1. Bufferless multiplexer
© Jörg Liebeherr, 2010

2. Internet Service Provider
A small internet service provider (ISP) sells internet access at rate R to its customers
The ISP has leased a link with capacity C to get access to the Internet
(P Mbps)
Question:
What is the maximum number of customers (Nmax) that can be supported on the link?
© Jörg Liebeherr

3. Bufferless Multiplexer
Assumptions:
Switch at ISP has no buffer. Traffic is dropped when rate from DSL customers exceeds
We only look at upstream traffic from DSL customers to Internet
If all users are transmitting all the time:
But customers are not always sending traffic.
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4. ON-OFF Sources
Each customer generates traffic at rate R and transmits packets of size L.
The customer waits until a full packet is available and then transmits at the subscribed rate P.
This results in an ON-OFF pattern of transmission
ON Phase:
Send at rate P
OFF Phase:
Wait until a full packet has been assembled ON ON ON
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5. ON-OFF Source ON ON ON
Arrival
time T1 T2
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6. Multiple ON-OFF Sources
Arrival
time
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7. Multiple ON-OFF Sources
Worst-case ON ON ON ON ON ON ON ON ON
Arrival
time
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8. CLT: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Computes P_loss at bufferless multiplexer
% for Greedy ON-OFF Sources
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% CLT
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
N=[1:1:1000]; % number of flows
C=100; % link capacity
P=1; % ON rate ("peak")
rho = 0.1; % average rate
epsilon=0.01;
subplot(2,1,1);
Ploss = 1- normcdf(C, N*rho, sqrt(N*(P*rho - rho * rho )));
semilogy(Ploss,'b');
xlabel('Number of flows (N)');
ylabel('P_{loss}');
axis([ ^-16 1]);
grid on
title('Application of CLT');
© Jörg Liebeherr

9. Chernoff Bound: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Computes P_loss at bufferless multiplexer
% for Greedy ON-OFF Sources
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Chernoff Bound
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
N=[1:10:1000];
a=C./N;
l=a./P.*log(a/rho.*(P - rho)./(P-a)) - log((P-rho)./(P-a));
Ploss=exp(-N.*l);
subplot(2,1,2);
semilogy(N,Ploss,'b');
xlabel('Number of flows (N)');
ylabel('P_{loss}');
axis([ ^-16 1]);
grid on;
title('Application of Chernoff Bound');
© Jörg Liebeherr

10. Effective bandwidth Slotted system Single source:
In each slot, source transmits with Prob. p
If source transmits, it sends at rate R
Bk={0,R} transmission in slot k
An are the transmissions in timer interval [0,n]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Effective bandwidth for Bernoulli On Off Source
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
p=0.4 % probability of transmit
R=1 % peak rate
C=0.75 % Capacity
theta=[0:.1:100] % Theta
semilogx(theta, log(1-p+p*exp(R*theta))./theta,theta,C, [ ], [0 1] )
© Jörg Liebeherr