14 th INFORMS Applied Probability Conference, Eindhoven July 9, 2007

14th INFORMS Applied Probability Conference,

EindhovenJuly 9, 2007

14th INFORMS Applied Probability Conference,

EindhovenJuly 9, 2007

Yoni NazarathyGideon Weiss

University of Haifa

Yoni NazarathyGideon Weiss

University of Haifa

Transient Fluid Solutions and

Queueing Networks withInfinite Virtual Queues

Transient Fluid Solutions and

Queueing Networks withInfinite Virtual Queues

Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 2

Overview:Overview:

MCQN model

Transient Fluid Solutions

Infinite Virtual Queues

Near Optimal Finite Horizon Control

MCQN model





Multi-Class Queueing Networks (Harrison 1988, Dai 1995,…)Multi-Class Queueing Networks (Harrison 1988, Dai 1995,…)

1 2

6

5 4

3

{1,..., }

{ ( ), 0}k

K

Q t t

Queues/Classes

6K

Routing Processes

(0)kQ kInitial Queue Levels

' ( ) , 'kk n k k

Resources

( )kS t k

Processing Durations

{1,..., }

{ } {0,1}I K ik ik

I

A A A

Resource Allocation (Scheduling)

( )

(0) 0 ( ) ( )

( ) ( ) 0

k

k ik k kk

k k

T t

T A T t T s t s s t

T t only when Q t

Network Dynamics

' ' ''

( ) (0) ( ( )) ( ( ( )))k k k k k k k kk

Q t Q S T t S T t

4I


Overview:Overview:

MCQN model




MCQN model





Sta

cked

Que

ue L

evel

s

time T

Q1

Q2Q3

Trajectory of a single job

Finished Jobs

Example NetworkExample Network

Server 1Server 2

1

23

3

10

( )T

kk

Q t dt

Attempt to minimize:


Fluid formulationFluid formulation

1 2 3

0

1 1 1 1

0

2 2 1 1 2 2

0 0

3 3 2 2 3 3

0 0

1 3

2

min ( ) ( ) ( )

( ) (0) ( )

( ) (0) ( ) ( )

( ) (0) ( ) ( )

( ) ( ) 1

( ) 1

( ), ( ) 0

T

t

t t

t t

q t q t q t dt

q t q u s ds

q t q u s ds u s ds

q t q u s ds u s ds

u t u t

u t

u t q t

(0, )t T

s.t.

This is a Separated Continuous Linear Program (SCLP)

Server 1Server 2

1

23


Fluid solutionFluid solution

•SCLP – Bellman, Anderson, Pullan, Weiss.•Simplex based algorithm, finds the optimal solution in a finite number of steps (Weiss).

The Optimal Solution:

•SCLP – Bellman, Anderson, Pullan, Weiss.•Simplex based algorithm, finds the optimal solution in a finite number of steps (Weiss).

The Optimal Solution:

0 10 20 30 40

0

5

10

15

20

3( )q t

2 ( )q t

1( )q t

The solution is

piece-wise linear with a

finite number

of “time intervals”


Overview:Overview:

MCQN model




MCQN model





INTRODUCING: Infinite Virtual QueuesINTRODUCING: Infinite Virtual Queues

( ) (0) ( ( ))R t R S T t t

5 10 15 20 25 30

-1

-0.5

0.5

1

1.5

2

2.5

Regular Queue

( ) : {0,1,2,...}kQ t

Infinite Virtual Queue

( )kQ t t

Example Realization

( )R t

NominalProduction

Rate

Relative Queue Length


IVQ’s Make Controlled Queueing Network even more interesting…IVQ’s Make Controlled Queueing Network even more interesting…

Some Resource

The Network PUSH

PULL

To Push Or To Pull? That is the question…

( ) (0) ( ( ))R t R S T t t

High Utilizatio

n

of ResourcesHigh and Balanced

Throughput

Stable and Low

Queue Sizes

Low variance of the

departure process

What does a “good” control achieve?

Fluid oriented Approach:Choose a “good” nominal production rate (α)…


Extend the MCQN to MCQN + IVQExtend the MCQN to MCQN + IVQ

0

( )

(0) 0 ( ) ( )

( ) ( ) 0

k

k ik k kk

k k

T t

T A T t T s t s s t

T t only when Q t for k

{1,..., }

{ ( ), 0}k

K

Q t t

1 2

6

5 4

3

Queues/Classes

Routing Processes

(0)kQ k

Initial Queue Levels

' 0( ) 'kk n k k

( )kS t k

Processing Durations

Resource Allocation (Scheduling)

Network Dynamics

0

0

{1,..., }

{ ( ), 0}

{ ( ), 0}k

k

K

Q t t k

R t t k

0(0)kQ k

' ' ' 0'

( ) (0) ( ( )) ( ( ( ))) 0( )

( ) (0) ( ( ))

k k k k k k k kk

k

k k k k k

Q t Q S T t S T t k KZ t

R t R S T t t k K

NominalProductio

nRates

0 {1,2,3,5}

{4,6}

K

K

Resources

{1,..., }

{ } {0,1}I K ik ik

I

A A A


Rates Assumptions of the Primitive SequencesRates Assumptions of the Primitive Sequences

' 0'

1

1

( )lim

'( )lim

0 '

1lim ( )

kk

t

kkkk

n

n

kn

l

S t

tP kn

kn

X l Cn

1

( ) max{ : ( ) }n

k kl

S t n X l t

Primitive Sequences:

' ' ' ' 0

{ ( ), 0} (0) 0 ( )

{ ( ), 0,1,2...} (0) 0 ( ) ( ) , 'k k k

kk kk kk kk

S t t S S t k K

n n n n n k K k K

May also define:

rates assumptions:

{ ( ), 1,2,..}kX l l k K


The input-output matrix (Harrison)The input-output matrix (Harrison)

( ) ( )TR I P diag

' 0

' ' ' 0

'

'

0

k

kk k k k

k k k K

R P k k k K

k K

A fluid view of the outcome of one unit of work on class k’:

is the average depletion of queue k per one unit of work on class k’.

'kkR

The input-output matrix:


0

, , 0

max 1

0i I i

k

x

for k K

The Static Equations

Rx

A

A feasible static allocation is the triplet , such that:( , , )x

1

1

1

,K K I K

K

I

K

R A

x

- MCQN model

- Nominal Production rates for IVQs

- Resource Utilization

- Resource Allocation

Similar to ideas from Harrison 2002


•Lyapunov function:•Find allocation that reduces it as fast as possible:

Maximum Pressure Policies (Tassiulas, Stolyar, Dai & Lin)Maximum Pressure Policies (Tassiulas, Stolyar, Dai & Lin)

•Reminder: is the average depletion of queue k per one unit of work on class k’. •Treating Z and T as fluid and assuming continuity:

•Reminder: is the average depletion of queue k per one unit of work on class k’. •Treating Z and T as fluid and assuming continuity:

( ) ( ) ( )f t Z t Z t

( ) 2 ( ) ( ) 2 ( ) ( )df t Z t Z t Z t RT t

dt

( ) ( )Z t RT t

'kkR

( )arg max ( )Ta A t

Z t R a

•An allocation at time t: a feasible selection of values of •At any time t, A(t) is the set of available allocations.

Intuitive Meaning of the Policy

( )T t

“Energy” Minimization

The Policy:

Choose:

Feasible Allocations


•MCQN + IVQ, Non-Processor Splitting, No-Preemption

•Nominal production rates given by a feasible static allocation.

•Primitive Sequences satisfy rates assumptions.

•Using Maximum Pressure, the network is stable as follows:

Rate Stability TheoremRate Stability Theorem

( )lim 0t

Z t

t

( )lim 0

(0, )

N

N

Z t

tuniformly on t T

( ) ( )

(0) ( )

Nk k

N

S t S N t

Z o N

(1) – Rate Stability for infinite time horizon:

(2) – Given a sequence :Where satisfies:( )NZ t

Proof is an adaptation of Dai and Lin’s 2005, Theorem 2.

( )NZ t


Overview:Overview:

MCQN model




MCQN model





Back to the example network:Back to the example network:

For each time interval, set a MCQN with Infinite Virtual Queues:

3

1

2

3

1

2

3

1

2

3

1

2

0 10 20 30 40

5

10

15

20

25

30

0 {} {} {2} {2,3}nK

31 1 10 0 1 0 14 4 4 4

{1,2,3} {1,2,3} {1,3} {1}nK

0 { | ( ) 0, }nk nk q t t

{ | ( ) 0, }nk nk q t t


Example realizations, N={1,10,100}Example realizations, N={1,10,100}

0 10 20 30 400

500

1000

1500

2000

0 10 20 30 400

500

1000

1500

2000

0 10 20 30 400

500

1000

1500

2000

0 10 20 30 400

500

1000

1500

2000

0 10 20 30 400

50

100

150

200

0 10 20 30 400

50

100

150

200

0 10 20 30 400

50

100

150

200

0 10 20 30 400

50

100

150

200

0 10 20 30 400

5

10

15

20

0 10 20 30 400

5

10

15

20

0 10 20 30 400

5

10

15

20

0 10 20 30 400

5

10

15

20

1N

10N

100N

seed 1 seed 2 seed 3 seed 4 seed 1 seed 2 seed 3 seed 4


(1) Let be an objective value for any general policy then:

- Scaling: speeding up processing rates by N and setting initial conditions:

Asymptotic Optimality TheoremAsymptotic Optimality Theorem

( )Q t

( )NQ t( ) (0)NQ t NQ

*fV

*1liminf N

fN

V VN

1lim ( ) ( ) (0, )N

NQ t q t uniformly on t T

N

*1lim N

fN

V VN

- Queue length process of finite horizon MCQN

- Value of optimal fluid solution.

NV

(2) Using the maximum pressure based fluid tracking policy:


How fast is the convergence that is stated in the asymptotic optimality theorem ???

How fast is the convergence that is stated in the asymptotic optimality theorem ???

*1lim N

fN

V VN


Empirical Asymptotics N = 1 to 106Empirical Asymptotics N = 1 to 106

0 200000 400000 600000 800000 1106

-2

0

2

4

6

8

10

0 200000 400000 600000 800000 1106

-2

0

2

4

6

8

10

0 200000 400000 600000 800000 1106

0

1000

2000

3000

4000

5000

0 200000 400000 600000 800000 1106

0

1000

2000

3000

4000

5000

0 200000 400000 600000 800000 1106

-2

0

2

4

6

8

10

0 200000 400000 600000 800000 1106

0

1000

2000

3000

4000

5000

0 200000 400000 600000 800000 1106

0

1000

2000

3000

4000

5000

0 200000 400000 600000 800000 1106

0

1000

2000

3000

4000

5000

0 200000 400000 600000 800000 1106

0

1000

2000

3000

4000

5000

0 200000 400000 600000 800000 1106

0

1000

2000

3000

4000

5000

0 200000 400000 600000 800000 1106

0

1000

2000

3000

4000

5000

0 200000 400000 600000 800000 1106

-2

0

2

4

6

8

10

10

1

{}

{1,2,3}

(0, 0, 1)

K

K

u

10

1

{}

{1,2,3}

(0, 1, 1)

K

K

u

10

1

{2}

{1,3}

(0.25, 1, 0.75)

K

K

u

10

1

{2,3}

{1}

(0.25, 0.25, 0.25)

K

K

u

3 3( ) ( )Nn nQ Nq

2 2( ) ( )Nn nQ Nq

1 1( ) ( )Nn nQ Nq

1Queue 1Queue


ThankYou

ThankYou

Documents

14 th INFORMS Applied Probability Conference, Eindhoven July 9, 2007