1. 1 Fairness Index Lee Cheng-Ta October 3, 2006

2. 2 Jain’s Fairness Index Fairness Index = where x i is the normalized throughput (in Kbps) of the i−th TCP flow and n is the number of connections. Example: A scheme gives 50, 30, 50 Mbps (throughput, t i ) when the optimal is 50, 10, 10 Mbps (fair throughput, o i ) – –Normalized Throughput: x i = t i /o i – –50/50, 30/10, 50/10  1, 3, 5 Fairness Index =(1+3+5) 2 / 3(1 2 +3 2 +5 2 ) =9 2 / 3(1+9+25)= 0.77 1, 4, 5  0.79 1, 2, 5  0.71

3. 3 Fairness Index: Properties Scale independent –Standard deviation (Throughput) = 10 Mbps = 10 4 kbps Bounded –between 0 and 1 or 0 and 100% variance, standard deviation, and relative distance are not bounded. Direct relationship: –Higher index  More Fair –Higher variance  Less fair Continuous –min/max is not continuous. –The index should be continuous. Any slight change in allocation should show up in the fairness index. In the above example, if the normalized throughput is 1, 4, and 5 respectively, the fairness should obviously be different, yet it is not reflected in the min-max ratio which remain at 1/5.

4. 4References 1. 1. R. Jain, W. Hawe, D. Chiu, “A Quantitative measure of fairness and discrimination for resource allocation in Shared Computer Systems,” DEC-TR-301, September 26, 1984, http://www.cis.ohio-state.edu/~jain/papers/fairness.htm 2. “,” IEEE Journal on Selected Areas in Communications, Volume 24, Issue 3, March 2006 Page(s):579 - 592 (2005/2/21) 2. Yu, O.; Saric, E.; Li, A.; “Fairly adjusted multimode dynamic guard bandwidth admission control over CDMA systems,” IEEE Journal on Selected Areas in Communications, Volume 24, Issue 3, March 2006 Page(s):579 - 592 (2005/2/21) 3.,” (2005/9/22) 3. Krishnamachari, B.; Ordonez, F,”Analysis of energy-efficient, fair routing in wireless sensor networks through non-linear optimization,” 2003 IEEE 58th Vehicular Technology Conference, 2003. VTC 2003-Fall. Volume 5, 6-9 Oct. 2003 Page(s):2844 - 2848 Vol.5 (2005/9/22) 4. (2005/9/22) 4. Baochun Li, “End-to-End Fair Bandwidth Allocation in Multi- Hop Wireless Ad Hoc Networks,” 25th IEEE International Conference on Distributed Computing Systems, June 2005 Page(s):471 – 480 (2005/9/22)