EFFICIENT RESOURCE ALLOCATION FOR

WIRELESS MULTICASTDe-Nian Young

Ming-Syan ChenIEEE Transactions on Mobile Computing

Slide content thanks in part to Yu-Hsun Chen, University of Taiwan

Introduction

Environment: Wireless Multicast Networks Heterogeneous Devices and Cells Differing “Costs” per Cell

Problem: Given a Heterogeneous Network, Select the

Lowest Cost Distribution Tree NOT STATED: From the perspective of the

network owner!

Heterogeneous Environment

Heterogeneous Network: Theory

Current mobile devices have multiple radios Can connect via:

Wi-Fi WiMax 3G EVDO Satellite Bluetooth (presumably tethered)

Heterogeneous Network: Theory cont’d

Devices (mobile hosts) can choose which radio and which “cell” to connect to with that radio to get Mobile-IP multicast messages

Different cells have different costs to both the distributor and mobile host

By aggregating individual mobile hosts appropriately, the provider can reduce overall bandwidth costs for multicasting

Concept: Shortest Path Tree

SPT Easy to build

(Dijkstra’s algorithm)

Not necessarily the most efficient in bandwidth usage

Concept: Minimum Cost Tree

MCT Finds the minimum

cost tree for a given graph

NP-hard!

Cell and Technology Selection Problem

CTSP – reformulation of Minimum Cost tree problem.

Contributions: For each technology – Clusters mobile hosts

and reduces the number of cells in the SPT. Takes into account bandwidth costs of links

(weighted edges). Transparent to the IP multicast protocols Supports dynamic group membership

(necessary for moving hosts)

CTSP – Assumptions

All wireless cells are multicast capable Paths from root to host are pre-given by

the multicast protocol Unwritten:

The root bears the bandwidth costs (questionable in practice)

The individual nodes have multiple cells and multiple technologies to choose from (again, questionable AND irrelevant – different technologies are the same as different cells when weighted!)

Notation

Integer Linear Programming

ILP, cont’d

Objective function for ILP formulation

Constraints

,

, ,minu v

c c u v u vc C e E

b b

,

,

, ,

, ,

1,

, ,

, ,

, , {0,1}

m

m cc C

m c c m

c u v u v c

m c c u v

m M

m M c C

c C e E

Minimum bandwidth

Each mobile host selects one cell

A cell is used in the shortest path tree if it is selected by any mobile host

A link is used in the shortest path tree if it is on the path from any selected cell

to the root of the tree

LAGRANGE Algorithm

Modification to ILP Relaxes a constraint to reduce complexity

(relaxation just sound better than cheating by approximation)

LAGRANGE

Relax the second constraint ( ) in the ILP New objective function

Lagrange multiplier : the cost of cell c for mobile host m

Constraints

,m c c

,

,

, , , ,

, , , , ,:

min ( )

min

u v m

m m u v

c c u v u v m c m c cc C e E m M c C

m c m c c m c c u v u vm M c C c C m c C e E

b b

b b

,m c

,

, ,

1,

, ,m

m cc C

c u v u v c

m M

c C e E

LAGRANGE - Properties

Properties For any feasible solution to the LRP that

contradicts the relaxed constraints ( ), the objective value is larger

Any feasible solution to CTSP is a feasible solution to the LRP

When adopting the optimal solution to CTSP, [the objective value of LRP] <= [the objective value of CTSP]

The objective value of the optimal solution to the LRP provides a lower bound to CTSP

,m c c

LAGRANGE – Subproblem 1

Objective function of the subproblem 1

Constraint

The runtime is The cost for cell c is stored in each

mobile host m

, ,minm

m c m cm M c C

, 1,m

m cc C

m M

Find the cell with the minimum costfor each mobile host m

( )O M C

,m c

LAGRANGE – Subproblem 2

Objective Function

Constraint

,

, , ,:

minm u v

c m c c u v u vc C m c C e E

b b

, ,, ,c u v u v cc C e E

Minimize the net cost of all selectedCells in the shortest path tree

LAGRANGE – Subproblem 2, cont’d

To find the minimum net cost of the whole shortest path tree, we consider each link in the bottom-up manner

: the minimum net cost of the subtree that includes link and the subtree rooted at v,u v

,u ve

, ,

,

, , ,:

min 0,u v u v

u v m

u v c u v m cm c C

b b

,

, , ,:

min 0,u v

u v u v v ww e E

b

LAGRANGE – Subproblem 2, cont’d

All cells in the subtree corresponding to a link are not selected if net cost is not negative

Each candidate cell c is selected in the second subproblem if the net cost of every link in the shortest path from c to the root of the tree is negative

,u ve,u v

,u v,u ve

LAGRANGE - Iterations

The selected cells may not be feasible to CTSP Each mobile host is not guaranteed to be covered

by a cell that is selected in the second subproblem Each member m in the LAGRANGE algorithm

selects the cell c according to the cost in the first subproblem

Adjust the cost iteratively with the subgradient algorithm and the solutions to the two subproblems of the LRP : the objective function of the LRP The subgradient of the LRP:

,m c

( )W

,,

( )( ) m c c

m c

WW

LAGRANGE - Iterations

The subgradient indicates the direction of adjusting to find the better feasible solution to CTSP : increase : decrease

The second subproblem tends to Select the cells cover more mobile hosts to save

wireless bandwidth Select the cells such that the shortest path from

the cells to the root share more common wireline links

,m c

, 0m c c

, 0m c c ,m c

,m c

Protocol Design

A distributed protocol based on the LAGRANGE algorithm Data tree: the shortest path tree for data

delivery Control tree: to solve the second subproblem

in a distributed manner Initially the control tree spans every candidate cell Incrementally prune the control tree to reduce the protocol

overhead

Each router and base station in the control tree maintains a node agent and cell agent

State

Each node agent stores the following states Multicast group address The address of the parent node agent in the control tree The bandwidth cost of the link with the parent node

agent The address of the child agent and a Join timer

Each cell agent stores the following states The bandwidth cost of the cell Control Flag (whether the cell is selected) Data Flag (whether the base station is in the data tree) The address of the mobile host The cost of the cell for the mobile host (Lagrange

multiplier) Join timer

Control Messages

Join Mobile hosts or node agents send Join to join the

control tree Join_Ack

Confirm the Join message Contain the Data Flag and the cost of the cell for

the mobile host (sent by cell agent) Leave

Sent by mobile hosts, cell agents, and node agents Request, Reply, and Inform

Update the cost of each cell in a distributed manner

Operations 1

Join a multicast group Mobile host sends a Join message to the cell

agent of each cell that covers the mobile host Handover to a new cell

Mobile host sends a Join message to the new cell and a Leave message to the original cell

Leave the multicast group Mobile host sends a Leave message to cell

agent

Operations 2

Update the cost of each cell Root periodically sends a Request message Cell agent first calculates the net cost → Set Control

Flag → send Reply message Node agent first calculates the net cost →

send Reply message to parent node agent If net cost = 0, send Inform

message to child node agent

Inform

Operations 3

Prune the control tree Cell agent or node agent obtains a zero net

cost for a period of time A node agent leaves the control tree if it

receives a Leave message from every child agent

Results for Small Wireless Networks

25 km × 25 km, 36 hexagon cells

Simulation results of small wireless networks.(a) total bandwidth cost. (b) number of cells in the tree.

Results for Large Wireless Networks 1

Simulation results of large wireless networks(a) original scenario (b) larger transmission range

Results for Large Wireless Networks 2

Simulation results of large wireless networks.(c) (d) zero bandwidth cost for each link.

Transient Behavior of the LAGRANGE Algorithm

Transient behavior of the LAGRANGE algorithm with different mobility(a) Probability = 0 percent (b) 0.1 percent (c) 0.5 percent (d) 2 percent

Conclusions

LAGRANGE provides a solution to the lowest cost spanning tree problem

The solution uses an iterative approximation approach

Problems: It really doesn’t address heterogeneous

networks The comparison choices in the experimental

results are dubious It assumes the root bears the cost (not likely)

or that it can be somehow transferred to the client

Details of the algorithm 1

assign a unit cost to each cell for each member

find the solution to the first subproblem

initial topology every cell is selected in the first subproblem

Details of the algorithm 2

, , 1u vc u vb b

, ,

,

, , ,:

min 0,u v u v

u v m

u v c u v m cm c C

b b

1+1-2=0

1+1-1=1

1+1-3=-1

find the solution to the second subproblem

Details of the algorithm 3

1+(-1)=0

no cell is selected in the second subproblem

Details of the algorithm 4

,

7, 2.8

1.4m c

optimal

threshold

Details of the algorithm 5

after the second iteration

Details of the algorithm 6

H3 handovers from C4 to C2

H5 moves out C4

H7 leaves the multicast group

Details of the algorithm 7

adjustment after the mobility