Flows and Networks Plan for today (lecture 5): Last time / Questions? Waiting time simple queue Little Sojourn time tandem network Jackson network: mean

Flows and Networks

Plan for today (lecture 5):

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

Last time: output simple queue; partial balance

• 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

• Global balance; partial balance;detailed balance,

traffic equations

kjk

J

kjkj

J

k

jkjkjk

jkjk

J

kjk

J

k

jk

J

jjk

J

kjk

J

j

J

k

nnTqnTnTnqn

nnTqnTnTnqn

nnTqnTnTnqn

11

11

1 11 1

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

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

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

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


• 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



Jackson network : Mean sojourn time

• Simple queues, FCFS,



• Open network


• Open network:

• Sojourn time in each queue:

• Sojourn time on path i,j,k:

kk

jj

jkjk

nTnq

nTnq

nTnq

))(,(

))(,(

))(,(

.

.

kjkk

jjkk

jj )(

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

nnTqnTnTnqn kj

J

jkj

J

kjk

J

j

J

k

)1(

1

jjkk

j

jEF

kjikji EFEFEFEF ,,

Flows and Networks



Blocking in tandem networks of simple queues (1)


• state

move

depart

arrive



• Solution

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

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

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

),...,(

10

10

111,

1

Jkk

Jjj

Jjjjj

J

nnnnT

nnnnT

nnnnnT

nnn

))(,(

))(,(

))(,(

01

0

1,

nTnq

nTnq

nTnq

JJ

jjj

Jj

Jj

jj

jjjj

,...,1,

,...,2,

11

11





• Solution

• Equilibrium distribution

• Partial balance

• PICTURE J=2

))(,(

))(,(

))(,(

01

0

1,

nTnq

nTnq

nTnq

JJ

jjj

Jj

Jj

jj

jjjj

,...,1,

,...,2,

11

11

}0:{)1()(1

nnSnn jnjj

J

j

kjjk

J

kjjjk

J

kj

jkjk

J

kjk

J

k

nTnTn

nnTqnTnTnqn

))(())(()(

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

10

1

00



• Suppose queue 2 has capacity constraint: n2<N2


• Partial balance?

• PICTURE J=2

• Stop protocol, repeat protocol, jump-over protocol

))(,(

))(,(

)(1))(,(

,...,2,))(,(

01

0

2212,1

1,

nTnq

nTnq

NnnTnq

JjnTnq

JJ

jjj

}0:{)1()(1

nnSnn jnjj

J

j

kjjk

J

kjjjk

J

kj

jkjk

J

kjk

J

k

nTnTn

nnTqnTnTnqn

))(())(()(

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

10

1

00

Flows and Networks



Summary / next:

Waiting times / sojourn times• Distribution• Little’s law• Mean

Blocking in Jackson network• Partial balance• Product form preserving blocking

protocols

NEXT: Optimization / applications

Exercises[R+SN] 1.3.3, 2.2.2, 2.2.4, 2.2.5, 2.2.6

Documents

Flows and Networks Plan for today (lecture 5): Last time / Questions? Waiting time simple queue Little Sojourn time tandem network Jackson network: mean