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Preconfigured k-edge-connected structures (p-kecs) against multiple link failures in optical networks
Optik 138 (2017) 214–222
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Original research article
Preconfigured k-edge-connected structures (p-kecs) against multiple link failures in optical networks Wei Zhang ∗ , Jie Zhang, Xin Li, Guangjun Luo, Yongli Zhao, Wanyi Gu, Shanguo Huang Beijing University of Posts and Telecommunications, State Key Laboratory of Information Photonics and Optical Communications, No. 10 Xi Tu Cheng Road, Beijing, 100876, China
a r t i c l e
i n f o
Article history: Received 18 October 2014 Received in revised form 12 November 2016 Accepted 4 March 2017 Keywords: WDM networks Network survivability Network connectivity Multiple-link failures Preconfigured k-edge-connected structures (p-kecs)
a b s t r a c t We study the capacity redundancy of WDM networks under multiple-link-failure environments and propose a protection scheme based on preconfigured k-edge-connected structures (p-kecs) against multiple link failures in WDM networks. The lower bound on redundancy and the upper bound on efficiency of WDM networks under multiple-linkfailure environments are provided, the reciprocal of lower bound of redundancy is the upper bound on efficiency in full connected network. We theoretically prove that the capacity redundancy of the p-kecs has the same lower bound as that of WDM networks under multiple-link failures, the efficiency of the p-kecs has the same upper bound as that of WDM networks under multiple-link failures, and the reciprocal of lower bound of redundancy of the p-kecs is the upper bound on efficiency of the p-kecs. Numerical results show that p-kecs can achieve the lower bound on redundancy and support protection against simultaneous (k-1)-link failures in static networks. Compared with k-regular and k-edgeconnected structures (k&k), p-kecs can reduce or meet the capacity redundancy and provide protection for handling simultaneous (k-1)-link failures in static and dynamic networks. © 2017 Published by Elsevier GmbH.
1. Introduction With annual growth of Internet traffic (e.g. web-based services, video on demand), optical networks are facing large important traffic protection demand, including dual or multiple failures scenarios. The existing theories on survivable protection are mainly elaborated in [1,2], these studies consider WDM-based wavelength routing optical networks. The author presents a detailed analysis of the resource efficiency and a detailed redundancy comparison of p-cycles with prevalent protection mechanisms, provides the well-known lower bound on redundancy of a link-restorable mesh network against single-link failure [1]. In this regard, the authors provide theoretical proof that 100% protection can be supported with an optimized set of closed cycles of spare capacity, while requiring little or no increase in spare capacity relative to a spanrestorable mesh network, it is proved that the Hamilton cycle is the optimal p-cycle against single-link failure, and the authors derive an upper limit to the number of paths that any (unspecified) pattern or combination of protection structures can provide per spare link consumed against single-link failure [2]. However, in WDM networks, they do not determine the lower bound on redundancy and the upper bound on the efficiency of preconfigured patterns in multiple-link-failure sce-
∗ Corresponding author. E-mail address: [email protected] (W. Zhang). http://dx.doi.org/10.1016/j.ijleo.2017.03.023 0030-4026/© 2017 Published by Elsevier GmbH.
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narios. The authors derive the lower bound on redundancy of link-restorable networks under multiple link failures [3]. The lower bound of redundancy and the upper bound on efficiency of preconfigured protection structures in flexible bandwidth optical networks are provided [4]. In this paper, we provide the lower bound on redundancy and the upper bound on the efficiency of WDM networks using preconfigured patterns under multiple-link-failure scenarios. Previous studies have been proposed to handle multiple failures. One kind of protection schemes is based on p-cycles, which achieves both fast protection speed and high capacity efficiency. For example, static p-cycles and p-cycles have been used to withstand dual sequential failures, where failures could be fixed one by one [5–9]. The important failure scenarios that may be considered in survivable networks design is dual-link-failure where dual-link are simultaneously failed in the network or new failure happens before repairing former occurred link failure. Dedicated-path protection, shared-path protection, and shared-link protection are proposed against simultaneous dual-link failures [10,11]. There are also several investigations on combination of the protection and restoration [12] or on path protection under multiple link failures [13]. However, the previous protection techniques cannot provide an effective tool to handle simultaneous three or more failures in WDM networks. The k-regular and k-edge-connected structures (k&k), the pre-configured polyhedron protection (PCP) and the K-dimensional protection structure (KDPS) based on k-regular and k-edge connected graphs are constructed, these structures support protection against k-1 simultaneous random link failures and sequential random link failures [3,14,15]. Considering that k&k have a strong regularity, some specific link do not exist in the physical topology, and the application of such structures in optical networks has been greatly limited, we apply two theorems of graph theory to construct p-kecs, and theoretically prove that the resource redundancy of the p-kecs has the same lower bound as that of flexible bandwidth optical networks under multiple link failures, the efficiency of the p-kecs has the same upper bound as that of flexible bandwidth optical networks under multiple link failures [4]. The k&k are special cases of p-kecs. The PCP and KDPS are special cases of k&k. In this paper, we theoretically prove that the capacity redundancy of the p-kecs has the same lower bound as that of WDM networks under multiple-link failures, the efficiency of the p-kecs has the same upper bound as that of WDM networks under multiple-link failures. The rest of the paper is organized as follows. In Sec. 2, the lower bound on redundancy and the upper bound on efficiency of WDM networks under multiple-link-failure environments are provided. In Sec. 3, we propose a construction algorithm for p-kecs design and analyse the capacity redundancy and the efficiency of the p-kecs under multiple-link failures in WDM networks, the integer linear programming (ILP) model is given for assigning capacities on p-kecs and only spare capacity is optimized. The numeric results in static and dynamic networks are introduced in Sec. 4. In Sec. 5, we conclude the paper by summarizing our contribution and discussing our future work in this area.
2. Theoretical analysis To normalize capacity value, redundancy is defined as the ratio of the spare (protection) capacity to the working capacity [2,16,17]. In this section, we derive the lower bound on redundancy and the upper bound on efficiency of WDM network under the condition of multiple-link failures. The redundancy is defined as the ratio of the spare capacity to the working capacity. The efficiency is defined as the upper limit for the number of capacities that protection structures can provide per spare link consumed. A given physical topology is modeled with graph G = (V, E), where V denote the set of optical switch nodes, where E indicate the set of bi-directional fiber between nodes in V, each link associated with working capacity wij and spare capacity sij with ∀ ij ∈ E. In worst-case scenarios, all failed links where k-1 denotes the number of failed links are incident to one node, it is necessary that the surviving links have enough spare capacities to carry the interrupted working capacities, we assume every link have same total capacities.
2.1. The lower bound on redundancy of WDM networks in multiple-link-failure environments The lower bound on redundancy of WDM networks in multiple-link-failure environments can be derived as follows. Let WSj denote the set of working capacities and PSj denote the set of spare capacities, ∀j ∈ E. Further, Sj denotes the set of the total capacities in link j, with s representing the number of the total capacities in each link. By similar proof reasoning in [4], we deduce the lower bound on the redundancy under N link failures in WDM networks as: d=
r≥
2|E| ≤ V −1 |V | N d−N
=
k−1 d − (k − 1)
(1)
≥
k−1 |V | − 1 − (k − 1)
(2)
where d denotes the average node degree, N = k-1 denotes the number of failed links, d = |V |-1 can be seen as the node degree in full connected network.
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Fig. 1. Simultaneous random triple link failures model.
2.2. The upper bound on efficiency of WDM networks using preconfigured pattern under multiple link failures Any potential capacities available in a spare capacity pattern can be used to handle failures. Let us consider a preconfigured pattern of an arbitrary nature that covers M nodes with M ≤ |V |. By similar proof reasoning in [4], we deduce the upper bound on the efficiency of WDM networks using preconfigured patterns under N link failures as: E bound ≤
M−1−N |V | − 1 − (k − 1) M − 1 − (k − 1) ≤ = N k−1 k−1
(3)
where N = k-1 denotes the number of failed links, the reciprocal of lower bound of redundancy is the upper bound on efficiency in full connected network. 3. Preconfigured k-edge-connected structures (p-kecs) The p-kecs are graphs of original optical physical topologies. The p-kecs must contain k link disjoint paths between any two distinct vertices, they can handle (k-1)-link failures. It has not a set of fewer than k links whose removal disconnects pkecs. When k ≥ 3, the p-kecs are essentially three-dimensional polyhedrons. The proposed p-kecs are based on the following Theorems. Theorem 2.1 ((Menger Theorem)). (1) Let G = (V, E) be a graph and {A, B}⊆V. Then the minimum number of edges separating A from B in G is equal to the maximum number of edge-disjoint A-B paths in G. (2) A graph is k-edge-connected if and only if it contains k edge-disjoint paths between any two vertices. Theorem 2.2. Let G be an undirected connected graph subject to the relational expression: connectivity of G ≤ edge connectivity of G ≤ minimum degree of G. It is worth noting that network traffic is unlikely to be symmetric in both directions between two nodes. It means that the number of working and spare wavelengths is not likely to be the same in both directions. We will consider unidirectional p-kecs in this paper without loss of generality and assume every link have same total capacities. For example, under simultaneous random triple link failures (AB, AD, AG) as presented in Fig. 1, all the previous structures could not effectively protect the simultaneous multiple failures. We construct p-4ecs presented in Fig. 2. The p-4ecs is essentially a three-dimensional polyhedron, which is composed of links on the structure (AB, AC, AD, AG, BC, BE, BH, CD, CF, DH, DJ, EF, EH, EK, FG, FI, GI, GJ, HK, IJ, IK, JK) and straddling links (CE, DG, HI). It is assumed that one unit of working capacity and three units of spare capacities are assigned for each link on the structure. Meanwhile, four units of working capacities and no spare capacity are assigned for each straddling link. When one unit of working capacity on AB, one unit of working capacity on AD and one unit of working capacity on AG simultaneously fail, the one unit of working capacity on AB switch to the one unit of spare capacity
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Fig. 2. Preconfigured 4-edge-connected structure.
on ACB, the one unit of working capacity on AD switch to the one unit of spare capacity on ACD, the one unit of working capacity on AG switch to the one unit of spare capacity on ACFG. With sufficient capacities, this structure can achieve 100% protection against simultaneous random triple link failures. Straddling links reduce the network redundancy while requiring little or no increase in spare capacity relative to the total capacity consumed in the protection process. Furthermore, the recovery time of on-structure links can be as fast as that for ring-based networks because there is no real-time work at any nodes other than the end nodes of failed links. The topological score (TS) and the a priori efficiency (AE) of a cycle are the preselection metrics of a cycle and are decided by the topology only. The efficiency ratio (ER) of a cycle is determined by both the topology and the working capacity that are actually protected by the cycle, hence it represents the posteriori efficiency of the cycle [18–20]. In this section we define the TS, the AE and the ER of p-kecs. The TS is defined as the total amount of working capacity p-kecs has the potential to protect. L is the set of links in the network, xij is the number of protection relationships available to link i from p-kecs j, xij = 1 if link i is on p-kecs j, xij = k if link i straddles p-kecs j and xij = 0 otherwise. TS (j) =
xij
(4)
∀i ∈ L
We define the AE of p-kecs as the total amount of working capacity p-kecs has the potential to protect divided by the total cost of p-kecs, Ci is the cost of a unit of capacity on link i.
AE (j) =
xij
∀i ∈ L
(5) Ci
(∀i ∈ L|xij =1) If all Ci = 1 the denominator is the raw hop-count of p-kecs, and this makes p-kecs’ AE (j) non-cost-weighted. In this case, the numerator represents the number of working link-hops that are protected, and the denominator is the total number of spare links used to do so. Equivalently, with all Ci = 1, we can express a priori efficiency as AE (j) =
|Lp,j | + k × |Ls,j | |Lp,j |
where Lp,j is the set of on p-kecs links and Ls,j is the set of straddling links in p-kecs j.
(6)
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Table 1 Notations used in the following algorithm. Symbol
Meaning
G(V, E) V(G) E(G) H k
Network topology Vertex set of G Edge set of G Hamiltonian cycle Edge connectivity of G(V, E) v∈G e∈G
v e
Table 2 Preconfigured k-edge-connected Structures (p-kecs) Construction Algorithm. Input: Network topology G (V, E) Output: Preconfigured k-edge-connected Structures (p-kecs) 1: for each e ∈ G (V(G), E(G)) do 2: m = min{Ford-Fulkerson(v, e)} 3: end for 4: if m < k then return (unsuccessful construction) 5: else find a Hamiltonian cycle of G (V, E) 6: Breadth First Search (G) 7: marked k nodes adjacent to V (G) 8: marked k edges incident to E (G) 9: all the marked nodes and edges construct k-edge-connected structures 10: return (successful construction) 11: end if
The ER of p-kecs is defined as the ratio of the number of working capacity that are actually protected by p-kecs to the number of spare capacity of p-kecs, yij = k-1 if link i is on p-kecs j and yij = 0 otherwise, wi is a unit of working capacity on link i at the present time, si is a unit of spare capacity on link i at the present time.
ER (j) =
xij wi
∀i ∈ L
(7) yij si
∀i ∈ L
We note that the ER of p-kecs is dependent not just on the number of links on p-kecs and straddling links, but also on the working capacities of those links and the spare capacities of links on p-kecs. This new quantity suggests not only a guess of a p-kecs’ ability to protect hypothetical working capacity, but also an indication of a p-kecs’ actual suitability in a specific working capacity state.
3.1. Design algorithm for constructing preconfigured k-edge-connected structures (p-kecs) By examining the relationships among the edge connectivity of the physical topology, minimum degree of the physical topology and number of link failures, we propose that p-kecs can resist k-1 random both sequential failures and simultaneous failures, while providing optimal protection of capacity efficiency. As shown in Tables 1 and 2, the algorithm to construct p-kecs is as follows: In the Preconfigured k-edge-connected Structures (p-kecs) Construction Algorithm, we use the Ford–Fulkerson algorithm to obtain the edge connectivity of the physical topology and use the breadth-first search algorithm to mark all the links and nodes. In the following discussion, one unit of capacity is defined as one wavelength throughout the paper. The physical topology of the network is COST239 network in Fig. 3(a). A Hamiltonian cycle is composed of red line as shown in Fig. 3(b). Preconfigured 3-edge-connected structure is composed of red line (links on the structure) and black line (straddling links) as shown in Fig. 3(c), it is assumed that one working wavelength and two spare wavelengths are assigned for each link on the structure, three working wavelengths and no spare wavelength are assigned for each straddling link in networks, it is against simultaneous random dual link failures. Preconfigured 4-edge-connected structure is composed of red line (link on the structure) and black line (straddling link) as shown in Fig. 3(d), it is assumed that one working wavelength and three spare wavelengths are assigned for each link on the structure, four working wavelengths and no spare wavelength are assigned for each straddling link in networks, it is against simultaneous random triple link failures.
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Fig. 3. Construction of k-edge-connected protection structure.
3.2. The lower bound on redundancy and the upper bound on efficiency of p-kecs in multiple-link-failure environments By similar proof reasoning in [4], we deduce the lower bound on the redundancy of p-kecs under N link failures in WDM networks as: N×(N+1)×M 2
r = lim
M→M ∗
P
N
=
d−N
=
k−1 d − (k − 1)
≥
k−1 |V | − 1 − (k − 1)
(8)
we obtain the upper bound on efficiency of p-kecs under N link failures in WDM networks as: E bound =
|V | − 1 − (k − 1) d−N d − (k − 1) ≤ = N k−1 k−1
(9)
where d denotes the average node degree, N = k-1 denotes the number of failed links, d = |V |-1 can be seen as the node degree in full connected network. The reciprocal of lower bound of redundancy is the upper bound on efficiency. We note that the resource redundancy of the p-kecs has the same lower bound as that of WDM networks under multiple link failures. This Equation suggests that in optimized preconfigured k-edge-connected-based networks, p-kecs can support 100% protection while requiring no increase in spare capacity relative to WDM networks. The failure number N = 1 represents the singlelink-failure scenario and the preconfigured pattern is a Hamilton cycle. Further, if the failure number N = 2 and node number M = 8, then the preconfigured pattern is a cube. 3.3. Integer linear programming model for assigning capacities on preconfigured k-edge-connected structures (p-kecs) The physical topology of optical mesh networks is described by the set of links E, indexed by i for failed links and by j for non-failed links. A set of p-kecs M is pre-computed, including the spare capacity matrix jm indicating on which links j each candidate p-kecs m lies. And im indicates how many units of working capacity on link i can be protected by a single p-kecs m (1 for links on p-kecs, k for straddling links, 0 otherwise). For a link j ∈ E, the cost of one unit of capacity is denoted by cj . The design solution is described by variables nm for the number of unit copies of p-kecs selected on p-kecs m, and by auxiliary variables sj for the number of units of spare capacity on link j. We assume the working capacities wj are given input parameters.
cj × sj
j∈E
wi ≤
(10)
im × nm , ∀i ∈ E
(11)
jm × nm , ∀j ∈ E
(12)
m∈M
sj =
m∈M
nm ≥ 0,∀m ∈ M
(13)
The objective function (10) minimizes the total spare capacity used to form p-kecs, constraints (11) and (13) ensures that all the working capacity of every link is protected for 100% protection for simultaneous random multiple failures, constraints (12) and (13) provides sufficient spare capacity on every link to form p-kecs.
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Table 3 The TS, AE and ER of preconfigured 3-edge-connected structure and preconfigured 4-edge-connected structure.
Redundancy
preconfigured 3-edge −connected structure preconfigured 4-edge −connected structure
2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0
TS
AE
ER
41 34
41/17 17/11
41/34 17/33
lower bound p-kecs k&k
1
2 Number of failures
3
Fig. 4. Comparison of the network redundancy of the lower bound, p-kecs, k&k in static networks.
Fig. 5. Comparison of the network redundancy of the lower bound, preconfigured 3-edge-connected structures and 3-regular and 3-edge-connected structures against simultaneous random dual link failures in dynamic networks.
4. Results and analysis In Table 3, the TS, AE and ER of preconfigured 4-edge-connected structure are all less than that of preconfigured 3-edgeconnected structure, these data accord with the fact that, in the same physical topology and under the identical capacity allocation strategy, to protect more work capacities require more spare capacities. As shown in Fig. 4, we compare the numeric results of p-kecs with that of the lower bound and k&k in static networks. Both of the structures are optimal protection structures against single link failure, thus, the redundancy can reach the lower bound (11/39) under single link failure. Both of the structures are not optimal protection structures against dual link failures, thus, the redundancy cannot reach the lower bound (11/14), while there is more straddling links in preconfigured 3-edgeconnected structure, the redundancy of p-kecs (34/41) is decreased than that of k&k (12/13). Because both of the structures are optimal protection structures against triple link failures, the redundancy can reach the lower bound (33/17). The numeric results in dynamic networks are introduced. The simulation topology is COST239. We assume that there are 80 wavelengths in each link. Wavelength consistency and continuity constraint are considered. And the faults are chosen randomly among all the links. As shown in Fig. 5, under dual link failures in dynamic networks, with increasing traffic load, the redundancy of preconfigured 3-edge-connected structure is stable and close to lower bound. Because there are fewer straddling links, the 3-regular and 3-edge-connected protection structure have higher redundancy than preconfigured 3-edge-connected structure. As shown in Fig. 6, under triple link failures in dynamic networks, with increasing traffic load, the redundancy of preconfigured 4-edge-connected structure is also stable and close to lower bound. Because the preconfigured 4-edge-connected structure is as same as the 4-regular and 4-edge-connected protection structure, they have the same redundancy. Thus, preconfigured protection structures with a large number of straddling links relative to their size have the highest potential efficiency as p-kecs. The idea behind this heuristic is to identify those p-kecs that can actually protect as many working capacities as possible, and hence to reduce the total spare capacities.
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Fig. 6. Comparison of the network redundancy of the lower bound, preconfigured 4-edge-connected structures and 4-regular and 4-edge-connected structures against simultaneous random triple link failures in dynamic networks.
5. Conclusions The lower bound on redundancy and the upper bound on efficiency of WDM networks in multiple-link-failure environments are provided. We define the TS, the AE and the ER of p-kecs, these data suggest to protect more work capacities require more spare capacities in the same physical topology and under the identical capacity allocation strategy. We design algorithm for building p-kecs and give ILP model, and theoretically prove that the capacity redundancy of the p-kecs has the same lower bound as that of optical networks under multiple-link-failure, the efficiency of the p-kecs has the same upper bound as that of optical networks under multiple-link-failure. Numerical results show that p-kecs can achieve or approximate the lower bound on redundancy and support protection against (k-1)-link-failure in networks. Compared with k-regular and k-edge-connected structures (k&k), p-kecs can reduce or meet the capacity redundancy and support protection against (k-1)-link-failure in networks. The lower bound of redundancy in WDM networks and flexible bandwidth optical networks under multiple-link-failure scenarios with homogeneous capacity (wavelength, slot) are the same, and the upper bound on efficiency of preconfigured protection structures in WDM networks and flexible bandwidth optical networks under multiple-link-failure scenarios with homogeneous capacity (wavelength, slot) are the same. Our future work includes two aspects. One is to apply protection scheme considering multiple node failures scenario. The other is to apply protection scheme considering multiple mixed failures scenario. Acknowledgement This work is supported partly by the National Basic Research Program of China (973 Program) (No. 2010CB328204). References [1] D.A. Schupke, Analysis of p-cycle capacity in WDM networks, Photonic Netw. Commun. 9 (8) (2006) 41–51. [2] D. Stamatelakis, W.D. Grover, Theoretical underpinnings for the efficiency of restorable networks using preconfigured cycles (p-cycles), IEEE Trans. Commun. 48 (8) (2000) 1262–1265. [3] S. Huang, B. Guo, X. Li, J. Zhang, Y. Zhao, W. Gu, Pre-configured polyhedron based protection against multi-link failures in optical mesh networks, Opt. Express 22 (3) (2014) 2386–2402. [4] W. Zhang, J. Zhang, X. Li, Theoretical analysis of preconfigured k-edge-connected structures in flexible bandwidth optical networks, Opt. Eng. 55 (6) (2016) 1–10. [5] D.A. Schupke, The tradeoff between the number of deployed p-cycles and the survivability to dual fiber duct failures, Proc IEEE Int. Conf. on Communications (2003) 1428–1432. [6] D.A. Schupke, W.D. Grover, M. Clouqueur, Strategies for enhanced dual failure restorability with static or reconfigurable p-cycle networks, Proc. IEEE Int. Conf. on Communications (2004) 1628–1633. [7] H. Wang, H.T. Mouftah, P-cycles in multi-failure network survivability, Proc The 7th Int. Conf. on Transparent Optical Networks (2005) 381–384. [8] A. Kodian, W.D. Grover, Multiple-quality of protection classes including dual-failure survivable services in p-cycle networks, Proc The 2nd Int. Conf. on Broadband Networks (2005) 231–240. [9] D.S. Mukherjee, C. Assi, A. Agarwal, Alternate strategies for dual failure restoration using p-cycles, Proc. IEEE Int. Conf. on Communications (2006) 2477–2482. [10] S. Ramamurthy, L. Sahasrabuddhe, B. Mukherjee, Survivable WDM mesh networks, J. Lightw. Technol. 21 (4) (2003) 870–883. [11] L. Guo, L. Li, J. Cao, H. Yu, X. Wei, On finding feasible solutions with shared backup resources for surviving double-link failures in path-protected WDM mesh networks, J. Lightw. Technol. 25 (1) (2007) 287–296. [12] L. Ruan, T. Feng, A hybrid protection/restoration scheme for two-link failure in WDM mesh networks, Proc. IEEE GLOBECOM (2010) 1–5. [13] X. Cheng, X. Shao, Y. Wang, Multiple link failure recovery in survivable optical networks, Photonic Netw. Commun. 14 (2) (2007) 159–164. [14] X. Li, S. Huang, J. Zhang, Y. Zhao, W. Gu, K-regular and k-(edge)-connected protection structures in optical transport networks, Proc. Nat. Fiber Opt. Eng. Conf. /Opt. Fiber Commun. Conf. Expo. (OFC/NFOEC) (2013) 1–3. [15] Y. Zhao, X. Li, J. Zhang, S. Huang, W. Gu, K-dimensional protection structure (KDPS) for multi-link failure in data center optical networks, Optik 125 (19) (2014) 5490–5493. [16] S. Aidarous, T. Plevyak, Distributed restoration of the transport network, in: Telecommunications Network Management into the 21 st Century: Techniques, Standards, Technologies, and Applications, IEEE Press, Piscataway, NJ, 1994, pp. 337–417.
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Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Original research article
Preconfigured k-edge-connected structures (p-kecs) against multiple link failures in optical networks Wei Zhang ∗ , Jie Zhang, Xin Li, Guangjun Luo, Yongli Zhao, Wanyi Gu, Shanguo Huang Beijing University of Posts and Telecommunications, State Key Laboratory of Information Photonics and Optical Communications, No. 10 Xi Tu Cheng Road, Beijing, 100876, China
a r t i c l e
i n f o
Article history: Received 18 October 2014 Received in revised form 12 November 2016 Accepted 4 March 2017 Keywords: WDM networks Network survivability Network connectivity Multiple-link failures Preconfigured k-edge-connected structures (p-kecs)
a b s t r a c t We study the capacity redundancy of WDM networks under multiple-link-failure environments and propose a protection scheme based on preconfigured k-edge-connected structures (p-kecs) against multiple link failures in WDM networks. The lower bound on redundancy and the upper bound on efficiency of WDM networks under multiple-linkfailure environments are provided, the reciprocal of lower bound of redundancy is the upper bound on efficiency in full connected network. We theoretically prove that the capacity redundancy of the p-kecs has the same lower bound as that of WDM networks under multiple-link failures, the efficiency of the p-kecs has the same upper bound as that of WDM networks under multiple-link failures, and the reciprocal of lower bound of redundancy of the p-kecs is the upper bound on efficiency of the p-kecs. Numerical results show that p-kecs can achieve the lower bound on redundancy and support protection against simultaneous (k-1)-link failures in static networks. Compared with k-regular and k-edgeconnected structures (k&k), p-kecs can reduce or meet the capacity redundancy and provide protection for handling simultaneous (k-1)-link failures in static and dynamic networks. © 2017 Published by Elsevier GmbH.
1. Introduction With annual growth of Internet traffic (e.g. web-based services, video on demand), optical networks are facing large important traffic protection demand, including dual or multiple failures scenarios. The existing theories on survivable protection are mainly elaborated in [1,2], these studies consider WDM-based wavelength routing optical networks. The author presents a detailed analysis of the resource efficiency and a detailed redundancy comparison of p-cycles with prevalent protection mechanisms, provides the well-known lower bound on redundancy of a link-restorable mesh network against single-link failure [1]. In this regard, the authors provide theoretical proof that 100% protection can be supported with an optimized set of closed cycles of spare capacity, while requiring little or no increase in spare capacity relative to a spanrestorable mesh network, it is proved that the Hamilton cycle is the optimal p-cycle against single-link failure, and the authors derive an upper limit to the number of paths that any (unspecified) pattern or combination of protection structures can provide per spare link consumed against single-link failure [2]. However, in WDM networks, they do not determine the lower bound on redundancy and the upper bound on the efficiency of preconfigured patterns in multiple-link-failure sce-
∗ Corresponding author. E-mail address: [email protected] (W. Zhang). http://dx.doi.org/10.1016/j.ijleo.2017.03.023 0030-4026/© 2017 Published by Elsevier GmbH.
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narios. The authors derive the lower bound on redundancy of link-restorable networks under multiple link failures [3]. The lower bound of redundancy and the upper bound on efficiency of preconfigured protection structures in flexible bandwidth optical networks are provided [4]. In this paper, we provide the lower bound on redundancy and the upper bound on the efficiency of WDM networks using preconfigured patterns under multiple-link-failure scenarios. Previous studies have been proposed to handle multiple failures. One kind of protection schemes is based on p-cycles, which achieves both fast protection speed and high capacity efficiency. For example, static p-cycles and p-cycles have been used to withstand dual sequential failures, where failures could be fixed one by one [5–9]. The important failure scenarios that may be considered in survivable networks design is dual-link-failure where dual-link are simultaneously failed in the network or new failure happens before repairing former occurred link failure. Dedicated-path protection, shared-path protection, and shared-link protection are proposed against simultaneous dual-link failures [10,11]. There are also several investigations on combination of the protection and restoration [12] or on path protection under multiple link failures [13]. However, the previous protection techniques cannot provide an effective tool to handle simultaneous three or more failures in WDM networks. The k-regular and k-edge-connected structures (k&k), the pre-configured polyhedron protection (PCP) and the K-dimensional protection structure (KDPS) based on k-regular and k-edge connected graphs are constructed, these structures support protection against k-1 simultaneous random link failures and sequential random link failures [3,14,15]. Considering that k&k have a strong regularity, some specific link do not exist in the physical topology, and the application of such structures in optical networks has been greatly limited, we apply two theorems of graph theory to construct p-kecs, and theoretically prove that the resource redundancy of the p-kecs has the same lower bound as that of flexible bandwidth optical networks under multiple link failures, the efficiency of the p-kecs has the same upper bound as that of flexible bandwidth optical networks under multiple link failures [4]. The k&k are special cases of p-kecs. The PCP and KDPS are special cases of k&k. In this paper, we theoretically prove that the capacity redundancy of the p-kecs has the same lower bound as that of WDM networks under multiple-link failures, the efficiency of the p-kecs has the same upper bound as that of WDM networks under multiple-link failures. The rest of the paper is organized as follows. In Sec. 2, the lower bound on redundancy and the upper bound on efficiency of WDM networks under multiple-link-failure environments are provided. In Sec. 3, we propose a construction algorithm for p-kecs design and analyse the capacity redundancy and the efficiency of the p-kecs under multiple-link failures in WDM networks, the integer linear programming (ILP) model is given for assigning capacities on p-kecs and only spare capacity is optimized. The numeric results in static and dynamic networks are introduced in Sec. 4. In Sec. 5, we conclude the paper by summarizing our contribution and discussing our future work in this area.
2. Theoretical analysis To normalize capacity value, redundancy is defined as the ratio of the spare (protection) capacity to the working capacity [2,16,17]. In this section, we derive the lower bound on redundancy and the upper bound on efficiency of WDM network under the condition of multiple-link failures. The redundancy is defined as the ratio of the spare capacity to the working capacity. The efficiency is defined as the upper limit for the number of capacities that protection structures can provide per spare link consumed. A given physical topology is modeled with graph G = (V, E), where V denote the set of optical switch nodes, where E indicate the set of bi-directional fiber between nodes in V, each link associated with working capacity wij and spare capacity sij with ∀ ij ∈ E. In worst-case scenarios, all failed links where k-1 denotes the number of failed links are incident to one node, it is necessary that the surviving links have enough spare capacities to carry the interrupted working capacities, we assume every link have same total capacities.
2.1. The lower bound on redundancy of WDM networks in multiple-link-failure environments The lower bound on redundancy of WDM networks in multiple-link-failure environments can be derived as follows. Let WSj denote the set of working capacities and PSj denote the set of spare capacities, ∀j ∈ E. Further, Sj denotes the set of the total capacities in link j, with s representing the number of the total capacities in each link. By similar proof reasoning in [4], we deduce the lower bound on the redundancy under N link failures in WDM networks as: d=
r≥
2|E| ≤ V −1 |V | N d−N
=
k−1 d − (k − 1)
(1)
≥
k−1 |V | − 1 − (k − 1)
(2)
where d denotes the average node degree, N = k-1 denotes the number of failed links, d = |V |-1 can be seen as the node degree in full connected network.
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Fig. 1. Simultaneous random triple link failures model.
2.2. The upper bound on efficiency of WDM networks using preconfigured pattern under multiple link failures Any potential capacities available in a spare capacity pattern can be used to handle failures. Let us consider a preconfigured pattern of an arbitrary nature that covers M nodes with M ≤ |V |. By similar proof reasoning in [4], we deduce the upper bound on the efficiency of WDM networks using preconfigured patterns under N link failures as: E bound ≤
M−1−N |V | − 1 − (k − 1) M − 1 − (k − 1) ≤ = N k−1 k−1
(3)
where N = k-1 denotes the number of failed links, the reciprocal of lower bound of redundancy is the upper bound on efficiency in full connected network. 3. Preconfigured k-edge-connected structures (p-kecs) The p-kecs are graphs of original optical physical topologies. The p-kecs must contain k link disjoint paths between any two distinct vertices, they can handle (k-1)-link failures. It has not a set of fewer than k links whose removal disconnects pkecs. When k ≥ 3, the p-kecs are essentially three-dimensional polyhedrons. The proposed p-kecs are based on the following Theorems. Theorem 2.1 ((Menger Theorem)). (1) Let G = (V, E) be a graph and {A, B}⊆V. Then the minimum number of edges separating A from B in G is equal to the maximum number of edge-disjoint A-B paths in G. (2) A graph is k-edge-connected if and only if it contains k edge-disjoint paths between any two vertices. Theorem 2.2. Let G be an undirected connected graph subject to the relational expression: connectivity of G ≤ edge connectivity of G ≤ minimum degree of G. It is worth noting that network traffic is unlikely to be symmetric in both directions between two nodes. It means that the number of working and spare wavelengths is not likely to be the same in both directions. We will consider unidirectional p-kecs in this paper without loss of generality and assume every link have same total capacities. For example, under simultaneous random triple link failures (AB, AD, AG) as presented in Fig. 1, all the previous structures could not effectively protect the simultaneous multiple failures. We construct p-4ecs presented in Fig. 2. The p-4ecs is essentially a three-dimensional polyhedron, which is composed of links on the structure (AB, AC, AD, AG, BC, BE, BH, CD, CF, DH, DJ, EF, EH, EK, FG, FI, GI, GJ, HK, IJ, IK, JK) and straddling links (CE, DG, HI). It is assumed that one unit of working capacity and three units of spare capacities are assigned for each link on the structure. Meanwhile, four units of working capacities and no spare capacity are assigned for each straddling link. When one unit of working capacity on AB, one unit of working capacity on AD and one unit of working capacity on AG simultaneously fail, the one unit of working capacity on AB switch to the one unit of spare capacity
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Fig. 2. Preconfigured 4-edge-connected structure.
on ACB, the one unit of working capacity on AD switch to the one unit of spare capacity on ACD, the one unit of working capacity on AG switch to the one unit of spare capacity on ACFG. With sufficient capacities, this structure can achieve 100% protection against simultaneous random triple link failures. Straddling links reduce the network redundancy while requiring little or no increase in spare capacity relative to the total capacity consumed in the protection process. Furthermore, the recovery time of on-structure links can be as fast as that for ring-based networks because there is no real-time work at any nodes other than the end nodes of failed links. The topological score (TS) and the a priori efficiency (AE) of a cycle are the preselection metrics of a cycle and are decided by the topology only. The efficiency ratio (ER) of a cycle is determined by both the topology and the working capacity that are actually protected by the cycle, hence it represents the posteriori efficiency of the cycle [18–20]. In this section we define the TS, the AE and the ER of p-kecs. The TS is defined as the total amount of working capacity p-kecs has the potential to protect. L is the set of links in the network, xij is the number of protection relationships available to link i from p-kecs j, xij = 1 if link i is on p-kecs j, xij = k if link i straddles p-kecs j and xij = 0 otherwise. TS (j) =
xij
(4)
∀i ∈ L
We define the AE of p-kecs as the total amount of working capacity p-kecs has the potential to protect divided by the total cost of p-kecs, Ci is the cost of a unit of capacity on link i.
AE (j) =
xij
∀i ∈ L
(5) Ci
(∀i ∈ L|xij =1) If all Ci = 1 the denominator is the raw hop-count of p-kecs, and this makes p-kecs’ AE (j) non-cost-weighted. In this case, the numerator represents the number of working link-hops that are protected, and the denominator is the total number of spare links used to do so. Equivalently, with all Ci = 1, we can express a priori efficiency as AE (j) =
|Lp,j | + k × |Ls,j | |Lp,j |
where Lp,j is the set of on p-kecs links and Ls,j is the set of straddling links in p-kecs j.
(6)
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Table 1 Notations used in the following algorithm. Symbol
Meaning
G(V, E) V(G) E(G) H k
Network topology Vertex set of G Edge set of G Hamiltonian cycle Edge connectivity of G(V, E) v∈G e∈G
v e
Table 2 Preconfigured k-edge-connected Structures (p-kecs) Construction Algorithm. Input: Network topology G (V, E) Output: Preconfigured k-edge-connected Structures (p-kecs) 1: for each e ∈ G (V(G), E(G)) do 2: m = min{Ford-Fulkerson(v, e)} 3: end for 4: if m < k then return (unsuccessful construction) 5: else find a Hamiltonian cycle of G (V, E) 6: Breadth First Search (G) 7: marked k nodes adjacent to V (G) 8: marked k edges incident to E (G) 9: all the marked nodes and edges construct k-edge-connected structures 10: return (successful construction) 11: end if
The ER of p-kecs is defined as the ratio of the number of working capacity that are actually protected by p-kecs to the number of spare capacity of p-kecs, yij = k-1 if link i is on p-kecs j and yij = 0 otherwise, wi is a unit of working capacity on link i at the present time, si is a unit of spare capacity on link i at the present time.
ER (j) =
xij wi
∀i ∈ L
(7) yij si
∀i ∈ L
We note that the ER of p-kecs is dependent not just on the number of links on p-kecs and straddling links, but also on the working capacities of those links and the spare capacities of links on p-kecs. This new quantity suggests not only a guess of a p-kecs’ ability to protect hypothetical working capacity, but also an indication of a p-kecs’ actual suitability in a specific working capacity state.
3.1. Design algorithm for constructing preconfigured k-edge-connected structures (p-kecs) By examining the relationships among the edge connectivity of the physical topology, minimum degree of the physical topology and number of link failures, we propose that p-kecs can resist k-1 random both sequential failures and simultaneous failures, while providing optimal protection of capacity efficiency. As shown in Tables 1 and 2, the algorithm to construct p-kecs is as follows: In the Preconfigured k-edge-connected Structures (p-kecs) Construction Algorithm, we use the Ford–Fulkerson algorithm to obtain the edge connectivity of the physical topology and use the breadth-first search algorithm to mark all the links and nodes. In the following discussion, one unit of capacity is defined as one wavelength throughout the paper. The physical topology of the network is COST239 network in Fig. 3(a). A Hamiltonian cycle is composed of red line as shown in Fig. 3(b). Preconfigured 3-edge-connected structure is composed of red line (links on the structure) and black line (straddling links) as shown in Fig. 3(c), it is assumed that one working wavelength and two spare wavelengths are assigned for each link on the structure, three working wavelengths and no spare wavelength are assigned for each straddling link in networks, it is against simultaneous random dual link failures. Preconfigured 4-edge-connected structure is composed of red line (link on the structure) and black line (straddling link) as shown in Fig. 3(d), it is assumed that one working wavelength and three spare wavelengths are assigned for each link on the structure, four working wavelengths and no spare wavelength are assigned for each straddling link in networks, it is against simultaneous random triple link failures.
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Fig. 3. Construction of k-edge-connected protection structure.
3.2. The lower bound on redundancy and the upper bound on efficiency of p-kecs in multiple-link-failure environments By similar proof reasoning in [4], we deduce the lower bound on the redundancy of p-kecs under N link failures in WDM networks as: N×(N+1)×M 2
r = lim
M→M ∗
P
N
=
d−N
=
k−1 d − (k − 1)
≥
k−1 |V | − 1 − (k − 1)
(8)
we obtain the upper bound on efficiency of p-kecs under N link failures in WDM networks as: E bound =
|V | − 1 − (k − 1) d−N d − (k − 1) ≤ = N k−1 k−1
(9)
where d denotes the average node degree, N = k-1 denotes the number of failed links, d = |V |-1 can be seen as the node degree in full connected network. The reciprocal of lower bound of redundancy is the upper bound on efficiency. We note that the resource redundancy of the p-kecs has the same lower bound as that of WDM networks under multiple link failures. This Equation suggests that in optimized preconfigured k-edge-connected-based networks, p-kecs can support 100% protection while requiring no increase in spare capacity relative to WDM networks. The failure number N = 1 represents the singlelink-failure scenario and the preconfigured pattern is a Hamilton cycle. Further, if the failure number N = 2 and node number M = 8, then the preconfigured pattern is a cube. 3.3. Integer linear programming model for assigning capacities on preconfigured k-edge-connected structures (p-kecs) The physical topology of optical mesh networks is described by the set of links E, indexed by i for failed links and by j for non-failed links. A set of p-kecs M is pre-computed, including the spare capacity matrix jm indicating on which links j each candidate p-kecs m lies. And im indicates how many units of working capacity on link i can be protected by a single p-kecs m (1 for links on p-kecs, k for straddling links, 0 otherwise). For a link j ∈ E, the cost of one unit of capacity is denoted by cj . The design solution is described by variables nm for the number of unit copies of p-kecs selected on p-kecs m, and by auxiliary variables sj for the number of units of spare capacity on link j. We assume the working capacities wj are given input parameters.
cj × sj
j∈E
wi ≤
(10)
im × nm , ∀i ∈ E
(11)
jm × nm , ∀j ∈ E
(12)
m∈M
sj =
m∈M
nm ≥ 0,∀m ∈ M
(13)
The objective function (10) minimizes the total spare capacity used to form p-kecs, constraints (11) and (13) ensures that all the working capacity of every link is protected for 100% protection for simultaneous random multiple failures, constraints (12) and (13) provides sufficient spare capacity on every link to form p-kecs.
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Table 3 The TS, AE and ER of preconfigured 3-edge-connected structure and preconfigured 4-edge-connected structure.
Redundancy
preconfigured 3-edge −connected structure preconfigured 4-edge −connected structure
2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0
TS
AE
ER
41 34
41/17 17/11
41/34 17/33
lower bound p-kecs k&k
1
2 Number of failures
3
Fig. 4. Comparison of the network redundancy of the lower bound, p-kecs, k&k in static networks.
Fig. 5. Comparison of the network redundancy of the lower bound, preconfigured 3-edge-connected structures and 3-regular and 3-edge-connected structures against simultaneous random dual link failures in dynamic networks.
4. Results and analysis In Table 3, the TS, AE and ER of preconfigured 4-edge-connected structure are all less than that of preconfigured 3-edgeconnected structure, these data accord with the fact that, in the same physical topology and under the identical capacity allocation strategy, to protect more work capacities require more spare capacities. As shown in Fig. 4, we compare the numeric results of p-kecs with that of the lower bound and k&k in static networks. Both of the structures are optimal protection structures against single link failure, thus, the redundancy can reach the lower bound (11/39) under single link failure. Both of the structures are not optimal protection structures against dual link failures, thus, the redundancy cannot reach the lower bound (11/14), while there is more straddling links in preconfigured 3-edgeconnected structure, the redundancy of p-kecs (34/41) is decreased than that of k&k (12/13). Because both of the structures are optimal protection structures against triple link failures, the redundancy can reach the lower bound (33/17). The numeric results in dynamic networks are introduced. The simulation topology is COST239. We assume that there are 80 wavelengths in each link. Wavelength consistency and continuity constraint are considered. And the faults are chosen randomly among all the links. As shown in Fig. 5, under dual link failures in dynamic networks, with increasing traffic load, the redundancy of preconfigured 3-edge-connected structure is stable and close to lower bound. Because there are fewer straddling links, the 3-regular and 3-edge-connected protection structure have higher redundancy than preconfigured 3-edge-connected structure. As shown in Fig. 6, under triple link failures in dynamic networks, with increasing traffic load, the redundancy of preconfigured 4-edge-connected structure is also stable and close to lower bound. Because the preconfigured 4-edge-connected structure is as same as the 4-regular and 4-edge-connected protection structure, they have the same redundancy. Thus, preconfigured protection structures with a large number of straddling links relative to their size have the highest potential efficiency as p-kecs. The idea behind this heuristic is to identify those p-kecs that can actually protect as many working capacities as possible, and hence to reduce the total spare capacities.
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Fig. 6. Comparison of the network redundancy of the lower bound, preconfigured 4-edge-connected structures and 4-regular and 4-edge-connected structures against simultaneous random triple link failures in dynamic networks.
5. Conclusions The lower bound on redundancy and the upper bound on efficiency of WDM networks in multiple-link-failure environments are provided. We define the TS, the AE and the ER of p-kecs, these data suggest to protect more work capacities require more spare capacities in the same physical topology and under the identical capacity allocation strategy. We design algorithm for building p-kecs and give ILP model, and theoretically prove that the capacity redundancy of the p-kecs has the same lower bound as that of optical networks under multiple-link-failure, the efficiency of the p-kecs has the same upper bound as that of optical networks under multiple-link-failure. Numerical results show that p-kecs can achieve or approximate the lower bound on redundancy and support protection against (k-1)-link-failure in networks. Compared with k-regular and k-edge-connected structures (k&k), p-kecs can reduce or meet the capacity redundancy and support protection against (k-1)-link-failure in networks. The lower bound of redundancy in WDM networks and flexible bandwidth optical networks under multiple-link-failure scenarios with homogeneous capacity (wavelength, slot) are the same, and the upper bound on efficiency of preconfigured protection structures in WDM networks and flexible bandwidth optical networks under multiple-link-failure scenarios with homogeneous capacity (wavelength, slot) are the same. Our future work includes two aspects. One is to apply protection scheme considering multiple node failures scenario. The other is to apply protection scheme considering multiple mixed failures scenario. Acknowledgement This work is supported partly by the National Basic Research Program of China (973 Program) (No. 2010CB328204). References [1] D.A. Schupke, Analysis of p-cycle capacity in WDM networks, Photonic Netw. Commun. 9 (8) (2006) 41–51. [2] D. Stamatelakis, W.D. Grover, Theoretical underpinnings for the efficiency of restorable networks using preconfigured cycles (p-cycles), IEEE Trans. Commun. 48 (8) (2000) 1262–1265. [3] S. Huang, B. Guo, X. Li, J. Zhang, Y. Zhao, W. Gu, Pre-configured polyhedron based protection against multi-link failures in optical mesh networks, Opt. Express 22 (3) (2014) 2386–2402. [4] W. Zhang, J. Zhang, X. Li, Theoretical analysis of preconfigured k-edge-connected structures in flexible bandwidth optical networks, Opt. Eng. 55 (6) (2016) 1–10. [5] D.A. Schupke, The tradeoff between the number of deployed p-cycles and the survivability to dual fiber duct failures, Proc IEEE Int. Conf. on Communications (2003) 1428–1432. [6] D.A. Schupke, W.D. Grover, M. Clouqueur, Strategies for enhanced dual failure restorability with static or reconfigurable p-cycle networks, Proc. IEEE Int. Conf. on Communications (2004) 1628–1633. [7] H. Wang, H.T. Mouftah, P-cycles in multi-failure network survivability, Proc The 7th Int. Conf. on Transparent Optical Networks (2005) 381–384. [8] A. Kodian, W.D. Grover, Multiple-quality of protection classes including dual-failure survivable services in p-cycle networks, Proc The 2nd Int. Conf. on Broadband Networks (2005) 231–240. [9] D.S. Mukherjee, C. Assi, A. Agarwal, Alternate strategies for dual failure restoration using p-cycles, Proc. IEEE Int. Conf. on Communications (2006) 2477–2482. [10] S. Ramamurthy, L. Sahasrabuddhe, B. Mukherjee, Survivable WDM mesh networks, J. Lightw. Technol. 21 (4) (2003) 870–883. [11] L. Guo, L. Li, J. Cao, H. Yu, X. Wei, On finding feasible solutions with shared backup resources for surviving double-link failures in path-protected WDM mesh networks, J. Lightw. Technol. 25 (1) (2007) 287–296. [12] L. Ruan, T. Feng, A hybrid protection/restoration scheme for two-link failure in WDM mesh networks, Proc. IEEE GLOBECOM (2010) 1–5. [13] X. Cheng, X. Shao, Y. Wang, Multiple link failure recovery in survivable optical networks, Photonic Netw. Commun. 14 (2) (2007) 159–164. [14] X. Li, S. Huang, J. Zhang, Y. Zhao, W. Gu, K-regular and k-(edge)-connected protection structures in optical transport networks, Proc. Nat. Fiber Opt. Eng. Conf. /Opt. Fiber Commun. Conf. Expo. (OFC/NFOEC) (2013) 1–3. [15] Y. Zhao, X. Li, J. Zhang, S. Huang, W. Gu, K-dimensional protection structure (KDPS) for multi-link failure in data center optical networks, Optik 125 (19) (2014) 5490–5493. [16] S. Aidarous, T. Plevyak, Distributed restoration of the transport network, in: Telecommunications Network Management into the 21 st Century: Techniques, Standards, Technologies, and Applications, IEEE Press, Piscataway, NJ, 1994, pp. 337–417.
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