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Optimal Tree Structures
for Large-Scale Grids
J. Palmer
I. Mitrani
School of Computing Science
University of Newcastle
NE1 7RU
[email protected]
[email protected]
Grid Performability, Modelling and Measurement AHM’04
Outline
 Introduction
 The model
 Computation of the optimal tree structure
 A simple heuristic
 Results
 Conclusions and future work
Grid Performability, Modelling and Measurement AHM’04
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Introduction
 In the provision of a Grid
 Within such a provision, it
service, a provider may have
will be desirable that the
heterogeneous clusters of
clusters are hosted in a
resources offering a variety of
cost effective manner
services
Job arrivals
Potential bottle-neck
Master Node
...
Server
Server
Server
Server
Server
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Job arrivals
additional decision-making process
Master Node
additional transfer delays
Master Node
Master Node
...
Server Server
...
Server
Server Server
Server

The problem of load-balancing considers how best to distribute
incoming jobs across a fixed tree structure

Instead, our approach considers the dynamic reconfiguration of the
underlying tree structure as load changes
Grid Performability, Modelling and Measurement AHM’04
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Job arrivals
dynamic network reconfiguration
Master Node
Job arrivals
Master Node
Master Node
Master Node
...
Server Server
...
Server
Server Server
Server
Master Node
...
...
...
Server Server
Master Node
Master Node
Server
Server Server
Server
Server Server
Grid Performability, Modelling and Measurement AHM’04
Server
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The model
 What value of k minimizes the overall average response
time of the system?

ck
level 1
master node
transfer delay T1
...
c k)
level 2


...
...
...



k master nodes

 
k sub-clusters of
N/k service nodes

Grid Performability, Modelling and Measurement AHM’04
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Job distribution policies
Different job
distribution
policies have
been
considered:
level i
ici ki
transfer delay Ti
...
level i+1
1. Each dependent has a separate queue; the master places new
jobs into
i. those queues in random order
ii. the queue which is currently shortest
iii. those queues in cyclic order
2. Dependents at the final service cluster level have a joint queue
Grid Performability, Modelling and Measurement AHM’04
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Computation of the optimal tree structure

The average response time at each level i master node is given
by:
1
Wi 
 i (1   i )

where
i 
ci

, i 
number of dependents
i
At the final service level, approximated by an M/M/n queue:

1

 (n  n   )   

W final  



2  
  j 1 ( j  1)! (n  1)!(n   )   j 0 j! (n  1)!(n   ) 
n 1
j
where
n
2
n 1
j
n

n  number of servers in each cluster ,  

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1
Computation of the optimal tree structure

For a flat structure ( c1>N for stability):
W  W1  W final

For a two level tree structure:
W  W1  T1  W2  W final

The objective is to minimise the latter with respect to k
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Computation of the optimal tree structure
 At each master node we require i  1
 So, for a given parameter set, k has upper and lower
bounds so that no master node becomes saturated:
N
c2
k
c1

 Average response times for each value of k within this range
have been evaluated and compared to find the minimum
 Hence, the optimal value of k has been determined numerically
 This gives the optimal network configuration with a single layer
of master nodes
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A simple heuristic
 Consider the total offered load at the level 1 master
node and one of the level 2 master nodes:
k
N

f (k ) 


2
c1 c2 k
N
 This total load can be minimized with respect to k to
find an initial value for k given N, c1 and c2:
2 Nc1
k 3
c2
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Results



Average response time as k varies
Parameters: N  100, c1  c2  100, T1  0.001,   8,  0.1
Load is 80%, flat structure not feasible
heuristic predicts k = 6
optimal k = 4
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Results


Optimal number of clusters as load increases
Parameters: N  100, c1  c2  100, T1  0.001,  0.1
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Conclusions and Future Work

Encouraging results suggest dynamic network
configuration will reduce long-term average response
times

A simple heuristic is available for initial network
configuration

Future work includes:
1. extension to include further tiers of master nodes
2. different modelling assumptions for how a master
node makes a routing decision
- shortest queue
- cyclic order
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Acknowledgment
 This work was carried out as part of the
collaborative project GridSHED,
funded by
North-East Regional e-Science
Centre
and
BT
 This project also aims to develop Grid middleware to
demonstrate the legitimacy of our models, providing a basis
for the development of commercially viable Grid hosting
environments
 Project web page:
http://www.neresc.ac.uk/projects/GridSHED/
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