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Construction of an optimal multicast tree for group communication in a cellular network using genetic algorithm
Computer Communications 29 (2006) 3306–3312 www.elsevier.com/locate/comcom
Construction of an optimal multicast tree for group communication in a cellular network using genetic algorithm C. Mala *, S. Selvakumar Department of Computer Science and Engineering, National Institute of Technology, Tiruchirappalli, Tamil Nadu 620015, India Received 10 March 2006; received in revised form 12 May 2006; accepted 12 May 2006 Available online 21 June 2006
Abstract The recent advances in wireless technology has enabled users of Mobile cellular networks in different parts of the world not only to communicate with each other but also to participate in real time applications, viz., video conferencing, multiparty games, online auctions, access to distributed databases, etc., on the fly. All these applications require a Multicast Tree (MT) to be constructed among the group users with the source being the root of the MT. Traditional methods used in a wired network to construct a MT take into account only the distance or delay between the nodes. These methods when extended to mobile networks fail because of the inherent dynamism in a mobile network. To overcome this problem and to give an optimized solution to this problem, a novel Genetic Algorithm (GA) based approach to construct an Optimal Multicast Tree (OMT) with four constraints viz., probability of delay over a path, queuing delay at a node, residual bandwidth of a link, and the speed of the user is proposed in this paper. Further, for a network of N nodes and E edges, with k independent constraints, it has been derived that the time complexity and space complexity for GA based algorithm are O (N2) and O (N), respectively, whereas the time complexity and space complexity for the non-GA based multi constrained algorithm are O (N2k) and O (N2). These results show that the GA based algorithm is insensitive to the number of constraints and it constructs OMT faster than the traditional algorithms. Ó 2006 Elsevier B.V. All rights reserved. Keywords: Mobile network; Genetic algorithm; Optimal Multicast Tree; Multi constraints; Time and space complexity
1. Introduction Due to the nomadic lifestyle of the mobile users in recent years, there is a demand to make use of the mobile networks not only for communication but also for participation in real time applications, viz., multi group conferencing, multi party games, online auctions, access to distributed databases, etc. [1,6]. All these applications need the construction of a MT rooted at the source of multicast. As the traditional methods [2,4,11] of constructing a MT are for a fixed wired network, these methods fail for a highly dynamic Mobile network [11]. Several factors such as Call admission control, Bandwidth allocation, Queuing delay, and QoS are more *
Corresponding author. Tel.: +91 9443559523. E-mail addresses: [email protected] (C. Mala), [email protected] (S. Selvakumar).
0140-3664/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.comcom.2006.05.011
significant to the wireless medium and add to the complexity of wireless medium. Therefore, it is required to construct an OMT taking into account certain constraints, viz., probability of delay over a path, queuing delay at a node, residual bandwidth of a link, and speed of the mobile user to improve the QoS of Multicast [2,4,12] in a Mobile environment. In this paper, a GA-based method [7] to construct an OMT with three different constraints for a Mobile network is proposed. The simulation results show that the proposed technique computes an OMT faster than traditional methods and also reduces the call-dropping rate [9] of Mobile users. The remainder of the paper is organized as follows: Section 2 gives an overview of the existing schemes. Section 3 introduces the proposed approach, the results of simulation, and the performance analysis. Finally, Section 4 concludes the paper.
C. Mala, S. Selvakumar / Computer Communications 29 (2006) 3306–3312
2. Existing techniques The three basic techniques used for multicast routing [5] are Source-based routing, Steiner trees, and Trees with Rendezvous Point. In Source based routing, a receiver of data initiates the calculation of routing information to forward the packet. Reverse path forwarding (RPF) algorithm is used to avoid duplication of data units in the network. It is a slow process as it makes use of unicast information to do multicast and hence not suitable for mobile networks as it may increase the call dropping rate [9]. In Steiner trees [3,5], a spanning tree with minimum overall cost covering all the group users and a few intermediate nodes is calculated. As it is more complex to construct a Steiner tree for multi groups and with dynamic change in group members, it is also not suitable for Mobile networks. The other technique for multicast/multipeer communication is to construct a MT with Rendezvous Points (RPs) [5,10]. A MT with RP eliminates flooding present in Source based routing, enables to conduct multi group conferencing comprising of multiple senders and receivers and is more suitable for the dynamic nature of group members. Because of the above advantages, the Trees with Rendezvous Points are selected to provide a backbone for mobile users participating in multicast communication. 3. Proposed approach The motivation for the proposed approach, the proposed model, and the proposed algorithm are discussed in this section. 3.1. Motivation Due to the inherent dynamism in a Cellular network, there are frequent changes in the position of the mobile users [8,9,11]. So, it is required to construct a MT as fast as possible satisfying the QoS requirements [4] to enable smooth conduct of multicast/multipeer communication and this motivated us to propose a GA-based OMT computation in this paper. 3.2. Proposed model To implement Multicast communication in a Cellular network [8], a graph G = (N, E) representing a Cellular network of N Base stations and E links is considered in this paper. At each base station, a RP is located. Each RP functions as a Rendezvous Server (RS) if the source of the MT is in its coverage, else as a router providing service to the mobile users. The roles of RS and router are mentioned in the following subsections: 3.2.1. Roles of a rendezvous server The role of RS is to receive information from the source of multicast and forward this information to all members of the multicast group by maintaining two data
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structures namely Group Table (GT) and a Group Membership table (GMT). The GT contains the names of all groups maintained by the RS. When a packet is received by the RS, it compares the group name in the packet with the entries of the GT and decides if the packet is intended for itself or to be forwarded. A Group Membership table is maintained for every group in the RS as shown in Table 1. RS updates entries in GMT as per users movements. For each source of Multicast in its coverage area, RS computes an OMT after receiving required information from the members of the group. RS loads into the packet a sub tree of the multicast tree with the next router as the root, and forwards the packet to the next router. 3.2.2. Roles of a router Each router maintains a table called Foreign Member Table (FMT) as shown in Table 2. This table contains the list of users in its coverage, their group names, and the address of the RS of that group. When the router receives data packet from the RS, it broadcasts it if it has some group members in its coverage area or forwards it to the neighboring router. This is shown in Fig. 1. As the entries in the FMT are updated when a user enters/leaves a group, even new users are serviced by the router. When the router receives a data packet from any group member, it forwards it to the group’s RS, which then distributes to all members of that group. Fig. 1 shows a typical network with mobile users. Router 1 hosts the multicast group A. Group A’s members are spread across the network under routers 2, 4, 5, and 6. The multicast tree formed in such a situation is shown in dark lines. Fig. 2 shows the movement of member MA9 from router 4’s coverage area to router 6. The newly generated multicast tree in such a scenario is also shown, where router 4 has been excluded from the multicast tree. 3.3. Proposed algorithm to construct an OMT Algorithm 1: There are three steps involved in this algorithm. The algorithm uses two variables N and i, where Table 1 GMT maintained by RS User name
Router name
Group A Group B
R1 R7
Table 2 FMT maintained by router 2 User name
Group name
RS name
MA1 MA2 MA3 MB1
Group Group Group Group
R1 R1 R1 R2
A A A B
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MA2
MB3
MA1
MA3
Router 8
MA5
Router 3
Router 2 Router 1 Parent of Group A
Router 7 Parent of Group B MB1
MA4
Router 5
Router 4 Router 6 MA9
MA6 MB2
M A7 MA8
Fig. 1. A typical mobile user network and the multicast tree used for data transmission.
MA2
M B3
MA3
M A1
Router 8 MA5
Router 3
Router 2 Router 1 Parent of Group A Router 7 Parent of Group B
MB1
M A4 Router 5
Router 4
Router 6 M A9 MA6 M B2
MA7 M A8
Multicast tree links of Group A Wireless link between router & user Fig. 2. Multicast tree after member A9 moves to Router 6’s coverage area.
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variable N refers to the number of base stations in a Cellular network and variable i is the index variable representing the id of the base station. The value of variable i ranges from 1 to N. Step 1: For all i in N, each i transfers information regarding constraints to RS. Step 2: This information is used as an input to the Genetic algorithm approach to compute the optimal path to each of the routers from the RS. Step 3: From the optimal paths computed from Step 2, an OMT is computed with RS as the root of the tree. To compute OMT using Genetic algorithm, a Fitness function (a maximization function) taking into account the constraints is formulated. With an initial population of size p, randomly created from the solution pool, the operations such as reproduction, crossover, and mutations are applied iteratively by the RS so that the algorithm converges to the optimal solution. 3.3.1. Formulation of fitness function The first constraint in computing the OMT is the probability of delay in a path P of length m. From the definition of Erlang-K distribution, with arrival rate c, the probability of delay over a path P of length m less than t is given by 1 6 m 6 N 1:
ð1Þ
The second constraint is the queuing delay at the individual nodes. If every node is assumed to have N buffers, the queuing delay at a node is given by F 2 ¼ minðdi tÞ for 1 6 i 6 N ;
ð2Þ
where t is the time required to load/free a buffer and di is the number of free buffers at node i. The third constraint is with respect to residual bandwidth available in each of the links. The residual bandwidth of a link in the network after allocating bandwidth bm for a link m is given by (cm bm), where cm is the capacity of a link m e P. The fraction of total bandwidth available as residual bandwidth for a path P is given as F 3 ¼ minðcm bm Þ for
1 6 m 6 N 1:
F ¼ F 1 þ 1=F 2 þ F 3 þ F 4:
ð4Þ
where b is the probability of influence of speed in the OMT computation and its value is in the range 0 < b < 1.From these four constraints, the fitness
ð5Þ
3.4. Simulation and performance analysis The simulation of a Cellular network with mobile members is done using JAVA. A probability of 0.6 for crossover and 0.01 for mutation is used in our simulation. We have considered up to 100 iterations in our simulation. From the simulations, it is found that the GA based approach converges faster to the optimized solution than the traditional algorithms. From the graph in Fig. 3, it is inferred that, with GA the Call service rate also increases with the increase in the Call arrival rate. But without GA, as it is a multi constrained routing, first the shortest paths from one source to all destinations are computed with one constraint using Dijktra’s algorithm. With this, second constraint is applied and then the third constraint, and so on. As it is a slow process, it is not suitable for a mobile network and so the Call service rate [9] reduces. The computational time is found to be of the order of milliseconds and it tends to reduce with enhancement in hardware. The time complexity of the proposed algorithm is independent of the processing overhead and is dependent only upon the steps of the proposed algorithm.The computational time in computing the OMT with GA and without GA is found out by simulation and the results are tabulated in Table 3. A graph plotted with these values is shown in Fig. 4. From the graph in Fig. 4, it is inferred that the computational overhead of GA approach is very less compared to the approach without GA. A comparison of the performance of a Cellular network in terms of call blocking rate for the MT constructed using Source based routing, Steiner trees, and OMT constructed with RPs using GA is shown in Fig. 5. From the graph in Fig. 5, it is inferred that the call-blocking rate in GA approach is drastically reduced compared to Steiner trees and Source based routing.
30
ð3Þ
The fourth constraint is the speed at which the mobile user is traveling. As the mobile user may travel at varying speeds based on the mode of transport, handoff between base stations take place based on speed. So, the frequency of computation of Optimal Multicast Tree depends on the speed of the user. The speed also decides whether the RP functions as a RS or as a router. F 4 ¼ b speed;
function which is a Maximization function is given by the following:
Without GA
25 Call Service Rate
F 1 ¼ ðcm tm1 ect Þ=ðm 1Þ! for
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With GA
20 15 10 5 0 0
4
8 12 16 20 24 28 Call Arrival Rate
Fig. 3. Call arrival rate vs call service rate.
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Table 3 Computational time (ms) for different nodes Time (in ms) without GA
Time (in ms) with GA (100 iterations)
Log (time) without GA
Log (time) with GA
6 9 14 15 17 200 400
16 32 32,958 209,418 10,990,670 >4 days >10 days
156 188 280 327 359 8639 66495
1.204 1.505 4.518 5.321 7.041 8.538 8.9365
2.193 2.274 2.447 2.515 2.555 3.936 4.823
log (Computational Overhead in msec)
No. of nodes
time complexity for one optimal path computation is O (p + p log p), which is O (p log p). 2.3 Time to compute N 1 optimal paths = N 1 * p log p = Np log p. Step3: From the N 1 optimal paths, the OMT is constructed by comparing the fitness functions of N 1 optimal paths. So, this step takes O (N2).
Without GA With GA
8 7 6 5 4 3 2 1 0
So the total time complexity by the GA-based algorithm to compute an OMT is 9
12 13 14 15 Number of Nodes
17
T ðN Þ ¼ N þ Np log p þ N 2 ¼ OðN 2 Þ:
Fig. 4. Number of nodes vs computational overhead.
Trees with RP
160
Steiner Trees
Call Blocking Rate
140
Source Based Routing
120 100 80 60 40 20 0 0
25
50
75
100
125
150
175
Call Arrival Rate
ð6Þ
Genetic Algorithm represents the different constraints as binary bit strings. As the crossover and mutation operations in Genetic Algorithm manipulates these strings to get the optimized solution, the time complexity of the proposed approach is less and is independent of the number of constraints. But the existing algorithms to compute the multicast tree based on multi constrained routing are iterative in nature. The constraints are applied one after the other in each iteration and the number of iterations to compute the multicast tree becomes equal to the number of constraints. Hence in iterative algorithms, the time complexity in computing shortest path from one source to one destination is O (Nk 1) for k constraints [4], the time to compute the OMT is O (N2k). 3.6. Analysis of space complexity of GA approach
Fig. 5. Call arrival rate vs call blocking rate.
3.5. Analysis of time complexity of GA approach The time complexity of the GA approach proposed in algorithm 1 is computed as follows: Step1: For all nodes in N, information is received by RS. This takes O (N). Step2: 2.1 GA approach makes use of a population size of p. So, to perform the operations selection, reproduction, crossover, and mutations, it takes O (p). 2.2 The path with the maximum fitness function is the optimal path and is selected from a population size of p. This takes O (p log p). Therefore,
In GA based approach, one GT with atmost m entries and one GMT with atmost N entries are maintained at the RS for computation of an OMT. So the space complexity of this approach is only O (N). But in multi constrained routing, a table of size atleast Nk 1 is maintained at each node resulting in a space complexity of O (N2). The Time complexity and the Space complexity of the proposed algorithm with GA is compared with that of the algorithm without GA as given in Table 4 and the plots Table 4 Comparison of time and space complexities Complexity
With GA
Without GA
Time complexity Space complexity
O (N2) O (N)
O (N2k) O (N2)
10
With GA Without GA
Without GA O(N^2k)
8
3311
120
With GA O(N^2)
100
6 Efficiency
log( Time complexity)
C. Mala, S. Selvakumar / Computer Communications 29 (2006) 3306–3312
4 2
80 60 40
0 9
12
15 18 21 24 No. of Nodes
27
30
20 0
Fig. 6. Time complexity.
0
10
30
50
70
100
120
140
Fig. 8. Speed vs efficiency.
With GA O(N)
1000
With out GA O(N^2)
800 600
Table 6 Comparison of simulated and derived time complexity
400
N
200 0 9
9 12 13 14 15 17
12 15 18 21 24 27 30 No. Of Nodes Fig. 7. Space complexity.
are shown in Figs. 6 and 7, respectively. From the graphs in Figs. 6 and 7, it is inferred that the GA based algorithm takes less time for computation and less space to store data compared to the algorithm without GA. The efficiency of the proposed algorithm is computed by the following formula: Efficiency ¼
Transmission delay 100 Transmission delay þ Computational Overhead ð7Þ
In this expression, transmission delay is the time required for transmission of data from each of the routers to RS. The computational overhead is the time required for the computation of an Optimal Multicast Tree when the user travels at different speed and is different for different algorithms/approaches. Hence the metric efficiency thus specifies how competent is the proposed approach using GA, in computing the Optimal Multicast Tree compared to the approach without GA and is tabulated in Table 5.
Table 5 Efficiency computation Speed
Efficiency with GA
Efficiency without GA
0 10 30 50 70 100 120 140
100 98 95 92 89 87 85 83
100 93 84 73 62 53 43 31
Log(Time Complexity)
Space Complexity
Speed
Simulation results
Theoretical results
Time in (ms)
Log (Time) with GA
O (N2)
Log (O (N2))
200 250 268 299 339 379
2.301029996 2.397940009 2.428134794 2.475671188 2.530199698 2.578639210
81 144 169 196 225 289
1.908485 2.158362 2.227887 2.292256 2.352183 2.460898
3 2.75 2.5 2.25 2 1.75 1.5 1.25 1 0.75 0.5 0.25 0
Simulation wit h GA Theoretical result
9
12
13 14 15 No. of Nodes
17
Fig. 9. Comparison of theoretical and simulated results.
The computed results are tabulated in Table 5. A plot of Speed vs Efficiency is shown in Fig. 8. From the graph in Fig. 8, it is inferred that the efficiency of the proposed algorithm tends to reduce with increase in speed but this decrease in efficiency is reasonable compared to the algorithm without GA. To improve the efficiency of the proposed algorithm with increase in speed, the processing power of RS and router should be increased with high speed hardware units while the transmission delay to get the vital information from the routers for the computation of OMT at the RS should be reduced by using high bandwidth links. Else the impact of the efficiency will be felt in terms of increase in the call dropping rate, which may disrupt the multicast communication. The time taken by the simulated algorithm is compared with the theoretical time complexity and the values are tabulated in Table 6. A graph plotted with these values is shown in Fig. 9. From the
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graph in Fig. 9, it is inferred that both the curves exhibit the same trend. Further, it is observed that the time taken by the simulation is slightly higher than the theoretical time as the processing and communication delay are significant in simulation. 4. Conclusion With the increasing demand for group communication in mobile networks, there is a growing need to construct an Optimal Multicast Tree as fast as possible, satisfying the QoS requirements. In this paper, a Genetic Algorithm based Optimal Multicast Tree construction algorithm using Rendezvous Points is proposed. The simulation results show that this algorithm performs better than the traditional algorithms for Multicast in terms of speed of computation and reduction in Call dropping rate. It is also proved that the time complexity and the space complexity of this algorithm are less compared to the traditional multi constrained algorithm.
[9] Nilanjan Banerje, Sajal K. Das, MODeRN: Multicast on-demand QoS-based routing in wireless networks, IEEE VTC 11 (2001) 2167– 2171. [10] Joshua Auerdach et al., Multicast group membership management, IEEE/ACM Trans. Network. 11 (2003) 166–175. [11] Charles E. Perkins, Mobile networking in the Internet, Mobile Netw. Appl. 3 (1999) 319–334. [12] R.H. Katz, Adaptation and mobility in wireless information systems, IEEE Pers. Commun. Mag. 1 (1994) 6–17.
C. Mala obtained her B.E. (HONS.) with University Rank in Electronics and Communication Engg. from Government College of Engg., Tirunelveli, Madurai Kamaraj University, Tamil Nadu, India in the year 1987 and completed her Post graduation in Computer Science and Engineering at National Institute of Technology, Tiruchirappalli (formerly Regional Engineering College) in the year 1990. She is working as a Faculty in the Department of Computer Science and Engineering in National Institute of Technology since 1991. Her research interests include Computer networks, Algorithms, Wireless and Mobile computing. She has published 13 papers in various International journals, International and National conferences.
References [1] Hrishikesh Gossain, Carlos de Morals Cordeiro, Dharma P. Agrawal, Multicasting in wireless environment, IEEE Commun. Mag. 40 (2002) 116–123. [2] D. Zapala, Multicast routing support for real-time applications, Ph.D. Dissertation, University of Southern California, Los Angeles, 1997. [3] Jinquan Dai, Touhai Angchuan, Hung Keng Pung, QROUTE: An QoS-guaranteed multicast routing, Comput. Commun. 27 (2004) 171–186. [4] Xin Yuan, Heuristic algorithms for multi constrained QoS routing, IEEE/ACM Trans. Network. 10 (2) (2002) 244–256. [5] Ralph Wittmann, Martina Zitterbart, Multicast communication – protocols and applications, Morgan Kaufmann Publishers, Los Altos, CA, 2001. [6] S. Deering, D. Estrin, An overview of quality of service routing for next-generation high-speed networks: problems and solutions, IEEE Network 12 (1998) 64–79. [7] David E. Goldberg, Genetic Algorithms in Search, Optimization, and Machine Learning, second ed., Pearson Education, 2004. [8] Jochen Schiller, Mobile Communications, second ed., Pearson Education, 2003.
S. Selvakumar obtained his degree in Electronics and Communication Engineering with Distinction in 1983 from Madurai Kamaraj University, Madurai and Postgraduate degree in Computer Science and Engineering in 1987 from Regional Engineering College (REC), Tiruchirappalli. He was awarded Ph.D. degree by the Indian Institute of Technology Madras, Chennai in 1999. Since 1983, he has been in the teaching profession and currently he is a Professor in the Department of Computer Science and Engineering, National Institute of Technology, Tiruchirappalli, Tamil Nadu, India. His research interests include group communication in high-speed networks, routing, multimedia communication, scheduling for QoS guarantee, mobile networks, and network computing. He has to his credit of publishing 7 research papers in International Journals, 3 research papers in National Journals, and 17 research papers in International and National Conferences. He is currently guiding three Doctoral Research Scholars. He has visited Spain during 15 Feb. to 17 Feb. 2006 to present a paper in an International Conference EHAC’06 organized by WSEAS in University of Alcala, Madrid.
Construction of an optimal multicast tree for group communication in a cellular network using genetic algorithm C. Mala *, S. Selvakumar Department of Computer Science and Engineering, National Institute of Technology, Tiruchirappalli, Tamil Nadu 620015, India Received 10 March 2006; received in revised form 12 May 2006; accepted 12 May 2006 Available online 21 June 2006
Abstract The recent advances in wireless technology has enabled users of Mobile cellular networks in different parts of the world not only to communicate with each other but also to participate in real time applications, viz., video conferencing, multiparty games, online auctions, access to distributed databases, etc., on the fly. All these applications require a Multicast Tree (MT) to be constructed among the group users with the source being the root of the MT. Traditional methods used in a wired network to construct a MT take into account only the distance or delay between the nodes. These methods when extended to mobile networks fail because of the inherent dynamism in a mobile network. To overcome this problem and to give an optimized solution to this problem, a novel Genetic Algorithm (GA) based approach to construct an Optimal Multicast Tree (OMT) with four constraints viz., probability of delay over a path, queuing delay at a node, residual bandwidth of a link, and the speed of the user is proposed in this paper. Further, for a network of N nodes and E edges, with k independent constraints, it has been derived that the time complexity and space complexity for GA based algorithm are O (N2) and O (N), respectively, whereas the time complexity and space complexity for the non-GA based multi constrained algorithm are O (N2k) and O (N2). These results show that the GA based algorithm is insensitive to the number of constraints and it constructs OMT faster than the traditional algorithms. Ó 2006 Elsevier B.V. All rights reserved. Keywords: Mobile network; Genetic algorithm; Optimal Multicast Tree; Multi constraints; Time and space complexity
1. Introduction Due to the nomadic lifestyle of the mobile users in recent years, there is a demand to make use of the mobile networks not only for communication but also for participation in real time applications, viz., multi group conferencing, multi party games, online auctions, access to distributed databases, etc. [1,6]. All these applications need the construction of a MT rooted at the source of multicast. As the traditional methods [2,4,11] of constructing a MT are for a fixed wired network, these methods fail for a highly dynamic Mobile network [11]. Several factors such as Call admission control, Bandwidth allocation, Queuing delay, and QoS are more *
Corresponding author. Tel.: +91 9443559523. E-mail addresses: [email protected] (C. Mala), [email protected] (S. Selvakumar).
0140-3664/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.comcom.2006.05.011
significant to the wireless medium and add to the complexity of wireless medium. Therefore, it is required to construct an OMT taking into account certain constraints, viz., probability of delay over a path, queuing delay at a node, residual bandwidth of a link, and speed of the mobile user to improve the QoS of Multicast [2,4,12] in a Mobile environment. In this paper, a GA-based method [7] to construct an OMT with three different constraints for a Mobile network is proposed. The simulation results show that the proposed technique computes an OMT faster than traditional methods and also reduces the call-dropping rate [9] of Mobile users. The remainder of the paper is organized as follows: Section 2 gives an overview of the existing schemes. Section 3 introduces the proposed approach, the results of simulation, and the performance analysis. Finally, Section 4 concludes the paper.
C. Mala, S. Selvakumar / Computer Communications 29 (2006) 3306–3312
2. Existing techniques The three basic techniques used for multicast routing [5] are Source-based routing, Steiner trees, and Trees with Rendezvous Point. In Source based routing, a receiver of data initiates the calculation of routing information to forward the packet. Reverse path forwarding (RPF) algorithm is used to avoid duplication of data units in the network. It is a slow process as it makes use of unicast information to do multicast and hence not suitable for mobile networks as it may increase the call dropping rate [9]. In Steiner trees [3,5], a spanning tree with minimum overall cost covering all the group users and a few intermediate nodes is calculated. As it is more complex to construct a Steiner tree for multi groups and with dynamic change in group members, it is also not suitable for Mobile networks. The other technique for multicast/multipeer communication is to construct a MT with Rendezvous Points (RPs) [5,10]. A MT with RP eliminates flooding present in Source based routing, enables to conduct multi group conferencing comprising of multiple senders and receivers and is more suitable for the dynamic nature of group members. Because of the above advantages, the Trees with Rendezvous Points are selected to provide a backbone for mobile users participating in multicast communication. 3. Proposed approach The motivation for the proposed approach, the proposed model, and the proposed algorithm are discussed in this section. 3.1. Motivation Due to the inherent dynamism in a Cellular network, there are frequent changes in the position of the mobile users [8,9,11]. So, it is required to construct a MT as fast as possible satisfying the QoS requirements [4] to enable smooth conduct of multicast/multipeer communication and this motivated us to propose a GA-based OMT computation in this paper. 3.2. Proposed model To implement Multicast communication in a Cellular network [8], a graph G = (N, E) representing a Cellular network of N Base stations and E links is considered in this paper. At each base station, a RP is located. Each RP functions as a Rendezvous Server (RS) if the source of the MT is in its coverage, else as a router providing service to the mobile users. The roles of RS and router are mentioned in the following subsections: 3.2.1. Roles of a rendezvous server The role of RS is to receive information from the source of multicast and forward this information to all members of the multicast group by maintaining two data
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structures namely Group Table (GT) and a Group Membership table (GMT). The GT contains the names of all groups maintained by the RS. When a packet is received by the RS, it compares the group name in the packet with the entries of the GT and decides if the packet is intended for itself or to be forwarded. A Group Membership table is maintained for every group in the RS as shown in Table 1. RS updates entries in GMT as per users movements. For each source of Multicast in its coverage area, RS computes an OMT after receiving required information from the members of the group. RS loads into the packet a sub tree of the multicast tree with the next router as the root, and forwards the packet to the next router. 3.2.2. Roles of a router Each router maintains a table called Foreign Member Table (FMT) as shown in Table 2. This table contains the list of users in its coverage, their group names, and the address of the RS of that group. When the router receives data packet from the RS, it broadcasts it if it has some group members in its coverage area or forwards it to the neighboring router. This is shown in Fig. 1. As the entries in the FMT are updated when a user enters/leaves a group, even new users are serviced by the router. When the router receives a data packet from any group member, it forwards it to the group’s RS, which then distributes to all members of that group. Fig. 1 shows a typical network with mobile users. Router 1 hosts the multicast group A. Group A’s members are spread across the network under routers 2, 4, 5, and 6. The multicast tree formed in such a situation is shown in dark lines. Fig. 2 shows the movement of member MA9 from router 4’s coverage area to router 6. The newly generated multicast tree in such a scenario is also shown, where router 4 has been excluded from the multicast tree. 3.3. Proposed algorithm to construct an OMT Algorithm 1: There are three steps involved in this algorithm. The algorithm uses two variables N and i, where Table 1 GMT maintained by RS User name
Router name
Group A Group B
R1 R7
Table 2 FMT maintained by router 2 User name
Group name
RS name
MA1 MA2 MA3 MB1
Group Group Group Group
R1 R1 R1 R2
A A A B
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MA2
MB3
MA1
MA3
Router 8
MA5
Router 3
Router 2 Router 1 Parent of Group A
Router 7 Parent of Group B MB1
MA4
Router 5
Router 4 Router 6 MA9
MA6 MB2
M A7 MA8
Fig. 1. A typical mobile user network and the multicast tree used for data transmission.
MA2
M B3
MA3
M A1
Router 8 MA5
Router 3
Router 2 Router 1 Parent of Group A Router 7 Parent of Group B
MB1
M A4 Router 5
Router 4
Router 6 M A9 MA6 M B2
MA7 M A8
Multicast tree links of Group A Wireless link between router & user Fig. 2. Multicast tree after member A9 moves to Router 6’s coverage area.
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variable N refers to the number of base stations in a Cellular network and variable i is the index variable representing the id of the base station. The value of variable i ranges from 1 to N. Step 1: For all i in N, each i transfers information regarding constraints to RS. Step 2: This information is used as an input to the Genetic algorithm approach to compute the optimal path to each of the routers from the RS. Step 3: From the optimal paths computed from Step 2, an OMT is computed with RS as the root of the tree. To compute OMT using Genetic algorithm, a Fitness function (a maximization function) taking into account the constraints is formulated. With an initial population of size p, randomly created from the solution pool, the operations such as reproduction, crossover, and mutations are applied iteratively by the RS so that the algorithm converges to the optimal solution. 3.3.1. Formulation of fitness function The first constraint in computing the OMT is the probability of delay in a path P of length m. From the definition of Erlang-K distribution, with arrival rate c, the probability of delay over a path P of length m less than t is given by 1 6 m 6 N 1:
ð1Þ
The second constraint is the queuing delay at the individual nodes. If every node is assumed to have N buffers, the queuing delay at a node is given by F 2 ¼ minðdi tÞ for 1 6 i 6 N ;
ð2Þ
where t is the time required to load/free a buffer and di is the number of free buffers at node i. The third constraint is with respect to residual bandwidth available in each of the links. The residual bandwidth of a link in the network after allocating bandwidth bm for a link m is given by (cm bm), where cm is the capacity of a link m e P. The fraction of total bandwidth available as residual bandwidth for a path P is given as F 3 ¼ minðcm bm Þ for
1 6 m 6 N 1:
F ¼ F 1 þ 1=F 2 þ F 3 þ F 4:
ð4Þ
where b is the probability of influence of speed in the OMT computation and its value is in the range 0 < b < 1.From these four constraints, the fitness
ð5Þ
3.4. Simulation and performance analysis The simulation of a Cellular network with mobile members is done using JAVA. A probability of 0.6 for crossover and 0.01 for mutation is used in our simulation. We have considered up to 100 iterations in our simulation. From the simulations, it is found that the GA based approach converges faster to the optimized solution than the traditional algorithms. From the graph in Fig. 3, it is inferred that, with GA the Call service rate also increases with the increase in the Call arrival rate. But without GA, as it is a multi constrained routing, first the shortest paths from one source to all destinations are computed with one constraint using Dijktra’s algorithm. With this, second constraint is applied and then the third constraint, and so on. As it is a slow process, it is not suitable for a mobile network and so the Call service rate [9] reduces. The computational time is found to be of the order of milliseconds and it tends to reduce with enhancement in hardware. The time complexity of the proposed algorithm is independent of the processing overhead and is dependent only upon the steps of the proposed algorithm.The computational time in computing the OMT with GA and without GA is found out by simulation and the results are tabulated in Table 3. A graph plotted with these values is shown in Fig. 4. From the graph in Fig. 4, it is inferred that the computational overhead of GA approach is very less compared to the approach without GA. A comparison of the performance of a Cellular network in terms of call blocking rate for the MT constructed using Source based routing, Steiner trees, and OMT constructed with RPs using GA is shown in Fig. 5. From the graph in Fig. 5, it is inferred that the call-blocking rate in GA approach is drastically reduced compared to Steiner trees and Source based routing.
30
ð3Þ
The fourth constraint is the speed at which the mobile user is traveling. As the mobile user may travel at varying speeds based on the mode of transport, handoff between base stations take place based on speed. So, the frequency of computation of Optimal Multicast Tree depends on the speed of the user. The speed also decides whether the RP functions as a RS or as a router. F 4 ¼ b speed;
function which is a Maximization function is given by the following:
Without GA
25 Call Service Rate
F 1 ¼ ðcm tm1 ect Þ=ðm 1Þ! for
3309
With GA
20 15 10 5 0 0
4
8 12 16 20 24 28 Call Arrival Rate
Fig. 3. Call arrival rate vs call service rate.
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Table 3 Computational time (ms) for different nodes Time (in ms) without GA
Time (in ms) with GA (100 iterations)
Log (time) without GA
Log (time) with GA
6 9 14 15 17 200 400
16 32 32,958 209,418 10,990,670 >4 days >10 days
156 188 280 327 359 8639 66495
1.204 1.505 4.518 5.321 7.041 8.538 8.9365
2.193 2.274 2.447 2.515 2.555 3.936 4.823
log (Computational Overhead in msec)
No. of nodes
time complexity for one optimal path computation is O (p + p log p), which is O (p log p). 2.3 Time to compute N 1 optimal paths = N 1 * p log p = Np log p. Step3: From the N 1 optimal paths, the OMT is constructed by comparing the fitness functions of N 1 optimal paths. So, this step takes O (N2).
Without GA With GA
8 7 6 5 4 3 2 1 0
So the total time complexity by the GA-based algorithm to compute an OMT is 9
12 13 14 15 Number of Nodes
17
T ðN Þ ¼ N þ Np log p þ N 2 ¼ OðN 2 Þ:
Fig. 4. Number of nodes vs computational overhead.
Trees with RP
160
Steiner Trees
Call Blocking Rate
140
Source Based Routing
120 100 80 60 40 20 0 0
25
50
75
100
125
150
175
Call Arrival Rate
ð6Þ
Genetic Algorithm represents the different constraints as binary bit strings. As the crossover and mutation operations in Genetic Algorithm manipulates these strings to get the optimized solution, the time complexity of the proposed approach is less and is independent of the number of constraints. But the existing algorithms to compute the multicast tree based on multi constrained routing are iterative in nature. The constraints are applied one after the other in each iteration and the number of iterations to compute the multicast tree becomes equal to the number of constraints. Hence in iterative algorithms, the time complexity in computing shortest path from one source to one destination is O (Nk 1) for k constraints [4], the time to compute the OMT is O (N2k). 3.6. Analysis of space complexity of GA approach
Fig. 5. Call arrival rate vs call blocking rate.
3.5. Analysis of time complexity of GA approach The time complexity of the GA approach proposed in algorithm 1 is computed as follows: Step1: For all nodes in N, information is received by RS. This takes O (N). Step2: 2.1 GA approach makes use of a population size of p. So, to perform the operations selection, reproduction, crossover, and mutations, it takes O (p). 2.2 The path with the maximum fitness function is the optimal path and is selected from a population size of p. This takes O (p log p). Therefore,
In GA based approach, one GT with atmost m entries and one GMT with atmost N entries are maintained at the RS for computation of an OMT. So the space complexity of this approach is only O (N). But in multi constrained routing, a table of size atleast Nk 1 is maintained at each node resulting in a space complexity of O (N2). The Time complexity and the Space complexity of the proposed algorithm with GA is compared with that of the algorithm without GA as given in Table 4 and the plots Table 4 Comparison of time and space complexities Complexity
With GA
Without GA
Time complexity Space complexity
O (N2) O (N)
O (N2k) O (N2)
10
With GA Without GA
Without GA O(N^2k)
8
3311
120
With GA O(N^2)
100
6 Efficiency
log( Time complexity)
C. Mala, S. Selvakumar / Computer Communications 29 (2006) 3306–3312
4 2
80 60 40
0 9
12
15 18 21 24 No. of Nodes
27
30
20 0
Fig. 6. Time complexity.
0
10
30
50
70
100
120
140
Fig. 8. Speed vs efficiency.
With GA O(N)
1000
With out GA O(N^2)
800 600
Table 6 Comparison of simulated and derived time complexity
400
N
200 0 9
9 12 13 14 15 17
12 15 18 21 24 27 30 No. Of Nodes Fig. 7. Space complexity.
are shown in Figs. 6 and 7, respectively. From the graphs in Figs. 6 and 7, it is inferred that the GA based algorithm takes less time for computation and less space to store data compared to the algorithm without GA. The efficiency of the proposed algorithm is computed by the following formula: Efficiency ¼
Transmission delay 100 Transmission delay þ Computational Overhead ð7Þ
In this expression, transmission delay is the time required for transmission of data from each of the routers to RS. The computational overhead is the time required for the computation of an Optimal Multicast Tree when the user travels at different speed and is different for different algorithms/approaches. Hence the metric efficiency thus specifies how competent is the proposed approach using GA, in computing the Optimal Multicast Tree compared to the approach without GA and is tabulated in Table 5.
Table 5 Efficiency computation Speed
Efficiency with GA
Efficiency without GA
0 10 30 50 70 100 120 140
100 98 95 92 89 87 85 83
100 93 84 73 62 53 43 31
Log(Time Complexity)
Space Complexity
Speed
Simulation results
Theoretical results
Time in (ms)
Log (Time) with GA
O (N2)
Log (O (N2))
200 250 268 299 339 379
2.301029996 2.397940009 2.428134794 2.475671188 2.530199698 2.578639210
81 144 169 196 225 289
1.908485 2.158362 2.227887 2.292256 2.352183 2.460898
3 2.75 2.5 2.25 2 1.75 1.5 1.25 1 0.75 0.5 0.25 0
Simulation wit h GA Theoretical result
9
12
13 14 15 No. of Nodes
17
Fig. 9. Comparison of theoretical and simulated results.
The computed results are tabulated in Table 5. A plot of Speed vs Efficiency is shown in Fig. 8. From the graph in Fig. 8, it is inferred that the efficiency of the proposed algorithm tends to reduce with increase in speed but this decrease in efficiency is reasonable compared to the algorithm without GA. To improve the efficiency of the proposed algorithm with increase in speed, the processing power of RS and router should be increased with high speed hardware units while the transmission delay to get the vital information from the routers for the computation of OMT at the RS should be reduced by using high bandwidth links. Else the impact of the efficiency will be felt in terms of increase in the call dropping rate, which may disrupt the multicast communication. The time taken by the simulated algorithm is compared with the theoretical time complexity and the values are tabulated in Table 6. A graph plotted with these values is shown in Fig. 9. From the
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graph in Fig. 9, it is inferred that both the curves exhibit the same trend. Further, it is observed that the time taken by the simulation is slightly higher than the theoretical time as the processing and communication delay are significant in simulation. 4. Conclusion With the increasing demand for group communication in mobile networks, there is a growing need to construct an Optimal Multicast Tree as fast as possible, satisfying the QoS requirements. In this paper, a Genetic Algorithm based Optimal Multicast Tree construction algorithm using Rendezvous Points is proposed. The simulation results show that this algorithm performs better than the traditional algorithms for Multicast in terms of speed of computation and reduction in Call dropping rate. It is also proved that the time complexity and the space complexity of this algorithm are less compared to the traditional multi constrained algorithm.
[9] Nilanjan Banerje, Sajal K. Das, MODeRN: Multicast on-demand QoS-based routing in wireless networks, IEEE VTC 11 (2001) 2167– 2171. [10] Joshua Auerdach et al., Multicast group membership management, IEEE/ACM Trans. Network. 11 (2003) 166–175. [11] Charles E. Perkins, Mobile networking in the Internet, Mobile Netw. Appl. 3 (1999) 319–334. [12] R.H. Katz, Adaptation and mobility in wireless information systems, IEEE Pers. Commun. Mag. 1 (1994) 6–17.
C. Mala obtained her B.E. (HONS.) with University Rank in Electronics and Communication Engg. from Government College of Engg., Tirunelveli, Madurai Kamaraj University, Tamil Nadu, India in the year 1987 and completed her Post graduation in Computer Science and Engineering at National Institute of Technology, Tiruchirappalli (formerly Regional Engineering College) in the year 1990. She is working as a Faculty in the Department of Computer Science and Engineering in National Institute of Technology since 1991. Her research interests include Computer networks, Algorithms, Wireless and Mobile computing. She has published 13 papers in various International journals, International and National conferences.
References [1] Hrishikesh Gossain, Carlos de Morals Cordeiro, Dharma P. Agrawal, Multicasting in wireless environment, IEEE Commun. Mag. 40 (2002) 116–123. [2] D. Zapala, Multicast routing support for real-time applications, Ph.D. Dissertation, University of Southern California, Los Angeles, 1997. [3] Jinquan Dai, Touhai Angchuan, Hung Keng Pung, QROUTE: An QoS-guaranteed multicast routing, Comput. Commun. 27 (2004) 171–186. [4] Xin Yuan, Heuristic algorithms for multi constrained QoS routing, IEEE/ACM Trans. Network. 10 (2) (2002) 244–256. [5] Ralph Wittmann, Martina Zitterbart, Multicast communication – protocols and applications, Morgan Kaufmann Publishers, Los Altos, CA, 2001. [6] S. Deering, D. Estrin, An overview of quality of service routing for next-generation high-speed networks: problems and solutions, IEEE Network 12 (1998) 64–79. [7] David E. Goldberg, Genetic Algorithms in Search, Optimization, and Machine Learning, second ed., Pearson Education, 2004. [8] Jochen Schiller, Mobile Communications, second ed., Pearson Education, 2003.
S. Selvakumar obtained his degree in Electronics and Communication Engineering with Distinction in 1983 from Madurai Kamaraj University, Madurai and Postgraduate degree in Computer Science and Engineering in 1987 from Regional Engineering College (REC), Tiruchirappalli. He was awarded Ph.D. degree by the Indian Institute of Technology Madras, Chennai in 1999. Since 1983, he has been in the teaching profession and currently he is a Professor in the Department of Computer Science and Engineering, National Institute of Technology, Tiruchirappalli, Tamil Nadu, India. His research interests include group communication in high-speed networks, routing, multimedia communication, scheduling for QoS guarantee, mobile networks, and network computing. He has to his credit of publishing 7 research papers in International Journals, 3 research papers in National Journals, and 17 research papers in International and National Conferences. He is currently guiding three Doctoral Research Scholars. He has visited Spain during 15 Feb. to 17 Feb. 2006 to present a paper in an International Conference EHAC’06 organized by WSEAS in University of Alcala, Madrid.