Flows and Networks Plan for today (lecture 4):

Flows and Networks

Plan for today (lecture 4):

• Last time / Questions?

• Output simple queue• Tandem network • Jackson network: definition• Jackson network: equilibrium

distribution• Partial balance• Kelly/Whittle network• Examples• Summary / Next• Exercises

Flows and Networks


• Last time / Questions?• Reversibility, stationarity• Truncation• Kolmogorov’s criteria• Poisson process• PASTA• Output simple queue• Tandem network• Summary / Next• Exercises

Poisson process• Definition : Poisson process :

Let S1,S2,… be a sequence of independent exponential() r.v. Let Tn=S1+…+Sn, T0=0 and N(s)=max{n,Tn≤s}. The counting process {N(s),s≥0} is called Poisson process.

• Theorem : If {N(s),s≥0} is a Poisson process, then(i) N(0)=0,(ii) N(t+s)-N(s)=Poisson( t), and(iii) N(t) has independent increments.Conversely, if (i), (ii), (iii) hold, then {N(s),s≥0} is a Poisson process

Flows and Networks



PASTA: Poisson Arrivals See Time Averages

• fraction of time system in state n

• probability outside observer sees n customers at time t

• probability that arriving customer sees n

customers at time t (just before arrival at time t there

are n customers in the system)

• in general

)(,' tP nn

)(0,' tP nn

)()( 0,',' tPtP nnnn

PASTA: Poisson Arrivals See Time Averages

• Let C(t,t+h) event customer arrives in (t,t+h)

• For Poisson arrivals q(n,n+1)= so that

• Alternative explanation;

• PASTA holds in general!

PASTA

)()1,(

)()1,(

)()]()1,([

)()]()1,([lim

}')0(|)(Pr{}')0(,)(|),(Pr{

}')0(|)(Pr{}')0(,)(|),(Pr{lim

}')0(),,(|)(Pr{lim)(

,'0

,'

,'0

,'

0

0

0

0

0,'

tPkkq

tPnnq

tPhohkkq

tPhohnnq

nXktXnXktXhttC

nXntXnXntXhttC

nXhttCntXtP

knk

nn

knk

nn

h

k

h

hnn

)()( 0,',' tPtP nnnn

Flows and Networks



Output simple queue

• Simple queue, Poisson() arrivals, exponential() service

• X(t) number of customers in M/M/1 queue:

in equilibrium reversible Markov process.

• Forward process: upward jumps Poisson ()

• Reversed process X(-t): upward jumps Poisson ()

= downward jump of forward process

• Downward jump process of X(t) Poisson () process

Output simple queue (2)

• Let t0 fixed. Arrival process Poisson, thus arrival process

after t0 independent of number in queue at t0.

• For reversed process X(-t): arrival process after –t0

independent of number in queue at –t0

• Reversibility: joint distribution departure process up to t0

and number in queue at t0 for X(t) have same distribution

as arrival process to X(-t) up to –t0 and number in queue

at –t0.

• In equilibrium the departure process from an M/M/1 queue

is a Poisson process, and the number in the queue at time

t0 is independent of the departure process prior to t0

• Holds for each reversible Markov process with Poisson

arrivals as long as an arrival causes the process to

change state

Flows and Networks



Tandem network of simple queues

• Simple queue, Poisson() arrivals, exponential() service

• Equilibrium distribution

• Tandem of J M/M/1 queues, exp(i) service queue i

• Xi(t) number in queue i at time t

• Queue 1 in isolation: simple queue.

• Departure process queue 1 Poisson,

thus queue 2 in isolation: simple queue

• State X1(t0) independent departure process prior to t0,

but this determines (X2(t0),…, XJ(t0)), hence X1(t0)

independent (X2(t0),…, XJ(t0)). Similar Xj(t0) independent

(Xj+1(t0),…, XJ(t0)). Thus X1(t0), X2(t0),…, XJ(t0) mutually

independent, and

1/,...},2,1,0{,)1()( Sjj j

1/,,)1(),...,(1

1

iijii

J

iJjj

Flows and Networks


• Last time / Questions?• Waiting time simple queue• Little• Sojourn time tandem network• Jackson network: mean sojourn time• Summary / Next• Exercises

Waiting time simple queue (1)

• Consider simple queue FCFS discipline– W : waiting time typical customer in

M/M/1(excludes service time)

– N customers present upon arrival

– Sr (residual) service time of customers present

PASTA

Voor j=0,1,2,…

tkj

k

r

j

r

j

ek

t

tSPjNtWP

jNtWPjtWP

!

)(

)()1|(

)1|()1()(

0

1

1

0

t

t

t

tt

kj

jk

kk

k

t

tkj

k

j

j

esdSstWPtFP

EXEWEF

EW

WWE

eWtWP

WP

e

ee

k

te

ek

ttWP

)1(

0

)1(

)1(

0

0

1

0

)()()(

1

)1(

111

)1(

1)0|(

)0|(

)0(

)1(

1)1(

!

)()1(

!

)()1()(

Waiting time simple queue (2)

• Thus

• is exponential (-)

Flows and Networks



Little’s law (1)

• Let– A(t) : number of arrivals entering in (0,t]– D(t) : number of departure from system

(0,t]– X(t) : number of jobs in system at time t

)()()0()( tDtRXtX

• Equilibrium for t∞

• In equilibrium: average number of arrivals per time unit = average number of departures per time unit

)(1

lim)(1

lim

0)(1

lim

tDt

tAt

tXt

tt

t

Little’s law (2)

)()0( tRX

• Fj sojourn time j-th departing job

• S(t) obtained sojourn times jobs in system at t

t

)(tD

)(uX

)()()(

10

tSFduuX j

tD

j

t

• Assume following limits exist(ergodic theory, see SMOR)

• Then

• Little’s law

Little’s law (3)

j

n

jn

tt

t

t

Fn

F

tDt

tAt

duuXt

X

1

0

1lim

)(1

lim)(1

lim

)(1

lim

)()()(

10

tSFduuX j

tD

j

t

)()0( tRX

t

)(tD

)(uX

FX

tSt

FtDt

tDduuX

t j

tD

j

t

)(1

)(

1)()(

1 )(

10

Little’s law (4)

• Intuition– Suppose each job pays 1 euro per time

unit in system– Count at arrival epoch of jobs: job pays

at arrival for entire duration in system, i.e., pays EF

– Total average amount paid per time unit EF

– Count as cumulative over time: system receives on average per time unit amount equal to average number in system

– Amount received per time unit EX

• Little’s law valid for general systems irrespective of order of service, service time distribution, arrival process, …

Flows and Networks



• Recall: In equilibrium the departure process from an M/M/1 queue is a Poisson process, and the number in the queue at time t0 is independent of the departure process prior to t0

• Theorem 2.2: If service discipline at each queue in tandem of J simple queues is FCFS, then in equilibrium the waiting times of a customer at each of the J queues are independent

• Proof: Kelly p. 38

• Tandem M/M/s queues: overtaking

• Distribution sojourn time: Ex 2.2.2

Sojourn time tandem simple queues

Flows and Networks


• Last time / Questions?• Output simple queue• Tandem network • Jackson network: definition• Jackson network: equilibrium

distribution• Partial balance• Kelly/Whittle network• Summary / Next• Exercises

Jackson network : Definition

• Simple queues, exponential service queue j, j=1,…,J

• state

move

depart

arrive

• Transition rates

• Traffic equations

• Irreducible, unique solution, interpretation, exercise

• Jackson network: open

• Gordon Newell network: closed

),...,1,...,()(

),...,1,...,()(

),...,1,...,1,...,()(

),...,(

10

10

1

1

Jkk

Jjj

Jkjjk

J

nnnnT

nnnnT

nnnnnT

nnn

kk

jj

jkjk

nTnq

nTnq

nTnq

))(,(

))(,(

))(,(

0

0

kjkk

jjkk

jj )(

Flows and Networks




Jackson network : Equilibrium distribution

• Simple queues,



• Closed network

• Open network

• Global balance equations:

• Closed network:

• Open network:

kk

jj

jkjk

nTnq

nTnq

nTnq

))(,(

))(,(

))(,(

.

.

kjkk

jjkk

jj )(

)),(())(())(,()(1 11 1

nnTqnTnTnqn kj

J

jkj

J

kjk

J

j

J

k

)),(())(())(,()(0 00 0

nnTqnTnTnqn kj

J

jkj

J

kjk

J

j

J

k

kjkk

jkk

j

closed network : equilibrium distribution



• Closed network


• Theorem: The equilibrium distribution for the closed Jackson

network containing N jobs is

• Proof

kk

jj

jkjk

nTnq

nTnq

nTnq

))(,(

))(,(

))(,(

.

.

)),(())(())(,()(1 11 1

nnTqnTnTnqn jk

J

jjk

J

kjk

J

j

J

k

}:{)(1

NnnSnBn jj

Nnj

J

jN

j

kjkk

jkk

j

kjjk

J

kjk

J

k

kj

J

jjk

J

kjk

J

j

J

k

jk

J

jjk

J

kjk

J

j

J

k

nTn

nTn

nnTqnTnTnqn

))(()(

))(()(

)),(())(())(,()(

11

1 11 1

1 11 1

Flows and Networks




Partial balance

• Global balance verified via partial balance

Theorem: If distribution satisfies partial balance, then it is

the equilibrium distribution.

• Interpretation partial balance

)),(())(())(,()(

))(()(

))(()(

)),(())(())(,()(

11

11

1 11 1

1 11 1

nnTqnTnTnqn

nTn

nTn

nnTqnTnTnqn

jkjk

J

kjk

J

k

kjjk

J

kjk

J

k

kj

J

jjk

J

kjk

J

j

J

k

jk

J

jjk

J

kjk

J

j

J

k

kjkk

jkk

j

Jackson network : Equilibrium distribution



• Open network


• Theorem: The equilibrium distribution for the open Jackson

network containing N jobs is, provided αj<1, j=1,…,J,

Proof

kk

jj

jkjk

nTnq

nTnq

nTnq

))(,(

))(,(

))(,(

.

.

}0:{)1()(1

nnSnn jnjj

J

j

kjjk

J

kjjjk

J

kj

jkjk

J

kjk

J

k

jk

J

jjk

J

kjk

J

j

J

k

nTnTn

nnTqnTnTnqn

nnTqnTnTnqn

))(())(()(

)),(())(())(,()(

)),(())(())(,()(

10

1

00

0 00 0

kjkk

jjkk

jj )(

)),(())(())(,()(0 00 0

nnTqnTnTnqn jk

J

jjk

J

kjk

J

j

J

k

Flows and Networks




Kelly / Whittle network


for some functions

:S[0,),

:S(0,)


• Open network

• Partial balance equations:

• Theorem: Assume that

then

satisfies partial balance,

and is equilibrium distribution Kelly / Whittle network

kk

jj

j

jkj

jk

n

nnTnq

n

nTnTnq

n

nTnTnq

)(

)())(,(

)(

))(())(,(

)(

))(())(,(

0

00

0

kjkk

jjkk

jj )(

)),(())(())(,()(00

nnTqnTnTnqn kjkj

J

kjk

J

k

jnj

J

jSn

nB 1

1 )(

SnnBn jnj

J

j

1

)()(

Examples

• Independent service, Poisson arrivals

• Alternative

kk

jjj

jjj

jkjj

jjjk

nTnq

n

nnTnq

n

nnTnq

))(,(

)(

)1())(,(

)(

)1())(,(

0

0

SnnBn jnjjj

J

j

)()(1

kk

jjjj

jkjjjk

nTnq

nnTnq

nnTnq

))(,(

)())(,(

)())(,(

0

0

Snr

Bnjn

j

r j

nj

J

j

1

1 )()(

Examples

• Simple queue

• s-server queue

• Infinite server queue

• Each station may have different service type

kk

jjjj

jkjjjk

nTnq

nnTnq

nnTnq

))(,(

)())(,(

)())(,(

0

0

1)( jj n

|},min{)( snnj

Flows and Networks




Summary / next:

Equilibrium distributions• Reversibility• Output reversible Markov process• Tandem network• Jackson network• Partial balance• Kelly-Whittle network

NEXT: Sojourn times

Exercises[R+SN] 2.1.1, 2.1.2, 2.3.1, 2.3.4, 2.3.5, 2.3.6,

2.4.1, 2.4.2, 2.4.6, 2.4.7

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