flows and networks plan for today (lecture 5): last time / questions? blocking of transitions kelly...

Flows and Networks

Plan for today (lecture 5):

• Last time / Questions?• Blocking of transitions• Kelly / Whittle network• Optimal design of a Kelly / Whittle network:

optimisation problem• Intermezzo: mathematical programming• Optimal design of a Kelly / Whittle network:

Lagrangian and interpretation• Optimal design of a Kelly / Whittle network:

Solution optimisation problem• Optimal design of a Kelly / Whittle network:

network structure• Summary• Exercises• Questions

Blocking in tandem networks of simple queues (1)

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

• state

move

depart

arrive

• Transition rates

• Traffic equations

• 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

Exercises

• Exercise BlockingConsider a tandem network of two simple queues. Let the arrival rate to queue 1 be Poisson , and let the service rate at each queue be exponential i , i=1,2. Let queue 1 have capacity N1. Queue 2 is a standard simple queue. For N1= , give the equilibrium distribution. For N1< formulate three distinct blocking protocols that preserve product form, indicate graphically what the implication of these protocols is on the transition diagram, and proof (by partial balance) that the equilbrium distribution is of product form.

Flows and Networks







Kelly / Whittle network


for some functions

:S[0,),


• Open network

• Partial balance equations:

• Theorem: Assume

then

satisfies partial balance,

and is equilibrium distribution Kelly / Whittle network

kk

jj

jj

jkj

jjk

pn

nnTnq

pn

nTnTnq

pn

nTnTnq

000

00

0

0

)(

)())(,(

)(

))(())(,(

)(

))(())(,(

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

nnTqnTnTnqn kjkj

J

kjk

J

k

j

j

nj

J

j

n

j

J

jSn

nB

11

1 1)(

SnnBn j

j

nj

J

j

n

j

J

j

11

1)()(

kjkk

kj

kjkkk

kjkkjj

pp

ppp

100

100

0

)(

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

Interpretation traffic equations


for some functions

:S(0,),


• Open network

• Theorem: Suppose that the equilibrium distribution is

then

and rate jk

• PROOF

kk

jj

jj

jkj

jjk

pn

nnTnq

pn

nTnTnq

pn

nTnTnq

000

00

0

0

)(

)())(,(

)(

))(())(,(

)(

))(())(,(

SnnBn j

j

nj

J

j

n

j

J

j

11

1)()(

kjkkk

kjkkjj ppp

1

000

)(

j

jj

j

n

nTE

)(

))(( 0

jkj

Flows and Networks







• Source

• How to route jobs, and • how to allocate capacity over the nodes?

• sink

Optimal design of Kelly / Whittle network (1)


for some functions

:S[0,),

• Routing rules for open network to clear input traffic

as efficiently as possible

• Cost per time unit in state n : a(n)

• Cost for routing jk :

• Design : b_j0=+ : cannot leave from j; sequence of queues

• Expected cost rate

kk

jj

jj

jkj

jjk

pn

nnTnq

pn

nTnTnq

pn

nTnTnq

000

00

0

0

)(

)())(,(

)(

))(())(,(

)(

))(())(,(

j

j

nj

J

jSn

nj

J

jSn

n

nna

A

1

1

)(

)()(

)(

jjkjkkj

bAC ,

)(

jkb

kp00



• Given: input traffic

• Maximal service rate

• Optimization problem :

minimize costs

• Under constraints

Jkjpn

nTnTnq jk

jjjk ,...,0,,

)(

))(())(,( 0

j

j

nj

J

jSn

nj

J

jSn

n

nna

A

1

1

)(

)()(

)(

kp00

jjkjkkj

bAC ,

)(

jjkjk

jkk

j p 00

fixed

,...,0,,...,1,0

1

,...,1,0

,...,1,

,...,1,

0

0

00

k

jk

j

jj

kjkk

jkjk

JkJj

Jj

Jj

Jj

Intermezzo: mathematical programming

• Optimisation problem

• Lagrangian

• Lagrangian optimization problem

• Theorem : Under regularity conditions: any point

that satisfies Lagrangian

optimization problem yields optimal solution

of Optimisation problem

mibxxg

xxf

ini

n

,...,1 ),...,( s.t.

),...,(min

1

1

)),...,(( ),...,( 11

1 niii

m

in xxgbxxfL

jxg

jxf

jxL

xxgbi

L

ii

m

i

nii

0),...,(

1

1

),...,,,...,(min 11 mnxxL

),...,,,...,( 11 mnxx

),...,( 1 nxx

Intermezzo: mathematical programming (2)


• Introduce slack variables

• Kuhn-Tucker conditions:

• Theorem : Under regularity conditions: any point

that satisfies Lagrangian optimization

problem yields optimal solution

of Optimisation problem

• Interpretation multipliers: shadow price for constraint. If

RHS constraint increased by , then optimal objective

value increases by i

mibxxg

xxf

ini

n

,...,1 ),...,( s.t.

),...,(min

1

1

mii

mixxgbi

njjxg

jxf

nii

ii

m

i

,...,1 ,0

,...,1 ,0)),...,((

,...,1 ,0

1

1

),...,( 1 nxx



• Lagrangian form

• Interpretation Lagrange multipliers :

jjkjkkj

jkj bAC ,

)(}),({min

fixed

,...,0,,...,1,0

1

,...,1,0

,...,1,

,...,1,0

0

0

00

k

jk

j

jj

kjkk

jkjk

JkJj

Jj

Jj

Jj

0

)()(

00000

0,0

0000

jkjkkj

jjj

jjkk

jj

jkjkjkjkj

CL

j

L

j

L

j

j

)0

(


• KT-conditions

• Computing derivatives:

j

Ac

bL

bcj

L

jj

jkjjkjjjjkjk

jjkkk

jjjkjkk

jj

)(1

0,,,

,...,0, ,0

,...,1 ,0

,...,1 ,0)(

,...,1 ,0)(

,...,0, ,0

,...,1 ,0

0

0

jkjjj

jkjk

jj

jjkk

j

jkjkjkjk

Jkj

Jj

Jj

Jj

Jkjjk

L

Jjj

L


• Theorem : (i) the marginal costs of input satisfy

with equality for those nodes j which are used in the

optimal design.

• (ii) If the routing jk is used in the optimal design the

equality holds in (i) and the minimum in the rhs is

attained at given k.

• (iii) If node j is not used in the optimal design then αj =0.

If it is used but at less that full capacity then cj =0.

• Dynamic programming equations for nodes that are used

0

,...,1),(min

0

Jjbc kjkkjj

0

)(min

0

kjkkjj bc


• PROOF: Kuhn-Tucker conditions :

0 if 0 and

(**) 0

0 if 0 and

(*) 0

jk

jjkjjjjk

j

jkkk

jjjkjkk

jj

b

bc

Flows and Networks







Exercise: Optimal design of Jackson network (1)

• Consider an open Jackson network

with transition rates

• Assume the service rates and arrival rates

are given. Let the costs per time unit for a job residing at

queue j be .Let the costs for routing a job from

station j to station k be

• (i) Formulate the design problem (allocation of routing

probabilities) as an optimisation problem.

• (ii) Consider the case of parallel simple queues, i.e. a

fresh job routes to station j with probability and

leaves the network upon completion at that station.

Provide the solution to the optimization problem for the

case for all j,k

kk

jjj

jkjjk

pnTnq

pnTnq

pnTnq

000

00

))(,(

))(,(

))(,(

ja

jkb

0j

0jkb

ojp

Exercise: Optimal design of Jackson network (2)

• Consider an open Jackson network

with transition rates

• Assume that the routing probabilities and arrival rates

are given

• Let the costs per time unit for a job residing at queue j be

• Let the costs for routing a job from station i to station j be

• Let the total service rate that can be distributed over the

queues be , i.e.,

• (i) Formulate the design problem (allocation of service rates) as

an optimisation problem.

• (ii) Now consider the case of a tandem network, and provide the

solution to the optimisation problem

kk

jjj

jkjjk

pnTnq

pnTnq

pnTnq

000

00

))(,(

))(,(

))(,(

jkp

ja

jkb

jj

0

flows and networks plan for today (lecture 5): last time / questions? blocking of transitions kelly...

Documents

interpretationoptimal

exponential service

standard simple queue

optimisation problemintermezzo

jsuppose queue

service rate

n2exercisesexercise

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