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Iterative Integer Programming Formulation for Robust Resource Allocation in Dynamic Real-Time Systems

Sethavidh Gertphol and Viktor K. Prasanna

University of Southern California

presented at WPDRTS

April 26, 2004

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Outline

• Introduction• Problem statement• Mathematical model• Linearization technique• IIP formulation• Experiments and results• Concluding remarks

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Introduction

• dynamic real-time systems

sensors actuators

environment

computing power

system

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Resource allocation

• issues in resource allocation– variations of parameters during run time

– performance requirements satisfaction• end-to-end latency

• throughput

– task and machine heterogeneities

– sharing of resources among tasks

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Initial allocation

• initial allocation– initial allocation of system resources

– utilizes “off-line” resources to determine an allocation

– can afford high time complexity

• necessity for our initial allocation technique– run-time parameter variations are taken into

account

– system is more robust with respect to varying run-time parameters

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Goal of initial allocation

• satisfy all performance requirements

• able to absorb variations of run-time parameters for the longest amount of time

• postpone the first dynamic re-allocation at run time

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Problem statement (1)

• system model– a set of heterogeneous machines connected by a

heterogeneous network– different machines have different compute power– different network links have different capacity

Machines

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Problem statement (2)

• application modelsensors actuators

1

9

6

3

11

8

ASDF process network

primary edge sensor communication and actuator Control

primary route

2

45

7

10

A

B

X

Y

Z

source Node

sink Node non-primary edge

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Problem statement (3)

• run-time parameters– load level of a task: amount of data received through the primary

edge

• run-time parameter variations– load level can vary during run time– variations affect computation and communication latency of a

task– assume linear function

between latency and

load level– consider task and

machine heterogeneitieslatency

load level

ETCF matrix

machines

tasks

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Problem statement (4)

• performance requirements– throughput requirement– end-to-end latency requirement

• performance metric– the amount of load level variations an allocation can absorb– let control load level of all tasks

– MAIL: maximum allowable increase in load level• maximum value of under condition that there is no performance

violation• this value is called max

initii ll ×+= )α1(

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• base latency– latency of a task when there is no resource sharing

• actual latency– latency of a task when resources are shared– assumption: resource is fairly shared among all tasks– is equal to

base latency * the number of tasks running on the same machine

Mathematical model (1)-- latency

latency

load level

base latency

current load level

an entry in ETCF matrix

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Mathematical model (2)-- normalized slackness

• normalized slackness - the percentage of “unused” latency– equal to

• must be greater than or equal to 0 to meet performance requirements

trequiremenlatency

zlatency requirement

initial latency

0

z

actual latency

-1 0

initii ll ×+= )α1(

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Objective

actual latency

i p0

i0 -1 0p )( max

• given an allocation– suppose p0 is a task or

route that violates its requirement first

– p0 determines max

– p0 also has the lowest normalized slackness

– the minimum normalized slackness is called cmin

• comparing different allocations– an allocation has its own

p0 and cmin

– an allocation that has higher cmin has higher max

• objective function: maximize cmin

i

i

i

i

REQ

z=

+α1

α

zi

REQi

REQp0

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Solving the problem

resources

mathematical

formulation

IIP formulation

initial allocation

tasks requirements objective function

linearization techniques

LP solver (LINDO)

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Linearization by substitution

• non-linear terms in the mathematical formulationbase latency * the number of tasks running on the same machine

• substitutes the product with an auxiliary variable and adding constraints

• introduces additional variables and constraints• increases complexity

cxx ji ≤ ×

cxQ ≤

0≤ - iQ xx0≤ - jQ xx

1≤ - Qji xxx +

cxQ ≤

0≤ - iQ xx0≤ - jQ xx

1≤ - Qji xxx +

cxx ji ≤ × cxx ji ≤ ×

cxQ ≤

0≤ - iQ xx0≤ - jQ xx

1≤ - Qji xxx +

cxQ ≤

0≤ - iQ xx0≤ - jQ xx

1≤ - Qji xxx +

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Given M, A, p, ETCF, ETKF, Linit, R, LREQ, THREQ

Find X where xi,l {0,1}

To Minimize

Subject to

Iterative Integer Programming (IIP) approach

1. Set threshold = 0

2. Solve the following Integer Program

Aa

ii

i

kc

MmjiAaayxx

MmjiAaaxy

MmjiAaaxy

Aax

Rrnsr

Aansk

Aansc

ljiljiljli

ljiljlji

ljililji

iMm

li

kk

ii

ii

l

,,,,0

,,,,0

,,,,0

,1

, threshold

, threshold

, threshold

,,,,

,,,

,,,

,

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IIP approach (2)

3. If the Integer Program is infeasible, stop and use the solution of the previous iteration as solution

4. If the Integer Program has an optimal solution, calculate cmin and save the solution

5. Set threshold = cmin + and go to step 2.

is a user-provided parameter between 0 and 1

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IIP approach (3)

• IIP maximizes cmin by solving Integer Program iteratively

• in each iteration, threshold value is the acceptable (minimum) cmin value of that iteration– threshold value starts at 0: the IIP approach is not restricted in the

first iteration

• threshold value increases in subsequent iterations, until the IP program is infeasible

• the last feasible allocation (with highest-known cmin value) is used as the solution

• cmin of the resulting allocation is within of the optimal value

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Experiments--problem generation

• task and machine heterogeneities– use Gamma distribution to generate ETCF and ETKF matrices– high task heterogeneity and high machine heterogeneity

environment

• initial load level– use uniform distribution from 10 to 100 to generate Linit vector

• performance requirements– use a parameter called f to adjust the tightness of requirements– higher f value means requirements are more lax

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Experiments --other approaches for comparison

• SMIP– linearizes the formulation by substitution

– encodes cmin directly into the formulation

– formulation is a mixed integer program– formulation is solved only once– is expected to produce an optimal allocation

• MIP(CBH)– linearizes the formulation by pre-selecting the value of variables

– encodes cmin directly into the formulation

– formulation is a mixed integer program– formulation is solved only once– may not produce an optimal allocation

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Experiment #1

• goal: evaluate the performance of IIP approach• problem sets

– 3-5 machines, 12 tasks, f = 1.5

• for each problem instance– enumerate all possible initial allocation

• for each allocation, find its max

• select the allocation with the highest max (the optimal allocation)

– formulate and solve the problem using IIP (with = 0.01), SMIP and MIP(CBH) approaches

– compare max of the allocations with the optimal solution– compare the execution time of each approach

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Results (1)

MAIL value of each approach (average over 40 problems)

0

20

40

60

80

100

120

3 4 5

number of machines

% o

f o

pti

ma

l v

alu

e

IIP

MIP(CBH)

SMIP

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Results (2)

Execution time of each approach (average over 40 problems)

0

200

400

600

800

1000

1200

1400

3 4 5number of machines

se

co

nd

s

enumerationIIPMIP(CBH)SMIP

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Experiment #2

• goal: evaluate the effect of on the performance of the IIP approach

• problem sets– 3-5 machines, 12 tasks, f = 1.5

• for each problem instance– enumerate all possible initial allocation

• for each allocation, find its max

• select the allocation with the highest max (the optimal allocation)

– solve the problem using the IIP approach with value equal 0.01, 0.05 and 0.1

– compare max of the resulting allocations with the optimal solution

– compare the execution time

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Results (3)

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Results (4)

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Concluding remarks

• IIP approach guarantees that cmin of the resulting allocation is within of the optimal value

• trade-off between the quality of results and execution time can be achieved by adjusting the value

• future work– explore alternate methods to adjust threshold value between

iterations to make IIP faster– formulate the IIP such that the resulting allocation is optimal– additional run-time parameters– different variation patterns of the run-time parameters

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Backup slides

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Problem statement (5)

• performance metric– the amount of load level variations an allocation can

absorb– let controls load level of all tasks

– MAIL: maximum allowable increase in load level• maximum value of under condition that there is no

performance violation

• this value is called max

initii ll ×+= )α1(

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Linearization

• the need for linearization: multiplication of two unknown variables in the same term

• recall: actual latency of a task is

base latency * the number of tasks running on the same machine

depend on where the task is allocated

depend on where the task is allocated and on other tasks allocated to the same machine

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Heuristic for Choosing Nl

• Capability Based Heuristic (CBH)– let slil be the aggregated slope of ETCFi,l and ETKFi,l

– define il

– let πil be the relative speed of ai on ml

– let CCl be the computation capability of ml

– Nl is calculated as

∑'

'απ

lil

ilil

β=

∑πi

illCC =

∑∑'

' ii

ll

ll n

CC

CCN ×=

ilil

sl

1β =

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Heuristic for Choosing Nl

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Our approach--overview

• define system and application models• define run-time parameters and their variations• define a performance metric for initial allocation • define mathematical model for latency and

throughput of tasks• formulate the initial resource allocation as a mixed

integer programming (MIP) problem• utilize available MIP packages to solve this

problem

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Assumptions

• applications are continuously running• multiple applications can share the same

machine and communication link– CPU cycles and communication link bandwidth are

fairly shared among all applications

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Load level

• LL(ai): load level (amount of data received

through the primary edge) of task ai

– LL: a vector of load level of all tasks

– LLinit: a vector of initial load level of all tasks

• ETCF: an estimated-time-to-compute-function matrix

• ETKF: an estimated-time-to-communicate-function matrix

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MmjiAaayxx

MmjiAaaxy

MmjiAaaxy

Aax

Rrnsr

Aansk

Aansc

ljiljiljli

ljiljlji

ljililji

iMm

li

kk

ii

ii

l

,,,,0

,,,,0

,,,,0

,1

, threshold

, threshold

, threshold

,,,,

,,,

,,,

,