F Kleinrock Jackson · 2009-09-09 · Dr. Cesar Vargas Rosales, Center for Electronics and...

Dr. Cesar Vargas Rosales, Center for Electronics and Telecommunications, 2006

NetworksNetworks


Birth-Death ProcessBirth-Death Process

( )( ),11)()()()( 1111 hhtphtphtphtp kkkkkkkk µλµλ −−++=+ ++−−

( ) ),()()()()()( 1111 hohtphtphtptphtp kkkkkkkkk ++−+=−+ ++−− µλµλ

( ) K,2,1),()()()(1111 =+−+= ++−− ktptptp

dttdp

kkkkkkkk µλµλ

).()()(0011

0 tptpdt

tdp λµ −=

0 1 2 3 k-1 k k+1

0λ

1µ 2µ 3µ kµ 1+kµ 2+kµ

1λ 2λ 1−kλ kλ 1+kλ


Birth-Death ProcessBirth-Death Process

.1

,2,1,

1

1 1

10

01

1

−∞

= =

−

=

−

+=

=

=

∑ ∏

∏

j

j

m m

m

k

m m

mk

p

kpp

µλ

µλ

K

Steady state solution


M /M /1 QueueM /M /1 Queue

Detailed balance equations

.1−= kk pp λµ

.021 ppppk

kkk

==== −− µ

λµλ

µλ

µλ

L

.10

=∑∞

=kkp

0 1 2 3 k-1 k k+1

0λ

1µ 2µ 3µ kµ 1+kµ 2+kµ

1λ 2λ 1−kλ kλ 1+kλ

KK ,3,2,1, and ,2,1,0,Let ==== kk kk µµλλ

,0111 =−=− −−− kkkkkk pppp µλµλ

System with infinite capacity for the waiting line, i.e., one user in service while others are waiting


M/M/1 QueueM/M/1 Queue

( )

( ).1

,11

,

,1

,1

0

1

00

100

10

ρρ

ρµλ

µλ

µλ

−=

−=

−=

=

=

+

=+

−∞

=

∞

=

∞

=

∑

∑

∑

kk

k

k

k

kk

k

p

p

p

pp

pp

Steady State Distribution

=Server Utilization< 1ρ


M/M/1 Queue

.

.1

.1

.1

,)(

2

0

λµρρ

ρ

λµλ

ρρ

−=

−=

−==

−=

== ∑∞

=

W

N

NT

N

pkKEN

Q

kk

Throughput = utilization / service time = ρ / TFor ρ=.5 and T=1ms Throughput is 500 packets/sec

Little’s formula TN λ=


Networks of Transmission LinesNetworks of Transmission Lines

Poisson Process

Interarrivals highly correlated with departures of

previous queue or packet lengths

M/M/1 ?

exponential server

Kleinrock Independence Assumption: It is often appropriate to adopt an M/M/1 queueing model for each communication link regardless of the interaction of traffic on this link with traffic on other links


Networks of Transmission LinesNetworks of Transmission Lines

1x

2x

3x

4x

Poisson process

i

j

( ){ }∑=

jissij x

,link traversesspath :λ

Total arrival rate at link (i, j), suitable for virtual circuit networks

( ){ }∑=

jissijij xsf

,link traversesspath :)(λ

Total arrival rate at link (i, j), suitable for datagram networks

fij(s) is the fraction of packets of stream s that goes through link (i, j)


Average packet transmission time on link (i, j) is

KIA Based on M/M/1KIA Based on M/M/1

∑∑

∑

=−

=

−=

−=

−=

ss

ji ijij

ij

ji ijij

ij

ijij

ij

ij

ijij

xT

N

N

γλµ

λγ

λµλ

λµλ

ρρ

,1

1

),(

),(

Average number of packets in queue or service at link (i, j)

Average number of packets in the network

ijµ1

By Little’s theorem, average delay per

packet in the network, neglecting processing and propagation delay

Total external arrival rate into

the network


KIA Based on M/M/1KIA Based on M/M/1

Considering the average processing and propagation delay dij, we have the average delay per packet as

∑

+

−=

),(

1ji

ijijijij

ij dT λλµ

λγ

( )∑∀

++

−=

pji

ijijijijij

ijp dT

path on ),(

1µλµµ

λ

The average delay per packet of a traffic stream traversing a path p is


Jackson NetworksJackson Networks


Network ModelNetwork Model

1

23

1r

3r

Network graph

Modeling nodes as M/M/1 queues

1

23

1r

3r

10p11p

13p12p

Routing probabilities for M nodes

10

=∑=

M

jijp

We require ri>0 and pi0>0 for at least one i, to have stability

Network of queues

Routing probabilities are the proportions of packets that are sent to other nodes. The state of the network is where niis ther number of packets in the i-th queue

( )Mnnn ,,, 21 K=n


Network ModelNetwork Model

j

rj

From other nodes

M/M/1

For a M/M/1 in steady state, what comes in comes out

To other nodes

jλ .,,2,1,1

MjprM

kkjkjj K=+= ∑

=

λλ

We can obtain the system of equations in matrix form


Service times of packets at i-th queue are assumed to be exponential with mean

Network of QueuesNetwork of Queues

.,,2,1,1

MjprM

kkjkjj K=+= ∑

=

λλ

λλ Pr +=

.,,

21

32313

22212

12111

2

1

2

1

=

=

=

MMMM

M

M

M

MM ppp

ppppppppp

r

rr

L

MOMM

L

L

L

MMPr

λ

λλ

λ

( ) rPI

rP1−−=

=−

λ

λλ

iµ/1


Jackson’s TheoremJackson’s Theorem

Assuming that , j=1, 2, …, M, we have for all1/ <= jjj µλρ ,0,,, 21 ≥Mnnn K

( ) ( )

( ) ( ) .0,1

1

≥−=

=∏=

jjnjjj

M

jjj

nnP

nPP

j ρρ

n

where and( )Mnnn ,,, 21 K=n

F Kleinrock Jackson · 2009-09-09 · Dr. Cesar Vargas Rosales, Center for Electronics and...

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Transcript of F Kleinrock Jackson · 2009-09-09 · Dr. Cesar Vargas Rosales, Center for Electronics and...