OPTIMAL INFORMATION GATHERING SCHEME OVER A
SCALABLE GRID INFORMATION SERVICES ARCHITECTURE
Nianjun Zhou, Dikran Meliksetian, Jean-Pierre Prost, Irwin Boutboul
150 Kettletown Road, Southbury, CT 06488, USA
Keywords: Information gathering, cost optimization, Markov Chain, information reduction, grid computing, dynamic
programming.
Abstract: A proactive model for gathering resource attribute values in large scale distributed systems is proposed and
analyzed within the context of resource discovery. This model, based on a tree topology, relies on
information reduction to limit the amount of information collected at each node of the tree structure and to
minimize information update and query cost. A Markov chain is used to model resource attribute value
changes. This model is solved using dynamic programming to determine the optimal reduction scheme that
minimizes the overall cost of updating and querying resource attributes.
1 INTRODUCTION
Grid computing (Kaufmann, 2004; Foster et al.,
2001; Foster et al., 2002) enables the virtualization
of distributed computing and data resources such as
CPU, network bandwidth and storage including
memory to create a single system image, granting
users and applications seamless access to vast IT
capabilities. One of the key issues is to maintain
accurate information about the entities constituting
the system. Entity state is typically represented by
attributes, exhibiting values, which change over
time. As the system size (i.e. the number of
resources in the system) grows, maintaining an
accurate representation of its state becomes a real
challenge, since the number of messages required to
maintain this state scales with the number of
resources. Typical models used for information
gathering, such as the Globus Toolkit MDS
(Czajkowski, et al., 2001; Zhang, Schopf, 2004),
take a reactive approach. In such models, it is only
once a query (explicit or implicit) for a resource
attribute value is submitted that the relevant
information is fetched. In order to improve the
latency incurred in answering queries, caching
techniques are used to store the information in
aggregator nodes until a pre-defined time-to-live
period expires. A tree topology is used, where the
leaves represent the system resources and upper
level nodes are aggregator nodes. In these models
however, the amount of information stored at each
aggregator node grows linearly with the number of
resources registered to report their attribute values to
them, and once cached information becomes stale,
querying it may lead to important latencies. This
method is efficient when the query to the system is
less frequent than the changes of the attribute value.
In other grid information systems, peer to peer
interactions are used among aggregator nodes to
improve scalability (Mastroianni et al., 2005), or
filtering techniques (Balaton et al, 2002) and
publish-subscribe mechanisms (Jie, 2004; Cooke et
al., 2004) allow for the selection of the information
to be gathered and thus limit the amount of
information aggregated. However, in these systems,
depending upon which aggregator is queried, the
resulting information can vary greatly.
Ganglia (
Massie et al., 2004) is a scalable distributed
monitoring system for high performance computing
environment. It is based on a hierarchical design
targeted at federations of clusters. It relies on a
multicast-based listen/announce protocol to monitor
state within clusters and uses a tree of point-to-point
connections amongst representative cluster nodes to
federate clusters and aggregate their state. Although
Ganglia is a monitoring system and while the
objective of the proposed scheme in this paper is
data acquisition and resource discovery; the two
share a common proactive approach for data
acquisition, and both are concerned with minimizing
the network load required for this operation. Since
the objectives of the two systems are different, the
approaches for load minimization are different.
While Ganglia uses standard data compression
techniques, we use an information reduction scheme
as described in the subsequent sections.
184
Zhou N., Meliksetian D., Prost J. and Boutboul I. (2006).
OPTIMAL INFORMATION GATHERING SCHEME OVER A SCALABLE GRID INFORMATION SERVICES ARCHITECTURE.
In Proceedings of WEBIST 2006 - Second International Conference on Web Information Systems and Technologies - Internet Technology / Web
Interface and Applications, pages 184-189
DOI: 10.5220/0001256801840189
Copyright
c
 SciTePress

In this paper, we propose a proactive model, which
triggers information updates up the tree structure
each time the attribute value of a resource changes.
In addition, we reduce information going up the tree
by consolidating at each node the values of a given
attribute from all the nodes reporting to it into a set
of attribute value intervals. Upon an explicit or
implicit user query, the reduction nodes are
consulted in a hierarchical fashion until the response
can be formulated. The introduction of the
information reduction provides us a mechanism of
not propagating the un-needed information for the
queries to the high levels of the topology. The key
issue becomes the determination of the optimal
number and ranges of the intervals at each tree level
to minimize the cost for updating and querying a
resource attribute.
This paper is organized as follows. In Section 2, we
formulate our optimization problem in the case of a
single attribute. In Section 3, we describe a
stochastic model where the process of the attribute
value change is represented as a Markov chain. In
Section 4, we solve the model to determine the
update and the query costs associated with a single
attribute. In Section 5, we demonstrate that the
determination of the optimal set of attribute value
intervals at each tree level to achieve least cost is a
dynamic programming problem, and we propose an
algorithm to determine the optimal solution. In
Section 6, we conclude our paper with future plans,
including the extension of this research into multiple
attribute scenarios and a comparison with the
reactive model.
2 PROBLEM FORMULATION
As described above, we have a multi-level tree
topology, in which leaves are system resources and
upper level nodes are reduction nodes. We assume a
fully balanced tree, in which each node has a limited
fixed number of children. The leaf nodes of our tree
structure are the system resources and are monitored
regularly for attribute value changes. Each time an
attribute value changes, some of the reduced
information maintained for this attribute by the
reduction nodes it reports to (directly or indirectly –
we will refer to these nodes as its reporting branch),
may have to be updated. In other words, some of the
numbers of resources in each attribute value interval
of the reporting nodes may have to be incremented
or decremented. We want to minimize the number of
updates up the reporting branch. Therefore, if we
were to only look at the update cost, the best
solution is to use coarse attribute value intervals.
We must also take in consideration the query cost.
Unlike updates, which are triggered from leaf nodes
to the root, queries start at the root. If we only use
coarse intervals, depending on the attribute value
query (assumed to be of the form “find all the
resources for which attr
≤
value”), we may end up
traversing most of the tree, since most top level
reduction nodes will have resources reported in each
attribute value interval. Therefore, the query cost
may be expensive.
The whole problem is to find the optimal set of
attribute value intervals that achieves the best trade-
off between the update cost and the query cost. We
define the cost as the number of messages
exchanged between nodes (either between a resource
and a reduction node, or between two reduction
nodes).
3 OUR MODEL
Each node has a fixed number of nodes reporting
directly to it, called n. K represents the number of
levels in the tree and N the number of leaf nodes (i.e.
system resources), Level 0 always represents the
leaf nodes and level K the root. Finally,
NK
n
log
=
(1)
3.1 Attribute Change Model
We assume that the attribute can only have discrete
numerical values. We let
M
represent the number
of attribute values and the set
{}
M
aaa ,...,,
21
represent the possible values of the attributes. We
model the attribute value change process using
Markov chain and the changes of the attribute at
different leaf nodes are identical independent
distributions. We further assume that the attribute
value can either stay the same, or increase and
decrease by one unit within a time interval as a
birth-and-death process (Leon-Garcia, 1994). This
assumption is valid if we limit the time interval to an
arbitrary small value. We denote by
)(
0
ip the
equilibrium probability that the value of the attribute
is
i
a , and by )(
0
if and )(
0
ir the forward and
backward transition probabilities from
i
a to
1+i
a
and
1−i
a respectively. The equilibrium probability
)(
0
ip can be deduced from the transition
probabilities. For our analysis, we assume that these
transition probabilities are known as prior
knowledge.
3.2 Update, Query and Cost Models
Updates occur at the same regular time intervals as
attribute changes. Updates are triggered up the tree,
starting from the system resource whose attribute
value either increased or decreased, along its
OPTIMAL INFORMATION GATHERING SCHEME OVER A SCALABLE GRID INFORMATION SERVICES
ARCHITECTURE
185

reporting branch. The number and identities of
resources falling into each attribute value interval of
each reduction node are adjusted when appropriate.
An update is generated by a leaf node only when the
change of the attribute would cause a change in the
attribute value interval where the value falls at the
parent node. Similarly, an update is generated by a
reduction node only if the change will cause a
change of the attribute value interval of its parent
node.
The number of attribute value intervals at a given
level
k , is denoted by
k
l , and the intervals are
indexed with
{}
k
li ,,1 L∈ . The attribute value of a
given leaf node will belong to one and only one
attribute value interval at level
k . We use
k
I to
represent the random variable corresponding to the
attribute value interval index that the attribute value
of a given resource belongs. It is obvious that the
changes of
k
I are Markov birth-and-death processes.
We use
)(ip
k
to denote the equilibrium probability
that
)( iI
k
= . The transition probabilities of
k
I
from interval
i to interval
()
1+i and
()
1−i during
a time interval are denoted by
)(if
k
and )(ir
k
respectively. Based on our earlier assumption, all
other transition probabilities are null.
We consider only simple queries of the form “find
all the system resources for which value of attribute
x is less or equal to value
j
a ”, where
j
a represents
the jth discrete numerical value that x can have.
In this study, we only consider the cost related to
network traffic. We consider the message payload
cost to be negligible compared to the message
assembly, transfer, and de-assembly, hence the cost
will be expressed as the total number of messages
exchanged between nodes, while accounting for all
potential attribute value updates and queries.
4 UPDATE AND QUERY COSTS
The total number of leaf nodes that are in the subtree
rooted by a node at level
k is
k
n . The number of
nodes at level
k is equal to
⎟
⎠
⎞
⎜
⎝
⎛
k
n
N
. The total
number of nodes at level k and above is denoted by
k
m and is equal to:
1
1
1
0
−
−
====
+−
−
==
−
=
∑∑∑
n
n
nn
n
N
m
kK
kK
j
j
K
kj
jK
K
kj
j
k
(2)
Given the attribute value intervals at all levels, and a
the Markov chain model for the attribute values
specified by
)(
0
ip , )(
0
if and )(
0
ir , the equilibrium
probability distribution and the transition
probabilities at all levels can be computed as
follows. Consider the i
th
attribute value interval
[
]
ii
yx
aa ,
at level k . The probability of the attribute
value belonging to the i
th
interval is equal to the sum
of the probabilities that the attribute has values
i
x
a
through
i
y
a
:
∑
=
=
i
i
y
xj
k
jpip )()(
0
(3)
The transition probabilities are determined similarly:
)(
)()(
)(
00
ip
yfyp
if
k
ii
k
=
(4) and
)(
)()(
)(
00
ip
xrxp
ir
k
ii
k
=
(5)
Each update generates one message, while a query to
a particular node consists of a request and a
response. We will take this factor into consideration
in the following subsections.
For the purpose of this analysis we are assuming a
given fixed reduction scheme that specifies the
attribute value intervals at all levels, a given attribute
value distribution by
)(
0
ip , )(
0
if and )(
0
ir , a query
activity model specified by
F
, the frequency of
query requests within a given time interval, and
j
q ,
the probability that the query of the form “find all
leaf nodes that have the attribute value less than or
equal to
j
a ” refer to the particular value
j
a .
Update Cost
Given
)(ip
k
, )(if
k
and )(ir
k
, we can determine the
update cost by determining the expected number of
messages exchanged for an update. A given leaf
node will cause an update to be generated from a
node at level
k to a node at level )1( +k , if the
attribute value crosses attribute value interval
boundaries at level
)1(
+
k . The probability of this
event is denoted as
k
u , where 10 −≤≤ Kk , and is
equal to:
[]
∑
+
=
+++
+=
1
1
111
)()()(
k
l
i
kkkk
irifipu
(6)
The probability that a node at level
k , 10
−
≤≤ Kk ,
will actually generate an update is equal to the
probability that at least one of the
k
n leaf nodes in
the subtree rooted at this node causes an update; this
probability,
k
U is:
(
)
k
n
kk
uU −−= 11
(7)
The expected number of updates, i.e. the update
cost
CU , is determined by adding these probabilities
at all nodes:
()
[
]
∑∑
−
=
−
=
−−==
1
0
1
0
11
K
k
K
k
n
k
k
k
k
k
u
n
N
U
n
N
CU
(8)
Query Cost
The query cost is equal to twice the number of nodes
that need to be visited to respond to the query. In the
following sections, we consider a simple query of
WEBIST 2006 - INTERNET TECHNOLOGY
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the form “find all leaf nodes that have the attribute
value less than or equal to
j
a
”, where
j
a
is one of
the numerical values of the attribute. We consider
two different variations for counting the visited
nodes.
Case I: It is obvious that if the query is
addressed to a node where
j
a is equal to the upper
bound of one of the attribute value intervals
],[
yx
aa
,
i.e.
yj
aa = , the node can definitely respond to the
query. Since the attribute value intervals are the
same for all nodes at a given level, the simplest way
of counting the nodes visited is to count all the
nodes down to the level where all the nodes have an
attribute value interval
],[
yx
aa
. Let
)(
j
ak
denote
this level, then the total number of nodes visited is
equal to the total number of nodes from the root,
level
K
, down to and including level
)(
j
ak
. Using
equation (2), this number is equal to:
1
1
1)(
)(
−
−
=
+−
n
n
m
j
j
akK
ak
(9)
Consequently, the query cost for searching the leaf
nodes satisfying
j
ax ≤ is:
1
1
22)(
1)(
)(
−
−
==
+−
n
n
mjcq
j
j
akK
ak
(10)
and the weighted total cost for all possible values of
j
a is
()
∑
=
+−
−
−
=
M
j
akK
j
j
nq
n
F
CQ
1
1)(
1
1
2
(11)
where
F
is the frequency of query requests within a
given time interval and
j
q is the probability that the
query refers to a particular
j
a
.
Case II: A more interesting and challenging
case occurs when we consider closely the conditions
under which we can prune the search space even
before we reach level
)(
j
ak . Obviously, if we
encounter a node
g
at level )()(
j
akgk > such that
the attribute value interval
],[
yx
aa
that contains
j
a
has no resources, node
g
can respond to the query
without consulting its children. Let the index of the
attribute value interval
],[
yx
aa
at this node be
denoted by
)(
)(
ji
gk
. Note that this index is a
function of only the level
)(gk of the node and not
of the node itself because all the nodes at a given
level have the same attribute intervals. The
probability of this attribute value interval having no
resources is equal to the probability that none of the
)( gk
n leaf nodes in the sub-tree rooted by
g
have an
attribute in this interval. In our notation this is equal
to
()()
)(
)(1
)()(
gk
n
gkgk
jip−
(12)
where
)(ip
k
is the probability that the attribute value
of a leaf node falls in the
i
th
interval of level k .
In this case, we want to determine the expected
number of visited nodes given the
)(ip
k
distributions at all levels. With the current notation,
the probability that the children of
g
are visited is:
(
)
(
)
)(
)(11
)()(
gk
n
gkgk
jip−−
(13)
and consequently, the expected number of the
children of
g
that are visited is
(
)
(
)
(
)
)(
)(11
)()(
gk
n
gkgk
jipn −−
(14)
Since there are
)(gk
n
N
nodes at level )(gk and there
are n children nodes for node
g
, and the probability
distributions of these nodes are independent from
each other, the expected number of visited nodes at
level
(
)
1)(
−
gk is:
()()
(
)
k
n
kk
k
jip
n
N
)(11
1
−−
−
(15)
Note that we have dropped the explicit dependence
of
)(gk on
g
since all nodes at level )(gkk = are
identical.
The query starts from the root. It is obvious that the
root has to be visited with probability one, while for
the lower levels the expected number of visited
nodes is given by the equation derived earlier.
Consequently the total expected number of visited
nodes is:
()()
(
)
∑
+=
−
−−+
K
akk
n
kk
k
j
k
jip
n
N
1)(
1
)(111
(16)
The corresponding expected cost for a given
j
a is:
()()
(
)
∑
+=
−
−−+=
K
akk
n
kk
k
j
k
jip
n
N
jcq
1)(
1
)(1122)(
(17)
and
()() (
)
∑∑
=+=
−
−−
+
=
M
j
K
akk
n
kk
k
j
j
k
jip
n
N
qF
FCQ
11)(
1
)(112
2
(18)
Total Cost
Case I:
From (8) and (11), we derive:
()
[
]
()
∑
∑
=
+−
−
=
−
−
+−−=+=
M
j
akK
j
K
k
n
k
k
j
k
nq
n
F
u
n
N
CQCUC
1
1)(
1
0
1
1
2
11
(19)
Case II: From (8) and (18), we derive:
OPTIMAL INFORMATION GATHERING SCHEME OVER A SCALABLE GRID INFORMATION SERVICES
ARCHITECTURE
187

()
(
)
()()
(
)
∑∑
∑
=+=
−
−
=
−−
++−−=+=
M
j
K
akk
n
kk
k
j
K
k
n
k
k
j
k
k
jip
n
N
qF
Fu
n
N
CQCUC
11)(
1
1
0
)(112
211
(20)
5 ANALYSIS OF THE
OPTIMIZATION PROBLEM
As described earlier, the optimization problem
considered in this paper is to determine the attribute
value intervals at all levels of the hierarchy given the
attribute value distribution and the query
probabilities. First, let us consider the possible
attribute value intervals, called partitions hereafter.
The total number of possible partitions of an
attribute with
M
discrete values is:
()
()
∑
=
−
=
−−
−
M
i
M
iiM
M
0
1
2
!!1
!1
. (21)
Let
Π denote the set of all partitions and let us
define a “coarser or equal to” relationship between
the partitions. A partition
Π∈
P
is coarser or equal
to another partition
Π∈
′
P
, denoted
P
P
′
f , if the
intervals of
P
are unions of one or more of
consecutive intervals of
P
′
.
A particular feasible solution consists of a sequence
of partitions
{}
011
,,,, PPPP
KK
L
−
where
k
P represents
the partition of level
k ,
0
P is always the partition
consisting of singleton intervals
P
1
, and
ji
PP f for
all
ji ≥ .
The cost functions
{}()
011
,,,, PPPPC
KK
L
−
that were
calculated in the previous sections are for a
particular feasible solution
{}
011
,,,, PPPP
KK
L
−
, and
the optimization problem is to determine a solution
{}
****
011
,,,, PPPP
K
K
L
−
that minimizes C .
5.1 Dynamic Programming
Formulation
In the following, we rewrite the cost equations to
highlight the dynamic programming (Ecker and
Kupferschmid, 1998) nature of the problem. We
will consider the results for Case II of the previous
section; however the same approach is also valid for
case I.
The cost for a particular feasible solution
{}
011
,,,, PPPP
KK
L
−
was derived as in equation (20).
The constant term in that equation has no
consequence on the solution. Let us consider the last
term
CQ
and change the order of summations:
()()
(
)
∑∑
=+=
−
−−=
M
j
K
akk
n
kk
k
j
j
k
jip
n
N
qFCQ
11)(
1
)(112
(22)
()()
()() (
)
∑
∑
=
=
−
−−×+−
×=
M
j
n
kkjj
K
k
k
k
jipakksq
n
N
FCQ
1
1
1
)(111)(
2
(23)
where
(
)
(
)
1)(
+
−
j
akks is the step function, i.e.:
()()
⎩
⎨
⎧
+≥
=+−
otherwise0
1 if1
1)(
)k(ak
akks
j
j
(24)
Let us define, for a given
k , the set )(kJ of all
indices
j
such that
j
a does not appear as a right
boundary at level
k or above.
)(kJ
is equal to:
{
}
)( and 1)(
j
akkMjjkJ >≤≤=
(25)
CQ can then be written as:
()()
(
)
∑∑
∈=
−
−−=
)(1
1
)(112
kJj
n
kkj
K
k
k
k
jipq
n
N
FCQ
(26)
With these transformations, the cost function can be
written as:
()
(
)
()()
(
)
∑∑
∑
∈=
−
=
−
−
−−
++−−=
−
)(1
1
1
1
1
)(112
211
1
kJj
n
kkj
K
k
k
K
k
n
k
k
k
k
jipq
n
N
F
Fu
n
N
C
(27)
Finally:
() (
)
()() (
)
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−−+−−
×+=
∑
∑
∈
−
=
−
−
)(
1
1
1
)(11211
2
1
kJj
n
kkj
n
k
K
k
k
kk
jipqFu
n
N
FC
(28)
By definition,
)(kJ
can be uniquely determined
knowing the partitions in the sub-
sequence
{
}
kKK
PPP ,,,
1
L
−
, )( ji
k
and other terms in
the second part are uniquely determined by the
partition
k
P . Finally, since:
[]
∑
+
=
+++
+=
1
1
111
)()()(
k
l
i
kkkk
irifipu
(6)
1−k
u is uniquely determined by the partition
k
P .
These considerations drive us to define the partial
cost function
{
}
(
)
iKK
PPPC ,,,
1
L
−
, for Ki ≤≤1 , as:
{}()
() (
)
()() (
)
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−−+−−
×+=
∑
∑
∈
−
=
−
+−
−
)(
1
1
11
)(11211
2,,,,
1
kJj
n
kkj
n
k
K
ik
k
iiKK
kk
jipqFu
n
N
FPPPPC L
(29)
or equivalently:
WEBIST 2006 - INTERNET TECHNOLOGY
188

{}(){}()
() (
)
()() (
)
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−−+−−
+=
∑
∈
−
−
+−+−
−
)(
1
1
1111
)(11211
,,,,,,,
1
kJj
n
iij
n
i
i
iKKiiKK
ii
jipqFu
n
N
PPPCPPPPC LL
(30)
with:
{}()
()
()
()()
()
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
−−
+−−
×+
=
∑
∈
−
−
−
)(
1
1
)(112
11
2
1
KJj
n
KKj
n
K
K
K
K
K
jipqF
u
n
N
FPC
(31)
where the second term is a function of
{}
iiKK
PPPP ,,,
11 +−
L and none of the partitions
k
P
for
ik < .
The following algorithm formalizes the derivation in
the previous subsections.
1. For every
Π∈
P
, calculate
{}()
PC .
2. For
1−= Kk to 1 do:
2.a For every
Π∈
P
, identify all partial solutions
{}
11
,,,
+− kKK
PPP L such that PP
k
f
1+
.
2.b For every partial solution in step 2.a do:
2.b.i Calculate
{}()
PPPPC
kKK
,,,
11 +−
L .
2.b.ii Determine the
{}
***
11
,,,
+− kKK
PPP L that
minimizes
{}()
PPPPC
kKK
,,,
11 +−
L .
2.b.iii Add
{}
PPPP
kKK
,,,,
***
11 +−
L
to the set of
potential solutions of level
k .
6 FUTURE WORK
In summary, we have introduced a new architecture
for a scalable grid information service and modeled
it in order to obtain the optimal parameters of the
system in the particular case of a single attribute
with known attribute model and query distribution
model. We are working on extending this approach
to multiple resource attributes. By assuming that
multiple attributes share the same index topology for
information update and query, and that the update
and query messages and their responses for all
attributes are merged together, we can calculate the
cost given the aggregation intervals for each
attribute. The aggregation intervals themselves can
be found by using the method given in Section 5 as a
first order of approximation.
Another avenue for future research is the overall
cost vs. accuracy comparison with reactive
information gathering scheme. We believe that there
is a threshold beyond which the proactive
information gathering scheme is better than the
reactive information gathering scheme. This
threshold depends on 1) how quickly the attribute
changes; 2) how often queries occur and 3) the time
to live values of the reactive caching.
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OPTIMAL INFORMATION GATHERING SCHEME OVER A SCALABLE GRID INFORMATION SERVICES
ARCHITECTURE
189