F. González, I. de Miguel, J. C. Aguado, P. Fernández, R. M. Lorenzo, E. J. Abril, M. López,, “Iterative linear programming formulation for routing and wavelength assignment in optical networks”, Signal Processing, Communications and Computer Science (N . Mastorakis, ed.), pp. 57-63, WSES-Press, 2000
Iterative Linear Programming Formulation for Routing and Wavelength Assignment in Optical Networks F. GONZÁLEZ (1), I. DE MIGUEL (2), P. FERNÁNDEZ, J.C. AGUADO, R.M. LORENZO, E.J. ABRIL, M. LÓPEZ Dpt. of Signal Theory, Communications and Telematics Engineering University of Valladolid E.T.S.I. de Telecomunicación, Campus “Miguel Delibes”, 47011 Valladolid SPAIN (1)
[email protected] (2)
[email protected] http://pesquera.tel.uva.es Abstract: - In this paper, we consider the Routing and Wavelength Assignment (RWA) problem in Wavelength Division Multiplexing (WDM) networks. Firstly, we propose separate linear programming (LP) formulations for the lightpath routing and the wavelength assignment subproblems. Then, an iterative algorithm that integrates both formulations is also proposed for solving the complete RWA problem. The objective is minimizing both the number of wavelengths required and the average distance traversed per traffic unit. Key-Words: - Wavelength division multiplexing (WDM), wavelength-routed optical network (WRON), lightpath, routing and wavelength assignment (RWA), wavelength reuse, linear programming.
1 Introduction Wavelength Division Multiplexing (WDM) has become the preferred technology for developing new high performance optical networks and for improving the present ones. When applying WDM to Wide Area Networks (WANs), the wavelength-routing approach presents a number of benefits [1−3]. In WavelengthRouted Optical Networks (WRONs), all-optical channels (called lightpaths [4]) are established between nodes that are not necessarily adjacent. For example, in Fig. 1, lightpaths between nodes A and C, and between nodes A and D have been established. At the intermediate node B, each channel is optically routed towards its destination according to its wavelength. Commonly, WRONs are multihop networks. In such kind of networks, there is not always a lightpath between every pair of nodes, so there is an amount of external traffic that needs to travel through several lightpaths to reach its destination. In this case, electrical layer devices (ATM switches, IP routers, etc) are in charge of extracting traffic from a lightpath and inserting it into another one. A key issue in WRONs is wavelength reuse, that is, to assign the same wavelength to several lightpaths in the network. The only restriction is that two
lightpaths must not occupy the same wavelength through the same fiber link. If wavelength conversion is available at intermediate nodes, a lightpath does not necessarily occupy the same wavelength at every fiber link it traverses. If wavelength conversion is not available, a single wavelength is assigned to every lightpath. In this case, two lightpaths that share a fiber link must use different wavelengths [5]. Lightpath communications allow a virtual (or logical) topology to be embedded into the physical one [5,6]. Fig. 2 shows the virtual topology for the network in Fig. 1. When solving the Virtual Topology Design (VTD) problem [2], the following subproblems (which are not necessarily independent) must be taken into account [6]: D C
B λ1 A
λ2
Fiber links Lightpaths
Fig. 1: Lightpath communications between nodes A and C using wavelength λ1, and nodes A and D in wavelength λ2. No wavelength converters are used.
.
C
D
A
Logical Links
Fig. 2: Logical topology embedded in Fig.1. i) Determine a good virtual topology, that is, the origin and destination node of every lightpath. ii) Route the lightpaths over the physical topology. iii) Assign wavelengths to the lightpaths. iv) Route the externally offered traffic on the virtual topology. In this study, we center in subproblems ii) and iii), which are known in joint as the Routing and Wavelength Assignment (RWA) problem. It can be defined as follows [2]: Given a network topology and a set of lightpath requests, determine a route and wavelength(s) for the requests. Some parameters to be optimized when solving the RWA problem are the number of wavelengths needed, the wavelength reuse, the physical distance traversed by the traffic, and the cost. We propose two separate formulations to solve the RWA problem when there is no wavelength conversion. The former routes the lightpaths in a way that the average distance traversed per traffic unit is minimized. Besides, it limits the number of lightpaths per fiber link (which is a heuristic for limiting the number of wavelengths needed). The latter assigns wavelengths to the lightpaths once they have been routed. In this case, the objective is to maximize the wavelength reuse. We also propose an iterative algorithm to solve the RWA problem by combining both formulations. Finally, a number of results are showed.
2 Previous Work In the last years, several papers about virtual topology design have been published [4−17]. Next, we will overview some of them concerning the routing and/or the wavelength assignment problem. The work in [4] proposes to allocate the wavelengths to the longest lightpaths first, as a heuristic for minimizing the number of wavelengths needed when solving subproblem iii). In [7], subproblems i), ii) and iii) are considered.
The lightpaths are routed using a shortest-path algorithm. Then, wavelengths are assigned to those lightpaths with a large traffic load first. The routing problem is also solved by means of a shortest-path algorithm in [8], but wavelengths are assigned sequentially beginning with the longest lightpath. In [9], subproblem ii) is formulated as an integer linear programming (ILP) problem. Several techniques, as randomized rounding, are employed to reduce the problem complexity. To solve subproblem iii), graph-coloring algorithms are used. A mixed integer linear programming (MILP) formulation for subproblems i) and ii), as well as a number of heuristics for solving subproblem i) in joint with the RWA problem are proposed in [5]. In [10], an ILP formulation and two heuristics (based on Dijkstra’s algorithm) are proposed for solving RWA. One of them routes lightpaths sequentially. The other one routes them in parallel. Both take into account the available wavelengths and their capacity at every fiber link. A cost model is stated to compute the distance used by the shortestpath algorithm. Finally, an improvement step is run. A genetic-algorithm/heuristic hybrid approach is proposed in [11] for solving RWA with the objective of minimizing the cost of the network.
3 Routing and Wavelength Assignment As stated before, the RWA problem consists in routing the lightpaths through the physical topology and assigning wavelengths to them. We suppose there is no wavelength conversion, so each lightpath is assigned a single wavelength. In the next paragraphs, we propose new separate ILP formulations for solving subproblems ii) and iii). By separating RWA into subproblems the total computational cost is reduced.
3.1 The Lightpath Routing Problem In this problem, a route over the physical topology is computed for every lightpath. As proposed in [6] and [11], we use flow conservation equations at each node to properly route the lightpaths. Note that the number of wavelengths that are needed for establishing the lightpaths, once they have been routed, depends on the maximum number of lightpaths that share a single fiber link (we call it ρmax). In fact, ρmax will be a lower bound on the number of wavelengths. In Fig. 1, the
.
fiber link between A and B supports two lightpaths, so (at least) two different wavelengths λ1 and λ2 are needed. As shown in Fig. 3, it is possible that the number of wavelengths needed exceeds ρmax. In this figure, every fiber link supports exactly two lightpaths, but a third wavelength is required to route the one from B to C, given that λ1 and λ2 are being used in the fiber links it traverses and there is not any wavelength converter. Consider an N-nodes network, with 2E unidirectional links (E bidirectional fiber links) numbered as e = 1, ... , 2E. Let us denote by s(e) the source node of physical link e, and by d(e) its destination node. Let we be the physical distance of link e. Consider H lightpath requests numbered as h = 1, ... , H. Let us denote by s(h), the source node of the lightpath h, and by d(h) its destination node. Let fh be the traffic to be routed over lightpath h. We define the variables p eh , with p eh = 1 if the lightpath h traverses through the physical link e and p eh = 0 otherwise. Then, we can formulate the lightpath routing problem as follows: 1 Minimize (1) fh δ h f total h subject to: (2) δ h = we p eh ∀ h
∑(
)
∑ e
f total =
∑p e d (e) = n
h e
∑p
=
∑f
(3)
h
h
∀ h, n = 1, L , N : n ≠ s( h), d ( h) (4)
h e'
e' s ( e ') = n
∑p
h e
=1;
e s (e) = s ( h)
∑p
h e
h e
=1 ∀ h
(5)
=0 ∀h
(6)
e d (e) = d ( h)
= 0;
e d (e) = s ( h)
B
∑p ∑p
h e
e s (e) = d ( h)
λ1 λ3
∑p
h e
≤M
∀e
(7)
h
We have called this problem BWR (for BoundedWavelength Routing). Note that it is an integer linear programming (ILP) problem. Equation (2) defines δ h as the total distance of the lightpath h. Equation (3) defines ftotal as the total traffic on the network. Therefore, the objective function (1) minimizes the average distance per traffic unit. Equations (4)−(6) are flow conservation restrictions. By means of equations (5) and (6), it is ensured that every lightpath starts at its source node and finishes at its destination node. Equation (4) ensures that the lightpath follows a continuous route from its source to its destination. The parameter M in (7) is an upper bound on the number of lightpaths supported by a single fiber link, and it must be introduced as a design parameter. Note that, as shown in Fig. 3, it does not ensure that only M different wavelengths are needed. As it will be shown later, BWR can be used in an iterative manner, beginning with an arbitrarily low value of M (obviously, greater or equal than one) and increasing it until a solution of BWR is feasible. Some variations can be applied to this formulation. The objective function (1) supposes the traffic flowing through every lightpath to be known or estimated. If it is not, the objective function δh
∑ h
can be used. Then, the minimum total lightpath length is achieved. On the other hand, if we set we = 1 ∀ e, the number of nodes crossed by lightpaths is minimized, which may be desirable for some networks. Furthermore, it is possible to model the nodes of the network like in [10] and [12] so that the node utilization can also be optimized. Another variation is to minimize M instead of the average distance per traffic unit, which would achieve the minimum ρmax for a given network. In the experiments that we have carried out, this variation resulted in an unacceptable increase of the computation time.
C
λ2 A
Fig. 3: Three wavelengths are needed although every fiber link only supports two lightpaths.
3.2 The Wavelength Assignment Problem Once the lightpaths have been routed, a wavelength must be assigned to each one. In this section, we will present an ILP formulation for the wavelength assignment problem. We have called it MRA (for Maximum-Reuse wavelength Assignment). Suppose a lightpath routing scheme, given by the p eh variables
.
previously defined. Let µ be the number of available wavelengths. Let c kh (h = 1, ... , H; k = 1, ... , µ) be the
same way, being R = { p eh } a lightpath routing scheme for the given requests.
wavelength assignment variables, with c kh = 1 if
IRWA algorithm:
c kh
wavelength k is assigned to lightpath h, and otherwise. Then, MRA is formulated as follows: Minimize (H + 1)k − 1 c kh
∑∑ k
=0 (8)
M = M + 1
∑c
h k
=1 ∀ h
(9)
k
∑p
h e
⋅ c kh ≤ 1 ∀ k , e
(10)
h
Equation (9) ensures that every lightpath is assigned a single wavelength. Equation (10) avoids that two lightpaths supported by the same fiber link e are assigned the same wavelength k. The objective function (8) maximizes the wavelength reuse, that is, it assigns the same wavelength (those corresponding to the lowest values of k are preferred) whenever possible. Hence, some of the µ wavelengths can be unnecessary. Next, we will prove that (8) maximizes the wavelength reuse. Note that the coefficient of each c kh in (8) is (H + 1)k – 1. Besides this, since at most H different lightpaths can be assigned the c kh ≤ H ∀ k . wavelength k, it is verified that
∑
∑ (H + 1) k
k−1
c kh
≤ (H + 1)
= (H + 1)
k−1
k−1
∑ h k
H < (H + 1)
(H + 1) is the coefficient of
c kh+ 1
End While
µ = M R = BWR(M) While MRA(R, µ) = ∅ Begin
µ = µ + 1 End While
Several authors have presented lower bounds on the number of wavelengths for different networks [2,8]. Next, we present a simple lower bound for estimating the goodness of IRWA. Given an N-nodes network with the same physical degree θ and the same logical degree T on each node, at most Nθ lightpaths of one-hop length can be established (supposing that there is at most one lightpath for each pair of nodes), Nθ (θ − 1) of two-hop length, and so on. Then, lightpaths of at least D hops must be established, being D an integer such as: D −2
h
h
While BWR(M) = ∅ Begin
h
subject to:
Then,
Initialization: Let M be some lower bound on the number of different wavelengths needed. If none is known, then M = 1.
c kh
∑ Nθ (θ − 1)
≤
i
+ N 'θ (θ − 1) D −1 = NT
(12)
i =0
(11)
∀ h, so (8) will be
lower if all the lightpaths in the network are assigned the wavelength k than if a single lightpath is assigned the wavelength k + 1.
where N ' is a real number such as 0 < N ' ≤ N . (Notice that N 'θ (θ − 1) D −1 is an integer number meaning the number of lightpaths of length D). A lightpath consists of a series of lightpath segments from its source to its destination (one segment for each fiber link it traverses). Then, the minimum number of lightpath segments in the network is: S=
3.3 An Iterative Algorithm for RWA (IRWA) In this section we propose an iterative algorithm (called IRWA) that integrates BWR and MRA to solve the RWA problem. The algorithm attempts to minimize the number of wavelengths needed to route a group of lightpath requests and to maximize the wavelength reuse. For a given network and for a set of lightpath requests, we denote by BWR(M) the solution obtained when solving BWR for that particular value of M. If the solution is unfeasible, then BWR(M) = ∅. We also define MRA(R, µ) in the
D−2
∑ (i + 1) Nθ (θ − 1)
i
+ DN 'θ (θ − 1) D −1 (13)
i =0
If we suppose these segments are uniformly distributed among the 2E unidirectional physical links (i. e., each physical link supports the same number of lightpaths), then the minimum number of lightpath segments per physical link (which is a lower bound on the number of wavelengths required) is: ⎡ S ⎤ µ min = ⎢ ⎥ (14) ⎢ 2E ⎥ Note that this is an optimistic lower bound. For each node, we are supposing that lightpaths are established
.
between its T nearest nodes, and that the number of lightpaths per link is the same for the entire network. Both assertions are unlikely for real networks, so more than µmin wavelengths will be needed in general.
4 Results We have used the algorithm explained above to solve RWA for a number of randomly generated networks. We have used the lp_solve package [18] to solve LP problems. Fig. 4 shows the number of wavelengths that were needed in 12-nodes networks, as a function of the number of bidirectional fiber links E, for logical degree T = 5. The lower bound stated in (14) is also shown. We carried out 10 experiments for each value of E. For choosing a virtual topology, random traffic patterns have been generated for each network, and logical links have been established in a manner that the largest amount of traffic can be routed in one hop [2,5,13], which is a heuristic for minimizing the congestion. Note that the results are slightly higher than the lower bound. We have also applied the algorithm to the NFSnet [5,6], which is shown in Fig. 5. The results are presented in Fig. 6 together with the results from [5] (algorithms HLDA, TILDA and MLDA). The joint use of BWR and MRA (IRWA) results in better values than HLDA, which chooses the virtual topology by maximizing the traffic in one-hop, and in similar values to those of TILDA and MLDA. Note that the latter algorithms establish lightpaths between adjacent nodes (they do not consider the traffic as HLDA and IRWA do) so the number of wavelengths required is expected to be low. Computation time is a key factor when choosing heuristic (sub-optimal) approaches instead of linear programming ones [2,5,6,9,10,13,16,17]. We have
10
executed the algorithm for series of five random generated networks with different number of nodes, and computed the mean time and the 95% confidence interval [19] that took to solve them. As shown in Fig. 7, computation time presents an exponential growth with the number of nodes, but it is smaller than thirty seconds for 20-nodes networks. We consider this time is good enough for the static approach (the one we have followed in this paper), in which lightpaths requests arrive simultaneously and they are routed off-line.
5 Conclusion In this paper, separate ILP formulations for routing lightpaths and assigning wavelengths to them have been proposed. The former limits the number of lightpaths per fiber link and minimizes the average distance traversed per traffic unit. The latter maximizes wavelength reuse. We have also proposed an iterative algorithm for the joint use of both methods to solve the whole RWA problem. This algorithm is computationally quite tractable and provides solutions where the number of wavelengths required is close to the optimal one.
IRWA
12
Lower bound
10
Wavelengths
Wavelengths
8
Fig. 5: The 14-nodes NFSnet backbone.
6 4 2 0
HLDA
8
TILDA
6
MLDA IRWA
4 2
20
24
27
30
33
Bidirectional Physical Links
0 2
3
4
5
6
Logical Degree
Fig. 4: Number of wavelengths required for randomly generated networks. Mode and intervals containing the whole of the results are showed.
Fig. 6: Number of wavelengths required for the NFSnet.
.
25
Time (s)
20
[10]
15 10 5 0 8
10
12
14
16
18
20
[11]
Nodes
Fig. 7: Computation time of IRWA (Pentium II 400 MHz, 128 MB RAM, Windows NT 4.0) References: [1] R. Ramaswami, Multiwavelength Lightwave Networks for Computer Communication, IEEE Communications Magazine, Feb. 1993, pp. 78-88. [2] R. Ramaswami, K. Sivarajan, Optical Networks: A Practical Perspective. Morgan Kaufmann Publishers, Inc., 1998. [3] O. Gerstel, On the Future of Wavelength Routing Networks, IEEE Network, Nov./Dec. 1996, pp. 14-20. [4] I. Chlamtac, A. Ganz, G. Karmi, Lightpath Communications: An Approach to High Bandwith Optical WAN’s, IEEE Transactions on Communications, Vol. 40, No. 7, 1992, pp. 11711182. [5] R. Ramaswami, K. Sivarajan, Design of Logical Topologies for Wavelength-Routed Optical Networks, IEEE Journal on Selected Areas in Communications, Vol. 14, No. 5, 1996, pp. 840851. [6] B. Mukherjee, D. Banerjee, S. Ramamurthy, A. Mukherjee, Some Principles for Designing a Wide-Area WDM Optical Network. IEEE/ACM Transactions on Networking, Vol. 4, No. 5, 1996, pp. 684-696. [7] Z. Zhang, A.S. Acampora, A Heuristic Wavelength Assignment Algorithm for Multihop WDM Networks with Wavelength Routing and Wavelength Re-Use, IEEE/ACM Transactions on Networking, Vol. 3, No. 3, 1995, pp. 281-288. [8] S. Baroni, P. Bayvel, Wavelength Requirements in Arbitrarily Connected Wavelength-Routed Optical Networks, Journal of Lightwave Technology, Vol. 15, No. 2, 1997, pp. 242-251. [9] D. Banerjee, B. Mukherjee, A Practical Approach for Routing and Wavelength Assignment in Large
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