158052sheetshc4.ppt

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

Last time: Reversibility and stationarity; various properties

• Definition: Reversible process: A stochastic process is reversible if for all t1,…,tn,

Skjjkqkkjqj ,),,()(),()(

))(),...,(),((~))(),...,(),(( 2121 nn tXtXtXtXtXtX

• Theorem: A stationary Markov chain is reversible if and only if there exists a collection of positive numbers π(j), jS, summing to unity that satisfy the detailed balance equations

When there exists such a collection π(j), jS, it is the equilibrium distribution

• Theorem 1.13: Kelly’s lemmaLet X(t) be a stationary Markov processwith transition rates q(j,k). If we can find a collection of numbers q’(j,k) such that q’(j)=q(j), jS, and a collection of positive numbers (j), jS, summing to unity, such that

then q’(j,k) are the transition rates of the time-reversed process, and (j), jS, is the equilibrium distribution of both processes.

),(')(),()( jkqkkjqj

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)

• PASTA

)(,' tP nn

)(0,' tP nn

)()( 0,',' tPtP nnnn

Flows and Networks



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

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()( Snn n

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

1

iinii

J

iJ

inn

Flows and Networks




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

Date post:	03-Dec-2014
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158052sheetshc4.ppt

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