Proportionally Fair Allocation of EndtoEnd Bandwidth in STDMA Wireless Networks

1
Proportionally Fair Allocation of End-to-End
Bandwidth in STDMA Wireless Networks

• Pablo Soldati, Björn Johansson and Mikael
  Johansson
• School of Electrical Engineering
• Royal Institute of Technology, KTH
• Edited and Presented by Tae-Seok Kim

2
Outline

• Resource allocation in spatial reuse TDMA
  networks
• Problem formulation
• Cross decomposition approach
• Distributed transmission scheduling
• Examples
• Conclusions

3
Resource allocation in wireless networks

• Scheduling method in wireless networks may be
  classified in
• Static (maintain schedule)
• Dynamic (opportunistic scheduling)
• We focus on static scheduling
• Key contributions distributed solution to
  jointly optimization of
• end-to-end communication rate
• transmission scheduling
• power control

4
Spatial reuse TDMA

• Spatial reuse TDMA (STDMA) is an extension of the
  TDMA where the network capacity is increased by
  spatially reuse of the time slots.

Note geographical separation of radio units is
not enough to guarantee interference-free
schedule
5
Network model

• Fixed network topology with nodes labeled n1N
  and wireless links labeled l1L
• Link quality measured by SINR
• Each link is viewed as a single-user Gaussian
  channel subject to the power constraint
  with capacity
• Rate allocation links offer a single rate
  when the SINR at the receiver exceeds a certain
  threshold

6
Network model

• The rate allocation leads to a finite number of
  achievable link-rate vectors
• Time-sharing between the achievable link-rate
  vectors we may achieve the following polyhedral
  rate region
• is the fraction of schedule where rate
  vector is activated.

7
Outline

• Resource allocation in spatial reuse TDMA
  networks
• Problem formulation
• Cross decomposition approach
• Distributed transmission scheduling
• Examples
• Conclusions

8
Problem formulation

• Inspired by mathematical model of TCP/IP by, e.g.
  Low and Lapsley (1999) and F.P. Kelly et al.
  (1998)
• The optimal network operation solves the utility
  maximization problem
• Utility of the pair p to communicate
  at the end-to-end rate (increasing and
  strictly concave).
• Routing matrix with
  entries

9
Problem formulation
end-to-end rate selection

• Distributed solution can be derived via dual
  decomposition.
• Dual function
• Where
• ( total congestion price
  along route p)
• The dual function can be evaluated by letting
  sources optimize their rates individually based
  on the total congestion price.
• Dual problem
• can be solved by the projected gradient iteration

Source side
Link side
congestion price updates
10
Problem formulation

• For wireless networks the previous problem may be
  rewritten as
• Links capacities are not fixed
• Centralized solution to this problem as been
  proposed by Johansson and Xiao 1 combining dual
  decomposition techniques with a column generation
  procedure
• 1 M. Johansson and L. Xiao Cross-Layer
  Optimization of Wireless Networks Using
  Nonlinear Column Generation IEEE Trans. on
  Wireless Comm. Feb. 2006

11
Centralized solution

• Full master problem
• Restricted master problem consider a subset
  of extreme points of where
• Optimization variables s and a

12
Centralized solution

• Lagrange Duality
• Lower bound given the optimal solution
• Upper bound

13
Outline

• Resource allocation in spatial reuse TDMA
  networks
• Problem formulation
• Cross decomposition approach
• Distributed transmission scheduling
• Examples
• Conclusions

14
Cross-decomposition approach

• The optimization problem may be rewritten as
• where
• For a fixed link capacity vector c the function
  can be evaluated via the dual decomposition
  method.
• Schedule update

15
Cross-decomposition approach

• Theorem Let be the optimal value of the
  centralized cross-layer design problem. The cross
  decomposition algorithm converges in the sense
  that
• the sample 10-node/46-link network

16
Cross-decomposition approach

• Distributed construction of a multiple time-slot
  schedule
• Key points
• Start with a schedule with a finite
• length and equal slot sizes
• Run the dual decomposition
• until the convergence
• Get the new links congestion levels
• Determine a new transmission group
• Augment the schedule
• Repeat the optimization with
• the new schedule

17
Outline

• Resource allocation in spatial reuse TDMA
  networks
• Problem formulation
• Cross decomposition approach
• Distributed transmission scheduling
• Examples
• Conclusions

18
Distributed Transmission Scheduling

• 2-Step procedure for distributed solution of the
  scheduling sub-problem
• Find a candidate transmission group that
  maximizes the objective function while satisfying
  the primary constraints
• Determine a feasible transmission group running
  distributed power control algorithms in order to
  satisfy the secondary interference constraints

At most one incoming or outgoing link can
be activated at each node
19
Candidate transmission group

• Greedy scheme links with the highest priority in
  a 2-hop neighborhood assigns itself membership of
  the candidate set
• Nodes exchange information about their link
  congestion levels.
• Each node n maintains the maximum congestion
  level on its incoming and outgoing links
• If corresponds to
  then node n should
  allow the link to join the candidate group.
• Example candidates
• 1, 3, 8

20
Candidate transmission group

• Performance analysis for primary interference
• Optimal candidates optimal power control
• Greedy candidates optimal power control

21
Feasible transmission group

• Links in the feasible transmission group must
  satisfy
• Links in the candidate transmission group contend
  for transmission rights running distributed
  algorithms in order to satisfy the SINR
  requirement.
• Some links might leave the candidate set

Candidate set 1, 3, 8
Feasible set 1, 8
22
Feasible transmission group

• We evaluate two distributed approaches
• Distributed link reservation scheme via RTS-CTS
  packet exchange (cf. Muqattash and Krunz, 2003)
• Fields of maximum allowed power and interference
  margin
• Overhearing and decoding RTS-CTS packets in a
  neighborhood
• Distributed power control with active link
  protection (DPC-ALP) algorithm (cf. Bambos et
  al., 2000)
• Note many other distributed protocol could be
  applied

23
Feasible transmission group

• DPC-ALP algorithm
• Exploits local measurements of SINR at the
  receiver and runs iteratively over a sequence of
  time slots
• (? gt1 is a control parameter)
• Links change status from inactive (I) to active
  (A) when their measured SINR exceeds the
  threshold.
• Inactive nodes that consistently fail to observe
  any SINR improvement enters a voluntary drop-out
  phase and go silent.

24
Outline

• Resource allocation in spatial reuse TDMA
  networks
• Problem formulation
• Cross decomposition approach
• Distributed transmission scheduling
• Examples
• Conclusions

25
Examples

• Performance RTS-CTS Vs DPC-ALP

26
Examples

• Comparison with alternative approaches
• Graph-based schemes

27
Conclusions

• Joint congestion control and resource allocation
  in STDMA wireless networks
• Utility maximization subject to link rate
  constraints
• Transmission scheduling
• Power allocation
• Decomposition method for convex optimization
• Convergence
• Distributed solution for TCP/IP fashion on a fast
  time scale
• Incremental updates of transmission scheduling on
  a slower time scale
• Two-step procedure for finding the schedule
  updates
• Two schemes for distributed channel reservation
  and power control under realistic interference
  model