Reference Analysis-Bianchi model for IEEE 802.11
Preliminary
Bianchi Model is based on the Bi-dimensional Markov Chain and is used for performance evaluation of DCF in 802.11. 802.11 is one Medium Access Control(MAC) protocol for Wireless LAN, DCF is its fundamental function, in which the Binary Exponential Backoff(BEB) rules and CSMA/CA based Access techniques are included.
- BEB algorithm
BEB algorithm is used for avoiding collision when multi-users access to the same wireless channel. Within this algorithm, when the user detect the busy channel, it will delay some exponential backoff times before it transmits next time. The Backoff Counter 'W' is adapted for the every exponential backoff time, i.e., the backoff time is relected randomly between 0 to 2^W. The Backoff Counter 'W' is variable in a range[Wmin,Wmax], where m*Wmin=Wmax, the W will increase to its double size when the user still detect the busy channel following a busy channel detection, otherwise it will decrease to the minimize size of W.
- CSMA/CA
Carrier Sensing and Multiple Access and Collision Avoiding(CSMA/CA) used two access techniques. Basic mechanism and RTS/CTS mechanism.
Basic mechanism:
RTS/CTS mechanism
2 mechanism in 802.11
If a user want to send uni-cast packets to AP, it must apart to the medium access process(competition for medium), in which two access mechanisms are applied.
- Basic mechanism
- Sender has to wait for at least DIFS(SIFS,PIFS) before sending data, after finishing that, if the channel is still busy, the BEB follows this process.
- Receiver acknowledged at once (after waiting for SIFS) if the packet was received correctly CRC.
- Sender retransmits data packets in case of transmission errors.
- Other stations defer access.
Noted that sender can transmit data immediately without waiting for DIFS once the backoff counter reached 0. Immediate access when medium is free.
- RTS/CTS mechanism
- Sender can send RTS with reservation parameter after waiting for >=DIFS(reservation determines amount of time the data packet needs the medium).
- Receiver acknowledgement via CTS after SIFS(if ready to receive).
- Sender can now send data at once, acknowledgement via ACK.
- Other stations defer access by setting NAV via RTS and CTS.
- Backoff Counter in RTS/CTS mechanism
The Idea is reverving channel via RTS/CTS and avoiding dta collision. The process is: DIFS+contention+RTS+SIFS+CTS+SIFS+DATA+SIFS+ACK
Bianchi's Model
Based on the topology of n links within one-hop network in wireless LAN,
Saturation state occurs when each node(link) always has a packet to transmit and the throughput in saturation state is called Saturation throughput, which means the limit of the system throughput as the offered load increases and also represents the maximum load that the system can carry in stable conditions. Therefore, Saturation throughput can measure the network performance in this scenario, so Bianchi's Model is used for modelling this scenario and aims at calculating the per-node saturation throoughput by calculating the probability of Collision and probability of Transmission.
Some assumptions are required:
- 1. All wireless nodes reside in single-cell network, i.e., all stations are in sensing range of each other.
- 2. The collosion processes of the nodes can be decoupled, i.e., at each transmission attempt, and regardless of number of retransmissions suffered, each packet collides with constant and independent probability.
- 3.Channel conditions are ideal, means transmission errors are a results of packet collision only amd no hiddden and exposed terminals.
Bi-dimensional Markov Chain model
The discrete and integer time slot is choosed as the time scale in the Model, let t(i) denotes the beginning of i-th backoff counter decrease and [t(i), t(i+1)] represents the interval between 2 consecutive backoff counter decrease, which is called a generic slot time and can be variable length. A practical scenario in 802.11 is as follows, 2 stations are competing with a common channel(i.e., two stations transmit information to station B), and RTS/CTS mechanism is adapted. As we can see, one generic slot is variable in it which can equal to Tc(Time of collision), Ts(Time to transmit a packet of size L) and 1 delta(A fixed idle time).
The size of the backoff time counter in stage i is Wi
= 2^
iWmin, where Wmin represents the minimun value of the backoff counter. Let m denotes the maximum backoff stage, then the maximum backoff counter is 2^m power of the minimum backoff counter, i.e.,
Wmax = 2^mWmin. Let s(t) denotes the backoff states with range in [0, m]，let b(t) denotes the backoff counter with range in [0, Wi], where i=s(t) . Then B(t)={s(t), b(t)} can denote the
Markov State in the Model
. The Markov State can be seen as the above figure, where BO represents the backoff counter. Then the state transition graph is as follows:
The saturation throughput can be defined by
.
Then the probability of transmission can be defined by
.
After a series deduction and computation, the collision probability p and the transmission probability have the following relationship:
Ok! Note that Now we can plot the relationship between collision probability and the number of stations n, the matlab codes are as follows:
%%%%%%%%%collision probability%%%%%%%%%%%%%%
l=1;t0=1;t1=1;t2=1;x=0:0.01:1;
y=@(x)2./(1+32+32*x.*(1+2.*x+4.*x.^2+8.*x.^3+16.*x.^4));
for n=1:1:30;
g=@(x)1-(1-y(x)).^(n-1)-x;
g0=@(x)g(x)-x;
g1=@(x)n.*y(x).*(1-y(x)).^(n-1)./g0(x);
%g1=@(x)n*y(x)*(1-y(x))^(n-1)./(1-(1-y(x))^n));
p(n)=fzero(g,0.5);
s(n)=(l*g0(p(n))*g1(p(n)))./(t0*g0(p(n))*g1(p(n))+t1*g0(p(n))*(1-g1(p(n)))+t2*(1-g0(p(n))));
end
plot(s);title('The collision probability with n increasing');
xlabel('The number of stations');ylabel('The collision probability');
%text();
grid on;
The output is just like this
Yeap, we can make it looks a little more beautiful by the drawing tools of matlab, just like this
Why I spend so much time to do this, cause it is one term of our exam.
用matlab代码计算Bianchi模型的冲突概率：
Wmin = 32, m = 5, and n=1:1:30
函数名用fzero表示
画出冲突概率随移动台个数的变化图
画出饱和吞吐量随移动台个数的变化图