QoS and Fairness Constrained Convex Optimization of Resource Allocation for Wireless Cellular and Ad

1
QoS and Fairness Constrained Convex Optimization
of Resource Allocation for Wireless Cellular and
Ad Hoc Networks

• David Julian, Mung Chiang, Daniel ONelill and
  Stemphen Boyd.
• Jo Woon Chong
• Communication Networks Research Lab.
• Dept. of EECS, KAIST
• 2002. 12. 11.

2
Contents

• Introduction
• Convex Optimization
• Wireless Cellular Networks
• Throughput
• Delay
• Wireless Ad Hoc Networks
• Throughput
• Delay
• Efficiency
• Simulation Results
• Conclusion

3
Introduction

• QoS is an important and commercial issue
• Different requirements for different services
• Not satisfied with best effort transmission
• QoS in a wireless network
• A difficult problem
• Physical channel is time varying and unreliable
• A computationally efficient tool for QoS is
  needed
• Convex Optimization
• Related with Power control !

4
Introduction

• Power control In a wireless network
• To control interference
• Indirectly control the QoS seen by users on the
  network
• SIR (Signal to Interference Ratio)
• Used to capture interference
• Co-channel interference
• Adjacent channel interference
• In this paper to characterize the QoS parameter
• Throughput of a particular link

5
Introduction

• QoS Problems to Solve
• In Wireless Cellular Network
• Determining feasibility of a set of SIR
  requirements
• Maximizing SIR for a particular class of users
  with lower bounds on the QoS of all other users
• Satisfying queuing delay requirements for users
  in various QoS class

6
Introduction

• In Ad Hoc Networks
• Finding the optimum power control to maximize
  overall system throughput consistent with QoS
  guarantees in a fading environment.
• Determining feasibility of a set of service level
  agreements (SLA) under network resource
  constraints.
• Solving for the minimum total transmission delay
  of the most time sensitive class of traffic by
  optimizing over powers, capacities, and SLA terms
• Maximizing the unused capacity of the network

7
Introduction

• Fairness Problems to Solve
• Proportional fairness
• Minmax fairness
• A joint optimization of the fairness parameters
  and the QoS criteria
• Other topics
• Admission control and pricing problem
• A computationally efficient Heuristics for
  optimization

8
Contents

• Introduction
• Convex Optimization
• Wireless Cellular Networks
• Throughput
• Delay
• Wireless Ad Hoc Networks
• Throughput
• Delay
• Efficiency
• Simulation Results
• Conclusion

9
Convex Optimization

• Optimal solution for nonlinear problems
• QoS and fairness problems are nonlinear.
• To solve efficiently
• Convex optimization
• Minimizing a convex objective function over
  convex constraint sets
• Ex. ) geometric program
• Primal dual interior point method

10
Convex Optimization

• Geometric program
• Focus on monomial and posynomial functions
• Definitions
• Monomial is a function ,where the
  domain contains all real vectors with
  non-negative components
• A posynomial is a sum of monomials

11
Convex Optimization

• Geometric program is an optimization problem with
  the following form
• Minimize
• Subject to
• Where and are posynomials and are
  monomials
• This is not a convex optimization problem
  
• A change of variables is needed
• ,

12
Convex Optimization

• Minimize
• Subject to
• Where are convex functions and are
  affine functions
• We have a convex optimization problem

13
Contents

• Introduction
• Convex Optimization
• Wireless Cellular Networks
• Throughput
• Delay
• Wireless Ad Hoc Networks
• Throughput
• Delay
• Efficiency
• Simulation Results
• Conclusion

14
Wireless Cellular Network

• Contents
• Problem formulations
• Interpretations of the QoS Constrained Power
  Control
• Proportional and Minmax Fairness Extensions
• SIR optimization simulation
• Admission control and Pricing

15
Wireless Cellular Network

• Problem formulation
• Propagation model
• received power
• transmitted power
• reference distance for the antenna
  far-field
• propagation path length
• path loss exponent
• normalization constant

16
Wireless Cellular Network

• SIR for link
• spreading factor in CDMA
• effect of normalization factor, the
  effect of beamforming, and other factors

17
Wireless Cellular Network

• Constraints Description
• Interference due to users, including base station
  and mobiles, in index set must be smaller
  than some positive constant because their
  assigned QoS values are relatively low.
• Interference due to users in index set has
  to be smaller than the received signal power for
  some mobile k so as to achieve a required SIR
• The received signal power for some mobile k needs
  to be exactly equal to a positive constant
• As in the special case of the classical power
  control scheme to solve the near-far problem in
  CDMA, the received signal power for one mobile
  needs to be equal to that of another mobile

18
Wireless Cellular Network

• Formulations
• SIR constrained optimization of power control
• To maximize SIR for a particular user under QoS
  constraints for other users in a cellular network
  
• A convex optimization problem
• If theres no objective function, then the above
  formulation is a SIR requirement feasibility
  problem

19
Wireless Cellular Network

• SIR constrained optimization for minimum power
• The minimum power vector under the QoS
  constraints can be determined
• A weighted sum or powers, or the maximum user
  power can be minimized.
• SIR constrained with proportional fairness
• SIR constrained with maxmin fairness

20
Wireless Cellular Network

• Admission Control and Pricing
• Spot pricing
• To charge more to users who consume more of the
  total system user capacity.
• Method
• The price could then be set as an linear function
  of resource reduction in system
• A user could experience different spot pricing
  at different times depending on the existing load
  on the system when the user sought to access the
  network.

21
Wireless Cellular Network

• Queuing delay is particularly important for
  bursty digital data.
• Example
• M/M/1 queue
• Link transmission rate
• Service rate
• SIR should be larger than

22
Wireless Cellular Network

• M/M/1 queue
• Average queuing delay D
• Link transmission rate
• Service rate
• QoS agreement
• Average delay bound and average maximum arrival
  rate
• This bound can be met by constraining the SIR on
  this link to exceed a minimum threshold, so that
  link transmission rate is larger than

Wireless Cellular Network
23
Contents

• Introduction
• Convex Optimization
• Wireless Cellular Networks
• Throughput
• Delay
• Wireless Ad Hoc Networks
• Throughput
• Delay
• Efficiency
• Simulation Results
• Conclusion

24
Wireless Ad Hoc Network

• Contents
• Multi-hop network model and Rayleigh fading
• Outage probability and system throughput
• Throughput optimization
• Admission control and pricing
• Pricing simulation

25
Wireless Ad Hoc Network

• Similar to wireless cellular network
• Except for Multi-hop
• Multi-hop network model and Rayleigh fading
• G path gain
• F fading factor
• P transmission Power
• Ex) G_ij j th transmitter to I th receiver
• Outage probability and system throughput

26
Wireless Ad Hoc Network

• Outage probability
• Aggregate data rate for system

27
Wireless Ad Hoc Network

• Throughput optimization
• Optimize power for throughput maximization
• Constraints Description (in order)
• The data rates demanded by existing system users
• Outage probability demanded by users using single
  hop
• Outage probability demanded by users using
  multi-hop
• regulatory or system limitation on transmitted
  power

28
Wireless Ad Hoc Network

• Admission Control and Pricing Pricing
  Simulation
• A user is admissible if a feasible solution of
  the problem exists.
• Consideration of multi-hop

29
Wireless Ad Hoc Network

• Throughput optimization
• Weighted Joint capacity and Delay Minimization
• Constraints Description (in order)
• Link capacity constraint
• Delay guarantee constraint
• Delivery prob. constraint
• Guaranteed data rate to each class of traffic

30
Wireless Ad Hoc Network

• Throughput optimization
• Weighted Joint capacity and Delay Minimization
• paramters
• K_j set of traffic using link j
• J_k set of links traversed by QoS class k
• n_k number of packets dynamically admitted in
  the kth class of traffic
• p_j prob. that this link will be maintained
• C_j C_j packets per sec
• b_k bandwidth
• d_k delay guarantee SLA

31
Contents

• Introduction
• Convex Optimization
• Wireless Cellular Networks
• Throughput
• Delay
• Wireless Ad Hoc Networks
• Throughput
• Delay
• Efficiency
• Simulation Results
• Conclusion

32
Wireless Cellular Network

• SIR optimization simulation
• Environment
• Five users in Formulation 1
• The five users are spaced at distance d of 1, 5,
  10, 15, and 20 units from the base station.
• Power drop off factor
• Noise power 0.5uW
• Max. power constraint 0.5W
• CDMA with 10

33
Simulation Results

• Threshold SIR (x-axis)
• Optimized SIR (y-axis)
• In high threshold SIR
• In moderated threshold SIR
• In low threshold SIR

34
Simulation Results

• 1st class(data)
• Path ABCD
• Rate 50 packets/s
• Max. delay 0.2
• 2nd class(data)
• Path DFEA
• Rate 50 packets/s
• Max. delay 0.2
• 3rd class(voice)
• Path ABFD
• Rate 250 packets/s

35
Simulation Results

• Minimizing a weighted sum of Voice traffic delay
  and the total capacity used.
• Capacity increase -gt delay decrease
• Capacity decrease
• -gt delay increase, and at some point
• Saturated!!!

36
Simulation Results

• Heuristic method gives similar optimal solution
  and has more computational efficiency
• Heuristic 1
• Heuristic 2
• Iterative method in getting outage probability

37
Contents

• Introduction
• Convex Optimization
• Wireless Cellular Networks
• Throughput
• Delay
• Wireless Ad Hoc Networks
• Throughput
• Delay
• Efficiency
• Simulation Results
• Conclusion

38
Conclusion

• By using convex optimization, the following
  problems solved efficiently.
• Various QoS provisioning problem
• Fairness problem
• Admission control and pricing problems
• 2 efficient heuristics for optimization