Networking Theory and Fundamentals - Lecture 1 potx

... Venkatesh c  19 97 TCOM 5 01: Networking Theory and Fundamentals University of Pennsylvania in queue. Consequently, N Q = ∞  n =1 (n − 1) π n = ∞  n =1 (n − 1) (1 − ρ)ρ n = (1 − ρ)ρ 2 ∞  n =1 (n − 1) ρ n−2 = (1 ... identity  ∞ n=0 π n = 1, whence we finally obtain π n = n  i =1 P i 1, i P i,i 1  ∞  m=0 m  i =1 P i 1, i P i,i 1 (n ≥ 0). In particular, if P n 1, n =...

Ngày tải lên: 22/07/2014, 18:22

... = == µ ρρρ == µρρρ ∑ ∑∑   F FF  Changing index 1, 0 ii i mm m ′ ′ = +≥ in the sum in the denominator: 1 1 1 11 1 1 1 1 1 1 10 11 11 1 1 0 () (1) i K iK iK i ii KK iK iK i n nn iK ij mmm iK mm ... , Ki iKλ=λ= =λ =µ ⇒ ρ= =…  1 1(0) (1) (1) 11 (1) , (1) 1 , (1) (1) (1) jjj jj j K j i i NTN TN K KT T = +µ == = =γ==µ µµ µ ∑  1( 1) (2) 11 / 11 2 2...

Ngày tải lên: 22/07/2014, 18:22

... np=λ moderate, binomial distribution converges to Poisson with parameter λ  Proof: {} (1) (1) (1) 1 ( 1) ( 1) 1 1 11 {} ! ! knk nk n k n n k n k k n n PX k p p k nk n n nn nk n n n e n n P k k Xk e λ λ λ λ λ λ λ − − →∞ − →∞ →∞ − →∞  == ... Processes λ pλ 1 p 1- p λ λ 1 + λ 2 λ ( 1- p) λ 2  A 1 ,…, A k independent Poisson processes with rates λ 1 ,…, λ k  Merged in...

Ngày tải lên: 22/07/2014, 18:22

... the 1st term of the iteration step n ij D 19 Proof of Bellman-Ford (1)  Proof is by induction on hop count h For h = 1 we have  D i 1 = d i1 for all i ≠ 1, so the result holds for i = 1 Assume ... After N -1 iterations, the algorithm terminates, and G' contains N nodes and N -1 arcs. {},n ′ ′ = =∅NA :{},:{(,)}jij ′′ ′′ =∪ =∪NN AA 10 Minimum Weight Spanning Tree 5...

Ngày tải lên: 22/07/2014, 18:22

... times and service times  Exponentially distributed service times Network model: Jackson network “Product-Form” stationary distribution 1 TCOM 5 01: Networking Theory & Fundamentals Lecture ... j = + =− =−+ 0 (, ) ( , ) { 0} , 1, , (, ) { 0} ii iiii ij i ij i qnAn qnDn r n i j K qnTn r n = γ =µ ⋅ > = =µ ⋅ > 1 1 8 -1 1 Non-Poisson Internal Flows Queue >> µ λ...

Ngày tải lên: 22/07/2014, 18:22

... tedious 2 λ 2 µ 2 λ 2 µ 2 λ 2 µ 1 λ 1 µ 2 λ 2 µ 2 λ 2 µ 2 λ 2 µ 1 λ 1 µ 1 λ 1 µ 1 λ 1 µ 2 λ 2 µ 2 λ 2 µ 2 λ 2 µ 02 12 1 λ 1 µ 01 11 21 1 λ 1 λ 1 µ 1 µ 00 10 20 30 1 λ 1 λ 1 µ 1 µ 03 13 22 31 12 1 2 {( , ) :( 1) ( 1) }Ennn ... Thus N 1 (t) and N 2 (t) independent:  Letting t→∞, the joint stationary distribution 1 11 1 1 1 ( ) (1 ) , 0...

Ngày tải lên: 22/07/2014, 18:22

... as in the M/M /1 queue:  Normalization constant: Generalize: Truncating a Markov chain 1 00 01 0 1 1 11 1 1 1 1 K KK n n nn K pp p p ρ ρ ρ ρ ρ + == + − =⇒ =⇒ = − − ⇒= − ∑∑ 0 , 1, 2, , n n ppn ... and interarrival times: independent {N(t): t ≥ 0} is a birth-death process Stationary distribution: 1 11 00 010 11 ,1 1 nn ii n ini ii pp n p λλ µµ − −∞− === ++   =≥=+  ...

Ngày tải lên: 22/07/2014, 18:22

... 1 n n n p ρ α − =≥ − 11 1 0 01 1 π 1 π 11 1( 1) (1) n n nn pp ρρ α α − − ∞∞ − ==    =⇒ =+ =+    −−−    ∑∑ ()()() ρα −= − = − − − − −=−− 1 )1( )1( )1( 11 1 q pq pq qppq pp 0 1 π 1 π (1 ) , 1 n n n ρ ραα − =−   =− ... 1 n n n ρ ραα − =−   =− ≥  3-2 0 Example: Discrete-Time Queue 0 1 n+1n2 (1 ) p − p (1 )pq − (1 ) (1 ) p qpq−−+ (1 )qp− (...

Ngày tải lên: 22/07/2014, 18:22

... -1 0 1 2 3 4 5 -2 .5 -2 -1 .5 -1 -0 .5 0 0.5 1 1.5 2 2.5 0 dB -1 0 dB -6 dB -4 dB -2 dB 10 dB 6 dB 4 dB 2 dB Ny quis t Diagram Real A x is Imaginary Axis -3 0 -2 0 -1 0 0 10 20 30 -5 0 -4 0 -3 0 -2 0 -1 0 0 10 20 30 40 50 0 ... Robust stability Let the linear system be given by the transfer function 1 110 1 110 11 11 ()...

Ngày tải lên: 20/06/2014, 04:20

... Bearings, Theory and Applications, Edited by Boštjan Polajžer p. cm. ISBN 97 8-9 5 3-3 0 7 -1 4 8-0 Magnetic Bearings, Theory and Applications4 3. System Identification 3 .1 System Identification and ... in equation (3). The closed-loop poles are at -4 .43 10 4 , -2 .73 10 3 , and -1 67±392j respectively. Figure 5 shows the Bode plot of the closed-loop s...

Ngày tải lên: 20/06/2014, 06:20