Improvement of multi-hop packet transmission scheduling in WDM optical star networks

Computer Communications 33 (2010) 706–713

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Computer Communications journal homepage: www.elsevier.com/locate/comcom

Improvement of multi-hop packet transmission scheduling in WDM optical star networks Hann-Jang Ho a,*, JungChun Liu b a b

Department of Computer Science and Information Engineering, Wufeng Institute of Technology, 117, Sec. 2, Jianguo Rd., Minsyong, Chiayi 621, Taiwan Department of Management Information System, Far East University, 49, Chung Hua Rd., Hsin-Shih, Tainan 744, Taiwan

a r t i c l e

i n f o

Article history: Received 2 July 2008 Received in revised form 20 November 2009 Accepted 20 November 2009 Available online 27 November 2009 Keywords: Wavelength division multiplexing (WDM) All-to-all broadcast (AAB) Multi-hop Schedule length Tuning latency

a b s t r a c t This paper addresses the design of multi-hop packet transmissions for the all-to-all broadcast (AAB) problem in a wavelength division multiplexing (WDM) optical star network with N nodes. We assume that each node is equipped with a tunable transmitter and a fixed-tuned receiver (TT-FR), and each tunable transmitter needs a non-negligible tuning latency d to switch between wavelengths. To reduce the number of O/E/O conversions in the multi-hop scheduling, the maximum hop distance of packet transmissions is limited to q. We propose an improved multi-hop scheduling algorithm to shorten the duration of scheduling periods. As q is odd, the number of tuning operations on each node is at most   e  1Þ qðq2þ1Þ þ d . d2N=ðq þ 1Þe  1 and the schedule length of our multi-hop scheduling is at most ðdq2N þ1 As q is even, the tuning time on each node is at most d2N=qe  1 and the schedule length is at most pffiffiffiffiffiffiffiffiffiffiffiffiffiffi   2 ðd2N=qe  1Þ qðq21Þ þ d . For general case, when q < 1þdþ 2d 6dþ5, the schedule length of our multi-hop scheduling algorithm will be less than that of the optimal single-hop scheduling algorithm. This also improves the previous result on the multi-hop schedule length of the AAB problem. Ó 2009 Elsevier B.V. All rights reserved.

1. Introduction Packet scheduling provides a way to reduce routing control and to guarantee overall throughput. An interesting special case of the packet scheduling is all-to-all broadcast (AAB) in which every transmitter/receiver pair has exactly one packet to be transferred. The AAB is one of the most common communication patterns in high performance computing. There are many high-performance applications, including matrix transpose, multidimensional convolution, and data redistribution [1]. In this paper, we consider the AAB problem in a wavelength division multiplexing (WDM) optical star network in which (1) nodes are connected to a passive star coupler via two-way fibers in order to broadcast from all inputs to all outputs, (2) each node is equipped with one tunable transmitter (tunable over all available WDM channels) and one fixedtuned receiver, and (3) the connectivity between nodes can be achieved by tuning wavelengths of transmitters to the receiving wavelengths of receivers [2]. Every node can select at a given time slot among the available wavelengths to perform transmission. However, the tuning time cannot be neglected with respect to the packet transmission time. With some of the optical components available today, the tuning latencies of tunable lasers need * Corresponding author. Tel.: +886 5 2267125x21711; fax: +886 5 2264087. E-mail addresses: [email protected] (H.-J. Ho), [email protected] (J. Liu). 0140-3664/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.comcom.2009.11.017

a few ms. It is much longer than the packet transmission time which is almost less than a fraction of ls [3]. Packet scheduling in WDM optical networks can be typically classified as either single-hop or multi-hop. In the single-hop scheduling, each packet transmits from the source to its destination completely in the optical domain without optical to electronic (O/E) and electronic to optical (E/O) conversions at intermediate nodes. Although single-hop transmission eliminates the O/E/O conversions at intermediate nodes, it requires significant amount of tuning latencies among the nodes in the network. Yeh et al. [4] proposed an optimal single-hop scheduling algorithm with schedule N e; N  1 þ wdg for the AAB problem, where N length maxfðN  1Þdw is the number of nodes, w is the number of available wavelengths, and d is the tuning latency of the tunable transmitter. Each node connects and communicates directly with every other node by appropriately tuning its transmitter. However,the schedule length of the optimal single-hop scheduling algorithm is seriously affected by the non-negligible tuning latency because each node must tune its transmitter w times in an all-to-all broadcast. For reducing the number of tuning operations and shorten the duration of schedule length, the multi-hop transmission scheduling is an alternative method for solving the AAB problem. In multi-hop scheduling, each packet transmission can be relayed though some intermediated nodes from source to destination to reduce the number of tuning operations on each node [5]. For example, we consider

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H.-J. Ho, J. Liu / Computer Communications 33 (2010) 706–713

ogies by our method is at most d2N=qe  1. When q ¼ 3, the nume of logical topologies by [6] is almost close to N  1. Our ber 2dN1 3 method constructs only dN2 e  1 logical topologies which can ape. proach to 1.5 times the lower bound dN1 3 In [7], Marsan et al. proposed a multi-hop scheduling algorithm for the general scheduling problem. Especially, when the tuning latency is large enough and the hop distance of packet transmissions is at most 3, their schedule length is shorter than that of single-hop scheduling algorithm. In this paper, as q ¼ 3, we will produce dN=2e  1 logical topologies that is 1.5 times the optimal solution of the q-LTC problem. In fact, our multi-hop scheduling needs dN=2e  1 tuning actions on each node for the 3-LTC problem, and its schedule length is ðdN=2e  1Þð6 þ dÞ. When d is larger than or equal to 4 slotsðd P 4Þ, the schedule length is smaller than the optimal schedule length Nd þ N  1 of single-hop scheduling. In pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 general case, when q < 1þdþ 2d 6dþ5, the multi-hop schedule length    dq2N e  1 qðq2þ1Þ þ d will be less than the optimal schedule þ1

the AAB problem in a WDM optical star network with seven nodes as Fig. 1(a) in which the receiving wavelength of node i is fixed as wi ; 0 6 i 6 6. By using the single-hop scheduling algorithm in [4], the optimal schedule length is 7d þ 6 time slots because the transmitter of each node needs to tune its wavelength to the receiving wavelength of each node for finishing an AAB operation. In contrast, the schedule length can be reduced by using the multi-hop scheduling. Assume that the maximum hop distance is restricted to 3. Firstly, each node i tunes its transmitting wavelength to wððiþ1Þmod7Þ for constructing a logical topology as Fig. 1(b). We schedule packet transmissions with one, two and three hop distance separately in Table 1. Next, each node i tunes its transmitting wavelength to wðði1Þmod7Þ for constructing another topology in Fig. 1(c). Packet transmissions can also be scheduled similarly as Table 1. The schedule length of multi-hop scheduling is 2d þ 12 which is shorter than the length of optimal single-hop scheduling when the tuning latency d is greater than 65 time slots. Although the adoption of multi-hop scheduling can reduce the number of tuning operations to provide an important throughput advantage over single-hop scheduling, the O/E/O conversion operations at intermediate nodes will lead to buffer overflows and increase transmission latencies. Decreasing the number of intermediate hops can reduce the transmission latencies of packets, but it will also increase the number of tuning operations of nodes and decrease the overall throughput. There is a tradeoff between throughput and delay in the multi-hop scheduling. Limiting the maximum number of hops will lead to a good compromise in terms of throughput vs delay with respect to single-hop and unbounded multi-hop scheduling. In this paper, we restrict the maximum hop distance to a constant q and propose an improved multi-hop scheduling algorithm for the AAB problem. The problem of logical topology construction with maximum q hop distance is called the q-LTC problem. The objective of the q-LTC problem is to minimize the number of logical topologies for reducing the number of tuning actions. In [6], we partitioned N nodes into groups with size either q or q þ 1, and designed 2dN1 q e logical rings However, the result is not close to at most for the q-LTC problem.  the optimal solution dN1 q e of the q-LTC problem. In this paper, we will improve previous results [6] by designing better logical topologies. We partition N nodes into x groups and the size of . Then we can simplify the degroups is reduced to either q2 or qþ1 2 sign of logical topologies by considering only these x groups. If x is odd, we transfer the problem to the Hamiltonian cycles problem on a complete directed graph with x vertices, and propose an algorithm to obtain x  1 logical rings. The number of logical topologies by our method is at most d2N=ðq þ 1Þe  1. On the other hand, if x is even, we transfer the problem to the round robin tournament problem with x players, and apply the tournament algorithm for constructing x  1 logical topologies. The number of logical topol-

1

N length maxfðN  1Þdw e; N  1 þ wdg of single-hop scheduling in [4]. This also improves previous results in [6].

2. Problem definition We consider the AAB problem in a WDM optical star network with N nodes and w wavelengths. For simplifying the design of logical topologies, we firstly consider the case when N ¼ w, and then extend it to the general case when N 6 w. To avoid the influence of transmission delay, we restrict the maximum hop distance of packet transmission to a constant q, and the problem is called the qAAB problem. In slotted schedule systems, packets have fixed length so that a packet transmission time equals one time slot, and we assume that the tuning latancy for each transmitter to tune from one wavelength to another needs d time slots. The design of multi-hop scheduling algorithm is generally divided into two stages: (1) design logical topology, and (2) schedule packet transmissions [5]. In the first part, to simplify the routing method and to decrease the affection of tuning delay, we assume that all transmitters do tuning operations synchronously in a multi-hop scheduling period. After a tuning operation of each transmitter, we can construct a logical topology by properly setting up the transmitting wavelength of each node. For example, we construct four logical topologies in Fig. 2 for the 3-AAB problem. The logical topology r 1 in Fig. 2(a) is created by that each node i tunes the transmitting wavelength to wððiþ1Þmod11Þ simultaneously. Similarly, logical topologies r2 ; r 3 and r 4 in Fig. 2 are constructed by that each node i separately tunes the transmitting wavelength to wððiþ2Þmod11Þ ; wðði1Þmod11Þ and wðði2Þmod11Þ . In the second step, we adopt the pipeline transmission mechanism [8] on each logical topology. The transmission period on each

1

1

0

2

0

2

0

2

6

3

6

3

6

3

5

4

5

4

Fig. 1. Logical topologies for the 3-AAB problem.

5

4

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H.-J. Ho, J. Liu / Computer Communications 33 (2010) 706–713

Table 1 Packet scheduling for the 3-AAB problem on logical topology as Fig. 1(b). Nodes 0 1 2 3 4 5 6

Slot 1

Slot 2

0!1 1!2 2!3 3!4 4!5 5!6 6!0

Slot 3

Slot 4

0!1!2 1!2!3 2!3!4 3!4!5 4!5!6 5!6!0 6!0!1

Slot 5

0 9

Slot 6

0!1!2!3 1!2!3!4 2!3!4!5 3!4!5!6 4!5!6!0 5!6!0!1 6!0!1!2

7

w7

w8 6

Tuning

7

9 w0

w2 w3

8 w9 w8

2

w4 3 w7

10

w6

6

5

(a) r 1

w5

4

7

3

w5

w8 6 w10

w5

w10 1

10

(b) r 2

w1

1

10

8

Lemma 1. If nodes in any two groups g i and g j are directly connected with a logical topology L, all nodes in g i can send packets through q hop distance at most to all nodes in g j on L, where i; j 2 f0; 1; ::; x  1g. Proof. If q is odd, each group has at most ðq þ 1Þ=2 nodes. Otherwise, each group has at most q=2 nodes. In an extreme case when the logical topology L is linear, the maximum hop distance between any two nodes is at most q. h To apply the condition of Lemma 1 on the q-LTC problem, we need to ensure that each group in Gx is adjacent to all other groups on some logical topologies. If the number of groups in Gx is even, we reduce the q-LTC problem to Hamiltonian Cycles (HC) problem.

Tuning 0 w10

10

w9 4

w8 6 w1

w3

Fig. 4. Pipeline scheduling packet transmissions with 3 hop distance on the logical topology r 2 in Fig. 2(b).

w6

w3

4

w7

3

w4

5 w7

w4 w6

5

2

w9

2

w2

w9

Tuning

1

w10

7

1

8

0

Tuning

9

w1

9

As d is large enough, the schedule length of the multi-hop scheduling is deeply affected by the number of logical topologies. This section focuses on the design of logical topologies, and the logical topologies construction problem with maximum q hop distance is called q-LTC problem. The lower bound of the q-LTC problem is at least dðN  1Þ=qe logical topologies [6]. For constructing logical topologies, we partition N nodes into x groups, g 0 ; g 1 ; . . . ; g x1 . If the maximum hop distance q is even, let x=dqN=2e ¼ d2N=qe. Nodes are divided into d2N=qe groups and Gx ¼ fg 0 ; g 1 ; . . . ; g d2N=qe1 g where g i ¼ ðiq=2; iq=2 þ 1; iq=2 þ 2; . . . ; ði þ 1Þq=2  1Þ, for all i 2 f0; 1; . . . ; d2N=qe  2g, and g d2N=qe1 ¼ ððd2N=qe  1Þq=2; ðd2N=qe  1Þq=2 þ 1; . . . ; N  1Þ. When q ¼ 4 e ¼ 5. Then g 0 ¼ f0; 1g; g 1 ¼ f2; 3g; g 2 ¼ f4; 5g; and N ¼ 9; x ¼ d29 4 g 3 ¼ f6; 7g and g 4 ¼ f8g. N e ¼ d2N=ðq þ 1Þe. We divide N nodes If q is odd, let x ¼ dðqþ1Þ=2 into d2N=ðq þ 1Þe groups and Gx ¼ fg 0 ; g 1 ; . . . ; g d2N=ðqþ1Þe1 g where g i ¼ ðiðq þ 1Þ=2; iðq þ 1Þ=2 þ 1; iðq þ 1Þ=2 þ 2; . . . ; ði þ 1Þðq þ 1Þ=2  1Þ, and g d2N=ðqþ1Þe1 ¼ for all i 2 f0; 1; . . . ; d2N=ðq þ 1Þe  2g, ðq þ 1Þe  1Þðq þ 1Þ=2 þ 1; . . . ; ððd2N=ðq þ 1Þe  1Þðq þ 1Þ=2; ðd2N= 217 e ¼ 6. So N  1Þ. For the case, q ¼ 5 and N ¼ 17; x ¼ dð5þ1Þ g 0 ¼ f0; 1; 2g; g 1 ¼ f3; 4; 5g; g 2 ¼ f6; 7; 8g; g 3 ¼ f9; 10; 11g; g 4 ¼ f12; 13; 14g and g 5 ¼ f15; 16g. Assume that there are two groups g i ¼ ða1 ; a2 ; . . . ; ap Þ and g j ¼ ðb1 ; b2 ; . . . ; bq Þ. If g i is arranged immediately before g j on a logical topology, then we define the connection of these nodes on the logical topology as a1 ; a2 ; . . . ; ap ; b1 ; b2 ; . . . ; bq .

w0

w3

Fig. 3. Pipeline scheduling packet transmissions with 2 hop distance on the logical topology r 2 in Fig. 2(b).

3. Logical topology design

10

w10

w5

w0

0 w2

4

w6

3

0 w1

w4

w9

5

logical topology is divided into q stages. On the logical topology r 2 , we schedule the pipeline transmission with one hop distance in Fig. 2(b), two hop distance in Fig. 3 and three hop distance in Fig. 4. Therefore, the schedule of pipeline transmissions on r2 is shown in Table 2. Obviously, packet transmissions with k hops are sent in the kth stage, and it will takes qðq2þ1Þ time slots to process all packets transmissions on each logical topology. If there are c logical topologies for the q-AAB problem, the schedule length is equal to cðqðq2þ1Þ þ dÞ.

2

w2

w0

8

9

1 w0

w8

w1

8 w7 7

w6

2

w2 3 w5

w4

6

5

w3

0 w9

9 w7

4

(c) r3

Fig. 2. Four logical topologies for the 3-AAB problem with 11 nodes.

7

2 w0

w5

w2

5 w3 3

w1

4

w4 6 w6 w10 1

w8

(b) r 4

10

8

709

H.-J. Ho, J. Liu / Computer Communications 33 (2010) 706–713 Table 2 Packet scheduling for the 3-AAB problem on the logical topology r 2 as Fig. 2(b). Nodes

0 1 2 3 4 5 6 7 8 9 10

1-st stage(1 hop)

2-nd stage(2 hop)

3-rd stage(3 hop)

Slot d þ 1

dþ2

dþ3

dþ4

dþ5

dþ6

0!2 1!3 2!4 3!5 4!6 5!7 6!8 7!9 8 ! 10 9!0 10 ! 1

0!2 1!3 2!4 3!5 4!6 5!7 6!8 7!9 8 ! 10 9!0 10 ! 1

2!4 3!5 4!6 5!7 6!8 7!9 8 ! 10 9!0 10 ! 1 0!2 1!3

0!2 1!3 2!4 3!5 4!6 5!7 6!8 7!9 8 ! 10 9!0 10 ! 1

2!4 3!5 4!6 5!7 6!8 7!9 8 ! 10 9!0 10 ! 1 0!2 1!3

4!6 5!7 6!8 7!9 8 ! 10 9!0 10 ! 1 0!2 1!3 2!4 3!5

We reduce the q-LTC problem to Round Robin Tournament (RRT) problem when x is odd. Our results show that x  1 logical topologies are sufficient to solve the q-LTC problem, i.e. we need d2N=qe 1 logical topologies when q is even and d2N=ðq þ 1Þe  1 logical topologies when q is odd. 3.1. Design by hamiltonian cycles(HC) When the number x of groups is odd, we reduce the q-LTC problem to the HC problem and design logical rings to satisfy the condition of Lemma 1. We transfer Gx to a complete directed graph D as follows. Each group g i in Gx is represented as a vertex in D and g i is adjacent to g j for 0 6 i; j 6 x  1 and i – j. In Fig. 5, there is a complete directed graph D with five groups for q ¼ 4 and N ¼ 9. In [9], Chartrand and Lesniak proposed several methods to create all Hamiltonian cycles on a complete undirected graph. They Hamiltonian cycles on also indicated that if x is odd, there are x1 2 a complete undirected graph. Since the logical topology in the multi-hop transmission should be directed, by applying methods in [9], we propose the Logical Topologies Construction (LTC) Algorithm to construct x  1 logical rings for the q-LTC problem as follows. LTC-Odd Algorithm 1. Ignore the direction of edges on D and apply methods in [9] to Hamiltonian cycles for x vertices. Assume that we create x1 2 g of all Hamiltonian cycles. obtain the set fHi j1 6 i 6 x1 2 2. Add the direction to all edges of all Hamiltonian cycles in     Hi j1 6 i 6 x1 . We have the logical ring set r i j1 6 i 6 x1 , 2 2 where we denote the logical ring r i as  g i0 ; g i1 ; g i2 ; g i3 ; . . . ; g iðx2Þ ; g iðx1Þ . 3. Reverse the direction of edges on each logical ring ri ¼ g i0 ; g i1 ; g i2 ; g i3 ; . . . ; g iðx2Þ ; g iðx1Þ  in fr i g, then we obtain another logical ring ri ¼ g iðx1Þ ; g iðx2Þ ; g iðx3Þ ; g iðx4Þ ; . . . ; g i1 ; g i0  for 1 6 i . 6 x1 2 In Fig. 5, we find two Hamiltonian cycles, H1 ¼ ðg 0 ; g 1 ; g 2 ; g 3 ; g 4 Þ and H2 ¼ ðg 0 ; g 2 ; g 4 ; g 1 ; g 3 Þ. We design r1 ¼ g 0 ; g 1 ; g 2 ; g 3 ; g 4 ¼

g0

0; 1; 2; 3; 4; 5; 6; 7; 8  by H1 , and constrcut r2 ¼ g 0 ; g 2 ; g 4 ; g 1 ; g 3 ¼ 0; 1; 4; 5; 8; 2; 3; 6; 7  by H2 . r1 is designed as  8; 7; 6; 5; 4; 3; 2; 1; 0  by r 1 ; r2 is constructed as  7; 6; 3; 2; 8; 5; 4; 1; 0  by r 2 . When x is odd, the LTC-Odd Algorithm creates x  1 logical  and topologies with two logical ring sets ri j1 6 i 6 x1 2   r i j1 6 i 6 x1 for the q-LTC problem. On each logical ring 2 ri ; 1 6 i 6 ðx  1Þ=2, nodes in group g ij are clockwise connected with nodes in g ið jþ1Þ for 1 6 i 6 x  1. On the other hand, nodes in groups g ij and g ið j1Þ are also clockwise connected on each logical ring ri ; 1 6 i 6 ðx  1Þ=2. 3.2. Design by round robin tournament(RRT) When the number x of groups is even, we reduce the q-LTC problem to the RRT problem with x players to produce logical topologies. In the RRT problem, we need to schedule a tournament such that each player plays once with every other person in finite rounds. If x is even, there are x  1 rounds for the RRT problem with x players [10]. Let Ri be a round fðg i0 ; g i1 Þ; ðg i2 ; g i3 Þ; . . . ; ðg iðx2Þ ; g iðx1Þ Þg of the RRT problem with x players, where i0 < i2 <    < iðx2Þ , and ði0 ; i1 ; . . . ; ix1 Þ is one permutation of ð0; 1; . . . ; x  1Þ. For the case, q ¼ 5 and N ¼ 17, we solve the RRT problem with six players and schedule the tournament in Table 3. There are five rounds and each round has three playing games. The round 3 is represented as R3 ¼ fðg 0 ; g 2 Þ; ðg 1 ; g 5 Þ; ðg 3 ; g 4 Þg. We translate these x  1 rounds of the RRT problem to logical topologies for the q-LTC problem as follows. On each round, each player pair ðg ip ; g iq Þ represents a small logical ring  g ip ; g iq  in which nodes in group g ip are clockwise connected with nodes in g iq . Let Li be the logical topology constructed by Ri . Since there are x=2 player pairs in each round, each logical topology should contain with x=2 small logical rings. Let Li ¼ f g i0 ; g i1 ;  g i2 ; g i3 ; . . . ;  g iðx2Þ ; g iðx1Þ g. For example, the logical topology of round R3 in Table 3 is represented as L3 ¼ f g 0 ; g 2 ;  g 1 ; g 5 ;  g 3 ; g 4 g ¼ f 0;1;2;6;7;8 ; 3;4;5;15;16 ; 9;10;11;12;13;14 g. In Fig. 6, there are three small rings on the logical topology L3 . Next we propose the LTC-Even Algorithm to construct x  1 logical rings for the q-LTC problem as follows. LTC-Even Algorithm 1. Establish ðx  1Þ rounds in the RRT problem with x player, and let fRi j1 6 i 6 ðx  1Þg be the set of finite rounds. 2. For each round Ri ; 1 6 i 6 x  1, construct a logical topology Li with x=2 small logical rings. Let fLi j1 6 i 6 ðx  1Þg be the set of x  1 logical topologies and Li ¼ f g i0 ; g i1 ;  g i2 ; g i3 ; . . . ;  g iðx2Þ ; g iðx1Þ g. 4. Packet scheduling In this section, we schedule packet transmissions of the AAB problem on logical topologies, and compute the schedule length of our multi-hop scheduling algorithm. 4.1. Packet scheduling for LTC-Odd

g4

g1

When the number x of groups is odd, the LTC-Odd Algorithm   and have designed x  1 logical topologies with ri j1 6 i 6 x1 2 Table 3 The RRT problem with six players.

g3

g2

Fig. 5. A complete directed graph with five groups.

Round 1 ðR1 Þ

Round 2 ðR2 Þ

Round 3 ðR3 Þ

Round 4 ðR4 Þ

Round 5 ðR5 Þ

ðg 0 ; g 3 Þ ðg 1 ; g 4 Þ ðg 2 ; g 5 Þ

ðg 0 ; g 1 Þ ðg 2 ; g 3 Þ ðg 4 ; g 5 Þ

ðg 0 ; g 2 Þ ðg 1 ; g 5 Þ ðg 3 ; g 4 Þ

ðg 0 ; g 5 Þ ðg 2 ; g 4 Þ ðg 1 ; g 3 Þ

ðg 0 ; g 4 Þ ðg 3 ; g 5 Þ ðg 1 ; g 2 Þ

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H.-J. Ho, J. Liu / Computer Communications 33 (2010) 706–713

0

3

8

1

g0

9 4

g1

14

10

g3

16 7

g2

g5 2

5

6

15

13

g4 11 12

Fig. 6. Three small rings on the logical topology L3 .

  r i j1 6 i 6 x1 for the q-LTC problem. Assume that nodes in each 2 logical ring r i are clockwise connected as  ni0 ; ni1 ; . . . ; niN1 . Similarly, nodes in ri are also clockwise connected as  niN1 ; niN2 ; . . . ; ni0 , where 1 6 i 6 x1 . We apply the pipeline transmission sched2 uling [8] to transmit packets on logical rings as follows. Scheduling Algorithm for LTC-Odd g, each node nij tunes its 1. On each logical ring ri in fri j1 6 i 6 x1 2 transmitting wavelength to the receiving wavelength of niððjþ1ÞmodNÞ for 0 6 j 6 N  1. Now each node nij transmits a packet through k hop distance to niððjþkÞmodNÞ for 0 6 j 6 N  1 and 1 6 k 6 ., where . ¼ q if q is odd and . ¼ q  1 otherwise.   2. On each logical ring r i in ri j1 6 i 6 x1 , each node nij tunes its 2 transmitting wavelength to the receiving wavelength of niððj1ÞmodNÞ for 0 6 j 6 N  1. Similarly, each node nij transmits a packet through k hop distance to niððjkÞmodNÞ for 0 6 j 6 N  1 and 1 6 k 6 ., where . ¼ q if q is odd and . ¼ q  1 otherwise. On each logical ring r i , the scheduling period is partitioned into q stages at most. Each packet transmission with k hop distance is sent in the kth stage, where 1 6 k 6 q. It takes at most qðq2þ1Þ time slots for sending packet transmissions on ri . Since the tuning time   qðqþ1Þ is d, the schedule length on fr i g is at most ðx1Þ þ d .) At the 2 2   qðqþ1Þ same reason, the schedule length on fri g is also ðx1Þ þ d at 2 2   most. Thus, the total is ðx  1Þ qðq2þ1Þ þ d at most when x is odd. 4.2. Packet scheduling for LTC-Even When the number x of groups is even, the LTC-Even algorithm already constructs a set fLi j1 6 i 6 x  1g of logical topologies, where Li ¼ f g i0 ; g i1 ;  g i2 ; g i3 ; . . . ;  g iðx2Þ ; g iðx1Þ g. On each logical topology, packet transmissions are scheduled on an individual group pair (small logical ring) independently. We assume that nodes in a group pair  g ij ; g ijþ1  are clockwise connected as  nij0 ; nij1 ; . . . ; nij.  to form a small logical ring, where if q is even then . ¼ q  1 else . ¼ q. Now, by applying the pipeline transmission scheduling, we describe the scheduling algorithm on these logical topologies as follows. Scheduling Algorithm for LTC-Even 1. On each logical topology Li in fLi j1 6 i 6 x  1g, each node nijt tunes its transmitting wavelength to the receiving wavelength of nijððtþ1Þmod.Þ for 0 6 t 6 .  1 and j 2 f0; 2; . . . ; x  2g. 2. On each small logical ring  nij0 ; nij1 ; . . . ; nij.  in Li , each node nijt transmits a packet through k hop distance to nijððtþkÞmod.þ1Þ for 0 6 t; k 6 . and j 2 f0; 2; . . . ; x  2g.

On  nij0 ; nij1 ; . . . ; nij. , each node transmits k packets through hop distance k to destinations separately. The transmission period on the scheduling is divided into k stages and each packet transmission with k hop distance is sent in the kth stage. It takes at most qðqþ1Þ time slots for sending packets. Although there are 2x small log2 ical rings on each logical topology Li in fLi g, the pipeline transmission on each ringis processed  independently. So, the total schedule length is ðx  1Þ qðq2þ1Þ þ d where x is even. 4.3. Performance No matter which scheduling algorithm we use, the maximum   schedule length is always ðx  1Þ qðq2þ1Þ þ d time slots at most. If

q is odd then x ¼ d2N=ðq þ 1Þe, and x ¼ d2N=qe otherwise. Therefore, the number of logical topologies is dðq2N e  1 if q is odd. þ1Þ 2N And, the number of logical topologies is d q e  1 if q is even. We improve the schedule length of multi-hop scheduling as follows:    1 qðq2þ1Þ þ d .   qðq1Þ  If q is even, the schedule length is ðd2N þd . 2 q e  1Þ  If q is odd, the schedule length is

l

m

2N ðqþ1Þ

The schedule length of our multi-hop scheduling algorithms is    qðqþ1Þ þ d in [6]. Moreshorter than the schedule length 2 dN1 2 q e over, when q ¼ 3, the number 2dN1 e of logical topologies by [6] is 3 almost close to N  1. Our method constructs only dN2 e  1 logical   . topologies, and can approach to 1.5 times the lower bound N1 3 When the tuning latency on each transmitter is greater than or equal to 4 time slots, the multi-hop schedule length ðdN=2e  1Þð6 þ dÞ should be shorter than that of optimal singlepffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 hop scheduling in [4]. For general case, when q < 1þdþ 2d 6dþ5, out multi-hop scheduling algorithm is better than the optimal single-hop scheduling algorithm. 5. The general case of AAB problem Generally, the number of nodes is grater than the number of wavelengths in WDM networks. In this section, we will propose an extend strategy which is scalable in terms of number of nodes and wavelengths for the AAB problem. Considering an WDM optical star network with N available wavelengths, we assume that there are at most h receivers with the same fixed receiving wavelength. Now, we aim at the AAB problem in WDM optical star networks with N available wavelengths and the number of nodes is at most hN. All nodes are partitioned into N groups G0 ; G1 ; . . ., and GN1 . For load balancing in each wavelength channel, we assume that node j is in the group GjmodN ; ð0 6 j 6 hN  1 and the receiving wavelength of each receiver in the group Gi is wi , where 0 6 i 6 N  1.

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wavelength to the receiving wavelength of nodes in

G1j

for

0 6 j 6 N  1. Each node n1j;k transmits a packet through one hop distance to n1j;l for 0 6 j 6 N  1 and 1 6 k–l 6 h.   , each node nij;k in Gij 2. On each logical ring ri in ri j1 6 i 6 x1 2 tunes its transmitting wavelength to the receiving wavelength of nodes in Giððjþ1ÞmodNÞ for 0 6 j 6 N  1. Each node nij;k in Gij transmits packets through

m hop distance to all nodes in

GiððjþmÞmodNÞ for 1 6 k 6 h and 1 6 m 6 ..   , each node nij;k in Gij 3. On each logical ring ri in ri j1 6 i 6 x1 2 tunes its transmitting wavelength to the receiving wavelength of nodes in Giððj1ÞmodNÞ for 0 6 j 6 N  1. Each node nij;k in Gij transmits packets through m hop distance to all nodes in GiððjmÞmodNÞ for 1 6 k 6 h and 1 6 m 6 .. In Step 1, it spends d slots for a tuning latency on each transmitter. Each node in each group must transmit a packet to another node in the same group in a time slot on the logical ring r 1 . It takes hðh  1Þ time slots for all nodes in Step 1. In Step 2 and Step 3, the schedule period is partitioned into m stages on each logical ring. In the .th stage, each node in a group sends h packets to all nodes in another group with

m hop distance, and it takes mh2 time slots, 2

where 1 6 m 6 . 6 q. Then it spends h qðq2þ1Þ time slots at most for packet transmissions on each logical ring. Therefore, the schedule length for the LTC-Odd scheduling algorithm is at most



2 qðq þ 1Þ hðh  1Þ þ d þ ðx  1Þ h þd : 2 On the other hand , when we apply the LTC-Even Algorithm to construct logical topologies as fLi g, these N groups are partitioned into x groups and x is even. On each logical topology Li in fLi j1 6 i 6 x  1g, we assume that groups in a group pair  g ij ; g ijþ1  are clockwise connected as  Gij0 ; Gij1 ; . . . ; Gij.  to form a small logical ring, where if q is even then . ¼ q  1 else . ¼ q. By applying the pipeline transmission scheduling, we describe the scheduling algorithm on these logical topologies as follows. Scheduling Algorithm for LTC-Even 1j 1. On the logical topology L1 , each node n1j t;k in Gt tunes its transmitting wavelength to the receiving wavelength of nodes in G1j t for 0 6 t 6 .; j 2 f0; 2; . . . ; x  2g and 1 6 k 6 h. 1j 1j 1j 2. On each small logical ring  G0 ; G1 ; . . . ; G.  in L1 , each node 1j 1j n1j t;k in Gt transmits a packet through one hop distance to nt;l for 0 6 t 6 .; j 2 f0; 2; . . . ; x  2g and 1 6 k–l 6 h. 3. On each logical topology Li in fLi j1 6 i 6 x  1g, each node nijt;k in Gijt tunes its transmitting wavelength to the receiving wavelength of nodes in Gijððtþ1Þmod.þ1Þ for 0 6 t 6 .; j 2 f0; 2; . . . ; x  2g and 1 6 k 6 h. 4. On each small logical ring  Gij0 ; Gij1 ; . . . ; Gij.  in Li , each node nijt;k

in Gijt transmits a packet through GijððtþmÞmod.þ1Þ . . . ; x  2g.

m hop distance to all nodes in for 0 6 t 6 .; 1 6 m 6 .; 1 6 k 6 h and j 2 f0; 2;

Moreover, in Step 4, the schedule period is partitioned into m stages on each small logical ring. In the mth stage, each node in a group transmits h packets to all nodes in another group with m hop dis2

tance, and it takes mh time slots, where 1 6 m 6 . 6 q. So it spends 2 h qðq2þ1Þ

time slots at most for packet transmissions on each logical topology. Thus the schedule length of the LTC-Even scheduling algorithm is at most



2 qðq þ 1Þ TSGSALT ¼ hðh  1Þ þ d þ ðx  1Þ h þd : 2 6. Simulation We implement the q-LTC scheduling algorithm and carry out an experimental simulation to compare the average scheduling length with related results [4,6,7]. In our simulations, the WDM optical star network is considered with 64 nodes and 32 wavelengths ðh ¼ 2Þ and the tuning latency d is restricted to between 20 to 100 time slots. The performance curves are plotted separately in Figs. 7–9. The vertical axis represents the schedule length and the horizontal axis represents the tuning latency. First of all, we compare the schedule length with the optimal single-hop scheduling algorithm. In [4], Yeh et al. proposed an optimal scheduling algorithm with schedule length  single-hop N   ; N  1 þ wd for the AAB problem, where d is the max ðN  1Þ w tuning latency of the tunable transmitter. In [7], Marsan et al. proposed a heuristic scheduling algorithm for the transmission scheduling problem. For logical topology design, their heuristic algorithm aims at finding an optimal single-hop traffic matrix such that the schedule length is minimized. In the scheduling stage,

Schedule length[slots]

1. On the logical ring r 1 , each node n1j;k in G1j tunes its transmitting

In Step 1, each transmitter tunes simultaneously its wavelength to another on the logical topology L1 , which takes d slots for a tuning latency. In Step 2, on each small logical ring in L1 , each node in each group sends a packet to another node in the same group in a time slot. Then it spends hðh  1Þ time slots for all nodes in Step 2. l m  2N In Step 3, there are q  1 tuning operations in scheduling.

3500 3000 2500 2000 1500 1000 500 0

SH 2H-LTC 3H-LTC 4H-LTC 5H-LTC 20

40

60

80

100

Tuning latency[slots] Fig. 7. Comparison of the schedule length with the optimal single-hop scheduling algorithm in [4] for N ¼ 32 and h ¼ 2.

Schedule length[slots]

In the network with N groups, we use the LTC-Odd Algorithm to create logical rings fr i g [ fri g, when x is odd. We suppose that groups in logical ring r i are clockwise connected as  Gi0 ; Gi1 ; . . . ; GiN1 . Similarly, groups in r i are also clockwise connected as . We denote that . ¼ q if  GiN1 ; GiN2 ; . . . ; Gi0 , where 1 6 i 6 x1 2 q is odd and . ¼ q  1 otherwise. We adopt the pipeline transmission scheduling for transmitting packets on each logical ring as follows. Scheduling Algorithm for LTC-Odd

4000 3000

2H-LRC 2H-LTC

2000

4H-LRC

1000 0

4H-LTC 20

40

60

80

100

Tuning latency[slots] Fig. 8. Comparison of the schedule length with the previous q-LRC scheduling algorithm in [6] when N ¼ 32; h ¼ 2 and q is even.

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Schedule length[slots]

3000 2500 3H-LRC

2000

3H-LTC

1500

5H-LRC

1000

5H-LTC

500 0

20

40 60 80 Tuning latency[slots]

100

Fig. 9. Comparison of the schedule length with the previous q-LRC scheduling algorithm in [6] when N ¼ 32; h ¼ 2 and q is odd.

they solved the single-hop scheduling by the Incremental Maximum Weight Matching (IMWM) algorithm. Since the traffic matrix of the AAB problem is uniform, the logical topology design in their heuristic algorithm is not suitable for the AAB problem. In this paper, we apply the optimal single-hop scheduling algorithm in [4] for creating a single-hop schedule (SH) to solve the AAB problem. We also implement the q-LTC algorithm with 2 hop distance (2H-LTC), 3 hop distance (3H-LTC), 4 hop distance (4H-LTC) and 5 hop distance (5H-LTC) for the problem see Fig. 7. The performance of our q-LTC scheduling algorithm is always better than the optimal single-hop scheduling algorithm when the tuning latency d P 40. If d < 20, the schedule length of q-hop LTC scheduling algorithm is affected by the packet delay for multi-hop transmissions, and it will be longer than the optimal single scheduling. However, the performances of q-hop LTC algorithms are better than the optimal single-hop scheduling algorithm when the number of hops is greater than or equal to 3. Next, we compare the schedule length with the previous q-LRC scheduling algorithm [6] in Figs. 8 and 9. These curves show that the schedule length of our q-LTC scheduling algorithm is always shorter than that of the q-LRC scheduling algorithm in [6]. When the hop distance is even, the curves in Fig. 8 show that the schedule length of our q-LTC scheduling algorithm is smaller but close to that of the q-LRC scheduling algorithm. When the hop distance is odd, the curves in Fig. 9 indicate that our q-LTC scheduling algorithm is performed better than the hop distance is even (see Fig. 8). However, our q-LTC scheduling algorithms can create shorter schedule periods in terms of arbitrary q and d. Finally, we focus the multi-hop scheduling algorithm on the maximum hop distance q to study the effect between packet delay and throughput for the AAB problem. The adoption of multi-hop scheduling can reduce the number of tuning operations to improve

Fig. 11. Comparison of the throughput with the optimal single-hop scheduling algorithm when N ¼ 32 and h ¼ 2.

network throughput. However, increasing the number of intermediate hops also increase the transmission latencies of packets and decrease the overall throughput. We carry out an experimental simulation to show that the limited number of hops will lead to a good compromise in terms of throughput vs delay with respect to single-hop and unbounded multi-hop scheduling. The WDM optical star network is still considered with 64 nodes and 32 wavelengths ðh ¼ 2Þ and the tuning latency d is restricted to between 20 to 220 time slots. For evaluating the overall throughput and the average packet delay in our simulations, we assume that the arrival process of new packets is Poisson distribution and the average arrival interval of an AAB operation is 500 slots. Therefore, the average arrival rate of packets is 64  63/500 = 8.064 packets per slot in the 64-node WDM optical star network. In order to investigate the effect between packet delay and throughput, we plot the curves of the average packet delay in Fig. 10, and the curves of the overall throughput in Fig. 11. The five curves again refer to the single-hop (SH), 2-hop (2LTC), 3-hop(3-LTC) and so on. Since the multi-hop packets must be transmitted several times in successive scheduling periods, in Fig. 10, the average packet delay of multi-hop scheduling is larger than that of single hop scheduling when the tuning latency d is small. In the q-LTC scheduling algorithm, the average number of hops for an AAB operation is ðq þ 1Þ=2 if q is odd, and the average number of hops is q=2 if q is even. In Fig. 11, the overall throughput of multi-hop scheduling is larger than that of single hop scheduling when the tuning latency d is large. Increasing the number of intermediate hops can improve the overall throughput as shown in Fig. 11, but it will also increase the average packet delay of multi-hop scheduling as shown in Fig. 10. We can observe that the overall throughput can be greatly reduced with 3-LTC approaches with minor losses in packet delay. Moreover, limiting the number of hops to 2 or 3 also provides the better performance in both the average packet delay and the overall throughput for the AAB problem.

7. Conclusion This paper concerns with the design of multi-hop packet transmissions for the AAB problem in a WDM optical star network. We partition N nodes into x groups and reduce the size of groups to

Fig. 10. Comparison of the average packet delay with the optimal single-hop scheduling algorithm for N ¼ 32 and h ¼ 2.

, where q is the maximum hop distance. Then the either q2 or qþ1 2 problem of logical topology design can be transferred to the Hamiltonian cycles problem if x is odd, and transferred to the round robin tournament problem if x is even. By reducing the number

H.-J. Ho, J. Liu / Computer Communications 33 (2010) 706–713

of designed logical topologies, we minimize the schedule length of multi-hop scheduling, and decrease the expensive cost of tunable pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 transmitters. When q < 1þdþ 2d 6dþ5, the schedule length  l m  qðqþ1Þ 2N þ d of our multi-hop scheduling algorithm will 2 qþ1  1  N be less than the optimal schedule length max ðN  1Þdw e; N  1 þ wdg of single-hop scheduling in [4]. This also improves previous results in [6]. Especially, when q ¼ 3, the number of logical topologies by [6] is almost close to N  1. Our method constructs only dN2 e  1 logical topologies which can approach to 1.5 times the lower bound. However, the number of logical topologies l m is still not close to the lower bound N1 q . The future work will be to keep on finding other properties for solving the AAB problem or to propose new logical topologies to create near optimal multi-hop scheduling.

Acknowledgment The authors are very grateful to anonymous referees whose helpful comments and suggestions have led to a substantially improved presentation of the paper. This work was supported in part by Taiwan NSC under grant no. NSC-97-2221-E-274-006 and NSC96-2221-E-274-002.

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