Networks of Queues Plan for today (lecture 6): Last time / Questions? Product form preserving...

Networks of QueuesPlan for today (lecture 6):

• Last time / Questions?• Product form preserving blocking• Interpretation traffic equations• 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• 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







Kelly / Whittle network


for some functions

ψ: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

kjj

kjkkk

kjkkjj

pp

ppp

100

100

0

)(

)(

Interpretation traffic equations


for some functions

ψ: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







• Source

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

• sink

Optimal design of Kelly / Whittle network (1)


for some functions

ψ: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







Exercise 1: insensitivity and Norton

Consider a network consisting of subnetworks. Each subnetwork is quasi reversible. The subnetworks are linked via Markov routing (as in the transparancy in class, but now state independent routing).

1. Formulate the transition rates at local and global level

2. Formulate and prove the decomposition theorem

3. Suppose that we want to aggregate a component into a single queue. Give the service rates for this queue.

4. Now assume that all queues in the network are infinite server queues. Give the service rate for the aggregated components. Show the relation between Norton’s theorem and the insensitivity property.

5. Investigate the relation between Norton’s theorem and insensitivity

Exercise 2: Optimal design of Jackson network (1)

• Consider an open Jackson network

with transition rates

• Assume that 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 i to station j be

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

probabilities) as an optimisation problem.

• (ii) Provide the solution to this problem

kk

jjj

jkjjk

pnTnq

pnTnq

pnTnq

000

00

))(,(

))(,(

))(,(

ja

jkb

0j

Exercise 3: Optimal design of Jackson network (2)

• Consider an open Jackson network


• 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) Provide the solution to this problem

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

the solution to the optimisation problem for the case

for all j,k

kk

jjj

jkjjk

pnTnq

pnTnq

pnTnq

000

00

))(,(

))(,(

))(,(

jkp

ja

jkb

jj

0

0jkb

Exercise 4 : routes vs Markovian routing

1. Show that an open Jackson network


Can equivalently be represented as a network with fixed routing

(i.e., the route is determined by the customer type).

2. Now consider an open Jackson network with queues with finite

capacity, i.e. ni<Ni, i=1,…,J. Consider the network under a

product form preserving blocking protocol. Investigate whether

or not the network can equivalently be represented as a

network with fixed routing as in 1.

kk

jjj

jkjjk

pnTnq

pnTnq

pnTnq

000

00

))(,(

))(,(

))(,(

Exercise 5

Consider an open tandem network of 3 single server queues (M/M/1 queues).

1. Formulate the transitions rates. 2. Give the equilibrium distribution, and show that this

distribution satisfies partial balance3. Show that a customer arriving to queue j, j=1,2,3

observes the equilibrium distribution4. Now assume that the total number of customers in

the network is restricted not to exceed N. Formulate the transition rates. Give the equilibrium distribution, and the distribution of the state observed by a customer arriving to queue 1

5. Now assume that each queue has a capacity restriction, say the number in queue i is restricted not to exceed Ni. A queue that has reached its capacity constraint is blocked, which means that no extra customer may enter the queue. This means that a customer cannot leave its predecessor, i.e., when queue 3 is blocked a customer cannot leave queues 2 (the server at queue 2 is stopped). Formulate the transition rates, and show that the equilibrium distribution cannot satisfy partial balance.

6. Again consider the tandem network of part 5. Consider the stop protocol, that is the following modification of the transition rates. The arrival process, and each non-blocked queue is stopped until a service completion has occurred at the blocked queue. Give the state space, and formulate the transition rates. Show that the equilibrium distribution has a product form and satisfies partial balance. Give the distribution of the state observed by a customer arriving to (and entering) queue 2.

7. Again consider the tandem network of part 5. Assume that a customer arriving to a queue that has reached its capacity restriction, jumps over the queue and moves to the next queue in the tandem. Formulate the transition rates. Give the equilibrium distribution (proof!).

Exercise 6

Consider a tandem network consisting of 4 queues. Queue 1 is a FCFS queue, queue 2 a PS queue, queue 3 a LCFS queue, and queue 4 an Infinite server queue. Let customers of type 1 and 2 arrive to queue 1 according to a Poisson process with rates λ1, and λ2.

1. Let the service requirements at each of the queues be exponentially distributed. Formulation the network as BCMP network. Also give the transition rates.

2. Show that the queues are quasi reversible (you may do this for the queues in isolation, i.e., for each queue consider only that queue [thus part of the network] with Poisson arrival process)

3. Give the equilibrium distribution for the tandem network.

4. Show that this distribution satisfies partial balance.5. Prove that the equilibrium distribution of queue 2, 3

and 4 is insensitive to the distribution of the service time except for its mean (you may do this for the queue in isolation).

6. Prove that the equilibrium distribution of the tandem network is insensitive to the distribution of the service time at each of the queues 2,3,4 except for their means.

7. Indicate why the distribution cannot be insensitive to the service time distribution in an FCFS queue.

Exercises: rules

In all exercises: give proof or derivation of results(you are not allowed to state something like: proof as given in class, or proof as in book…) motivate the steps in derivation

hand in exercises week before oral exam, that is based on these exercises

All oral exams (30 mins) for this part on the same day, you may suggest the date…

Hand in 4 exercises: 1 and $ and select 2 or 3 and 5 or 6

Each group 2 or 3 persons: hand in a single set of answers

G1: 1,2,4,5G2: 1,2,4,6G3: 1,3,4,5G4: 1,3,4,6

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