Content uploaded by Anita Raja
Author content
All content in this area was uploaded by Anita Raja on Feb 16, 2015
Content may be subject to copyright.
DLB-SDPOP: A Multiagent Pseudo-tree Repair
Algorithm for Load Balancing in WLANs
Shanjun Cheng, Anita Raja
Department of Software and Information Systems
The University of North Carolina at Charlotte
Charlotte, NC 28223
{scheng6, anraja}@uncc.edu
Jiang(Linda) Xie, Ivan Howitt
Department of Electrical and Computer Engineering
The University of North Carolina at Charlotte
Charlotte, NC 28223
{jxie1, ilhowitt}@uncc.edu
Abstract
The traffic load of wireless local area networks (WLANs) is often dis-
tributed unevenly among access points. In addition, interference from col-
located wireless devices operating in the same unlicensed frequency band
may cause WLANs to become unstable, leading to temporary failures of
access points. This paper addresses the questions of how to dynamically
balance the load and how to quickly respond to instability in WLANs1. We
present a new decentralized multi-agent load balancing framework for WLANs
that uses DLB-SDPOP, a distributed optimization algorithm, to dynamically
load balance the WLAN. The algorithm implements a less expensive pseudo-
tree repair mechanism instead of complete pseudo-tree reconstruction when
instability problems occur. It also leverages an efficient communication
mechanism to control communication overhead in heavily loaded situations.
We empirically show that DLB-SDPOP improves WLAN load balancing
performance significantly.
1. Introduction
Wireless local area networks (WLANs) have become one
of the most popular wireless technologies due to their low
cost, simple installation, and great capability to support high
speed data communications. However, research studies on
operational WLANs have shown that the traffic load is often
distributed unevenly among access points (APs) [1]. Also, in a
dynamic operational environment like a WLAN, interference
may significantly impact the signal quality, and hence, impact
the decision-making related to network management. The
ability of the system to handle such dynamic changes and
quickly move from a previous stable state to a new optimal
stable state is a critical issue.
We show that the WLAN load balancing issue can be
handled using a multiagent system (MAS) [2]. An agent
is located inside each AP within the WLAN and interacts
with agents within its neighborhood. The neighborhood of
a certain AP is the set of those APs with whom it has
frequent interactions. These interactions include sharing of
data and negotiating about resource assignments. Individual
agents act as coordinators and cooperate with agents in their
neighborhood to take care of resource management across
the WLAN. We assume that all the agents in this domain
are cooperative. The dynamics in WLAN includes agent (AP)
failure [3] and the movement of mobile stations (MSs).
1. This work is supported in part by the US National Science Foundation
(NSF) under Grant No.CNS-0855200, CNS-0915599 and CNS-0953644.
In this paper, we map the WLAN load balancing issue to
a distributed constraint optimization problem (DCOP) [4] and
define a decentralized framework that leverages DLB-SDPOP,
a distributed optimization algorithm with a self-stabilizing
mechanism, to solve the problem.
We discuss this framework by first describing DLB-DPOP,
a Dynamic Load Balancing-Distributed Pseudo-tree Optimiza-
tion Procedure. We designed DLB-DPOP to be an improve-
ment of the DPOP algorithm [5] in that it reduces the total
message size by adding a communication filtering mechanism
in its utility propagation phase. We then extend DLB-DPOP to
Dynamic Load Balancing-Self-stabilizing Distributed Pseudo-
tree Optimization Procedure (DLB-SDPOP) by augmenting it
with a self stabilization mechanism. The main data structure in
the DPOP family of algorithms is the pseudo-tree. A pseudo-
tree of a graph Gis a rooted tree with the same vertices as G
and has the property that adjacent vertices from the original
graph fall in the same branch of the tree [5]. The key feature of
DLB-SDPOP is that once it finds initial assignments to reach
steady state, it uses self-stabilizing [6] pseudo-tree localized
repair mechanisms instead of complete pseudotree reconstruc-
tion in order to load balance the WLAN dynamically. Self
stabilization in distributed systems [7] is the ability of a system
responding to transient failures, eventually reaching a legal
state, and maintaining it afterwards. Self-stabilizing systems
are particularly fault tolerant and able to cope with dynamic
environments [8]. DLB-SDPOP pseudo-tree localized repair
mechanism helps to maintain the current tree structure and
avoid the retransmission of redundant messages. In this paper,
we empirically show that DLB-SDPOP is a scalable approach
and significantly improves WLAN load balancing performance
in dynamic environments when compared to other state-of-the-
art algorithms.
The following is an example scenario to motivate our
approach. Fig 1 is a WLAN scenario with 8 APs and several
MSs. All the APs are static. Some MSs are static and some
are moving (as shown in Fig 1). Each AP has a coverage
radius of 80 meters (shown as the dashed circles) which means
the effective distance to associate a MS is 80 meters. If two
APs have overlapping coverage areas, they are defined as
neighboring APs (e.g., AP5and AP4are neighboring APs,
while AP5and AP3are not.). APs can only hand off MSs to
their neighboring APs.
Fig. 1. 8-APs
WLAN scenario.
Fig. 2. Pseudo-tree
for the WLAN sce-
nario in Fig 1.
Fig 2 is the whole pseudo-tree representing the global view
for AP3. All the neighboring APs in Fig 1 are connected by
tree-edges (solid lines) or back-edges (dashed lines) so that
they can send messages to each other.
Fig. 3. Perturbation
at AP1.
Fig. 4. Repaired
pseudo-tree where
AP1−AP4edge is
deleted.
Suppose at time tk, a new source of perturbation causes AP1
to cease working temporarily (Fig 3). We apply our pseudo-
tree localized repair algorithm as shown in Fig 4 where AP2
and AP1have switched parent/child roles. This allows for AP1
to be removed from the tree without detrimentally affecting the
other nodes.
Fig. 5. Perturbation moves into AP6.
Now suppose at time tk+1, the previous perturbation moves
out of AP1and interferes with the functionality of AP6
(Fig 5). AP1is active again and AP6ceases working. AP6
would be deleted resulting in Fig 6. AP1is then added back
Fig. 6. Repaired
Pseudo-tree (AP6
is deleted).
Fig. 7. Repaired
Pseudo-tree
(Adding back
AP1).
to the tree and the final repaired pseudo-tree is depicted in
Fig 7.
The rest of the paper is organized as follows. First, we
discuss the mapping of the WLAN load balancing problem
as a DCOP. Our DLB-SDPOP algorithm and related work are
presented and the performance results of the algorithms are
discussed later, followed by the conclusions and future work.
2. WLAN load balancing
The goal of the WLAN load balancing problem is to
dynamically assess the associations of MSs at time tk, find the
optimal set of MSs under each AP based on the estimates of
the states of the MSs at time tk+1, and change the associations
of specific MSs from one AP to another neighboring AP when
required and complete handoffs by tk+1 (the term time refers
to discrete time in this paper). We define tdelay =tk+1 −tk, as
the maximum time required to handoff one MS from one AP to
a neighboring AP. We formulate the load balancing task as an
optimization problem. The optimization satisfies the following
criteria:
Criterion I: The received signal strength (RSS) of each MS
associated with an AP is above the minimum received power
threshold γ. In this paper, γis set to be -82 dBm.
Criterion II: Maximize the minimum received power by
each MS in order to minimize the likelihood of packet loss.
We implement this criterion as:
max
i,j min(Rj
i(tk)) (1)
where Rj
i(tk)denotes the reward of MSibeing handed off to
APjat time tk.
Criterion III: Distribute the load amongst viable APs in
order to increase fairness as well as the overall network-wide
resource utilization. In this paper, we assume each MS has the
same load to each AP and implement this criterion as:
min{max X
j,l,j6=l
|MSN um
j(tk)−MSN um
l(tk)|} (2)
where MSN um
j(tk)denotes the number of MSs assigned to
APjat time tk.
Criterion II and III may not be satisfied simultaneously. In
our work, load balancing is more critical to maintain capacity
availability across the network. We give Criterion III higher
priority than Criterion II.
In our MAS based decentralized approach for WLAN load
balancing, a distributed load balancing (DLB) agent is located
inside each AP. Each DLB agent cooperates with other DLB
agents in its neighborhood to ensure load balancing across the
entire WLAN. A DLB agent’s neighborhood consists of those
DLB agents with whom it has frequent interactions. We map
the WLAN load balancing problem to a DCOP [4] model in
the following way. The model is a tuple hA,X,D,Ri, where
•A={A1, ..., An}is the set of agents interested in the
optimal solution; in the WLAN context, each access point
APjis assigned a DLB agent.
•X={X1, ..., Xm}is the set of variables; in the WLAN
context, each APjhas a variable Xifor MSi, which
represents the new associated AP after a handoff.
•D={d1, ..., dm}is a set of domains for the variable
set X, where each domain djis a set of APs in APj’s
neighborhood.
•R={r1, ..., rp}is a set of relations where a relation
riis a utility function that provides a measure of the
value associated with a given combination of variables.
In WLAN, Rrepresents the objective functions, which
are the three criteria defined above for load balancing.
Our goal is to find a complete instantiation X∗for the set X
that maximizes the sum of the utilities of individual relations
in the multi-agent system, in other words, to find which AP
each MS should be associated with so that all the criteria for
load balancing can be achieved.
DLB agent interaction is initiated by two event triggers: (1)
a handoff event and (2) the need for load balancing among
APs. A handoff event occurs when the RSS of one MS begins
to drop below the threshold. When a AP’s DLB agent Ai
recognizes that it is over-loaded, some MSs associated with
this AP need to be handed off to its neighboring APs so as
to increase fairness as well as the overall resource utilization
of the WLAN. Upon receiving a handoff event trigger, Ai
initiates agent interactions within its neighborhood by sending
request messages to its neighbors. The request messages
contain the information about which MSs should be handed
off and the deadlines of such handoffs. Aihas the pseudo-tree
which is made up of its neighborhood and itself. Similarly,
the neighboring agents also have partial views of the global
pseudo-tree and send back response messages to announce the
possible new assignments based on local evaluations of the
three aforementioned criteria. The response messages contain
the information about the possible time durations in which
handoffs can be initiated. Each DLB agent calculates the utility
values of its own assignment choices, and runs the DLB-
SDPOP algorithm using the local utility values as initial inputs.
After the optimal assignment is found, the handoff decisions
are implemented by the appropriate target DLB agents.
3. Solution
In this section, we discuss our approach to handle the
dynamics of the load balancing problem by first describing
DLB-DPOP, a variation of the DPOP algorithm. It regenerates
a new pseudo-tree that excludes the fault nodes every time a
perturbation occurs.
3.1. DLB-DPOP Algorithm
The DLB-DPOP includes 3 phases:
Phase 1-DFS (Depth-First Search) Traversal: DLB-
DPOP performs a distributed depth-first traversal of the net-
work to establish a pseudo-tree structure [9]. This is similar
to the pseudo-tree creation phase in DPOP.
We have defined two heuristic functions here: Num(v)
and Low(v). For each node v, we call its preorder num-
ber Num(v).Low(v)is defined as:
Low(v) = min{N um(v),min{N um(w),∀back−edge(v, w)},
min{Low(w),∀tree −edge(v, w)}} (3)
Low(v)is the minimum value of the preorder number
of node v, the lowest preorder number of node wfrom all
the back-edges connecting node vto node wand the lowest
Low(w)from all the tree-edges connecting node vto node w.
This definition ensures that all the nodes in the same pseudo-
tree “loop” (A pseudo-tree “loop” is formed by tree-edges and
back-edges as a close circle.) are assigned the same value of
Low(v). They help us efficiently represent and identify the
triggered AP when a handoff trigger happens, thus obviating
the need to traverse the DFS tree in search of the triggered AP.
Building the DFS tree takes O(|E|+|V|)time, where |E|and
|V|are the number of edges and vertices in the pseudo-tree,
respectively.
Phase 2-Utility Propagation: DLB-DPOP propagates util-
ity messages (called U T IL messages) which contain utility
vectors sent bottom-up along the pseudo-tree starting from
the leaves, only through tree edges. This step too is similar to
DPOP, except that (a) the values propagated in the U T I L
messages of DLB-DPOP are the reward values and (b) a
communication filtering mechanism is used to reduce the size
of U T IL messages. We define Uj
i(tk)as the normalized signal
strength above the threshold of MSifrom APjper unit time:
Uj
i(tk) = Pte
tsPi,j
ratio
te−ts
(4)
where Pi,j
ratio =Pi,j
receive −Pthres ,Pi,j
receive denotes the
received power (dBm) of MSifrom APjat a certain time
unit, Pthres denotes the threshold value (If Pi,j
receive is lower
than Pthres,M Sishould be handed off to another powerful
AP so as to remain working. In our WLAN problem, Pthres
is set to be -82 dBm.), Pi,j
ratio measures how much Pi,j
receive
is above Pthres at a certain time unit. tsand tedenote the
earliest and last time at which the signal goes above Pthres.
Cj
i(tk)is defined as the estimated handoff cost function:
Cj
i(tk) = Ch
te−tsif te−ts> tdelay
Cmax otherwise (5)
Chand Cmax are constant values (dBm) representing the
handoff cost. If the time duration of good signal is not long
enough (te−ts≤tdelay), the handoff decision would be
unnecessary. We use the function Rj
i(tk)to denote the reward
of MSibeing handed off to APjat time tk, which can be
expressed as:
Rj
i(tk) = Uj
i(tk)−Cj
i(tk)(6)
The reward value Rj
i(tk)is used to reduce the possibility of
reaching myopic solutions. The handoff decision is viable only
if the value of Rj
i(tk)is positive.
In DPOP, the child node sends its U T I L message as a
hypercube to its parent node. The largest message is expo-
nential in the induced width along the particular pseudo-tree
chosen [5]. Sometimes the hypercube transmitted contains a
large portion of uninformative reward values (In our problem,
the value of Rj
i(tk)is zero or negative). This always happens
when a MS is moving away and continually loses signals
from its nearby APs. This MS has few potential APs to be
handed off to which results in a large number of uninformative
reward values in the hypercube. Instead of sending the whole
hypercube, the child agent of DLB-DPOP only sends the set
of reward tuples to its parent agent that contain the positive
reward values. The filtering process continues bottom-up until
the root agent receives its UT I L message. This reduces
the total message size substantially. For example, Consider
a very dense network with pseudo-tree induced width 8,
neighborhood size 9 and each reward value in the hypercube
uses 1byte of storage. So the total size of the hypercube
is calculated as: (1 byte)∗98≈41 M B. In a heavily
loaded situation, most APs would have reached their maximum
capacities and will be incapable of handling any new MSs,
meaning their information does not have to be transmitted in
the hypercube. Suppose only 2of the 9neighbors can take
on additional MSs. Each reward tuple contains 8bytes for
information about domain combinations and 1byte for the
reward value. Using the filtering strategy, the total size is
calculated as: (9 bytes)∗28≈2KB. The filtering strategy
filters huge amounts of useless information in this scenario.
Phase 3-Optimal VALUE Propagation: The optimal value
assignments are then propagated top-down from the root
node [5]. The root agent chooses the optimal assignment and
sends V ALUE message to its children agents (a V ALU E
message represents this assignment). Each child agent deter-
mines its optimal assignment based on the messages from
Phase 2 and the V ALUE message and repeats the propagation
process. When all the nodes finish choosing an assignment, the
algorithm is completed.
The domain size (dom) and induced width (w) of the
pseudo-tree increase with the propagation of the neighborhood
size of the whole WLAN. The increase of neighborhood size
would bring in a larger set of involved APs and a local
view of pseudo-tree for each AP with more nodes, leading
to a larger domain size and an equal or larger induced
width separately. In the worst case (All the APs are in the
same neighborhood and no fault nodes exist.), the complexity
converges at O(|dom∗|w∗
), where dom∗=the set of all APs
in the WLAN, w∗=induced width of the pseudo-tree of the
whole WLAN.
Changes in the pseudo-tree structure will adversely affect
the performance of DLB-DPOP since some of the U T IL
messages will have to be recomputed and retransmitted. This
is sometimes wasteful, since some of the faults have limited,
localized effects, that do not need to propagate through the
whole problem. It is therefore desirable to maintain as much
of the current DFS tree as possible.
3.2. DLB-SDPOP Algorithm
We now describe DLB-SDPOP, an extended version of
DLB-DPOP, designed to respond to the dynamic failures
of APs in WLANs quickly, repair the original pseudo-tree
efficiently, and make the optimal hand-off decisions for load
balancing in the whole networks.
Algorithm 1 DLB-SDPOP
1: DLB-SDPOP (X,D,R)
Each agent Xiexecutes:
Phase 1:DFS Traversal
2: root ←electedleader
3: assignNum(root)
4: assignLow(root)
5: afterwards, Xiknows Num (Xi)and Low (Xi).
Phase 2:Self-stabilizing Utility Propagation
6: store all new U T IL messages Xi, U T ILj
i
7: if any perturbation is detected in WLAN then
8: case 1: agent Xistops working
9: for all constraints Rk
iof Xiin the pseudo-tree do
10: deleteEdgeXi, Rk
i
11: delete the single agent Xifrom the pseudo-tree
12: case 2: agent Xiresumes working
13: connect Xito existing agent Xj
14: send MessageXi, U T I Lj
i
15: for all original constraints Rk
iof Xido
16: addEdgeXi, Rk
i
Phase 3:Optimal Value Propagation
17: Xi←v∗
i=choose Optimal (agent view)
18: send V ALUEl
ito all Xl∈C(Xi)
END ALGORITHM
DLB-SDPOP extends DLB-DPOP by incorporating a self-
stabilizing mechanism. It dynamically modifies/repairs the af-
fected nodes in the original pseudo-tree retaining the topology
and states of unaffected nodes when inconsistency is detected
(e.g., one or several APs stop working for a moment or
previously fault APs resume functionalities). We define the
following variables: UT I Lj
iis the U T IL message that Xi
sends to Xj;Rk
iis the constraint relationship between Xi
and Xk;V ALUEk
iis the V ALUE message that Xisends to
Xk;Sep(Xi)is the set of ancestors of Xiin the pseudo-
tree; P(Xi)is the parent of Xi(the single node higher
in the hierarchy of the pseudo-tree that is connected to Xi
directly through a tree-edge.); C(Xi)is the children of Xi
(the set of nodes lower in the pseudo-tree that are connected
to Xidirectly through tree-edges.); P P (Xi)is the pseudo-
parents of Xi(the set of nodes higher in the pseudo-tree
that are connected to Xidirectly through back-edges.); and
P C(Xi)is the pseudo-children of Xi(the set of nodes lower
in the hierarchy of the pseudo-tree that are connected to Xi
directly through back-edges.). The DLB-SDPOP algorithm
is described in Algorithm 1. It starts with a DFS traversal.
After this phase, each node is assigned Num() and Low().
The Self-stabilizing Utility Propagation process (line 6−16,
Algorithm 1) is initialized and then run continuously. DLB-
SDPOP ends with the propagation of optimal values to each
node.
Main functions in DLB-SDPOP are described in Pro-
cedure 1. The functions assignNum(vertex)and assign-
Low(vertex)respectively generate N um(v)and Low(v)
for each node vin the pseudo-tree. In Phase 2 of DLB-
SDPOP, deleting interfered agents and adding back previ-
ous agents cleared of interference are two main concerns.
deleteEdge(Xi, Rk
i)deletes a relation/constraint Rk
idepend-
ing on the type of the edge (whether they are tree-edge or
back-edge), addEdge(Xi, Rk
i)adds a new relation/constraint
Rk
ibetween two existing agents based on their relative po-
sition (whether they are ancestor-descendant or siblings).
initiateU T IL(Xj, P (Xj)) runs the communication filtering
mechanism and sends the set of reward tuples from Xjto
P(Xj).
In order to prove that DLB-SDPOP algorithm is complete,
we have to prove its correctness and liveness. We use the
two heuristics Num(v)and Low(v)to generate a pseudo-
tree. Num(v)helps to sort the nodes and Low(v)is used to
distinguish all the nodes to different groups that each group
forms a pseudo-tree “loop”. Adding each node according to
Num(v)and Low(v)leads to a pseudo-tree that guarantees
completeness as described below. We extend the optimality
proof for DPOP [5] to DLB-SDPOP. A pseudo-tree has no cy-
cles. This implies that all messages come from unrelated parts
of the tree and these messages are hence accurate evaluations
of the utility that can be obtained by the sub-trees belonging
to each sender node for each value of the node. The upper
bound of the utility obtained from the whole problem at a node
can be accurately computed by summing up the messages, for
each possible value of the node. The value that produces the
maximum utility is then assigned to the node. This shows
correctness of the algorithm. In addition, both the absence
of cycles in the pseudo-tree and the fact that all the leaves
initiate the message propagation guarantee that each node will
eventually receive m−1messages (with mas the number of
Procedure 1 Main Functions in DLB-SDPOP
1: assignNum(vertex)
2: vertex.num ←counter++
3: for all Xido
4: if vertex.relation[i] == true then
5: ifXihas not been visited then
6: Xi.parent ←vertex
7: assignNum(Xi)
8: return true
9: assignLow(vertex)
10: vertex.low ←vertex.num
11: for all Xido
12: if vertex.relation[i] == true then
13: if Xi.num > vertex.num then
14: assignLow(Xi)
15: vertex.low ←min (vertex.low, Xi.low)
16: elseifvertex.parent 6=Xi
17: vertex.low ←min (vertex.low, Xi.num)
18: return true
19: deleteEdgeXi, Rk
i
20: if back-edge(i, k) == true then
21: remove Rk
ifrom the pseudo-tree
22: for all lower agents Xjinvolved in Rk
ido
23: initiateU T IL (Xj, P (Xj))
24: if tree-edge(i, k) == true then
25: let Xk←P(Xi)
26: if (∀Xl∈C(Xi), Sep(Xl) = {Xk})
27: && (Sep(Xi) = {APk})then
28: Xi←root
29: initiateV ALUE (Xi, subtree (Xi))
30: initiateU T IL (Xk, P (Xk)) ←previous UT I Ls
31: else
32: let Xm←highest agent in Sep(Xi)
33: sendV ALUE (Xm, C (Xm)∪P C (Xm))
34: for all Xs∈subtree (Xm)in right-hand side do
35: traversal ←U T ILC(Xs)
s
36: switchRole ←valueChange(Xs, C (Xs))
37: addEdgeXi, Rk
i
38: if Xiand Xkare in an ancestor-descendant relation
39: then
40: connect Xi, Xkas a back-edge ( Suppose Xiis
41: descendant)
42: initiateU T IL (Xi, P (Xi)∪P P (Xi))
43: if Xiand Xkare siblings
44: let Xk←P(Xi)
45: let Xl←lowest common ancestor of Xiand Xk
46: for all agents Xson the tree-path from Xito Xl
47: switchRole ←valueChange(Xs, C (Xs))
48: P C (Xl)←C(Xl)
neighbors) and hence it will be able to send its mth message.
This also means that each node will receive a message from
its last neighbor, thus terminating the algorithm. This proves
liveness. Hence, DLB-SDPOP algorithm is complete.
We now describe DLB-SDPOP’s self-stabilizing pseudo-
tree repair mechanism in the context of the motivating example
presented in Section 1.
Given that WLAN scenario in Fig 1, Fig 2 is the whole
pseudo-tree generated by Phase 1 (line 2−5, Algorithm 1)
as a global view for AP3. If we assume that a new source
of perturbation causes AP1to cease working at time tk
as described in the example, then AP1should be deleted
(line 8−11, Algorithm 1) from the pseudo-tree in Fig 2. AP1
has two neighboring agents: AP4and AP2. The algorithm
first considers removing tree-edge AP1−AP4(line 24, Pro-
cedure 1). Removing the edge AP1−AP4does not disconnect
the problem, but disrupts the structure of the pseudo-tree
(line 31 −36, Procedure 1). The nearest ancestor of AP1
and AP4in the pseudo-tree is AP3(line 32, Procedure 1).
Thus the pseudo-tree repair begins from AP3and proceeds
as follows (line 34 −36, Procedure 1): AP3→AP2→
AP1→AP4→AP3. It should be noted that AP2and
AP1have switched parent/child roles. The UT I L message
between AP2and AP1has to be recomputed as well as
that between AP4and AP3, while other U T IL messages
can be reused. The algorithm considers removing tree-edge
AP1−AP2(line 26−30, Procedure 1). AP1becomes a single
node (line 28, Procedure 1) that can be deleted directly, and
AP2begins a new UT I L propagation by re-computing its
U T IL message which does not include the previous message
that sent from the sub-tree of AP1(line 30, Procedure 1). The
result is depicted in Fig 4.
At time tk+1, the perturbation affects AP6and AP1resumes
its activity (Fig 5). The algorithm deletes AP6from the
pseudo-trees by doing the following: First, the back-edge
AP6−AP4is removed (line 20−21, Procedure 1). Then tree-
edge AP6−AP5is removed just as AP1−AP2was removed
in previous scenario. AP1is added back to the pseudo-
tree implying it is connected as a child of AP2(line 13,
Algorithm 1). Fig 6 shows the repair results so far. AP1begins
propagation by sending AP2U T IL messages that considering
all the influence of tree-edges and back-edges between them
(line 14, Algorithm 1, in this case, only consider the tree-edge
AP1−AP2). AP1and AP4are siblings (they lie in different
branches of the pseudo-tree [9]). Adding AP1−AP4violates
the required property that agents in different branches of the
pseudo-tree should be disconnected [9]. So the pseudo-tree
is not valid any more and has to be repaired (line 43 −48,
Procedure 1). The final repaired pseudo-tree is depicted in
Fig 7. AP4becomes AP1’s parent and AP1and AP2switch
their roles.
4. Related Work
Distributed algorithms such as DSA/DBA [10], ADOPT [4]
and DPOP [5] have previously been proposed to solve dis-
tributed constraint optimization (DCOP) problems. These al-
gorithms have been applied to problems such as graph col-
oring and meeting scheduling. However, there are only few
attempts to address real world scenarios using this formalism,
mainly because of the complexity associated with these al-
gorithms [11]. Atlas [12] presented a complete mapping to
DCOP for large-scale team coordination problems that offers
fast convergence to high quality solutions. However, their
algorithm did not address the issue of handling agent failures.
Choxi and Modi [13] proposed an approach to reposition
wireless routers to maximize signal strength in the network.
In their work, small robots act as wireless routers and can
reposition themselves. In our WLAN problem, APs are fixed
and can not be repositioned. DSA and DBA do not guarantee
completeness. We show in this paper that the DLB-SDPOP
algorithm is complete. ADOPT requires polynomial memory,
but it may produce a very large number of small messages, re-
sulting in large communication overheads which would occupy
the bandwidth used by MSs. Among these algorithms, DPOP
is a complete algorithm based on dynamic programming. It
is a utility-propagation method that extends tree propagation
algorithms to work on arbitrary topologies using a pseudo-tree
structure. It can generate only a linear number of messages.
SDPOP [8] is the first self stabilization mechanism for
multi-agent combinatorial optimization. SDPOP has the com-
plexity of O(domw), where dom bounds the domain size and
w=induced width along the particular pseudo-tree chosen.
The induced width is the maximum number of parents of any
node in the induced graph [9]. SDPOP has been implemented
to solve the meeting scheduling problem with up to 10% of
the agents having simultaneous perturbations.
Our algorithm DLB-SDPOP is different from SDPOP in
that (a) We introduce a self-stabilizing pseudo-tree repair
mechanism to handle the perturbations in WLAN efficiently.
(b) It can solve the complicated load balancing situations
with up to 50% agents simultaneously failing. (c) It has the
complexity of O(domw), where dom bounds the domain size
and w=the maximum induced width of all the localized
pseudo-trees involved in the perturbations in WLAN.
5. Empirical Evaluation
All experiments in this section were performed using the
Matlab optimization toolbox for simulation, running in an
Intel Centrino Core Duo with 1.6GHz, 1G RAM memory,
under Windows XP. Values reported here are averages over
at least 10 repetitions of the simulation. We assume that all
the APs are cooperative. We consider the dynamics of both
agent (AP) failure and movement of MSs. In the experiments,
our baselines are ADOPT, DSA and DPOP, the existing state
of the arts. We show the computational advantage of DLB-
SDPOP over others with a low overhead.
5.1. Metrics for Evaluation
We evaluate the performance of ADOPT, DSA, DPOP and
DLB-SDPOP based on the following variables:
•#of APs - the total number of APs in the scenario.
•Neighborhood size -the number of neighboring APs of
each associated AP.
•#-changes - the number of both pseudo-tree nodes which
are added and deleted simultaneously.
•#of MS/AP - the average number of MSs associated with
each AP in the scenario.
We compare the algorithms by measuring the following met-
rics when perturbation occurs: DLB-Value,Messages,Message
size and Solving time.DLB-Value measures the Criterion III in
Section 2. The lower DLB-Value is, the better the load balanc-
ing performance is. Messages is defined as the total number
of messages exchanged between DLB agents to respond to
the perturbations and stabilize in a state corresponding to the
optimal solution. Message size is defined as the total amount
of information bytes exchanged among DLB agents to solve
a problem scenario. Solving time is defined as the time (sec.)
it takes for the DLB agent architecture to recover from the
perturbations and stabilize in an optimal solution.
We make the assumptions in our experiments. When per-
turbation occurs (DLB agents stop working or previous fault
DLB agents resume functionalities), the DFS trees in ADOPT,
DSA and DPOP which represent the communication topology
among DLB agents would be regenerated according to the
changes. In DSA, the probability p[10] which controls how
frequently neighboring agents changing values is set to 0.6(If
pis too high, the phase transition occurs that would decrease
the performance sharply).
5.2. Scenario Setup
For the experiments reported here, we use four different
scenarios or grids with different sizes (3×3,4×4,6×6and
9×9). Thus the variable #of APs is set to 9, 16, 36, and
81. Each grid in our simulation is wrapped around, i.e., if a
MS moves out of one boundary of the simulation scenario,
it moves into the scenario through the opposite boundary.
The distance between APs is 80 meters. The link quality is
based on the expected received power over a transmission
distance of dij (tk)between APjand MSiat time tkgiven
by PR(dij (tk)) = PT−(20 log10 fc+ 10κlog10 (dij (tk)) −
28)(dBm)[14]. In our simulation, 10% of MSs belonging to
each AP are randomly chosen to move in a random direction
with a constant speed 0.5 m/s (MSs mobility percentage
= 10%). We conduct simulation experiments with increasing
Neighborhood size of 5, 9, 16 and 25. We set the #-changes to
be 1, 2, 4, 5, 8, 16 and 36 to see the trends of all the algorithms
(We set the perturbation areas in the scenario randomly to
influence the functionalities of APs). Ch= 30dBm and
Cmax = 25dBm. Fig 8 shows the simulation scenario.
5.3. Discussion
Table 1 provides the performance comparison of the algo-
rithms. ADOPT spends substantial time and messages to reach
a solution. This is mainly because ADOPT is more susceptible
Fig. 8. Simulation Scenario.
#of APs Algorithm Solving time(sec.) Messages Message size
9
ADOPT 0.480 ±0.016 306 ±57 16.5K
DSA 0.037 ±0.023 129 ±20 3.3K
DPOP 0.053 ±0.008 38 ±5 2.4K
DLB-SDPOP 0.024 ±0.006 72 ±9 1.7K
16
ADOPT 1.341 ±0.302 2393 ±430 1665.9K
DSA 0.984 ±0.153 473 ±87 123.6K
DPOP 0.109 ±0.012 96 ±27 366.2K
DLB-SDPOP 0.028 ±0.009 147 ±41 58.0K
36
ADOPT 14.456 ±2.482 9531 ±967 3.5M
DSA 2.846 ±0.634 2420 ±592 2.8M
DPOP 0.458 ±0.037 162 ±42 8.3M
DLB-SDPOP 0.041 ±0.013 253 ±72 248.9K
81
ADOPT 273.833 ±58.484386246 ±4958 611.0M
DSA 19.473 ±3.298 49857 ±6447 93.8M
DPOP 8.522 ±0.122 384 ±83 294.4M
DLB-SDPOP 0.849 ±0.105 540 ±127 1.8M
TABLE 1. DLB-SDPOP vs. other DCOP algorithms
(Neighborhood size= 5,#of MS/AP= 5,#-changes= 1)
to the variations in the construction of the constraints. DLB-
SDPOP performs significantly better than others on Solving
time and Message size (compared with DPOP, the pvalues
from t-tests are 0.00478, 0.00359, 0.0370 and 0.000846 on
Solving time and 0.0221, 0.00351, 0.00613 and 0.000289 on
Message size). This shows the effectiveness of pseudo-tree
repair and communication filtering in DLB-SDPOP. DLB-
SDPOP consumes more Messages only than DPOP which
mainly occur in the V ALUE and U T IL initiation processes
(Phase 2, Algorithm 1). Given the importance of fast response
time to perturbations in real-time WLAN scenarios, DLB-
SDPOP is preferable.
In Fig 9, DLB-SDPOP performs a little worse than ADOPT
and DPOP on load balancing, but much better than DSA when
#of APs increases to 36 and 81. However, DLB-SDPOP
significantly outperforms (p < 0.05) the others on Solving time
on most cases (DPOP spends less time than DLB-SDPOP on
small-scale scenarios (#of AP s = 9 and 16)). Solving time
of DLB-SDPOP decreases when #of APs increases to 36
and 81. This is mainly because the percentage of #-changes
in the whole scenario becomes much smaller which helps
the self-stabilizing algorithm to discover the fault points and
recover from the inconsistent state. DLB-SDPOP significantly
outperforms (p < 0.05) ADOPT and DSA on Messages.
DPOP uses fewer messages than DLB-SDPOP. The increase in
(a) DLB-Value (b) Solving time (log scale)
(c) Messages (log scale) (d) Message size (Kbytes) (log
scale)
Fig. 9. DLB-SDPOP vs. other DCOP algorithms (Neigh-
borhood size=5, #of MS/AP=5, #-changes=4), for #of
APs to be 9,16, 36 and 81.
messages comes from the pseudo-tree repair process of DLB-
SDPOP. DLB-SDPOP consumes substantially (p < 0.05) less
Message size than the others when the scenario scales up.
(a) DLB-Value (b) Solving time (log scale)
Fig. 10. DLB-SDPOP vs. other DCOP algorithms (#of
APs=36, Neighborhood size=9, #of MS/AP=5), for #-
changes to be 1,4, 8 and 16.
When the #-changes parameter is tweaked as in Fig 10,
DLB-SDPOP takes extremely short time (compared with other
algorithms) in the price of not very large decrease of load
balancing. In real-time WLAN environments, recovering from
perturbations and making handoff decisions quickly are given
higher priority. So DLB-SDPOP outperforms the other algo-
rithms in response to dynamism in WLAN environments.
6. Conclusion and Future Work
We have developed a multi-agent approach for decentralized
load balancing in WLANs. This approach uses DLB-SDPOP,
a constraint optimization algorithm to determine the optimal
allocation of MSs under each AP. DLB-SDPOP dynamically
repairs the affected nodes in the original pseudo-tree retaining
the topology and states of unaffected nodes when incon-
sistency is detected. Empirical evaluation of DLB-SDPOP
shows that the self-stabilizing, scalable mechanism leverages
pseudo-tree localized repair efficiently and handles up to 50%
perturbations in real-time WLAN scenarios. It outperforms
ADOPT, DSA and DPOP as the problem scales (up to 81 APs).
The experiments also show that the communication filtering
mechanism improves communication efficiency especially in
heavily loaded scenarios. As future work, we plan to extend
our algorithm to handle situations where the interference
prediction is probabilistic in keeping with real-world scenarios.
References
[1] A. Balachandran, G. M. Voelker, P. Bahl, and P. V. Rangan, “Character-
izing user behavior and network performance in a public wireless LAN,”
in Proceedings of ACM SIGMETRICS, 2002, pp. 195–205.
[2] S. Cheng, A. Raja, L. Xie, and I. Howitt, “A distributed constraint
optimization algorithm for dynamic load balancing in WLANs,” in Pro-
ceedings of Eleventh International Workshop on Distributed Constraint
Reasoning to be held in conjunction with IJCAI 2009, pp. 31–45.
[3] M. N. Sahoo, P. M. Khilar, and B. Majhi, “A redundant neighborhood
approach to tolerate access point failure in IEEE 802.11 WLAN,” in
Fourth International Conference on Industrial & Information Systems,
December 2009.
[4] P. J. Modi, W. Shen, M. Tambe, and M. Yokoo, “ADOPT: Asynchronous
distributed constraint optimization with quality guarantees,” in AI Jour-
nal, vol. 161, 2005, pp. 149–180.
[5] A. Petcu and B. Faltings, “DPOP: A scalable method for multiagent con-
straint optimization,” in Proceedings of International Joint Conference
on Artificial Intelligence (IJCAI), 2005, pp. 266–271.
[6] S. Dolev, Self-stabilization. MIT Press, 2000.
[7] E. W. Dijkstra, “Self-stabilizing systems in spite of distributed control,”
Commun. ACM, vol. 17, no. 11, pp. 643–644, 1974.
[8] A. Petcu and B. Faltings, “S-DPOP: Superstabilizing, fault-containing
multiagent combinatorial optimization,” in Proceedings of the National
Conference on Artificial Intelligence (AAAI-05), Pittsburgh, Pennsylva-
nia, July 2005, pp. 449–454.
[9] R. Dechter, “Constraint processing.” Morgan Kaufmann, 2003.
[10] W. Zhang and Z. Xing, “Distributed breakout vs. distributed stochas-
tic: A comparative evaluation on scan scheduling,” in Proceedings of
AAMAS-02 Workshop on Distributed Constraint Reasoning, Bologna,
Italy, 2002, pp. 192–201.
[11] R. Junges and A. L. C. Bazzan, “Evaluating the performance of DCOP
algorithms in a real world, dynamic problem,” in AAMAS (2), 2008, pp.
599–606.
[12] J. Atlas and K. Decker, “Coordination of agent schedules using dis-
tributed neighbor exchange,” in Proceedings of International Joint
Conference on Autonomous Agents and Multi Agent Systems (AAMAS),
Toronto, Canada, May 2010.
[13] H. Choxi and P. J. Modi, “A distributed constraint optimization approach
to wireless network optimization,” in Proceedings of AAAI07 Workshop
on Configuration, July 2007.
[14] J. Xie, I. Howitt, and A. Raja, “Framework for decentralized wireless
LAN resource management,” in Emerging Wireless LANs, Wireless
PANs, and Wireless MANs. Wiley, 2008.
