A knowledge-based inference multicast protocol using adaptive fuzzy Petri nets

Expert Systems with Applications 36 (2009) 8115–8123

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Expert Systems with Applications journal homepage: www.elsevier.com/locate/eswa

A knowledge-based inference multicast protocol using adaptive fuzzy Petri nets Tzu-Chiang Chiang a,b, Cheng-Feng Tai a, Ting-Wei Hou a,* a b

Department of Engineering Science, National Cheng Kung University, Tainan 70101, Taiwan, ROC Department of Information Management, Hsing Kuo University of Management, Taiwan, ROC

a r t i c l e

i n f o

Keywords: Fuzzy Petri nets Wireless networks Multicast routing protocol KIMP

a b s t r a c t Multicast routing protocols need a new path discovery algorithm for a newly joining node (receiver) in an ad hoc network. One issue of the approach to find the nearest forwarding node for a new node is that it may increase the distance between the source node and the new members, which results in an increase in latency time and packet loss, as compared with the shortest path algorithms. This issue is important in a high collision network. In this paper, we propose a knowledge-based inference approach for a new path discovery for multicasting. A fuzzy Petri net agent, which is a special expert system, is introduced at each node to learn and to adjust itself to fit the dynamic conditions in a multicast ad hoc network. The simulation results show that the proposed approach is up to 67.17% more efficient in the packet delivery ratio as compared with a bandwidth effective multicast routing protocol. Ó 2008 Elsevier Ltd. All rights reserved.

1. Introduction A mobile ad hoc network (MANET) is a collection of mobile nodes that communicate with each other, but they have no fixed links such as wireless infrastructure networks (Kim & Kim, 2005; Li & Li, 2006; Shen, Huang, & Jaikaeo, 2006). Each node acts as a router and is responsible for dynamically discovering other nodes with which it can directly communicate. Most importantly, multicasting plays an important role in applications of ad hoc networks to transmit datagrams to a group of nodes in group or distributed computing. One example of the applications is multiplayer online games, in which players from different locations participate with their handheld devices in a local area (El-Sayed, Roca, & INRIA Rhone-Alpes, 2003; Khisti, Erez, & Wornell, 2006). Think about the scenario where a game player wants to join an online game group or a multicast group, and he/she initiates JOIN (broadcasting) control packets. Conventional multicast protocols try to establish the link between the nearest forwarding node and the newly joining member in the link initialization phase. According to the type of operations for link initializations, multicast protocols for MANETs are classified into two types: source-initiated and receiver-initiated. There are also two types of multicast topology: tree-based and mesh-based as shown in Fig. 1 (Huang, Chiang, & Hou, 2006; Siva Ram Murthy & Manoj, 2004). In a tree-based approach, there is only a single-path, which is also the shortest path between the

* Corresponding author. E-mail addresses: [email protected] (T.-C. Chiang), max.dai@msa. hinet.net (C.-F. Tai), [email protected], [email protected] (T.-W. Hou). 0957-4174/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2008.10.045

source-receiver pair. But tree-based protocols are not robust in a highly mobile network because of the single-path feature. Meshbased protocols provide multiple paths in a source-receiver pair to enhance the robustness of link connections at the cost of network overhead and efficiency. Bandwidth efficient multicast routing protocol (BEMRP) is a receiver-initiated and tree-based protocol that finds the nearest forwarding group to join instead of the shortest path in a sourcereceiver pair. Although BEMRP can save bandwidth due to a reduction in the number of data transmissions by joining the nearest forwarding node, it still suffers from transmission delay, link robustness and a reduction in the packet delivery ratio (Cheng, Cao, & Wang, 2006; Ozaki, Kim, & Suda, 2001; Viswanath, Obraczka, & Tsudik, 2006). This article presents a dynamic knowledge inference agent to adjust the learning ability of multicast receivers using adaptive fuzzy Petri nets for MANETs. The proposed multicast routing protocol is derived based on BEMRP protocol and incorporates the knowledge inference with fuzzy reasoning automatically. The rest of this paper is organized as follows: Some related works are given in Section 2. In Section 3 a list of knowledge representation of fuzzy logic based rules for multicast routing is proposed and described. Section 4 proposes a new knowledge inference based bandwidth-efficient multicast protocol using adaptive fuzzy Petri nets and focuses on the tree discovery and maintenance mechanisms in ad hoc networks. In Section 5, we compare the performances of the transmission delay, link robustness and reduction in the packet delivery ratio with Bandwidth efficient multicast routing protocol (BEMRP) using simulations. Conclusions are drawn in Section 6.

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Initialization Approach of Multicast Routing protocols in MANET

Receiver-Initiated Tree-bases

BEMRP PLBM WBM Mesh-bases

CAMP FGMP

a

Source-Initiated Tree-bases

AMRIS ABAM MZRP

b

Mesh-bases

ODMRP DCMP NSMP

c

Fig. 1. Classifications of multicast routing protocols in ad hoc networks.

2. Mapping fuzzy production rules into fuzzy Petri nets for representation of a network topology Fuzzy Petri nets can be used to represent structured knowledge and to describe the procedure for supporting fuzzy reasoning automatically in a rule-based expert system. As proposed by Chen, Ke, and Chang (1990), Lee (2007) and Li and Lara-Rosano (2000), a generalized fuzzy Petri net is defined as an 8-tuple:

FPN ¼ ðP; T; D; I; O; f ; a; bÞ; where P = {p1, p2, . . ., pn} denotes a finite nonempty set of places, T = {t1, t2, . . ., tm} denotes a finite nonempty set of transitions, D = {d1, d2, . . ., dn} denotes a finite set of propositions, P \ T \ D = £, and |P| = |D|  I = {T ? P} is the input mapping function from transitions to bags (?) of places, and O = {P ? T} is the output mapping function from places to transitions. f : T ? [0, 1] is an association function which maps from transitions to real values between zero and one. a : P ? [0, 1] is an association function which maps from places to real values between zero and one, and b : P ? D is also a ejective association function mapping from places to propositions. Fuzzy Petri nets may depict the fuzzy relationships between many propositions. A fuzzy production rule is given as follows:

Ri : IFdj THENdk ðCF ¼ li Þ; where R = {R1, R2, . . ., Rn} is a set of fuzzy production rules, and dj and dk are propositions with some fuzzy variables. The certainty factor (CF) is a value between zero and one, li 2 [0, 1]. The relationships from place to place by firing through transitions are represented by directed arcs in a fuzzy Petri net. The certainty factor, a probability, represents the strength of the certainty in the network rule. An example fuzzy Petri net model for network knowledge representation is shown in Fig. 2. Assuming the degree of truth of the network proposition ‘‘the bandwidth is scarce” is 0.8, a transition t1 fires from its input place (p1) into an output place (p2). The certainty factor (CF) is 0.9. The value of the degree of truth in an output place (p2) of t1 is calculated as 0.72 (by 0.8  0.9). According to the value, the probability of the truth of the network propositions firing from p1 to p2 is high. From a network point of view, the route form from place p1 to place p2 is unsuitable for network routing because the route is bad. A multicast routing representation in ad hoc networks using fuzzy Petri nets was proposed in our previous research (Chiang & Huang, 2004), so each topology of the multicast in a wireless ad hoc network can be represented as a marked fuzzy Petri net. As an example shown in Fig. 3, a network topology graph generated by multicast source node 2 is a structured representation of net-

Fig. 2. An example fuzzy Petri net model for network knowledge representation. (a) Before firing transition; (b) after firing transition; (c) fuzzy Petri net representation.

work topology which has a fuzzy reasoning algorithm for finding multicast tree. 3. Knowledge representation of fuzzy logic based rules for multicast routing In many situations, it is hard to determine the best route from the source to a new member of a multicast group within a MANET. Different fuzzy logic production rules for knowledge representation are considered in this research. The correlative factors of different inputs of the determination for the certainty factor (CF) are also discussed to predict the degree of the truth of the multicast tree discovery phase for a newly joining receiver. The best route found by the multicast protocol relies on the efficiency of discovering the links between forwarding nodes and the receiver. Three correlative factors, hop count, number of neighbor nodes, and link bandwidth, are discussed: (1) The smaller the hop count, the more efficient throughput in a source-receiver pair because the smaller hop count means a shorter distance, which implies fewer overheads are required in order for the forwarding nodes to relay the multicast packets. (2) The less neighbor hosts the forwarding group has, the better throughput because the less neighbors the node has, the less interference there is between wireless communications. Therefore, the throughput is better. (3) The more bandwidth the forwarding node has, the better throughput is. The forwarding node provides not only the multicast communication but also provides other

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a

b 1

2 t1

t2

1

8

2

t10

8

16

16 t6

3

3

4

6

6

14

t11

t4

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14 t9

4 5

5

15

13

t5

9

9

t12

13 7

11

11

t8

7

15 12

t7

12 10

10

Fig. 3. (a) A network topology graph according to multicast source node 2. (b) A marked fuzzy Petri net.

communications for any transmission routed by this forwarding node. Therefore, the performance of the throughput is worse when the capacity of bandwidth becomes smaller. The certainty factor is also a function with multiple variables which can be affected by more than one situation. The larger the value of certainty factor, the more the rule is believed in. Since the certainty factor represents the strength of the belief in the fuzzy rules, it should not be a constant, but the traditional certainty factor is a constant determined by an expert. Instead of using a constant, we choose to use the degree of truth between two propositions to automatically estimate fuzzy reasoning. A typical form of fuzzy production rules is exemplified by the following rule by which the hop count and the number of neighbor hosts are considered, and the adjustment of certainty factor is determined by the proposed fuzzy reasoning scheme. Box 1. An example of fuzzy production rule for routing information. IF the hop count is very large AND the number of neighbor hosts is slightly more THEN the strength of the feasibility for certainty factor should be adjusted to slightly smaller

the fuzzy Petri net model and the controlled processes, a fuzzy rule base, an inference engine, and fuzzifier/defuzzifier, are shown in Fig. 4. The basic steps involved in determining the certainty factor value that applies an adaptive fuzzy Petri net for multicast routing in MANET are as follows: Step 1.





First, input and output variables of the fuzzy controller are identified. The fuzzy quantization for each variable is then determined. The hop counts, the number of neighbor hosts and the capacity of the bandwidth are considered. The trapezoid sharps for the membership functions, rather than triangular sharps, are used to balance the variations of dynamic situations in wireless networks (Lao & Cui, 2006). The input fuzzy variable ‘‘the hop count” has three fuzzy sets – large, normal and short. One fuzzy quantization of the range [1, 15] for hop counts is suggested (Sandrasegaran & Prag, 1999) by different research results. Fuzzy quantization is exemplified in Fig. 5. The input fuzzy variable ‘‘the number of neighbor hosts” has three fuzzy sets – many, medium and few. The possible fuzzy quantization of the range [1, 10] for the number of neighbor hosts is an experimental result. The fuzzy quantization is exemplified in Fig. 6.

The linguistic terms very large, slightly more, and slightly small can be described by fuzzy sets defined on the discourse universes of hop count values, the number of neighbor hosts, and the strength of feasibility, respectively. The interconnections among

Fig. 4. Fuzzy reasoning controller with the strength of belief (CF).

Fig. 5. Membership function of the fuzzy variable ‘‘hop-count”.

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Step 2. Step 3.

The input fuzzy variable ‘‘capacity of bandwidth” has two fuzzy sets – heavy and low. The possible fuzzy quantization of the range [0, 1] for the capacity of the bandwidth is also an experimental result in which we can classify the range of two linguistic labels which are low and heavy (Dolev, Schiller, & Welch, 2006; Liang, 2006; Wei & Zakhor, 2004). Since the capacity of the bandwidth is the ratio of the available bit rate of the bandwidth, the more capacity has, the more efficiency is. The fuzzy quantization is exemplified in Fig. 7. The output fuzzy variable ‘‘strength of the feasibility for certainty factor” has five fuzzy sets – weakest, weak, normal, strong and strongest. The fuzzy quantization is exemplified in Fig. 8. A fuzzifier performs a mapping from each input variable of the fuzzy set to express the associated measurement. The fuzzy rule base consists of IF-THEN fuzzy rules of knowledge pertaining from routing information that are determined from experiential results.(Li, Li, & Lau, 2006). Since each input variable has 2 or 3 linguistic states, the total number of possible fuzzy inference rules is 3  3  2 = 18. Table 1 shows the fuzzy rule base in the fuzzy controller.

In many situations, it may be difficult to determinate the optimal route among candidate routes for a join request of a new multicast receiver. In order to properly represent real-world situations with variable constraints, we use the certainty factor for production rules to represent the strength of the probability in the rule for choosing the optimal solution. The value of the certainty factor . . . 0, 0.15, 0.30, 0.5, 0.65, 0.85, 1.0 with the truth scales having the corresponding numerical intervals table seen in Table 2.

Fig. 8. Membership function of the fuzzy variable ‘‘certainty factor”.

Table 1 Fuzzy logic system rules. Input

Fig. 6. Membership function of the fuzzy variable ‘‘neighbor-host”.

Output

H (hop count)

N (# of neighbors)

C (traffic load)

CF

Large Normal Short Large Normal Short Large Normal Short Large Normal Short Large Normal Short Large Normal Short

Many Many Many Medium Medium Medium Few Few Few Many Many Many Medium Medium Medium Few Few Few

Low Low Low Low Low Low Low Low Low Heavy Heavy Heavy Heavy Heavy Heavy Heavy Heavy Heavy

Weak Weak Weak Weak Normal Normal Weak Strong Strongest Weakest Weakest Weak Weak Normal Strong Weak Normal Strong

Table 2 Truth scales for certainty factors and corresponding numerical intervals.

Fig. 7. Membership function of the fuzzy variable ‘‘bandwidth load”.

Truth scales for certainty factors

Numerical intervals

Always true Very true Considerably true Moderately true More or less true Minimally true Not true

[1.00, 1.00] [0.85, 0.99] [0.65, 0.84] [0.45, 0.64] [0.30, 0.44] [0.01, 0.29] [0.00, 0.00]

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4. Knowledge Inference based bandwidth-efficient multicast protocol (KIMP) Our proposed scheme, KIMP, is receiver-tree-based. KIMP uses automatic fuzzy reasoning to find the best routing solution in MANET. The BEMRP tries to find the nearest forwarding group for a new receiver to join a multicast communication, instead of the shortest source-receiver path. Therefore, the distance between multicast source and receivers will increase, which decreases the efficiency of the original multicast links in the forwarding group. In the BEMRP scheme, the routing initial state is triggered by a new node that wants to join a multicast group. The initial phase finds the nearest forwarding node and assigns a new path for the new one in the multicast group. As the Fig. 9 shows, if R5 wants to join the multicast group, R5 sends Join Request control packets to find all the possible forwarding group nodes. All receiver nodes will receive the Join Request control packet and fill the Hop Count number relay back to R5. R5 selects the path of the minimum hop count number and sets the path for multicast communication. In Fig. 9, R5 selects the path between R5, I2, I5 and F4 because its hop count number is the minimum, three. Additionally, when R5 wants to leave the multicast group, it also sends a QUIT packet to I5. Although the BEMRP quickly finds the path for multicast, the selected path may be the worst path due to other considerations, such as many neighbor hosts or less bandwidth capacity. In KIMP, when a new multicast receiver wants to join a multicast group, a route setup phase is triggered, and the newly joining receiver broadcasts a Join Request control packet to find all the possible forwarding group nodes or multicast group members. Join Re-

F3 s

R4 R3

F2

F1

I11 F4

R2

R1

I10

I8 I5

I7

Join Request packet I6

I4 I2 I3 I1

Reply packet Ri

Receiver node

Fi

Receiver node

quest packets are flooded until they reach a forwarding group node or a multicast group member (including the multicast source). Before each node relays a Join Request packet from the receiver, it records its ID number and increases the hop count number in the forwarding Join Request packet. Therefore, after a while, every candidate forwarding node of the target multicast group node receives at least one Join Request packet sent from the want-to-join receiver. We suggest enhancing the RREP (reply) packet, whose format includes the hop count number, with the number of neighbor hosts and capacity of bandwidth shown the Fig. 10. Each forwarding node or multicast group node sends an RREP back to the new node. As the RREP packets are accepted by the new node, all reverse paths are available, not just the path of the smallest path count (Das S. Perkins & Belding-Royer, 2003). An example of this case is shown in Fig. 9. For choosing the smallest hop count path, four candidate nodes (R1, R2, F1 and F4) only reply four Reply packets with the smallest hop count in each path back to the receiver R5. After R5 receives all the Reply packets, F4 will be chosen and establish the new route with R5. For choosing all possible paths, four candidate nodes (R1, R2, F1 and F4) reply five Reply packets with all possible paths, and an optimal path is selected for receiver R5. On receiving the Join Request, each forward node of the multicast tree responds with Reply control packet backward to the newly joining receiver. Each Reply control packet travels along the reverse path of a Join Request control packet. Each node in the reverse path keeps the correlative factors of different inputs for fuzzy logic basic rules which are discussed in Section 3. After collecting all Reply control packets from the forwarding nodes, the new receiver constructs a marked fuzzy Petri net model (FPN) using fuzzy production rules with dynamic certainty factors which were determined by the fuzzy controller shown in Fig. 11. Fuzzy reasoning algorithm: INPUT: the degree of truth yr of the proposition for the newly joining node dr, where yr 2 [0, 1]. OUTPUT: the degree of truth of the proposition for each candidate forwarding node dj. Step 1: Initially, build the set of the forwarding nodes and the root node (the new receiver) as input places for the fuzzy Petri net. The place of the root node contains a token. Step 2: Build the set of transitions for all the input places. Step 3: Connect all possible paths for both input places and transitions according to the reverse paths found in the route initial phase. Step 4: Fire all possible transitions one by one and calculate the dynamic certainty factor. Each path from the root node to each succeeding forwarding node presents a reasoning path. The

R5 Fig. 9. Route setup diagram.

Fig. 10. Proposed RREQ packet format.

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Fig. 11. Route optimization process fuzzy Petri net.

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Fig. 12. Routing decision process sprouting tree.

degree of truth of the proposition for each candidate forwarding Q node dj is calculated by dj ¼ dr nk¼1 yr where n is the number of the forwarding nodes in a reasoning path. Step 5: If there are no successful node exits, then STOP. After performing the knowledge inference, the receiver node creates a sprouting tree starting from the receiver node to each candidate forwarding node shown in Fig. 12. To scale up the efficiency of the knowledge inference for all possible links, we define the degree of the truth of the proposition of receiver dr as a constant, and we are to find the degree of truth of the proposition of each candidate forwarding node. Since there are n forwarding nodes, there are R1, R2, . . ., Rn, totally n possible routes from the receiver node to each candidate forwarding node. Assuming Ri traverses intermediate nodes n1, n2, . . ., nm, totally m possible intermediate nodes to relay the packets from the candidate forwarding group node to the newly join receiver. Assuming the current CF of the h node in the Rk route is dkh, then the degree of truth of proposition for the k route, (DTk), is defined:

DT k ¼ dr

m Y

Path Path Path Path

2: 3: 4: 5:

R5 ? I2 ? I5 ? F4 R5 ? I1 ? I4 ? I8 R5 ? I1 ? I4 ? I8 ? F1 R5 ? I1 ? I4 ? I4 ? R1

Furthermore, we compared the packet delivery ratio with the original BEMRP method and other redundancy paths. Arrival rates were 10, 20, 30, 40, 50 and 60 (packets/s) with 5000 packets respectively to evaluate the average latency for the whole networks. When applying KMIP, path 2 was selected because the probability of the fuzzy inference result was the best solution, with a value 0.62. Path 4 was selected by BEMRP because it had the smallest the hop count number. The simulation result in Fig. 14 shows that the average waiting time increases as the

dkh :

k¼1

Therefore, the optimal desired route can be obtained from DT opt ¼ maxi2S DT i , where S is the set containing all possible routes. After finding the optimal route, a Reserve packet is sent along the reverse path taken by the chosen Reply packet. On receiving path it updates relevant fuzzy inputs and multicast routing table. 5. Simulation results An NS2 (Network Simulator – version 2) (Fall & Varadhan, 2000) simulator was used to deploy and design our mechanism. The environment for the simulation is shown in Fig. 13. In this scenario, a network environment with 19 nodes and the exponential distribution arrival rate were used. Each packet size was 1Mb. To simulate the environment in a different scheme as seen in the Fig. 9, we design five paths in between, from path 1 to path 5, respectively. The each path include nodes as the follow: Path 1: R5 ? I3 ? I6 ? I10 ? R2

Fig. 13. The environment for simulation.

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packet delivery rate increases continuously for each case due to the limitation of link bandwidth caused by the bottleneck of some forwarding nodes. Hence, when the arrival rate is more than 50 packets/per second, the average waiting time increases dramatically. This is because too many packages are waiting in the queue. Hence, from the simulation result, path 2 chosen by KIMP as the best solution among other paths, including the path chosen by BEMRP method. In Fig. 15, we compared the average waiting time of BEMRP to that of KIMP. Following the increase of the arrival rate, KIMP is increasing better than MEMRP. The simulation results show the

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performance improvements of KIMP over BEMRP are as much as 5.22%, 5.70%, 6.56%, 14.80%, 18.69% and 67.17%, with the arrival rate from 10 to 60 packets/per second, respectively. We also simulate the packet loss rate in a busy network. When the network traffic is heavy, we can find from Fig. 16 that both the paths of KIMP and BEMRP lose the packets. The simulation results show the packet losses of our KIMP are 11, 25, 28, 24, 23, 22, 29, 21, 32 and 25, respectively. The packet losses of BEMRP are 17, 33, 37, 45, 38, 38, 38, 36, 37 and 39, respectively. Hence, it can be concluded that KIMP selects a better path more effectively and KIMP also improves the overall.

Fig. 14. The average waiting time in each path.

Fig. 15. Comparison of the average waiting time between KIMP and BEMRP.

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Fig. 16. Comparison of packet losses between KIMP and BEMRP.

Fig. 17. Compare the improvement between all paths.

In the Fig. 17, the simulation results imply that KIMP selects the best path by the packet information of the two fields with the number of neighbor hosts and the capacity of the bandwidth. Using the arrival rate of 10 packets per second for an example, the improvement ratio with paths 1, 3, 4 and 5 are 30.68, 19.50, 6.04, and 42.04, respectively. When the arrival rates are up to 50 packets per second, the improvement ratios are up to 66.84, 18.17, 22.98 and 69.23, respectively. Although KIMP introduces an additional 8 bytes of overhead in the two new fields, the simulation results show that KIMP selects a better path than BEMRP, and KIMP remarkably improves loss rate and reduces average waiting time effectively, as compared with BEMRP protocol.

6. Conclusions In this paper, we proposed a Knowledge Inference based bandwidth-efficient multicast protocol (KIMP) scheme which uses automatic fuzzy reasoning to discover the best routing path for multicast routing protocols in a highly bandwidth-scarce environment. KIMP has both features of fuzzy inference reasoning and learning ability that adapt each node with the dynamic conditions of wireless networks. In the discovery phase, each forwarding node or multicast group node sends Reply packets with the proposed RREP packets back to the newly joining receiver with all possible reverse paths instead of only the path of the smallest hop count.

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Conventional multicast schemes try to find the nearest forwarding node for a newly joining member, which might increase the distance between a multicast source and the multicast members and consequently give rise to a longer delay and reduction in packet loss. We propose to add two new fields, the number of neighbor hosts and the capacity of the bandwidth, to the RREP packets. From the simulation results, we can find the improvement will rise up to 67.17%. Simulations show that our proposed scheme can improve the link stability and packet delivery ratio of wireless networks. References Chen, S.-M., Ke, J.-S., & Chang, J.-F. (1990). Knowledge representation using fuzzy Petri nets. IEEE Transactions on Knowledge and Data Engineering, 2(3), 311–319. Cheng, H., Cao, J., & Wang, X. (2006). A heuristic multicast algorithm to support QoS group communications in heterogeneous network. IEEE Transactions on Vehicular Technology, 55(3), 831–838. Chiang, T.-C., & Huang, Y.-M. (2004). Multicast routing representation in ad hoc networks using fuzzy Petri nets. In The 18th international conference on advanced information networking and applications (pp. 1–4). Das S. Perkins, C. E., & Belding-Royer, E. M. (2003). Ad-hoc on-demand distance vector (AODV) routing RFC 3561. IETF Network Working Group. Dolev, S., Schiller, E., & Welch, J. L. (2006). Random walk for self-stabilizing group communication in ad hoc networks. IEEE Transactions on Mobile Computing, 5(7), 893–905. El-Sayed, A., Roca, V., & INRIA Rhone-Alpes (2003). A survey of proposals for an alternative group communication service. IEEE Network, 46–51. Fall, K., & Varadhan, K. (2000). NS notes and documents. The VINT Project, UC Berkeley, LBL, USC/ISI, and Xerox PARC, February. . Huang, Y.-M., Chiang, T.-C., & Hou, T.-W. (2006). A partition network model for ad hoc networks in overlay environments. Wireless Communications and Mobile Computing, 6, 711–725.

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