TCOM 501: Networking Theory & Fundamentals. Lecture 7 February 25, 2003 Prof. Yannis A. Korilis

Transcription

1 TCOM 501: Networkng Theory & Fundamentals Lecture 7 February 25, 2003 Prof. Yanns A. Korls 1

2 7-2 Topcs Open Jackson Networks Network Flows State-Dependent Servce Rates Networks of Transmsson Lnes Klenrock s Assumpton

3 8-3 Networks of./m/1 Queues k γ 1 r k r j j γ r 0 Network of K nodes; Node s./m/1-fcfs queue wth servce rate µ External arrvals ndependent Posson processes γ : rate of external arrvals at node Markovan routng: customer completng servce at node s routed to node j wth probablty r j or exts the network wth probablty r 0 =1- j r j Routng matrx R=[r j ] rreducble external arrvals eventually ext the system

4 8-4 Networks of./m/1 Queues Defnton: A Jackson network s the contnuous tme Markov chan {N(t)}, wth N(t)=(N 1 (t),, N K (t)) that descrbes the evoluton of the prevously defned network Possble states: n=(n 1, n 2,, n K ), n =1,2,, =1,2,..,K For any state n defne the followng operators: An= n+ e arrval at Dn= n e departure from Tn= n e+ e transton from to j j j Transton rates for the Jackson network: qnan (, ) = γ whle q(n,m)=0 for all other states m qndn (, ) =µ r 1{ n> 0}, j= 1,..., K 0 qntn (, ) =µ r 1{ n> 0} j j

5 8-5 Jackson s Theorem for Open Networks λ : total arrval rate at node Open network: for some node j: γ j >0 Lnear system has a unque soluton λ 1, λ 2,, λ K K λ =γ + λ, 1,..., j 1 jrj = K = Theorem 13: Consder a Jackson network, where ρ =λ/µ <1, for every node. The statonary dstrbuton of the network s pn ( ) = p( n), n,, n 0 = 1 where for every node =1,2,,K K 1 K n p ( n ) = (1 ρ ) ρ, n 0 k γ 1 r k γ r j r 0 j

6 8-6 Jackson s Theorem (proof) Guess the reverse Markov chan and use Theorem 4 Clam: The network reversed n tme s a Jackson network wth the same servce rates, whle the arrval rates and routng probabltes are λ r γ γ =λ r, r =, r = * * j j * 0 j 0 λ λ Verfy that for any states n and m n, p mq mn pnqnm * ( ) (, ) = ( ) (, ) Need to prove only for m=a n, D n, T j n. We show the proof for the frst two cases the thrd s smlar q ( An, n) = q ( An, D An) =µ r =µ ( γ / λ ) * * * 0 * = µ γ λ = γ =ρ p( Anq ) ( Ann, ) pnqnan ( ) (, ) pan ( ) ( / ) pn ( ) pan ( ) pn ( ) q ( Dn, n) = q ( Dn, ADn) = γ =λ r * * * 0 * 0 0 pdnq ( ) ( Dnn, ) = pnqndn ( ) (, ) pdn ( ) λ r = pn ( ) µ r1{ n > 0} ρ pdn ( ) = pn ( )1{ n> 0}

7 8-7 Jackson s Theorem (proof cont.) Fnally, verfy that for any state n: * qnm (, ) = q( nm, ) m n m n qnm (, ) = γ + µ r1{ n> 0} + µ r1{ n> 0} j 0 m n, j = γ + µ [ r + r ] 1{ n > 0} j 0 j = γ + µ 1{ n > 0} q ( n, m) = γ + µ 1{ n > 0} = λ r + µ 1{ n > 0} * * 0 m n Thus, we need to show that γ = λ r 0 λ r = λ λ r = λ λr 0 j j j j = λ ( λ γ ) = γ j j j j j

8 8-8 Output Theorem for Jackson Networks Theorem 14: The reversed chan of a statonary open Jackson network s also a statonary open Jackson network wth the same servce rates, whle the arrval rates and routng probabltes are λ r γ γ =λ r, r =, r = * * j j * 0 j 0 λ λ Theorem 15: In a statonary open Jackson network the departure process from the system at node s Posson wth rate λ r 0. The departure processes are ndependent of each other, and at any tme t, ther past up to t s ndependent of the state of the system N(t). Remark: The total arrval process at a gven node s not Posson. The departure process from the node s not Posson ether. However, the process of the customers that ext the network at the node s Posson.

9 8-9 Arrval Theorem n Open Jackson Networks The composte arrval process at node n an open Jackson network has the PASTA property, although t need not be a Posson process Theorem 16: In an open Jackson network at steady-state, the probablty that a composte arrval at node fnds n customers at that node s equal to the (uncondtonal) probablty of n customers at that node: p ( n) = (1 ρ ) ρ n, n 0, = 1,..., K k Proof s omtted λ j

10 8-10 Non-Posson Internal Flows Jackson s theorem: the numbers of customers n the queues are dstrbuted as f each queue s an solated M/M/1 wth arrval rate λ, ndependent of all others Total arrval process at a queue, however, need not be Posson Loops allow a customer to vst the same queue multple tmes and ntroduce dependences that volate the Posson property Internal flows are Posson n acyclc networks Smlarly. the departure process from a queue s not Posson n general The process of departures that ext the network at the node s Posson accordng to the output theorem

11 8-11 Non-Posson Internal Flows Queue λ Posson λ µ >> λ λ 1 p p λ Posson Example: Sngle queue wth µ >> λ, where upon servce completon a customer s fed back wth probablty p 1, jonng the end of the queue The total arrval process does not have ndependent nterarrval tmes: If an arrval occurs at tme t, there s a very hgh probablty that a feedback arrval wll follow n (t, t+δ] At arbtrary t, the probablty of an arrval n (t, t+δ] s small snce λ s small Arrval process conssts of bursts, each burst trggered by a sngle customer arrval Exact analyss: the above probabltes are respectvely λδ + µδ p+ o( δ), λδ + (1 p ) µδ p+ o( δ) 0

12 8-12 Non-Posson Internal Flows (cont.) λ Posson λ µ λ 1 p p λ Posson Example: Sngle queue, exponental servce tmes wth rate µ, Posson arrvals wth rate λ. Upon servce completon a customer s fed back at the end of the queue wth probablty p or leaves wth probablty 1-p Composte arrval rate and steady-state dstrbuton: λ =λ+λ r11 =λ+λ p λ =λ/(1 p) n pn ( ) = (1 ρ) ρ, n 0; ρ=λ / µ =λ/(1 p) µ Probablty of a composte arrval n (t, t+δ]: λ λδ+ (1 p0 ) µδ p+ o( δ ) =λδ+ pδ+ o( δ ) =λδ+ o( δ) (1 p) µ Probablty of a composte arrval n (t, t+δ], gven that a composte arrval occurred n (t-δ, t]: λδ+µδ p+ o( δ ) >λδ+ o( δ)

13 8-13 State-Dependent Servce Rates Servce rate at node depends on the number of customers at that node: µ (n ) when there are n customers at node./m/c and./m/ queues Theorem 17: The statonary dstrbuton of an open Jackson network where the nodes have state-dependent servce rates s pn ( ) = p( n), n,, n 0 1 K where for every node =1,2,,K n 1 λ p( n) =, n 0 G µ (1) µ ( n) wth normalzaton constant n λ G = < = 0 µ (1) µ ( n ) K = 1 n Proof follows dentcal steps wth the proof of Theorem 13

14 8-14 Network of Transmsson Lnes Real Networks: Many transmsson lnes (queues) nteract wth each other Output from one queue enters another queue, Mergng wth other packet streams departng from the other queues Interarrval tmes at varous queues become strongly correlated wth packet lengths Servce tmes at varous queues are not ndependent Queueng models become analytcally ntractable Analytcally Tractable Queueng Networks: Independence of nterarrval tmes and servce tmes Exponentally dstrbuted servce tmes Network model: Jackson network Product-Form statonary dstrbuton

15 8-15 Klenrock Independence Assumpton 1. Interarrval tmes at varous queues are ndependent 2. Servce tme of a gven packet at the varous queues are ndependent Length of the packet s randomly selected each tme t s transmtted over a network lnk 3. Servce tmes and nterarrval tmes: ndependent Assumpton has been valdated wth expermental and smulaton results Steady-state dstrbuton approxmates the one descrbed by Jackson s Theorems Good approxmaton when: Posson arrvals at entry ponts of the network Packet transmsson tmes nearly exponental Several packet streams merged on each lnk Densely connected network Moderate to heavy traffc load