View
214
Download
2
Category
Tags:
Preview:
Citation preview
ORSIS Conference,Jerusalem Mountains, Israel
May 13, 2007
ORSIS Conference,Jerusalem Mountains, Israel
May 13, 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
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 k Initial 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
Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 3
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
Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 4
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)…
Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 5
AN EXAMPLE: A Push-Pull Queueing System (Weiss, Kopzon 2002,2006)AN EXAMPLE: A Push-Pull Queueing System (Weiss, Kopzon 2002,2006)
1 1
22
Server 2Server 1
PUSH
PULL
PULL
PUSH
1 1 2 2, 1 1 2 2, “Inherently Stable” “Inherently Unstable”
0,0 1,0 2,0 3,0 4,0 5,0
0,1
0,2
0,3
0,4
n1
n2
1 1 1 1 1 1
1 1 1 1 1 1
2
2
2
2
2 2
2
2
2
2
0,0
1,3
2,0 3,0 4,0 5,0
0,1
0,2
0,3
0,4
n1
n2
11 1 1 1 1
1 1
1 1 1 1
2
2
2
2
2
2,1
2,2
2,3
2
2
22
2
2
3,1
3,2
3,3
2
2
2
2
2
2
4,1
4,2
4,3
2
2
2
2
2
2
5,1
5,2
5,3
2
2
22
2
2
1 1
1,0
1,4
1
1 1
2,4
0,5
2
1,5
1
1 1
2,5
2
2
2
1
1 1
4,5
2For Both Cases,
Positive Recurrent Policies Exist
1 1 2 1 2 2
1 2 1 2
( )
2 2 1 2 1 1
1 2 1 2
( )
1 2 1 Require:
Low variance of the departure process?
PROBABLY
NOT WITH THESE
POLICIES?
Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 6
OUR MODEL:MCQN+IVQ
OUR MODEL:MCQN+IVQ
Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 7
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
Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 8
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
Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 9
Static Fluid FormulationStatic Fluid Formulation
Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 10
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
Given, the rates assumptions , 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
( , )P
The input-output matrix:
Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 11
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 (and much older ideas).
Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 12
Rate Stable Controls
Rate Stable Controls
Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 13
•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 ( ) ( )d
f t Z t Z t Z t RT tdt
( ) ( )Z t RT t
'kkR
( )arg max ( )T
a A tZ 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.• , so there is always some allocation.0 ( )A t
Intuitive Meaning of the Policy
( )T t
“Energy” Minimization
The Resulting Policy
Choose:
Feasible Allocations
Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 14
Given a MCQN with IVQs defined with nominal production rates that are given by a feasible static allocation. The non-processor splitting, no-preemption Maximum Pressure Policy is stable for any primitive sequences that satisfy the rates assumptions in the following two senses:
Given a MCQN with IVQs defined with nominal production rates that are given by a feasible static allocation. The non-processor splitting, no-preemption Maximum Pressure Policy is stable for any primitive sequences that satisfy the rates assumptions in the following two senses:
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 for Finite time horizon, T:
Where satisfies:( )NZ t
Proof is an adaptation of Dai and Lin’s 2005, Theorem 2.
( )NZ t
Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 15
Work in progress….Work in progress….
How fast does the queue (virtual queue) sizes grow?
How fast does the queue (virtual queue) sizes grow?
How do simpler policies (randomized), that follow the static fluid equations compare?
How do simpler policies (randomized), that follow the static fluid equations compare?
General Applications…General Applications…
Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 16
Possible Applications of the Theorem
Steady State Systems
Systems with Time Varying Parameters
Tracking Transient Fluid Solutions
of a MCQN
Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 17
Transient Fluid Solutions
Transient Fluid Solutions
Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 18
Sta
cked
Que
ue L
evel
s
time T
Q1
Q2Q3
Trajectory of a single job
Finished Jobs
Example NetworkExample Network
Stacked Queue level representation:
Stacked Queue level representation:
Server 1Server 2
1
23
3
10
( )T
kk
Q t dt
Attempt to minimize:
Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 19
10
( )T K
kk
Q t dt
Sta
cked
Que
ue L
evel
sT
J12
J11
J10
J9
J8
J7
J6
J5
J4
J3
J2
J1
Sta
cked
Que
ue L
evel
s
T
Sta
cked
Que
ue L
evel
s
T
J12
J11
J10
J9
J8
J7
J6
J5
J4
J3
J2
J1
Minimizing inventory costs.
Minimizing the total job waiting time.(truncated to time horizon).
Maximizing the total time from job completion to the time horizon. (maximizing “useful life”)
Corresponds to:
Minimization of
Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 20
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
Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 21
Fluid solutionFluid solution
•SCLP – Bellman, Anderson, Pullan, Weiss.•Piecewise linear solution. •Simplex based algorithm, finds the optimal solution in a finite number of steps (Weiss).
The Optimal Solution:
•SCLP – Bellman, Anderson, Pullan, Weiss.•Piecewise linear solution. •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 3
2 2
1 1
1 3
2
(0) (0) 15
(0) (0) 1
(0) (0) 8
1.0
0.25
40
Q q
Q q
Q q
T
3( )q t
2 ( )q t
1( )q t
Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 22
0 10 20 30 40
0
5
10
15
20
0 10 20 30 40
0
5
10
15
20
Structure of the optimal solution – comparison to LBFSStructure of the optimal solution – comparison to LBFS
Last Buffer First Server (LBFS):Last Buffer First Server (LBFS):
Improve: Don’t wait with the emptying of buffer 1 until 3 is empty…
The optimal solution:
352Obj
376Obj
Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 23
Fluid TrackingPolicy
Fluid TrackingPolicy
Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 24
Near Optimal Control over a Finite Time HorizonNear Optimal Control over a Finite Time Horizon
Approximation Approach:1) Approximate the problem using a fluid system.2) Solve the fluid system (SCLP).3) Track the fluid solution on-line (Using MCQN+IVQs).4) Under proper scaling, the approach is asymptotically optimal.
Approximation Approach:1) Approximate the problem using a fluid system.2) Solve the fluid system (SCLP).3) Track the fluid solution on-line (Using MCQN+IVQs).4) Under proper scaling, the approach is asymptotically optimal.
Solution is intractable
10
( )T K
kk
Min Q t dt
Finite Horizon Control of MCQN
Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 25
4 Time Intervals4 Time Intervals
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
Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 26
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 ( ) ( ) 0N
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
Using our maximum pressure based fluid tracking policy:
Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 27
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
Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 28
Work in progress…. Continued….Work in progress…. Continued….
How fast does the queue (virtual queue) sizes grow?
How fast does the queue (virtual queue) sizes grow?
How fast is convergence stated in the asymptotic optimality theorem???
How fast is convergence stated in the asymptotic optimality theorem???
*1lim N
fNV V
N
Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 29
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
1 Queue 1 Queue
Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 30
0 10 20 30 40
5
10
15
20
25
30
Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 31
ThankYou
ThankYou
Recommended