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Algorithms for Wireless Network Design: A Cell Breathing Heuristic

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**1. ** Algorithms for Wireless Network Design: A Cell Breathing Heuristic

**2. ** Application of Market Equilibrium in Distributed Load Balancing Wireless devices
Cell-phones, laptops with WiFi cards
Referred as clients or users interchangeably
Demand connections to access points
Uniform for cell-phones (voice connection)
Non-uniform for laptops (application dependent)

**3. **Application of Market Equilibrium in Distributed Load Balancing Access points
Cell-towers, Base stations, Wireless routers
Capacities
Total traffic they can serve
Integer for cell-towers
Variable transmission power
Capable of operating at various power levels
Assume levels are continuous real numbers

**4. **Clients to APs assignment Assign clients to APs in an efficient way
No over-loading of APs
Assigning the maximum number of clients and thus satisfying maximum demand

**5. **One Heuristic Solution A client connects to the AP with reasonable signal and then the lightest load
Requires support both from AP and Clients
APs have to communicate their current load
Clients have WiFi cards from various vendors running legacy software
Overall it has limited benefit in practice

**6. **Ideal Case We would like a client connects to the AP with the best received signal strength
If an AP j transmitting at power level Pj then a client i at distance dij receives signal with strength
Pij = a.Pj.dij-c
where a and c are constants capturing various models of power attenuation

**7. **Cell Breathing Heuristic An overloaded AP decreases its communication radius by decreasing power
A lightly loaded AP increases its communication radius by increasing power
Hopefully an equilibrium would be reached
Will show that an equilibrium exist
Can be computed in polynomial time
Can be reached by a tatonnement process
Let’s start with economics and game theory

**8. **Market Equilibrium – A distributed load balancing mechanism. Fisher setting with linear Utilities:
m buyers (each with budget Bi) and n goods for sale
(each with quantity qj)
Each buyer has linear utility ui, i.e. utility of i is
sumj uij xij where uij>= 0 is the utility of buyer i for good j and xij is the amount of good j bought by i.
A market equilibrium or market clearance is a price vector p that
maximizes utility sumj uij xij of buyer i subject to his budget sumj pj xij <= Bi
The demand and supply for each good j are equal
sumj xij = qj (and thus the budgets are totally spent).

**9. **Fisher Setting with Linear Utilities

**10. **Market Equilibrium – A distributed load balancing mechanism. Static supply
corresponding to capacities of APs
Prices
corresponding to powers at APs
Utilities
Analogous to received signal strength function
Either all clients are served or all APs are saturated
Analogous to the market clearance(equiblirum) condition
Thus our situation is analogous to Fisher setting with linear utilities

**11. **Clients assignment to APs

**12. **Analogousness Is Only Inspirational
Get inspiration from various algorithms for the Fisher setting and develop algorithms for our setting
Though we do not know any reduction – in fact there are some key differences

**13. **Differences from the Market Equilibrium setting Demand
Price dependent in Market equilibrium setting
Power independent in our setting
Is demand splittable?
Yes for the Market equilibrium setting
No for our setting
Market equilibrium clears both sides but our solution requires clearance on either client side or AP side
This also means two separate linear programs for these two separate cases

**14. **Three Approaches for Market Equilibrium Convex Programming Based
Eisenberg, Gale 1957
Primal-Dual Based
Devanur, Papadimitriou, Saberi, Vazirani 2004
Auction Based
Garg, Kapoor 2003

**15. **Three Approaches for Load Balancing Linear Programming
Minimum weight complete matching
Primal-Dual
Uses properties of bipartite graph matching
Auction
Useful in dynamically changing situation

**16. **Another Application of Market Equilibria in Networking Fleisher, Jain, Mahdian 2004 used market equilibrium inspiration to obtain Toll-Taxes in Multi-commodity Selfish Routing Problem
This is essentially a distributed load balancing i.e., distributed congestion control problem

**17. **Linear Programming Based Solution Create a complete bipartite graph
One side is the set of all clients
The other side is the set of all APs, conceptually each AP is repeated as many times as its capacity (unit demand)
The weight between client i and AP j is
wij = c.ln(dij) – ln(a)= -ln(Pij/Pj )
Find the minimum weight complete matching

**18. **Theorem Minimum weight matching is supported by a power assignment to APs
Power assignment are the dual variables
Two cases for the primal program which is known at the beginning
Solution can satisfy all clients
Solution can saturate all APs

**19. **Case 1 – Complete matching covers all clients

**20. **Case 1 – Pick Dual Variables

**21. **Write Dual Program

**22. **Optimize the dual program Choose Pj = e pj
Using the complementary slackness condition we will show that the minimum weight complete matching is supported by these power levels

**23. **Proof Dual feasibility gives:
-?i = pj – wij= ln(Pj) – c.ln(dij) + ln(a) = ln(a.Pj.dij-c)
Complementary slackness gives:
xij=1 implies -?i = ln(a.Pj.dij-c)
(Remember if an AP j transmitting at power level Pj then a client i at distance dij receives signal with strength Pij = a.Pj.dij-c)
Together they imply that i is connected to the AP with the strongest received signal strength

**24. **Case 2 – Complete matching saturates all APs

**25. **Case 2 – The rest of the proof is similar

**26. **Optimizing Dual Program Once the primal is optimized the dual can be optimized with the Dijkstra algorithm for the shortest path

**27. **Primal-Dual-Type Algorithm Previous algorithm needs the input upfront
In practice, we need a tatonnement process
The received signal strength formula does not work in case there are obstructions
A weaker assumption is that the received signal strength is directly proportional to the transmitted power – true even in the presence of obstructions

**36. **Simple Observation Deficiency of a Set S = Deficiency of the Maximum Matching
Maximum Deficiency over Sets = Minimum Deficiency over Matching

**37. **Generalization of Hall’s Theorem Maximum Deficiency over Sets = Minimum Deficiency over Matching
Maximum Deficiency over Sets = Deficiency of the Maximum Matching

**56. **Unsplittable Demand

**57. **Unsplittable Demand The integer program is APX-hard in general (because of knapsack)
Assuming that the number of clients is much larger than the number of APs, a realistic assumption, we can obtain a nice approximation heuristic.
First we compute a basic feasible solution

**58. **Analysis of Basic Feasible Solution

**59. **Approximate Solution All xij’s but a small number of xij’s are integral
Theorem: Number of xij which are integral is at least the number of clients – the number APs
Most clients are served unsplittably
Clients which are served splittably – do not serve them
The algorithm is almost optimal

**60. **Discrete Power Levels Over the shelf APs have only fixed number of discrete power levels
Equilibrium may not exist
In fact it is NP-hard to test whether it exists or not
If every client has a deterministic tie breaking rule then we can compute the equilibrium – if exists under the tie breaking rule

**61. **Discrete Power Levels Start with the maximum power levels for each AP
Take any overloaded AP and decrease its power level by one notch
If an equilibrium exist then it will be computed in time mk, where m is the number of APs and k is the number of power levels
This is a distributed tatonnement process!

**62. **Proof Suppose Pj is an equilibrium power level for the jth AP.
Inductively prove that when j reaches the power level Pj then it will not be overloaded again.
Here we use the deterministic tie breaking rule.

**63. **Conclusion Theory of market equilibrium is a good way of synchronizing independent entity’s to do distributed load balancing.
We simulated these algorithm and observed meaningful results.

**64. **Thanks for your attention…