This document discusses Jain's Fairness Index, which is a metric used to measure fairness in distributed systems. It is defined as the ratio of the squared sum of individual throughput values to the product of the total throughput and number of flows. A higher index indicates more fairness, with a maximum of 1. The document provides an example calculation and discusses properties of the fairness index such as being scale independent, bounded between 0 and 1, and directly related to fairness and variance. Finally, it lists four references for further information.

1. Fairness Index
Lee Cheng-Ta
October 3, 2006
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2. Jain’s Fairness Index
Fairness Index =
where xi 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, ti) when the
optimal is 50, 10, 10 Mbps (fair throughput, oi)
– Normalized Throughput: xi = ti/oi
– 50/50, 30/10, 50/10  1, 3, 5
Fairness Index =(1+3+5)2 / 3(12+32+52)
=92 / 3(1+9+25)= 0.77
1, 4, 5  0.79
1, 2, 5  0.71
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3. Fairness Index: Properties
Scale independent
– Standard deviation (Throughput) = 10 Mbps = 104 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.
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4. References
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. 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. 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. 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)
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