplittable Flow", IEEE Symposium on Founations of Computer Science, J. Kleinberg, Y. Rabani, É. Taros, "Fairness in Routing an Loa Balancing", IEEE Symposium on Founations of Computer Science, J. Lawrence, "Designing Multiprotocol Label Switching Networks", IEEE Communication Magazine, pp , Vol. 39, No. 7, July Q. Ma, P. Steenkiste, H. Zhang, "Routing High-banwith Traffic in Max-min Fair Share Networks", SIGCOMM 97, pp , August, L. Massoulie, J. W. Roberts, "Banwith Sharing an Amission Control for Elastic Traffic", International Teletraffic Specialist Seminar, Yokohama, L. Massoulie, J. W. Roberts, "Banwith Sharing: Objectives an Algorithms", IEEE INFO- COM 99, M. Pioro, "On Some Dimensioning Tasks Associate with the Notion of Proportional Fairness", Technical Report, Lun Institute of Technology at Lun University, CODEN:LUTEDX(TETS-7181)/1-6/(2001)&local 18, October S. Plotkin, "Competitive Routing of Virtual Circuits in ATM Networks", IEEE Journal of Selecte Areas in Communications, Vol. 13, No. 6, pp , Aug H. Qingyanga, D.W. Petr, "Global Max-Min Fairness Guarante for ABR Flow Control", IEEE INFOCOM 98, B. Teitelbaum, S. Hares, L. Dunn, R. Neilson, V. Narayan, F. Reichmeyer, "Internet2 QBone: Builing a Testbe for Differentiate Services", IEEE Network, Vol. 13, No. 5, pp. 8-16, September/October W. K. Tsai, M. Iyer, "Constraint Preceence in Max-Min Fair Rate Allocation", IEEE International Conference on Communications, ICC, G. e Veciana, T-J. Lee, T. Konstantopoulos, "Stability an PerformanceAnalysis of Networks Supporting Elastic Services", IEEE/ACM Transactions on Networking, Vol. 9, No. 1, pp. 2-14, Feb Anreu Mas-Collel, Michael D. Whinston, Jerry R. Green, "Microeconomic Theory" New York, Oxfor University Press 1995.

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