A Relaxing Multi-Constraint Routing Algorithm by Considering QoS Metrics Priority for Wired Network Teerapat Sanguankotchakorn
Surabodi Maneepong
Nobuhiko Sugino
Senior member IEEE Information and Communications Technology Interdisciplinary Graduate School of Engineering and Technology School of Engineering and Technology School of Science and Engineering Asian Institute of Technology, Thailand Asian Institute of Technology, Thailand Tokyo Institute of Technology, Japan
[email protected] [email protected] [email protected]
Abstract—To find a path satisfying multi-constrained QoS metrics in packet switching network is a very challenging problem. The problem of finding such feasible paths is known as an NP-complete problem. In this work, we propose a solution to the afore-mentioned problem by relaxing the multi-constrained QoS routing using the significance level of QoS metrics. Our proposed algorithm adopts the nonlinear cost function and relaxing look-ahead concept: to put into account the QoS metrics’ priority. The performance of our proposed algorithm is evaluated by simulation using MATLAB in terms of Success Ratio of finding the feasible paths and Computational Complexity. According to the simulation results, it is obvious that our proposed algorithm is superior to the existing algorithm (H_MCOP) in terms of Success Ratio, but inferior in terms of Computational Complexity. However, the Computational Complexity of our proposed algorithm is still within the acceptable level (1-2 ms for 100-node network).
aims to satisfy QoS metrics, is called QoS routing [16]. QoS is a set of technical requirements illustrating the capability of network on satisfying the users’ some requirements, namely the minimum delay or the maximum bandwidth. Since QoS routing always has to deal with multiple requirements, in this study, we are focusing on the multiconstrained QoS routing problem, which is considered as an NP-complete (Nondeterministic Polynomial time) problem [9], [11]. Several heuristics or even polynomial time approximation algorithms such as TAMCRA [8], SAMCRA [12], H_MCOP [14], A*Prune [7] have been proposed in the literature. However, most of them consider each QoS metrics with equal weight in the path calculation. In realistic, there might be different demand for each QoS constraint to be satisfied. For example, voice and videos transmission are real-time applications, which require very low delay while the level of loss is more tolerable [13]. However, the file transfer application is more sensitive to loss than the other parameters [3]. Therefore, the paths satisfying multi-constrained QoS should be computed by putting into account this fact. In other words, the appropriate paths satisfying multiconstrained QoS for applications should be calculated based on each QoS metrics’ significance. In this work, we propose the algorithm to solve the afore-mentioned problem by adopting the nonlinear cost function and relaxing the look-ahead concept and then verify it by simulation on various scenarios. The major contribution of this work is the routing algorithm that can find the feasible path somewhat satisfying multi-constrained QoS by relaxing post-path look-ahead algorithm. In addition, such that feasible path can be found with higher success ratio than H_MCOP [14]. This paper is structured as follows: Section 2 describes the related works while Section 3 illustrates the proposed algorithm in detail. Section 4 describes simulation model and parameters while the simulation results and discussion are shown in Section 5. Section 6 concludes the work.
Keywords-Quaity of Service (QoS), Multi-constrainted QoS Routing, Nonlinear cost function, Look-ahead concept.
I.
INTRODUCTION
Recently, communication technologies have been growing very fast. It makes people to be able to communicate everywhere and every time via mobile devices, such as notebook computer, smart phone, etc. The demand of internet-based applications as well as the number of users have been increasing everyday and do not seem to stop. Of course, the requirements of users or organizations are different depending on the applications and networks. To accommodate these kinds of applications appropriately, the service provider has to provide the interconnection more efficiently. That is, routing which is the key process has to be designed in an efficient manner. In general, routing consists of two modules: routing protocols and algorithms. Routing protocols is used to collect information of the nodes in the network while routing algorithms will use this information to find suitable paths for sending applications [2]. The routing algorithm, which simply finds the path without satisfying any QoS metrics, is called best-effort routing [16]. On the other hands, the routing algorithm, which
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II.
A. Definitions Let G = (V, E) is the network, where V is the set of nodes and E is the set of links. Each link between node u and v illustrated as (u,v) has K non-negative additive QoS metrics wk (u, v) and K constraints ck, where k = 1, 2,…, K.
RELATED WORKS
TAMCRA[8] and SAMCRA[12] were proposed to solve multiple constraints routing problem in year 2000 and 2001, respectively. Both algorithms use the nonlinear cost function, the concepts of k-shortest path and nondominated path to increase the possibility of finding an exact shortest path, which is tunable by setting the value of k, where k is the number of shortest paths stored at each node. In TAMCRA, the value of k is fixed, while in SAMCRA, it is self-adaptively. This makes SAMCRA to be more exact algorithm than TAMCRA. Even though the performance of SAMCRA seems better than TAMCRA, but its worst-case complexity grows exponentially. Almost at the same time, an algorithm called H_MCOP was proposed [14]. It aims to find a feasible path while minimizing the primary cost simultaneously. This algorithm also uses the nonlinear function and look-ahead concept, which is similar to TAMCRA. However, it proposes the optimization technique to select the path minimizing the primary cost first, and then the nonlinear cost function. Moreover, the k-shortest path is not considered in this algorithm. Some other similar algorithms proposed to handle the multi-constrained multicast routing problem are such as MAMCRA[5], HCMC[6], A*Prune[7], etc. In 2004, another heuristic algorithm called H_MCP (Heuristic Multi-Constrained Path) was also proposed to solve the MCP problem. The simulation results illustrate that the success ratio of finding feasible path is much higher than the other ones. In year 2010, Xin Jin [17] considered an MCP problem with imprecise additive link state information and proposed pre-computation algorithm called Limited Selective Flooding (LFS) routing algorithm with imprecise additive link state information. In year 2012, the MPLMR [18] (Multi-Postpath-based Look-ahead Multi-constraint QoS Routing) was proposed. This algorithm stores a limited number of sub-paths between the source node and each intermediate node, and extends these sub-paths toward the destination node using the improved look-ahead method to estimate the path length of the full path to which each sub-path is extended. It is shown that MPLMR has a higher success ratio of finding a feasible path than competing schemes in the literature, i.e.,TAMCRA [8], H_MCOP [14]. III.
Let s denotes the source node, d is the destination node, u is an intermediate node. The paths from s to every node u and the paths from every node u to d are called pre-path and post-path, respectively.
B. Proposed Algorithm The proposed algorithm is developed based on H_MCOP[14]. It consists of two paths: pre-paths and post-paths calculation. In this algorithm, the nonlinear cost function for any path p, as shown in (1), is adopted to find the feasible pre-paths (the paths satisfying QoS constraints from source node s to all intemediate nodes u).
! w ( p) $ gλ ( p) = ∑# i & ci % i=1 " K
λ
(1)
Where λ ≥ 1 . To increase the probability of finding a feasible path, we set λ as large as possible, that is, we set λ → ∞ . Then, we can replace the cost function in (1) by the following cost function:
! w ( p) w ( p) w ( p) $ h( p) = max " 1 , 2 ,, K % c2 cK & # c1
(2)
This cost function obviously does not involve λ explicitly. In addition, by using this cost function, the same order of candidate paths as (1) can be achieved [14]. In finding the post-paths, we propose the prioritybased linear cost function for any path p as follows: K ! w ( p) $ f1 ( p) = ∑ ai # i & " ci % i=1
(3)
Where ai ∈ R + (i=1,2,…,K) is the weight allocated to each additive QoS metric based on its priority in the application. In this work, we aim to find the feasible paths satisfying different priority level of K QoS attributes. Thus, we allocate the highest weight to the highest priority QoS attribute in the post-path calculation process. For example, if K=3 (there are 3 constraints), we will allocate the weight a1 > a2 > a3 for the case of the priority
PROPOSED ALGORITHM
In this research, we propose a modified algorithm solving multi-constrained QoS routing by putting into account the significance of QoS metrics of each application in the network. We use two elements of multiconstrained QoS routing: nonlinear cost function and lookahead concept, which is similar to H_MCP. We modify the algorithm by relaxing the multi-constrained QoS requirements in the look-ahead part to put into account only the most significant one based on each application’s requirement. By doing this, the success ratio of finding feasible path increases significantly comparing to H_MCOP algorithm.
of each link weight in descending order is w1 ( p) , w2 ( p) and w3 ( p) , respectively.
Firstly, we find the best path w.r.t. to f1 (p) from each intermediate node u to destination node d. Then, we find the paths minimizing gλ ( p*) using Dijkstra’s algorithm [4], where p* is the path from source s to intermediate node u.
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If the concatenating segment from source node s to intermediate node u and from u to destination node d gives the link cost lower than the QoS constraints, then that complete path from source s to destination d through node u will be declared as a feasible path. If the link cost of the path s-d is greater than the number of constraints ( g1 ( p) > K ) , the algorithm will
All networks’ topologies and details are illustrated as follows: • NSFNET Network Topology
return no feasible path and be terminated. The problem here is the way to find the appropriate values of ai , i = 1, 2,.., K . However, in this work, in order
to relax the multi-constrained QoS requirements, we consider only the highest priority QoS metric in the postpath computation process and the lower priority QoS attributes can be neglected. That is, we consider a1 = 1 and ai = 0 (i = 2 ~ K ) .
Figure 1. NSFNET Network Topology [10]
• Pan-European Network Topology
C. Pseudo Code of the Proposed Algorithm The pseudo code of the proposed algorithm can be illustrated as follows: Proposed Algorithm (G=(V, E), s, d, ck, k=1,…,K) Pre_path (G=(V, E), d) for i=1 : K, Post_path (G=(V, E), s) if Gk[d] ≤ ck return feasible path pk end if end for return failure Pre_path (u,v) (u and v are arbitrary nodes) for i = 1: K if Rk[u] > Rk[v]+wk(u,v) then Rk[u] = Rk[v]+wk(u,v) Rk[v] = Rk[v]+wk(u,v) Prev[v] = u; end if end for
Figure 2. Pan-European Network Topology [15]
• Random Waxman Network In Random Waxman Network [1], the location of each node is randomly generated within the area of the graph defined by value on the x and y-axes. The link probability of node u and node v is determined by the probability function as follows:
Post_path (i,j) tmpk = temporary node k, k=1,…,K g[tmp]= max[(Gk [u] +wk (u,v)+Rk[v])/ck], k=1,…,K if Gk[tmp]+Rk[tmp]≤ck Gk [tmp] = Gk[u]+wk(u,v), k=1,…,K Rk[tmp]= Rk[v], k=1,…,K end if IV.
" −δ (u, v) % P(u, v) = α exp $ ' # βL &
(4)
Where α (α > 0) represents the maximum link probability, β (β ≤ 1) represents the parameter to control the length of the link in the network, δ (u, v) is the distance between node u and node v and L is the maximum distance between any two nodes in the network. By defining the number of nodes, we can generate the desired random network for our simulation. In this work, we set 𝛼𝛼 = 0.5 and 𝛽𝛽 = 0.2. The basic parameters of the network consisting of 50 and 100 nodes are illustrated in Table 1, and the examples of Random Waxman Networks are illustrated in Fig.3.
SIMULATION MODEL AND PARAMETERS
A. Simulation Networks In this work, we want to investigate the performance of our proposed algorithm in all kinds of networks. Therefore, we simulate our proposed algorithm using three types of networks: (a) NSFNET (a fixed network with small number of nodes and links) (b) Pan European Network (a fixed network with large number of nodes and links) (c) Random Waxman Network (a randomly generated network with random nodes’ position and number of links).
TABLE 1 BASIC PARAMETERS OF RANDOM WAXMAN NETWORK
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Network Topology
Number of Nodes
Number of Links
Random Waxman Network
50 100
≈ 100 ≈ 500
V.
SIMULATION RESULTS AND DISCUSSION
In this work, we compare the performance of our proposed algorithm with H_MCOP based on two types of Link QoS Metrics Distribution: Uniform and Normal Distribution. The Mean and Standard Deviation of both distributions are set equally in each simulation scenario. The simulation results are illustrated with 95% confidence interval. (a) 50 nodes
A. Uniform Distributed Link QoS Metrics
(b) 100 nodes
• Success Ratio
Figure 3. Example of Random Waxman Network
B. Simulation Parameters
1.00
Parameters Number of Nodes S-D pair
0.80
No. of Simulations per one data point
Success ratio
Link QoS Metrics between any node u and v Constraints
Value 14, 28, 50, 100 Randomly selected with minimum 3-hop distance wk(u,v)= [a,b] for Uniform Distribution, where a and b are real numbers. wk(u,v)=(E[x];σ2) for Normal Distribution ck = min (wk(p))*[1,1.5] where p is the complete path from source s to destination d 100
0.60
II
III Constraints ck
0.00
0.14
14
28
0.10 0.02
50
100
Figure 4. Success Ratio of Scenario I
Uniform Distribution w1(u,v)= [1,100] w2(u,v)= [1,100] w3(u,v)= [1,100] w1(u,v)= [1,100] w2(u,v)= [200,400] w3(u,v)= [1,100] w1(u,v)= [200,400] w2(u,v)= [200,400] w3(u,v)= [200,400] min(wk(p))*[1,1.5]
1.00
Normal Distribution w1(u,v)= (50;28.6) w2(u,v)= (50;28.6) w3(u,v)= (50;28.6) w1(u,v)=(50;28.6) w2(u,v)=(300;57.7) w3(u,v)=(50;28.6) w1(u,v)=(300;57.7) w2(u,v)=(300;57.7) w3(u,v)=(300;57.7) min(wk(p))*[1,1.5]
0.80
0.84
0.77 0.66
0.60
H_MCOP
0.59
0.60
Proposed algorithm 0.29
0.40 0.20 0.00
0.20 0.07
14
28
50
100
Number of nodes Figure 5. Success Ratio of Scenario II 1.20
Success ratio
Success Ratio: the capability of finding feasible path in the network satisfying QoS requirements. It is defined as
1.00
0.99 0.95
0.95
0.98
0.930.98
0.80
0.95 0.80 H_MCOP Proposed algorithm
0.60 0.40 0.20 0.00
14
28
50
100
Number of nodes
𝑁𝑁𝑁𝑁. 𝑜𝑜𝑜𝑜 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝑝𝑝𝑝𝑝𝑝𝑝ℎ𝑠𝑠 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 = 𝑁𝑁𝑁𝑁. 𝑜𝑜𝑜𝑜 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖
Figure 6. Success Ratio of Scenario III
From Fig.4, 5 and 6, it is obvious that the Success Ratio decreases when the density of nodes is large enough in both algorithms. This is because the interference between each node increases when the node’s density increases. In addition, the Success Ratio of our proposed algorithm is much better than H_MCOP in all type of networks since the multi-constrained QoS metrics are relaxed based on its priority. However, when the link QoS metric varies with higher standard deviation (scenario III), the Success Ratio of our proposed algorithm becomes closed to H_MCOP. This is because both algorithms may be able (or unable) to find the feasible paths regardless of the priority of QoS metrics.
Computational Complexity: the time consumed by the algorithm to find the feasible paths satisfying QoS requirements. This can be used to evaluate indirectly the complexity of the algorithm. In this work, we perform the computational complexity measurement under the following platform and environment: TABLE 3 SIMULATION ENVIRONMENT
RAM
Proposed algorithm
0.40
Number of nodes
C. Performance Evaluation Metrics The performance of our proposed algorithm is evaluated using the following metrics:
Simulation Tool Operating System Processor
0.47
0.20
Success ratio
I
H_MCOP
0.52
0.40
TABLE 2 SIMULATION SCENARIO AND LINK QOS METRIC DISTRIBUTION Scenario
0.73
0.65
MATLAB R2010a, MATLAB compiler Windows 7, Ultimate 64-bit Intel(R) Core(TM)2 Duo CPU P8700 2.53 GHz 4 GB
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• Computation Complexity
Time (10-3s)
1.66
1.50
H_MCOP
1.1
Proposed algorithm
0.75
1.00 0.50
Success ratio
1.00
2.00
0.11 0.04
0.00
14
0.10
0.29
0.27
28
50
0.80
0.82 0.66
0.43 0.13
0.20 14
Success ratio
Time (10-3s)
1.20
H_MCOP Proposed algorithm
1.13 0.74 0.04 0.09
0.26 0.10
14
28
0.00
1.00
0.940.99
Time (10-3s)
1.53
1.50
0.030.09
0.00
14
0.08
0.41 0.15
28
50
H_MCOP
0.59
100
0.40
14
Time (10-3s)
0.50 0.00
0.00
Time (10-3s)
Success ratio
0.21
Proposed algorithm
28
50
0.04
0.10
14
0.25 0.09 28
0.29 50
100
2.05 H_MCOP Proposed algorithm
1.50 1.00 0.50
0.25
0.00
0.05 14
0.75
Figure 13. Computational Complexity of Scenario I
H_MCOP
0.20
Proposed algorithm 1.1
Number of nodes
0.80
0.40
H_MCOP
1.00
2.00
0.53
2.06
1.50
2.50
0.60
100
2.00
• Success Ratio 0.62
50
2.50
B. Normal Distributed Link QoS Metrics
0.55
28
• Computational Complexity
From Fig.7, 8 and 9, it is apparent that the Computational Complexity of both algorithms increases exponentially with the number of nodes. In addition, our proposed algorithm has approximately 3 times of complexity of that of H_MCOP since the complexity of this algorithm is K times of H_MCOP, where K is the number of constraints. Note that the Complexity in all scenarios of our proposed algorithm is within 2.0 ms for a large network (100-node). Therefore, the Complexity of our proposed algorithm is acceptable in practice.
0.80
Proposed algorithm
As shown in Fig.10, 11 and 12, the simulation results follow the same trend as the case of Uniform Distributed link QoS metrics. That is, the Success Ratio decreases when the node density becomes large enough. Additionally, it is obvious that our proposed algorithm is superior to H_MCOP in terms of Success Ratio. The reason is the same as provided previously.
Figure 9. Computational Complexity of Scenario III
0.72
H_MCOP
Figure 12. Success Ratio of Scenario III
Number of nodes
1.00
0.95
Number of nodes
Proposed algorithm
0.21
0.98
0.60
100
2.00
0.50
0.92
0.78
Figure 8. Computational Complexity of Scenario II
1.00
0.94 0.98
0.80
0.00
Number of nodes
100
0.20
0.26 50
50
Figure 11. Success Ratio of Scenario II
2.00
0.50
28
Number of nodes
100
2.00
1.00
Proposed algorithm
0.32
0.40
0.00
Figure 7. Computational Complexity of Scenario I
1.50
H_MCOP
0.64
0.60
Number of nodes
2.50
0.87 0.72
0.62 0.24 0.04 0.11 0.09 14
28
0.85
0.26 50
100
Number of nodes
100
Figure 14. Computational Complexity of Scenario II
Number of nodes Figure 10. Success Ratio of Scenario I
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Time (10-3s)
2.00
1.58
1.50
Proposed algorithm
1.00 0.50 0.00
finding feasible paths, the proposed algorithm provides the option to the users to receive QoS guarantee on their applications effectively.
H_MCOP
0.44 0.030.09 14
0.20 0.09 28
ACKNOWLEDGMENT
0.68
The authors would like to thank National Institute of Information and Communication Technology (NICT), Tokyo, Japan for providing the funding of this research.
0.17 50
100
REFERENCES
Number of nodes [1]
Figure 15. Computational Complexity of Scenario III
As illustrated in Fig. 13, 14 and 15, it is obvious that the Computational Complexity of this case for all scenarios has the same trend as the case of Uniform Distributed link QoS metrics. Therefore, we can conclude that the Computation Complexity depends on the number of nodes and is irrelevant to the distribution of link QoS metrics. VI.
[2] [3] [4] [5]
CONCLUSIONS
[6]
Recently, Quality of Service (QoS) in the network has really become an important issue in the network operation. In this work, we propose the relaxing multiconstrained routing using priority of QoS metrics. This proposed algorithm relaxes the multi-constrained QoS requirement satisfaction by adopting the priority of QoS metric on the post-paths computation. Therefore, the feasible paths found will not fully, but somewhat satisfy the multi-constrained QoS requirements. We perform the simulations on three types of network: NSFNET, Pan-European and Randomly generated Waxman Network, to verify the superiority of our proposed algorithm. We consider two types of link QoS metric distribution: Uniform and Normal distribution. It is obvious that by adopting the proposed algorithm, the Success Ratio of finding the feasible paths increases significantly comparing to H_MCOP. It is found that the success ratio depends on the node’s density and link QoS metric distribution. According to the simulation results, the Success Ratio increases when the number of node increases to some certain level (approximately 28 nodes in this work) and then decreases drastically when the number of node increase further. Moreover, the Success Ratio in case of Normal Distributed link QoS metric is higher than the case of Uniform Distribution. The Computational Complexity also increases, but in acceptable manner (within 2 ms for 100-node network). The Computational Complexity is found irrelevant to the link QoS metric distribution but depending only on the number of nodes. It is found that the Computational Complexity increases approximately exponentially with the number of nodes. By trading off the fully satisfaction of multiconstrained QoS requirements with the Success Ratio of
[7] [8] [9] [10] [11] [12] [13]
[14] [15]
[16] [17]
[18]
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B.M. Waxman: “Routing of multipoint connections”, Institute of Electrical and Electronics Engineers (IEEE) Journal on Selected Areas in Communications, Vol. 6, 1988, pp. 1617-1622. Cisco System: “Routing Basics”, Internetworking Technology 4th edition, Chapter5, 2003. Derek C.W. Pao, S.P. Lam: “Cell scheduling for ATM switch with delay-sensitive and loss-sensitive traffic”, Computer Communications, Vol.21, Issue 13, 1998, pp. 1153-1164. E.W. Dijkstra: "A note on two problems in connexion with graphs", Numerische Mathematik , 1959, pp. 269–271. F. Kuipers and P.Mieghem: “MAMCRA: a constrained-based multicast routing algorithm”, Computer Communications, Vol.25, 2002, pp.802-811. Gang Feng: “A heuristic for multi-constrained multicast routing”, Proceeding of High Performance Switching and Routing (HPSR) Workshop, 2004, pp.309-313. Gang Liu, K.G. Ramakrishnan: “A*Prune: an algorithm for finding K shortest paths subject to multiple constraints”, Proceedings of IEEE INFOCOM , Vol. 2, 2001, pp. 743–749. H.D. Neve and P.V. Mieghem: “TAMCRA: a tunable accuracy multiple constraints routing algorithm”, Computer Communications, Vol. 23, Issue 7, 2000, pp. 667-679. J.M. Jaffe: “Algorithms for finding paths with multiple constraints”, Networks, Vol. 14, 1984, pp. 95-116. Keyao Zhu: “Network topologies used in my study”, 2003, http://networks.cs.ucdavis.edu/~zhuk/toplogies.html M.R. Garey and D.S. Johnson: “A Guide to the Theory of NPCompleteness”, Computers and Intractability, 1979. P. V. Mieghem, H. D. Neve and F.A. Kuipers: “Hop-by-hop quality of service routing” Computer Networks, Vol. 37, No.3-4, Nov. 2001, pp.407-423. T. Henderson and S. Bhatti: “Networked games - a QoS-sensitive application for QoS-insensitive users?” Proceedings of ACM SIGCOMM Revisiting IP QoS (RipQoS) Workshop, 2003, pp 141147. T. Korkmaz and M. Krunz: “Multi-constrained optimal path selection” Proceedings of IEEE INFOCOM, 2001, pp. 834-843. T. Stevens, M. De Leenheer, F. De Turck, B.Dhoedt and P. Demeester: “Distributed Job Scheduling based on Multiple Constraints Anycast Routing”, Proceedings of 3rd International Conference on Networks and Systems (BROADNETS), 2006, pp.1–8. Wei Sun: “QoS/Policy/Constraint Based Routing”, 1999, http://www.cse.wustl.edu/~jain/cis78899/ftp/qos_routing/index.ht ml Xin Jin: “Routing for Multi-Constrained Path Problem with Imprecise Additive Link State Information” Proceedings of International Conference on Computer Application and System Modeling (ICCASM), Vol. 3, 2010, pp.472-475. Dong-won Shin, Edwin K.P. Chong and Howard Jay Siegel: “Multi-postpath-based Lookahead Multiconstraint QoS Routing” Journal of the Franklin Institute, Vol.349, 2012, pp.1106-1124.
