1. 1 On the Performance of Slotted Aloha with Capture Effect in Wireless Networks Arash Behzad and Julan Hsu Professor Mario Gerla CS218 Project UCLA December 1, 2003 {abehzad,julan}@ee.ucla.edu@ee.ucla.edu

2. 2 System Model Assumptions:  Slotted ALOHA  Omni directional antennas  Half-duplex radios  Immobile nodes and fixed topological configuration (single access net) Assumptions:  Slotted ALOHA  Omni directional antennas  Half-duplex radios  Immobile nodes and fixed topological configuration (single access net) Objective:  A nalysis of the throughput performance of the Slotted Aloha medium access control for an arbitrary topology under variations of transmission probability (q) and transmission power level (P) Objective:  A nalysis of the throughput performance of the Slotted Aloha medium access control for an arbitrary topology under variations of transmission probability (q) and transmission power level (P)

3. 3 Highlights of Presentation Highlights of this presentation: i.Major interference models in wireless networks and their features ii.Some asymptotic theoretical results iii. Power control and capture effect iv. Preliminary simulation results v.Conclusions Highlights of this presentation: i.Major interference models in wireless networks and their features ii.Some asymptotic theoretical results iii. Power control and capture effect iv. Preliminary simulation results v.Conclusions

4. 4 I. Two Core Interference Models in Wireless Networks and their Features

5. 5 1. Protocol Interference Model  Assuming all nodes employ a common transmission range r c, transmission from node i 1 to node j 1 is successful if and for every other node i 2 transmitting in the same time slot  r c and r i are commonly known as communication range and interference range, respectively.  Assuming all nodes employ a common transmission range r c, transmission from node i 1 to node j 1 is successful if and for every other node i 2 transmitting in the same time slot  r c and r i are commonly known as communication range and interference range, respectively. j1j1 j2j2 j3j3 i3i3 i1i1 i2i2

6. 6  Let be the subset of nodes simultaneously transmitting at some time instant (time slot) employing an identical transmit power level. Then the transmission from a node,, is successfully received by a node,, if and only if whereby is the distance between nodes and, N is the ambient noise power level, and is the path loss exponent.  Let be the subset of nodes simultaneously transmitting at some time instant (time slot) employing an identical transmit power level. Then the transmission from a node,, is successfully received by a node,, if and only if whereby is the distance between nodes and, N is the ambient noise power level, and is the path loss exponent. 2. Physical Interference Model j1j1 j2j2 j3j3 i3i3 i1i1 i2i2

7. 7 Disadvantages of Protocol Interference Model: Aggregate Effect of Interference -Illustration of a transmission scenario, which is feasible based on Protocol Interference Model and is infeasible based on Physical Interference Model. Note that all receivers are out of the “interference range” r i of non-associated transmitters. This problem can be resolved by Protocol Interference Model via considering a larger interference range in the expense of losing some spatial reuse. j1j1 j2j2 j3j3 i3i3 i1i1 i2i2 Feasible based on Protocol Interference Model, but infeasible based on Physical Interference Model Feasible based on Protocol Interference Model, but infeasible based on Physical Interference Model

8. 8 -Illustration of a transmission scenario, which is feasible based on Physical Interference Model (assuming i 2 is sufficiently close to j 2 ) and is infeasible based on Protocol Interference Model, since j 2 is in the interference range of i 1. j1j1 i1i1 i2i2 j2j2 Infeasible based on Protocol Interference Model, but feasible based on Physical Interference Model Infeasible based on Protocol Interference Model, but feasible based on Physical Interference Model Disadvantages of Protocol Interference Model: Capture Effect

9. 9 KEY ASSUMPTIONS: I. Protocol Interference Model II. Identical transmit power level (no power control) III. Identical probability of transmission for all nodes KEY ASSUMPTIONS: I. Protocol Interference Model II. Identical transmit power level (no power control) III. Identical probability of transmission for all nodes Conventional Slotted Aloha in Wireless Networks 4 AP/BN/BS 2 6 3 5 1

10. 10 Interference Model Assumptions: Protocol Model  Based on the Protocol Interference Model, transmission from node k to AP is successfully received if and only if it is the only transmission in the underlying slot. Why? 4 AP 2 6 3 5 1 Is it fair?

11. 11 Interference Model Assumptions: Physical Model  What if we consider the Physical Interference Model? The throughput under Protocol Interference Model is higher or under the Physical Interference Model? Why? 4 AP 2 6 3 5 1 Does q=1/n still work?

12. 12 II. Some Asymptotic Theoretical Results

13. 13  Consider an arbitrary (symmetric/asymmetric) topology. Based on the Physical Interference Model, the probability of success for transmission from i k to AP (C k ) is equal to where Y r is a 0-1 Bernoulli variable defined as follows:  Consider an arbitrary (symmetric/asymmetric) topology. Based on the Physical Interference Model, the probability of success for transmission from i k to AP (C k ) is equal to where Y r is a 0-1 Bernoulli variable defined as follows: Probability of Successful Transmission

14. 14 Estimation of Aggregate Interference by L-F Central Limit Theorem  Equation (1) can be written as  The term (i.e. the aggregate interference) is a linear combination of independent Bernoulli variables. Conclusively, based on a generalization of the Central Limit Theorem (the Lindeberg- Feller Central Limit Theorem) we havethe Lindeberg- Feller Central Limit Theorem where n is a sufficiently large number.  Equation (1) can be written as  The term (i.e. the aggregate interference) is a linear combination of independent Bernoulli variables. Conclusively, based on a generalization of the Central Limit Theorem (the Lindeberg- Feller Central Limit Theorem) we havethe Lindeberg- Feller Central Limit Theorem where n is a sufficiently large number.

15. 15 The Lindeberg-Feller Central Limit Theorem If the independent random variables satisfy the Lindeberg condition, then for all a < b, where is the normal distribution function. Note: In most practical cases the Lindberg condition is satisfied. If the independent random variables satisfy the Lindeberg condition, then for all a < b, where is the normal distribution function. Note: In most practical cases the Lindberg condition is satisfied.

16. 16 Theorem 1  Theorem 1. Consider an arbitrary (symmetric/asymmetric) topology with large number of nodes operating under a Slotted Aloha medium access control. Based on the Physical Interference Model, the probability of success for transmission i k  AP can be calculated as where Q(.) is the Q-function, n is an arbitrarily large number and  Theorem 1. Consider an arbitrary (symmetric/asymmetric) topology with large number of nodes operating under a Slotted Aloha medium access control. Based on the Physical Interference Model, the probability of success for transmission i k  AP can be calculated as where Q(.) is the Q-function, n is an arbitrarily large number and

17. 17 Derivation of Aggregate Throughput  The aggregate throughput (per slot) can be calculated as the following: where X r is a 0-1 Bernoulli variable defined as  Clearly, (why?; are X i ’s independent?)  The aggregate throughput (per slot) can be calculated as the following: where X r is a 0-1 Bernoulli variable defined as  Clearly, (why?; are X i ’s independent?)

18. 18 Theorem 2  Theorem 2. Consider an arbitrary (symmetric/asymmetric) topology with large number of nodes operating under a Slotted Aloha medium access control. Based on the Physical Interference Model, the aggregate throughput can be calculated as where Q(.) is the Q-function, n is arbitrarily large number and  Theorem 2. Consider an arbitrary (symmetric/asymmetric) topology with large number of nodes operating under a Slotted Aloha medium access control. Based on the Physical Interference Model, the aggregate throughput can be calculated as where Q(.) is the Q-function, n is arbitrarily large number and

19. 19 Based on Theorem 2, what happens to TH as q  1? Would it be still fair? Intuitive Interpretation of Theorem 2 Based on equalities (3) and (4), it can be easily shown that 4 AP 2 6 3 5 1

20. 20 III. Power Control and Capture Effect

21. 21 For a given topology, every node i k can individually select the transmit power such that its received signal power at AP becomes a constant, say A mW. Clearly, this method supports fairness. What is the drawback of this approach? Conventional Power Control  Under the former approach the probability of capture becomes zero. 4 AP 2 6 3 5 1

22. 22  Assume there are m different transmit power levels. Every node randomly and independently selects a power level for its transmission in the underlying slot.  This method supports fairness and increases the channel utilization. Why?  Assume there are m different transmit power levels. Every node randomly and independently selects a power level for its transmission in the underlying slot.  This method supports fairness and increases the channel utilization. Why? Power-Controlled Approach for Symmetric Topologies

23. 23 Key Observation  As noted before, the aggregate throughput can be calculated as  Key observation: Based on Chebychev’s inequality and Wald’s lemma we have proven that the upper bound for probability of capture is inversely proportional with the variance of power distribution. This is also intuitively correct. Why?  As noted before, the aggregate throughput can be calculated as  Key observation: Based on Chebychev’s inequality and Wald’s lemma we have proven that the upper bound for probability of capture is inversely proportional with the variance of power distribution. This is also intuitively correct. Why?

24. 24 IV. Preliminary Simulation Results

25. 25 Simulation Assumptions  Noise power (N) = -90 dBm  Transmit power (P) = 50 mW (unless otherwise specified)  Communication range (r c ) = 250 m Minimum required SINR = 10 dB  Path loss exponent = 4  Number of nodes = 50  Noise power (N) = -90 dBm  Transmit power (P) = 50 mW (unless otherwise specified)  Communication range (r c ) = 250 m Minimum required SINR = 10 dB  Path loss exponent = 4  Number of nodes = 50

26. 26 Throughput per Node associated with Approach 1 (Symmetric)  Note that throughput of node k is equal to

27. 27 Optimal Probability of Transmission associated with Approach 1

28. 28 Throughput per Node associated with Approach 1

29. 29 Aggregate Throughput associated with Approach 2 (Asymmetric Topology) P1 = 1 mW; P2 = 100 mW

30. 30 Throughput per Node associated with Approach 2 (Asymmetric Topology) P1 = 1 mW; P2 = 100 mW

31. 31 Conclusions  We introduced two methods for increasing the aggregate throughput of a single access net based on the Slotted Aloha MAC: - In approach one no power control were used. The only control knob considered was q, probability of transmission. - In approach two, two knobs were takes in consideration simultaneously: P (transmit power level) and q (probability of transmission)  Approach one seems to be more promising, as the results are not asymptotic and a minimum fairness is guaranteed. However, not all systems possess the power control capabilities.