Journal ArticleDOI

# Scalable service migration in autonomic network environments

01 Jan 2010--Vol. 28, Iss: 1, pp 84-94

TL;DR: The migration policies proposed in this work are analytically shown to be capable of moving a service facility between neighbor nodes in a way that the cost of service provision is reduced and the service facility reaches the optimal (cost minimizing) location, and locks in there as long as the environment does not change.

AbstractService placement is a key problem in communication networks as it determines how efficiently the user service demands are supported. This problem has been traditionally approached through the formulation and resolution of large optimization problems requiring global knowledge and a continuous recalculation of the solution in case of network changes. Such approaches are not suitable for large-scale and dynamic network environments. In this paper, the problem of determining the optimal location of a service facility is revisited and addressed in a way that is both scalable and deals inherently with network dynamicity. In particular, service migration which enables service facilities to move between neighbor nodes towards more communication cost-effective positions, is based on local information. The migration policies proposed in this work are analytically shown to be capable of moving a service facility between neighbor nodes in a way that the cost of service provision is reduced and - under certain conditions - the service facility reaches the optimal (cost minimizing) location, and locks in there as long as the environment does not change; as network conditions change, the migration process is automatically resumed, thus, naturally responding to network dynamicity under certain conditions. The analytical findings of this work are also supported by simulation results that shed some additional light on the behavior and effectiveness of the proposed policies.

Topics: , Network topology (56%), Routing protocol (51%)

### Introduction

• The problem of determining the optimal location of a service facility is revisited and addressed ina way that is both scalable and deals inherently with network dynamicity.
• For such environments, a new policy (referred to 2 as Migration Policy E) is proposed that moves the facility until the end of a monotonically cost decreasing path, provided that tentative movements to the one-hop neighbor nodes are allowed.
• In order to deal with the increased complexity, several near-optimal approaches have been proposed that can be categorized as either centralized or distributed [6].
• Let T xt be the shortest path tree in Tt(x) over which data corresponding to the nodes’ service demands are forwarded towards the facility nodex.

### A. Service Migration for a Single Facility

• Consider the case of one facility located at nodex at timet.
• Note that in general shortest path trees are different for different roots (i.e., T yt+1 6= T x t ), except for the special case of topologies with unique shortest path trees [40].
• The following lemmas are the basis for the migration policy presented later.
• In view of Lemma 1, two interesting observations can be made regarding the difference when the facility is located at neighbor nodes.
• Second, it depends on the difference between the aggregate service demands.

### B. Multiple Facilities

• The following theorem shows that under Migration Policy S, overall cost reduction is always achieved (i.e.,Ct+1 < Ct), whatever the number of facilities in the network.
• In a network consisting of a unique shortest path tree, a single service facility always arrives at the optimal location under Migration PolicyS, assuming a static environment, also known as Theorem 3.
• The results of Theorem 3 apply to many real cases.
• Migration PolicyS is also useful for environments that do not comply with the previous conditions, since it allows for cost reduction (even though not necessarily for cost minimization) based on strictly local information.
• In that example, nodes3 and 5 have chosen a shortcut to implement the new shortest path towards the (new) facility node (node3 is a neighbor of the new facility node and node 5 utilizes the path through node3, instead of the one through node6).

### A. Cost Reduction and Tentative Facility Movements

• On the other hand,Λ ( Ix(T yt+1) ) , which is available at node y at time t + 1, could be made available after a tentative facility movement to nodey and then moving back to the previous facility nodex before deciding if a facility movementx t+2 −−→ y should take place.
• Eventually, the condition of Lemma 2 requires information that is not locally available.

### C. Hybrid Policy

• As the tentative movements associated with Migration Policy E introduce overhead (two facility movements per neighbor node), a hybrid policy is proposed here that combines Migration Policy S and E so that tentative movements are avoided whenever possible.
• It is interesting to see that under Migration Policy H , the facility moves to the optimal position in less time (a20 = 1) than under Migration PolicyE (a30 = 1).
• By comparing the results depicted in Figure 5.a and Figure 5.b, it is also interesting to observe that for two facilities: (a) for geometric random graphs,at slightly increases; (b) for Erdős-Rényi random graphs,at remains about the same; and (c) for Albert-Barabási random graphs,at decreases.
• “An improved approximation algorithm for the metric uncapacitated facility location problem,” In W.J. Cook andA.S. Schulz, editors, Integer Programming and Combinatorial Optimization, Volume 2337 of Lecture Notes in Computer Science, pages 240-257, Springer, Berlin, 2002. [14].

### D. Proof of Theorem 3

• The cost corresponding to the facility located at some node v and the facility located at the optimal position (e.g., nodeu), can be shown as monotonically increasing by the number of hops away from the optimal position in a topology of a unique shortest path tree, [3].
• This is due to the fact that the aggregate service demands – corresponding to the particular neighbor (of the facility) node that is towards the optimal position – monotonically increase.
• Since there is a unique shortest path tree, the facility will always move to nodes of lower overall cost (Theorem 1).

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1
Scalable Service Migration in Autonomic Network
Environments
Konstantinos Oikonomou, Member, IEEE, and Ioannis Stavrakakis, Fellow, IEEE
AbstractService placement is a key problem in communica-
tion networks as it determines how efﬁciently the user service
demands are supported. This problem has been traditionally
approached through the formulation and resolution of large opti-
mization problems requiring global knowledge and a continuous
recalculation of the solution in case of network changes. Such
approaches are not suitable for large-scale and dynamic network
environments. In this paper, the problem of determining the
optimal location of a service facility is revisited and addressed
in a way that is both scalable and deals inherently with network
dynamicity. In particular, service migration which enables service
facilities to move between neighbor nodes towards more commu-
nication cost-effective positions, is based on local information.
The migration policies proposed in this work are analytically
shown to be capable of moving a service facility between neighbor
nodes in a way that the cost of service provision is reduced
and under certain conditions the service facility reaches the
optimal (cost minimizing) location, and locks in there as long as
the environment does not change; as network conditions change,
the migration process is automatically resumed, thus, naturally
responding to network dynamicity under certain conditions. The
analytical ﬁndings of this work are also supported by simulation
results that shed some additional light on the behavior and
effectiveness of the proposed policies.
Index TermsService Placement, Service Migration, Auto-
nomic Networks, Scalability.
I. INTRODUCTION
I
NTERNET globalization and expansion make the service
placement problem a challenging one and necessitate a
careful selection of the location of the service facilities (a
facility being a service provisioning infrastructure), aiming
at bringing the service provision points close to the demand
in order to minimize communication costs (i.e., resource
consumption) and enhance the Quality of Service (QoS) of
the provided service. Due to the recent technological changes
(e.g., powerful machines and services have proliferated), the
traditional problem of placing relatively few big services in
one of the few (powerful) potential service provider facilities
(big network elements) is increasingly being transformed into
a problem of placing the one or more service facilities in one
of the numerous network nodes that are now capable of hosting
services. Peer-to-peer networks, cloud computing, content dis-
This work has been supported in part by the IST-FIRE project Autonomic
Network Architecture (ANA) (IST-27489) and the IST-FET project SOCIAL-
NETS (IST-217141), funded by the European Commission.
Konstantinos Oikonomou is with the Department of Informatics, Ionian
University, Corfu, Greece. Address: Tsirigori Square 7, 49100 Corfu, Greece.
Phone: +30 26610 87708, Fax: +30 26610 48491, E-mail: okon@ionio.gr.
Ioannis Stavrakakis is with the Department of Informatics and Telecommu-
nications, National and Kapodistrian University of Athens, Athens, Greece.
Address: Panepistimiopolis, Ilissia, 157 84, Athens, Greece. Phone: +30 210
7275315, Fax: +30 210 7275333, E-mail: ioannis@di.uoa.gr.
tributions networks, software updates and patches and sensor
networks are examples of such modern environments.
The problem of determining the optimal service placement
has been studied in the past in areas such as transportation
and supply networks [1], and has been approached through
the formulation and solution of large optimization problems
(NP -hard) requiring global knowledge, as for instance is
the case with the k-median problem [2]. Such approaches
requiring global knowledge and a continuous recalculation of
the solution in case of network changes; do not scale and are
not suitable for dynamic network environments, such as those
considered in this work. Instead, approaches based on local
information should be adopted, despite the fact that they might
not be able to guarantee optimality all the time (near-optimal
solutions).
In this paper, the problem of determining the optimal
location of a service facility is revisited and addressed in a
way that is both scalable and deals inherently with network
dynamicity. The approach advocated in this paper parts of it
initially presented in [3] and [4] is that of moving a service
facility among neighbor nodes by utilizing local information,
in a such way that the cost of service provision is reduced
and the service facility reaches under certain conditions the
optimal (cost minimizing) location, and locks in there as long
as the environment does not change; as network conditions
change, the migration process is automatically resumed, thus,
naturally responding to network dynamicity.
The ﬁrst proposed policy (referred to as Migration Policy
S) is analytically shown to be capable of moving the facilities
along a monotonically cost decreasing path in the network.
For special cases of topologies such as trees, it is analytically
shown that under Migration Policy S a single facility moves
until it reaches the optimal position (i.e., the node at which
the overall cost is minimized). In the general case, service
facility migration under Migration Policy S, allows for overall
cost reduction, but it may fail to move the facilities until the
end of a monotonically cost decreasing path, mostly due to
unforeseen shortcuts (i.e., alternative shortest paths utilized
by some nodes to reach a certain facility after a facility
movement). The potential cost reduction that is due to the
aforementioned shortcuts is not taken into account under
Migration Policy S, thus certain facility movements that would
allow for further cost reduction are not detected and thus not
implemented.
The aforementioned limitation is overcome in the case of a
single facility and for topologies of equal link weights, while
still utilizing only local information. Note that topologies
with equal link weights and a single service facility are not
uncommon. For such environments, a new policy (referred to

2
as Migration Policy E) is proposed that moves the facility
until the end of a monotonically cost decreasing path, provided
that tentative movements to the one-hop neighbor nodes are
allowed.
ments under Migration Policy E (when compared to Migration
Policy S), motivates the introduction of a hybrid policy
(referred to as Migration Policy H) that efﬁciently combines
the Migration Policies S and E, in the case of equal link
topologies and a single facility. Under Migration Policy H,
a single facility moves until no further movement is possible
under Migration Policy S. Then, Migration Policy E takes
over and moves the facility until no further movement is
possible. The ﬁrst part of facility movement beneﬁts from the
guaranteed cost reduction under Migration Policy S and the
almost negligible overhead. The second part moves the facility
towards positions of even smaller overall cost (i.e., the end of
the monotonically cost decreasing path), at the expense of the
extra overhead due to the tentative facility movements.
The analytical ﬁndings of this work are also supported by
simulation results. In addition, simulation results provide for
further insight on the behavior of the proposed policies and,
particularly, illustrate their effectiveness for cases not captured
by the analysis. More speciﬁcally, it is shown that in realistic
topologies, the overhead (due to tentative facility movements)
under Migration Policies E and H can be prohibitively large
and therefore, Migration Policy S becomes the obvious alter-
native. This is also the case in dynamic environments due to
the latters capability to move almost immediately service
facilities to more effective positions.
Section II presents related work and in Section III the
service placement problem is described in detail. Migration
Policy S is presented and studied in Section IV while Migra-
tion Policies E and H are presented and studied in Section
V. Section VI presents simulation results and the conclusions
are drawn in Section VII.
II. RELATED WORK
The service placement problem has been addressed in the
past particularly in the area of transportation, supply networks
etc., most of these works surveyed in [1]. It has been shown
that the optimal solution of placing k service facilities in a
network (i.e., the k-median problem) is an N P -hard problem
for general graphs [2]. Even in the case of undirected tree
topologies, the complexity remains as high as O(kN) [5] (N is
the number of nodes in the network). In order to deal with the
increased complexity, several near-optimal approaches have
been proposed that can be categorized as either centralized
or distributed [6]. The centralized approaches focus on greedy
heuristics [7], [8], [9], [10], on linear programming [11], [12],
[13], on primal-dual [14], [15], local search [16], [17], and
other techniques [18], [19], [20], [21], [22], [23] that have been
proposed in the past to deal with the increased complexity of
the service placement problem. However, all of them require
global knowledge of the network topology and demands, and
this information is generally not available in the large-scale
network environment considered in this paper.
The focus in this work is on distributed approaches being
based on local information (as opposed to the aforementioned
centralized local-search-based approaches attempting to as-
sume local minima of performance) in order to avoid scalabil-
ity problems introduced by global knowledge requirements. To
the best of the authors’ knowledge, the ﬁrst distributed facility
location was described by Jain and Vazirani [14]. Moscibroda
and Wattenhofer [24] proposed the ﬁrst distributed approach.
Their work focuses on primal-dual techniques in order to
derive worst-case performance bounds which are difﬁcult to
implement (e.g., impractical communication model) compared
to the work presented in this paper. A recent work by Krivitski
et al. [25] proposes a distributed hill-climbing algorithm based
on local majority votes and used by nodes to agree on the
next step of the algorithm. The overall overhead is kept low
by avoiding unnecessary votes. The algorithm converges to
the optimal solution, as shown using simulation results. This
approach is different from the one proposed here (evaluated
both through analysis and simulations) since no majority
voting is considered and the need for local information is
almost negligible.
The closest works to the one presented here are [26] and
[27]. Ragusa et al. [26] propose a heuristic approach for
partitioning the network into a number of k clusters self-
managed through mobile agents. Each agent migrates to more
efﬁcient positions within the cluster after deriving the median
position in a centralized manner. Clusters may be further
partitioned or merged in response to dynamic network changes
based on empirically selected threshold values. The second
approach, proposed by Laoutaris et al. [27] solves the service
placement problem in a distributed manner reusing existing
near-optimal centralized approaches inside suitably deﬁned r-
balls (i.e., network areas of r hops away from each service
facility). As service facilities move inside the network, r-
balls are partitioned or merged based on the outcome of the
centralized approaches. The main difference between [26],
[27] and the work presented here, amounts to the requirement
for local information in order to select more efﬁcient positions
for the mobile agents in [26] and to incorporate classical
centralized solution approaches in [27]. Instead, the proposed
service migration approach relies on strictly local information;
this local information is passively collected by the service
facility itself, as opposed to requiring the deployment of
a mechanism that provides both the demand and topology
information within each cluster as in [26] and [27].
Several other approaches have also been proposed, spe-
ciﬁc to the considered applications, including intermediate
caching [28], reﬂectors’ deployment [29], online version [30],
placement based on mobility [31], content distribution [32],
replication in overlays [33], service discovery in mobile ad
hoc networks [34], gateway placement [35], sensor networks
[36], [37], replica placement updates [39], distributed shared
platform [38].
III. PROBLEM STATEMENT
The network topology is represented by a connected undi-
rected graph G(V, E), where V is the set of nodes and E

3
the set of links among them. Let |X| denote the size of
a particular set X. Let N be the number of nodes in the
network or N = |V |. Let S
v
denote the set of neighbor
nodes of node v (i.e., nodes having a link with node v).
Let (u, v) denote the link between two neighbor nodes u and
v; each such link is assigned a positive value referred to as
weight and denoted as w(u, v). For a given graph G(V, E),
let d(x, y) denote the distance between node x and node y,
corresponding to the summation of the weights of the links
along a shortest path between these nodes (for the same node
v, d(x, x) = w(x, x) = 0); alternatively, d(x, y) is referred to
as the traveling cost between nodes x and y. In this paper, a
facility is considered to be hosted by one of the network nodes.
It is also possible that the service provisioning is replicated
to cope with a higher demand or for increased reliability
and, thus, more that one facility may be deployed. Without
any loss of generality, only one service and the associated
one or multiple service facilities is considered in this paper.
Let the mean rate at which data packets associated with
a particular node v are transferred through the network be
denoted by λ
v
(i.e., service demands). Let K
t
denote the set
of facility nodes (i.e., nodes hosting a facility corresponding
to a certain service) at time t. In the sequel, t will take discrete
values corresponding to facility movements.
Between a given facility node and any other node in the
network, a shortest path route is assumed to be established by
the underlying employed routing protocol, [40]. Eventually, a
shortest path tree is created, rooted at the particular facility
node and including all network nodes. Such a tree (depicted
with dense lines) rooted at the facility node 0 (marked by
a dotted hexagonal) is shown in Figure 1.a, where the ser-
vice demands and link weights are set to 1 to facilitate the
discussion. In general, a shortest path tree associated with a
given root node is not unique. When the number of facilities is
greater than one, then a forest of shortest path trees is created,
with each tree rooted at the corresponding facility node (e.g.,
as depicted in Figure 1.b). Let T
t
denote the set of all possible
shortest path trees in a network at time t. Let T
t
(x) denote
the subset of T
t
(i.e., T
t
(x) T
t
) containing the shortest path
trees rooted at node x. Let T
x
t
be the shortest path tree in
T
t
(x) over which data corresponding to the nodes’ service
demands are forwarded towards the facility node x. v T
x
t
will indicate that node v is served by facility node x.
Let x
t
y denote the facility movement from node x to
its neighbor node y initiated at time t; at time t (t + 1) the
facility is located at node x (y). Assume that K
0
= {0}
and T
0
0
is the shortest path tree depicted in Figure 1.a.
Assume that K
1
= {1} and T
1
1
is the shortest path tree
depicted in Figure 1.c after a facility movement 0
0
1. It
is reasonable to assume that any routing protocol would try to
minimize the overhead introduced by such a facility movement
by preserving previously established shortest paths and not
switching to new ones, provided that the previous ones are not
worse than any new one. This migration rule will be adopted
here and ensures that the new shortest path tree T
1
1
would be
the one depicted in Figure 1.c and not the one depicted in
Figure 1.d (both belong to T
1
(1)) or any other. To illustrate
this rule further, consider node 7 that forwards data towards
the facility node 0 over path {(7, 8), (8, 0)}, as depicted in
Figure 1.a. After the service moves to node 1, node 7 would
based on the above rule utilize path {(7, 8), (8, 0), (0, 1)},
instead of the equal cost path {(7, 6), (6, 0), (0, 1)}.
Let C
t
(x) denote the cost incurred by facility x for serving
nodes v at time t, for all v T
x
t
. Clearly,
C
t
(x) =
X
vT
x
t
λ
v
d(v, x). (1)
The overall cost over all facilities in the network at time
t, denoted as C
t
, is given by, C
t
=
P
xK
t
C
t
(x) =
P
xK
t
P
vT
x
t
λ
v
d(v, x). Assuming ﬁxed network topol-
ogy and service demands, it is evident that the (optimal) set
of facility locations for which cost minimization is achieved
(denoted by K) does not depend on time t, and the same
holds true for the corresponding minimum cost deﬁned as C.
Let a
t
=
C
t
C
be deﬁned as the approximation ratio of the cost
induced at time t when the set of facility nodes is K
t
, over
the minimum (optimal) one; the closer the value of a
t
to 1,
the closer the induced cost at time t to the optimal one.
The optimal set K and the resulting minimal cost C can
be determined by solving the previously mentioned k-median
problem, (k = |K|). This difﬁcult and large optimization prob-
lem cannot be afforded in the large-scale and dynamic network
environments considered here, where the network topology
is subject to frequent changes requiring the recalculation of
the (expensive) k-median solution. In the sequel, services are
migrated in order to exploit information locally available at
the facility nodes.
IV. SERVICE MIGRATION BASED ON STRICTLY LOCAL
INFORMATION
Strictly local information refers to information that is avail-
able only at a particular node. Based on such information, a
migration policy is proposed in this section to reduce the cost
after moving the facilities to neighbor nodes.
Assume a service facility is located at node x at time t.
There exist a number of neighbor nodes S
x
over which data
associated with service demands of all nodes v T
x
t
\{x}, are
forwarded to the particular facility. In the example depicted in
Figure 1.a, data associated with the service demands towards
the facility node 0, are forwarded over link (0, 1) for nodes
1 and 2, over link (0, 6) for nodes 3, 4, 5 and 6, and over
link (0, 8) for nodes 7, 8 and 9. That is, nodes of a certain
subtree of T
x
t
forward data associated with their own service
demands through some node y, y S
x
. Let I
y
(T
x
t
) denote
the particular subtree, which is also a tree of root node y. In
the previous example, there are three such subtrees denoted
by I
1
(T
0
t
), I
6
(T
0
t
) and I
8
(T
0
t
), as shown in Figure 1.a.
Let Λ (I
y
(T
x
t
)) denote the aggregate service demands that
are forwarded to the facility node x through link (x, y) (for
some neighbor node y S
x
, and y T
x
t
) over subtree
I
y
(T
x
t
). Λ (I
y
(T
x
t
)) is equal to the summation of the service
demands of the individual nodes of the corresponding subtree,
or:
Λ (I
y
(T
x
t
)) =
X
vI
y
(T
x
t
)
λ
v
. (2)

4
0
1
5
8
6
7
3
4
9
2
)(
08
t
TI
)(
01
t
TI
)(
06
t
TI
0
1
5
8
6
7
3
4
9
2
0
1
5
8
6
7
3
4
9
2
0
1
5
8
6
7
3
9
a. b. c. d.
Fig. 1. Shortest path trees and subtrees for an example network.
Λ (I
y
(T
x
t
)) can be available to facility node x using a
monitoring mechanism that captures the incoming and out-
going packets or, in case λ
v
is known to node v, by com-
municating these values to x (e.g., through piggybacking).
It will be assumed that each facility node x has knowledge
of Λ(I
y
(T
x
t
)) for all neighbor nodes y S
x
. This locally
available information will be utilized by the proposed service
migration policy.
A. Service Migration for a Single Facility
Consider the case of one facility located at node x at time t.
The key question for service migration is to establish whether
a cost reduction is achieved by moving the facility to node
y. Since according to Equation (1) the previous requires
global information, the main challenge is to derive a condition
that would be based on information that is locally available at
the facility node x, such as Λ(I
y
(T
x
t
)).
Let C
T
x
t
t+1
(y) denote a hypothetical cost assuming that (a) the
facility moves to node y at time t+1; and (b) the corresponding
shortest path tree over which data are forwarded towards the
facility node y (which should have been T
y
t+1
, if facility
movement x
t
y had actually taken place) remains the current
one (i.e., T
x
t
). For this hypothetical cost, let the distance
between any node v that is served by facility y over the shortest
path tree T
x
t
be denoted by d
T
x
t
(v, y) instead of d(v, y).
Equivalently, C
T
x
t
t+1
(y) =
P
vT
x
t
λ
v
d
T
x
t
(v, y). Note that in
general shortest path trees are different for different roots
(i.e., T
y
t+1
6= T
x
t
), except for the special case of topologies
with unique shortest path trees [40]. Unique shortest path tree
topologies are those for which T
t
(x) = T
t
(y), for all pairs of
nodes x, y V , at any time t. The following lemmas are the
basis for the migration policy presented later.
Lemma 1: For a single service facility in a network located
at some node x at time t and some neighbor node y S
x
,
C
T
x
t
t+1
(y) C
t+1
(y) is satisﬁed (the equality holds for unique
shortest path tree topologies). In addition, the difference be-
tween cost C
T
x
t
t+1
(y) and cost C
t
(x) is given by:
C
T
x
t
t+1
(y) C
t
(x) =
Λ
T
x
t
\ I
y
(T
x
t
)
Λ
I
y
(T
x
t
)
w(x, y).
(3)
The proof of Lemma 1 can be found in Appendix A.
The right part of Equation (3) depends on the link weight
w(x, y), the aggregate service demands that are forwarded to
node x through node y (i.e., Λ
I
y
(T
x
t
)
) and the rest of the ag-
gregate service demands that arrive through the other neighbor
nodes of x (i.e., set S
x
\{y}) plus the service demands of node
x itself (i.e.,
P
vS
x
\{y}
Λ
I
v
(T
x
t
)
+λ
x
= Λ
T
x
t
\I
y
(T
x
t
)
,
since
vS
x
\{y}
I
v
(T
x
t
){x} = T
x
t
\I
y
(T
x
t
)). As mentioned
before, both Λ
T
x
t
\ I
y
(T
x
t
)
and Λ
I
y
(T
x
t
)
are locally
available at node x (i.e., strictly local information).
In view of Lemma 1, two interesting observations can be
made regarding the difference when the facility is located at
neighbor nodes. First, the difference does not depend on the
weights of the links of the network, apart from the weight
of the link among them, i.e., w(x, y). Second, it depends
on the difference between the aggregate service demands.
Consequently, global knowledge of the network (i.e., knowl-
edge of the weights of each link and the service demands
of each node in the network) is not necessary to determine
differences in costs associated with neighboring facility nodes
and, eventually, determine the facility node that induces the
lowest cost among all neighboring nodes. Even knowledge
of w(x, y) is not necessary, as it is shown later in Theorem
1. What is actually required is information regarding the
aggregate service demands, which can be available at the
facility node.
Theorem 1: For a single service facility in a network lo-
cated at some node x at time t and some neighbor node
y S
x
, if the facility moves to node y, then cost reduction is
achieved, i.e., C
t+1
< C
t
, provided that Λ
T
x
t
\ I
y
(T
x
t
)
<
Λ
I
y
(T
x
t
)
.
The proof of Theorem 1 can be found in Appendix B.
Motivated by Theorem 1, the following migration policy
is proposed, referred to hereafter as Migration Policy S to
emphasize the use of strictly local information (i.e., S) and to
distinguish it from other migration policies proposed later in
this paper.
Deﬁnition of Migration Policy S: For a facility located at
some node x at time t, the facility is moved from node x
to some neighbor node y at time t iff Λ
T
x
t
\ I
y
(T
x
t
)
<
Λ
I
y
(T
x
t
)
.
According to Migration Policy S and in view of Theorem
1, it is easy to conclude that every movement of the facility
results in cost reduction.
B. Multiple Facilities
The following theorem shows that under Migration Policy
S, overall cost reduction is always achieved (i.e., C
t+1
< C
t
),
whatever the number of facilities in the network.
Theorem 2: In a network of more than one facilities, if
a facility located at some node x at time t moves under

5
Migration Policy S to some neighbor node y, then C
t+1
< C
t
.
The proof of Theorem 2 can be found in Appendix C.
Moving a facility under Migration Policy S and achieving
overall cost reduction does not necessarily mean that the
facility will eventually reach the optimal position (i.e., the
solution of the k-median problem) that minimizes the overall
cost. This is guaranteed (as shown next) for unique shortest
path tree topologies (e.g., trees) and for a single facility.
Theorem 3: In a network consisting of a unique shortest
path tree, a single service facility always arrives at the op-
timal location under Migration Policy S, assuming a static
environment.
The proof of Theorem 3 can be found in Appendix D.
Unique shortest path tree topologies (e.g., trees) are not
uncommon; in fact, trees are formed frequently as a result
of routing protocols in dynamic environments (e.g., mobile ad
hoc networks [6]). In addition, the presence of a single service
facility within certain network boundaries is also frequent.
Consequently, the results of Theorem 3 apply to many real
cases. Migration Policy S is also useful for environments
that do not comply with the previous conditions, since it
allows for cost reduction (even though not necessarily for cost
minimization) based on strictly local information.
V. SERVICE MIGRATION BASED ON ONE-HOP LOCAL
INFORMATION
Suppose that at time t a single facility is located at node x
and moves to neighbor node y. If the shortest path tree of root
node y is different from that of root node x (i.e., T
y
t+1
6= T
x
t
),
then this indicates that some nodes have preferred a shortcut,
i.e., a shortest path towards the new facility node y that is
shorter than that towards node x plus the weight w(x, y) (fur-
ther details about shortcuts are given next). Migration Policy
S fails to capture the potential cost reduction caused by the
aforementioned shortcuts and decide on a facility movement
that would eventually allow for further cost reduction. The
aim of this section is to overcome this limitation of Migration
Policy S using information that is available at the neighbor
nodes (one-hop) of a facility node by permitting tentative
movements of the facility to these nodes and reducing that
way the impact of the (unknown) shortcuts.
Consider the facility movement 0
t
1 depicted in Figures
1.a and 1.c. It easy to see that there exist nodes whose distance
from the facility remains the same, or increases, or decreases.
In that example, nodes 3 and 5 have chosen a shortcut to
implement the new shortest path towards the (new) facility
node (node 3 is a neighbor of the new facility node and node
5 utilizes the path through node 3, instead of the one through
node 6). Note that when the facility was located at node 0,
the existence of these shortcuts was not known to node 0; this
is basically the reason why Migration Policy S is unable to
foresee the corresponding cost reduction (it assumes that node
3 will utilize path {(3, 6), (6, 0), (0, 1)} instead of link (3, 1),
and that node 5 will utilize path {(5, 6), (6, 0), (0, 1)} instead
of path {(5, 3), (3, 1)}, which is a safe worst-case assumption).
In view of the above discussion, let Φ
x
t
y
denote the set
of nodes of T
x
t
\ I
y
(T
x
t
) utilizing a shortcut towards the new
facility node (i.e., node y at time t + 1). Basically, the reason
for some node v to utilize a shortcut is that the distance
towards the new facility node y over this shortcut (i.e., d(v, y))
is smaller than that utilizing the previous path towards the
(previous facility) node x (i.e., d(v, x)) plus the weight of
link (x, y) (i.e., w(x, y)). More formally:
Φ
x
t
y
= {v : v T
x
t
\I
y
(T
x
t
) and d(v, y) < d(v, x)+w(x, y)} .
(4)
In the previous example, Φ
0
t
1
= {3, 5}.
A. Cost Reduction and Tentative Facility Movements
Lemma 2: For a single facility in a network and facility
movement x
t
y, C
t+1
< C
t
is satisﬁed iff:
Λ
I
y
(T
x
t
)
Λ
I
x
(T
y
t+1
)
w(x, y)
>
X
vΦ
x
t
y
λ
v
d(v, y) d(v, x)
. (5)
The proof of Lemma 2 is similar to the proof of Lemma 1.
Based on the paradigm of Migration Policy S that was
proposed after Theorem 1, another policy could be proposed
based on Lemma 2, requiring knowledge of Λ
I
y
(T
x
t
)
,
Λ
I
x
(T
y
t+1
)
, w (x, y) and
P
vΦ
x
t
y
λ
v
d(v, y) d(v, x)
.
Λ
I
y
(T
x
t
)
and w(x, y) are available at time t at node x.
On the other hand, Λ
I
x
(T
y
t+1
)
, which is available at node
y at time t + 1, could be made available after a tentative
facility movement to node y and then moving back to the
previous facility node x before deciding if a (permanent)
facility movement x
t+2
y should take place. However,
P
vΦ
x
t
y
λ
v
d(v, y) d(v, x)
is not known either at node
x or node y at any time. Eventually, the condition of Lemma
2 requires information that is not locally available.
B. Service Migration in Topologies of Equal Link Weights
The main objective of the analysis next is to simplify the
right term of Lemma 2 using information that is available
at least one hop away from the facility node (i.e., term
P
vΦ
x
t
y
λ
v
d(v, y) d(v, x)
). The objective is not to
determine Φ
x
t
y
or
d(v, y) d(v, x)
, which are based on
non-local information, but rather to exploit the aforementioned
tentative facility movements in such a way that the right term
of Lemma 2 becomes obsolete (i.e., 0 ). This is possible in
topologies with equal link weights (normalized to 1 here), as
shown analytically next.
Topologies with equal link weights (e.g., w(u, v) = 1,
(u, v) E) may be considered as a worst case scenario
with respect to shortcuts (provided that the topology is not a
tree). In particular, equal link weights allow for an increased
number of alternative paths in the network, which on the event
of a facility movement are likely to be utilized (i.e., shortcuts),
resulting in Φ
x
t
y
6= , for some facility movement x
t
y.
In these topologies, Φ
x
t
y
can be analyzed further as follows.
Lemma 3: For a single facility in a network of link weights
equal to 1 and a facility movement x
t
y, for some node y

##### Citations
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Abstract: Ingredients of Locational Analysis (J. Krarup & P. Pruzan). The p-Median Problem and Generalizations (P. Mirchandani). The Uncapacitated Facility Location Problem (G. Cornuejols, et al.). Multiperiod Capacitated Location Models (S. Jacobsen). Decomposition Methods for Facility Location Problems (T. Magnanti & R. Wong). Covering Problems (A. Kolen & A. Tamir). p-Center Problems (G. Handler). Duality: Covering and Constraining p-Center Problems on Trees (B. Tansel, et al.). Locations with Spatial Interactions: The Quadratic Assignment Problem (R. Burkard). Locations with Spatial Interactions: Competitive Locations and Games (S. Hakimi). Equilibrium Analysis for Voting and Competitive Location Problems (P. Hansen, et al.). Location of Mobile Units in a Stochastic Environment (O. Berman, et al.). Index.

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TL;DR: A novel cloud computing network architecture is introduced that allows for the formulation of the optimization as an Uncapacitated Facility Location (UFL) problem, where a facility corresponds to an instantiation of a particular service (e.g. a virtual machine).
Abstract: The growth of cloud computing and the need to support the ever increasing number of applications introduces new challenges and gives rise to various optimization problems, such as calculating the number and location of virtual machines instantiating cloud services to minimize a well-defined cost function. This paper introduces a novel cloud computing network architecture that allows for the formulation of the optimization as an Uncapacitated Facility Location (UFL) problem, where a facility corresponds to an instantiation of a particular service (e.g. a virtual machine). Since UFL is not only difficult (NP-hard and requires global information), but also its centralized solution is non-scalable, the approach followed here is distributed and elastic, and relays local information to improve scalability. In particular, virtual machine replication and merging are proposed and analyzed ensuring overall cost reduction. In addition, a policy that employs virtual machine replication and merging along with migration is proposed to reduce the overall cost for using a service. The efficiency of this policy and its limitations are analyzed and discussed, with simulation results supporting the analytical findings and demonstrating a significant overall cost reduction when the proposed policy is implemented.

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TL;DR: The proposed iterative service migration algorithm, called cDSMA, is extensively evaluated over both synthetic and real-world network topologies and achieves remarkable accuracy and robustness, clearly outperforming typical local-search heuristics for service migration.
Abstract: As social networking sites provide increasingly richer context, user-centric service development is expected to explode following the example of User-Generated Content. A major challenge for this emerging paradigm is how to make these exploding in numbers, yet individually of vanishing demand, services available in a cost-effective manner; central to this task is the determination of the optimal service host location. We formulate this problem as a facility location problem and devise a distributed and highly scalable heuristic to solve it. Key to our approach is the introduction of a novel centrality metric. Wherever the service is generated, this metric helps to a) identify a small subgraph of candidate service host nodes with high service demand concentration capacity; b) project on them a reduced yet accurate view of the global demand distribution; and, ultimately, c) pave the service migration path towards the location that minimizes its aggregate access cost over the whole network. The proposed iterative service migration algorithm, called cDSMA, is extensively evaluated over both synthetic and real-world network topologies. In all cases, it achieves remarkable accuracy and robustness, clearly outperforming typical local-search heuristics for service migration. Finally, we outline a realistic cDSMA protocol implementation with complexity up to two orders of magnitude lower than that of centralized solutions.

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TL;DR: The evaluation results showed that the framework is capable of managing resources according to the requirements given by administrator, even during in the event of multiple consecutive resources failure.
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TL;DR: A distributed service migration heuristic that iteratively solves instances of the 1-median problem pushing progressively the service to more cost-effective locations and demonstrating the effectiveness of the heuristic over synthetic and real-world topologies as well as its advantages against comparable local-search-like migration schemes are proposed.
Abstract: The optimal placement of service facilities largely determines the capability of a data network to efficiently support its users' service demands. As centralized solutions over large-scale distributed environments are extremely expensive, inefficient or even infeasible, distributed approaches that rely on partial topology and demand information are the only credible approaches to the service placement problem, even at the expense of non-guaranteed optimality. In this paper, we propose a distributed service migration heuristic that iteratively solves instances of the 1-median problem pushing progressively the service to more cost-effective locations. Key to our algorithm is a traffic-aware centrality metric, called weighted conditional betweenness centrality (wCBC), that captures the ability of a node to act as service demand concentrator and is employed in both selecting the nodes and setting their weights for the 1-median problem instance. The assessment of our heuristic proceeds in two steps. First, assuming (ideal) knowledge of the invoked wCBC metric, we carry out a proof-of-concept study that demonstrates the effectiveness of the heuristic over synthetic and real-world topologies as well as its advantages against comparable local-search-like migration schemes. Next, we devise practical protocol implementations that approximate the heuristic using local measurements of transit traffic and preserve the excellent accuracy and fast convergence properties of the algorithm for different routing policies. Our solution applies to a broad range of networking scenarios, and is very relevant to the emerging trends for in-network storage and involvement of the end-user in the creation and distribution of lightweight (autonomic) service facilities.

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##### References
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Abstract: The emergence of order in natural systems is a constant source of inspiration for both physical and biological sciences. While the spatial order characterizing for example the crystals has been the basis of many advances in contemporary physics, most complex systems in nature do not offer such high degree of order. Many of these systems form complex networks whose nodes are the elements of the system and edges represent the interactions between them. Traditionally complex networks have been described by the random graph theory founded in 1959 by Paul Erdohs and Alfred Renyi. One of the defining features of random graphs is that they are statistically homogeneous, and their degree distribution (characterizing the spread in the number of edges starting from a node) is a Poisson distribution. In contrast, recent empirical studies, including the work of our group, indicate that the topology of real networks is much richer than that of random graphs. In particular, the degree distribution of real networks is a power-law, indicating a heterogeneous topology in which the majority of the nodes have a small degree, but there is a significant fraction of highly connected nodes that play an important role in the connectivity of the network. The scale-free topology of real networks has very important consequences on their functioning. For example, we have discovered that scale-free networks are extremely resilient to the random disruption of their nodes. On the other hand, the selective removal of the nodes with highest degree induces a rapid breakdown of the network to isolated subparts that cannot communicate with each other. The non-trivial scaling of the degree distribution of real networks is also an indication of their assembly and evolution. Indeed, our modeling studies have shown us that there are general principles governing the evolution of networks. Most networks start from a small seed and grow by the addition of new nodes which attach to the nodes already in the system. This process obeys preferential attachment: the new nodes are more likely to connect to nodes with already high degree. We have proposed a simple model based on these two principles wich was able to reproduce the power-law degree distribution of real networks. Perhaps even more importantly, this model paved the way to a new paradigm of network modeling, trying to capture the evolution of networks, not just their static topology.

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### "Scalable service migration in auton..." refers methods in this paper

• ...A simulation tool was written in programming language C for creating network topologies (trees, grids, geometric random graphs [41], Erdős-Rényi random graphs [42], and Albert-Barabási graphs [43]) and implementing the migrat ion policies....

[...]

• ...In the seque l, migration is studied considering morealistic topologieslike geometric random graphs (suitable for studying mobile ad hoc networks [41]), Erdős-Rényi random graphs (suitable for comparison reasons [42]), and Albert-Barabási graphs (po werlaw graphs that model many modern networks including the Internet [43]), in dynamically changing environments....

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TL;DR: Undergraduate and graduate classes in computer networks and wireless communications; undergraduate classes in discrete mathematics, data structures, operating systems and programming languages.
Abstract: Undergraduate and graduate classes in computer networks and wireless communications; undergraduate classes in discrete mathematics, data structures, operating systems and programming languages. Also give lectures to both undergraduate-and graduate-level network classes and mentor undergraduate and graduate students for class projects.

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### "Scalable service migration in auton..." refers background in this paper

• ...Between a given facility node and any other node in the network, a shortest path route is assumed to be established b y the underlying employed routing protocol, [40]....

[...]

• ..., T y t+1 6= T x t ), except for the special case of topologies with unique shortest path trees [40]....

[...]

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TL;DR: This chapter discusses probabilistic ingredients, the largest component for a binomial process, and connectedivity and the number of components in a graph-like model.
Abstract: 1. Introduction 2. Probabilistic ingredients 3. Subgraph and component counts 4. Typical vertex degrees 5. Geometrical ingredients 6. Maximum degree, cliques and colourings 7. Minimum degree: laws of large numbers 8. Minimum degree: convergence in distribution 9. Percolative ingredients 10. Percolation and the largest component 11. The largest component for a binomial process 12. Ordering and partitioning problems 13. Connectivity and the number of components References Index

2,161 citations

### "Scalable service migration in auton..." refers methods in this paper

• ...A simulation tool was written in programming language C for creating network topologies (trees, grids, geometric random graphs [41], Erdős-Rényi random graphs [42], and Albert-Barabási graphs [43]) and implementing the migrat ion policies....

[...]

• ...In the seque l, migration is studied considering morealistic topologieslike geometric random graphs (suitable for studying mobile ad hoc networks [41]), Erdős-Rényi random graphs (suitable for comparison reasons [42]), and Albert-Barabási graphs (po werlaw graphs that model many modern networks including the Internet [43]), in dynamically changing environments....

[...]

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TL;DR: An algorithm is presented which finds a p-median of a tree (for $p > 1$) in time $O(n^2 \cdot p^2 )$.
Abstract: It is shown that the problem of finding a p-median of a network is an $NP$-hard problem even when the network has a simple structure (e.g., planar graph of maximum vertex degree 3). However, results leading to efficient algorithms are presented when the network is a tree: In particular, we first show that a 1-median of a tree is identical to its w-centroid, and obtain Goldman’s $O(n)$ algorithm for finding a 1-median of a tree out of more general considerations. Then, we present an algorithm which finds a p-median of a tree (for $p > 1$) in time $O(n^2 \cdot p^2 )$.

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### "Scalable service migration in auton..." refers background in this paper

• ..., thek-median problem) is anNP -hard problem for general graphs [2]....

[...]

• ...The problem of determining the optimal service placement has been studied in the past in areas such as transportation and supply networks [1], and has been approached through the formulation and solution of large optimization problem s (NP -hard) requiringglobal knowledge , as for instance is the case with thek-median problem [2]....

[...]