Photon Netw Commun (2006) 12:153–160 DOI 10.1007/s11107-006-0033-2

O R I G I NA L A RT I C L E

A heuristic solution to SONET ADM minimization for static traffic grooming in WDM uni-directional ring networks Kuntal Roy · Mrinal K. Naskar

Received: 27 December 2005 / Accepted: 27 July 2006 / Published online: 9 September 2006 © Springer Science+Business Media, LLC 2006

Abstract In recent years, minimization of add-drop multiplexers (ADMs) in wavelength division multiplexing (WDM) optical networks has gained lots of attention in both the research and commercial areas. This motivates the research presented in this paper. A heuristic algorithm is formulated for static traffic grooming in WDM uni-directional ring networks with an eye to minimize the number of required ADMs. The distinguished feature of the proposed heuristic is that it pairs up the calls of a given static traffic to approach the solution. The proposed heuristic is compared with the previous approach with same network configuration and traffic matrix to establish its effectiveness. Keywords WDM uni-directional ring networks · Static traffic grooming · SONET add-drop multiplexer · Heuristic algorithm

Introduction Wavelength division multiplexing (WDM) [1, 2] coupled with synchronous optical network (SONET) [3, 4] has emerged as a promising technology for use in backbone networks. Multiple signals distinguished by their wavelengths can be carried over through a fiber using WDM technology [5–7]. The reason behind the bandwidth-division of a fiber is that its bandwidth is K. Roy (B) · M. K. Naskar Department of Electronics and Tele-Communication Engineering, Jadavpur University, Kolkata 700032, India e-mail: [email protected] M. K. Naskar e-mail: [email protected]

too high to carry a single signal. As the technology progresses, transmission speed of fiber is also increasing from OC-48 (2.5 Gbps) to OC-192 (10 Gbps). As there are constraints (e.g., power consumption) in increasing the number of wavelength channels without limit using WDM technology, the recent trend is to employ TDM slots in the wavelength channels. The resulting network configuration is known as WDM-TDM network or WDM grooming network [8–10]. At each node in the network, there are SONET adddrop multiplexers (ADM) for each wavelength to add or drop signal streams. An ADM has the capability to sum up lower-rate signals into a higher-rate signal. For example, four OC-48 s can be multiplexed into an OC192. But, the cost of ADMs dominates the total cost of designing optical network. As the nodes in the optical network increase, the number of ADMs required is also increased by an amount equal to the number of wavelength channels in the network per node. Fortunately, it is not necessary for every node to be equipped with ADMs for all wavelengths. An ADM corresponding to a wavelength are required only to transmit or receive signals at that wavelength. Therefore, tremendous efforts have been exploited to minimize the number of ADMs in a SONET/WDM network. Previous approach There are different approaches found in literature to optimize the performance of WDM networks on the basis of SONET-ADM minimization. The approaches proposed in Refs. [11–13] are the first-stage works on this field. In the next stage, there are some theoretical approaches to compute the lower-bound of ADMs for uniform all-to-all traffic, proof of NP-completeness as in

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Refs. [14, 15]. Some good-illustration of traffic grooming in optical networks is presented in Refs. [16, 17]. In Ref. [18], greedy heuristic algorithms are proposed for arbitrary static traffic to minimize the number of ADMs. There are several other schemes [19–21] for the same. But, in the recent progress [22, 23], Integer Linear Programing (ILP) and simulated-annealing are used as an optimization tool to solve the problem. In Ref. [23], there is detailed comparison among greedy heuristic, ILP, and simulated-annealing for different groomingcapability and different types of traffic. But, in Ref. [23], the ILP solver fails to reach a conclusion for the assumed non-uniform traffic. Simulated-annealing algorithm does not seem to be efficient as running different times, it outputs different results and sometimes its result is poor compared to the greedy heuristic algorithm. Proposed approach In this article, a heuristic algorithm based on intuitive reasoning is proposed to approach an appreciable solution for the same. For a uni-directional ring network, a single path exists between the source–destination pairs and accordingly the specific path from node i to node j (i
= j) is complementary with the path from node j to node i. Based on this fact, the corresponding complementary calls have been paired up. Thus, if there are m calls between the node-pair (i, j) and ncomplementary calls correspondingly, Number of different wavelength channels needed, 
max(m, n) W(i,j) = Ceil C where, C is the grooming-capability of a wavelength channel in the network. The ‘Ceil’ function ceils the value to next integer. In this way, the maximum number of ADMs required can be calculated. For a network having N nodes, there are N(N − 1)/2 number of different combination of node-pairs. Subsequently, the maximum number of wavelength channels to be involved is Wmax =

N  N  i=1 j=1,i



max(Ti,j , Tj,i ) Ceil C



where, T is the traffic matrix and Tm,n denotes the number of calls corresponding to the node-pair (m, n). Since, for a connection to be established between a source–destination node-pair along a particular wavelength channel, it requires two ADMs corresponding to source and destination nodes, the maximum number of required ADMs is

ADMmax = 2Wmax . Depending on the traffic matrix and grooming-capability of the network, actual ADM required, ADMactual might be less than ADMmax and might not be a multiple of two. Thus, a figure of merit (χ ) is defined that determines the percentage saving of ADMs with respect to ADMmax and is defined as follows: χ=

ADMmax − ADMactual × 100%. ADMmax

As illustrated in literature, number of wavelengths and ADMs both cannot be optimized simultaneously. According to the proposed algorithm, the primary concern is to minimize the number of ADMs required. There may exist few situations where the required wavelength channels according to the proposed algorithm is greater than the minimum possible number of wavelengths required, but it is always intended to minimize the number of ADMs. The article is organized as follows: Section “Proposed algorithm” presents the proposed algorithm with stepwise pseudocode. The time-complexity analysis of the same is done in Section “Time-complexity analysis”. Experimental results along with comparison with previous approach are in Section “Results”. Finally, Section “Conclusions” concludes the paper. Proposed algorithm Inputs: traffic ← the given static traffic matrix; N ← number of nodes in WDM uni-directional ring network; W ← number of different wavelength channels in the network.; C ← number of TDM slots in each wavelength channel. Output: noADM ← number of ADMs required. Step1: Calculate the maximum number of ADMs required, ADMmax for the given traffic matrix. ADMmax ← calculateMaxADMs(traffic, N) The pseudocode is as follows: Procedure calculateMaxADMs (traffic, N, AD Mmax ) Begin For i = 1 to N For j = 1 to N If i < j then ADMmax = ADMmax + 2 * Ceil(max(traffic (i, j ), traffic(j, i))/C) End End End End. // end procedure

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Step2: Form traffic sequence matrix ‘trafficSequence’ pairing up (i, j)-th calls with (j, i)-th calls of the “traffic” matrix. It is of dimension [N(N − 1)/2] × 4. The entries in a particular row denote source node, destination node, number of requests from source node to destination node, and number of requests from destination node to source node, respectively. All the source and destination nodes correspond to the upper triangular portion of “traffic” matrix. The pseudocode for the same is as follows: trafficSequence ← generateTrafficSequence (traffic, N) Procedure generateTrafficSequence (traffic, N, trafficSequence) Begin x←0 For I = 1 to N For j = 1 to N If i < j then x←x+1 trafficSequence(x, 1) = i trafficSequence(x, 2) = j trafficSequence(x, 3) = traffic(i, j) trafficSequence(x, 4) = traffic(j, i) End End End End. // end procedure Step3: Assign ADMs. [ADM, wv] ← assignADM (trafficSequence, N, W, C) // ADM is an N × W boolean matrix in which a ‘1’ in (i, j)-th entry denotes the assignment of an ADM at node i for wavelength j. wv is an N × N × W × C boolean matrix signifying the established connections between the nodes at a wavelength channel and a particular time-slot of that channel. A ‘1’ denotes established connection and ‘0’ denotes no connection. // Procedure assignADM (trafficSequence, N, W, C, ADM, wv) Begin For nCall = 1 to N(N-1)/2 S ← trafficSequence(nCall, 1) // S denotes the source node. // D ← trafficSequence(nCall, 2) // D denotes the destination node.// STOD ← trafficSequence(nCall, 3) // STOD denotes number of requests from S to D. // DTOS ← trafficSequence(nCall, 4) // DTOS denotes number of requests from D to S.//

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pathSTOD ← path (route) from S to D. pathDTOS ← path (route) from D to S. sourceADM ← set of ADMs (corresponding to certain wavelengths) already allocated to the source node, S. destinationADM ← set of ADMs (corresponding to certain wavelengths) already allocated to the destination node, D. minSD ← minimum of STOD and DTOS. If minSD = 0 then If STOD= 0 then path ← pathDTOS trafficSD ← DTOS Else path ← pathSTOD trafficSD ← STOD End If sourceADM and destinationADM both are empty then WVSlots ← Find ‘trafficSD’ number of wavelengths and consequent slots for all wavelengths along ‘path’. Else if sourceADM is empty then // no possibility of having common ADMs between source and destination node // xWVSlots ←Find ‘trafficSD’ number of wavelengths and consequent slots for the wavelengths corresponding to destination ADM along “path”. If free wavelengths and slots for all traffic not found then remTrafficSD ← trafficSD – noXW VSlots. // noXWVSlots ← no. of entries in xWVSlots. // WVSlots ← Find ‘remTrafficSD’ number of wavelengths and consequent slots for all wavelengths along “path”. End Else if destinationADM is empty then // no possibility of having common ADMs between source and destination node // xWVSlots ← Find ‘trafficSD’ number of wavelengths and consequent slots for the wavelengths corresponding to source ADM along “path”. If free wavelengths and slots for all traffic not found then remTrafficSD ← trafficSD – noXWVSlots.

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// noXWVSlots ← no. of entries in xWVSlots. // WVSlots ← Find ‘remTrafficSD’ number of wavelengths and consequent slots for all wavelengths along “path”. End Else // may have common ADMs between source and destination node // commonADM← intersection of the sets sourceADM and destinationADM. If commonADM is empty If noSourceADM > noDestination ADM then xWVSlots ← Find ‘trafficSD’ number of wavelengths and consequent slots for the wavelengths corresponding to sourceADM along “path”. Else xWVSlots ← Find ‘trafficSD’ number of wavelengths and consequent slots for the wavelengths corresponding to destinationADM along “path”. End Else xWVSlots ← Find ‘trafficSD’ number of wavelengths and consequent slots for the wavelengths corresponding to commonADMalong “path”. End If free wavelengths and slots for all traffic not found then remTrafficSD ← trafficSD – noXWVSlots. // noXWVSlots ← no. of entries in xWVSlots. // WVSlots ← Find ‘remTrafficSD’ number of wavelengths and consequent slots for all wavelengths along “path”. End Else // minSD
= 0 //

If STOD = DTOS then commonWVSlots←Find ‘STOD’ number of common usable wavelengths and slots along path STOD and path DTOS with an eye to minimize the number of required ADMs. Else If STOD > minSD then extraSD ← STOD – minSD. Else extraSD ← DTOS – minSD. End [commonWVSlots, extraWVSlots] ← Find ‘minSD’ number of usable wavelengths and slots that is common to both source-to-destination and destinationto-source requests and ‘extraSD’ number of usable wavelengths and slots, which is required for remaining sourceto-destination requests or destinationto-source requests along pathSTOD and pathDTOS. End End If free wavelengths and slots for at least one request found then Update the matrices wv and ADM accordingly. End End End. //end procedure Step 4: noADM ← number of ‘1’s in the matrix ADM. Time-complexity analysis To analyze the time-complexity of the proposed algorithm, the following notations have been used. N = Number of nodes in the network. W = Number of wavelength channels in the network. C = Number of slots in a wavelength channel. P = Average number of hops for a source-destination pair. Time required for ‘maxADM’ function = O(N (N − 1)/2) = O(N 2 ). Time required for ‘trafficSequence’ function = O(N (N − 1)/2) = O(N 2 ).

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Time required to search WADM wavelength channels for free TDM-slots = O(PWADM C). Since, the solution of the problem is not real-time and due to the mismatch in working environment, the timecomplexity of the proposed heuristic is not compared with that of the previous approach (in Ref. [23]).

respectively), whereas the algorithm in Ref. [23] cannot reach it for nodes greater than 11. The busy links in the network for the case N = 8, C = 3 are shown in Fig. 1 below. The ADMs required at the nodes are shown by ellipse. The number of busy links is 84, as it can be verified from the traffic matrix. The busy links in the network for the case N = 4, C = 4 are shown in Fig. 2 below. The ADMs required at the nodes are shown by an ellipse. In this case, number of busy links is ten that is according to the traffic matrix.

Results The proposed algorithm has been experimented with different types of traffic matrix as given below:

B. Non-uniform Traffic

A. All-to-All uniform traffic. B. Non-uniform traffic. C. A set of traffic where requests between node-pairs is uniformly distributed between 0 and C (Where, C is the grooming-capability of the WDM optical network). D. A set of traffic where requests between node-pairs is uniformly distributed between 0 and 2C.

As in Ref. [23], the algorithm has been tested for a nonuniform traffic matrix to show its effectiveness.

All the results correspond to single-hop connection. A. All-to-all uniform traffic This type of traffic has been well studied in literature. In Ref. [23], it has been also experimented extensively. For different algorithms, the results are compared in Ref. [23]. The result of the proposed heuristic is compared with the best results found in Ref. [23] as follows (Table 1). Except the result for the case N = 16 and C = 12, all the other results are same or better than the previous results. The best possible solutions have been reached for C = 64 and below node 17 (It needs 24 and 30 ADMs for the two cases N = 17, C = 64 and N = 18, C = 64

Fig. 1 Established Connections between the nodes for all-to-all uniform traffic (N = 8, C = 3)

Fig. 2 Established Connections between the nodes for all-to-all uniform traffic (N = 4, C = 4)

Table 1 Comparison of the results corresponding to the proposed heuristic with the best results in [23] for all-to-all uniform traffic N = 4 N = 5 N = 6 N = 7 N = 8 N = 9 N = 10 N = 11 N = 12 N = 13 N = 14 N = 15 N = 16 C=3 C=4 C = 12 C = 16 C = 48 C = 64

Previous result Our result Previous result Our result Previous result Our result Previous result Our result Previous result Our result Previous result Our result

7 6 7 4 4 4 4 4 4 4 4 4

12 9 10 8 5 5 5 5 5 5 5 5

17 14 15 12 9 6 6 6 6 6 6 6

21 18 21 16 12 7 11 7 7 7 7 7

31 26 28 23 16 13 14 8 8 8 8 8

36 34 36 29 18 16 18 13 9 9 9 9

48 40 45 37 24 20 20 18 10 10 10 10

67 50 55 45 30 25 26 21 16 11 11 11

69 63 66 54 36 31 33 27 19 12 15 12

78 72 78 64 39 39 37 31 22 13 19 13

95 83 91 74 49 47 42 38 24 16 22 14

105 100 105 85 57 56 46 45 31 22 25 15

124 111 120 102 64 65 57 52 34 27 28 16

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The corresponding network configuration and traffic matrix are stated here to maintain continuity and hence for convenience of readers. Number of nodes in the network, N = 4. Number of wavelength channels in the network, W = 15. Number of time-slots in a wavelength channel, C = 3. Traffic matrix is characterized by the two dimensional matrix, Traffic = {0, 1, 8, 4; 12, 0, 3, 9; 1, 2, 0, 2; 4, 1, 7, 0}. The corresponding established connections are shown below in Fig. 3. The four nodes are denoted as Node1,

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Node2, Node3, and Node4. Since it is ring network, Node1 again comes after Node 4. The number of busy links between the nodes is 123, which can be verified from the traffic matrix for a uni-directional ring. In Fig. 3, wavelength 7 holds a total of eight connections. Table 2 presents the corresponding busy slots along wavelength 7. The number of maximum ADMs that may be required according to ‘maxADM’ function is 32. So, in this case, the defined figure of merit, χ = 3.125%.

C. Uniform traffic with one wavelength capacity For different uniform matrix of one wavelength capacity, the ADM requirements and corresponding figure of merit in terms of ADM savings have been calculated. Results are given in Table 3 and Table 4.

Table 2 Established connections through different slots of wavelength 7

Slot-1 Slot-2 Slot-3

Node 1–2

Node 2–3

Node 3–4

Node 4–1

Node 1–3 Node 1–3 Node 1–4

Node 1–3 Node 1–3 Node 1–4

Node 3–4 Node 3–4 Node 1–4

Node 4–1 Node 4–1 Node 4–1

Table 3 Results for required ADMs with uniform traffic (one wavelength capacity) for the case N = 4, C = 3 Traffic matrix

Fig. 3 Established Connections between the nodes for a nonuniform traffic

0 3 1 2

0 0 3 2

2 1 0 3

1 0 3 0

0 3 3 3

2 0 2 1

2 2 0 2

3 3 2 0

0 0 2 2

2 0 3 0

2 1 0 0

1 1 1 0

0 0 2 1

3 0 3 0

3 0 0 2

1 2 2 0

0 0 2 0

1 0 3 1

0 0 0 0

3 1 2 0

Maximum ADM

Required ADM

Figure of merit, χ

12

11

8.33

12

12

0.00

12

8

33.33

12

10

16.67

12

10

16.67

Photon Netw Commun (2006) 12:153–160

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Table 4 Results for required ADMs with uniform traffic (one wavelength capacity) for the case N = 5, C = 3

Table 6 Results for required ADMs with uniform traffic (two wavelength capacity) for the case N = 5, C = 3

Traffic matrix

Figure of merit, χ

Traffic matrix 0 3 1 4 3

1 0 3 6 6

6 6 0 1 0

4 2 2 0 6

6 3 4 0 0

0 1 6 3 5

0 0 6 1 2

3 6 0 0 2

1 0 2 0 1

0 4 0 0 5

6 0 0 5 3

4 1 0 1 1

1 4 1 0 2

Maximum ADM

0 0 0 0 1

1 0 0 3 0

1 3 0 0 3

2 2 1 0 1

0 0 1 1 0

0 0 3 0 0

2 0 3 3 1

1 2 0 0 3

1 2 2 0 3

2 0 1 1 0

0 2 2 3 0

1 0 2 3 1

3 2 0 1 3

1 2 2 0 2

2 2 2 2 0

Required ADM

18

14

22.22

20

18

10.00

20

20

0.00

Maximum ADM

Required ADM

Figure of merit, χ

36

32

11.11

5 3 3 3 0

26

24

7.69

6 4 0 0 0

30

25

16.67

D. Uniform traffic with two wavelength capacity

Conclusions

For different uniform matrix of two wavelength capacity, the ADM requirements and corresponding figure of merit in terms of ADM savings have been calculated. Results for the same are given in Table 5 and Table 6.

In this article, a heuristic algorithm is proposed to solve the ADM minimization problem for WDM optical networks. Since in a uni-directional ring, there is only one path from one node to another node and the path of (i, j)-th source–destination pair is complementary to the path of (j, i)-th source–destination pair, the pair-up of the corresponding calls as proposed is quite intuitively justified. The overall results are quite better than that of previously proposed approach as in Ref. [23].

Table 5 Results for required ADMs with uniform traffic (two wavelength capacity) for the case N = 4, C = 3 Traffic matrix 0 6 4 3

6 0 5 0

2 0 0 5

1 6 1 0

0 6 1 0

3 0 0 3

2 2 0 0

2 0 3 0

0 4 4 4

4 0 1 4

1 4 0 4

4 1 3 0

0 5 4 0

6 0 0 1

0 2 0 3

2 6 0 0

0 1 0 2

5 0 0 3

4 2 0 6

6 1 5 0

Maximum ADM

Required ADM

Figure of merit, χ

22

21

4.55

14

11

21.43

24

19

20.83

18

16

11.11

20

17

15.00

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