Shared Backup Path Protection in Elastic Optical Networks: Modeling and Optimization Krzysztof Walkowiak, Member, IEEE and Mirosław Klinkowski
Abstract— Elastic Optical Network (EON) architectures are considered as a very promising solution for both huge bandwidth and flexible connection provisioning in next generation optical networks. In EON, a basic problem in network design and operation is the problem of Routing and Spectrum Allocation (RSA). In this article, we focus on offline RSA in a survivable EON scenario with shared backup path protection (SBPP). We formulate RSA/SBPP as an Integer Linear Programming (ILP) problem. Since RSA is a difficult problem itself, we propose several heuristic algorithms including both new proposals called Adaptive Frequency Assignment with Shared Backup Path Protection (AFA/SBPP) and Most Subcarriers and Average Longest Path First (MSALPF) as well as existing RSA methods adapted to the SBPP scenario. We investigate the efficiency of all algorithms for a set of network scenarios and we show that the proposed new algorithms outperform other reference algorithms. Moreover, numerical experiments show that the shared backup path protection approach enables reduction of the spectrum usage up to 28% comparing to the dedicated path protection approach without sharing of backup capacity. However, the difference between these scenarios strongly depends on the network topology and other parameters.
I. INTRODUCTION
T
HE evolution of optical communication networks leads toward elastic optical networks (EONs) in which advanced single-carrier modulation formats (such as m-PSK, m-QAM) and multi-carrier modulation techniques (such as O-OFDM) are applied for adaptive and mixed-line-rate transmission and where spectrum resources are allocated within flexible frequency grids [1]. These components will allow EON to utilize the spectrum more efficiently and will support elastic and on-demand bandwidth provisioning [2]. Due to space limitations, we refer to [3]-[5] for more details on EON architectures and proof-of-concept EON experiments. ITU-T has recently revised the G.694.1 recommendation and included the definition of a flexible DWDM grid (we call it flexgrid) [6]. According to [6], the frequency spectrum in an optical fiber link is divided into narrow frequency segments (we refer to them as slices). The optical path (lightpath) is determined by its routing path and a channel, which consists of a flexibly (ad-hoc) assigned subset of slices around a nominal Krzysztof Walkowiak is with Wroclaw University of Technology, Wybrzeże Wyspianskiego 27, PL-50-370 Wroclaw, Poland (phone +48713203539; e-mail:
[email protected]). Mirosław Klinkowski is with National Institute of Telecommunications, 1 Szachowa Street, 04-894 Warsaw, Poland. (e-mail:
[email protected]). The work of K. Walkowiak was supported in part by the statutory funds of the Department of Systems and Computer Networks, Wroclaw University of Technology. The work of M. Klinkowski was supported in part by NCN under Grant DEC-2011/01/D/ST7/05884.
central frequency. The channel covers both the frequency range occupied by the optical signal and a guard band required for the roll-off filters. In EON, the problem of finding unoccupied spectrum resources so that to establish a lightpath is called the Routing and Spectrum Allocation (RSA) problem. RSA concerns assigning a contiguous fraction of frequency spectrum to a connection request subject to the constraint of no frequency overlapping in network links. The RSA optimization problem is NP-hard [7] and it is more difficult than the Routing and Wavelength Assignment (RWA) problem in fixed grid wavelength division multiplexing (WDM) networks. Offline RSA has been addressed with both Integer Linear Programming (ILP) [7]-[10], meta-heuristics [8], [11], and heuristic algorithms [7], [8]. So far, there have been proposed several solutions for survivable EON in the literature. Regarding offline network design, survivable RSA algorithms for dedicated path protection (DPP) and shared backup path protection (SBPP) in a ring network have been studied in [12]. DPP in a network with generalized connectivity has been addressed with both ILP formulation [13], metaheuristic [14], and heuristic algorithms [13], [15]. Concurrently, an ILP formulation [16], a metaheuristic algorithm [17], and an heuristic algorithm [16] have been proposed for SBPP. Besides, in [18] a MILP formulation and an heuristic algorithm have been presented for the so-called squeezed protection scheme in which the protection of a lightpath may not cover its entire bandwidth but it is allowed to be partial. Eventually, dynamic survivable EON scenarios have been studied in [19]. All these solutions assure network survivability under single link failures. In this work, we focus on optimization of routing and spectrum assignment in survivable EON which is protected using the shared backup path protection approach called RSA/SBPP. The objective is to minimize the width of spectrum resources required in the network. As a failure scenario we consider a single link failure. We address two possible cases of SBPP, namely with and without stub release. We propose a novel ILP formulation of the problem, which is less complex than the one presented in [16]. Moreover, we propose effective heuristic algorithms to solve the problem including an extension of our original proposal for solving RSA called AFA (Adaptive Frequency Assignment) and MSALPF (Most Subcarriers and Average Longest Path First) as well as a set of heuristics developed for the classical RSA and modified for the RSA/SBPP problem. Obtained results show that AFA provides the best results – on average the
results are about 4.71% and 2.14% from optimal solutions yielded by CPLEX for two reference networks. However, the AFA needs significantly lower execution time and provides radically better scalability (CPLEX solves only small problem instances). Using the heuristic, we conduct extensive numerical experiments to investigate the problem of SBPP in EON. The remainder of the paper is organized as follows. In Section 2, we present the details of our SBPP scenario and we formulate the optimization problem. In Section 3, we describe heuristic algorithms. In Section 4, we present numerical results. Finally, in Section 5 we conclude the work. II. OPTIMIZATION MODEL In this Section, we present an ILP model of an offline problem of Routing and Spectrum Allocation with Shared Backup Path Protection (RSA/SBPP) in an EON with static traffic demands and subject to single-link failures. First, we discuss the considered path protection scenario and then we formulate the model. The model will be used as a reference in the evaluation of heuristic algorithms proposed in Section III. A. SBPP scenario Among network survivability schemes, connection recovery through path protection, in which backup network resources are provisioned in advance for each connection, is preferred due to its quick recovery time. In EON, network resources correspond to spectrum resources in optical fiber links and path protection is equivalent with provisioning backup lightpaths for working lightpath connections. In SBPP, backup resources can be shared between the demands whose primary paths are not likely to fail at the same time, on the contrary to DPP, in which each connection has its own backup resources. Thanks to this property, resource requirements in SBPP are usually lower than in DPP [20]. Regarding SBPP, the resources on the failure-affected paths can be either: a) left without use or b) released and used as spare resources for backup connections - the latter concept is called stub release. In this paper, we focus on both scenarios, that is without and with stub release; they are denoted, respectively, as NS and SR. In [13] and [15], there are distinguished two alternative scenarios for EON with path protection capability, namely: a) with Same Channel (SC) allocation and b) with Different Channel (DC) allocation. Case SC is a cost-effective scenario in which the transponders are shared between primary (working) and backup connections and a traffic demand has allocated the same segment of optical frequency spectrum (i.e., channel) on its primary and backup path. Such a solution reduces the network cost and alleviates the connection switching time [15]. In the DC case, the SC constraint of having the same operating channel in both primary and backup connections is relaxed, at the cost of installing either tunable or dedicated (for both working and backup lightpaths) transponders. In [13], it is shown that DC allows to reduce
spectrum requirements in the network when comparing to SC. Therefore, in this work, we focus on modeling and performance assessment of SBPP in the more efficient DC scenario. However, presented below algorithms can be easily adapted to the SC approach. In the following, we present ILP formulations of RSA/SBPP with NS and SR under the DC constraint, which are denoted, respectively, as RSA/SBPP/NS/DC and RSA/SBPP/SR/DC. B. ILP formulation The considered network is modeled as a directed graph. Links are labeled with e, where e = 1,2,…,E. In EON, the number of slices to be allocated to a connection is a function of the requested bandwidth, the modulation technique applied, the slice width in the flexgrid, and the guard band introduced to separate two spectrum adjacent connections, among others. For a given network scenario, where both the transmission parameters and flexgrid definitions are given, there is a direct relation between the requested spectrum and the requested bandwidth (e.g., see Sec. II in [8]). For instance, assuming the QPSK modulation format with spectral efficiency 2 bit/s/Hz and the slice width equal to 6,25GHz, a 100Gbit/s demand will occupy a 50GHz spectrum segment consisting of 8 slices. Therefore, without loss of generality, in this work the volume of demands is expressed in terms of the number of slices. Apart from that, we assume that the requested number of slices is the same for both working and backup connections. It corresponds to an EON scenario in which either a pair of dedicated working and backup transponders operates with the same modulation level, or a tunable transponder without modulation adaptation capability is used. RSA/SBPP model extensions which account for selection of different modulation levels on working and backup paths are left for further study. Taking the above into account, demands are labeled with d, where d = 1,…,D, and each demand corresponds to a lightpath request with two end nodes and the volume nd expressed in terms of the number of slices. We apply the link-path modeling approach [21] and we consider that for each demand d a set of candidate pairs of link disjoint routing paths indexed p = 1,2,…,Pd is given. In more detail, for each pair of end nodes we have k candidate pairs of link disjoint paths given. Thus, there are k candidate pairs of paths for each demand (i.e., Pd = k). We take a similar approach as in [10] and [13] for formulating the RSA/SBPP problem as an ILP problem. In particular, slices are labeled with s, where s = 1,2,…,S. Then, for demand d a set of candidate optical channels c = 1,2,…,Cd is considered, where each channel consists of a subset of adjacent slices of size nd slices, and RSA concerns selecting a path and assigning a channel to a demand. As shown in [10], this modeling approach is simple and computationally more efficient when comparing with alternative ILP formulations of RSA [7]-[9]. Consequently, the ILP formulation of RSA/SBPP presented in this paper is more efficient than the previous formulation in [16], since [16] makes use of [7]. In total, there are seven sets of binary decision variables in
the presented models. First, xdpc and zdpc denote the selection of channel c for demand d, respectively, on a working and a backup routing path p. The third variable is wes, which is 1, if slice s on link e belongs to a working lightpath and 0 otherwise. Next, wegs says if slice s is occupied on link e for a working lightpath and the selected working lightpath is not using link g. Similarly, begs denotes if slice s is occupied on link e for restoration in the case of link g failure. The sixth variable is yes, which is 1, if slice s is occupied on link e and 0 otherwise. Finally, variable ys denotes if slice s is occupied on any network link. The objective is to minimize the width of spectrum resources (denoted as Φ) required in the network, similarly as in [7]-[9], [13], and [16]. In particular, Φ corresponds to the largest slice index among all slices allocated in network links and it determines the width of spectrum that the network should support in a green-field network design [9]. To present the models we use notation as in [21]. RSA/SBPP/NS/DC (Without Stub Release) indices s = 1,2,…,S slices d = 1,2,…,D demands p = 1,2,…,Pd candidate pairs of link disjoint paths for demand d c = 1,2,…,Cd candidate channels for demand d e,g = 1,2,…,E network links constants δedp = 1, if link e belongs to working path p realizing demand d; 0, otherwise βedp = 1, if link e belongs to backup path p realizing demand d; 0, otherwise nd number of slices required for demand d γdpcs = 1, if channel c associated with demand d on working path p uses slice s; 0, otherwise (calculated according to nd) αdpcs = 1, if channel c associated with demand d on backup path p uses slice s; 0, otherwise (calculated according to nd) variables xdpc = 1, if channel c on working path p is used to realize demand d; 0, otherwise (binary) zdpc = 1, if channel c on backup path p is used to realize demand d; 0, otherwise (binary) wes = 1, if slice s is occupied on link e for working flows; 0, otherwise (binary) begs = 1, if slice s is occupied on link e for restoration in the case of link g failure; 0, otherwise (binary) yes = 1, if slice s is occupied on link e; 0, otherwise (binary) ys = 1, if slice s is occupied on any network link; 0, otherwise (binary) objective minimize Φ = ∑s ys subject to
(1)
∑p∑c xdpc = 1, d = 1,2,…,D
(2)
∑d∑p∑c γdpcsδedpxdpc = wes e = 1,2,…,E s = 1,2,…,S
(3)
∑d∑p∑c δgdpαdpcsβedpzdpc = begs e = 1,2,…,E g = 1,2,…,E g ≠ e s = 1,2,…,S
(4)
wes + begs ≤ yes e = 1,2,…,E g = 1,2,…,E g ≠ e s = 1,2,…,S
(5)
∑e yes ≤ Eys s = 1,2,…,S
(6)
∑c xdpc = ∑c zdpc d = 1,2,…,D p = 1,2,…,Pd
(7)
The objective (1) is to minimize the width of spectrum, in terms of the number of slices, required in the network. Equation (2) assures for each demand d exactly one candidate path and exactly one candidate channel are selected. To find the allocation of slices to working paths and to meet the guarantee that a slice on a particular link can be allocated to at most one lightpath, we add equation (3) to the model. Constraint (4) denotes the allocation of slices to backup paths. A single link failure (denoted as index g) is considered. Only demands affected by the failure (i.e., δgdpzdpc = 1) are taken into account. Next, the slice allocation following from activation of backup paths is included in the definition of begs. To meet the guarantee that a slice on a particular link can be allocated to at most one lightpath considering both working and backup paths as well as sharing of backup resources, we add equation (5) to the model. Constraint (6) defines that slice s is used in the network (ys = 1) only when there is at least one link on which the slice s is allocated. Constraint (7) allows to assign different channels to working and backup path of a particular demand, however the same pair of paths (associated with index p) must be used for the demand. RSA/SBPP/SR/DC (With Stub Release) variables (additional) wegs = 1, if slice s is occupied on link e for working paths and the selected working path is not using link g; 0, otherwise (binary) objective (1) subject to (2), (4), (6-7) and ∑d∑p∑c γdpcs(1–δgdp)δedpxdpc = wegs e = 1,2,…,E g = 1,2,…,E g ≠ e s = 1,2,…,S (8) wegs + begs ≤ yes e = 1,2,…,E g = 1,2,…,E g ≠ e s = 1,2,…,S
(9)
Equation (8) defines variable wegs, which is 1, if slice s is occupied on link e by working paths excluding working path that use link g. Therefore – comparing to (3) – on the left-hand side of (8) an additional term (1 – δgdp) is added. Consequently, when link g fails, the slices allocated to serve working paths of demands affected by the failure (i.e., selected working paths of these demands use link g) are released and can be used to serve for backup paths. Constraint (9) is a new definition of variable yes and together with (8) guarantees the stub release mode. Note that, similarly as in the RSA/DPP model [13], the above RSA/SBPP models can be extended to the SC scenario
by substituting constraint (7) with the following constraint: xdpc = zdpc d = 1,2,…,D p = 1,2,…,Pd c = 1,2,…,Cd (10) However, the evaluation of the SC model is out of scope of this paper. III. HEURISTIC ALGORITHMS In this Section, we describe heuristic algorithms for solving the RSA/SBPP problem. The algorithms are based on previously proposed RSA algorithms: − FF a Fixed-alternate routing and First-fit frequency assignment algorithm [22]; − LPF (Longest Path First) algorithm based on a greedy sequential processing of demands according to the decreasing values of the length of routing paths [8]; − MSF (Most Subcarriers First) algorithm based on a greedy sequential processing of demands according to the decreasing number of requested slices [8]; − LPMSF (Longest Path and Most Subcarriers First) algorithm based on a greedy sequential processing of demands according to the decreasing length of a routing path, as an additional criterion the decreasing number of requested slices (demand size) is taken into account [16]; − AFA (Adaptive Frequency Assignment) algorithm adaptively selects a sequence of processed demands in order to minimize Φ [7]. Moreover, we introduce a following algorithm similar to LPMSF, however using another sorting method: − MSALPF (Most Subcarriers and Average Longest Path First) algorithm is based on a greedy sequential processing of demands according to decreasing number of requested slices, in the case of a tie, demands are sorted according to decreasing value of an average length of candidate paths. Note that all algorithms except AFA are based on the same scheme consisting in sequential processing of demands which are sorted according to a selected criterion. Now we introduce several functions and operators necessary to describe all algorithms. Let MinFS_W(d,p) return the lowest indexed and accessible slice when demand d is assigned to working path with index p. In analogous way, MinFS_B(d,p) returns the lowest indexed and accessible slice when demand d is allocated to a backup path with index p. Both functions work in a residual network, i.e., all previously allocated demands are taken into account and only available slices (not allocated) are available for a new demand. In more detail, in the case of working paths all previous allocations of working and backup paths must be considered to find the residual network. In the case of backup paths – according to the SBPP approach – the residual network is calculated as follows. Without loss of generality, let assume that we consider that demand d is assigned to path pair p. First, all slices allocated to working paths are excluded from the residual network (in the case of the stub release scenario, the slices allocated to working paths of all failed demands are available for backup
paths). Next, all allocated demands for which the selected working path shares at least one link with path p are identified and slices allocated to backup paths of these demands are also excluded from the residual network. Let Allocate_W(d,p,s) denote a function that allocates in the network demand d to working path p starting from slice s. Similarly, Allocate_B(d,p,s) allocats the demand to a backup path. Let Sort_ALG(B) denote a sorting operation applied to a set of demands B according to the selected ordering method ALG, e.g., in the case of the LPF method demands are sorted according to the decreasing values of the length of routing paths. Each algorithm presented in this Section is formulated in two versions. In the former one – called SA (separate assignment) – first only working paths of each demand are allocated in the network and next backup paths are analyzed. In the latter version – named JA (joint assignment) in the same run both working and backup paths are allocated jointly. Below, we present the pseudocode of FF/SBPP/SA and FF/SBPP/JA algorithms, which simply process all demands without any special sorting and assigns the demand to the first candidate path. Algorithm 1 FF/SBPP/SA 1: for d = 1 to D do begin 2: s ← MinFS_W(d, 1) 3: Allocate_W(d, 1, s) 4: end 5: for d = 1 to D do begin 6: s ← MinFS_B(d, 1) 7: Allocate_B(d, 1, s) 8: end Algorithm 2 FF/SBPP/JA 1: for d = 1 to D do begin 2: s ← MinFS_W(d, 1) 3: Allocate_W(d, 1, s) 4: s ← MinFS_B(d, 1) 5: Allocate_B(d, 1, s) 6: end
Now, we focus on methods LPF, MSF, LPMSF and MSALPF. All four methods process demands in a similar greedy way, however, various ordering of demands is applied. As an example pseudocode, we describe the LPF methods both for SA and JA scenarios. Algorithm 3 LPF/SBPP/SA 1: B ← {d : d = 1,2,…,D} 2: B ← Sort_LPF(B) 3: for each demand d ∈ B do begin 4: smin ← Large_number 5: for p = 1 to Pd do begin 6: s ← MinFS_W(d, p) 7: if s < smin then path(d) ← p and smin ← s 8: end 9: Allocate_W(d, path(d), smin)
10: end 11: for each demand d ∈ B do begin 12: p ← path(d) 13: s ← MinFS_B(d,p) 14: Allocate_B(d,p,s) 15: end
First, algorithm LPF/SBPP/SA sorts demands according to selected ordering (line 2). Since working and backup paths are examined in separate runs, there are two main loops in the algorithm: lines 3-10 and lines 11-15. In the first loop, for each demand d all candidate paths are analyzed to find the one guaranteeing selection of the lowest slice. Since the candidate path is chosen in the context of the working path, for the backup path only slice selection is made (line 13). Algorithm 4 LPF/SBPP/JA 1: B ← {d : d = 1,2,…,D} 2: B ← Sort_LPF(B) 3: for each d ∈ B do begin 4: smin ← Large_number 5: for p = 1 to Pd do begin 6: s1 ← MinFS_W(d, p) 7: s2 ← MinFS_B(d, p) 8: s ← max(s1, s2) 9: if s < smin then path(d) ← p, smin ← s, sw ← s1, sb ← s2 10: end 11: Allocate_W(d, path(d), sw) 12: Allocate_B(d, path(d), sb) 13: end
The JA version of the LPF/SBPP algorithm analyzes all demands in a single run, thus there is only one main loop (lines 3-13). Again all candidate pair of paths are examined to find the one with the lowest slice index. When we compare SA approach against JA approach the key difference is the number of function MinFS_B(d, p) calls. In the former algorithm this function is called D times, while in the second case it is called kD, where k denotes the number of candidate paths. Recall that function MinFS_B(d, p) among others calculates the residual network including available slices according to the SBPP approach. Therefore, the JA approach on average needs more computational time to find the solution comparing to the SA approach. Note that in the case of algorithms LPMSF and MSALPF in line 6 of SA version and line 8 of JA version as the tie-breaker the summary length of working and backup path is applied as in [16]. Consequently, algorithm LPMSF/SBPP/JA is the same as the algorithm proposed in [16]. Finally, we present the AFA algorithm based on the method that we have proposed in [7]. The key idea behind the AFA algorithm is to adaptively divide demands into subsets and process each subset individually. Moreover, a special collision metric is used to select the best paths. In particular, we want to select paths that do not include links that can be potentially selected in a large number of demands and consequently to
avoid potential collisions. Let B denote a set all demands. Set Bn contains all demands from B with requested nd equal to n, i.e., Bn = {d : d ∈ B, nd = n}. For each link e = 1,2,…,E, the collision metric is defined as ce = ∑d∑p δedpnd. Notice that ce estimates the number of slices that might be allocated to link e taking into account all candidate paths (both working and backup). Let lp = ∑e∈p ce denote the length of path pair p (both working and backup) calculated according to metric ce. Metric ld = |Pd|-1 ∑p lp denotes the average length of candidate path pairs of demand d according to link metric ce. For the sake of notation simplicity, we introduce the following functions. Let MinDemandFS_W(d) return the lowest accessible slice for allocation of demand d and the index of a working path guaranteeing the lowest slice alocation. The processing of MinDemandFS_W(d) is analogous to lines 4-8 of the LPF/SBPP/SA algorithm. Let MinDemandFS(d) return the lowest accessible slice for allocation of demand d to both working and backup paths and the index of the candidate path pair – similarly to lines 4-10 of LPF/SBPP/JA algorithm. Algorithm 5 AFA/SBPP/SA 1: B ← {d : d = 1,2,…,D}, n ← max{nd : d ∈ B}, i = 1 2: Bn ← {d : d ∈ B, nd = n} 3: for each d ∈ Bn do (sd, path(d)) ← MinDemandFS_W(d) 4: d* ← arg min (sd). If more than one demand yields the minimum value of sd, use metric ld as a tie-breaker 5: Ci ← d*, i ← i + 1 6: Allocate_W(d*, path(d*), sd*) 7: Bn ← Bn \ {d*}; if Bn = ∅, go to step 8, otherwise go to step 3 8: n ← n – 1; if n < 1 go to step 9, otherwise go to step 2 9: for i = 1 to D do begin 10: d ← Ci 11: p ← path(d) 12: s ← MinFS_B(d,p) 13: Allocate_B(d,p,s) 14: end
The AFA/SBPP/SA algorithm analyzes demands divided into subsets according to the value of the number of slices required for a demand. The process of generating the subsets is given in lines 1, 2 and 8. Each subset of demands is processed to allocate working paths (lines 3-7), next using the same order of demands the backup paths are selected (lines 9-14). In a currently analyzed subset, all remaining (not allocated) demands are processed to find the lowest slice allocation (lines 3 and 4). Afterwards, the selected demand d* is assigned to a path guaranteeing a selection of the lowest possible slice. The procedure to assign slices for backup paths is comparable to algorithm LPF/SBPP/SA. Algorithm 6 AFA/SBPP/JA 1: B ← {d : d = 1,2,…,D}, n ← max{nd : d ∈ B 2: Bn ← {d : d ∈ B, nd = n} 3: for each d ∈ Bn do begin
(sdw ,sdb ,path(d)) ← MinDemandFS(d) sd ← max(sdw ,sdb) end d* ← arg min (sd). If more than one demand yields the minimum value of sd, use metric ld as a tie-breaker 8: Allocate_W(d*, path(d*), sd*w) 9: Allocate_B(d*, path(d*), sd*b) 10: Bn ← Bn \ {d*}; if Bn = ∅, go to step 11, otherwise go to step 3 11: n ← n – 1; if n < 1 stop, otherwise go to step 2
4: 5: 6: 7:
Algorithm AFA/SBPP/JA works analogously to AFA/SBPP/SA, however both working and backup paths are processed in the same run. As in the case of algorithms LPF, MSF, LPMSF and MSALPF, also the JA version of AFA is requires larger number of residual network constructions comparing to the SA version. All heuristic algorithms presented in this section are polynomial time algorithms. Due to space limitations, we skip the detailed discussion on their complexity. IV. RESULTS In this Section, we present and discuss the results of computational experiments. Four networks topologies are examined, namely: SIMPLE6 (6 nodes, 16 links), INT9 (9 nodes, 26 links), NSF15 (15 nodes, 46 links) and UBN24 (24 nodes, 86 links), all shown in Fig. 1. We compare heuristic algorithms presented in Sec. III and, besides, for small networks, we provide optimal results obtained by solving ILP models presented in Sec. II. Our main focus is on the occupied spectrum width (Φ), in terms of the number of slices, and computation time (T). The numerical experiments were performed on an Intel i5 3.3GHz 16GB computer. SIMPLE6
INT9
UBN24 NSF15
Fig. 1. Network topologies used in simulations: SIMPLE6, INT9, NSF15, and UBN24.
Candidate pairs of primary-backup paths are link disjoint and they are calculated (and ordered) as shortest paths, taking into account the overall length of both paths; we consider k∈{2,3,5,10,30}. As discussed in Sec. II.B, demands are expressed in terms of the number of requested slices. In details, the requested spectrum nd is an even number generated with a uniform distribution and for randomly selected source and destination pair of nodes. We assume the ITU flexgrid definition [6], which requires to have the spectrum allocated symmetrically around a central frequency. Therefore, in the evaluation, nd is considered to be an even number, in particular, nd∈{2,4,..., nmax}, where nmax∈{8,16}.
We have used the same set of network and traffic scenarios as in the evaluation of dedicated path protection in our recent paper [13]. In particular, for all networks but INT9, the results are averaged over 100 randomly generated demand sets. The results for INT9 are averaged over 88 demand sets since the ILP solver could not attain the optimality for some of DPP scenarios (see [13]). The use of common scenarios allows us to compare the efficiency of DPP and SBPP in EON. All results presented below in subsections A, B and C refer to the scenario without stub release. Only subsection D includes results with stub release. A. Comparison of Algorithms The first goal of experiments is to evaluate performance of heuristic algorithms. For smaller networks (SIMPLE6 and INT9) optimal results were provided by IBM ILOG CPLEX 12.4 (with default settings) [23]. When solving ILP problems, it was more difficult to find the solution of SBPP than of the corresponding DPP problem (see the relevant results for DPP in [13]). To speed up the computation of SBPP, we set the solution of DPP as an upper bound (UB) on Φ. As a result, the computation time of SBPP has been reduced significantly, from more than tens of minutes to some seconds (see Table I). In Table I, we report performance of heuristics in comparison to optimal results. We present results of SA and JA approaches for each heuristic. Average value of optimality gap and corresponding values of lengths of 95% confidence intervals are presented (the results are averaged over 100 demand sets for SIMPLE6 and 88 demand sets for INT9). The execution time of each heuristic was always below 30ms in small networks. We can easily notice that AFA outperforms other algorithms for both tested networks and provides results close to optimal ones. Moreover, the JA approach yields better results comparing to the SA approach for each method except the FF algorithm. For larger networks (NSF15 and UBN24) the CPLEX is not able to provide optimal results in reasonable times, therefore we compare only heuristics. In Table II, we show the average values of Φ, average execution time, average distance to minimum results and corresponding lengths of 95% confidence intervals (the results are averaged over 100 demand sets for each topology) To find the distance to minimum results, for each unique demand set we run all algorithms and next for each algorithm we calculate a percentage gap to the best (minimum) obtained result among all tested methods. Again, the AFA method provides the best results, while the MSALPF algorithm always is the second. Recall that both these methods are original algorithms proposed in this paper. The potential drawback of the AFA method is much larger execution time comparing to other methods, especially in the JA approach. The MSALPF algorithm needs computation time similar to other greedy methods LPF, MSF, LPMSF, while yields better results. As in the case of smaller networks, the JA approach provides better results than the SA approach. However – as it was pointed out in the previous section – the JA approach requires more operations what is reflected in higher values of
Scenario Network Heuristic SA SIMPLE6 JA SA INT9 JA SA JA SA JA
SIMPLE6 INT9
Scenario Net. Heur. SA NSF15 JA SA UBN24 JA NSF15 UBN24
SA JA SA JA
Scenario Net. k 2 3 NSF15 5 10 30 2 3 UBN24 5 10 30
Φ 22.00 22.00 31.75 31.75 -
TABLE I OPTIMALITY GAP OF HEURISTICS FOR SMALLER NETWORKS ILP FF MSF LPF MSALPF Time [sec] Average optimality gap 3.7 12.81% 10.00% 10.80% 9.95% 3.7 14.46% 6.55% 11.12% 5.76% 6.3 7.06% 4.22% 4.92% 4.31% 6.3 7.45% 3.30% 5.21% 2.78% Lengths of 95% confidence intervals 2.34% 2.20% 2.11% 2.18% 2.54% 1.97% 2.23% 1.89% 1.91% 1.51% 1.43% 1.44% 1.86% 1.36% 1.63% 1.27%
LPMSF
AFA
10.45% 9.82% 4.48% 4.84%
7.90% 4.71% 3.40% 2.15%
2.13% 2.29% 1.37% 1.56%
1.91% 1.68% 1.38% 1.22%
28.64% 38.90% 36.16% 44.79%
TABLE II COMPARISON OF HEURISTICS FOR LARGER NETWORKS Execution time [s] Φ MSF LPF MSALPF LPMSF AFA FF MSF LPF MSALPF LPMSF 308.2 318.6 293.0 301.8 283.4 0.2 0.2 0.2 0.2 0.2 272.9 294.5 263.3 283.4 256.2 0.2 5.4 5.3 5.3 5.4 651.4 640.8 632.3 632.9 585.4 2.6 2.0 2.0 2.0 1.9 567.0 559.5 542.9 550.3 521.3 2.0 49.2 47.7 48.3 47.4 Average distance to minimum result Lengths of 95% confidence intervals 9.13% 12.05% 4.47% 7.18% 1.30% 1.02% 0.86% 0.92% 0.78% 0.94% 6.66% 13.50% 3.31% 10.11% 0.60% 0.69% 0.63% 0.75% 0.55% 0.75% 10.12% 8.68% 7.44% 7.51% 0.11% 0.73% 0.81% 0.71% 0.82% 0.65% 8.20% 7.11% 4.20% 5.59% 0.41% 0.51% 0.97% 0.63% 0.81% 0.57%
FF 417.9 417.9 417.9 417.9 417.9 942.6 942.6 942.6 942.6 942.6
TABLE III PERFORMANCE OF HEURISTICS AS A FUNCTION OF THE NUMBER OF CANDIDATE PATHS Execution time [s] Φ MSF LPF MSALPF LPMSF AFA FF MSF LPF MSALPF LPMSF 371.4 392.0 368.2 381.9 366.7 0.2 0.6 0.6 0.6 0.6 364.5 382.8 358.2 372.1 356.2 0.2 0.8 0.8 0.8 0.9 348.2 360.3 342.1 351.4 340.6 0.2 1.2 1.2 1.1 1.2 310.6 316.7 299.4 305.7 292.6 0.2 2.0 2.1 2.0 2.1 272.9 294.5 263.3 283.4 256.2 0.2 5.4 5.3 5.3 5.4 813.4 826.9 794.4 817.3 792.6 2.2 6.8 6.6 6.8 6.6 756.1 762.9 741.4 755.7 730.5 2.3 9.0 8.7 9.0 8.8 681.7 684.0 664.7 675.1 639.3 2.3 12.7 12.4 12.7 12.6 605.8 604.9 586.6 595.1 560.1 2.3 21.3 20.9 21.2 21.1 567.0 559.5 542.9 550.3 521.3 2.0 49.2 47.7 48.3 47.4
FF 393.7 417.9 918.6 942.6
execution time. B. Impact of the Number of Candidate Paths Our next goal was to analyze the impact of parameter k (number of candidate paths) on the objective function Φ. In Table III, we report the average results and execution time of all heuristics as a function of k for large networks NSF15 and UBN24. Obviously, for the FF algorithm there is no impact of different values of k since the algorithm always selects the first candidate path. The improvement in Φ for other algorithms between k = 2 and k = 30 is in the range 24%-30% and 30%34% for NSF15 and UBN24, respectively. However, the execution time of algorithms grows approximately in a linear way with the increase of k what is in harmony with the greedy approach applied in the algorithms. The execution times are still on acceptable levels and we can conclude that it is beneficial to use a large set of candidate paths since it has a great impact on Φ.
AFA 0.6 70.9 7.8 1048.9 0.49% 0.27% 0.09% 0.18%
AFA 5.7 8.9 13.5 25.3 70.9 103.0 151.0 237.3 429.7 1048.9
C. SBPP versus DPP The last goal of experiments is to examine the benefits following from the sharing of backup path resources. Therefore, we compare the SBPP approach against DPP (Dedicated Path Protection) scenario. For comparison we use the results of the DPP evaluation in our recent paper [12]. In Table IV, we present the results obtained with both approaches, for smaller networks the results are optimal, for larger networks the results of AFA/SBPP/JA are reported. We can easily notice that the gap between DPP and SBPP depends on network topology and the number of candidate paths. The general trend observed for larger networks is that with the increase of the number of candidate paths, the gap between both approaches also increases. D. Stub Release Up to now, all presented results were obtained for the scenario without stub release. However, ILP solver as well as all heuristics were also run for the stub release approach.
Nevertheless, the results with stub release were almost the same as without stub release. In more detail, only in two of 388 demand sets considering all four tested networks, there was a slight difference between two investigated scenarios. In our opinion such results follow mainly from two facts. First, the objective function Φ is a min-max function, what means that in many cases some changes in network routing and slice allocation can have no impact on the objective. Second, comparing to classical survivability models formulated in the context of MPLS and similar protocols, our EON models include additional slice continuity constraint, which makes quite difficult using of slices released by working paths after a single link failure.
with constructive methods proposed in this work. Besides, we would like to examine other objective functions, e.g., to minimize the total maximum spectrum index of all links. REFERENCES [1] [2]
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TABLE IV SBPP VS. DPP
[6] Network
k
DPP
SBPP
Average Gap
SIMPLE6 INT9
3 2 2 3 5 10 30 2 3 5 10 30
25.28 33.61 449.6 432.5 432.5 382.2 358.1 970.0 882.1 882.1 737.8 677.8
22.00 31.75 366.7 356.2 340.6 292.6 256.2 792.6 730.5 639.3 560.1 521.3
12.64% 5.56% 18.4% 17.6% 21.2% 23.4% 28.3% 18.2% 17.1% 27.4% 24.0% 22.9%
NSF15
UBN24
95% Conf. Int. 2.4% 1.8% 0.9% 0.9% 1.0% 1.0% 0.9% 0.7% 1.0% 0.8% 0.6% 1.0%
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V. CONCLUSIONS In this paper, we have focused on optimization of shared protection in elastic optical networks. New ILP models of routing and spectrum assignment in EONs protected against single link failures were introduced. To the best of our knowledge, the models are the first ones that use the notion of channels in the context of SBPP protection in EONs. Since the ILP modelling can be applied efficiently only for relatively small problem instances, we proposed several heuristic algorithms. Through extensive numerical experiments we showed that the best heuristic (original algorithm called AFA) provides results very close to optimal results for small network topologies. Furthermore, we run tests for larger networks to verify performance of algorithms as a function of number of candidate paths and compare SBPP protection against nonsharing DPP. The main conclusions are: (i) increasing the number of candidate paths can significantly improve the objective function and (ii) benefit of SBPP over DPP is about 28% and 22% for NSF15 and UBN24 networks, respectively. Additionally, we discovered that SBPP with stub release approach provides practically no additional savings comparing to the SBPP without stub release. In future work, we plan to develop soft optimization algorithms (e.g., evolutionary, tabu search, simulated annealing) to obtain results closer to optimal when comparing
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