Berkeley ELENG 228A  Networks  Some Analysis (17 pages)
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 University of California, Berkeley
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 Eleng 228a  High Speed Communications Networks
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Unformatted text preview:
Networks Some Analysis Jean Walrand www eecs berkeley edu wlr Outline Performance Evaluation How well does TCP work Simple Model and Fixed Point Model Distributed Optimization User and Network Optimization Stability of closed loop system Some basic models Walrand Classical queuing results 2 Performance Evaluation Objective Understand behavior of TCP AIMD Motivation Compare with other protocols Compatibility Understand limitations how to improve Simple Model Express throughput as function of loss rate Fixed Point Model Walrand Calculate throughput 3 Performance Evaluation Simple Model Walrand 4 Performance Evaluation Simple Model cont 6 5 RT packets 4 3 2 1 0 1 0 0 2 0 4 0 6 0 8 1 1 2 L Walrand 5 Performance Evaluation Fixed Point Key Ideas Sources adjust rate based on observed losses Losses depend on rate of source Rate Losses Instantaneous rate x t AIMD Controlled by Poisson Losses Analysis E x t R Losses Rate Walrand Router queue fed by arrivals Loss rate increasing in queue length Analysis L R 6 Performance Evaluation Fixed Point cont Summary Walrand 7 Performance Evaluation Fixed Point cont Analysis dW t dt T W t 2 dN t E dW t dt T E W t dN t 2 dt T E W t E dN t W s s t 2 dt T E W t t dt 2 t f qm t d qm t E q t dx t W t T Cdt dqm t dt axm t 1 a qm t dqm t dt axm t 1 a qm t Numerical integration Wm t dxm t Wm t T Cdt Walrand 8 Stability of Systems Lyapunov Function Deterministic Nonlinear system dx t dt f x t Want to show that x t x Show that there is some function V x t such that dV x t dt aV x t whenever x t x For all 0 there is 0 s t x x if V x V x 0 for x x V x 0 Then x t x Note that dV x t dt V x t dx t dt V x t f x t Walrand 9 Stability of Systems Lyapunov Function Stochastic Stochastic system P x n 1 j x n i P i j i j in 0 1 Assume irreducible Want to show that x n is positive recurrent i e that the fraction of time in any state is positive Assume there is some V x such that E V x n 1 V x n x n x a 0 whenever x is not in some finite set A E V x n 1 V x n x
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