1
A Multiobjective Off-line Routing Model for MPLS Networks Selin Cerav Erbas
∗a
, Cagkan Erbasb
a
Department of Stochastics Aachen University of Technology W¨ ullnerstrasse 3, 52056 Aachen, Germany
[email protected] b
Department of Computer Science University of Amsterdam Kruislaan 403, 1098 SJ Amsterdam, The Netherlands
[email protected] This study focuses on the multiobjectivity in the off-line routing of QoS traffic in MultiProtocol Label Switching (MPLS) Networks. The routing problem is formulated as a multiobjective mixed-integer programming. It aims at exploring the trade-offs between three objectives, namely minimal routing delay, optimal load-balance in the network, and minimal splitting of traffic trunks. For the multiobjectivity analysis, we first decompose the model into sub-problems. We then apply the lexicographic weighted Chebyshev metric method to these sub-problems to find the Pareto optimal solutions and visualize the tradeoff between the objective functions. The study is finalized with a case study to analyse the basic properties of the model. 1. Introduction As QoS and policy requirements of network traffic bring more complexity to the network administration, traffic engineering tools are increasingly gaining more importance. Internet service providers need to take multiple criteria into consideration for the optimal utilization of network resources shared between traffic with different QoS requirements. With the increasing need of convenient traffic engineering in autonomous systems, MPLS has been introduced [1]. The basic idea in an MPLS network is to forward the packets through Label Switched Paths (LSPs) by making use of the labels which are attached to packets at the ingress router of the network. The labels are assigned to the packets according to their Forwarding Equivalence Class (FEC) which are then sent through one of the LSPs associated with FEC. The idea behind the classification of the packets into FEC groups is to enable the network manager to distinguish the packets according to their QoS and policy requirements. The core routers throughout the domain use these labels as an index to look up their forwarding tables. The label is removed at the egress ∗
This research is supported by German Science Foundation (DFG).
2 router. Another concept, called traffic trunk, is associated with FEC such that a trunk is an aggregation of traffic flows with the same FEC and ingress-egress router. A major problem in traffic engineering is to distribute the traffic trunks on the network by LSPs [1]. Both on-line and off-line approaches to this problem are possible. This paper is based on an off-line traffic engineering model of which the initial ideas appeared in [2]. The complete model is developed in this study where we employ the multiobjectivity theory to the problem and do the complexity analysis. The multiobjective model targets at showing the existing trade-offs between the following objective functions: minimizing the routing cost, balanced distribution of the traffic, minimum traffic splitting. The study also includes a case study based on a realistic network for the verification of the model. Multiobjective off-line routing problems in telecommunication have been studied previously in [3] and [4]. Knowles et. al. [3] select the following objectives: minimization of the routing cost, minimization of the deviation from the target utilization of each link and minimization of over-utilization of links. Resende et. al. [4] develop a model for permanent virtual circuit routing. They try to minimize the weighted objective function consisting of both a delay component and a load balancing component. In both studies the traffic is not allowed to be split. Our study however considers the minimization of traffic splitting as the third objective which causes a dramatic change in the nature of the problem. The outline of the paper is as follows: next section defines the multiobjective network problem and introduces the proposed modeling, Section 3 and Section 4 analyse the problem according to the multiobjective optimization theory and complexity theory, respectively. In Section 5, a case study is carried out. Section 6 concludes the study with a discussion on future work. 2. Problem Definition & Modeling The basic problem is to select the optimal LSPs for traffic trunks with QoS requirements in a capacitated network. The whole network traffic consists of both QoS and Best-Effort traffics. Within the QoS context, it is reasonable to put these traffic trunks into priorities so that the QoS traffic is given precedence. We assume that the set of all QoS traffic trunks is denoted by T , and that they have the following attributes: • Each traffic trunk has an expected bandwidth demand dt . These bandwidth demands can be computed from the customer contracts and/or the statistics collected between the ingress and egress routers. • The routing performance of QoS traffic is highly dependent on the jitter, delay and reliability. As the traffic moves through a LSP, the waiting times at each hop will have a negative effect on the routing performance in terms of the jitter and delay. Moreover, using fewer hops increases the transmission reliability of the traffic trunks, since the probability of a failure on the LSP decreases. Therefore, the traffic trunks with QoS requirements have a constraint on the number of hops on their LSP(s). In order to implement this constraint, an admissible path set Pt = {p1t , ..., pLt t } with Lt paths is defined for each traffic trunk.
3 2.1. Model Formulation In the mathematical model the network is represented as a directed graph where V = {1, 2, ..., N } and E = {1, 2, ..., M } define the set of the routers and links, respectively. The directed link m has capacity um (in units/sec). Three objectives are taken into consideration for this model. The following sections explain these objective functions. 2.1.1. Minimizing the routing cost The first objective in the model aims at minimizing the routing cost experienced by the traffic trunks. A value cm is assigned to link m to represent the routing cost on the link which may depend on some parameters, namely its speed, length, and reliability. The cost P of the path plt is denoted by Ctl and is equal to the sum of its links’ costs: Ctl = m∈plt cm . Let xlt represent the bandwidth that is routed on the admissible path plt of the traffic trunk t. Then, the following constraints are related to this objective: Lt X
xlt = dt
∀ t ∈ T,
(1)
xlt ≥ 0,
∀ t ∈ T and ∀ l ∈ {1, . . . , Lt }.
(2)
l=1
Constraints (1) ensure the demand satisfaction. The first objective function is as follows: P P t min t∈T Ll=1 Ctl xlt . 2.1.2. Balancing the load The second objective aims at avoiding high utilization of some of the links while leaving others less utilized. A slightly modified version of the function that was proposed in [5] is suggested in our model. In their study, a piece-wise linear cost function is defined for each link based on its utilization rate which is defined as the proportion of the total traffic load on the link to its capacity. The idea behind the function is to penalize sending packets over a link as its utilization increases. The load balancing function used in this study differs from the original function in a way that the links are not allowed to be utilized more than their capacities. Moreover, it has different break points. In fact the exact shape of the function is not critical; the more important point is that it is a piece-wise linear increasing and convex function. The complete definition of the function depends highly on the network and traffic demand. The load balancing cost function used in this study is illustrated in Figure 1. The mathematical formulation corresponding to this objective is as follows: fm =
Lt XX
l am t,l xt
∀ m ∈ E,
(3)
t∈T l=1
fm ≤ um φm ≥ f m 1 φm ≥ 2fm − um 2 23 φm ≥ 5fm − um 10 93 φm ≥ 15fm − um 10
∀ m ∈ E, ∀ m ∈ E,
(4) (5)
∀ m ∈ E,
(6)
∀ m ∈ E,
(7)
∀ m ∈ E,
(8)
4
Load Balancing Cost
20
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Load
Figure 1. The Load Balancing Cost Function for a link with a capacity of 1 units/sec
453 um 10 2613 um ≥ 300fm − 10
φm ≥ 60fm −
∀ m ∈ E,
(9)
φm
∀ m ∈ E.
(10)
In constraints (3) fm refers the total load on the link m, where alt,m is equal to 1 if path plt uses link m, and 0 otherwise. Constraints (4) are to ensure that the link does not get over-capacitated. Constraints (5) - (10) are used to determine the value of the load balancing cost function φm for the link m. The break points of the functions are at the following utilization rates of the links: 0.5, 0.6, 0.7, 0.8 and 0.9. Our second objective is, P min m∈E φm . Balancing the QoS traffic load on the network also improves the negative impact of QoS traffic over the Best-Effort traffic. When the network is in operation, the QoS packets have the precedence while being served at the routing nodes. If the QoS traffic rate on the link increases, the Best-Effort traffic may suffer from large waiting times in the queues. 2.1.3. Minimizing the number of LSPs Our third objective is related to the number of LSPs utilized by the traffic trunks. The network management will be more complex with the increasing number of established LSPs. Splitting the traffic trunks over multiple paths will bring more messaging and labeling overhead. What is more, if the traffic flows are also split, the packets may experience more variant delays from each other and need to be reordered. Thus, the model aims at minimizing the number of LSPs assigned to the traffic trunks. To accommodate the third objective in the model, in constraints (11) decision variables ytl are introduced which are equal to 1 if the path plt is utilized, and 0 otherwise. xlt ≤ dt ytl
and
ytl = {0, 1}
∀ t ∈ T and ∀ l ∈ {1, . . . , Lt }.
Hence, our third objective function has the following form: min
P
t∈T
(11)
PLt
l=1
ytl .
5 3. Multiobjective Optimization The salient feature of the model in this study is its ability to capture the trade-offs between the objective functions. Instead of targeting to find a single optimal solution, one is interested in finding the set of alternative solutions, namely the Pareto optimal solutions [6]. A multiobjective optimization with Q ≥ 2 objectives minimizes a vector of valued functions f (v) = (f1 (v), . . . , fQ (v)) where the decision vector v has to be a member of the feasible set S. A decision vector v ∗ ∈ S is Pareto optimal, if there exists no v ∈ S such that fi (v) ≤ fi (v∗ ) for all i = 1, . . . , Q and fj (v) < fj (v∗ ) for at least one j. The study in this section requires the introduction of another concept in multiobjective optimization, namely weak Pareto optimality. A decision vector v ∗ ∈ S is called weakly Pareto optimal if there exists no v ∈ S such that fi (v) < fi (v∗ ) for all i = 1, . . . , Q. The multiobjective programming targets at plotting the Pareto front which is obtained by the images of the Pareto optimal solutions in the objective space. Our initial study regarding this problem is to visualize the Pareto front by using exact methods. One of the most interesting attributes of our model is that while two of its objective functions are continuous, the third objective consists of a totally discrete function. The counting property of the third objective function allows us for a nice decomposition of the model. The third objective function is removed from the original problem and replaced as a constraint into the original problem. So, we substitute the original problem with a series of decomposed multiobjective problems, P (N )’s,: P (N ) : min
(f1 (v) =
Lt XX
Ctl xlt , f2 (v) =
φm )
m∈E
t∈T l=1
subject to
X
(1) − (11), Lt XX
ytl ≤ N.
(12)
t∈T l=1 L
L
Note that v = [x11 . . . x|T|T| | φ1 . . . φ|E| y11 . . . y|T|T| | ] for our problem. The most traditional approach in multiobjective optimization is to combine the objective functions into a single function by using a weighted mean [2]. The main drawback of this weighted sum method is that it cannot generate the Pareto optimal solutions which are located in the non-convex part of the front [6]. For linear programming problems, the Pareto front is always convex [6]. Since the constraint (12) brings discreteness to the model, there exists a possibility that the trade-off curves may include non-convex parts. The results obtained from the case studies in Section 5 also proves the possibility for the non-convexity of the Pareto front. Instead of the weighted sum method, the lexicographic weighted Chebyshev metric method, introduced by Steuer [7], is employed to plot the Pareto front of each subproblem P (N ). This method makes it possible to compute the whole Pareto front [7] regardless of its shape, by minimizing the distance to a reference point. The reference point selection is based on the ideal point, zI where ziI = minx∈S fi (v). Then the reference point is given by ziref = ziI − i where i > 0 for all i = 1, . . . , Q. In this study, each i is selected such that they decrease each ziref to the nearest integer less than its ideal point value. To measure the
6 distance, the weighted Chebyshev norm is used, k z − zref kδ∞ = maxi=1,...,Q δi (zi − ziref ) P where Q i=1 δi = 1.0. The lexicographic weighted Chebyshev metric method proceeds in two steps which are explained in the following theorems. The first step deals with the problem T P (N, δ) : subject to
min
α
α ≥ δ1 (
Lt XX
Ctl xlt − z1ref ),
(13)
t∈T l=1
α ≥ δ2 (
X
φm − z2ref ),
(14)
m∈E
(1) − (12), where δ1 , δ2 > 0 and δ1 + δ2 = 1. Theorem 1 A feasible point v ∗ is a weakly Pareto optimal solution of the problem P (N ), if and only if there is a weight vector δ, δ1 , δ2 > 0 and δ1 + δ2 = 1.0 such that v∗ is an optimal solution of the problem T P (N, δ). In [6, pp 93], a proof for the general case of this theorem is given. In this paper, we look into a special case where the weights sum up to unity. Although we are interested in plotting the Pareto optimal solutions, according to Theorem 1, solving the problem T P (N, δ) only ensures the weak Pareto optimality. Since a Pareto optimal point is also weakly optimal, it is among the optimal solutions set of the problem T P (N, δ). Finding the Pareto optimal points among this set requires an additional step, which is explained in Theorem 2. Theorem 2 A feasible point v ∗ is a Pareto optimal solution of the problem P (N ), if v ∗ is an optimal solution of the problem T P 0 (N, δ). T P 0 (N, δ) : subject to
min
Lt XX t∈T l=1 ∗
Ctl xlt +
X
φm
m∈E
α = α , (1) − (14),
(15)
where α∗ refers to the optimal value of T P (N, δ). Proof: Assume that v∗ is an optimal solution but not Pareto optimal. We know that v ∗ is weakly Pareto optimal by Theorem 1. Therefore, the following relationship applies for at least one solution from optimal solutions set of the first problem and v ∗ : fi (v) < fi (v∗ ) P P for one i ∈ {1, 2} and fi (v) = fi (v∗ ) for the other i. Then, 2i=1 fi (v) < 2i=1 fi (v∗ ) which is a contradiction to the optimality of v ∗ . 4. Complexity Analysis For the complexity analysis of the model, the decomposed model is generalized to a single objective problem and called as “Number of Paths Restrained Multicommodity Flow Problem” . If the single objective version of a problem is NP-complete, we can conclude that the same problem with multiple objectives is also NP-complete.
7 Let G = (V, E) be a connected digraph. Suppose that the tth commodity for the network is given as Tt = (st , gt , dt ), 0 < t ≤ n; st , gt ∈ V , dt ∈ R+ where st , gt , and dt represent the associated source node, target node and demand, respectively. Pt denotes the set of all possible paths for the tth commodity. Notice that this is a generalization of the problem introduced in Section 2 of this study. We define functions u : E → R+ and c : E → R+ which assign each edge on the network a capacity and a cost, respectively. The cost of a path, C(p) is simply the sum of the edges’ costs along the path. Let x(p) be the flow on the path p and S(Pt ) be the set of the paths which are assigned a positive flow for the commodity t. The Number of Paths Restrained Multicommodity Flow Problem looks for a flow assignment satisfying the below two constraints, (16) and (17), additional to capacity and demand satisfaction constraints. N and B denote the upperbounds on the number of used paths and flow assignment cost, respectively. | S(P1 ) | + . . . + | S(Pn ) | ≤ N
(16)
where S(Pt ) ⊆ Pt and S(Pt ) 6= ∅. X
C(p)x(p) ≤ B
(17)
p∈S(P1 )∪...∪S(Pn )
Theorem 3 The Number of Paths Restrained Multicommodity Flow Problem is NPcomplete. Proof: The Number of Paths Restrained Multicommodity Flow Problem can be restricted to Integer Multicommodity Flow Problem, by limiting the bound on the number of paths in the first constraint of the Path Restrained Multicommodity Flow Problem to the number of commodities to be transmitted on the network (i.e. allowing only instances having N = n). The Integer Multicommodity Flow Problem tries to minimize the flow assignment where each flow can only use a single path and it is known to be NP-complete [8]. 5. Case Study The multiobjective model is implemented in the network illustrated in Figure 2, which is also studied in [9]. We aim at visualizing the trade-offs between the objective functions, by solving T P 0 (N, δ) for various weight vectors and N values. In this case study, each link is assigned to a routing cost of unity. The links are unidirectional and have a capacity of 50 units/sec. A full traffic demand matrix is assumed, so there exist 90 traffic trunks. The traffic demand matrix for QoS traffic is given in Figure 2. The traffic trunks are allowed to use the paths which have at most three hops. The optimal solutions are obtained by solving the problems with the Cplex 6.6 optimizer [10]. In Figure 3, we show the trade-off curves for the problem P (N ) when N is taken as 90, 92, 94, and 96. The Pareto optimal solutions, obtained by solving the problem T P 0 (N, δ) with various δ for each N value, are shown in the figure. The points are connected by lines in order to increase the visual perception of the trade-off. Our first observation in Figure 3 is that, the trade-off curves exhibit worse performance regarding the load balancing and routing costs, as the constraints (12) become more restricted. For a specific value of the
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Figure 2. Example Network Topology and Traffic Demand Matrix
load balancing cost (routing cost), the routing costs (load balancing costs) increase as the number of LSPs decreases. The two objectives, minimizing the routing cost and minimizing the number of LSPs, support each other in the sense that both objectives can be minimized to their optimal values concurrently. However at this solution we obtain a very high load balancing cost. This solution corresponds to the values of the routing and load balancing costs of 843.5 (optimal routing cost) and 1185.2, respectively. Interestingly, the following general observation is true for all of the curves in Figure 3. When the routing costs are kept at low values (especially for values less than 852), the load balancing cost suffers dramatically. Moreover, the trade-off curves become closer as the number of LSPs increases. This is because of the fact that we loosen the number of 1200 "number of paths = 90" "number of paths = 92"
1180
Load Balancing Cost
"number of paths = 94" 1160
"number of paths = 96"
1140 1120 1100 1080 1060 840
845
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Figure 3. The Trade-off Curves
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9 LSPs constraints of the problems. As a last remark on Figure 3, we observe that the shape of (some of) the trade-off curves may include non-convex parts. This supports for our choice of the lexicographic weighted Chebyshev metric method over the weighted sum method for the trade-off analyses. Additionally, we have made the following observation regarding the effect of load balancing on the network. The shape of the load balancing function allows us a classification of the links into regions according to their utilization rates. As seen in Figure 1, we have six regions: [0, 0.5] – (0.5, 0.6] – (0.6, 0.7] – (0.7, 0.8] – (0.8, 0.9] – (0.9, 1]. During the case study we have observed that the average utilization rates (range between 0.527 and 0.541) and the maximum utilization rates (ranges between 0.7 to 0.74) don’t change dramatically through the Pareto optimal solutions. However, the load balancing function has a more striking effect on the distribution of the links into the regions. When the second region is defined as the target (region of the average utilization rates), the balanced distribution of the load will imply having as many links as possible in the second region and having as few links as possible far away from the second region. The following weighting function gives a general idea about the distribution of the links around the target, i∗ = 2. W =
6 X
| i − i ∗ | ni
(18)
i=1
where ni denotes the number of the links in the ith region. Notice that the weights increase as the distance from the target region becomes larger. Figure 4 shows the values of W versus load balancing function. In the figure, we observe that W tends to increase as the load balancing function increases. 34 32 30 28 26
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Figure 4. Impact of the Load Balancing Function
6. Conclusion & Future Work In this study a multiobjective mathematical model is introduced for the selection of the LSPs for the QoS traffic in MPLS networks. Three objectives are taken into consid-
10 eration, namely minimization of the routing cost, balancing the load over the network and minimization of traffic splitting. Within the context of this study, we were primarily interested in the exact methods in order to plot the trade-off curves. We have chosen the lexicographic weighted Chebyshev metric method due to its ability to find the solutions located both in the convex and non-convex parts of the Pareto front. With the trade-off curves, we were able to acquire a very clear picture about the relationship between the objectives. According to the complexity analysis, the problem becomes NP-complete with the inclusion of the third objective. As a future study, instead of using exact methods we aim at the development and evaluation of powerful heuristic methods for the problem. The approximation methods for multiobjective problems can be grouped into two primary classes [11]: methods of local search in objective space by updating the direction of the search and population based methods where all the population contributes to the evolution process to build the Pareto front. However, these methods are usually studied for combinatorial optimization problems. We are especially interested in the adaptation of these methods for our mixed-integer multiobjective problem, which has a very different nature and difficulties from the combinatorial optimization problems. REFERENCES 1. D. Awduche, J. Malcolm, J. Agogbua, M. O’Dell, J. McManus, “Requirements for traffic engineering over MPLS,” RFC 2702, September 1999. 2. S.C. Erbas, R. Mathar, “An off-line traffic engineering model for MPLS networks”, In Proceedings of IEEE LCN 2002, Tampa, Florida, November 2002. 3. J. Knowles, M. Oates, D. Corne, “Advanced multi-objective evolutionary algorithms applied to two problems in telecommunications”, In BT Technology Journal, 18(4), pp 51-65, October 2000. 4. M. G. C. Resende, C. C. Ribeiro, “A grasp with path-relinking for permanent virtual circuit routing”, In Proceedings of Networks, 2002. 5. B. Fortz, M. Thorup, “Internet traffic engineering by optimizing OSPF weights,” In IEEE INFOCOM 2000, pp 519-528, Tel-Aviv, Israel, March 2000. 6. M. Ehrgott, Multicriteria Optimization, Lecture Notes in Economics and Mathematical Systems 491, Springer-Verlag, 2000. 7. R. E. Steuer, Multiple Criteria Optimization: Theory, Computation and Application, John Wiley and Sons, New York, 1985. 8. W. J. Cook, W. H. Cunningham, W. R. Pulleyblank, A. Schrijver, Combinatorial Optimization, John Wiley and Sons, New York, 1998. 9. C. Scoglio, JC de Oliveira, I. F. Akyildiz, G. Uhl, “A new threshold-based policy for label switching path (LSP) setup in MPLS networks”, In Proceedings ITC17, Brazil, 2001. 10. Cplex 6.6, http://www.ilog.com/. 11. M. Ehrgott, X. Gandibleux, “A survey and annotated bibliography of multiobjective combinatorial optimization”, OR Spektrum, 2000.
