Networking Theory and Fundamentals - Lecture 4 & 5 doc

... iiijiiiiiqnm r n r nrr nn≠=γ+µ >+µ >=γ+µ + ⋅ >=γ+µ >∑∑∑ ∑∑∑∑∑∑**0(,) 1{0} 1{0}iii ii iimn i i i iqnm n r n≠=γ+µ >=λ+µ >∑∑∑ ∑∑ 8-1 5 Kleinrock Independence Assumption1. ... times and service times Exponentially distributed service timesNetwork model: Jackson network“Product-Form” stationary distribution1TCOM 50 1: Networking Theory & Fundamentals Lecture ... i j KqnTn r n=γ=µ ⋅ > ==µ ⋅ >11 8-1 1Non-Poisson Internal FlowsQueue>>µλ′λp1 p−′λλλPoissonPoisson Example: Single queue with µ >> λ, where upon service...

... MST, and F extended by arc e and node j is a fragment.AECDFBG 4 4316723 4 e'eF6Spanning Tree Algorithm1. Select arbitrary node n∈N, and initialize: G' = (N ', A' ... DE DE= 4 + DG DG= 3 + DB = 3  DD= 2 + DG DC= 1 + DD DF= 6 + DB = 6 AB = A-E-G-B B A C D E F G 3 4 2 7 3 6 4 1 4 34 Complexity of Floyd-Warshall ... graph G' = (N ', A' ), with N '=N and (i, j)∈A' if (i, j)∈A, or (j, i)∈A A digraph G = (N, A) is called connected if its associated undirected graph G' = (N ',...

... G(M ,5) based on the iterative algorithm using these values12 34 51 53 1 3 51 3 1,2rrλ=λ λ=λ=⇒λ=λλ=λ+λ=λ121 34 3 1 55 1 1/2 , / 2/2 /4/ , with /ρ=ρ=λ µρ=ρ=λ µ=ρρ=λ λ=λ λ=ρ ρ ρ≡λµ12 34 5 1, ... denote the set of all such states1µ2λ3µ 4 µ 5 µ1λ2µ 4 λ3λ 5 λ 51 r 53 rM1,1, ,Kijjijri K=λ= λ =∑10 and | |KiiinnnM=≥≡=∑ 8-1 4 Marginal Distribution Proposition 1: In ... 1 2 3 4 5 0 1 1 1 1 11 1 2 4 6 6 +4/ ρ 2 1 3 11 23 23+(6 +4/ ρ) (4/ ρ) 3 1 4 26 72 72+[23+(6 +4/ ρ) (4/ ρ)] (4/ ρ) km12 34 5 1, 2, 4 /ρ=ρ= ρ=ρ= ρ= ρ1( ,1) , 0,1, ,(0, ) 1, 1,2, ,mGm m MGk...

... j→∞===== 6-1 0Continuous-Time Markov Chains {X(t): - < t <∞} irreducible aperiodic Markov chain with transition rates qij, i≠j Unique stationary distribution (pi> 0) if and only ... π /jjjjjiijiiipqjννν≠=≡=∑∑1TCOM 50 1: Networking Theory & Fundamentals Lecture 6February 19, 2003Prof. Yannis A. Korilis 6-1 7Kolmogorov’s Criterion (Continuous Chain) ... Discrete-time birth-death processes are reversible, since they satisfy the DBEππ, , 0,1, iij j jiPPij== 7-3 2Queues in Tandem Theorem 11: Network consisting of K single-server queues in tandem....

... 0P=()111π , , 0.310, 0. 3 45 , 0. 3 45 333pppp−==−−−0.310 0. 3 45 0. 3 45 lim 0.310 0. 3 45 0. 3 45 ( 150 )0.310 0. 3 45 0. 3 45 nnPn→∞=≈ 3-3 4 M/M/1 Queue Arrival process: ... wet} π3pPppp−==−1TCOM 50 1: Networking Theory & Fundamentals Lecture 3 January 29, 2003Prof. Yannis A. Korilis 3 -4 Discrete-Time Markov Chain Discrete-time stochastic process {Xn: ... iS iS jSpq pqS∈∉ ∉∈=⊆∑∑ ∑∑ 3-2 5 Limit TheoremsTheorem 3: Irreducible semi-Markov process, E[Tii] < ∞ For any state j, the following limit exists and is independent of the initial...