A Lagrangean relaxation based near-optimal algorithm for advance lightpath reservation in WDM networks

DOI 10.1007/s11107-009-0215-9

A Lagrangean relaxation based near-optimal algorithm

for advance lightpath reservation in WDM networks

Received: 23 October 2008 / Accepted: 31 August 2009 / Published online: 17 September 2009 © Springer Science+Business Media, LLC 2009

Abstract Advance lightpath reservation is a new research topic for connecting high-speed computer servers in lambda grid applications and for dynamic lightpath provisioning in the future optical internet. In such networks, users make call requests in advance to reserve network resources for commu-nications. The challenge of the problem comes from how to jointly determine call admission control, lightpath routing, and wavelength assignment. In this paper, we propose an efficient Lagrangean relaxation (LGR) approach to resolve advance lightpath reservation for multi-wavelength optical networks. The task is first formulated as a combinatorial optimization problem in which the revenue from accept-ing call requests is to be maximized. The LGR approach performs constraint relaxation and derives an upper-bound solution index according to a set of Lagrangean multipli-ers generated through subgradient-based iterations. In par-allel, using the generated Lagrangean multipliers, the LGR approach employs a new heuristic algorithm to arrive at a near-optimal solution. By upper bounds, we assess the per-formance of LGR with respect to solution accuracy. We fur-ther draw comparisons between LGR and three heuristic S. S. W. Lee (

B

Department of Communications Engineering, National Chung Cheng University, Taiwan, ROC

e-mail: [email protected] A. Chen· M. C. Yuang

Department of Computer Science, National Chiao Tung University, Taiwan, ROC

e-mail: [email protected] M. C. Yuang

e-mail: [email protected] A. Chen

Information and Communications Research Laboratories, Industrial Technology Research Institute, Taiwan, ROC

algorithms—Greedy, First Come First Serve, and Deadline First, via experiments over the widely-used NSFNET net-work. Numerical results demonstrate that LGR outperforms the other three heuristic approaches in gaining more revenue by receiving more call requests.

Keywords Advance lightpath reservation· Call admission control· Routing and wavelength assignment · Lagrangean relaxation

1 Introduction

With advances in optical Wavelength Division Multiplex-ing (WDM) technologies and its potential of providMultiplex-ing vir-tually unlimited bandwidth, optical WDM networks have been widely recognized as the dominant transport infrastruc-ture for fuinfrastruc-ture Internet backbone networks. Applications like lambda grid and virtual private optical networks usually need many high-speed lightpaths for connecting computer servers in diverse enterprise campuses. A major feature of such appli-cations is that traffic demands are requested to the network in advance before the connections are set up [1–4].

The advance lightpath reservation problem is in short referred to as ALR in this paper. One major challenge aris-ing in ALR has been to jointly determine call admission control, scheduling, and routing and wavelength assignment (RWA) [5]. Particularly, for optical network without wave-length conversion capability, the problem deals with RWA between source and destination nodes subject to the wave-length-continuity constraint [6]. It has been shown that RWA is an NP-complete problem [6]. Therefore, the ALR problem is also NP-complete since an RWA problem is a special case of the ALR problem.

Several algorithms for resolving the ALR problem have been proposed in the literature. In [1], the authors present

a basic framework for automated provisioning of advance reservation service based on GMPLS protocol suites. In [7], the ALR problem is classified into several types depending on the flexibility of call arrival time and call duration. Heuristic to RWA algorithms are also demonstrated for the problems. In [2], a simulated annealing based algorithm is proposed to find a solution on predetermined k-shortest paths. For lambda grid networks, Miyagi et al. [3] consider how to reserve a wavelength for deadline-aware application. Performance for blocking probability is evaluated under Greedy based and Deadline First based heuristic algorithms.

Lagrangean relaxation (LGR) based method has been shown to be an effective method in solving WDM network problems, for instance in [8]. In this paper, we propose a new LGR algorithm, which is used for the first time to our best knowledge to precisely and efficiently solve the advance lightpath reservation problem. In this paper, ALR is first for-mulated as a combinatorial optimization problem in which the revenue from admitting call requests is maximized. The LGR approach performs constraint relaxation and derives an upper-bound solution according to a set of Lagrangean multipliers generated through subgradient-based iterations. In parallel, using the generated Lagrangean multipliers, the LGR approach employs a new primal heuristic algorithm to arrive at a near-optimal solution. By upper bounds, we delin-eate the performance of LGR with respect to accuracy and convergence speed under different parameter settings and termination criteria. We further draw comparisons between LGR and a set of heuristic approaches via experiments over the widely-used NSFNET network. Numerical results dem-onstrate that LGR outperforms the other heuristic approaches in both accuracy and call blocking probability.

The remainder of this paper is organized as follows. In Sect.2, we first give the ALR problem formulation. In Sect.3, we present the LGR approach and its primal heuristic algo-rithm. In Sect.4, we demonstrate numerical results of the performance study and comparisons under benchmark of the NSFNET network. Finally, concluding remarks are made in Sect.5.

2 Problem formulation

We consider a WDM network where each WDM link con-sists of a pair of unidirectional fiber links with a number of wavelengths on each fiber. The network is under central-ized control. There is a central controller responsible for call admission control, RWA so as to establish lightpaths for all connection requests on behalf of all network nodes.

The ALR problem is formulated as an integer linear programming problem stated as follows. Given a physical topology and each call information (start time, end time, rev-enue), determine the scheduling of routes and wavelengths of

Call 1 Call 2 Call 3

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Time slot index

16

Call 1: start time =1, end time =13 Call 2: start time =3, end time =15 Call 3: start time =5, end time =11 Time event set T={1,3,5,11,13,15} σ1*={1,1,1,1,1,0} σ2*={0,1,1,1,1,1} σ3*={0,0,1,1,0,0} Call 1 Call 2 Call 3 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Time slot index

16

Call 1: start time =1, end time =13 Call 2: start time =3, end time =15 Call 3: start time =5, end time =11 Time event set T={1,3,5,11,13,15} σ1*={1,1,1,1,1,0}

σ2*={0,1,1,1,1,1} σ3*={0,0,1,1,0,0}

Fig. 1 Example of ALR problem

lightpaths, such that the total revenue from admitting calls is maximized under the wavelength continuity constraint. The bandwidth demand is one wavelength in the context of this paper. Throughout the paper, we use connection and call interchangeably. For ease of illustration, we assume in the sequel that the number of available wavelengths on each link is the same.

Before describing the model, we first give an example for the ALR problem. In Fig.1, there are three calls requests. Call 1 goes from time slot 1 to time slot 13. Call 2 and call 3 start from time slots 3 and 5, and end at time slots 15 and 11, respectively. Hence we have six time events (1, 3, 5, 11, 13, and 15) that need to be taken care of. Those six event points form the six members of the set T . Let us denoteσkt as a binary index to represent if call k includes event time t. Since, in this example, call 1 goes over event index 1, 2, 3, 4, and 5, we could derive that σ11 = 1, σ12 = 1, σ13 = 1, σ14= 1, σ15= 1, and σ16= 0.

We summarize the notation used in the formulation as follows:

Input values: L: set of optical links;

N : set of optical cross-connects;

W : set of wavelengths on each link (same for all links);

|W|: number of wavelengths available on each fiber link;

K : set of connection requests;

|K |: number of call requests;

rk: revenue for accepting call request k;

Pk: candidate path set for call k;

δpl: = 1, if path p includes link l; = 0, otherwise;

T : set of time events;

σkt:= 1, if call k goes through event time t; = 0, otherwise;

Decision variables:

xpw:= 1, if lightpath p uses wavelength w; = 0, otherwise;

yk:= 1, if call k is accepted by the network; = 0, other-wise;

Problem (P): max
k_∈K rkyk subject to: yk =
p∈Pk
w∈W xpw ∀k ∈ K (1)
p∈Pk
w∈W xpw ≤ 1 ∀k ∈ K (2)
k∈K
p∈Pk xpwδplσkt≤ 1 ∀w ∈ W, l ∈ L, t ∈ T (3) xpw = 0 or 1 ∀p ∈ Pk, k ∈ K, w ∈ W (4) yk = 0 or 1 ∀k ∈ K (5)

The objective function is to maximize the total revenue. Usually the revenue is proportional to the call duration. If we set rk to be one for all requests k, the problem becomes to maximize the number of accepted calls. In that case, the prob-lem is also equivalent to minimize call blocking. Constraints (1) and (2) require that at most one lightpath to be selected for each request. If the connection of call k is rejected, in which case the corresponding variable xpwis 0, a zero revenue con-tributes to the objective function. Constraint (3) guarantees no over-booking on any wavelength channel at any time slot. It requires that for any wavelength on a link, there is at most one lightpath using it. Constraint (4) states the 0/1 binary constraint on routing variable xpw. Please note that, we use time event T in our model, instead of directly using time slot index. The reason to use set T is to reduce the problem size. There are at most 2|K | members in T . That is usually far smaller than the total number of time slots. For exam-ple, in Fig.1, the total number of time slots is 16 while the total number of events is six. By using this technique, we can reduce the total number of constraints significantly. Finally, whether a call request is accepted or not is determined by constraint (5).

If we set all call requests with the same duration, the above ALR problem is reduced to a general RWA problem which has been proved to be NP-complete. Therefore, it is unlikely to obtain an exact solution for realistic networks in real-time. The problem is approximated using the LGR approach pre-sented in the next section.

3 Lagrangean relaxation based heuristic algorithm LGR [8,9] has been successfully employed to solve complex mathematical problems by means of constraint relaxation and problem decomposition. Particularly for solv-ing a linear integer problem, unlike the traditional linear programming approach that relaxes integer into non-integer constraints, the Lagrangean-based method generally leaves

the integer constraints in the constraint sets while relaxing complex constraints such that the relaxed problem can be decomposed into independent manageable subproblems. Through such a relaxation and decomposition, the LGR met-hod is shown to provide tighter bounds and shorter com-putation time on the optimal values of objective functions than those provided by the linear programming relaxation approach in many instances [9].

Essentially, the original primal problem is first simpli-fied and transformed into a dual problem after some con-straints are relaxed. If the objective of the primal problem is a maximization or minimization function, the solution to the dual problem is a respective upper or lower bound to the original problem. Such Lagrangean bound is a use-ful by-product in resolving the LGR problem. Next, due to constraint relaxation, the upper bound solutions generated during the computation might be infeasible for the original primal problem. However, these solutions and the generated Lagrangean multipliers can serve as a base to develop effi-cient primal heuristic algorithms for achieving a near-optimal solution to the original problem.

3.1 Dual problem and upper bound

In the relaxation process, constraint (3) is first relaxed from the constraint set. The expression corresponding to the con-straints, is multiplied by Lagrangean multipliers u_wlt, and then summed with the original objective function. Problem (P) is thus transformed into a dual problem, called Dual_P, given as follows: Problem (Dual_P): Zdual(u) = max ⎧ ⎨ ⎩
k_∈K rkyk−
w∈W
l_∈L
t_∈T u_wlt × ⎛ ⎝
k∈K
p∈Pk xpwδplσkt− 1 ⎞ ⎠ ⎫ ⎬ ⎭ = max ⎧ ⎨ ⎩
k∈K ⎛ ⎝rkyk−
p∈Pk
w∈W
l∈L
t∈T u_wltxpwδplσkt ⎞ ⎠ +
w∈W
l∈L
t∈T u_wlt ⎫ ⎬ ⎭ (6)

subject to Constraints (1), (2), (4), and (5) where vector u (with component u_wlt) is the non-negative Lagrangean mul-tiplier vector. Problem (Dual_P) in Eq.6can be decomposed into |K | independent sub-problems (one for each call k). Problem (Dual_P) is then expressed as Zdual(u) =

 k∈K

Zsub_k (u) +_w∈Wl_∈L 

t_∈Tuwlt, where Zksub(u) is as follows.

Z_ksub(u) = max ⎧ ⎨ ⎩rkyk−
p∈Pk
w∈W
l∈L
t∈T u_wltxpwδplσkt ⎫ ⎬ ⎭ subject to: yk =
p∈Pk
w∈W xpw (s1)
p∈Pk
w∈W xpw ≤ 1 (s2) xpw = 0 or 1 ∀p ∈ Pk, w ∈ W (s3) yk = 0 or 1 (s4)

The above sub-problem for call k is to determine the deci-sion variable, ykand xpwfor all p∈ Pkandw ∈ W. By care-fully observing the problem, one could find that it includes a shortest path problem. We propose the following algorithm to solve the problem optimally. For each wavelengthw on link l, we assign cost to be _t∈Tu_wltσkt, then we apply Dijkstra’s algorithm to obtain the shortest path p∗ and the best wavelengthw∗. Let ck denote the cost of path p∗. If

rk − ck is non-negative, then yk is set to 1 and xp∗w∗ is 1; otherwise, ykand all xpware 0 for every p∈ Pkandw ∈ W. The computational complexity for each Z_ksub(u) is O(m+nlog

n), where m= |L|, n = |N|.

By solving all the|K |Z_ksub(u) subproblems, we can obtain the value of Zdual(u). According to the weak Lagrangean duality theorem [9], Zdualin Eq.6is an upper bound of the original Problem (P) for any non-negative Lagrangean mul-tiplier vector u. Clearly, we are to determine the lowest upper bound. Equation6can be solved by the subgradient method, as shown as a part of the LGR approach delineated in Fig.2, as which shows that the algorithm is run for a fixed num-ber of iterations (Iteration_Numnum-ber). In every iteration, the sub-problems are solved (as described above), resulting in the generation of a new Lagrangean multiplier vector value. Then, according to Eq.6, a new upper bound is generated. If the new upper bound is tighter (lower) than the current best achievable upper bound (UB), the new upper bound is designated as the UB. Otherwise, the UB value remains unchanged. Significantly, if the UB value does not improve for a number of iterations that exceeds a threshold, called

Quiescence_ Threshold (QT), the step size coefficientλ of the

subgradient method is halved, in an attempt to reduce oscil-lation possibility. Specifically, in the update-step-size and update-multiplier procedures in Fig.2, the Lagrangean mul-tiplier vector u is updated as uk+1= uk+θkbk,whereθkis the step size, determined byθk = [λk(Zdual(u) − LB)]/bk2, in whichλk is the step size coefficient, LB is the current achievable largest lower bound obtained from the primal heuristic algorithm described next, and bkis a subgradient of

Zdual(u) with vector size|L + W + T |.

3.2 Primal heuristic algorithm and upper bound

The primal heuristic algorithm in the LGR approach is used to find a near optimal solution. Since our problem is a max-imization problem, a near optimal solution is clearly also a lower bound solution. Similar to the upper bound case, as given in Fig.2, if the new lower bound (lb) is tighter (larger) than the current best achievable upper bound (LB), the new lower bound is designated as the LB.

To obtain a near-optimal solution that is the highest lower bound, at the end of a subgradient iteration, the LGR solu-tion is verified whether or not it satisfies those relaxed con-straints. If it does, the solution is feasible and is thus used to calculate a lower bound of the primal problem (P). If the solution is infeasible, we employ the following LGR-based heuristic algorithm, which takes advantage of Lagrangean multipliers. As shown in Fig.3, the LGR algorithm sequen-tially accepts connections based on the rk− ckvalues. Calls with higher rk− ckhold higher priority in the sequence. The

begin

initialize Lagrangean multiplier vector u := 0

UB :=

∑

r

step size coefficient λ:= 2

for each k := 1 to Iteration_Number do begin

solve sub-problem for each k∈ K

∑∑∑

∑

else quiescence_age := quiescence_age + 1 if quiescence_age ≥ Quiescence_Threshold then begin

λ:=λ/2

quiescence_age := 0 end

run Primal Heuristic Algorithm to get lb /* Sec. 3.2 */

if lb > LB then

LB := lb /* lb is the new lower bound */ run update-step-size

run update-multiplier end

end

begin

Sorting Zsub(u)

k for all calls k and put their index in

priority Q

/* Q[1] is the call with the largest Zsub(u)

k value*/

/* Q[|K|] is the call with the smallest Zsub(u)

k

value*/

for each link l∈ L, w ∈ W

alw:= 1 /*all wavelength channels available*/

for (i = 1; i K; i++) begin

k = Q(i) /*DeQueue the highest priority call from Q*/

ck:=

accept := False

for each wavelength w∈ W do begin

for each link l∈L do if alw= 1 then

link cost costl:= _kt T t wlt

u

σ

∑

begin

accept := True;

/* denote cwk as path cost of p’ */

If cwk < ckthen begin ck:= cwk p* := p’ w* := w end end end If accept = True begin

Accept call k and p* is the routing path,

alw*:= 0 for those links used by path p* end

else

Reject call k

end end

update total revenue and return as a lower bound lb

end

w* is the wavelength

Fig. 3 Primal heuristic algorithm

routing is determined by Dijkstra’s shortest path algorithm based on the link cost,_t∈Tu_wltσkt, as those used in the previous section except that those links cost are set to infi-nite for wavelengths are taken by previous calls. It prevents those calls with lower priority to use the wavelength channel taken by previous high priority one. If there are not enough resources for the request, the call is rejected. The algorithm run repeatedly until all requests are satisfied or rejected.

4 Experimental results

We have carried out a performance study on the LGR appr-oach, and drawn comparisons between LGR and some heuristic algorithms via experiments over the well-known NSFNET Network. In the simulations, the start time and end time of call requests are generated randomly following uni-form distribution in one day. Each time slot is five minutes in the experiments. Consequently, the mean call duration is 450 min. The call revenue rk is set exactly equal to the call duration. Therefore, a call with longer duration receives more revenue than those with shorter durations.

In the computation using our LGR approach, we adopted

Iteration_Number= 3000 and Quiescence_Threshold = 50.

The LGR algorithm can obtain near optimal results within 10 min of computation time operated on a PC running Windows XP with a 2 GHz CPU power.

Three other heuristics are also considered in the study. The Greedy method sequentially allocates lightpaths accord-ing to connection’s rkvalue. Calls with larger revenues hold higher priority in call setup process. We also consider two timing related heuristics in our experiments. The First Come First Serve (FCFS) method schedule the requests according to call arrival time while the Deadline First (DF) method schedule the requests according to call finish time instead. The numerical results on NSFNET ranging from 150 to 275 calls are plotted in Fig.4. Figure4b shows the total reve-nue. The LGR achieves highest total revenue followed by the Greedy method. The FCFS method and the DF method results in lower output due to lack of taking call revenue into account. We use Percentage Gap (Gap%) to be the perfor-mance metric to evaluate the quality of those algorithms to a legitimate upper bound. The Percentage Gap (Gap%) is defined as the percentage of (Lagrangean UB—total revenue of the considered algorithm)/Lagrangean UB. As shown in Fig.4c, the percentage gap between the LGR and the UB are within 7% for all cases.

We further make comparisons of performance with respect to call blocking. As shown in Fig.4d, the LGR outperforms the other three methods. It is interesting that the Greedy heuristic algorithm is the one with largest number of call

0 5 10 15 20 25 30 Gap % LGR Greedy FCFS Deadline First

14 nodes, 42 links, 8 wavelengths (a) (b) (d) (c) 0 10 20 30 40 50 60 70

Number of Reject Calls

LGR Greedy FCFS Deadline First 150 175 200 225 250 275 14000 16000 18000 20000 22000 24000 Re ve nue Number of calls 150 175 200 225 250 275 Number of calls 150 175 200 225 250 275 Number of calls UB LGR Greedy FCFS Deadline First

Fig. 4 Experimental Results. a NSFNET network. b Performance comparisons—Revenue. c Performance Comparisons—Percentage Gap. d Performance Comparisons—Call blocking

rejection. By closely examining the results we find that those calls rejected by the Greedy algorithm are with small call durations.

5 Conclusions

In this paper, we have resolved an ALR problem using a LGR based approach augmented with an efficient primal Heuristic algorithm. The primal heuristic algorithm of LGR achieves a near-optimal lower-bound solution. With upper and lower bounds, we assess the performance of LGR with respect to solution accuracy. We have drawn comparisons of accuracy among LGR, Greedy, FCFS, and DF algorithms. Experimen-tal results demonstrate that LGR outperforms the other three heuristic approaches in gaining more revenue on receiving more call requests over the widely-used NSFNET Network. References

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Author Biographies

Steven S. W. Lee received his Ph.D. degree in Electri-cal Engineering from National Chung Cheng University, Tai-wan, in 1999. From 1999 to 2008, he was with Industrial Technology Research Institute (ITRI), Taiwan, where he was the leader of Intelligent Optical Networking project and a section manager of Optical Communica-tions Department. He was also a Research Associate Professor of National Chiao Tung Univer-sity, Taiwan, during 2004–2007. In 2008, he joined National Chung Cheng University, Taiwan, where he is currently an Associate Professor of the Department of Communica-tions Engineering. His research interests include optical and broadband networking, network planning, and network optimization.

Alice Chen received her B.S. degree in Electronics Engineer-ing from the National Chiao Tung University, Taiwan in 1984 and the M.S. degrees in Com-puter Science and Information Engineering from the National Chiao Tung University, Tai-wan in 1992. Currently, she is the deputy director of Opti-cal Communication Division at Information and Communica-tions Research Laboratories of Industrial Technology Research Institute, Taiwan, where she works on network control and management of optical networks.

Maria C. Yuang received her Ph.D. degree in Electrical Engi-neering and Computer Science from the Polytechnic Univer-sity, Brooklyn, New York, in 1989. From 1981 to 1990, she was with AT&T Bell Labo-ratories and Bell Communica-tions Research (Bellcore), where she was a member of technical staff working on Broadband Net-works and Protocol Engineer-ing. In 1990, she joined National Chiao Tung University (NCTU), Taiwan, where she is currently a Professor of the Department of Computer Science. Prof. Yuang has served as a Guest Editor for a Special Issue of IEEE Journal of Selected Areas in Communications on Next-Generation Broadband Optical Access Network Technologies in 2009. She has served on the technical program committee of many technical conferences includ-ing IEEE ICC and GLOBECOM, and has been invited to give invited talks at many technical conferences. Her main research interests include broadband optical networks, wireless networks, multimedia communi-cations, and performance modeling and analysis. She is a senior member of IEEE, and a member of OSA. She holds 17 patents in the field of broadband networking, and has over 100 publications, including a book chapter.