A Heuristic Ant Algorithm for Solving QoS Multicast Routing Problem Chao-Hsien Chu

JunHuaGu

Xiang Dan Hou

Qijun Gu

School oflnfo. Sci. & Tech Pennsylvania State Univ. Univ. Park, PA 16802 USA

School oflnfonnation Hebei Univ. of Technology Tianjin, P.R.C. (China)

School oflnfonnation Hebei Univ. of Technology Tianjin, P.R.C. (China)

School oflnfo. Sci. & Tech Pennsylvania State Univ. Univ. Park, PA 16802 USA

Abstract - In this paper, we present an ant colony based heuristic to solve the QoS constrained multicast routing problems. Our algorithm considers multiple QoS metrics, such as bandwidth, delay, delay jitter, and packet loss rate, to find the multicast tree that minimizes the total cost. We also explore the scalability of the ant algorithm. Our tests show that the algorithm can find optimal (or near optimal) solution quickly and has a good scalability.

I.

INTRODUCTION

The demand for real time and quality of services (QoS) in the network increases as Internet expands. However, the service level is sensitive to the characteristics of network transmission, such as delay, delay jitter, bandwidth, packet loss rate, and cost. When services such as highly abrupt FTP and HTTP with images and/or videos are transmitted through the network, real time services will be affected. Therefore, QoS measures should be introduced to the network so that quality of real time services can be guaranteed. The QoS multicast routing (QMR) problem concerns the search of optimal routing trees in the distributed network, where messages or information are sent from the source node to all destination nodes, while meeting all QoS requirements. This problem is NP complete [1]. Over the past decades, many unconstrained or simple constrained multicast routing algorithms have been developed. Typical approaches include (1) applying Dijkstra algorithm to find the shortest path, (2) seeking the minimum network cost using Steiner tree routing algorithm, and (3) finding multicast trees that the paths between source node and the destination nodes are connected and their cost is minimized. A state of the art review and analysis can be found, for example, in [1]-[3]. Recently, with the high demand of fast and better quality of services, a number of rigid QoS criteria, such as bandwidth, delay, delay constraint, and packet loss rate, have been considered. A number of efficient heuristic [4]-[6] or nature-based algorithms [7]-[13] have been proposed. An extensive review of QMR problem can be found in [2]. Worth noting is that the number of studies that apply genetic algorithms (GA) and ant algorithms to solve the QMR problems (with different types ofQoS constraints) is increasing. In [8] and [9] a heuristic GA is used to solve the QMR problems. The algorithm acquires the solution by representing a multicast tree as a chromosome so as to save the coding spaces and reduce the decoding operations (compared with

0-7803-7282-4/021$10.00 ©2002IEEE

the binary coding mechanism). However, these approaches cannot be expanded. If one or more nodes are added into the network, the system needs to scan all nodes again to acquire the optimum solution. That is, previous network information cannot be transferred to the expanded network. The main purpose of this study is to develop an efficient heuristic algorithm that can solve the QMR problems. We consider both genetic algorithm and ant algorithm in our tests as they have recently been emerged as new heuristics that can efficiently solve large-scale optimization problems. However, the former suffers from the deficiency that it cannot transfer previous network information to the expanded network. Ant algorithm performs well in solving the Traveling Salesman Problem (TSP) [14]. TSP seeks to find a closed path in n cities, which is the shortest while passing through every city only once and returning to the starting city. The characteristics of QMR seems similar to those of TSP, thus the ant algorithm may be a viable candidate for solving the QMR problems. Our study differs from existing literature in the following aspects. In [10] and [11] only a couple of QoS measures are considered. In [12] only bandwidth and cost are included in a function ere). In [13] only delay, loss rate, and delay variation are considered. In this study we adopt a model that treats these constraints separately, add more constraints such as delay jitter and packet loss rate, and take network expansion into account. II.

BACKGROUND

A. Ant Algorithm

Ant algorithm is a simulated evolution algorithm based on population and ant colony behaviors. The behavior of a single ant is simple, however, a population of ants may behave very complicated, which can complete complex tasks and even adapt to the change of environment. For example, when an obstacle appears on the moving path of an ant population, ants can find a new optimal path quickly. By observing and studying ant colony behaviors, researchers found that information exchange and collaboration among units play an important role [14]. An ant can excrete a material, called pheromone, along the path on which it moves. Ants can sense this material and detect its intensity. They can then use pheromone intensity as a guide to move and tend to move toward the direction of higher intensity, thus the ants can find the food by this kind of information exchange.

1630

The key features of an ant algorithm include distributed computation, positive feedback, and constructive greedy heuristic [14], [15]. These features can help to avoid premature convergence and find a very good solution in a shorter period of time. Since it inception, the algorithm has emerged as a new heuristic to solve many stochastic combinatorial optimization problems [14], [15].

Packetjoss (pT(S, t»=I-

n (1- packet _loss(n»

(5)

nEPT(s,t)

Where pr{s, t) is the routing path from source s to destination t in the multicast tree T(s, M). The objective of the QoS multicast routing problem is to find a multicast tree T(s, M), which satisfies: (1) delay constraint:

B. QoS Multicast Routing (QMR)Model

delay(pT(s, t»:::=;Dt

(2) bandwidth constraint: bandwidth/p-Is, t»?B (3) delay jitter constraint: delayjittertp-ts, t»:::=;DJt

The objective of QMR is to find the optimal path, which starts from the source node and passes through all destination nodes, that meets all QoS constraints with minimum cost or reaches a specific service level in a distributed network. For convenience of analysis, the network is considered as a connected, undirected and weighted graph. Let N represents a network, where V denotes the set of network nodes and E denotes the set of bi-directional links. SE V is the source node in multicast, Ms={V-(s)} is the set of destination nodes in multicast, R+ denotes the set of positive real numbers and R+ denotes the set of non-negative real numbers. Assume that there are four QoS measures associated with each edge: network delay refers to the average time an IP packet needs to be transmitted through the network; network jitter refers to the range of time an IP packet needs to be transmitted through the network (these two criteria are the factors that can cause the decrease of network transmission quality); network bandwidth is the determinative factor that reduces the end-to-end delay. Thus, for any link eEE, we denote: delay function, delay(e): E~ R+; delay jitter function, delayjitter(e): E~ R+, bandwidth function, bandwidth(e): E ~ R+, and cost function, cost(e): E ~ R+. Similarly, for each node nEV, the four measures can be denoted as: delay function, delay(n): V~ R+, cost function, cost(n): V~ R+, packet loss rate function, packetIosst»): V~ R+, and delay jitter function, delayjitter(n): V~ R+. IP packet may be damaged or lost in the transmission. If the loss rate is too high, obviously the data will be damaged. Given a source node SE V and a set of destinations M, the following relationships exist in the multicast tree T(s, M) composed ofs and M: Delay (pT(S, t»=

I.delay(e) eE

PT(s,t)

+

I.delay(n) nE

(4) packet loss rate constraint: packetIossip-ts, t»~P~

Such that cost(T(s, M) is minimized for all multicast trees that satisfy the above constraints. Where B is the bandwidth constraint, D" DJ, and PL, are the delay, delay jitter and packet loss rate constraints of destination node t respectively. In this model, all bandwidth of the multicast destination nodes are assumed to be identical, however delay, delay jitter and packet loss rate constraints can be different.

III.

The proposed procedure was adapted from the Ant-cycle algorithm as it was shown to perform better than the Ant-density and Ant-quantity algorithms [14] and [15]. Given the value of (d; dj; pl; Cj) for all nodes, (dij' diy, bij' cij) for all edges, and the value of the constraints D, DJ, B, PL, the procedure can be divided into the following steps: 1) Initialize network nodes. Set T: =0 (t is a timer and can be omitted.). NC: =0 (NC

is a loop counter.); Assign an initial value 'iij(t)=c to the pheromone intensity of every edge (i,j) and L1'lij=O; Put m ants to the source node.

2)

Check PL (packet loss rate) of all nodes, deletes the

3)

straint. Check B (bandwidth) of all edges, deletes those edges

edges linking those nodes that do not meet the PL con-

(1)

that do not satisfy the bandwidth requirements.

PT(S,t)

4)

Cost (T(s, M»=

Lcost(e) eEPT(s,t)

+

Lcost(n)

(2)

For k: =1 to m Put the value of source node into tabuus)

(3)

Delayjitter (pT(S, t»= 'LdelaY_iitter(e) eEPT(s,t)

+

I. delay _jitter(n) ne PT (s,t)

(4)

Setup tabu table.

Sets: =1

n€PT(s,t)

Bandwidth (pT(S, t»=min{bandwidth(e), eEpT(s, t)}

THE ALGORITHM AND IMPLEMENTATION

5)

Here tabu is used to save the nodes that were reached before t. tabuits) denotes the s-th node visited by the k-th ant in the current route and s is the index of tabu table. Repeat this step until the tabu is full. Set s: =s+I

0-7803-7282-4/021$10.00 ©2002mEE

1631

if (NC
For k: =1 to m Choose a node (or the next node) j according to the following probability: [T··(t)]a lJ

pb(t)=

* [17··]P lJa

2.kEallOwedko[Tij(t)]

p , if jE allowedi,

* [17ij]

1

(6) otherwise

Compute the delay and delay jitter to reach node j, and compare the result with delay D and delay jitter DJ. If the result exceeds the constraints, choose a new node; otherwise move the k-th ant to node j. Put j into tabuifs}. Where 'rij(t) is the pheromone intensity of edge (i ,j) at t, and a; f3 denote the information accumulated during the movement of ants and the different effects of factors in the path selection. allowedv-it), 1, "', n-Ts-tabu; denotes the node that the k-th ant can select in the next step. When using ant algorithm to solve TSP, Tfij= lIdij' where dij is the distance between nodes i and j. However, in this study it is not the length of path that will affect the probability, but the next node, the delay of edges linking them and the delay jitter. Thus, we set rz ij=lI(d ij+dj;). (6)Compute and Arij:

IV. RESULTS AND ANALYSIS This paper analyzes the advantages of using an ant algorithm to solve the QMR problems from two perspectives performance and scalability. In order to compare our proposed ant algorithm with an existing genetic algorithm [8], we used the same data (network) for testing. Figure 1 shows the network and constraints. In this example, node 1 is the source node and nodes 2, 4, 5 and 7 are the destination nodes that the information should reach. Constraints considered are: B=70, D=46, DJ=8, PL=O.OO1. As suggested in [14], we used the following parameters: £F1, {F1, p=0.5 in our ant algorithm.

Mt

Fork: =1 tom For every edge(i ,j) For k: =1 to m

Set

k

Mij

=

IJL Lk

if (i,j)E tabuj.

O

Mt

otherwise

is the pheromone amount left by the k-th ant at

(2,O,1~,7)

(7,31,10- 3,3)

edge (i, j) during the period from t to t+n. M ij is the Fig. 1. The numerical example

sum of pheromone amount left at edge (i, j) in this loop. In the original ant algorithm L k is the total path length of the k-th ant. Here L k is the summary of d and dj of all nodes and edges passed by the k-th ant through its path, and Q is a constant. (7) Compute r;/t+n) for every edge (i, j): Set 'rij(t+n)=p* r;/t)+L1 r;j, t: =t+n; NC: =NC+1; set L1 'rij: =0 to every edge (i, j).

A. Comparison ofAnt and Genetic Algorithms

Where parameter p must be set to a value less than 1 to avoid the infinite accumulation of pheromone. (8) Check stop condition.

0-7803-7282-4/021$10.00 ©2002 IEEE

Figure 2 (a) shows the original links (edges) and positions of the nodes. Figure 2(b) depicts the tree, which meets bandwidth requirements. As shown, the edge linking nodes 3 and 6 was deleted, as it does not satisfy the constraint. Figure 2( c) shows the tree after considering the packet loss rate constraint. Figure 2(d) shows the final tree after 20 iterations.

Figures 3(a) and (b) show the cost, delay, and delay jitter (iteration by iteration) obtained by the ant algorithm and genetic algorithm [8] respectively. From these figures, we can see that the proposed ant algorithm has a cost curve much stabilizer than that of the genetic algorithm. In addition, it finds optimal (or suboptimal) solution faster. Although the

1632

delay curve of the ant algorithm vibrates more in the beginning, the stabilized value is lower than that of GA. The delay jitter curves of the two algorithms are similar. Clearly, in this example, the proposed ant algorithm performs better than the genetic algorithm in [8].

B. Scalability ofthe Ant Algorithm

Another advantage of the ant algorithm is its expandability (scalability). Our experiment involves the use of three different sizes of networks (8 nodes, 16 nodes, and 20 nodes). By adding new nodes and providing or without providing previous information to the network, we can observe how the al(b) gorithm performs. We use the same data as in Figure 1 for the 8 nodes case. Figure 4(a) shows the curves of cost and delay of the network, in which the information before expansion is not used. In Figure 4(b), ant algorithm is used first to select the optimum path in the 7-node network. Then the network is expanded, by 7 : 7"l----{ adding a node, to become the same as the network in Figure ----(:;;:t-;:;~;t-;;,;s-------------·r-~;;;;;~-.;;s:~;------------------4(a), where the algorithm computes on this expanded network with the routing information of the previous 7-node network. I These figures show that the optimal solution was found and turned into steady state more quickly by using the information before expansion.

! Meet Bandwidth Constraint

(a) Original

I

~

!

!

! i ~

®!

®

I

(a) Do not use pre-information

Fig. 2. Sketch illustration of search tree

100 - , - - - - - - - - - - - - - - - - - - ,

(a) Performance of Ant Algorithm

: r<:.

Cost

40

Delay

20

100

O-f--r-,--r-;--r---,---r-r--r-.,.....,......,......,...-,-,--r-r--;-....,......-i

80

C st

o

60

2

4

6

Dela

40

8 10 12 14 16 18 20 Iterations

20 0

(b) Use pre-information 6

4

2

0

8

10

12

14

16

18

20

Iterations

100 - , - - - - - - - - - - - - - - - - ,

80..---------<.-------------+ Cost 60 ..-__---<.---__

(b) Performance of Genetic Algorithm

--4

Delay

40 100 80 60 40 20 0

• ~

------. -

_ _ + -_ _- _ - .

20

""". • ............

•

.CO!j .Dell)'

• • •

0-+--'''''''''''''''''''''''''''-'''''''''''''''''''''''''''-''''''''''''''''''''''''''''''''---'--''''''''''''''''''''---'--....,......-1

024

•

6

8 10 12 14 16 18 20 Iterations

Delay Jitter

• • • • • • • • • •

0

2

4

6

8

10 12 14

Generations

Fig. 3. Performance of ant system and genetic algorithm

0-7803-7282-4/021$10.00 ©2002 IEEE

Fig. 4. Scalability of ant algorithm - 8 nodes case

16 18 20

For the lti-node and 20-node cases, we first use the computer to generate the lti-node and 20-node networks. I to 8 nodes (for the 16-node case) and 1 to 10 nodes (for the 20-node case) are then added to the network. Due to the stochastic nature, we solve the problems by using the ant algo-

1633

rithm for 10 times. Each time we use a different random number seed. We then calculate the average iterations it converged. Figures 5 and 6 show the performance curves (in iteration times), one of which considers previous information and the other one doesn't.

12 , . . - - - - - - - - - - - - - - - - - - , 10 Do not use Pre-infonI!lIRllHt-"-

8 6 4 2

leaves information on the path for the sake of next routing during the search process. The information can be kept when more nodes are added. With this information the expanded network can be routed quickly. Applying ant algorithm to solve the QoS multicast routing problem is a new attempt. A number of researches has appeared in the conferences, however, most of them have not been tested extensively (including this one). In the future, we would like to conduct more tests using more data samples. Meanwhile, in this study, one node is added to the network in the analysis of expandability. In the future, more nodes will be added at once so that the relationship of the amount of initial nodes and added nodes can be explored.

Use Pre-information

REFERENCES

0+--...,..-----,----,-----,-----,--,-----1

1

2

4

3

6

5

8

7

[I]

L. H. Sahasrabuddhe, and B. Mukherjee, "Multicast Routing Algo-

[2]

ary/February 2000 B. Wang, and J. C. Hou, "Multicast Routing and Its QoS Extension: Problems, Algorithms, and Protocols," IEEE Network, pp. 22-36,

rithms and Protocols: A Tutorial," IEEE Network, pp. 90-102, Janu-

Fig. 5. Scalability of ant algorithm - 16 nodes case

JanuarylFebruary 2000 [3]

H. F. Salama, D. S. Reeves, and Y. Viniotis, "Evaluation of Multicast Routing Algorithms for Real-Time Communication on High-Speed

24 20 16 12

Networks," IEEE Journal on Selected Areas in Communications, Vol. 15:3, pp. 332-345,1997 [4]

L. Guo and I. Matta, "QDMR: An Efficient QoS Dependent Multicast

Routing Algorithm," Proceedings of the Fifth IEEE Real-Time Technology and Applications Symposium, pp. 213-222, 1999

8

[5]

Z. Wang, and J. Crowcroft, "Quality of Service for Supporting Multimedia Applications," IEEE Journal on Selected Areas in Communications, Vol. 14:7, pp. 1228-1234, 1996

[6]

W. Wu, "Multicast routing Algorithm Based 011 Multiple Qualities of

4 0

1 Fig. 6.

2

3

4

5

6

7

8

9

10

Services," Applied Electronic Technologies, Vol. 266:8, 2000 (In Chinese)

Scalability of ant algorithm - 20 nodes case

[7]

As can be seen from the figures, the expanded transmission, which uses previous information, takes less time to find the optimal or suboptimal solutions than the one who doesn't. V. CONCLUSIONS

E. Gelenbe, A. Ghanwani, and V. Srinivasan, "Improved Neural Heuristics for Multicast Routing," IEEE Journal on Selected Areas in

[8]

Communications, Vol. 15:2, pp. 147-155, February 1997 Z. Wang, and B. Shi, "Solution to QoS Multicast Routing Problem

[9]

Vol. 24:1, pp. 55-61, 2001 (in Chinese) X. Zhou, C. Chen, and
Based on Heuristic Genetic Algorithm," Journal of Computer, 2001,

Routing Problem," International Conference on Communication

In this paper, a heuristic algorithm based on ant colony is proposed to solve the QoS multicast routing problems. We consider multiple QoS metrics such as delay, delay jitter, bandwidth, and packet loss rate in our study. The proposed ant algorithm has the following characteristics: (l) The cost curve of this algorithm is somewhat stable, and the optimum or suboptimum can be found quickly; although the delay curve vibrates a little bit more, the steady delay value is less; the delay jitter curve can turn to stability quickly. Therefore, the algorithm is able to effectively improve the transmission quality of the data packets in the network. (2) The algorithm has good scalability. This is because that the ant algorithm 0-7803-7282-4/02/$10.00 ©2002 IEEE

Technology, Vol. 2, pp. 1248-1253,2000 [10]

G Lu, Z. Liu and Z. Zhou, "Multicast Routing Based on Ant Algorithm for Delay-Bounded and Load-Balancing Traffic," 25th Annual IEEE Conference on Local Computer Networks, pp. 362-368, 2000

[II]

G Lu, and Z. Liu, "Multicast Routing Based on Ant Algorithm with Delay and Delay Variation Constraints," The 2000 IEEE Asia-Pacific Conference on Circuits and Systems, pp. 243-246, 2000

[12]

[13]

1634

Y. Wang, and J. Xie, "Ant Colony Optimization for Multicast Rout-

ing," The 2000 IEEE Asia-Pacific Conference on Circuits and Systems, pp. 54-57, 2000 S. Zhang, and Z. Liu, "A QoS Routing Algorithm Based on Ant Algorithm," 25th Annual IEEE Conference on Local Computer Networks,

pp. 574-578,2000 [14]

M. Dorigo, V. Maniezzo, and A. Colomi, "The Ant System: Optimization by a Colony of Cooperation Agents," IEEE Transactions on Systems, Man, and Cybernetics - Party B, Vol. 26:1, pp. 1-13, 1996.

[15] M. Dorigo, and L. M. Gambardella, "Ant Colony System: A Cooperative Learning Approach to the traveling Salesman Problem," IEEE Transactions on Evolutionary Computation, Vol. 1:I, pp. 53-66, April

1997

ACKNOWLEDGEMENT This study was partially supported by grants from the Natural Science Foundation of Hebei Province, China.

0-7803-7282-4/021$10.00 <02002 IEEE

1635