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DWDM networks
The Journal of China Universities of Posts and Telecommunications December 2008, 15(4): 95–100 www.sciencedirect.com/science/journal/10058885
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QoS multicast routing scheme using QGA in IP/DWDM networks XING Huan-lai ( ), BAI Lin, JI Yue-feng Key Laboratory of Optical Communications and Lightwave Technologies, Beijing University of Posts and Telecommunications, Beijing 100876, China
Abstract
This article studies multi-constraints least-cost multicast routing problem in internet protocol over dense wavelength division multiplexing (IP/DWDM) networks. To address this problem, an individual-difference-based quantum genetic algorithm (IDQGA) is proposed. This algorithm considers individual differences among chromosomes by introducing an adaptive rotation angle step determination scheme and a grouping-based quantum mutation operation. Simulations are conducted over network topologies. The results indicate that compared with other heuristic algorithms, IDQGA has better optimal performance on solving quality of service (QoS) multicast routing problem in IP/DWDM networks and is characterized by strong robustness, high success ratio and excellent capability on global searching. Keywords IP-over-DWDM, multicast, QoS, quantum genetic algorithm
1
Introduction
Since more and more multimedia multicast applications (such as video conference and distance education) appear, QoS multicast routing is required to ensure service quality in modern communication networks. Thus, as one of the future solutions for the next generation Internet (NGI) backbone [1], IP/DWDM networks that transmit IP packets directly over optical layer by using the dense wavelength division multiplexing (DWDM) technologies, should have the basic capability of providing QoS multicast. The QoS multicast routing (QoSMR) problem is proved to be NP-complete [2], and it can be solved by finding a least-cost multicast tree with QoS parameters guaranteed (also called the Steiner-tree problem). To solve the QoSMR problem, some heuristic algorithms have been proposed [3–11]. Peng et al. proposed a genetic algorithm (GA) based overlay multicast routing algorithm [3]. Yue et al. presented a chaotic genetic algorithm to solve the QoSMR problem [4]. However, sometimes pre-maturity phenomena might appear in the above two algorithms because of inherent shortcomings of GA, such as pre-maturity, slow convergence speed, weak global searching capability, etc. Algorithms based on ant colony algorithm were also adopted [5–6]. Nevertheless, they are complex to implement. Sun et al. Received date: 19-05-2008 Corresponding author: XING Huan-lai, E-mail: [email protected]
offered a QoSMR algorithm based on a combination of GA and the ant colony algorithm [7]. Although it has better performance on convergent speed, the algorithm consumes much more computing time. An optical-internet-oriented multicast routing scheme based on neural networks (NN) is provided [8]. However, poor optimal performance might be obtained, because the values of correlative coefficients are hard to be determined. Another shrinking-chaotic-mutation evolutionary algorithm is presented to construct a least cost light-tree for multicast services [9]. However, the performance of mutation operation is not stable. Recently, the quantum genetic algorithm (QGA) has been applied to electric-layer QoSMR problem [10–11]. The QGA is a combination of quantum computation and evolutionary algorithm. By adopting quantum bit (qubit) representation, each chromosome is able to represent a linear superposition of solutions. The QGA evolves chromosomes by using suitable quantum gates, and makes binary solutions by observing the states of chromosomes. Rapid convergence and robustness can be provided only if suitable evolution parameter values are selected. However, the existing algorithms usually lack efficient strategies to determine suitable evolution parameters for all chromosomes, since the differences among chromosomes are not taken into consideration [10–22]. In Refs. [12–16], the fixed rotation angle step (FRAS) scheme is used in lookup table. If two chromosomes are under the same case according to the lookup table, they use the same rotation angle step value to
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update. Dynamic rotation angle step (DRAS) schemes are proposed in Refs. [10–11,17–20], where the rotation angle step (RAS) scheme is changed according to the current generation. But all chromosomes of the same generation still share the same RAS scheme. These two RAS schemes usually result in bad optimization, because they might not be fit for all chromosomes. Mutation is an effective operation to prevent pre-maturity and enhance global search capability. However, it is not adopted as a basic evolution operation in conventional QGA. Although quantum mutation operations are introduced in Refs. [10,15,17,21–22], they neglect individual differences and thus the algorithms may sometimes be trapped in local search. To enhance the optimal performance by considering individual difference of the population and providing an effective QoSMR scheme in IP/DWDM networks, this article offers an individual-difference-based QGA (IDQGA), which introduces an adaptive RAS (ARAS) determination scheme and a grouping-based quantum mutation (GQM) operation. By adopting the ARAS scheme, a suitable RAS value is assigned to each chromosome of each evolutionary generation according to its current situation, and therefore each chromosome has more chances to evolve itself to a better position and thus the convergent speed of the IDQGA is accelerated. Using the GQM operation, a mutation probability set is established at the beginning of the IDQGA. The chromosomes whose fitness values belong to the same numerical range are allocated with the same mutation probability selected from a mutation probability set. Therefore, local optimization can be avoided efficiently. Simulation results indicate that the IDQGA outperforms the conventional GA (CGA) and conventional QGA (CQGA) in aspects such as robustness, success ratio and global searching ability, etc.
2 Problem description In this section, the model of the IP/DWDM network and the mathematical formulation of the QoSMR problem are described respectively. 2.1
IP/DWDM network model
The IP/DWDM network can be modeled as an undirected graph G = (V,E), where V and E are the set of optical nodes and the set of links representing optical fibers respectively. For each optical node n V , it is assumed that n has unlimited multicast capacity using optical components such as splitters and compensators. Some nodes among V possess wavelength conversion capability so that they can use another output wavelength to carry the input optical signals to prevent collision. The wavelength conversion (WC) equipment is
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expensive and its cost cannot be neglected. In this article, the WC cost is considered. Given an arbitrary n V , it has the following QoS parameters: RD(n), RDJ(n), RPL(n), C(n) representing its delay, delay jitter, packet loss ratio and node cost respectively, where RD(n) consists of the processing delay RPD(n) and WC delay RWCD(n) (if there is no WC, set RWCD(n) = 0), and C(n) is composed of node usage cost CNU(n) and WC cost CWC(n). For each edge e E , the following parameters
RB(e), RD(e), RDJ(e), C(e), ȁ(e) are considered and they denote its available bandwidth, propagation delay, delay jitter, cost and the set of available wavelengths respectively, where / (e) / {O1 , O2 ,..., OW } and / is the set of available wavelength Oi (i 1,2,...,W ) in IP/DWDM networks and W |/|. Let s V be the source, and D={d1,d2,…,dm} { V {s} } be the destinations. The established multicast tree covering s and D is denoted by T(s,D). The path from s to d i (di D) is represented by p(s, di). The relations are as follows: RD ( p ( s, di )) ¦ RD (e) ¦ RD (n) e p ( s , di )
RDJ ( p ( s, di ))
¦
e p ( s , di )
RDJ (e)
¦
n p ( s , di )
RDJ (n)
RB ( p( s, di )) min ^ RB (e), e p( s, di )` RPL ( p ( s, d i )) 1 C (T ( s, D ))
¦
n p ( s , di )
2.2
(1 RPL (n))
C (e)
eT ( s , D )
(1)
n p ( s , di )
¦
C (n)
(2) (3) (4) (5)
nT ( s , D )
Mathematical formulation
Given a QoS multicast request R(s,D, :D , :DJ ,:PL , :B ), where s is the source, D is the set of destination nodes, :D is the constraint of delay, :DJ is the constraint of delay jitter,
:B is the bandwidth constraint, and :PL is the packet loss ratio constraint. The objective of QoS multicast routing is to minimize the cost of the established multicast tree with the QoS requirement guaranteed as follows: Obj. minimize C(T(s,D)) (6) s.t. RD ( p ( s, di ))İ:D (7) RDJ ( p ( s, di ))İ:DJ
(8)
RB ( p ( s, di ))ı:B
(9)
RBL ( p ( s, d i ))İ:PL
3
(10)
The proposed algorithm
This section depicts an efficient QoSMR algorithm (IDQGA) based on the QGA with ID-based evolution schemes. First, two new schemes: the ARAS scheme and the GQM operation are introduced. Then, important relevant
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XING Huan-lai, et al. / QoS multicast routing scheme using QGA in IP/DWDM networks
operations are described. Finally, the structure of the proposed algorithm is described in detail. 3.1
ARAS scheme
As the RAS has a great influence on the convergent speed of the QGA, it is an imperative to select a suitable RAS. Chromosomes might fail to reach optimum solutions in short generations if the RAS is excessively small, or they might miss optimum solutions if the RAS is excessively large. However, the existing QGA algorithms adopt either the FRAS or the DRAS as its RAS determination strategy, which completely ignore whether the RAS scheme is fit for all chromosomes, which might result in sub-optimal performance eventually. To improve optimization efficiency, this article proposes an ARAS scheme, which adaptively allocates each chromosome of each generation a suitable RAS value to update. Under this scheme, every chromosome is treated separately according to its own situation. For arbitrary chromosome of the generation t, if its fitness value is larger than that of the last generation, the current RAS value is set larger than its last RAS value; if its fitness value is smaller than that of the last generation, the current RAS value is set smaller than its last RAS value; otherwise, the current RAS value is set to be equal with its last RAS value. Let Q(t ) {q1t , q2t ,..., qNt } be the population colony of the generation t, where q tj is the jth ( j 1,2,..., N ) chromosome, and N is the population size. The lookup table of the rotation angle Ti of q tj of the tth generation is given in Table 1, where the ARAS scheme is applied and Ti
S (D i , E i )'Ti .
Table 1 Lookup table of the ith rotation angle of the jth chromosome of generation t xi bi f ( x )ıf (b)
'Ti
S (D i , Ei )
D i Ei ! 0 D i Ei ! 0 D i Ei ! 0 D i Ei ! 0
0
0
False
0
0
0
True
0
0
1
False
G tj
+ 1
1
0
±1
0
1
True
G tj
1
+ 1
±1
0
False
G
t j
1
+ 1
±1
0
t j
+ 1
1
0
±1
1
0
1
0
True
G
1
1
False
0
1
1
True
0
In Table 1, 'Ti and S (D i , E i ) are the RAS and the sign of Ti respectively, f(x) is the profit, and bi and xi are the ith bit of the best solution b and the current solution x, respectively. The expression of G tj is defined as follows:
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* ; t 1 °G t 1; f ( X t ) f ( X t 1 ) ° j j j G tj ® t 1 (11) t ; f ( X ) f ( X tj 1 ) G * ' j ° j °G tj 1 '* ; f ( X tj ) ! f ( X tj 1 ) ¯ where * and '* are two constants initialized at the beginning of the algorithm and influence the convergent speed, and '* is smaller than * . We set '* = 0.05 * in this article. G tj 1 is the RAS of q tj1 ; X tj and X tj 1 are the observing states of the jth chromosome of the generation t and t 1 respectively. If G tj 0 , set G tj M '* , where M is a positive integer. Based on the ARAS scheme, all the chromosomes use the same RAS value to update at the beginning of the IDQGA. With the generation growing, excellent chromosomes get larger RAS values while the bad ones get smaller ones. Therefore, the distinguished chromosomes will speed up the process of evolution while the bad ones will have enough time to change their searching directions to better ones. Thus, fast convergence and high optimization efficiency will be achieved. 3.2
GQM operation
In this article, the GQM operation is based on the quantum mutation operation proposed in Ref. [22]. Instead of using only one mutation probability for the entire population, the IDQGA classifies all the chromosomes into 2H different groups according to their fitness values, where H is a positive integer. Each group has a corresponding mutation probability Pi [0,1] (i=1,2,…,2H), which is selected from a mutation probability set PM={P1,P2,…,P2H}(1>P1>P2>…>P2H>0). And PM is generated at the beginning of the IDQGA. If H is determined properly, the GQM operation can balance the complexity and efficiency of the algorithm and achieve better performance than the traditional quantum mutation operation. Here, the quantum mutation pm(i) of the ith individual is defined as: f P ; 0İf i f min < 1 H ° ° kf (k 1) f (12) pm (i ) ® Pk ; İf i f min < H H ° °¯ P2 R ; otherwise where k
2,3,...,2 H 1 and f i , f
and f min represent
the ith fitness value, the average fitness value, and the minimum fitness value of the current generation respectively. Based on the GQM operation, the excellent chromosomes have more opportunities to reserve, while the bad ones are more likely to mutate. Hence, pre-maturity is avoided and global searching ability is enhanced. Moreover, this operation
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'
is easy to implement.
W PL
3.3
'D (I ) ®
Encoding and decoding strategies
The k shortest path algorithm in Ref. [23] is used to find at most k candidate shortest paths from the source s to each destination di D ={d1,d2,…,dw}, where D is the destination set, i =1,2,…,w. All the selected candidate paths have to satisfy their QoS constraints. Suppose there are mi (0
Fitness evaluation
The evaluation function Ffitness(pi) of the observation state pi of the ith chromosome qit at generation t is designed as below, where penalty function is adopted to deal with chromosomes that violate QoS constraints. SW DW DJW PL Ffitness ( pi ) (13) C (T ( s, D ))
WD W DJ
'D (RD ( p(s, di )) :D )
'DJ ( RDJ ( p(s, di )) :DJ )
(14) (15)
PL
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( RPL ( p ( s, di )) :PL )
(16)
1; Iİ0 ¯tD ; I ! 0
(17)
1; Iİ0 ¯tDJ ; I ! 0
(18)
1; Iİ0 ¯tPL ; I ! 0
(19)
'DJ (I ) ®
'PL (I ) ®
Here S is a variable, tD (0,1), tDJ (0,1) and tPL (0,1) . If the wavelengths are pre-assigned to the corresponding tree T(s,D) successfully, set S = 1, otherwise, set S = 0. W D , W DJ and W PL are the delay penalty factor, the delay jitter penalty factor, and the packet loss ratio penalty factor respectively. If a chromosome satisfies all QoS constraints, W D , W DJ and
W PL are set to 1. Otherwise, the product of these three penalty factors is smaller than 1. For instance, a chromosome violates QoS constraints after decoding. Two paths do not meet the delay constraint and one path does not satisfy the packet loss ratio constraint. Then the product of the three penalty factors is (tD ) 2 tPL according to Eqs. (13)–(19). Since tD , tDJ and tPL influence the performance of the algorithm, their values
must be selected properly. In this article, tD
0.7 , tDJ
tPL
0.7 , respectively.
3.6
The QGA based QoS multicast routing scheme
0.7,
In this section, the QoS multicast routing problem can be solved via the following steps: Procedure IDQGA begin tĸ 0 initialize Q(t) make P(t) by observing Q(t) states evaluate P(t) with wavelength assignment store the best individual among P(t) and its fitness while (not termination condition) do begin tĸt+1 make P(t) observing Q (t 1) states evaluate P(t) with wavelength assignment update Q(t 1) with ARAS scheme store the best individual among P(t) and its fitness carry out GQM operation end end Here,
Q(t )
^q , q ,..., q ` t 1
t 2
t N
is a population of qubit
chromosomes at the generation t, where qit is the ith (i=1, 2,…,N) individual defined as shown in Eq. (20), N is the
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XING Huan-lai, et al. / QoS multicast routing scheme using QGA in IP/DWDM networks
population size, and P (t )
^ x , x ,..., x ` t 1
t 2
t N
is a set of binary
ªD irt º ªD irt 1 º ªcos Tirt sin T irt º ªD irt 1 º t (22) « t » U (Tir ) « t 1 » « t t » « t 1 » ¬ Eir ¼ ¬ Eir ¼ ¬ sin Tir cos T ir ¼ ¬ E ir ¼ The ARAS scheme is used to update Q(t 1) . Then the
(20)
best solution among P(t) is selected. And if the solution is fitter than the best stored solution, the stored solution is replaced by this best solution. At the end of each loop, the GQM operation is adopted to avoid pre-maturity.
solutions of observation states qit . qit
t ªD it1 D it2 D im º « t t » t E im ¼» ¬« Ei1 E i 2
In the ‘initialize Q(t)’ step, each pair of qubit probability amplitudes, D irt and Eirt , r = 1,2,…,m, are initialized with 1
2 for arbitrary qit Q (t ) . After initialization, the next
step is to make a set of binary solutions, P(t), of Q(t), where P (t ) ^ x1t , x2t ,..., x Nt ` is formed by means of selecting each bit using the probability of qubit, either D
t 2 ir
or E
t 2 ir
, r = 1,
t i
2,…,m and each solution x is a binary string of length m. When the observation is completed, each xit of P(t ) is evaluated to give some measure of its fitness. The wavelength assignment algorithm (Sect. 3.4) and fitness function (Sect. 3.5) are both adopted in the step of ‘evaluate P(t) with wavelength assignment’. Then the initial best solution is selected and stored among the binary solutions P(t). In the while loop, P(t) is observed and evaluated in the same way as before. The quantum rotation gate is used to update Q(t 1) so that the fitter states of the qubit chromosomes are generated. The quantum rotation gate U (T ) is described as: U (T )
ªcos T « sin T ¬
99
sin T º cos T »¼
(21)
where T is the rotation angle. The rth qubit value (D irt , Eirt )
4
Simulation and analysis
To evaluate the performance of the algorithm, comparisons of the IDQGA, the CGA, and the CQGA with ARAS (called ARAS-CQGA below) are made over three network topologies (20-nodes, 40-nodes and 60-nodes) constructed by Waxman’s algorithm [24]. Suppose 50% of the nodes have the WC function and the WC delay between any two wavelengths at all nodes vi V is the same. The maximum number of the available wavelengths |ȁ| = M in the experimental IP/DWDM network is set to 24, and | / (e) | (e E ) is a number between 8 and 24. All the QoS parameters of optical nodes and fibers are uniformly distributed, where the delay is from 1 ms to 5 ms, the delay jitter is from 0.1 ms to 1 ms, the bandwidth is selected from 20 Mb/s to 100 Mb/s, the packet loss ratio is from 0.01% and 0.1%, and the cost is from 1 to 10. Each multicast group is randomly selected from the graph and its size is considered as 20% to 40% of the number of network nodes. For each of the four algorithms, the simulation results obtained in 5 000 random trials are shown in Table 2.
of qit is updated as: Table 2 Performance comparisons of CGA, CQGA, ARAS-CQGA and IDQGA
Algorithms
CGA CQGA ARAS-CQGA IDQGA
Case1: 20-nodes network (ȍDİ20 ms, ȍDJİ6 ms, ȍBı30 Mbps, ȍPLİ0.2%) MSR/% BTC ABTC 100 154 161.6 100 154 156.2 100 154 154.8 100 154 154.2
Multicast scenarios Case2: 40-nodes network (ȍDİ35 ms, ȍDJİ10 ms, ȍBı40 Mbps, ȍPLİ0.6%) MSR/% BTC ABTC 92.7 341 376.4 94.3 338 371.7 98.6 335 352.2 99.4 335 345.4
In Table 2, ȍD, ȍDJ, ȍB, and ȍPL refer to the QoS constraints of delay, delay jitter, bandwidth and packet loss ratio, respectively; MSR denotes the successful ratio; BTC represents the best-tree cost; ABTC refer to the average best-tree cost, which is the mean cost of 5 000 best founded trees of each algorithm. Table 2 indicates that the algorithm outperforms the other algorithms. For example, in case1, the four algorithms can build QoS multicast trees at a MSR of 100% and all of them are able to find the best tree with a cost of 154. However, the IDQGA achieves the smallest ABTC value, 154.2, compared
Case3: 60-nodes network (ȍDİ65 ms, ȍDJİ15 ms, ȍBı40 Mbps, ȍPLİ1%) MSR/% BTC ABTC 87.9 653 695.6 92.3 638 683.1 97.0 632 660.5 98.6 627 649.6
with 154.8 (by ARAS-IDQGA), 156.2 (by CQGA) and 161.6 (by CGA). In other words, the IDQGA has more chances to hit the best multicast tree. In the other two cases (40-nodes network and 60-nodes network), the IDQGA clearly outperforms the CGA, the CQGA, and the ARAS-CQGA in all aspects, because it gets the best MSR values, 99.4% and 98.6% and the best BTC values, 335 and 627, and the best ABTC values, 345.4 and 649.6. According to Table 2, with the network size growing, the difference of ABTC between the IDQGA and any other algorithm becomes clearer. For example, in case1, the ABTC values achieved by the IDQGA,
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The Journal of China Universities of Posts and Telecommunications
the ARAS-QGA, the CQGA and the GA are 154.2, 154.8, 156.2, and 161.6 respectively. Thus, the difference between the IDQGA and the ARAS-QGA (CQGA and GA) is 0.6 (2.0 and 7.4); In case 2, this difference is enlarged to 6.8 (26.3 and 31). And in case 3, this difference is up to 10.9 (33.5 and 46). It means that compared with other algorithms, the IDQGA is the strongest in robustness. Hence, it is obvious that the IDQGA is the best among these algorithms, and the ARAS-CQGA performs better than the CGA and the CQGA, and the CQGA is better than the CGA. As better characteristic of population diversity is achieved by QGA algorithms, broader solution space is explored and exploited simultaneously. Hence, these three QGA algorithms can find better multicast light-trees than the CGA can do, especially when the solution space is large. The individual differences are paid enough attention to and the different RAS values are assigned to different chromosomes for evolution, and thus the IDQGA and the ARAS-CQGA result in high successful ratio and excellent BTC and ABTC. Nevertheless, the ARAS-CQGA will be trapped in local search, since it does not adopt mutation operation. As the ARAS scheme and the GQM operation are applied at the same time, fast convergence and global search are obtained simultaneously so that the IDQGA does the best among the four algorithms.
5
Conclusions
An improved QGA algorithm has been proposed to solve the QoS multicast routing problem in IP/DWDM networks. This proposed algorithm considers individual difference and introduces an adaptive rotation angle step scheme and a grouping-based quantum mutation scheme to enhance its optimal efficiency. The simulation results demonstrate its superiority to other algorithms such as the CGA and the CQGA. Acknowledgements This work was supported by the National Natural Science Foundation of China (60572021, 90704006), the National Basic Research Program of China (2007CB310705), the Hi-Tech Research and Development Program of China (2007AA01Z247), PCSIRT (IRT0609), 111 Project (B07005), ISTCP (2006DFA11040).
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QoS multicast routing scheme using QGA in IP/DWDM networks XING Huan-lai ( ), BAI Lin, JI Yue-feng Key Laboratory of Optical Communications and Lightwave Technologies, Beijing University of Posts and Telecommunications, Beijing 100876, China
Abstract
This article studies multi-constraints least-cost multicast routing problem in internet protocol over dense wavelength division multiplexing (IP/DWDM) networks. To address this problem, an individual-difference-based quantum genetic algorithm (IDQGA) is proposed. This algorithm considers individual differences among chromosomes by introducing an adaptive rotation angle step determination scheme and a grouping-based quantum mutation operation. Simulations are conducted over network topologies. The results indicate that compared with other heuristic algorithms, IDQGA has better optimal performance on solving quality of service (QoS) multicast routing problem in IP/DWDM networks and is characterized by strong robustness, high success ratio and excellent capability on global searching. Keywords IP-over-DWDM, multicast, QoS, quantum genetic algorithm
1
Introduction
Since more and more multimedia multicast applications (such as video conference and distance education) appear, QoS multicast routing is required to ensure service quality in modern communication networks. Thus, as one of the future solutions for the next generation Internet (NGI) backbone [1], IP/DWDM networks that transmit IP packets directly over optical layer by using the dense wavelength division multiplexing (DWDM) technologies, should have the basic capability of providing QoS multicast. The QoS multicast routing (QoSMR) problem is proved to be NP-complete [2], and it can be solved by finding a least-cost multicast tree with QoS parameters guaranteed (also called the Steiner-tree problem). To solve the QoSMR problem, some heuristic algorithms have been proposed [3–11]. Peng et al. proposed a genetic algorithm (GA) based overlay multicast routing algorithm [3]. Yue et al. presented a chaotic genetic algorithm to solve the QoSMR problem [4]. However, sometimes pre-maturity phenomena might appear in the above two algorithms because of inherent shortcomings of GA, such as pre-maturity, slow convergence speed, weak global searching capability, etc. Algorithms based on ant colony algorithm were also adopted [5–6]. Nevertheless, they are complex to implement. Sun et al. Received date: 19-05-2008 Corresponding author: XING Huan-lai, E-mail: [email protected]
offered a QoSMR algorithm based on a combination of GA and the ant colony algorithm [7]. Although it has better performance on convergent speed, the algorithm consumes much more computing time. An optical-internet-oriented multicast routing scheme based on neural networks (NN) is provided [8]. However, poor optimal performance might be obtained, because the values of correlative coefficients are hard to be determined. Another shrinking-chaotic-mutation evolutionary algorithm is presented to construct a least cost light-tree for multicast services [9]. However, the performance of mutation operation is not stable. Recently, the quantum genetic algorithm (QGA) has been applied to electric-layer QoSMR problem [10–11]. The QGA is a combination of quantum computation and evolutionary algorithm. By adopting quantum bit (qubit) representation, each chromosome is able to represent a linear superposition of solutions. The QGA evolves chromosomes by using suitable quantum gates, and makes binary solutions by observing the states of chromosomes. Rapid convergence and robustness can be provided only if suitable evolution parameter values are selected. However, the existing algorithms usually lack efficient strategies to determine suitable evolution parameters for all chromosomes, since the differences among chromosomes are not taken into consideration [10–22]. In Refs. [12–16], the fixed rotation angle step (FRAS) scheme is used in lookup table. If two chromosomes are under the same case according to the lookup table, they use the same rotation angle step value to
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update. Dynamic rotation angle step (DRAS) schemes are proposed in Refs. [10–11,17–20], where the rotation angle step (RAS) scheme is changed according to the current generation. But all chromosomes of the same generation still share the same RAS scheme. These two RAS schemes usually result in bad optimization, because they might not be fit for all chromosomes. Mutation is an effective operation to prevent pre-maturity and enhance global search capability. However, it is not adopted as a basic evolution operation in conventional QGA. Although quantum mutation operations are introduced in Refs. [10,15,17,21–22], they neglect individual differences and thus the algorithms may sometimes be trapped in local search. To enhance the optimal performance by considering individual difference of the population and providing an effective QoSMR scheme in IP/DWDM networks, this article offers an individual-difference-based QGA (IDQGA), which introduces an adaptive RAS (ARAS) determination scheme and a grouping-based quantum mutation (GQM) operation. By adopting the ARAS scheme, a suitable RAS value is assigned to each chromosome of each evolutionary generation according to its current situation, and therefore each chromosome has more chances to evolve itself to a better position and thus the convergent speed of the IDQGA is accelerated. Using the GQM operation, a mutation probability set is established at the beginning of the IDQGA. The chromosomes whose fitness values belong to the same numerical range are allocated with the same mutation probability selected from a mutation probability set. Therefore, local optimization can be avoided efficiently. Simulation results indicate that the IDQGA outperforms the conventional GA (CGA) and conventional QGA (CQGA) in aspects such as robustness, success ratio and global searching ability, etc.
2 Problem description In this section, the model of the IP/DWDM network and the mathematical formulation of the QoSMR problem are described respectively. 2.1
IP/DWDM network model
The IP/DWDM network can be modeled as an undirected graph G = (V,E), where V and E are the set of optical nodes and the set of links representing optical fibers respectively. For each optical node n V , it is assumed that n has unlimited multicast capacity using optical components such as splitters and compensators. Some nodes among V possess wavelength conversion capability so that they can use another output wavelength to carry the input optical signals to prevent collision. The wavelength conversion (WC) equipment is
2008
expensive and its cost cannot be neglected. In this article, the WC cost is considered. Given an arbitrary n V , it has the following QoS parameters: RD(n), RDJ(n), RPL(n), C(n) representing its delay, delay jitter, packet loss ratio and node cost respectively, where RD(n) consists of the processing delay RPD(n) and WC delay RWCD(n) (if there is no WC, set RWCD(n) = 0), and C(n) is composed of node usage cost CNU(n) and WC cost CWC(n). For each edge e E , the following parameters
RB(e), RD(e), RDJ(e), C(e), ȁ(e) are considered and they denote its available bandwidth, propagation delay, delay jitter, cost and the set of available wavelengths respectively, where / (e) / {O1 , O2 ,..., OW } and / is the set of available wavelength Oi (i 1,2,...,W ) in IP/DWDM networks and W |/|. Let s V be the source, and D={d1,d2,…,dm} { V {s} } be the destinations. The established multicast tree covering s and D is denoted by T(s,D). The path from s to d i (di D) is represented by p(s, di). The relations are as follows: RD ( p ( s, di )) ¦ RD (e) ¦ RD (n) e p ( s , di )
RDJ ( p ( s, di ))
¦
e p ( s , di )
RDJ (e)
¦
n p ( s , di )
RDJ (n)
RB ( p( s, di )) min ^ RB (e), e p( s, di )` RPL ( p ( s, d i )) 1 C (T ( s, D ))
¦
n p ( s , di )
2.2
(1 RPL (n))
C (e)
eT ( s , D )
(1)
n p ( s , di )
¦
C (n)
(2) (3) (4) (5)
nT ( s , D )
Mathematical formulation
Given a QoS multicast request R(s,D, :D , :DJ ,:PL , :B ), where s is the source, D is the set of destination nodes, :D is the constraint of delay, :DJ is the constraint of delay jitter,
:B is the bandwidth constraint, and :PL is the packet loss ratio constraint. The objective of QoS multicast routing is to minimize the cost of the established multicast tree with the QoS requirement guaranteed as follows: Obj. minimize C(T(s,D)) (6) s.t. RD ( p ( s, di ))İ:D (7) RDJ ( p ( s, di ))İ:DJ
(8)
RB ( p ( s, di ))ı:B
(9)
RBL ( p ( s, d i ))İ:PL
3
(10)
The proposed algorithm
This section depicts an efficient QoSMR algorithm (IDQGA) based on the QGA with ID-based evolution schemes. First, two new schemes: the ARAS scheme and the GQM operation are introduced. Then, important relevant
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XING Huan-lai, et al. / QoS multicast routing scheme using QGA in IP/DWDM networks
operations are described. Finally, the structure of the proposed algorithm is described in detail. 3.1
ARAS scheme
As the RAS has a great influence on the convergent speed of the QGA, it is an imperative to select a suitable RAS. Chromosomes might fail to reach optimum solutions in short generations if the RAS is excessively small, or they might miss optimum solutions if the RAS is excessively large. However, the existing QGA algorithms adopt either the FRAS or the DRAS as its RAS determination strategy, which completely ignore whether the RAS scheme is fit for all chromosomes, which might result in sub-optimal performance eventually. To improve optimization efficiency, this article proposes an ARAS scheme, which adaptively allocates each chromosome of each generation a suitable RAS value to update. Under this scheme, every chromosome is treated separately according to its own situation. For arbitrary chromosome of the generation t, if its fitness value is larger than that of the last generation, the current RAS value is set larger than its last RAS value; if its fitness value is smaller than that of the last generation, the current RAS value is set smaller than its last RAS value; otherwise, the current RAS value is set to be equal with its last RAS value. Let Q(t ) {q1t , q2t ,..., qNt } be the population colony of the generation t, where q tj is the jth ( j 1,2,..., N ) chromosome, and N is the population size. The lookup table of the rotation angle Ti of q tj of the tth generation is given in Table 1, where the ARAS scheme is applied and Ti
S (D i , E i )'Ti .
Table 1 Lookup table of the ith rotation angle of the jth chromosome of generation t xi bi f ( x )ıf (b)
'Ti
S (D i , Ei )
D i Ei ! 0 D i Ei ! 0 D i Ei ! 0 D i Ei ! 0
0
0
False
0
0
0
True
0
0
1
False
G tj
+ 1
1
0
±1
0
1
True
G tj
1
+ 1
±1
0
False
G
t j
1
+ 1
±1
0
t j
+ 1
1
0
±1
1
0
1
0
True
G
1
1
False
0
1
1
True
0
In Table 1, 'Ti and S (D i , E i ) are the RAS and the sign of Ti respectively, f(x) is the profit, and bi and xi are the ith bit of the best solution b and the current solution x, respectively. The expression of G tj is defined as follows:
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* ; t 1 °G t 1; f ( X t ) f ( X t 1 ) ° j j j G tj ® t 1 (11) t ; f ( X ) f ( X tj 1 ) G * ' j ° j °G tj 1 '* ; f ( X tj ) ! f ( X tj 1 ) ¯ where * and '* are two constants initialized at the beginning of the algorithm and influence the convergent speed, and '* is smaller than * . We set '* = 0.05 * in this article. G tj 1 is the RAS of q tj1 ; X tj and X tj 1 are the observing states of the jth chromosome of the generation t and t 1 respectively. If G tj 0 , set G tj M '* , where M is a positive integer. Based on the ARAS scheme, all the chromosomes use the same RAS value to update at the beginning of the IDQGA. With the generation growing, excellent chromosomes get larger RAS values while the bad ones get smaller ones. Therefore, the distinguished chromosomes will speed up the process of evolution while the bad ones will have enough time to change their searching directions to better ones. Thus, fast convergence and high optimization efficiency will be achieved. 3.2
GQM operation
In this article, the GQM operation is based on the quantum mutation operation proposed in Ref. [22]. Instead of using only one mutation probability for the entire population, the IDQGA classifies all the chromosomes into 2H different groups according to their fitness values, where H is a positive integer. Each group has a corresponding mutation probability Pi [0,1] (i=1,2,…,2H), which is selected from a mutation probability set PM={P1,P2,…,P2H}(1>P1>P2>…>P2H>0). And PM is generated at the beginning of the IDQGA. If H is determined properly, the GQM operation can balance the complexity and efficiency of the algorithm and achieve better performance than the traditional quantum mutation operation. Here, the quantum mutation pm(i) of the ith individual is defined as: f P ; 0İf i f min < 1 H ° ° kf (k 1) f (12) pm (i ) ® Pk ; İf i f min < H H ° °¯ P2 R ; otherwise where k
2,3,...,2 H 1 and f i , f
and f min represent
the ith fitness value, the average fitness value, and the minimum fitness value of the current generation respectively. Based on the GQM operation, the excellent chromosomes have more opportunities to reserve, while the bad ones are more likely to mutate. Hence, pre-maturity is avoided and global searching ability is enhanced. Moreover, this operation
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'
is easy to implement.
W PL
3.3
'D (I ) ®
Encoding and decoding strategies
The k shortest path algorithm in Ref. [23] is used to find at most k candidate shortest paths from the source s to each destination di D ={d1,d2,…,dw}, where D is the destination set, i =1,2,…,w. All the selected candidate paths have to satisfy their QoS constraints. Suppose there are mi (0
Fitness evaluation
The evaluation function Ffitness(pi) of the observation state pi of the ith chromosome qit at generation t is designed as below, where penalty function is adopted to deal with chromosomes that violate QoS constraints. SW DW DJW PL Ffitness ( pi ) (13) C (T ( s, D ))
WD W DJ
'D (RD ( p(s, di )) :D )
'DJ ( RDJ ( p(s, di )) :DJ )
(14) (15)
PL
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( RPL ( p ( s, di )) :PL )
(16)
1; Iİ0 ¯tD ; I ! 0
(17)
1; Iİ0 ¯tDJ ; I ! 0
(18)
1; Iİ0 ¯tPL ; I ! 0
(19)
'DJ (I ) ®
'PL (I ) ®
Here S is a variable, tD (0,1), tDJ (0,1) and tPL (0,1) . If the wavelengths are pre-assigned to the corresponding tree T(s,D) successfully, set S = 1, otherwise, set S = 0. W D , W DJ and W PL are the delay penalty factor, the delay jitter penalty factor, and the packet loss ratio penalty factor respectively. If a chromosome satisfies all QoS constraints, W D , W DJ and
W PL are set to 1. Otherwise, the product of these three penalty factors is smaller than 1. For instance, a chromosome violates QoS constraints after decoding. Two paths do not meet the delay constraint and one path does not satisfy the packet loss ratio constraint. Then the product of the three penalty factors is (tD ) 2 tPL according to Eqs. (13)–(19). Since tD , tDJ and tPL influence the performance of the algorithm, their values
must be selected properly. In this article, tD
0.7 , tDJ
tPL
0.7 , respectively.
3.6
The QGA based QoS multicast routing scheme
0.7,
In this section, the QoS multicast routing problem can be solved via the following steps: Procedure IDQGA begin tĸ 0 initialize Q(t) make P(t) by observing Q(t) states evaluate P(t) with wavelength assignment store the best individual among P(t) and its fitness while (not termination condition) do begin tĸt+1 make P(t) observing Q (t 1) states evaluate P(t) with wavelength assignment update Q(t 1) with ARAS scheme store the best individual among P(t) and its fitness carry out GQM operation end end Here,
Q(t )
^q , q ,..., q ` t 1
t 2
t N
is a population of qubit
chromosomes at the generation t, where qit is the ith (i=1, 2,…,N) individual defined as shown in Eq. (20), N is the
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XING Huan-lai, et al. / QoS multicast routing scheme using QGA in IP/DWDM networks
population size, and P (t )
^ x , x ,..., x ` t 1
t 2
t N
is a set of binary
ªD irt º ªD irt 1 º ªcos Tirt sin T irt º ªD irt 1 º t (22) « t » U (Tir ) « t 1 » « t t » « t 1 » ¬ Eir ¼ ¬ Eir ¼ ¬ sin Tir cos T ir ¼ ¬ E ir ¼ The ARAS scheme is used to update Q(t 1) . Then the
(20)
best solution among P(t) is selected. And if the solution is fitter than the best stored solution, the stored solution is replaced by this best solution. At the end of each loop, the GQM operation is adopted to avoid pre-maturity.
solutions of observation states qit . qit
t ªD it1 D it2 D im º « t t » t E im ¼» ¬« Ei1 E i 2
In the ‘initialize Q(t)’ step, each pair of qubit probability amplitudes, D irt and Eirt , r = 1,2,…,m, are initialized with 1
2 for arbitrary qit Q (t ) . After initialization, the next
step is to make a set of binary solutions, P(t), of Q(t), where P (t ) ^ x1t , x2t ,..., x Nt ` is formed by means of selecting each bit using the probability of qubit, either D
t 2 ir
or E
t 2 ir
, r = 1,
t i
2,…,m and each solution x is a binary string of length m. When the observation is completed, each xit of P(t ) is evaluated to give some measure of its fitness. The wavelength assignment algorithm (Sect. 3.4) and fitness function (Sect. 3.5) are both adopted in the step of ‘evaluate P(t) with wavelength assignment’. Then the initial best solution is selected and stored among the binary solutions P(t). In the while loop, P(t) is observed and evaluated in the same way as before. The quantum rotation gate is used to update Q(t 1) so that the fitter states of the qubit chromosomes are generated. The quantum rotation gate U (T ) is described as: U (T )
ªcos T « sin T ¬
99
sin T º cos T »¼
(21)
where T is the rotation angle. The rth qubit value (D irt , Eirt )
4
Simulation and analysis
To evaluate the performance of the algorithm, comparisons of the IDQGA, the CGA, and the CQGA with ARAS (called ARAS-CQGA below) are made over three network topologies (20-nodes, 40-nodes and 60-nodes) constructed by Waxman’s algorithm [24]. Suppose 50% of the nodes have the WC function and the WC delay between any two wavelengths at all nodes vi V is the same. The maximum number of the available wavelengths |ȁ| = M in the experimental IP/DWDM network is set to 24, and | / (e) | (e E ) is a number between 8 and 24. All the QoS parameters of optical nodes and fibers are uniformly distributed, where the delay is from 1 ms to 5 ms, the delay jitter is from 0.1 ms to 1 ms, the bandwidth is selected from 20 Mb/s to 100 Mb/s, the packet loss ratio is from 0.01% and 0.1%, and the cost is from 1 to 10. Each multicast group is randomly selected from the graph and its size is considered as 20% to 40% of the number of network nodes. For each of the four algorithms, the simulation results obtained in 5 000 random trials are shown in Table 2.
of qit is updated as: Table 2 Performance comparisons of CGA, CQGA, ARAS-CQGA and IDQGA
Algorithms
CGA CQGA ARAS-CQGA IDQGA
Case1: 20-nodes network (ȍDİ20 ms, ȍDJİ6 ms, ȍBı30 Mbps, ȍPLİ0.2%) MSR/% BTC ABTC 100 154 161.6 100 154 156.2 100 154 154.8 100 154 154.2
Multicast scenarios Case2: 40-nodes network (ȍDİ35 ms, ȍDJİ10 ms, ȍBı40 Mbps, ȍPLİ0.6%) MSR/% BTC ABTC 92.7 341 376.4 94.3 338 371.7 98.6 335 352.2 99.4 335 345.4
In Table 2, ȍD, ȍDJ, ȍB, and ȍPL refer to the QoS constraints of delay, delay jitter, bandwidth and packet loss ratio, respectively; MSR denotes the successful ratio; BTC represents the best-tree cost; ABTC refer to the average best-tree cost, which is the mean cost of 5 000 best founded trees of each algorithm. Table 2 indicates that the algorithm outperforms the other algorithms. For example, in case1, the four algorithms can build QoS multicast trees at a MSR of 100% and all of them are able to find the best tree with a cost of 154. However, the IDQGA achieves the smallest ABTC value, 154.2, compared
Case3: 60-nodes network (ȍDİ65 ms, ȍDJİ15 ms, ȍBı40 Mbps, ȍPLİ1%) MSR/% BTC ABTC 87.9 653 695.6 92.3 638 683.1 97.0 632 660.5 98.6 627 649.6
with 154.8 (by ARAS-IDQGA), 156.2 (by CQGA) and 161.6 (by CGA). In other words, the IDQGA has more chances to hit the best multicast tree. In the other two cases (40-nodes network and 60-nodes network), the IDQGA clearly outperforms the CGA, the CQGA, and the ARAS-CQGA in all aspects, because it gets the best MSR values, 99.4% and 98.6% and the best BTC values, 335 and 627, and the best ABTC values, 345.4 and 649.6. According to Table 2, with the network size growing, the difference of ABTC between the IDQGA and any other algorithm becomes clearer. For example, in case1, the ABTC values achieved by the IDQGA,
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The Journal of China Universities of Posts and Telecommunications
the ARAS-QGA, the CQGA and the GA are 154.2, 154.8, 156.2, and 161.6 respectively. Thus, the difference between the IDQGA and the ARAS-QGA (CQGA and GA) is 0.6 (2.0 and 7.4); In case 2, this difference is enlarged to 6.8 (26.3 and 31). And in case 3, this difference is up to 10.9 (33.5 and 46). It means that compared with other algorithms, the IDQGA is the strongest in robustness. Hence, it is obvious that the IDQGA is the best among these algorithms, and the ARAS-CQGA performs better than the CGA and the CQGA, and the CQGA is better than the CGA. As better characteristic of population diversity is achieved by QGA algorithms, broader solution space is explored and exploited simultaneously. Hence, these three QGA algorithms can find better multicast light-trees than the CGA can do, especially when the solution space is large. The individual differences are paid enough attention to and the different RAS values are assigned to different chromosomes for evolution, and thus the IDQGA and the ARAS-CQGA result in high successful ratio and excellent BTC and ABTC. Nevertheless, the ARAS-CQGA will be trapped in local search, since it does not adopt mutation operation. As the ARAS scheme and the GQM operation are applied at the same time, fast convergence and global search are obtained simultaneously so that the IDQGA does the best among the four algorithms.
5
Conclusions
An improved QGA algorithm has been proposed to solve the QoS multicast routing problem in IP/DWDM networks. This proposed algorithm considers individual difference and introduces an adaptive rotation angle step scheme and a grouping-based quantum mutation scheme to enhance its optimal efficiency. The simulation results demonstrate its superiority to other algorithms such as the CGA and the CQGA. Acknowledgements This work was supported by the National Natural Science Foundation of China (60572021, 90704006), the National Basic Research Program of China (2007CB310705), the Hi-Tech Research and Development Program of China (2007AA01Z247), PCSIRT (IRT0609), 111 Project (B07005), ISTCP (2006DFA11040).
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