Generalities The star network Algorithm Simulations Conclusion Fairness in a Static Wireless Network Marc GILG 1 Pascal LORENZ 1 Abderrahim MAKHLOUF 2 1 University of Haute-Alsace 2 University of Pierre and Marie Curie ResCom 2008
Generalities The star network Algorithm Simulations Conclusion Outline Generalities 1 Network model Fairness The star network 2 Definition Fairness in a star network Algorithm 3 Simulations 4 The 8 star network A no-star network Conclusion 5
Generalities The star network Algorithm Simulations Conclusion Network model Ad-hoc networks Ad-hoc networks An Ad-Hoc network is made of wireless nodes which establish channel communications between themselves. A node can be in two states : transmission or reception. The limitation of radio communication implies that the communications of channel are limited in distance and that they act in half duplex mode.
Generalities The star network Algorithm Simulations Conclusion Network model Network model Network Model the network is packet-switched the nodes are in half-duplex mode only nodes in some distance are receiving the packet the time is divided in time slots packets are sent in a time slot
Generalities The star network Algorithm Simulations Conclusion Fairness Fairness Definition The fairness index of a shared resource x i is given by � 2 �� n i = 1 x i f ( x i ) = (1) n � n i = 1 x 2 i A network is fair if f ( x ) = 1.
Generalities The star network Algorithm Simulations Conclusion Fairness Fairness Definition The fairness index of a shared resource x i is given by � 2 �� n i = 1 x i f ( x i ) = (1) n � n i = 1 x 2 i A network is fair if f ( x ) = 1. Remark for Ad-Hoc networks A node with a high degree of connectivity in transmission will interfere with a high number of nodes. One of the consequences of this is that each node will not have the same transmission and reception rate. The fairness of an Ad-Hoc network is not obvious.
Generalities The star network Algorithm Simulations Conclusion Definition Definition of a star network Definition The star network SN n is composed of n + 1 nodes { X 0 , X 1 , . . . , X n } where { X 1 , . . . , X n } are neighbors of the node X 0 , and there is no connexion between X i , X j for i , j > 0.
Generalities The star network Algorithm Simulations Conclusion Definition Definition of a star network Definition The star network SN n is composed of n + 1 nodes { X 0 , X 1 , . . . , X n } where { X 1 , . . . , X n } are neighbors of the node X 0 , and there is no connexion between X i , X j for i , j > 0. 4-star network graph X 1 X 4 X 0 X 2 X 3
Generalities The star network Algorithm Simulations Conclusion Fairness in a star network Reception rate in a star network. Notations The following notations will be used : Let x i be the reception rate of node X i Let S i j be the transmission rate of node X j which is a neighbor of node X i
Generalities The star network Algorithm Simulations Conclusion Fairness in a star network Reception rate in a star network. Notations The following notations will be used : Let x i be the reception rate of node X i Let S i j be the transmission rate of node X j which is a neighbor of node X i Reception rate We have x i = S 0 for 1 ≤ i ≤ n We have x 0 = � n i = 1 S 0 i
Generalities The star network Algorithm Simulations Conclusion Fairness in a star network Fairness in a star network Lemma For a star network SN n , the fairness for the reception rates hold only and only if : n � S 0 S 0 − i = 0 (2) i = 1
Generalities The star network Algorithm Simulations Conclusion Algorithm Algorithm computes the sum of the neighbors transmission rates A i 1 compares the sum A i to the node Y i transmission rate S i . 2 if S i − A i > s then reduce S i , if S j becomes negative, then 3 set it to 0. if S i − A i < s then increase S i if it is possible. 4 if S j = 0 for a long time, then increase it. 5 go to step 1 6
Generalities The star network Algorithm Simulations Conclusion The 8 star network Fairness of a SN 8 star network 0.435 ’ex.txt’ 0.43 0.425 Fairness index 0.42 0.415 0.41 0 50 100 150 200 250 300 350 400 450 500 Time Slots
Generalities The star network Algorithm Simulations Conclusion The 8 star network Rate difference to fairness -2 ’ex-1.txt’ -4 -6 Diffrence rate to fairness -8 -10 -12 -14 -16 0 50 100 150 200 250 300 350 400 450 500 Time Slots
Generalities The star network Algorithm Simulations Conclusion The 8 star network Fairness of a SN 8 star network with modified access algorithm 0.51 ’ex.txt’ 0.5 0.49 0.48 Fairness index 0.47 0.46 0.45 0.44 0.43 0.42 0 50 100 150 200 250 300 350 400 450 500 550 Time Slots
Generalities The star network Algorithm Simulations Conclusion The 8 star network Rate difference to fairness 2 ’ex-1.txt’ 1.5 1 0.5 Diffrence rate to fairness 0 -0.5 -1 -1.5 -2 -2.5 0 50 100 150 200 250 300 350 400 450 500 550 Time Slots
Generalities The star network Algorithm Simulations Conclusion A no-star network Network topology X 1 X 2 X 3 X 0 X 5 X 6 X 4
Generalities The star network Algorithm Simulations Conclusion A no-star network Fairness of a no-star network 0.8 ’ex.txt’ 0.78 0.76 Fairness index 0.74 0.72 0.7 0.68 0.66 20 40 60 80 100 120 140 160 Time Slots
Generalities The star network Algorithm Simulations Conclusion A no-star network Rate difference to fairness 65 ’ex-1.txt’ 60 55 50 Diffrence rate to fairness 45 40 35 30 25 20 15 0 20 40 60 80 100 120 140 160 Time Slots
Generalities The star network Algorithm Simulations Conclusion A no-star network Fairness of a no star network with modified access algorithm 0.8 ’ex.txt’ 0.78 0.76 0.74 Fairness index 0.72 0.7 0.68 0.66 0.64 20 40 60 80 100 120 140 160 Time Slots
Generalities The star network Algorithm Simulations Conclusion A no-star network Rate difference to fairness 40.5 ’ex-1.txt’ 40 39.5 Diffrence rate to fairness 39 38.5 38 37.5 20 40 60 80 100 120 140 160 Time Slots
Generalities The star network Algorithm Simulations Conclusion Conclusion Conclusion We study the fairness of a Ad-Hoc network A N -star network was introduced. An expression of transmission rates was established to solve the fairness problem. An algorithm was given to get the fairness in a star network. Simulations was done for star networks and no star networks.
Generalities The star network Algorithm Simulations Conclusion Conclusion Conclusion We study the fairness of a Ad-Hoc network A N -star network was introduced. An expression of transmission rates was established to solve the fairness problem. An algorithm was given to get the fairness in a star network. Simulations was done for star networks and no star networks. Further works In some further work, we propose to study multiple star network to approach a general Ad-Hoc network.
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