Novel approach to jointly optimize working and spare capacity of survivable optical networks

Ashutosh Kumar Singh; Vanya Arun; Pallavi Singh; Kamal Kishore Upadhayay

doi:10.1515/joc-2020-0228

Article Open Access

Novel approach to jointly optimize working and spare capacity of survivable optical networks

Ashutosh Kumar Singh , Vanya Arun , Pallavi Singh and Kamal Kishore Upadhayay

Published/Copyright: October 23, 2020

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From the journal Journal of Optical Communications Volume 44 Issue s1

Abstract

As technology advancing day by day, the data rate of optical network is moving towards Tb/s speed. The minimum capacity utilization and survivability are the crucial requirement in such high speed optical networks. This research work presents a new approach to calculate both working and spare capacity with the help of single mathematical programming model named as joint capacity planning model. The working traffic and restored traffic are routed jointly in proposed joint capacity planning model. Therefore the joint capacity planning model required minimum capacity in as compare to other optimization models. To evaluate our model, three example networks are proposed i.e., network A (6 node), network B (8 node) & National science foundation network (14 node). Results of these networks are analyzed and compared. The capacity utilization is optimized by increasing the backup paths of the optical networks. It has also been proved in this manuscript that capacity requirement is dependent on the backup path. The proposed joint capacity model provides fast restoration speed and guaranteed protection for optical network.

Keywords: capacity planning; optimization methods; optical network; survivability

1 Introduction

1.1 Background

The optical networks are high-capacity telecommunications networks based on optical technologies and component that offer routing, restoration and grooming at the wavelength level as well as wavelength-based services. These networks are based on the emergence of the optical layer in transport networks, provide higher capacity and reduced cost for new applications such as the internet, multimedia interaction and advanced digital services. Survivability is a crucial requirement in optical networks because today’s networks carries enormous amount of data, widely used in large application such as high-speed supercomputing, real-time medical imaging, scientific visualization and high-definition video distribution. If a cut occurs in a fiber it will result in a huge loss of data. The network will be called survivable network when it continues to work even when a link failure occurs [1], [2], [3]. The techniques that have been used for survivable network can be classified by two general categories: protection & restoration [2], [4], [5]. If backup resources are pre-analyzed and pre-reserved before the connection setup, it is called a protection scheme. In a restoration model, the new route and available wavelength have to be dynamically assigned for each failed link. The protection/restoration scheme is further classified in two categories: path protection/restoration and link protection/restoration.

In the path protection/restoration scheme, if link failure occurs then backup path establishes new connections between all affected source and destination nodes, which were using the failed link. For path protection model, the backup path for each failed link is pre-analyzed and reserved at the time of connection setup [1], [2], [6]. While in path restoration model, the backup paths for each failed link are dynamically assigned. On the other hand in link protection/restoration, if link failure occur then backup path establishes the connection between nodes adjacent to failed link only. For link protection model, the backup resources around each failed link are reserved at the time of connection setup. While in link restoration model, the previous node of each failed link dynamically discover a different route around the link with the help of distributed algorithm [1]. The main difference between path restoration and link restoration is that, the path restoration model uses entirely different route from source node to destination node in case of fiber cut or failure, whereas in link restoration the data is saved at the previous node of failed link and turns away from that failed link. Finding of a new routes and allocation of appropriate wavelengths to the lightpath are an additional challenge as technology is advancing day by day. The method of finding a new route and assigning a wavelength for a light is known as routing and wavelength allocation (RWA) problem [7], [8].

1.2 Related work

Recently numerous works have been done in the field of survivable optical network and capacity allocation planning. In [1], the authors have classified various protection and restoration technique for survivable WDM optical network. Here authors have briefly described the path based and link based protection technique. A joint lightpath routing approach to select the working and protection paths for the dynamic traffic in survivable optical network was proposed in Ref. [9]. The different types of fault recovery schemes in WDM mesh based networks have been investigated in Ref. [2]. They reviewed various protection and restoration schemes, primary and backup route computation methods, dynamic restoration and sharability optimization. They also discussed different parameters that can measure the quality of service provided by a WDM mesh network to upper protocol layers, such as service reliability, service availability, restoration time and service restorability. In Ref. [10], the authors have mathematically programmed the working Capacity allocation, spare capacity allocation and joint capacity allocation techniques to optimize the capacity utilization in optical network. In this paper the authors have calculated working capacity, spare capacity and joint capacity utilized by six-node network.

An integer linear programming (ILP) for saving total spare capacity was described in Ref. [11]. The authors have proposed spare capacity allocation models for protection at the bottom layer only, protection at the top layer only with cross layer sharing of spare capacity and common pool cross layer protection. In Ref. [12], the authors have presented the design of logical survivable topologies for service recovery against link failures. A general classification of the existing research works on disaster survivability in optical networks and a survey on relevant works have been presented in Ref. [13]. In Ref. [14], the authors have proposed the ILP model to find the primary and secondary backup paths sequentially in order to minimizing the total spare capacity of optical network. In Ref. [15], the authors have presented an arc-path working capacity allocation strategy for protection and restoration from single link failure. In this paper the authors have calculated working capacity utilized by six-node, eight-node and National science foundation network (NSFNET). A novel survivability technique called dynamic and hybrid with multiple backup selection criteria (DHMBC) for WDM optical networks has been described in Ref. [5]. They also proposed an integer linear programming model to reduce computational time and complexity. An integer linear program to control the propagation of high-power jamming attacks in transparent optical networks has been presented in [8].

1.3 Our proposal

A joint capacity planning model is a single model which calculates both working capacity and spare capacity of optical networks. In the past, the optimal working capacity was calculated by one model and then another model was used to calculate the spare capacity needed to protect the calculated optimal working capacity. The working traffic and restored traffic are routed jointly in the proposed joint capacity planning model. Therefore the capacity required is minimized in this model as compared to other optimization models. Minimum spare capacity requirement is very important issue in optical network to support Tb/s speed. In this study, we proposed joint capacity planning model to optimize the total capacity (working + spare) used by optical network. In our proposal, the spare capacity allocation planning has been performed with the help of three example problem network i.e., network A, network B & National Science Foundation Network (NSFNET). Also, we present a mathematical programming model to optimize the capacity utilization in optical networks. With the help of this mathematical programming model, the total spare capacity utilization for network A, network B & NSFNET has been analyzed. Paper also proves that capacity requirement is closely related to backup paths of a network. The proposed joint capacity planning model provides guaranteed protection against a link failure. The rest of paper is organized as follows. In Section II, we present a mathematical programming model named as joint capacity planning model to calculate both working and spare capacity of optical networks. Section III, presents result analysis of network A, network B & NSFNET. Finally, section IV concludes the paper.

2 Problem formulation

In this section, a mathematical programming model for joint capacity allocation planning to protect link failure has been discussed [10], [15]. The fiber optic based telecommunication networks consists set of nodes and set of links. The connection between two nodes is called a link. The node represents the access point through which data traffic is transferred or received. The number of links connected to the node is called degree of a node. In this paper, every node of network has minimum degree 2. For joint capacity allocation model, a link denotes bi-directional connection between a node pair. Hence, node can communicate in both directions (i to j) and (j to i). A graph is represented by [N, L]. Here N is a set of node and L denotes the link corresponding to node. The directed graph for given network is denoted as G = [N, E]. An arc corresponding to a link is denoted by E. Arc (i, j) represents flow is from node i to node j and arc (j, i) represent flow is from node j to node i.

To solve the example problem, we assume the demand for all the links of the network and generate a matrix corresponding to these demands which is called the demand matrix or traffic matrix. The demand with source node i and destination node j is denoted by d _ij. The set of demand pair corresponding to link is denoted by D. From node s to node t, the directed path is a sequence of nodes and arcs p = { i 1 , ( i 1 , i 2 ) , i 2 , ( i 2 , i 3 ) , i 3 , … , i l , ( i l , i l + 1 ) , i l + 1 } , where i ₁ = s and i _l+1 = t. In the case of directed path all arc and node are distinct. The set of directed paths from node i to node j in a network is denoted by Q _ij. Let T = U ( i , j ϵ D ) Q i j , here U denotes the set of cycle. In a network, the set of path from node i to node j is denoted by A _ij and flow on path p is denoted by y _p. Let c denotes the working capacity utilized by a network and variable c _ij denotes the working capacity on link (i, j). The working capacity for all {i, j} ∈ L are fixed. Let h denotes the spare capacity utilized by the network and the variable h _ij denotes the spare capacity on link (i, j). The restoration flow on path p is denoted by w p s t in case of link {s, t} failure. Let the set V i j s t denotes the set of directed paths, which are available for restoration from node i to node j in the case if link {s, t} fails. These directed paths are entirely different path from node i to node j and it does not use the failed link { s , t } . V ‾ i j s t denotes the set of directed paths from node i to node j, which are not available for working traffic.

To calculate joint capacity (working capacity plus spare capacity) utilized by network, the joint capacity allocation model is given as:

Optimize working capacity plus spare capacity:

(1) ∑ { i , j } ∈ L ( c i j + h i j )

Subject for demand:

(2) ∑ p ∈ Q i j y p = d i j ∀ ( i , j ) ∈ D

Normal direction capacity:

(3) ∑ p ∈ A i j y p ≤ c i j ∀ ( i , j ) ∈ L

Reverse direction capacity:

(4) ∑ p ∈ A j i y p ≤ c i j ∀ ( i , j ) ∈ L

Non-negativity and integrality:

(5) c i j ≥ 0 ∀ ( i , j ) ∈ L ,

Subject for spare demand 1:

(6) ∑ p ∈ V i j s t w p s t = ∑ p ∈ V ‾ i j s t y p ∀ { s , t } ∈ L ∀ ( i , j ) ∈ D

Subject for spare demand 2:

(7) ∑ p ∈ V j i s t w p s t = ∑ p ∈ V ‾ j i s t y p ∀ { s , t } ∈ L ∀ ( i , j ) ∈ D

Spare capacity in normal direction:

(8) ∑ p ∈ A i j w p s t ≤ h i j ∀ { s , t } ∈ L ∀ ( i , j ) ∈ L \ { { s , t } }

Spare capacity in reverse direction:

(9) ∑ p ∈ A j i w p s t ≤ h i j ∀ { s , t } ∈ L ∀ ( i , j ) ∈ L \ { { s , t } }

Non-negativity:

(10) h i j ≥ 0 ∀ ( i , j ) ∈ L ,

(11) w p s t ≥ 0 ∀ { s , t } ∈ L ∀ p ∈ T .

In this model all the nodes are capable of detecting a link failure and discovering a different route across the failed link with the help of distributed algorithm. Suppose if link {s, t} fails, then node-arc link restoration model proposed in this manuscript allows all the data that uses failed link {s, t} to be followed different route on reduced graph [N, L{s, t}].

Joint Model for Spare Capacity Allocation Planning has been performed with the help of three example network i.e., network A, network B & National Science Foundation Network (NSFNET). Figure 1 shows network A, it consists of six nodes and nine links. Table 1 represents the demand matrix corresponding to network A. Table 2 represents the demand matrix corresponding to network B shown in Figure 2. Network B has eight nodes 13 links. Figure 3 shows National science foundation network, it consists of 14 nodes and 22 links. Table 3 represents the demand matrix corresponding to NSFNET.

Figure 1:

Network A.

Table 1:

Demand matrix for network A.

Node	1	2	3	4	5	6
1	–	10	0	10	10	0
2	0	–	10	10	10	10
3	0	0	–	0	10	10
4	0	0	0	–	10	0
5	0	0	0	0	–	10
6	0	0	0	0	0	–

Table 2:

Demand matrix for network B.

Node	1	2	3	4	5	6	7	8
1	–	10	10	10	0	10	0	10
2	0	–	10	10	10	0	10	0
3	0	0	–	10	10	10	0	0
4	0	0	0	–	10	10	10	0
5	0	0	0	0	–	10	10	10
6	0	0	0	0	0	–	10	10
7	0	0	0	0	0	0	–	10
8	0	0	0	0	0	0	0	–

Figure 2:

Network B.

Figure 3:

NSFNET.

Table 3:

Demand matrix for NSFNET.

Node	1	2	3	4	5	6	7	8	9	10	11	12	13	14
1	–	0	0	0	10	10	0	10	10	10	0	0	10	10
2	0	–	10	10	0	10	0	0	10	10	10	10	10	0
3	0	0	–	10	0	10	0	0	10	10	10	10	10	0
4	0	0	0	–	10	10	10	0	0	10	10	0	0	0
5	0	0	0	0	–	10	10	0	10	0	0	0	10	10
6	0	0	0	0	0	–	10	10	0	0	0	0	10	10
7	0	0	0	0	0	0	–	10	10	0	0	10	10	10
8	0	0	0	0	0	0	0	–	10	10	0	10	10	0
9	0	0	0	0	0	0	0	0	–	10	0	10	0	0
10	0	0	0	0	0	0	0	0	0	–	10	10	10	0
11	0	0	0	0	0	0	0	0	0	0	–	10	10	10
12	0	0	0	0	0	0	0	0	0	0	0	–	10	10
13	0	0	0	0	0	0	0	0	0	0	0	0	–	10
14	0	0	0	0	0	0	0	0	0	0	0	0	0	–

3 Results and discussions

CPLEX 12.2 solver with AMPL language is used to solve the mathematical programming model for the given example networks. The network A uses 30 numbers of working path. To compare the effect of increasing backup paths on spare capacity utilized by network A, we provide increased sets of back path which are 60 (2 × 30), 90 (3 × 30) and 120 (4 × 30). If 60 number of backup path are assigned to network A, then it is represented by |T| = 60. Here the backup paths are generated according to the shortest path between source node and destination node of failed link. These shortest backup paths or restoration paths cover minimum link of networks. An optimal solution of joint capacity planning model for network A with |T| = 60 is shown in Figure 4. For network A, 110 units of spare capacity are required to protect 130 units of working capacity. The working and spare capacity utilized by each link of problem network A has been analyzed and shown in Figure 4. For |T| = 60 the total capacity (working capacity plus spare capacity) utilized by network A are 240 units.

Figure 4:

Solution of joint capacity planning model for network A with |T| = 60.

(Working capacity = 130 units and spare capacity = 110 units).

On increasing the backup path from |T| = 60 to |T| = 90, 98 units of spare capacity are required to protect 130 units of working capacity. An optimal solution of joint capacity planning model for network A with |T| = 90 is shown in Figure 5. The total capacities utilized by network A are 228 units. On further increasing the backup path from |T| = 90 to |T| = 120, 95 units of spare capacity are required to protect 130 units of working capacity. An optimal solution of joint capacity planning model for network A with |T| = 120 is illustrated in Figure 6. For |T| = 120, the total capacity utilized by network A are 225 units.

Figure 5:

Solution of joint capacity planning model for network A with |T| = 90.

(Working capacity = 130 units and spare capacity = 98 units).

Figure 6:

Solution of joint capacity planning model for network A with |T| = 120.

(Working capacity = 130 units and spare capacity = 95 units)

The network B uses 56 numbers of working path. To compare the effect of increasing backup paths on spare capacity utilized by network B, we provide increased sets of back path which are 168 (3 × 56), 224 (4 × 56) and 280 (5 × 56). An optimal solution of joint planning model for network B with |T| = 168 is illustrated in Figure 7. In this case, 390 units of spare capacity are required to protect 300 units of working capacity. For |T| = 168, the total capacity utilized by network B are 690 units. On increasing the backup path to |T| = 224, then 199 units of spare capacity are required to protect 336 units of working capacity. An optimal solution of joint capacity planning model for network B with |T| = 224 is shown in Figure 8. In this case, the total capacities utilized by network B are 535 units. On further increasing the backup path from |T| = 224 to |T| = 280, 190 units of spare capacity are required to protect 345 units of working capacity. An optimal solution of joint capacity planning model for network B with |T| = 280 is illustrated in Figure 9. For |T| = 280, the total capacity utilized by network B are 535 units.

Figure 7:

Solution of joint capacity planning model for network B with |T| = 168.

(Working capacity = 300 units and spare capacity = 390 units).

Figure 8:

Solution of joint capacity planning model for network B with |T| = 224.

(Working capacity = 336 units and spare capacity = 199 units).

Figure 9:

Solution of joint capacity planning model for network B with |T| = 280.

(Working capacity = 345 units and spare capacity = 190 units).

The NSFNET uses 182 numbers of working path. To compare the effect of increasing backup paths on spare capacity utilized by NSFNET, we provide increased sets of back path which are 728(4 × 182), 910 (5 × 182) and 1092 (6 × 182). An optimal solution of joint capacity planning model for NSFNET with |T| = 728 is illustrated in Figure 10. In this case, 840 units of spare capacity are required to protect 730 units of working capacity. For |T| = 728, total capacity utilized by NSFNET are 1570 units. On increasing the backup path from |T| = 728 to |T| = 910, then 297 units of spare capacity are required to protect 763 units of working capacity. An optimal solution of joint capacity planning model for NSFNET with |T| = 910 is illustrated in Figure 11. For |T| = 910, the total capacity utilized by NSFNET are 1060 units. On further increasing the backup path from |T| = 910 to |T| = 1092, then 276 units of spare capacity are required to protect 769 units of working capacity. An optimal solution of joint capacity planning model for NSFNET with |T| = 1092 is illustrated in Figure 12. For |T| = 1092, the total capacity utilized by NSFNET are 1045 units. The capacity utilized by each link of problem NSFNET has been analyzed.

Figure 10:

Solution of joint capacity planning model for NSFNET with |T| = 728.

(Working capacity = 730 units and spare capacity = 840 units).

Figure 11:

Solution of joint capacity planning model for NSFNET with |T| = 910.

(Working capacity = 763 units and spare capacity = 297 units).

Figure 12:

Solution of joint capacity planning model for NSFNET with |T| = 1092.

(Working capacity = 769 units and spare capacity = 276 units).

The comparison of spare capacity utilized by network A, network B and NSFNET is illustrated in Figure 13. Red bar shows the total spare capacity utilized by network A; green bar shows the total spare capacity utilized by Network B and purple bar shows the total spare capacity utilized by NSFNET. For network A, 180 units of spare capacity are required for 60 backup paths. If total number of backup path is increased to 90, then 120 units of spare capacity are required for the same network. If total number of backup path is further increased to 120, then 110 units of spare capacity are required for network A. In the case of network B, 290 units of spare capacity are required for 168 backup paths. If total number of backup path is increased to 224, then network B utilized the 270 units of spare capacity. If total number of backup path is further increased to 280, then 245 units of spare capacity are required for the same network. For NSFNET, 585 units of spare capacity are required for 728 backup paths. If total number of backup path is increased to 910, then 505 units of spare capacity are required for NSFNET. If total number of backup path is further increased to 1092, then 475 units of spare capacity are required for NSFNET. The comparison of joint capacity (working capacity + spare capacity) utilized by network A, network B and NSFNET is illustrated in Figure 14. Obviously it is found that on increasing the number of backup paths, the spare capacity utilized by the network decreases and on reducing the number of backup path the spare capacity utilized by the network increases. Hence it is found that for minimum spare capacity utilization, maximum backup path should be given.

Figure 13:

Comparison of spare capacity utilized by network A, network B and NSFNET.

Figure 14:

Comparison of joint (working + spare) capacity utilized by network A, network B and NSFNET.

The most important things is that although the backup paths are increased for a particular network but the assigned restoration path covers the shortest path to minimize spare capacity utilization. The advantage of generating shortest backup paths is the reduction in the installation cost. On each link failure, these shortest paths is dynamically discovered and assigned to a failed link. With the help of shortest restoration path, the recovery of packets against the link failure becomes more efficient. It increases the speed of optical networks and makes it suitable for Tb/s class speed. Fast recovery of packet is very necessary requirement to manufacture high speed optical networks.

4 Conclusions

Optical network is widely used in large application such as high-speed supercomputing, real-time medical imaging, scientific visualization and video network conferencing. If a fiber cut occurs it will result in a huge loss of data. Therefore, all service provider companies ensure about the survivability and protection/restoration via analysis of recovery mechanism, backup routers and backup capacity of their network. Minimum spare capacity requirement is very important issue in optical networks to support tb/s high speed. In this paper, the joint capacity allocation model for network A, network B & NSFNET has been thoroughly analyzed and compared. It is found that 13.6% less spare capacity is required for network A, if backup path is increased from 60 to 120 and 51.5% less spare capacity is required for network B, if backup path is increased from 168 to 280. In the case of NSFNET, if backup path is increased from 728 to 1092 then 67.2% less spare capacity is required. The spare capacity required for each link of network A, network B & NSFNET has been also analyzed.

For joint capacity planning model, the paper proves that network requires minimum spare capacity on increasing the backup path. Therefore for a survivable network if maximum numbers of backup paths are provided, the recovery of packet against the link failure is fast. The result obtained proves that the proposed model provides fast restoration speed for optical network. The optimized spare capacity model will also help in reducing the network cost and increasing the speed of the optical network considerably, since the network utilizes minimum capacity. The model presented in this manuscript is for restoration at the physical layer. In future if this model is applied at the ATM layer there can be a significant improvement.

Corresponding author: Ashutosh Kumar Singh, Department of Electronics & Communication Engg., F.E.T., M.J.P. Rohilkhand University, Bareilly, India, E-mail: aksinghelectronics@gmail.com

Funding source: World Bank assisted Technical Education Quality Improvement Programme 3 (TEQIP-III)

Acknowledgments

This work was supported in part by the World Bank assisted Technical Education Quality Improvement Programme 3 (TEQIP-III) under Seed grant for the Minor Research Project dated 15 June 2019.

Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.
Research funding: World Bank assisted Technical Education Quality Improvement Programme 3 (TEQIP-III).
Conflict of interest statement: The authors declare no conflicts of interest regarding this article.

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Received: 2020-09-11

Accepted: 2020-09-23

Published Online: 2020-10-23

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/joc-2020-0228

Keywords for this article

capacity planning; optimization methods; optical network; survivability

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BY 4.0

Node	1	2	3	4	5	6
1	–	10	0	10	10	0
2	0	–	10	10	10	10
3	0	0	–	0	10	10
4	0	0	0	–	10	0
5	0	0	0	0	–	10
6	0	0	0	0	0	–

Node	1	2	3	4	5	6	7	8
1	–	10	10	10	0	10	0	10
2	0	–	10	10	10	0	10	0
3	0	0	–	10	10	10	0	0
4	0	0	0	–	10	10	10	0
5	0	0	0	0	–	10	10	10
6	0	0	0	0	0	–	10	10
7	0	0	0	0	0	0	–	10
8	0	0	0	0	0	0	0	–

Node	1	2	3	4	5	6	7	8	9	10	11	12	13	14
1	–	0	0	0	10	10	0	10	10	10	0	0	10	10
2	0	–	10	10	0	10	0	0	10	10	10	10	10	0
3	0	0	–	10	0	10	0	0	10	10	10	10	10	0
4	0	0	0	–	10	10	10	0	0	10	10	0	0	0
5	0	0	0	0	–	10	10	0	10	0	0	0	10	10
6	0	0	0	0	0	–	10	10	0	0	0	0	10	10
7	0	0	0	0	0	0	–	10	10	0	0	10	10	10
8	0	0	0	0	0	0	0	–	10	10	0	10	10	0
9	0	0	0	0	0	0	0	0	–	10	0	10	0	0
10	0	0	0	0	0	0	0	0	0	–	10	10	10	0
11	0	0	0	0	0	0	0	0	0	0	–	10	10	10
12	0	0	0	0	0	0	0	0	0	0	0	–	10	10
13	0	0	0	0	0	0	0	0	0	0	0	0	–	10
14	0	0	0	0	0	0	0	0	0	0	0	0	0	–

Node	1	2	3	4	5	6
1	–	10	0	10	10	0
2	0	–	10	10	10	10
3	0	0	–	0	10	10
4	0	0	0	–	10	0
5	0	0	0	0	–	10
6	0	0	0	0	0	–

Node	1	2	3	4	5	6	7	8
1	–	10	10	10	0	10	0	10
2	0	–	10	10	10	0	10	0
3	0	0	–	10	10	10	0	0
4	0	0	0	–	10	10	10	0
5	0	0	0	0	–	10	10	10
6	0	0	0	0	0	–	10	10
7	0	0	0	0	0	0	–	10
8	0	0	0	0	0	0	0	–

Node	1	2	3	4	5	6	7	8	9	10	11	12	13	14
1	–	0	0	0	10	10	0	10	10	10	0	0	10	10
2	0	–	10	10	0	10	0	0	10	10	10	10	10	0
3	0	0	–	10	0	10	0	0	10	10	10	10	10	0
4	0	0	0	–	10	10	10	0	0	10	10	0	0	0
5	0	0	0	0	–	10	10	0	10	0	0	0	10	10
6	0	0	0	0	0	–	10	10	0	0	0	0	10	10
7	0	0	0	0	0	0	–	10	10	0	0	10	10	10
8	0	0	0	0	0	0	0	–	10	10	0	10	10	0
9	0	0	0	0	0	0	0	0	–	10	0	10	0	0
10	0	0	0	0	0	0	0	0	0	–	10	10	10	0
11	0	0	0	0	0	0	0	0	0	0	–	10	10	10
12	0	0	0	0	0	0	0	0	0	0	0	–	10	10
13	0	0	0	0	0	0	0	0	0	0	0	0	–	10
14	0	0	0	0	0	0	0	0	0	0	0	0	0	–

Node	1	2	3	4	5	6
1	–	10	0	10	10	0
2	0	–	10	10	10	10
3	0	0	–	0	10	10
4	0	0	0	–	10	0
5	0	0	0	0	–	10
6	0	0	0	0	0	–

Node	1	2	3	4	5	6	7	8
1	–	10	10	10	0	10	0	10
2	0	–	10	10	10	0	10	0
3	0	0	–	10	10	10	0	0
4	0	0	0	–	10	10	10	0
5	0	0	0	0	–	10	10	10
6	0	0	0	0	0	–	10	10
7	0	0	0	0	0	0	–	10
8	0	0	0	0	0	0	0	–

Node	1	2	3	4	5	6	7	8	9	10	11	12	13	14
1	–	0	0	0	10	10	0	10	10	10	0	0	10	10
2	0	–	10	10	0	10	0	0	10	10	10	10	10	0
3	0	0	–	10	0	10	0	0	10	10	10	10	10	0
4	0	0	0	–	10	10	10	0	0	10	10	0	0	0
5	0	0	0	0	–	10	10	0	10	0	0	0	10	10
6	0	0	0	0	0	–	10	10	0	0	0	0	10	10
7	0	0	0	0	0	0	–	10	10	0	0	10	10	10
8	0	0	0	0	0	0	0	–	10	10	0	10	10	0
9	0	0	0	0	0	0	0	0	–	10	0	10	0	0
10	0	0	0	0	0	0	0	0	0	–	10	10	10	0
11	0	0	0	0	0	0	0	0	0	0	–	10	10	10
12	0	0	0	0	0	0	0	0	0	0	0	–	10	10
13	0	0	0	0	0	0	0	0	0	0	0	0	–	10
14	0	0	0	0	0	0	0	0	0	0	0	0	0	–