MPLS Traffic Engineering Using Enhanced Minimum Interference Routing: An Approach Based On Lexicographic Max-Flow

Transcription

1 MPLS Traffic Engineering Using Enhanced Minimum Interference Routing: An Approach Based On Lexicographic Max-Flow Koushik Kar Murali Kodialam T. V. Lakshman ECE epartment University of Maryland College Park Maryland, M Bell Labs Lucent Technologies 101 Crawfords Corner Road Holmdel, NJ 07733, USA Abstract The main contribution of this paper is a new algorithm for dynamic routing of bandwidth guaranteed tunnels. The algorithm is a generalization of the minimum-interference based algorithm presented in [6]. The routing objective is to satisfy as many potential future requests as possible in an environment where requests arrive one-by-one, and there is no knowledge of future arrivals other than their potential ingress-egress points. An application of these algorithms is for the routing of MPLS bandwidth-guaranteed label-switched paths (LSPs). The presented algorithm performs better than the scheme presented in [6], and other previously proposed algorithms, on several metrics like the number of rejected demands and successful re-routing of demands upon link failure. Also, the new algorithm is more robust in performance to implementations where most requests are routed with a shortest-path computation, and the computations required for determining minimuminterference are performed only infrequently. I. INTROUCTION The main contribution of this paper is an algorithm that generalizes the minimum-inteference scheme for dynamic routing of bandwidth guaranteed tunnels presented in [6]. The main new ideas and results are: (1) The lexicographic max-flow based formulation of minimum interference routing. This addresses shortcomings of the formulation in [6] and results in a routing algorithm with much improved performance. (2) The notion of routing based on identification of -critical links which is a generalization of the critical link idea introduced in [6]. (3) An approximate algorithm for computing -critical links. evelopment of a fast exact algorithm is an open problem. (4) Performance studies showing that most requests can be routed using only a shortest-path computation and that minimuminterference computations need be done only relatively infrequently. The algorithm only uses residual bandwidth information and is implementable in a distributed manner. Residual bandwidth information can be obtained in a distributed manner by link-state routing protocol extensions. The need for routing bandwidth guaranteed tunnels arises from the need to provision services with quality-of-services guarantees. Typically, this has been done in an offline manner with the assumption that all demands are known apriori. However with the advent of services that permit dynamic and frequent requests for capacity change, it cannot be assumed that all requests are known apriori. The routing algorithm must handle requests that arrive one-by-one and must satisfy as many requests as possible without re-routing of paths that have already been set-up. The only dynamic information available to the tunnel routing algorithms are the link residual capacities which can be obtained from routing protocol extensions such as in [3], [7], [8]. Quasi-static information such as the the ingressegress nodes in the network are known. This knowledge of the ingress-egress nodes should be exploited to reduce the number of request rejections due to insufficient network capacity. Even when all nodes are ingress-egress nodes, it is likely that some subset of ingress-egress nodes are more important and so tunnel requests between them will be required to have a lower probability of being rejected. The algorithm should be able to protect such important ingress-egress pairs. An important application of routing of bandwidth guaranteed tunnels and of the algorithm developed is the routing of bandwidth-guaranteed label switched paths (hereafter refered to as LSPs) in Multi-Protocol Label Switched (MPLS) networks. For conciseness, we focus only on the MPLS application in this paper. The algorithm presented in this paper can be used for efficient routing of LSPs in MPLS networks. It improves on the performance of the minimum interference routing algorithm presented in [6] and also on the performance of several previously proposed schemes. Furthermore, the proposed algorithm is amenable and more robust in performance to an implementation where most LSPs are routed using only a shortest-path computation. The presented algorithm is an on-line algorithm whose objective is to accomodate as many future requests as possible. It does not make any assumptions about future arrivals other than knowlege of the network ingress-egress nodes of arrivals. II. PRELIMINARIES We consider a network of routers. A subset of these routers are assumed to be ingress-egress routers between which LSPs can be set-up. However, it is not necessary that there be a potential LSP between every ingress and every egress. Instead, from a certain ingress, LSPs may be allowable only to certain egresses. We consider the request for an LSP to be defined by a triplet. The first field, specifies the ingress router, the second field specifies the egress router and the third specifies the amount of bandwidth required for LSP. We assume that requests for LSPs come in one at a time and there is no knowledge of the characteristics of future demands. The objective is to determine a path (if one exists) along which each demand for

2 an LSP is routed so as to make optimal use of network infrastructure. We define the residual bandwidth along a link to be the difference between the bandwidth of the link and the sum of the LSP demands that are routed on that link. First note that a new LSP can be routed along a given link only if the residual bandwidth on that link exceeds the bandwidth requested by the new LSP. These links will be referred to as feasible links (with respect to the given LSP demand). The network consisting of all the routers and just the feasible links will be referred to as the feasible network. Therefore, when performing the routing we can restrict our attention to paths in the feasible network. Note that if the ingress and the egress routers are disconnected in the feasible network then there is no path that has the desired bandwith and the LSP request is rejected. The objective of any routing scheme is to reject as few demands as possible. The most commonly used algorithm for routing LSPs is the min-hop algorithm. In this algorithm, the path from the ingress to the egress with the least number of feasible links is chosen. This algorithm though simple uses the same information that the proposed new algorithms use. Its performance in terms of efficient network usage can be easily improved upon with a little more computation. With the rapid rise in processor speeds and with traffic trunks not expected to be set-up and torn-down at high rates, it is justifiable to trade-off increased computation for more efficient network usage. The min-hop algorithm does not take information on ingress-egress pairs into account nor does it adapt routing to increase chances of successful re-routing upon link failure. A min-hop like algorithm which attempts to load-balance the network traffic is proposed in [4]. This widest-shortest path algorithm () finds a feasible min-hop path between ingress and egress such that the chosen min-hop path has the maximum residual path bottleneck link capacity. This algorithm too does not take ingress-egress information into account. With every node being assumed to be a potential ingress and egress point, a widest path algorithm without a min-hop restriction does not work well since long paths which increase network usage get chosen. Hence, a widest-shortest path hop heuristic performs better. However, if the ingress-egress pairs are known then an algorithm which picks paths longer than min-hop works well provided it avoids unnecessarily loading certain critical links as we shall see later. III. REVIEW OF KEY IEAS FOR MINIMAL INTERFERENCE ROUTING ALGORITHM We first review the key concepts in minimum interference routing since the new ideas in this paper are generalizations of the minimum intereference routing ideas in [6], The next section has a mathematical description of our new algorithm and also a proof of NP-hardness of this formulation of the MPLS routing problem. The NP-hardness justifies use of a heuristic algorithm in the absence of a known approximation algorithm. Illustrative Example If routing is done oblivious to the location of ingress-egress nodes in the network then the current routing may interfere with the routing of some future demands. We illustrate this with a simple example. Consider the network shown in Figure 1.!"#%$&'$"")('(*!+ There are three potential source destination pairs, Assume that all links have a residual bandwidth ( of ( 1 unit. We now have a request for an LSP between and with a bandwith request of 1 unit. If min-hop routing is used, the route )$ will be '$ Note that this route blocks the paths between and as well as and. In this example it is better to pick route even though the path is longer S S S 3 Fig. 1. Illustrative Example Interference The key idea is to pick paths that do not interfere too much with potential future LSP set-up requests (demands) between other source destination pairs. We first have to make this concept of interference more concrete. Consider the maximum flow (maxflow) value [1], between a given ingress-egress pair. This maxflow value is an upper bound on the total amount of bandwidth # that can be routed between that ingressegress pair. Note that maxflow value, decreases whenever - a bandwidth demand of units is routed between and. Note that the value of, will also decrease when an LSP is routed between some other ingress-egress pair. We define the amount of interference on a particular ingress-egress pair say, due to routing an LSP between some other ingress-egress pair as the decrease in the value of, Minimum Interference Paths With interference defined as above, we can think of a minimum interference path for an LSP between say, as that explicit route which maximizes the minimum maxflow between all ingress-egress pairs. Intuitively, this can be thought of as a choice of path between #! that maximizes the minimum open capacity between every ingress-egress pair. Though this formulation has intuitive appeal it has the drawback that it is the minimal maxflow that impacts the routing irrespective of values of the other maxflows. Note that another objective function that we might want to maximize is the sum of the maxflows. However, this objective has the drawback that some ingress-egress pairs may be cut-off or have to close to zero max-flow values because only the sum of the max-flows is now important. The minimum interference routing problem with these two objective criteria was mathematically formulated in [6]. The paper also describes a heuristic that finds and approximate solution to the problem with the sum criteria and studies its performance through simulations. An objective, which we show to be better than the above two objectives is to maximize the open capacities in a lexicographical order i.e. maximize the minimum. 1 3

3 I I 9 J l J I I J J J maxflow, then amongst all solutions for which the minimum maxflow is maximized, find the ones for which the second minimum maxflow is maximized, and amongst those, find the ones for which the third minimum maxflow is maximized, and so on. This forms the basis of our routing algorithm. This problem, which we would call LEX-MAX, is mathematically formulated in the next subsection. The formulation as described above, not only makes capacity available for the uncertain but possible arrival of future demands but also makes capacity available for re-routing of LSPs in case of link failures (this is illustrated by experimental results in Section VI). Critical Links To progress from the notion of minimum interference paths to a viable routing algorithm that uses familiar max-flow and shortest path algorithms we need the notion of critical links. These are links with the property that whenever an LSP is routed over those links the maxflow values of one or more ingress-egress pairs decreases. The next section gives an algorithm to determine critical links with a reasonable amount of computation (i.e., remains within a small target time budget for networks with hundreds of routers and thousands of ingress-egress pairs). Path Selection by Shortest Path Computation Once the critical links are identified, we would like to avoid routing LSPs on critical links to the extent possible. Also, we would like to utilize the well-used ijkstra or Bellman-Ford algorithms to compute the actual explicit route. We do this by generating a weighted graph where the critical links have weights that are an increasing function of their criticality (see next section for the actual link weight functions). The increasing weight function is picked to defer loading of critical links whenever possible. The actual explicit route is calculated using a shortest path computation as in other routing schemes. IV. MATHEMATICAL FORMULATION OF LEX-MAX We will give a couple of definitions that are needed in order to develop the LEX-MAX formulation. efinition 1: Given two vectors. and /, vector. is defined to be lexicographically not less than / if the first non-zero component in.102/ is non-negative. We will denote that. is lexicographically not less than / by.34/. efinition 2: Given an 5 -vector., an non-decreasing ordering of., denoted by 687.:9 is a renumbering of the components of. such that.<;>=?.a@b=c"c C:=?.AE. In this problem, the objective is to find a route such that the smallest maxflow value is as high as possible. Among all the solutions with the same smallest maxflow value, the secondary objective is to find the solution such that the second lowest maxflow is as high as possible and so on. More formally let F be the set of feasible G -vector of maxflow values between the G ingress-egress pairs after the current demand is routed. The objective IKJ IKJ of LEX-MAX is to find the H F such that 687*H 9L3M6N7 for all F. Before we describe the problem formulation, we define some of the notation used. Let OK7 PRQSTQUV9 describe the given network, where P is the set of routers (nodes) and S the set of links (arcs) and U the bandwidth of the links. Let 5 denote the number of nodes and W the number of links in the network. We assume that all bandwidths and demands for bandwidths are integral. Assume that there are a set of distinguished node (router) pairs X. These can be thought of as the set of potential ingressegress router pairs. We denote a generic element of this set by 7Y&QZ[9. Let G denote the cardinality of the set XC All LSP set-up requests (demands) are assumed to occur between these pairs. Let \ represent the node-arc incidence matrix. Each row in this matrix corresponds to a node in the graph and each column matrix corresponds to an arc. Each column has exactly two nonzero entries. The column corresponding to arc 7^]8Q_`9 has a +1 in the row ] and a -1 in row _ and a zero corresponding to all other rows. Assume that the arcs are numbered sequentially in any arbitrary order. Let a%bc be an W -vector corresponding to a pair 7Y&QZ[9 X I. Let represent a scalar that is the maximum bc flow that can be sent between nodes Y and Z in the residual network. Let d be an W -vector of residual capacities. Entry e in vector corresponds to the residual capacity of arce. The value of d is initialized to U. Let fbc represent an 5 vector with a +1 in position Y and -1 in position Z. We assume that demands arrive one at a time. The current demand is assumed to be between routers. and /, where 7.8Q/9 X. The demand is assumed to be for g (integral) units of bandwidth. Note that at this point other demands may already have been routed and the residual capacities of the arcs have been updated to reflect these routed demands. Assume that the maximum flow (maxflow) problem is now solved between an ingress-egress pair 7Y&QZA9 using the current residual capacity on the arcs as the arc capacity. This maximum flow value represents an upper bound on the total amount of bandwidth that can be routed from Y to Z and this bound is tight in the case of unit demands. We define the interference between a given path between. and /, and the ingress-egress pair 7Y&QZ[9, as the reduction in maxflow value between that ingressegress pair due to the routing of bandwidth on that path. In the LEX-MAX problem, since we do not know what the order of source-destination pairs, according to their (increasing) maxflow values, will be after the demand is routed, we introduce additional variables, h&i j bc (k`lnm to G, 7#Y&QZA9 X ), to represent that order in the solution. Thus in the solution to the integer program, if h i^j bc lom for some particular k and 7#YpQZ[9, that would imply that the pair 7Y&QZA9 is the k th smallest maxflow in the solution of the integer program, hai^j bc lrq would imply that it is not. The variables h i j bc can be thought of as elements of a GtsuG matrix v, such that there would be only a single m in all rows and columns, the rest being all q s. Thus the matrix v in the solution would just represent a permutation matrix. The integer program is stated below. In order to represent the LEX-MAX problem, the constants w i (k`lrm to G ) need to satisfy some constraints, which we will state later. xkyz { w i h i^j i~} ; b j bc bc c ƒ \ a bc bc f bc 7#Y&QZA9 X (1) \ a8 ˆ l g f ˆ (2) a bc Š a ˆ = d 7#YpQZ[9 X (3)

4 Á Á Œ Œ š Û Æ Á Ý Œ Ž š œ š&žÿ:ž" ~ (4) Ž & Ž š œ &žä ª «(5) & Ž Ž Œ Ž Ž Ž p Ž p ± œ ²š&žŸ<ž ~ ³ 'Ḿš* (6) µ Ž œ &žä ª «(7) µ8 ¹ ª º ž»u¼*½ (8) Ž ª º ž"š¼ œ š&žÿ:ž" ~ -ž œ &žä ª «(9) Equation (1) is the maximum flow problem for each of the ingress-egress pairs in «. Note that [Ž represents the maximum flow value for the pair &žä. Equation (2) states that» units of flow have to be sent between nodes ¾ and. Equation (3) ties up the variables for these problems. It ensures that the maximum flow problems only utilize the arc capacities that are left over after routing the current demands. Equations (4) and (5) state that the rows and the columns of the permutation matrix À each sum upto 1, as already explained before. Equation (6) ensures that the maxflow value multiplied by Á in the objective function is indeed the th smallest maxflow in the solution. The non-negativity restrictions are specified in equation (7) and equation (8) ensures that the demand is routed along a single path. Equation (9) simply states that the elements of the permutation matrix À can be either 0 or 1. Instead of solving the LEX-MAX problem simply in the form described above, we might want to solve a weighted version of it, in which each ingress-egress pair is given a weight. Let  Ž be the weight given to pair # pž [ Tªu«. The integer program for this weighted version can be obtained simply be replacing the term Ž in the objective function and in Equation (6) by ÃÄ#Å. Now we state some bounds on the constants Á so that the ÄÅ integer program as stated above indeed represents the LEX-MAX problem. We want to choose Á such that a difference of one unit in the th smallest maxflow should outweigh the effect of the maximum difference in all of the ^ %Ç?š th, ^ %ÇÈŸ& th,... th smallest maxflows. Then we have, Á Ç Á ÇÈ ~Ç Á ÌË for œ ² Í Î Ïš* ž" ³ ΠП&!ž" ~ ~š& 2É ~± ±)Ê (10) where Ë is the largest change in maxflow value for any ingressegress pair. Since Ë represents the change in maxflow value after routing a demand of» units, ËÑ ÒÔÓu» (11) The inequality comes from the fact that on routing a demand of value», the value of any cut can be reduced by at most» times the size of the cut, which is bounded by ÒÕ». From (10) and (11) we get, Ö Ì Ø ÒÕ»R šùçøòu»ú Û Ì Ü ³ Ḿš* ž ~ š (12) Now we will show that the problem LEX-MAX is NP-hard. Consider a problem MIN-MAX, where we want to route a demand along a path such that the minimum maxflow amongst all ingress-egress pairs is maximized. Thus MIN-MAX is a special case of LEX-MAX, since LEX-MAX maximizes the minimum maxflow too. Below we state a result on the complexity of MIN- MAX, without the proof. The proof can be found in [6]. Theorem 1: If all the capacities are integral, then the problem MIN-MAX is NP-hard. Since the problem MIN-MAX is a special case of the problem LEX-MAX, it immediately follows that: Corollary 2: If all the capacities are integral, then the problem LEX-MAX is NP-hard. V. SOLUTION APPROACH Since the problem LEX-MAX is NP-hard, one option is to solve the linear programming relaxation of this problem. However, the solution provided by the linear programming relaxation may have two problems : i) it may split the demand of» units between ¾ and over multiple paths, and ii) it may assign fractional values to the variables Ž (instead of 0 or 1). Thus the solution provided by the linear A programming relaxation represents only an upper bound on the actual solution. However, some heuristics can be applied to convert the linear programming solution to a feasible solution of the integer program, which may not be optimal, though. But the main drawback of the LP approach is the fact that it is not possible to incorporate any path constraints (for example, if there is a restriction on the number of hops that the demand can take, then it is not possible to incorporate it into the linear programming formulation). Further, we would also need a good rounding approach to convert the solution provided by the linear programming relaxation to a good feasible solution of the integer program. The approach that we take is to determine appropriate weights for the links in the network and route the demand along the weighted shortest path. Additional path constraints can be incorporated when the shortest path problem is being solved. The shortest path problem can be solved using the well-known ijkstra or Bellman-Ford algorithms. The problem now is to determine appropriate link weights so that the current flow does not interfere too much with potential future demands. The link weights are estimated by the following procedure. First we relax constraint (2), i.e., we remove the requirement that» units of flow have to be routed between nodes ¾ and. Equivalently this can be viewed as setting the value of» to zero. This results in the decoupling of LEX-MAX into independent maxflow problems, one for each ingress-egress pair in «. These prob- Ž represent the maxflow value for lems are now solved and let Ý ingress-egress pair &žä. Now, let # Þ ž &Þ!ž* *Þ ž &Þ!ž" ~ # *Þ ž &Þ be the sequence of the ingress-egress pairs arranged Ê Ê in the ascending order of their maxflow values, Ý Ž Þß Þß. The optimal solution to LEX-MAX when»à is given by á Ž ~ â Þß Þß. Now if some non-zero demand between ¾ and is routed on a link, the residual capacity of the link decreases. This results in a decrease in the current optimal solution value of LEX-MAX. The weight of a link is now estimated to be the rate of change in the optimal solution of LEX-MAX with respect to changing the residual capacity of the link. Let ã*å ã8ä à Ä#Å ÍæÍ represent the rate of change in maxflow value between ingress-egress pair &žä if

5 òú òú òú òú î õ ò ô the residual capacity of link ç is changed infinitesimally. Therefore the partial derivative represents the reduction in the è#épêë[ì maxflow value when an incremental amount of the current demand is routed on link ç. The weight íîèçì of a link ç is set to where è#é êë íîèçìïî ñò óôõ ð ò*öt øùúû ü úû ö8ý èçìnþ (13) ì is the ingress-egress pair with ÿ th smallest maxflow. The weight of a link represents the change in the objective function value of LEX-MAX if an incremental amount of the current demand is routed on that link. Note that the weight of the link íîèçì is a heuristic since it ignores the dependencies between the different links and also the fact that units of flow have to be routed from to. The approximation that we are making here is to assume that the ordering of the maxflow values after the current demand is routed, will be the same as the current ordering of the maxflow values. This approximation, though clearly not always true, is usually accurate considering that the demand to be routed is small. Before we proceed further, we will define the notion of critical links which we will use in describing our heuristic approach. A. Critical Links Roughly speaking, a critical link for an ingress-egress pair is a link with the property that if the capacity of the link decreases by a small amount, then the maxflow value between that ingressegress pair decreases by the same amount. An ingress-egress pair may have multiple critical links. By linear programming duality, associated with each maximum flow is a minimum cut. The maximum flow value of a particular ingress-egress pair decreases whenever the capacity of any of the arcs in any min-cut for that pair is decreased infinitesimally. In fact the maximum flow value will not decrease the capacity of any other link (not in a mincut) is decreased infinitesimally. efinition 3: An arc is defined as critical for a given ingressegress pair, if that arc belongs in any mincut for that ingressegress pair. The mincut is computed with the current residual capacities ùü on the links. Let represet the set of critical links for the ingress-egress pair èé&êëaì. From the maxflow-mincut theorem, ö ø ùü ùü ö ý if ç otherwise Therefore, from (13), where è#é êë è çì ívè çìïî ò ñ úûƒúû (14) ì is the ingress-egress pair with ÿ th smallest maxflow. Therfore the problem of computing the weights of the arcs is now reduced to determining the set of critical arcs for all ingress-egress pairs. The maximum flow between two nodes in a network can be computed in time 'è uì by the Goldberg Tarjan highest label preflow push algorithm [2] or in time 'è!"#%$&('1ì using an excess scaling algorithm. The maxflow algorithm will give us a mincut. However, there may be several alternate mincuts for a given ingress- egress pair. The critical links for the ingress-egress pairs are arcs belonging to the union of all these mincuts. In [6], it was shown that the set of critical links for a given ingress-egress pair can be determined by one execution of the maxflow algorithm between that ingress-egress pair. Theorem 3: Let )Ôîoè* ê,+ê,- ì be the given directed graph. Let é represent the source node and ë the destination node. Assume that a maximum flow between é and ë has been computed. Let. be the set of nodes reachable from é in the flow residual graph. Let/ represent the nodes that can ùü reach the sink ë in the flow residual graph. An arc è ÿê10aì( if 2 Arc è^ÿ ê30aì is filled to capacity and ÿ74 8/. 2 There no path between ÿ and0 in the flow residual graph. The proof of the above theorem is straightforward, and can be found in [6]. The set of critical links can be determined from the flow residual graph [1] by using the ùü above theorem. The procedure for determining all the arcs in takes 'è ì time in addition to the maxflow computation. Note that the asymptotic complexity of computing the maxflow ('è9"ì ) can be less than the subsequent procedure for determining the critical links ('è:*ì ) (for dense networks). However, most real networks are sparse, and therefore, in a realistic scenario, we would expect that the maxflow computation to dominate the overall running time. B. ; -Critical Links Recall that a link is critical for a particular ingress-egress pair if routing one unit of flow (of the current demand) on that link reduces the maxflow value for the ingress-egress pair. This is the same as saying that the maxflow value for the ingress-egress pair decreases when the capacity of the critical link is decreased by one unit. This also implies that if the capacity of a noncritical link for an ingress-egress pair is decreased by one unit, then the maxflow value for that ingress-egress pair does not decrease. However, if the capacity of a non-critical link for a given ingress-egress pair is decreased by more than one unit, the maxflow value for that ingress-egress pair may decrease. This brings us to a generalization of the notion of a critical link, a ; -critical link. efinition 4: A ; -critical link for an ingress-egress pair is a link such that if the capacity of the link is decreased by ;, the maxflow value for the ingress-egress pair decreases. Note that the critical links as defined in the previous subsection are 1-critical links, because if the capacity of a link is decreased by one unit, then the maxflow decreases (by one unit). In the definition of ; critical links we just insist that the maxflow value decreases (not necessarily by ; units). The notion of a link being; critical captures the notion that there may be links that are close to being critical that we may want to identify. This is especially important if the computation of critical links is done periodically, i.e., once every< demands. In this case it is important to protect links that are currently not critical but are close to being critical because they may become critical before = Note that we are dealing with capacities and demands all of which have integral values

6 w F b t w A K A A the computation is performed again. Thus the weight function K (15) would be >@?ACBEGFIH J KLMNOPQRSTQRVU%WYX[Z where \^] QR3_ QR?GF is the set of?3` Q K B,a Q K F -critical links for, the ingress-egress pair withb th smallest maxflow. Next comes the question of finding the set of -critical links for a given value of. One naive way to find whether a link is -critical for an ingress-egress pair is to reduce the capacity of the link by an amount, and compute the maxflow all over again, and check if the maxflow value decreases. However, this approach is prohibitively expensive, since it requires c times the time for a single maxflow computation to find links that are critical. A faster approach of finding -critical links is an open algorithmic problem. Instead of determining the set of critical links exactly, we determine the approximate set of critical links. INPUT: A graphd A threshold \ ] _?GF?`hB,aiF APPROX?e6B,f(F OUTPUT:, the set of pair PROCEURE: 1. \ ] _?3GFImon?e6BEfsF -CRITICAL LINKS?`hB,aiF^jlk and a setg of all residual link capacities and a source-destination pair -critical links for source-destination?`hb,aif 2. Compute maxflow p for source-destination pair Let drq be the flow residual graph after the maxflow computation, and trq be the residual capacity of anya"hv? Auj8f?e6BEfsF linkb3wif(j find@q. 3. For each link, include in\ ] _?GF if and only if all the following two conditions hold: x?a3f(yz x t7q There?e6BEfsF is no path betweenb and inv{ graph?af with capacity greater than d q Note that if the flow residual capacity of a link is not less than or if there is a path of capacity from nodeb to node in the flow-residual graph, then the link cannot be -critical. However this is only a necessary condition. Thus, although we would not miss out any links which are actually -critical, we may determine some links to be -critical even if they are not. The }? time-complexity of the APPROX -CRITICAL LINKS is c ~, in addition to the time for maxflow computation. One important issue in this case is how to choose. We will address this issue later.?af { C. Minimum Interference Routing Algorithm Now we are in a position to state the complete routing algorithm. Once the weights of the links are determined, the idea is to route the traffic along the shortest weighted path from?af t q >?ACBVGF to. Before this is done, all links having a residual capacity of less than units are eliminated. When computing the shortest weighted path, the weight of link is set to. In order to ensure that among arcs with zero weights, i.e., arcs that are not critical to any ingress-egress pair, we choose the one where the number of hops is minimal, we set the weight of these arcs to some small positive number. The box below gives a high level view of the{ ˆŠ Lexicographic Minimum Interference Routing Algorithm (ƒ ). { ˆŠ MINIMUM INTERFERENCE ROUTING ALGORITHM (ƒ )?e6b,f(f INPUT: A graphd and a setg of all residual link capacities An ingress node and an egress node between which a flow of units have to routed. OUTPUT: A route between and having a capacity of units of bandwidth. ALGORITHM: 1. Choose?3`hB,aiF^jŠk an appropriate 2. For all, use APPROX compute the maxflow and the set?`hb,aif^jlk of critical links\ ] _?GF 3. Order the ingress-egress pairs?` Q order of their maxflow values. LetH Œ K B,a Q K F be theb th ingress-egress pair in that order,b tož. 4. Compute >@?A BEGFIH the weights KLMNOPQRSTQRVU%WYX Z Auj f ( Z K are chosen according to the expression in (12)) 5. Eliminate all links which have residual bandwidth less than and form a reduced network. 6. Using ijkstra s algorithm >@?ACBV F compute shortest path in -CRITICAL LINKS to in the ascending reduced network using as the weight on link 7. Route the demand of units from to along this shortest path and update the residual capacities. Note that the box describes a heuristic for the unweighted version of LEX-MAX. For the weighted case, we just have to modify step 3 of the algorithm and order the ingress-egress pairs according to their weighted maxflow values, instead of their actual maxflow values. An appropriate choice of is very important for a good performance of the algorithm. If is chosen to be larger than the maximum capacity of the links, then our algorithm reduces to the min-hop algorithm. On the other hand, if is set to zero, then we may miss out some near-critical links, and thus the demand may be routed on those links thus reducing the maxflows of some ingress-egress pairs considerably. If the new demand that is to be routed is units, then may be a reasonable choice for, since no link whose index of criticality is more than can turn critical by routing a demand of units. One interesting case is when we are not computing the maxflows (and hence the critical links) on every demand arrival. Note that.

7 the maxflow calculation is computationally expensive, and when the demands are much smaller than the capacities of the links, we do not expect the maxflow values and the critical links to change very drastically. Therefore, we may not degrade the performance of the system significantly by doing the maxflow and the critical link calculations after, say, a few demand arrivals, or after a fixed time, or after the residual capacity of some link has changed beyond a certain threshold. In that case, we may be able to make a rough estimate of how much of bandwidth demand can arrive and be routed between the current maxflow computation and the next one, and use that information to determine. are done for the static case, i.e., arriving LSPs stay in the network forever. In this case, we will look at the flows between various ingress-egress pairs after a certain number of LSPs have arrived. Figure 3 plots the decrease in max-flow between ingressegress pair S1,1 as more and more LSPs are routed between other ingress-egress pairs. We see that with S-MIRA there is no interference at all till more than 2000 LSPs have been set up, whereas with L-MIRA there is no interference till 3000 LSPs are routed. However, every LSP set-up using the other two algorithms causes interference to potential future LSPs between S1 and 1. VI. PERFORMANCE STUIES In this section, we compare the performance of our routing algorithm, L-MIRA, with that of S-MIRA (MIRA using the sum criteria as presented in [6]), min-hop and widest shortest path () ([4]) routing algorithms, on some sample networks. The performance of each algorithm is measured by the proportion of LSPs rejected by the algorithm. For all the results shown in this section, the requests are uniformly distributed over all ingressegress pairs. We will also study the effect of the parameter and the frequency of maxflow computation on the performance of L-MIRA. decrease in available flow between S1 and 1 due to interference demand number 3 2 s 2 5 s 4 12 Fig. 3. ecrease in available flow between S1 and 1 due to interference s s Fig. 2. An Example Network total flow available between S1 and The performance comparisons are done using the network shown in Figure 2, and two other networks. The ingress-egress pairs are shown in the figure. For the illustrative example below, the capacity of the light links is 12 units and the dark link is 48 units (taken to model the capacity ratio of OC-12 and OC-48 links) and each link is bidirectional (i.e., acts like two unidirectional links of that capacity). In all the simulation experiments described in this paper, LSPs arrive randomly, and at the same average rate for all ingress-egress pairs. Interference We first show experimentally that the interference, as defined in Section IV, is indeed reduced by both L-MIRA and S-MIRA in comparison to min-hop and. We then show that this reduced interference translates to better LSP acceptance. In the experiments of the subsection, we will use the network of Figure 2, but we will scale all the capacities by 100. This larger capacity network is used for the performance studies because this permits us to experiment with thousands of LSP set-ups. The LSP bandwidth requests are taken to be uniformly distributed between 1 and 3 units. All the experiments of this subsection demand number Fig. 4. Total available flow between S1 and 1 after every LSP is routed Figure 4 shows the total available flow between S1 and 1 after every LSP is routed. Note that for L-MIRA, the decrease in the available flow between S1 and 1 is entirely due to the LSPs routed between S1 and 1, since as seen from Figure 3, there is no interference at all when the number of LSPs is less than 3000 (the same is true for S-MIRA when number of LSPs is less than 2000). The available capacities are much lower for min-hop and. This means that min-hop and should experience more blocking between S1 and 1. (S1 and 1 were picked for illustrative purposes only. The same applies for other ingress-egress pairs.) Figures 5 and 6 show the minimum and the sum of the available flows, respectively, between every ingress-egress pair after each LSP is routed. Again we see that the available flows for L- MIRA are consistently higher than those for S-MIRA as well as those for min-hop and. Next we see how these increased

8 minimum available flow number of rejects out of 5000 demands demand number trial number Fig. 5. Minimum available flow between all ingress-egress pairs after every LSP is routed Fig. 7. Static Case: Number of LSP set-up requests rejected for 20 experiments total available flow 2 x demand number too large). The rejection ratio is calculated over a window of approximately 1,000,000 LSP setup requests. Again, we see that both L-MIRA and S-MIRA perform much better than min-hop and. Between L-MIRA and S-MIRA, L-MIRA performs better, as we would expect. Figures 9 and 10 show the proportion of LSPs rejected for 20 experiments, for the dynamic case, under the same conditions as above, but for two other networks, network 2 (18 nodes, 30 links), and network 3 (20 nodes, 35 links), both for 4 ingressegress pairs. In all the cases, we see that S-MIRA and L-MIRA perform much better than and. Note that in Figure 10, L-MIRA and S-MIRA perform identically. Fig. 6. Total available flow between all ingress-egress pairs after every LSP is routed available flows translate to better performance. LSP acceptance In the first LSP acceptance experiment, we assume that all LSPs are long lived ( static case). We load the network with 5000 LSPs and observe the number of LSPs rejected by the different algorithms. We conducted 20 trials. This result is shown in Figure 7. Observe the noticeable degradation in performance of min-hop and due to interference. Also note that L-MIRA performs much better than S-MIRA; till 5000 LSP arrivals, the number of rejects in L-MIRA is zero for all the experiments. Note that in S-MIRA, in an attempt to maximize the total available maxflow over all ingress-egress pairs, the available capacity between some ingress-egress pair may become very small, causing a large number of rejects. L-MIRA, however, avoids this problem by giving the maximum priority to the ingress-egress pair with the least maxflow, and hence performs better. In the second experiment we tried to determine the dynamic behavior of the three algorithms. Figure 8 shows the proportion of LSPs rejected for 20 expreriments, under the following scenario. LSPs arrive between each ingress-egress pair according to a Poisson process with an average rate, and the holding times are exponentially distributed with mean. For our experiments, 6 œ[ Ÿž š. In this case too, bandwidth demands for LSPs are uniformly distributed between 1 and 3 units. For our experiment in this case, we take the same network as in Figure 2, but the capacities are scaled only by 10 (and not by 100 as for the static case - this is done so that the system warmup time is not rejection ratio trial number Fig. 8. ynamic Case: Rejection ratio for 20 experiments - network 1 rejection ratio trial number Fig. 9. ynamic Case: Rejection ratio for 20 experiments - network 2 Effect of critical-link computation frequency on performance

9 trial 1 trial 2 trial 3 trial 4 trial rejection ratio rejection ratio trial number interval between successive maxflow computations Fig. 10. ynamic Case: Rejection ratio for 20 experiments - network 3 Fig. 11. Rejection ratio vs. interval (in terms of LSP request arrivals) between successive critical link computations In the experiments described so far, the computation of the maxflow, and hence the computation of the critical links, is done on every LSP arrival. However, it is worthwhile examining whether it is necessary to recompute the critical links after each LSP is routed. Note that without critical link computation, each LSP routing involves only a shortest-path computation. Ideally, one would like to avoid frequent computation of critical links. We carried out a number of experiments to determine the effect of the critical link computation frequency on the system performance. Five representative trials are shown in Figure 11, when the routing algorithm is L-MIRA. The network is the same as the one shown in Figure 2 (but with the capacities scaled by 10) and the simulation conditions are the same as those for the dynamic case described in the last subsection. In the figure, the y-axis shows the rejection ratio, and the x-axis shows the interval between two successive critical link computations, in terms of LSP request arrivals. Note that the computed critical link set, and hence the path weights, do not change between successive critical link computations. The plots show, that in general, as we increase the interval between successive critical link computations, the rejection ratio show an increasing trend, as we would expect. However, the rejection ratio, even when the critical link computation frequency has been reduced by a factor of 50 or more, does not differ vastly from the case when the critical link computations are done on every LSP request arrival. This means that 49 out of 50 LSPs can be routed using only a shortest path computation. We would like to keep the frequency of critical link computation as small as possible. Note that the gradient of the plots are steeper when the interval between successive critical link computations is small, and become flattened out as the the interval increases, as we would intuitively expect. Effect of on performance Now we will study the effect of the parameter (which is used for determining the critical link set) on the performance of the algorithm. Figure 12 shows some representative examples to illustrate this. The network is the same as the one in Figure 2 (with the capacities scaled by 10), the routing algorithm is L-MIRA, and the simulation conditions are the same as those for the dynamic case for the last two subsections. However, in this case the size of each demand ( ) is kept fixed. In the figure, the results are shown for four different demand sizes ( ). Moreover, for demand size, the average load is kept fixed at 300 units. The figure shows, as we would intuitively expect, that the rejection ratio, in general, increases as we increase. A careful inspection shows that for three out of the four cases, namely ª «E, as we start increasing from 1, the number of rejects decreases very slightly, after which it increases monotonically (for ±«², the number of rejects always increase with increasing ). For ³ «[ ª V Ÿ, the minimum is attained when µ. For «² Ÿ, the number of rejects remain the same even as we increase. We have also verified that for ¹ ±«² Ÿ, the number of rejects is the same as that for min-hop. Note that our algorithm reduces to min-hop when is greater than the maximum link capacity (then all the links would be considered critical links and would be given the same weight). Note that in our network, all but five links have a capacity of 120 units, and the remaining five have 480 units. Thus we would expect that our algorithm would perform similar to min-hop when º»½¼. In this case, however, the performance is the same for º ¾«[ Ÿ. Figure 12 also demonstrates that when the size of the demand is increased while keeping the average load constant, the number of rejects increase. This can be intuitively explained in the following way. Consider two demand sizes, and Ÿ. Firstly, an arrival of a demand of size Ÿ is equivalent to a burst of two arrivals of size. Secondly, since a flow cannot be split, we may have to reject a demand of size Ÿ because of the non-availability of a path with Ÿ units of bandwidth, even though we may be able to route one (or more) demands of size. For most of the experiments we have carried out, we have observed that the minimum rejection ratio is achieved either at, oro ¹«, when the demand size is kept fixed at. However, in a practical scenario, we would expect that the demand size would not be fixed, and would vary over a range. What value of achieves the minimum rejection ratio in that case remains to be seen. However, one would expect that a good choice of would have to take into account the distribution of the demand size. Note that when the demand sizes are small compared to the link capacities (which is typically the case), setting «may not be a bad choice, if the critical links are computed on every request arrival. This is also evident from Figure 12. However, if the maxflow computation, and hence the determination of the critical links, is done less frequently, then settingà Á«may be a bad choice, even if the demand sizes are small. In this case, a lot of flow might be pushed into the network between two suc-

10 rejection ratio =5 =10 =15 = delta Fig. 12. Rejection ratio vs. Â, for various demand sizes it should have better rerouting performance. To test this we first route some randomly generated (static) LSPs in the network shown in Figure 2 (capacities scaled by 100), under the four different algorithms, S-MIRA, L-MIRA, and min-hop. We then cut a randomly chosen link and then use the same algorithm for re-routing the LSPs that had been routed on the link that is cut. The results, for 20 different trials, are shown in Figure 14. For the results shown, the cut link is (1,3) and the number of LSPs routed initially (before the link is cut) is It can be seen from the figure that, as expected, S-MIRA and L- MIRA have much better re-routing performance than min-hop and. Moreover, in the results shown, L-MIRA performs better than S-MIRA. cessive maxflow computations, and hence links that were not the most critical during a particular maxflow computation might become the most critical links before the next maxflow computation occurs. Thus we would like to set à to some larger value, such that it identifies the near-critical links, i.e., the links that can potentially become bottlenecks before the next maxflow computation. Figure 13 illustrates this on a network 2 (18 nodes, 30 links) with 3 ingress-egress pairs. The traffic is poisson with Ä Å!Æ Ç²ÈŸÉ and Ê is uniformly distributed between 1 and 3 units, and the routing algorithm is L-MIRA. The plots are shown for the cases when the interval between successive maxflow computations is 30, 40 (in terms of LSP request arrivals). The plots demonstrate that when the interval between successive maxflow computations is large, rejection ratio initially decreases with increase in Ã, after which it starts increasing again. When à is too small, then we are missing out too many near-critical links, while when à is too large, then too many links are being included in the set of critical links, reducing the algorithm to minhop. The value of à which achieves the minimum rejection ratio, would, in general, depend on the demand size distribution and the interval (in terms of LSP request arrivals) between successive maxflow computations. rejection ratio interval = 30 interval = 40 number of LSPs that cannot be rerouted when link (1,3) is cut trial number Fig. 14. Rerouting performance for 20 trials REFERENCES [1] R. K. Ahuja, T. L. Magnanti, J. B. Orlin Network Flows: Theory, Algorithms, and Applications, Prentice Hall, [2] A. V. Goldberg, R. E. Tarjan, Solving Minimum Cost Flow Problem by Successive Approximation, Proceedigns of the 19th ACM Symposium on the Theory of Computing, pp.7-18, [3] R. Guerin,. Williams, A. Przygienda, S. Kamat, A. Orda, QoS Routing Mechanisms and OSPF Extensions, Internet raft draft-guerin-qosrouting-ospf-04.txt, ecember [4] R. Guerin,. Williams, A. Orda, QoS Routing Mechanisms and OSPF Extensions, Proceedings of Globecom [5] M. Kodialam, T. V. Lakshman On-line Routing of Guaranteed Bandwidth Tunnels, Seventh IFIP Workshop on Performance Modelling and Evaluation of ATM/IP Networks, June [6] M. Kodialam, T. V. Lakshman Minimum Interference Routing with Applications to MPLS Networks, INFOCOM, March [7]. Katz,. Yeung, Traffic Engineering Extensions to OSPF, work in progress, Internet raft, [8] H. Smit, T. Li, IS-IS Extensions for Traffic Engineering, work in progress, Internet raft, [9] S. Plotkin, Competitive Routing of Virtual Circuits in ATM Networks, IEEE J. Selected Areas in Comm., 1995 pp Special Issue on Advances in the Fundamentals of Networking delta Fig. 13. Rejection ratio vs. Â, for different critical link compuatation intervals Link Failure Rerouting Performance Next we compare the performance of the algorithms in terms of how good the algorithms are in re-routing already set-up LSPs when a link failure happens. Note that to successfully reroute an LSP, there must be capacity available between the corresponding ingress-egress pair after the link failure has happened. Since MIRA tries to leave capacity open between ingress-egress pairs