Improved AIMD-A Mathematical Study

: One of the crucial elements in the Internet is the ability to adequately control Congestion. AIMD (Additive Increase Multiplicative Decrease) is the best algorithm among the set of liner algorithms because it reflects good efficiency as well as good fairness. Our Control model is based on original approach of AIMD. In this paper we introduce improved version of AIMD. We call our approach improved AIMD. We are also including various inherent properties of Congestion Control i.e. Fairness, Responsiveness, Smoothness and efficiency.


INTRODUCTION
Congestion Control in the Internet was introduced in the late 1980s by Van Jacobson [1] . A network is considered congested when too many packets try to access the same route, resulting in an amount of packets being dropped. In this state, the total load exceeds the capacity of the network. During congestion, actions are taken both by transmission protocols and network router in order to avoid a congestive collapse ensure network stability, efficiency and fair resources allocation of bandwidth. During a time of collapse, only a fraction of bandwidth is utilized and remaining is wasted.
In the last few years, many congestion control algorithms have been introduced [1][2][3][4] . Since the dominant Internet flow is TCP based [5] , it is widely accepted that new algorithm should be TCP friendly. A System is said to be TCP friendly if Non TCP and TCP flow have approximately the same data-transferring rate (in terms of packets per second) under same conditions [6,7] . The following are the basic properties of congestion control protocol.
Efficiency: It is the average flows throughput per round trip time (RTT) when system is in equilibrium. System is said to be in equilibrium when each flow shares same window.
Smoothness: It is magnitude of oscillations during decrease step [8] .
Responsiveness: It is number of RTTs required for the system to achieve equilibrium [8] .
Fairness: Every flow uses equal share of bandwidth.
There are a number of linear algorithms introduced till now. In linear algorithm increasing factor and decreasing factor varies linearly. e.g. AIMD [8] (Additive increase/ Multiplicative decrease) MIMD [9] (Multiplicative increase/ Multiplicative decrease), MIAD [9] (Multiplicative Increase/ Additive Decrease) and AIAD [9] (Additive Increase/ Additive Decrease). But long-term fairness is achieved by AIMD [9] . Our proposed work is related to AIMD family wherein we present an improvement of AIMD algorithms that improves fairness as well as efficiency.
AIMD congestion control basic technique and system model: Chiu and Jain provide a theoretical justification for favoring AIMD [3] : according to their analysis of linear adjustment algorithm for a simple feedback model, AIMD yields the quickest Convergence to efficiency -fair states [9] .
Within the class of increase decrease method; we specifically focus on the class of Additive Increase and Multiplicative Decrease (AIMD). In AIMD when system responds to congestion, used Bandwidth (Window) is multiplied by some factor (Decrease step) and in the absence of Congestion used Window is increased by some factor (Increase Step). Suppose these factors are a and b respectively. Many researchers have proved 1 = a for best utilization of channel.
Obviously we follow these factors. But in our proposed work these are implemented in such a way that system gives better efficiency than that of previous works.
Our system is binary and synchronized. System is synchronized because every user has same RTT and the system gives feedback simultaneously for each user. The system feedback is 1 when window is available. Our system model is defined in following figure that is based on assumption of Chiu and Jain model [3] .  Initially let flows f1 and f2 contain x and y window respectively. With out loss of generality we assume that y x < and W y x < + furthermore, we are assuming that system converges to 'fair' in 'm' cycle. In 1 st cycle Pseudocode is given by:

Flow f1
Flow f2 x y It is clear in 1 st cycle that system has 1 k +1 Round Trip Time (RTTs) or steps. Let W k y x ≥ + + 1 2 then there is Congestion and system gives 0 feedback. Now we will use decrease step. In 2 nd cycle Pseudocode is given by: Thus total flow is k y Obviously 2 nd cycle contains k 2 +1 RTT. Let feedback. Obviously we will use decrease step. In 3 rd cycle Pseudocode is given by: It gives Thus total flow is Suppose m th cycle points to equilibrium that is all flows share fair allocation of resources.
The algorithmic approach when initial window size of 2 flows and Window size are x , y and W respectively, is given by: Total number of packets in various cycles: In 1 st Cycle, total number of packets is given by: In 2 nd Cycle, total number of packets is given by: After solving the equation we have: Thus total number of packets is given by : Similarly in 3 rd cycle, total number of packets is given by: Similarly m th cycle, total number of packets is given by: Thus total number of packets in all cycles is given by:  Similarly at the end 2 nd cycle, fairness ratio is given by 1  . Similarly we can find fairness ratio for remaining cycle. According to these results we can say that our system converge to monotonic fairness. There is one interested question here how much cycles are required for fairness. We have following reasoning for it.
Since every time both x and y are divided by 2 of its previous value and equal constant are added in both flows. Thus system can never reach equilibrium if we assume float arithmetic. In Integer arithmetic we are assuming that system reaches fairness in m cycle.
Obviously Convergence to fairness of Improved AIMD is faster than that of AIMD.
Responsiveness: Numbers of RTTs required for equilibrium (Responsiveness) is measured as:

CONCLUSION AND FUTURE WORK
In this paper we presented and evaluated a new algorithm of AIMD family of congestion management, called Improved AIMD. It generalizes during increasing step k x x + = and on decreasing step x=x+k/2. It converges to fairness in 1+log (n) approximately. This is the best result in AIMD family. Responsiveness is reflected as very good . From above numerical figure it gives more than 99% efficiency. It is compare to the improved [10] . The issue that we have not included is the impact of different arrival time of each flow. It means that any flow can join the Network at any time. But in our work we assumed that arrival time of each flow is same. We will consider different arrival issue for future study.