Fine two-phase routing over shortest paths with traffic matrix

Computer Networks 54 (2010) 3223–3231

Contents lists available at ScienceDirect

Computer Networks journal homepage: www.elsevier.com/locate/comnet

Fine two-phase routing over shortest paths with traffic matrix Eiji Oki *, Ayako Iwaki Department of Information and Communication Engineering, The University of Electro-Communications, 1-5-1 Chofugaoka, Chofu, Tokyo 182-8585, Japan

a r t i c l e

i n f o

Article history: Received 30 August 2009 Received in revised form 11 June 2010 Accepted 17 June 2010 Available online 30 June 2010 Responsible Editor: J.C. de Oliveira Keywords: Routing IP MPLS Traffic engineering Load balancing Optimization Linear programming

a b s t r a c t This paper presents an IP finely-distributed load-balanced routing scheme based on twophase routing over shortest paths, where the traffic matrix is given. It is called the fine two-phase routing (F-TPR) scheme. F-TPR more finely distributes traffic from a source node to intermediate nodes than the original TPR. F-TPR determines the distribution ratios to intermediate nodes for each source–destination node pair independently. To determine an optimum set of distribution ratios, a linear programming (LP) formulation is derived. We compare the F-TPR scheme against the TPR scheme and the sophisticated traffic engineering (TE) scheme of Multi-Protocol Label Switching (MPLS-TE). Numerical results show that F-TPR greatly reduces the network congestion ratio compared to TPR. In addition, F-TPR provides almost the same network congestion ratios as MPLS-TE, the difference is surprisingly less than 0.1% for the various network topologies examined. In addition, considering the practical implementation of F-TPR for routers, we also investigate the case that traffic from a source node to a destination node is not allowed to be split over multiple routes. The non-split problem is formulated as an integer linear programming (ILP) problem. As it is difficult to solve the ILP problem within practical time, two heuristic algorithms are presented: Largest Traffic Demand First (LTDF) and a Random Selection (RS). The applicability of LTDF and RS are presented in terms of network size. We find that non-split F-TPR also matches the routing performance of MPLS-TE within an error of 1%, when network size is large enough. Ó 2010 Elsevier B.V. All rights reserved.

1. Introduction Adopting an appropriate routing scheme can increase the network resource utilization rate and network throughput of Internet Protocol (IP) networks [1]. Since it optimizes the assignment of traffic resources, additional traffic can be supported. One useful approach to enhancing routing performance is to minimize the maximum link utilization rate, also called the network congestion ratio, of all network links. Minimizing the network congestion ratio increases the admissible traffic. Several routing strategies have been extensively studied [2–5,8]. Wang et al. [2] formulate a general traffic engineering problem, where traffic demands are assumed to * Corresponding author. Tel.: +81 42 443 5195; fax: +81 42 443 5926. E-mail address: [email protected] (E. Oki). 1389-1286/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.comnet.2010.06.012

be flexibly split among source and destination nodes. This sophisticated operation is performed by the Multi-Protocol Label Switching (MPLS) Traffic-Engineering (TE) technology [9]. However, legacy networks mainly employ shortest path-based routing protocols such as Open Shortest Path First (OSPF) and Intermediate System to Intermediate System (IS-IS). This means that already deployed IP routers in the legacy networks would need to be upgraded, which would significantly increase capital expenditure. Therefore, it is desirable that any new strategy be compatible with a current IP routing protocol. Traffic engineering schemes that set optimum link weights in OSPF-based networks were addressed in [3–5]. When traffic demands are changed, optimum link weights are re-calculated and network operators configure the updated link weights. The link weights may be updated every day, every week, or every month depending on traffic and

3224

E. Oki, A. Iwaki / Computer Networks 54 (2010) 3223–3231

the network’s operation policies. According to the updated weights, IP routes are changed. Changing routes frequently causes network instability, which leads to packet loss and the formation of loops. Load-balanced routing increases network resource utilization efficiency [8,10–12], under a given set of traffic conditions. The hose model [13–15] is assumed as a traffic model in [1,8,10,11]. In the hose model, just the total outgoing traffic and incoming traffic for each node are specified, the traffic demand between each source–destination pair does not need to be specified. It is beneficial for network operators to specify a set of traffic conditions in the hose model, especially when network size is large. The hose model well handles highly variable traffic conditions. Routing that is robust to changing and uncertain traffic demands is called oblivious routing [11,16–18,7]. Oki et al. presented an IP load-balanced routing scheme based on two-phase routing over shortest paths for the hose model [19]; it is an extended version of the original shortest path-based two-phase routing (TPR) [8]. It is called fine two-phase routing (F-TPR). F-TPR more finely distributes traffic from a source node to intermediate nodes than the original TPR. F-TPR performs load balancing and each flow is routed according to the OSPF protocol, which is an existing IP routing protocol, in two stages across intermediate nodes. Since F-TPR solves a non-linear programming problem [19], it is applied only to networks with fewer than 10 nodes so as to yield practical levels of computation complexity. In the hose model, it is observed that F-TPR outperforms TPR at the cost of higher implementation complexity for the forwarding functions. Although the studies in [19] were developed assuming the hose model, it is desirable to specify the traffic matrix of T = {dpq}, where dpq is the traffic demand from node p to node q, even for large-scale networks, so as to enhance the routing performance. There are several studies that estimate the traffic matrix efficiently and accurately [20–22]. To estimate the traffic matrix, network operators do not need to measure each traffic demand dpq from node p to node q, where the number of dpq is proportional to the network size. Instead, they have only to measure traffic loads on each link in the network. Using the measured link loads, the traffic matrix can be estimated. The traffic model that is specified by T = {dpq} is called a pipe model [13] [14], to contrast it with the hose model. The traffic matrix estimation includes some errors, but the range of errors can be estimated by simulation studies [20,22,23]. One report states that the traffic matrix estimation based on link traffic measurements includes error of 20% or more [23], which is acceptable because the network resource allocation includes sufficient margin. One question arises: If F-TPR, which was originally designed for the hose model, is applied to the pipe model, where the traffic matrix is known, what is the performance of F-TPR relative to that of the sophisticated MPLS-TE scheme and TPR? The answer to this question gives network operators beneficial information by allowing them to choose the best routing scheme given the trade-offs among performance, operational cost, and deployment cost. This paper presents the F-TPR scheme for the pipe model, where the traffic matrix is assumed to be known [24].

F-TPR determines the distribution ratio to node m for each pq source–destination pair of (p, q), km independently. To pq determine an optimum set of km for the pipe model, a linear programming (LP) formulation is derived. We compare F-TPR against TPR and MPLS-TE. Numerical results show that F-TPR greatly reduces the network congestion ratio compared to TPR. In addition, F-TPR provides comparable routing performance to MPLS-TE. In addition, considering a practical implementation of F-TPR for routers, we also consider the case that traffic from a source node to a destination node is not allowed to be split over multiple routes. The problem for the non-split case is formulated as an integer linear programming (ILP) problem. As it is difficult to solve the ILP problem within a practical time, two heuristic algorithms are introduced. Non-split F-TPR provides comparable routing performance to MPLS-TE, when network size is large enough. The remainder of this paper is organized as follows. Section 2 uses a network model to introduce the terminology of this paper. Section 3 describes the TPR scheme. Section 4 presents F-TPR with the pipe model. Section 5 presents FTPR without traffic splitting. Section 6 evaluates the performance of F-TPR in a comparison with the TPR and MPLS-TE schemes. Section 7 evaluates the performance of F-TPR without traffic splitting. Finally, Section 8 summarizes the key points. 2. Network model The network is represented as a directed graph G(V, E), where V is the set of vertexes (nodes) and E is the set of links. A link from node i 2 V to node j 2 V is denoted as (i, j) 2 E. cij is the capacity of (i, j) 2 E. Lij is the link load of (i, j) 2 E. T = {dpq} is the traffic matrix, where dpq is the traffic demand from node p to node q. N is defined as the number of nodes in the network. The network congestion ratio, which refers to the maximum value of all link utilization in the network, is n rates o L denoted as r, where r ¼ maxði;jÞ2E cij . Minimizing r means ij that admissible traffic is maximized. The admissible traffic volume is accepted up to the current traffic volume multiplied by 1/r. Minimizing r through routing control is the objective of this paper. 3. Two phase routing The original TPR [8,10] assumes the hose model. ap is the traffic that node p can send into the network, where P q2V dpq ¼ ap . This is a constraint at node p in the hose model. bq is the traffic that node q can receive from the netP work, where p2V dpq ¼ bq . This is a constraint at node q in the hose model. In TPR, the traffic from node p to node q is not sent directly. It is split in portions that are directed to intermediate node m 2 V. For all source–destination pairs (p, q), the portion of a flow dpq that is balanced across a node m P equals km, where 0 6 km 6 1 and m2V km ¼ 1. Then, every intermediate node m forwards the received traffic to its final destination node q. Traffic from node p to node m and from node m to node q is routed along the shortest paths.

3225

E. Oki, A. Iwaki / Computer Networks 54 (2010) 3223–3231

TPR assumes the use of an existing IP protocol that offers the configuration of IP tunnels, such as IP-in-IP and Generic Routing Encapsulation (GRE) tunnels, between all source nodes and intermediate nodes in the network. We briefly present the formulation that determines an optimum set of km so as to minimize the network congestion ratio, r. In deducing this formulation, we assume both pipe and hose models, but we show that using the hose model is enough to determine the optimum set, or that we do not need the additional traffic information specified by the pipe model. Note that, to compare the formulations of TPR and F-TPR easily later, our formulation of TPR differs from that of [8], but the meaning is the same. Consider traffic from node p to node q across node m. Let bpm be the traffic between node p and node m. The traffic between node p and node m consists of two components. The first one is the traffic generated by node p and ð1Þ balanced across node m, which is defined as bpm . The second one is the traffic for m balanced across node p, which ð2Þ is defined as bpm . Therefore, bpm is given by: ð1Þ

ð2Þ

bpm ¼ bpm þ bpm :

ð1Þ

The traffic generated by node p and balanced across node m is given by: ð1Þ

bpm ¼

X

km dpq ¼ km ap :

ð2Þ

q2V

The traffic for m balanced across node p is given by: ð2Þ bpm

¼

X

kp dum ¼ kp bm :

ð3Þ

An optimal routing formulation with TPR to determine the distribution ratio km is as follows.

min

r X

s:t:

m2V X m2V X

ð9aÞ km ¼ 1;

ð9bÞ

wijm km 6 cij  r

ði; jÞ 2 E;

ð9cÞ

xjm km ¼ bj  aj j 2 V;

ð9dÞ

m2V

km P 0 m 2 V; 0 6 r 6 1:

ð9eÞ ð9fÞ

The objective function in Eq. (9a) minimizes the network congestion ratio. Eq. (9b) states that the sum of km over all intermediate nodes m is equal to 1. Eq. (9c) indicates that the sum of the fractions of traffic demands transmitted over (i, j) is equal to or less than the network congestion ratio times the total capacity cij for all links. The constraints of the hose model are incorporated into Eq. (9c). The left term of Eq. (9c) is derived by Eqs. (5) and (6), which use the hose model constraints of Eqs. (2) and (3). Eq. (9d) is a constraint for flow conservation. It states that the difference in traffic flows incoming to node j and outgoing from j is equal to  P  P bj  aj ¼ p2V dpj  q2V djq . Eqs. (9a)–(9f) are an LP problem and can be solved optimally with a standard LP solver. In the formulation expressed by Eqs. (9a)–(9f), the only traffic information that need be known is ap and bq, which are specified by the hose model. The formulation does not require any element of the traffic matrix T = {dpq}.

u2V

In Eqs. (2) and (3), the equalities are obtained by using the parameters of the hose model. Thus,

bpm ¼ km ap þ kp bm :

ð4Þ

Let variable F ijpm set to 1 if (i, j) belongs to the shortest path between the nodes p and m and 0 otherwise. The link load Lij of (i, j) is given by:

Lij ¼

XX

F ijpm bpm ¼

p2V m2V

¼

XX

F ijpm ðkm ap þ kp bm Þ

p2V m2V

 X ij X X  ij F pm ap þ F ijmp bp km ¼ wm km ; p2V m2V

ð5Þ

m2V

ð1Þ

ð6Þ

The difference of the traffic flows incoming to node j and outgoing from j is given by: o X X n ij ðLij  Lji Þ ¼ wm km  wjim km i2V

 X X X  ij F pm ap þ F ijmq bp  F jipm ap  F jimq bp km ¼ p2V i2V m2V

¼

X

xjm km ;

ð7Þ

where xjm is defined as:

 X X  ij x ¼ F pm ap þ F ijmq bp  F jipm ap  F jimq bp : p2V

i2V

pq

km dpq :

ð10Þ

The traffic for m balanced across node p is given by: ð2Þ bpm

¼

X

um

kp dum :

ð11Þ

u2V

The link load Lij of (i, j) is obtained as:

Lij ¼

XX

F ijpm bpm ¼

p2V m2V

¼

XX

XX

  ð1Þ ð2Þ F ijpm bpm þ bpm

p2V m2V

F ijpm

p2V m2V

X

pq km dpq

q2V

þ

X

! um kp dum

u2V

 X X X  ij pq ¼ F pm þ F ijmq dpq km

m2V

j m

X q2V

p2V

i2V

F-TPR assumes the pipe model, where the traffic matrix T = {dpq} is known. F-TPR distributes traffic more finely from a source node to intermediate nodes than the original TPR. The distribution ratio to node m for each source–despq tination pair of (p, q) is introduced as km . To determine a pq set of optimum km that minimizes the network congestion ratio, a general programming formulation is presented in this section. In the same way as TPR, the traffic generated by node p and balanced across node m is given by:

bpm ¼

where wijm is defined as:

 X  ij wijm ¼ F pm ap þ F ijmp bp :

4. Fine two-phase routing with pipe model

p2V q2V m2V

ð8Þ

¼

XXX p2V q2V m2V

pq

wijpqm km ;

ð12Þ

3226

E. Oki, A. Iwaki / Computer Networks 54 (2010) 3223–3231

where wijpqm is defined as:

wijpqm

  ¼ F ijpm þ F ijmq dpq :

ð13Þ

The difference in traffic flows incoming to node j and outgoing from j is given by: ( ) X X X X X ij pq X X X ji pq ðLij  Lji Þ ¼ wpqm km  wpqm km i2V

i2V

¼ ¼

p2V q2V m2V

p2V q2V m2V

5.1. ILP formulation

p2V q2V i2V m2V

The optimal solution in Eqs. (16a)–(16f) provides, theoretically, the best routing performance in F-TPR. It requires a source node to unevenly distribute traffic to the intermediate nodes by using the optimized distribution ratios. However, practically speaking, most available routers may not be able to set uneven traffic distributions. In this section, we consider that traffic from a source node to a destination node is not split over multiple routes, in other words, that it goes via only one intermediate node to a destination node. We call the corresponding F-TPR non-split F-TPR. For non-split F-TPR for the pipe model, an optimal routpq ing formulation to determine the distribution ratio km is as follows:

XXX

xjpqm kpq m;

ð14Þ

where xjpqm is defined as:

 X  ij F pm þ F ijmq  F jipm  F jimq dpq :

ð15Þ

i2V

An optimal routing formulation with F-TPR for the pipe pq model to determine distribution ratio km is as follows.

min s:t:

r X

ð16aÞ pq km

¼ 1 p; q 2 V;

ð16bÞ

m2V

XXX

pq

wijpqm km 6 cij  r;

p2V q2V m2V

XXX p2V q2V m2V

06

pq km

5. Non-split fine two-phase routing with pipe model

 X X X X  ij pq F pm þ F ijmq  F jipm  F jimq dpq km

p2V q2V m2V

xjpqm ¼

memory constraint is more significant than the computational time to solve the LP problem. In our examination to solve the LP problems, the memory constraint becomes a bottleneck before the issue of the computational time is faced with.

xjpqm kpq m ¼

X p2V

6 1 p; q; m 2 V;

0 6 r 6 1:

ði; jÞ 2 E;

dpj 

X

djq ;

ð16cÞ j 2 V; ð16dÞ

q2V

ð16eÞ ð16fÞ

The objective function in Eq. (16a) minimizes the network congestion ratio. Eq. (16b) states that the sum of pq km over all intermediate nodes m for each source–destination node pair of (p, q) is equal to 1. Eq. (16c) indicates that the sum of the fractions of traffic demands transmitted over (i, j) is equal to or less than the network congestion ratio times the total capacity cij for all links. Eq. (16d) is a constraint for flow conservation, where the difference in traffic flows incoming to node j and outgoing from j is P P equal to p2V dpj  q2V djq . Eqs. (16a)–(16f) are also an LP problem and can be solved optimally with a standard LP solver. The noticeable difference between the formulations of TPR and F-TPR is as follows. The TPR formulation expresses the constraints by ap and bq, and dpq does not appear in the formulation. On the other hand, in the F-TPR pq formulation, dpq remains in the constraints, because km depends on node pair (p, q). Note that a network size that can be handled in an LP optimization is generally limited due to a memory constraint and a computational time. If the interior point projective algorithm presented by Karmarkar [25] is adopted to solve the LP problem, the overall complexity of the algorithm is O(n3.5 L2lnLlnlnL), where n is the number of variables and L is the number of bits in the input. The LP problem defined in Eqs. (16a)–(16f) has O(N3) variables. Therefore, the overall complexity of our problem is O(N10.5L2lnLlnlnL), where (N3)3.5 = N10.5 is used. In addition, O(N4) memories are required to set the LP problem. Although the computational complexity is higher than the memory complexity, in practice, we observe that the

min s:t:

r X

ð17aÞ pq

km ¼ 1 p; q 2 V

ð17bÞ

m2V

XXX

pq

wijpqm km 6 cij  r;

p2V q2V m2V

XXX

xjpqm kpq m ¼

p2V q2V m2V

X p2V

ði; jÞ 2 E

dpj 

X

djq ;

ð17cÞ j 2 V; ð17dÞ

q2V

pq

km ¼ f0; 1g p; q; m 2 V

ð17eÞ

0 6 r 6 1:

ð17fÞ pq km

Eq. (16e) specifies a constraint that is limited to values of 0 or 1. The formulation for non-split F-TPR presented in Eqs. (17a)–(17f) constitutes an integer linear programming (ILP) problem. As a traffic demand for each node pair has at most O(N) possible routes and there are N2 source– destination pairs, the time complexity to search all combi2 nations of possible routes is OðN N Þ, which is an NP-hard [2,6]. As the number of nodes in the network increases, it becomes harder to solve the ILP problem within a practical time. Therefore, heuristic algorithms are required. 5.2. Heuristic algorithms We introduce two heuristic algorithms against the ILP problem. In both algorithms, first, the corresponding LP pq problem is solved by assuming km is a real number, where pq 0 6 km 6 1 [2,6]. Based on the LP solution, if any node pair (p, q) whose traffic demands are not split exists, the correpq sponding km is considered as a part of the solution in the ILP problem. Second, for other node pairs, by selecting each (p, q) one by one, the corresponding traffic demands are allocated in a sequential manner which makes the congestion ratio as small as possible. The algorithms differ on how to prioritize node pairs for selection. The first algorithm, called the Largest Traffic Demand First (LTDF) algorithm gives highest priority to the

E. Oki, A. Iwaki / Computer Networks 54 (2010) 3223–3231

source–destination pair that has the largest traffic demand when the traffic demand is allocated. The priority is assigned in decreasing order of traffic demands. The second algorithm, called the Random Selection (RS) algorithm, gives highest priority to a randomly selected source–destination pair. To solve the ILP problem, which has binary integer form, possible approaches include depth first search or breadth first search along with the branch strategy of choosing either the variable with minimum integer feasibility or that with maximum integer infeasibility. The time complexity of these algorithms can be adjusted with parameters that limit the number of search iterations. However, these algorithms need to remember which combinations were already searched. As a result, the memory requirements increase with the number of nodes. LTDF and RS relax the memory constraint, because they do not have to remember the combinations. S is defined as a set of source and destination node pairs whose traffic demands are not split. T is defined as a set of source and destination node pairs whose traffic demands are split over multiple routes. 5.2.1. Largest Traffic Demand First (LTDF) algorithm Let u be the source and destination node pair whose traffic demand is the largest among those in T, where u 2 T.  Step 1: Solve the corresponding LP problem and obtain pq pq the optimal solution for km and r by assuming km is a pq real number with 0 6 km 6 1. If no feasible solution exists, the algorithm stops. pq  Step 2: Using km obtained at Step 1, each source–destination pair (p, q) is classified as either S or T. Node pair (p, q) whose traffic demand is the largest in T is set to u. pq The values of km of (p, q) in S will be used in Step 3, but pq those of km of (p, q) in T are not.  Step 3: Traffic demands of (p, q) in S are allocated pq according to km obtained at Step 1.  Step 4: Find intermediate node m that minimizes the maximum link utilization after the traffic demand of u is routed without splitting, or minimizes r ¼ maxði;jÞ2 P P pq pq E ðp;qÞ2Sþu m2V wijpqm km , where km ¼ f0; 1g.  Step 5: If r 6 cij, u moves from T and U to S. Otherwise, no feasible solution is found. The algorithm stops.  Step 6: (p, q) whose traffic demand is the largest in T is set to u. If T is empty, the solution is obtained. Otherwise, return to Step 4. Step 1 requires the overall complexity of O(N10.5L2lnLlnln L) to solve the LP problem, as described in Section 2. LTDF pq requires to check r for O(N3) routing patterns on km for Steps 2–5. In practice, as presented in Section 6, the computation time for Steps 2–5 is more significant than that of Step 1. The order of the memory requirement of LTDF is the same as that of the LP problem, which is O(N4). 5.2.2. Random selection (RS) algorithm In the RS algorithm, multiple iterations are employed with different seeds for the generation of random numbers. The minimum r obtained by multiple iterations considered as a solution.

3227

Let I be the number of iterations. i is an iteration index, where i 6 I. Si is a set of source and destination node pairs whose traffic demands are not split at the ith iteration. Ti is defined as a set of source and destination node pairs whose traffic demands are split over multiple routes at the ith iteration. ui is a source and destination node pair that is randomly selected from among those in Ti at the ith iteration, where ui 2 Ti. ri is a solution at the ith iteration.  Step 1: Solve the corresponding LP problem and obtain pq pq optimal solution for km and r by assuming km is a real pq number with 0 6 km 6 1. If no feasible solution exists, the algorithm stops. pq  Step 2: Using km obtained at Step 1, each source–destination pair (p, q) is classified as either S or T. Node pair (p, q) whose traffic demand is the largest in T is set to u1. pq The values of km of (p, q) in S will be used in Step 3, but pq those of km of (p, q) in T are not used.  Step 3: Traffic demands of (p, q) in S are allocated pq according to km obtained at Step 1.  Step 4: i = 1 is set as an initial value.  Step 5: ith iteration starts if i 6 I. Set a seed to generate a random value. Set Si = S and Ti = T. If i > I, go to Step 6. – Step 5a: Find an intermediate node m that minimizes the maximum link utilization after the traffic demand of ui is routed without splitting, or miniP P pq mizes ri ¼ maxði;jÞ2E ðp;qÞ2Sþui m2V wijpqm km , where pq km ¼ f0; 1g. – Step 5b: If ri 6 cij, ui moves from Ti to Si. Otherwise, no feasible solution is obtained at the ith iteration and Step 5 is reentered after i is incremented by one. – Step 5c: (p, q) that is randomly selected from Ti is set to ui. If Ti is empty, the solution at the ith iteration is obtained. Otherwise, return to Step 5a.  Step 6: If at least one solution ri exists after I iterations, r = miniri is a solution yielded by the RS algorithm. Otherwise, no feasible solution is found. Step 1 requires the same arithmetic operations as the LP problem. RS requires to check ri for O(N3I) routing patterns pq on km for Steps 2–6. The order of memory requirement of RS is the same as that of the LP problem, which is O(N4).

6. Performance evaluation for F-TPR The performance of the F-TPR scheme is compared to those of the TPR scheme and the MPLS-TE scheme with the pipe model by solving their LP problems. This section considers that traffic demands can be split. CPLEX [27] is used as LP solver. The performance measure is the network congestion ratio, r. We use six sample networks to determine the basic characteristics of these schemes; the randomly generated network topologies identify the dependency of the performance on the number of nodes and the average node degree. For the given network topologies, link capacities are randomly generated with uniform distribution in the range of (80UC, 120UC), where UC [Gbit/ s] is given a constant unit value. dpq is also randomly generated with uniform distribution in the range of (0, 100Ud), where UD [Gbit/s] is a given constant unit value. UD/UC is

3228

E. Oki, A. Iwaki / Computer Networks 54 (2010) 3223–3231

fixed for each network condition1 In F-TPR and TPR, the link weights that are used for shortest path computation are set to be inversely proportional to the link capacities. These conditions are also applied to the performance evaluation for non-split F-TPR, which will be presented in Section 7. To compare the r values of the different schemes, we normalize the network congestion ratios of F-TPR and MPLS-TE by that of TPR. The normalized network congestion ratios are denoted as rF-TPR, rMPLS-TE, and rTPR( = 1.0), respectively. The congestion ratios of MPLS-TE are obtained by using the ideas of [2].

(a) Network 1

(b) Network 2

(c) Network 3

(d) Network 4

6.1. Routing performances The network congestion ratios are compared by using the sample networks, as shown in Fig. 1. We consider that F-TPR is applied to a single OSPF-area network in an Internet Service Provider’s (ISP’s) backbone network. Networks 1–4 in Fig. 1(a)–(d) are used as typical backbone networks [5]. Networks 5 and 6 in Fig. 1(e) and (f) are presented in [6], which are the Cable and Wireless network and CRL network, respectively, available at [26]. The characteristics of the networks are shown in Table 1. The maximum network size in our evaluation is restricted, due to the memory constraint to solve the LP problem, as described in Section 4, First, we examine the degree of difference between rP and rH, using five sample networks that were used as reference backbone networks in [5,6] to evaluate routing performance, as shown in Fig. 1. Networks 1–4 are used as reference typical backbone networks in [5]. Network 5, which is a Cable and Wireless backbone network [26], was used in [6]. The network congestion ratios are compared in Fig. 2. We obtained the average value of the normalized network congestion ratios for 100 randomly generated combinations of link capacities and traffic demands. rF-TPR values are dramatically lower than those of rTPR, where the values of rF-TPR are in the range of 0.46–0.59 for the sample networks. In TPR, a source node distributes traffic to all nodes in the network as transit or destination nodes with a set of optimum distribution ratios that are set to be the same values among all source–destination pairs. On the other hand, in F-TPR, the distribution ratio to transit or destination nodes is determined for each each source–destination pair. As a result, F-TPR offer higher routing flexibility than TPR. To more carefully compare rF-TPR with rMPLS-TE to elucidate the difference, the deviation between rMPLS-TE and rFTPR is defined as

dFTPR ¼

rFTRP  r MPLSTE : r MPLSTE

(e) Network 5

(f) Network 6 Fig. 1. Sample networks.

Table 1 Characteristics of networks. Network type

No. of nodes

No. of links

Average node degree

Network Network Network Network Network Network

6 12 12 15 20 35

24 36 48 56 68 100

4.00 3.00 4.00 3.73 3.40 2.86

1 2 3 4 5 6

ð18Þ

All deviations for the sample networks were less than  = 106.  is the optimality convergence tolerance that is set in the LP solver that we used. Thus, the results indicate that F-TPR provides comparable routing performance to MPLS-TE in the sample networks examined. 1 UD/UC is determined so that we can get feasible solutions. UD/UC does not affect the discussion of the performance comparisons, because we focus only on normalized congestion ratios as described later.

We analyzed the dependency of performance on network topology; the toplogies were generated in a random manner under the condition that average node degree D is satisfied for a given number of nodes N and at least one path exists between every source–destination node pair. D is the average number of other nodes to which individual nodes are connected by links. D is defined as

PN PN D¼

i

j

N

aði; jÞ

;

ð19Þ

3229

E. Oki, A. Iwaki / Computer Networks 54 (2010) 3223–3231

Normalized congestion ratio

TPR

F-TPR

Table 3 Computation time in sample networks. *Times are normalized against that of Network 1.

MPLS-TE

1.0 0.8

Network type

Computation time [sec]

Normalized time*

0.6

Network Network Network Network Network Network

0.004 0.005 0.007 0.015 0.025 0.256

1.00 1.25 1.75 3.75 6.25 42.67

0.4 0.2 0.0

1

2

3

4

5

1 2 3 4 5 6

6

Network # to randomly generate sparse topologies with D = 2 when N is large. The results show that rF-TPR lies in the range of 0.45–0.54 for the random networks, and does not strongly depend on N or D. Table 2 shows the deviations of network congestion ratio, as defined by Eq. (18), in random networks. In Table 2, ‘‘ 6 ” indicates that the value is less than  = 106. The results show that all deviations are significantly less than 103( = 0.1%). This means that F-TPR provides almost the same routing performance as MPLS-TE in the random networks examined. Why can F-TPR match the performance of MPLS-TE? MPLS-TE can freely choose any route and provides the best routing performance by using the explicit routing mechanism. On the other hand, F-TPR cannot explicitly choose any route, but can choose freely intermediate nodes; routes from a source node to an intermediate node and those from an intermediate node to a destination node follow shortest path routing. In F-TPR, the flexible choice of intermediates nodes offsets the limitation of shortest path routing and so minimizes the congestion ratio. The flexible choice of intermediates nodes is almost equivalent to the flexible choice of routes from source node to destination node.

Fig. 2. Congestion ratios in sample networks.

Normalized congestion ratio

1.0 0.8 TPR 0.6

N=8 N=16 N=24 N=32 N=36

0.4 0.2 0.0

2

F-TPR

3

4

5

6

Average node degree, D Fig. 3. Congestion ratios in random networks.

where a(i, j) is element (i, j) of the network adjacency matrix:

8 0 > > > <

when there is no link

6.2. Computation time of F-TPR

from node i to node j: aði; jÞ ¼ > 1 when there is a link > > : from node i to node j:

ð20Þ

The computation times for F-TPR to solve the routing problems were measured on a Linux computer with 3.00 GHz Intel Ò Core™ 2 Duo CPU E8400 and 3 GB memory. Average computation times for F-TPR are presented for 100 randomly generated networks in Tables 3 and 4. The results indicate that the computation times of F-TPR are less than 1.2 s. For reference, those of TPR are equal to or less than 0.004 s, which is the minimum measurable time in the solver.

In random network generation, links are assumed to be bi-directional, i.e., the condition of aij = aji is considered. Fig. 3 compares the average values of network congestion ratios for F-TPR and TPR for 100 randomly generated networks with 8 6 N 6 36 and D = {2,3,4,5,6}, except for D P 3 with N P 16. The exception is because it is difficult

Table 2 Deviation of network congestion ratios for F-TPR, dF-TPR, in random networks. indicates that the value is less than . D

2 3 4 5 6

 = 106 is the optimality convergence tolerance in the LP solver. ‘‘6”

N 8

12

16

20

24

28

32

36

6 6 3.36  105 6 6

6 6 6 6 6

6.87  104 6 6 6

6 6 2.49  104 6

6 6 6 5.23  105

6 6 6 7.53  105

6.84  105 6 6 1.30  105

6 9.44  106 6 7.21  106

3230

E. Oki, A. Iwaki / Computer Networks 54 (2010) 3223–3231

Table 4 Computation time in random networks [sec]. D

N

2 3 4 5 6

12

16

20

24

28

32

36

0.005 0.005 0.006 0.007 0.009

0.013 0.018 0.022 0.030

0.029 0.049 0.067 0.092

0.073 0.127 0.239 0.428

0.152 0.352 0.673 1.012

0.311 0.824 0.788 0.594

0.454 0.882 1.153 1.199

Table 5 Deviation of network congestion ratios for non-split F-TPR, dnon-split

F-TPR,

in sample networks.

Network type

LTDF

RS (I = 1)

RS (I = 25)

RS (I = 50)

RS (I = 75)

RS (I = 100)

Network Network Network Network Network Network

0.395 0.070 0.169 0.134 0.029 0.009

0.451 0.076 0.198 0.154 0.032 0.014

0.342 0.056 0.121 0.095 0.020 0.005

0.338 0.055 0.113 0.089 0.020 0.005

0.338 0.055 0.111 0.087 0.020 0.005

0.338 0.055 0.108 0.084 0.020 0.004

1 2 3 4 5 6

Table 6 Computation time for non-split F-TPR in sample networks [sec]. Network type

LTDF

RS (I = 1)

RS (I = 25)

RS (I = 50)

RS (I = 75)

RS (I = 100)

Network Network Network Network Network Network

0.006 0.112 0.299 1.069 1.721 60.772

0.005 0.112 0.300 1.068 1.724 60.773

0.120 2.681 7.347 26.349 42.384 1513.272

0.241 5.355 14.687 52.684 84.666 3024.557

0.362 8.032 22.024 78.999 126.861 2388.957

0.482 10.707 29.358 105.359 169.342 1750.334

1 2 3 4 5 6

7. Performance evaluation for non-split F-TPR Performances of non-split F-TPR were investigated by using the two heuristic algorithms of LTDF and RS. For RS, the dependency on the number of iterations, I, was also investigated. The network congestion ratio of non-split FTPR is denoted as rnon-spilt F-TPR In the same way as rF-TPR, the performance of non-split F-TPR is assessed by determining the deviation between rMPLS-TE and rnon-split F-TPR, which is defined as

dnonsplit

FTPR

¼

r nonsplit

FTRP  r MPLSTE : rMPLSTE

ð21Þ

Note that MPLS-TE allows splitting of traffic demands to obtain rMPLS-TE. Table 5 shows the average deviations of network congestion ratios, dnon-split F-TPR, LTDF and RS in the sample networks in Fig. 1 for 100 randomly generated combinations of link capacities and traffic demands, as described in Section 6. Table 6 shows average computation times for nonsplit F-TPR. In RS, I is set to 1, 25, 50, 75, and 100. For the networks examined, dnon-split F-TPR of LTDF is less than that of RS (I = 1). As I increases, dnon-split F-TPR decreases and falls under that of LTDF when I = 25. However, the improvement in dnon-split F-TPR by increasing I is marginal when I is large. The computation times of RS are approximately proportional to I, as shown in Table 6. For a given I for RS, the minimum deviation value among two network congestion

ratios computed by LTDF and RS is considered as a solution for non-split F-TPR. As network size becomes large, dnon-split F-TPR is decreased. For example, dnon-split F-TPR is less than 1% for network 6. As network size becomes large, non-split FTRF has enough routing flexibility that it can match the performance of MPLS-TE. In addition, for large networks, LTDF, which has lower computation complexity than RS, provides almost the same performance as RS with large I. Therefore, LTDF can provide a reasonable solution for large networks (e.g. network 6), while RS with an appropriate I suits small networks (e.g. networks 1–5).

8. Conclusions This paper presented the fine-TPR scheme over shortest paths for the pipe model, where the traffic matrix is assumed to be given. F-TPR more finely distributes traffic from a source node to intermediate nodes than the original TPR. F-TPR determines the distribution ratio to node m for pq each source–destination pair of (p, q), km independently. To pq determine an optimum set of km , an LP formulation is derived. We compare F-TPR against TPR and MPLS-TE. Numerical results showed that F-TPR greatly reduces the network congestion ratio compared to TPR. In addition, F-TPR provides almost the same network congestion ratio as MPLS-TE, the difference is less than 0.1% for the various

E. Oki, A. Iwaki / Computer Networks 54 (2010) 3223–3231

network topologies examined. Furthermore, we addressed the practical implementation of F-TPR for routers by examining the case that traffic from a source node to a destination node is not allowed to be split over multiple routes. As it is difficult to solve the ILP problem within a practical time, two heuristic algorithms, LTDF and RS, were presented. The applicability of LTDF and RS were presented in terms of network size. We found that non-split F-TPR also matches the routing performance of MPLS-TE within an error of 1%, when network size is large enough. Acknowledgment This work was supported in part by the Okawa Foundation and the Support Center for Advanced Telecommunications Technology Research (SCAT). References [1] R. Zhang-Shen, N. McKeown, Designing a fault-tolerant network using valiant load balancing, in: IEEE Infocom 2008, April 2008. [2] Y. Wang, Z. Wang, Explicit routing algorithms for internet traffic engineering, in: IEEE International Conference on Computer Communications and Networks (ICCCN), 1999. [3] Y. Wang, M.R. Ito, Dynamics of load sensitive adaptive routing, in: IEEE International Conference on Communications (ICC), 2005. [4] B. Fortz, M. Thorup, Optimizing OSPF/IS-IS weights in a changing world, IEEE J. Sel. Areas Commun. 20 (4) (2002) 756–767. [5] J. Chu, C. Lea, Optimal link weights for maximizing QoS traffic, in: IEEE ICC 2007, 2007, pp. 610–615. [6] A.K. Mishra, A. Sahoo, S-OSPF: A traffic engineering solution for OSPF-based on best effort networks, in: IEEE Globecom 2007, 2007, pp. 1845–1849. [7] D. Applegate, E. Cohen, Making routing robust to changing traffic demands: algorithms and evaluation, IEE/ACM Trans. Networking 14 (6) (2006) 1193–1206. [8] M. Antic´, A. Smiljanic´, Oblivious routing scheme using load balancing over shortest paths, in: IEEE ICC 2008, 2008. [9] D. Awduche et al., Requirements for traffic engineering over MPLS, RFC 2702, September 1999. [10] M. Kodialam, T.V. Lakshman, J.B. Orlin, S. Sengupta, Pre-configuring IP-over-optical networks to handle router failures and unpredictable traffic, in: IEEE Infocom 2006, April 2006. [11] M. Kodialam, T.V. Lakshman, J.B. Orlin, S. Sengupta, Oblivious routing of highly variable traffic in service overlays and IP backbones, IEEE/ ACM Trans. Networking 17 (2) (2009) 459–472. Apr.. [12] M. Kodialam, T.V. Lakshman, S. Sengupta, Traffic-oblivious routing for guaranteed bandwidth performance, IEEE Commun. Mag. 45 (4) (2007) 46–51. Apr.. [13] A. Kumar, R. Rastogi, A. Silberschatz, B. Yener, Algorithms for provisioning virtual private networks in the hose model, in: The 2001 Conference on Applications, Technologies, Architectures, and Protocols for Computer Communications, 2001, pp. 135–146. [14] N.G. Duffield, P. Goyal, A. Greenberg, P. Mishra, K.K. Ramakrishnan, J.E. van der Merwe, Resource management with hoses: point-tocloud services for virtual private networks, IEEE/ACM Trans. Networking 10 (5) (2002) 679–692. [15] A. Juttner, I. Szabo, A. Szentesi, On bandwidth efficiency of the hose resource management model in virtual private networks, in: IEEE Infocom, 2003 March/April 2003, pp. 386–395. [16] B. Towles, W.J. Dally, Worst-case traffic for oblivious routing functions, IEEE Comput. Archit. Lett. 1 (1) (2002). [17] D. Applegate, E. Cohen, Making intra-domain routing robust to changing and uncertain traffic demands: understanding fundamental tradeoffs, in: Proceedings of SIGCOMM’03, 2003. [18] M. Bienkowski, M. Korzeniowski, H. Räcke, A Practical algorithm for constructing oblivious routing schemes, in: Proceedings of SPAA’03, 2003. [19] E. Oki, A. Iwaki, F-TPR: Fine two-phase IP routing scheme over shortest paths for hose model, IEEE Commun. Lett. 13 (4) (2009) 277–279. [20] Y. Zhang, M. Roughan, N. Duffield, A. Greenberg, Fast accurate computation of large-scale IP traffic matrices from link loads, in: ACM SIGMETRICS 2003, June 2003, pp. 206–217.

3231

[21] Y. Ohsita, T. Miyamura, S. Arakawa, S. Ata, E. Oki, K. Shiomoto, M. Murata, Gradually reconfiguring virtual network topologies based on estimated traffic matrices, in: INFOCOM 2007, May 2007, pp. 2511–2515. [22] A. Nucci, R. Cruz, N. Taft, C. Diot, Design of IGP link weight changes for estimation of traffic matrices, in: INFOCOM 2004, vol. 4, March 2004, pp. 2341–2351. [23] A. Medina, N. Taft, K. Salamatian, S. Bhattacharyya, C. Diot, Traffic matrix estimation: existing techniques and new directions, in: ACM SIGCOMM 2002, August 2002, pp. 161–174. [24] E. Oki, A. Iwaki, Fine two-phase routing with traffic matrix, in: 18th International Conference on Computer Communications and Networks (ICCCN 2009), August 2009. [25] N. Karmarkar, A New Polynomial Time Algorithm for Linear Programming, Combinatorica 4 (4) (1984) 373–395. [26] Mapnet, online, . [27] ILOG CPLEX, , 2009, online, .

Eiji Oki is an Associate Professor of The University of Electro-Communications, Tokyo Japan. He received B.E. and M.E. degrees in Instrumentation Engineering and a Ph.D. degree in Electrical Engineering from Keio University, Yokohama, Japan, in 1991, 1993, and 1999, respectively. In 1993, he joined Nippon Telegraph and Telephone Corporation’s (NTT’s) Communication Switching Laboratories, Tokyo Japan. He has been researching IP and optical network architectures, traffic-control methods, high-speed switching systems, and communications protocols. From 2000 to 2001, he was a Visiting Scholar at Polytechnic University, Brooklyn, New York, where he was involved in designing tera-bit switch/router systems. He joined The University of Electro-Communications, Tokyo Japan, in July 2008. He has published more than one hundred peer-reviewed journal, transaction articles, and international conference papers. He is active in standardization of Path Computation Element (PCE) and GMPLS in the IETF. He wrote more than 10 IETF RFCs and drafts. He served as a Guest Co-Editor for the Special Issue on ‘‘Multi-Domain Optical Networks: Issues and Challenges,” June 2008, in IEEE Communications Magazine, Guest Co-Editor for Special Issue on Routing, ‘‘Path Computation and Traffic-Engineering in Future Internet,” December 2007, in Journal of Communications and Networks, and Co-Chair of Technical Program Committee for Workshop on High-Performance Switching and Routing in 2006 and 2010, Track Co-Chair on Optical Networking for ICCCN 2009, and Co-Chair of Technical Program Committee for International Conference on IP + Optical Network (iPOP 2010). Oki was the recipient of the 1998 Switching System Research Award and the 1999 Excellent Paper Award presented by IEICE, and the 2001 AsiaPacific Outstanding Young Researcher Award presented by IEEE Communications Society for his contribution to broadband network, ATM, and optical IP technologies. He co-authored two books, ‘‘Broadband Packet Switching Technologies,” published by John Wiley, New York, in 2001 and ‘‘GMPLS Technologies,” published by RC Press, Boca Raton, in 2005. He is an IEEE Senior Member and an IEICE Senior Member.

Ayako Iwaki received B.E. and M.E. degrees in Information and Communications Engineering from The University of Electro-Communications, Tokyo Japan, in 2007 and 2009, respectively. She has been engaged in research on optical signal processing, optical networks, and traffic engineering. She is currently with NTT Network Innovation Laboratories.