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Channel Assignment for Maximum Throughput in Multi-Channel Access Point Networks

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Channel Assignment for Maximum Throughput in Multi-Channel Access Point Networks Xiang Luo, Raj Iyengar and Koushik Kar Rensselaer Polytechnic Institute – PowerPoint PPT presentation

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Title: Channel Assignment for Maximum Throughput in Multi-Channel Access Point Networks


1
Channel Assignment for Maximum Throughput
in Multi-Channel Access Point Networks
  • Xiang Luo, Raj Iyengar and Koushik Kar
  • Rensselaer Polytechnic Institute
  • WCNC 2007

1
1
2
Outline
  • Introduction
  • System Model
  • Throughput Analysis in the High SINR Regime
  • Throughput Analysis in the Low SINR Regime
  • Performance Analysis
  • Conclusion

2
2
3
Introduction
  • Future generation wireless systems are likely to
    provide user with simultaneous access to multiple
    channels
  • These channels could be a consequence of dynamic
    spectrum allocation and deallocation
  • In such system, a multiple channel model is a
    useful abstraction to study allocation problems
  • This paper
  • Consider the uplink channel assignment problem
    for a multi-channel access point system
  • Develop solutions that maximize the overall
    system throughput

4
Optimal Channel Assignment and Power Allocation
Problem
  • The optimal channel assignment problem is a
    challenging problem
  • For any given channel allocation, a user splits
    its total power across all channels allocated to
    it so as to maximize the overall user throughput
  • The optimum power allocation for a user
    corresponds to a "water-?lling" type solution
  • This results in the user throughputs being
    complex non-linear functions of the channel
    allocations
  • We develop solutions result in a performance that
    is close to optimal

5
System Model
  • Our system consists of a set of L users sharing a
    set of M channels to communicate with an access
    point (AP)
  • Each user is capable of using multiple channels
    simultaneously
  • But a single channel cannot be used
    simultaneously by multiple users
  • Time is slotted and focus on the channel
    allocation problem across users for a given time
    slot
  • channel conditions or user population do not
    change over the duration of a time slot

6
Poly-matching
  • A valid assignment of channels to users
    corresponds to a one-to-many mapping from users
    to channels
  • We refer to such an assignment as a poly-matching
    in the user-channel bipartite graph

7
Problem Formulation (1)
  • The throughput of user i on a channel j is
  • Bj and ? constants
  • nij The noise power seen by user i on channel j
  • pij The transmission signal power corresponding
    to user i on channel j

8
Problem Formulation (2)
  • The throughput maximization problem for the
    entire system can be posed as
  • F the set of all poly-matchings in the
    user-channel graph
  • For a given poly-matching f, the above problem
    reduces to the optimal power allocation problem
    for each user
  • whose solution corresponds to a water-?lling
    across the different channels assigned to the user

9
Classical Water-filling Allocation
10
Naive Solution
  • The problem corresponds to a joint channel and
    power allocation problem
  • It requires us to find the channel assignment
    (poly-matching) that will yield the best system
    throughput under optimal power allocations for
    that channel assignment
  • A naive approach
  • Enumerate all poly-matchings
  • Compute the attainable throughput for a
    poly-matching by running the water-filling
    algorithm
  • Pick the poly-matching that yields the maximum
    throughput value
  • Our goal is to obtain optimal or near-optimal
    channel assignments in a computationally
    efficient manner

11
Throughput Analysis in the High SINR Regime (1)
  • We analyze the throughput attained by a user i in
    the high SINR regime
  • Let fi j (i, j) ? F denote the set of
    channels assigned to user i, and ki fi
  • In the high SINR regime, Pi gtgt nij ?j ? fi
  • Water-filling solution
  • Summing over all the ki channels, we obtain
  • where Pi is the aggregate transmission power of
    user i, and Ni, the aggregate noise power of user
    i, is defined as

12
Throughput Analysis in the High SINR Regime (2)
  • The throughput attained by user i
  • where the approximation comes from the fact that
    in the high SINR regime, Pi gtgt Ni

13
Incremental Utility
  • Consider the incremental utility of allocating
    channel j to user i, when k - 1 channels have
    already been allocated to it
  • The incremental utility expression does not
    depend on the exact set of channels, but only on
    the size of that set (k)
  • This allows us to set up graph formulation of the
    throughput maximization problem in the high SINR
    regime

(
)
14
Constructed Bipartite Graph (1)
  • The L nodes representing the users are split up
    into M sub-nodes
  • The channels are represented separately using M
    nodes, as usual
  • All possible edges between the user sub-nodes and
    channels are drawn, with edge weights computed
    using (12)
  • (i, j, k) denotes the edge between the kth
    sub-node of user i and the channel j
  • A matching in the constructed bipartite graph
    corresponds to a poly-matching in the original
    graph

15
Constructed Bipartite Graph (2)
  • The edge-weights exhibit a decreasing property in
    k
  • i.e., aijk gt aij(k1) for any k 1
  • The decreasing property of the edge-weights imply
    that a maximum weight matching will prefer edges
    that correspond to a lower k, for the same i and
    j
  • Thus in a maximum weight matching, for any user
    i, there will be a ki such that
  • sub-nodes 1, ..., ki, will be matched
  • sub-nodes ki 1, ...,M, would not be matched
  • It can be extended further to show that a maximum
    weight matching maximizes the sum of user
    throughputs
  • The complexity of this approach is O(L3M3) using
    the classical Hungarian algorithm 8

8 H. W. Kuhn, The Hungarian Method for the
assignment problem, Naval Research Logistic
Quarterly, 283-97, 1955.
16
High-SINR-Optimal (HSO) Algorithm
17
Throughput Analysis in the Low SINR Regime
  • In the low SINR regime, we approximate the
    objective function as
  • using the approximation log(1 x) x when 0 lt x
    ltlt 1
  • If all nij values are distinct, then for small
    enough SINR, each user will allocate all its
    power in a single channel
  • The one with the smallest nij among all channels
    assigned to the user
  • The channel assignment policy in the low SINR
    regime corresponds to a maximum weight matching
    in the complete bipartite graph of users and
    channels, with edge-weights

18
Low-SINR-Optimal (LSO) Algorithm
19
Simulation Setting
  • Comparison
  • Incremental Max-Throughput (IMT) Heuristic
  • Assign channels (to users) one by one, with the
    user chosen such that the assignment yields the
    maximum additional throughput across all users
  • Incremental SINR-Balancing (ISB) Heuristic
  • Assign channels (to users) one by one, with the
    user chosen such that the ratio of the total
    power and the total noise is balanced across all
    users, as much as possible
  • Parameter
  • vnij from Gaussian distribution N(0, s2)
  • the maximum power Pi is chosen from U(0.5, 1.5)
  • Performance Ratio
  • The ratio of the average throughput attained by
    an algorithm/heuristic and the maximum throughput
    attainable

20
Performance Ratio
21
Conclusion
  • Consider the impact of the channel and power
    allocation across a set of users to maximize sum
    throughput across all users
  • Analyze the system in the two extreme SINR
    regimes (very high and very low SINR)
  • Show how the optimal solutions can be obtained in
    these regimes in a computationally efficient
    manner
  • Demonstrate that the best of the optimal
    solutions obtained for the two extremes show
    excellent performance over the entire SINR range

22
Max-Weight Matching (1)
  • A Perfect Matching is an M in which every vertex
    is adjacent to some edge in M
  • A vertex labeling is a function l V ? R
  • A feasible labeling is one such that
  • l(x) l(y) w(x, y), ?x ? X, y ? Y
  • The Equality Graph (with respect to l) is G (V,
    El) where
  • El (x, y) l(x)l(y) w(x, y)

23
Max-Weight Matching (2)
  • Theorem Kuhn-Munkres
  • If l is feasible and M is a Perfect matching in
    El then M is a max-weight matching
  • Algorithm for Max-Weight Matching
  • Start with any feasible labeling l and some
    matching M in El
  • While M is not perfect repeat the following
  • 1. Find an augmenting path for M in El this
    increases size of M
  • 2. If no augmenting path exists, improve l to
    l such that El ? El Go to 1
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