Efficient QoS Partition and Routing in Multiservice IP Networks

Efficient QoS Partition and Routing in Multiservice IP Networks I. Atov H. T. Tran R. J. Harris RMIT University Melbourne Box 2476V Vic. 3001, Australia Abstract This paper considers the combined problem of QoS partitioning and routing (Problem OPQR-G) for a QoS framework in which a performance dependent cost function is associated with each network element and the QoS metric is additive (e.g., delay, itter). This problem has been addressed in the context of unicast and multicast only. Here we consider the problem for a more general case of an arbitrary topology network. Also, it is considred that the performance dependent cost functions are of a general integer type. The goal is to determine primary paths between the OD pairs and QoS partitions on the links so that the overall cost in the network is minimized while all OD pair QoS requirements are satisfied. As the problem is NP complete, we concentrate on development of an efficient heuristic algorithm. In addition, two LP-based algorithms were developed, that use the optimization tool ILOG TM CPLEX 7.1 LP for solving the Problem OPQR- G. The results obtained for various network scenarios are very close to the optimal. The problem addressed in this paper provides the basis for the solution of many interesting and practical engineering problems, such as dimensioning and admission control/resource reservation in multiservice IP networks. 1 Introduction The development of various technologies, such as Diff- Serv and MPLS, has enabled support of various traffic classes with different QoS requirements on an integrated IP network [8] [9]. For such multiservice IP networks, the dominant factor in the cost of provisioning a particular service class is dependent on the performance requirement (QoS) of that class. Accordingly, a key issue in the design of these networks is to determine the minimum amount of resources needed to be allocated in the network, so that the QoS requirements for the various traffic classes can be satisfied. This requires development of several mechanisms. One is the QoS routing mechanism, which has been the subect of many papers [4] [5]. The aim of QoS routing is to provide a path for a traffic demand between an origin and destination (OD) pair, based on the QoS requirement of the traffic and some optimization criteria. The second mechanism is resource provisioning, that provides the QoS guarantees. Based on a given topology and routing information this mechanism determines the minimum required levels of resources that must be allocated to each network element in terms of bandwidth and buffer space, so that the end-to-end QoS obectives of the various classes can be satisfied. With the increasing demands for delay critical applications (i.e., real-time) it is important that delay QoS metric be considered in the design and planning functions of these networks. The delay QoS metric is additive (i.e., end-to-end QoS along a path is a sum of the QoS of its constituent links) and such a QoS class may cover end-to-end delay bound, or random variation of delay (itter). In [6] a methodology for capacity planning of multiservice IP networks has been developed that takes account of the varying delay (QoS) requirements for the classes. This methodology exploits a framework in which capacity provisioning is based on the partition of the end-to-end delay QoS requirements of each class-based traffic demand between each OD pair into local QoS requirements at each link in the network. In this context, it is important to study the combined problem of QoS partitioning and routing (Problem OPQR- G) for a QoS framework in which a performance dependent cost function is associated with each network element and the QoS metric is additive. The solution of this problem will provide the basis for the solution of many interesting and practical engineering problems, such as dimensioning and admission control/resource reservation in multiservice IP networks subect to varying QoS constraints for the classes, as considered in [6]. The combined routing and QoS partitioning problem has been addressed in the literature in the context of unicast connections and multicast trees only, see [1] [2] [7]

and references therein. In this paper, we consider the problem for a more general case of arbitrary network topology. Also, it is considered that the performance dependent cost functions are non-increasing and of general integer type. The goal is to determine routes between the OD pairs and QoS partitions on the links so that the overall cost in the network is minimized while all OD pair QoS requirements are satisfied. This problem is NP complete as it has been shown this to be the case even in the (simpler) case of unicast connections. As a result, we concentrate on development of an efficient heuristic algorithms and we provide a pseudopolinomial solution for the problem. The proposed greedy algorithm OPQR-G first considers each OD pair in isolation by exploiting an algorithm for optimal routing and QoS partitioning for unicast developed by Lorenz et al [1] and then perfoms re-adustment of the delay partition in the network in order to reduce the overall network cost. In addition, two LP-based algorithms were developed, that use optimization tool ILOG TM CPLEX 7.1 LP for solving the Problem OPQR-G. Using the lower bound of the optimal cost as a benchmark, the performance of the three proposed heuristic algorithms has been examined. The results obtained for various network scenarios are very close to the optimal. In Section 2, we first provide mathematical formulation of the Problem OPQR-G and we review the routing and QoS partitioning problem in the context of unicast, as it can be viewed as a subproblem of the Problem OPQR-G. Section 3 describes in detail the proposed heuristic algorithms. Analysis of the results is provided in Section 4 and finally Section 5 identifies issues for future work and concludes the paper. 2 Problem Formulation In this section, we give a formal definition of the Problem OPQR-G (Optimal QoS Partition and Routing in General Topology Networks). In the following, the network is represented as a graph G(V, E), where V = n is the number of nodes and E = m is the number of links (edges) in the network. We consider a QoS framework in which, each link may offer different delay guarantees, d l, each associated with a cost c l (d l ). Furthermore, the link delay/cost functions are considered to be non-increasing and of general integer type (e.g., delays and costs are integers). We assume that there are u uv possible routes between each OD pair (u, v V ) and a decision variable is introduced that has value 1 if route is chosen for the OD pair (u, v) and 0 otherwise. The set of candidate routes for each OD pair may be obtained through various route generation procedures. Problem QPQR-G: Given a network G(V, E), a delay/cost function for each link {c l (d)} l E, and end-toend delay requirements for all OD pairs in the network {D uv } u,v V. Find a partition d = {d l } l E and set of routes for all OD pairs r = {r uv } u,v V, such that the corresponding network cost is minimized while the end-to-end delay requirements for all OD pairs are satisfied. Mathematically, the Problem OPQR-G can be formulated as follows: Problem OPQR-G Minimize: a uv l c l (d l ) With respect to: Subect to: l u,v l {d l } l, { }, u,v d l D uv + (1 )M (1) = 1 = {0, 1}, d l 0 l E where the symbol a uv l is an indicator parameter with value 1 if link l lies on route and is zero otherwise. The constant M in the above formulation should be sufficiently large and therefore we set it to the sum of all end-to-end delay requirements, e.g., M = uv Duv. The routes, for which = 1 determine the optimal set of routes r = {r uv } u,v V in the network. In the above formulation, a link (local) delay partition represents a maximum allowable delay on a link, so that the end-to-end delay constraints of all traffic flows traversing that link are still satisfied. The network cost represents a sum of all link costs where a total cost of a link is a function of the delay allocation on that link and the number of flows that traverse the link. 1 2.1 Subproblem: OPQR for Unicast In the following, we state the Problem OPQR (Optimal QoS Partition and Routing for Unicast Connection) (see [1]), as it can be viewed as a special case of our problem. Each link may offer different (integer) delay guarantees, d l, each associated with a (integer) cost c l (d l ). The cost of a route r with a given delay partition {d l } l r is defined as c(r) = l r c l(d l ). Problem OPQR: Given a network G(V, E), a delay/cost function for each link {c l (d)} l E, and end-to-end delay requirement D between a source and destination (s, t). Find the minimal cost route r and partition {d l } l r that satisfies the end-to-end delay requirement. Lorenz et al solve this problem by the algorithm OPQR presented in Fig. 1 (for details, see [1]). The OPQR Problem is a generalization of the RSP (Restricted Shortest Path) Problem for integer cost functions [3]. The general 1 If the order of the sums in the cost function is changed, the sum u,v xuv auv l represents the total number of flows traversing link l.

OPQR (G(V, E), {c l (d)} l E, D, U) for all v s D(v, 0) D(s, 0) 0 for i = 1, 2,..., U for v V D(v, i) D(v, i 1) for l {(u, v) v V } for = 1, 2,..., i d l () = min{d c l (d) } D(v, i) min{d(v, i), d l () + D(u, i )} if D(t, i) D return the corresponding path and partition. return FAIL Figure 1: Algorithm OPQR [1] idea that algorithm OPQR (Fig. 1) exploits is to represent each link, based on the given link delay/cost functions, as a set of links corresponding to all possible costs on the link. Thus, each link in this set is associated with a single cost/delay pair. Once, the set of links is created for each link in the network one can then run the restricted shortest path algorithm to find the optimum solution [2]. The complexity of the OPQR algorithm is O(mU(U +log D)), where U is an upper bound on the cost of the solution [1]. In [1] the authors also present an efficient approximation scheme for the OPQR problem that uses approximating techniques based on sampling and scaling. The approximate solution is ɛ-optimal in a sense that its cost is within a factor of 1 + ɛ of the optimal cost. As, there is a tradeoff between the accuracy of the solution and the computational cost to find the solution, ɛ-opqr may be more useful for on-line applications. Our heuristic algorithm OPQR-G, as discussed in the next section, can make use either of OPQR or ɛ-opqr based on its application. For off-line tools used for example for network planning, where the computational complexity is not as much an issue, the heuristic algorithm shall make use of the OPQR algorithm. 3 Heuristic Algorithms This Section presents three heuristic algorithms. First, in Subsection 3.1 the proposed algorithm OPQR-G is defined in detail together with a discussion on its complexity. Subsection 3.2 and Subsection 3.3 respectively, describe two LP-based algorithms that use ILOG TM CPLEX 7.1 LP solver for solving the Problem OPQR-G. These two LP based algorithms are used later in Section 4 for performance analysis of the proposed algorithm. Finally, this section concludes with discussion on the usefulness and applicability of the algorithm OPQR-G. 3.1 Algorithm OPQR-G In our approach for the solution of the Problem OPQR- G, we first compute an optimal route and delay partitions on its links for each origin-destination (OD) pair in isolation by using one of the algorithms for the unicast problem e.g., OPQR or ɛ-opqr provided in [1]. We then apply a reallocation heuristic algorithm STRETCH to determine a single near-optimal set of routes for all OD pairs and delay partitions on all the links, so that the network cost is minimized while the end-to-end delay requirements for every OD pair demand in the network is still satisfied. The general idea of the heuristic algorithm can be explained as follows. When for a given OD pair demand with an end-to-end delay requirement OPQR (or ɛ-opqr) algorithm is run, it returns an optimal route and optimal delay partitions on the links along that route. After OPQR algorithm is run for all OD pairs in isolation, this will produce a set of routes in the network, where a single unique route is allocated to each OD pair, and a set of delay partitions for each link in the network, where the elements in this set represent delay partitions on the link from all the routes in the network that traverse that link. As, each link may be traversed by multiple routes, the delay partition on a link must be set to the minimum value of all delay partitions associated with the link, in order to satisfy the end-to-end delay requirements of all routes that traverse that link. A set of delay partitions on the links in the network obtained in this way will result in feasible solution. However, it will be too stringent in a sense that it may be possible to increase the delay partitions on some links in order to further minimize the network cost while still satisfy the end-to-end delay requirements for all OD pairs in the network. In order to achieve this, a reallocation heuristic algorithm STRETCH is applied. Thus, the heuristic algorithm OPQR-G consists of two main parts, the initialisation step and iteration step, which employs the reallocation algorithm STRETCH. In the following, we discuss these two main algorithmic steps and a detailed description of them is given in Fig. 2 and Fig. 3, respectively. Initialisation In this step OPQR (or ɛ-opqr) algorithm is run for each OD pair (u, v) with given end-to-end delay requirement D uv in isolation, to determine an optimal path r uv and optimal delay partitions on the links along that path d l ruv for each OD pair in the network. This will produce a set of routes in the network r = {r uv } u,v V, and a set of delay partitions for each link in the network d l = {d l ruv } u,v V (l = 1, 2,... E). In order to satisfy all end-to-end delay requirements of all routes that traverse a given link the initial delay allocation for a link is set to the minimum of all delay partitions from the routes that tra-

Step 1 INITIALISATION for all pair (u, v) run OPQR (G(V, E), {c l (d)} l E, D uv, U) r {r uv} u,v V, d l r {d l ruv } u,v V for all link l E set the initial delay allocation: d l = min{d l ruv } u,v V RETURN r = {r uv} u,v V and d l = {d l } l E Figure 2: Algorithm - Initialisation Step verse that link, i.e., d l = min{d l ruv } u,v V ( l E). This set of initial delay allocations on the links and the set or routes for all OD pairs in the network defines an input for the next algorithmic step. Iteration STRETCH In this step we go systematically through all the routes between all OD pairs in the network to determine those which are tight and those which are slack. A tight route is considered one for which the sum of the allocated delay partitions on its links is equal to its end-to-end delay requirement and thus the allocated delay on this route cannot be further stretched (or relaxed) i.e., d l = D uv. A slack route is considered one for which the sum of the allocated delay partitions on its links is less than its end-to-end delay requirement and thus the allocated delay can be further stretched i.e., d l < D uv. Performing delay reallocation on the slack routes results in increased overall delay allocation in the network and thus the network cost is minimized. Specifically, in Step 2a, the tight and slack routes are determined based on the value of the slack factor for each route, which is calculated as follows: s ruv = D uv d l. The slack routes are ordered in LSF (Least Slack First) order, that is in an increasing order based on the slack factors for the routes. All the links that belong to routes for which the slack factor is equal to zero (e.g., tight route) are removed from the graph, as delay partitions on those links cannot be further relaxed. Finally, this step updates the link cost functions (as the residual link cost functions) to account for the delays that are already allocated on the links. Step 2b runs OPQ algorithm for each slack route in LSF order, where the target delay that the algorithm takes at input is now actually the route slack factor instead. The OPQ algorithm performs delay partition on a given route and end-to-end delay requirement. The implementation of OPQ algorithm requires a simple modification of the OPQR algorithm to allow for a route information to be Step 2 ITERATION STRETCH Input: (G(V, E), {r uv} u,v V, {d l } l E, {D uv } u,v V ) 2a for all pair (u, v) p uv 0, s l LSF 0, for all route r uv r s uv D uv d l if (s uv > 0) p uv {s uv} else G(V, Ẽ) G(V, E) \ (l ruv) for all l E c l (d) c l (d + d l ) 2b p LSF {r uv (u, v) = arg min p uv} s LSF arg min p uv while (s LSF > 0) do {s l LSF } l plsf OPQ(G(V, Ẽ), p LSF, {c l (d)} l E, s l LSF ) for all l p LSF d l d l + s l LSF s LSF s LSF l p LSF s l LSF for all route {r uv l p LSF } s uv s uv s l LSF p uv {s uv} 2c for all l p LSF G(V, Ẽ) G(V, E) \ (l p LSF) for all l E c l (d) c l (d + d l ) p uv p uv \ {s l LSF } if (p uv 0) go to Step 2b else RETURN {d l } l E Figure 3: Algorithm STRETCH specified at input and thus to perform delay partition on a specified route. When OPQ is run for the LSF route p LSF it will return slack factor partition along this route {s l LSF } l plsf. These slack factor partitions on the links belonging to the LSF route indicate that the delay allocation on these links can be increased for the amount of their respective slack factor partitions and all OD pair demands that traverse these links will be still satisfied. After, this adustment is made for the link delays belonging to the LSF route, the slack factors for the LSF route, as well as, for all routes that traverse any of the constituent links of the LSF route should be also updated. Finally, the LSF list is updated. This step is performed until the slack factor for the LSF route becomes zero. In Step 2c all the links that belong to the tight routes, which were created in Steb 2b are removed from the graph

(Step 2b produces at least one tight route, that is the LSF route). It also updates the link cost functions for the next iteration. The Steps 2b and 2c are repeated until all slack factors become zero or there are no links left in the graph. Complexity of the Algorithm OPQR-G The initialisation step requires the OPQR algorithm to be run for each OD pair in the network. Thus, the overall running time of this step is O(n 2 mu(u + log D)). The parameter U is an upper bound on the cost of the solution. The iteration STRETCH step first examines all routes to determine those which are slack and then for each slack route the algorithm OPQ is run to perform a re-adustment of the delay partitions. The required running time of this operation is determined by the number of links on a slack route and the running time of the OPQ algorithm. The worst case running time for this step is O(m(n 1)U(U + log D)). Hence, the complexity of the heuristic algorithm OPQR-G is O((n 2 + n 1)mU(U + log D)). 3.2 Algorithm LP1 This LP-based heuristic first determines the routes between OD pairs as in INITIALISATION. Knowing the routes, it re-formulates OPQR-G as a LP problem as shown below, and consequently solves the LP problem to determine the delay partition 2. Problem OPQR-G-LP1 Minimize: Subect to: 3.3 Algorithm LP2 u,v V c l (d l ) d l D uv d l 0 l E u, v V A limitation to both of the previous heuristics is that the routes are kept unchanged after the INITIALISATION step. In the case where these routes have many common links and/or the cost function of some common links are steeply decreasing then keeping the routes fixed may not be a good idea. The LP2 heuristic manages to solve the overall OPQR- G problem with LP techniques. In the original formulation, both and d l are unknowns, therefore, this formulation is not linear (i.e., quadratic), and cannot be solved by LP solvers. This problem is overcome by employing an iterative algorithm as described in Fig. 4. The variable n l in the formulation below represents the total number of routes that traverse link l. 2 Note that LP1 will find optimal solution for the delay partitions in the network only (e.g. Problem OPQ) provided the routing information as an additional input. (2) LP2 1 for all l E 2 n l 1 3 for all u, v V 4 generate a set of routes {ruv} 5 determine { } and {d l} by solving OPQR-G-LP2 6 for all u, v V 7 r uv ruv for = 1 8 for all l E 9 n l 0 10 for all u, v V 11 for all l r uv 12 n l n l + 1 13 d = l E n2 l n,2 l 14 if d > ɛ 15 for all l, n l n l 16 go back to line 5 17 return the routes and partition Figure 4: Algorithm LP2 Problem OPQR-G-LP2 Minimize: n l c l (d l ) Subect to: l E l d l D uv + (1 )M = 1 = {0, 1}, d l 0 l E 3.4 Discussion It is easy to see that the lower bound on the optimal cost, LB, can be computed by u,v V c l (d l r uv ), where r uv and {d l r uv } l ruv are the route and delay partition found by running OPQR for a OD pair (u, v). The heuristic can always provide a feasible solution (if feasible solutions exist) as mentioned before. However, like other heuristics, the heuristic cannot guarantee optimal solutions. When the routes found in INITIALISATION are link disoint, the heuristic will not change either the routes or the delay partitions, hence, the results are guaranteed to be optimal. If the routes have common links, the solution may not be optimal. The solution is guaranteed to be within l L c l(d l ) LB, where L is the set of common links and {d l } l E is the delay partition computed by the heuristics. According to the above analysis, the heuristic is expected to not work very well when the routes found by running OPQR separately for each OD pair have many common links or when c l (d l ) c l(d l ruv ) is large. For ex- (3)

ample, the heuristic may have trouble with a network that consists of two parts connected via a single link and the (non-increasing cost function of that link is very steep. Finally, it should be noted that the heuristic is intended for employment primarily as a static algorithm as part of a network planning procedure. In this regard, it could be envisaged as an algorithm for a single domain network when the planning process involves a core network (e.g., service provider s domain), as well as, for multi-domain network when enterprise networks are considered. In the latter case, many remote offices have to be connected in a most costefficient way provided the costs per service class (e.g. delay or QoS requirement) between various access points offered by the core and access network providers. 4 Numerical Results The algorithms presented in Section 3 were implemented in C + + using the Graph Template Library [10]. We have performed two tests (e.g., test-1 and test-2), where we have compared the performance of the proposed algorithm OPQR-G against the performance of the LP-based algorithms, LP1 and LP2, respectively. For testing of the heuristic algorithm we have considered a 15 node network, which is an ATM VP network obtained from [11] and is shown on Fig. 5. Five dif- Table 1: Difference between the cost of algorithm OPQR- G, LP1 and LB Test-1 LB OPQR-G LP1 Sce 1 334.22 374.94 364.94 (12.10 %) (9.18 %) Sce 2 213.78 234.01 229.43 (9.46 %) (7.32 %) Sce 3 171.27 185.32 179.19 (8.20 %) (4.62 %) Sce 4 104.46 120.71 110.60 (15.50 %) (5.87 %) Sce 5 57.15 61.70 59.08 (7.96 %) (3.37 %) Table 2: Difference between the cost of algorithm OPQR- G, LP1, LP2 and LB Test-2 LB OPQR-G LP1 LP2 Route % Sce 1 225.33 239.66 237 239.66 N/A (6.35 %) (5.17 %) (6.35 %) Sce 2 163.51 172.17 167.84 171.60 N/A (5.29 %) (2.64 %) (4.94 %) Sce 3 121.49 127.52 126.77 126.77 N/A (4.96 %) (4.34 %) (4.34 %) Sce 4 107.95 113.95 110.96 113.08 8.33 % (5.56 %) (2.79 %) (4.75 %) 3 1 2 4 5 6 7 Figure 5: 15 Node Test Network 8 9 10 11 14 15 12 13 ferent test scenarios were considered for test-1. For each test scenario delay requirements ({D uv } u,v V ) for all OD pairs were randomly assigned from a given range of values, as follows: Scenario 1 = (10 15) ms, Scenario 2 = (15 25) ms, Scenario 3 = (25 40) ms, Scenario 4 = (40 70) ms and Scenario 5 = (70 100) ms. Links were randomly assigned a cost function from a set of three different cost functions. We tested the heuristic algorithm with the test network and the settings as described above on a Pentium 1.7 GHz PC with 512 MB RAM and the results obtained from test- 1 and test-2 are summarized in Table 1 and Table 2, respectively. Table 1 shows the difference between the cost solutions obtained by algorithm OPQR-G and LP1 against the lower bound (LB) of the optimal cost for all five test scenarios. The relative error (RE), defined as: RE = heuristic LB LB 100% for the various scenarios is shown immediately below the cost value for OPQR-G and LP1, respectively. Note that this RE actually represents the worst error of the cost solution obtained by the heuristic algorithms. As the amount of memory required by LP2 for solving the scenarios in test-1 is very large, a new set of input scenarios was created for test-2 involving fewer nodes and delay requirements of smaller values. Specifically, the target delays were drawn from four different sets, as follows: Scenario 1 = (8 12) ms, Scenario 2 = (12 16) ms, Scenario 3 = (16 20) ms, Scenario 4 = (20 25) ms. Again, links were randomly assigned a cost function from a set of three different cost functions. Table 2 shows the results obtained from test-2, that is the cost solutions of the algorithms OPQR-G, LP1 and LP2 against the lower bound of the optimal cost. Despite the relative error of the cost solutions, the sixth column in the table shows the discrepancy of route allocation obtained from the algorithm OPQR-G in terms of number of routes out of the total number of routes in the network that are different than the routes obtained

from LP2. For most scenarios we did not record discrepancy on the route allocation, except in Scenario 4 where there was only one route out of twelve that was different from the ones found by LP2. It can be seen, that in terms of optimality LP1 performs best, followed by OPQR-G and LP2. LP2 at best, provides the same cost solution as LP1 when there is no route discrepancy recorded (provided the obtained delay partitions are the same as in LP1, which may not always be the case). However, when there was route discrepancy recorded LP2 failed to provide a beter solution then LP1. Moreover, for smaller number of routes LP2 works well, but for larger number of routes and particularly when the routes between the OD pairs share many links in common (which we were not able to test due to shortage of memory) cyclic problem might occur which requires further investigation. In addition, we have performed tests for networks that were randomly generated and of larger size. For networks of size bigger than 30 nodes the heuristic algorithm OPQR- G was able to obtain good solutions in about 30 seconds whereas the LP1 failed after running for 30 minutes due to shortage of memory. In conclusion, the heuristic algorithm OPQR-G provides the fastest solution of all the algorithms presented in this paper and performs well in terms of cost solution when compared to the lower bound for the optimal cost. 5 Conclusion In this paper, we have addressed the combined problem of routing and QoS partitioning in general topology networks for a framework in which performance dependent cost function is associated with each network element and the QoS metric is additive (e.g., delay or itter). Moreover, the delay/cost functions associated with the links in the network are assumed to be non-increasing an of general integer type. An efficient heuristic algorithm OPQR-G, which is a greedy algorithm has been proposed, which essentially is an extension of the algorithm OPQR for unicast connections, developed by Lorenz et al [1]. In addition, two LPbased algorithms were developed that use optimization tool ILOG TM CPLEX 7.1 LP for solving the Problem OPQR- G. The performance of the three proposed heuristic algorithms has been examined using the lower bound of the optimal cost as a benchmark. The results obtained for a 15 node test network and various scenarios represent very close match to the optimal. For randomly generated networks of size bigger than 30 nodes the heuristic OPQR- G was able to obtain good solutions in about 30 seconds whereas the LP1 failed after running for 30 minutes due to memory shortage. Future work will consider improvements to allow the STRETCH algorithm to perform re-adustment to the routes between OD pairs, as obtained from the OPQR algorithm, in addition to the delay re-adustment. Further investigation of LP2 heuristic is also required. In addition, future work will involve integration of the OPQR-G algorithm with the algorithms described in [6] for optimal design of multiservice IP networks. References [1] Lorenz, D.H., Orda, A., Raz. D., Shavitt, Y. Efficient QoS Partition and Routing of Unicast and Multicast. Proceedings of IWQoS 2000, Pittsburgh, PA, USA. [2] Lorenz, D.H., Orda, A. Optimal Partition of QoS Requirements on Unicast Paths and Multicast Trees. IEEE/ACM Transactions on Networking, vol.10, pp.102-114, 2002. [3] Hassin, R. Approximation Schemes for the Restricted Shortest Path Problem. Mathematics of Operations Research, vol.17, no.1, 1992. [4] Wang, Z., Crowcroft, J. Quality-of-Service Routing for Supporting Multimedia Applications. IEEE Journal of Selected Areas on Communications, vol.14 no.7, pp.1228-1234, 1996. [5] Crawley, E., Nair, R., Raagopalan, B., Sandick, H. A Framework for QoS-based Routing in the Internet. IETF RFC 2386, 1998. [6] Atov, I., Harris, R. J. Dimensioning Method for Multiservice IP Networks to Satisfy Delay QoS Constraints. Proceedings of IFIP-TC6 Interworking 2002 Symposium, Perth, Australia, October 2002. [7] Tran, H. T., Harris, R. J. Near-Optimal Allocation of Delay Requirements on Multicast Trees. Proceedings of IFIP-TC6 Interworking 2002 Symposium, Perth, Australia, October 2002. [8] Blake, S., Black, D., Carlson, E., Davies, E., Wang, Z., Weiss, W. An Architecture for Differentiated Services. IETF RFC 2475, 1999. [9] Rosen, E., Viswanathan, A., Collon, R. Multiprotocol Label Switching Architecture. IETF RFC 3031, 2001. [10] Forester, M. et al. Gtl - Graph Template Library. http://www.infosun.fmi.uni-passau.de/gtl/ [11] Lee, H. et al. Preplanned Rerouting Optimisation and Dynamic Path Rerouting for ATM Restoration. Telecommunications Systems, vol.14, 2000.