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

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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

environment
system
computing power
sensors
actuators
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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

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Problem statement (2)

• application model

actuators
sensors

ASDF process network
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X
3
1

4
A
5

6
8
Y
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B
9

10
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Z
sensor communication and

primary edge

source Node
actuator Control

primary route
non-primary edge
sink Node
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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 heterogeneities

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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

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Mathematical model (1)-- latency

• 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

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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

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Objective

• 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

p
0

i

REQi
REQp0
zi
a
0

-
1

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Solving the problem
resources
tasks
requirements
objective function
mathematical formulation
linearization techniques
IIP formulation
LP solver (LINDO)
initial allocation
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Linearization by substitution

• non-linear terms in the mathematical formulation
• base 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

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Iterative Integer Programming (IIP) approach

1. Set threshold 0
2. Solve the following Integer Program

Given M, A, ?p, ETCF, ETKF, Linit, R, LREQ,
THREQ Find X where xi,l ?0,1 To Minimize
Subject to
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IIP approach (2)

• If the Integer Program is infeasible, stop and
  use the solution of the previous iteration as
  solution
• If the Integer Program has an optimal solution,
  calculate cmin and save the solution
• 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)
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Results (2)
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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

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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 pil be the relative speed of ai on ml
• let CCl be the computation capability of ml
• Nl is calculated as

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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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