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Nearly Optimal Bounds for Distributed Wireless Scheduling in the SINR Model Magn´ us M. Halld´ orsson and Pradipta Mitra

Halld´ orsson and Mitra ()

Nearly Optimal Bounds for Distributed Wireless Scheduling in the SINR Model

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Wireless communication



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Path loss



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Modeling interference

“Links” are intended transmissions. How do we capture interference between links?



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Modeling interference

Physical model of signal transmission Signal received at r from s =


P d(s,r )α ,

if s transmits with power P.


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Modeling interference (contd.)

Physical model of interference Pv d(sv ,rv )α Pu d(su ,rv )α

(Good) Signal at rv = Interference at rv = Ambient noise N

Signal to Interference + Noise Ratio SINR =

Pv d(sv ,rv )α Pu +N d(su ,rv )α

Link succeeds iff SINR ≥ β. Halld´ orsson and Mitra ()


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Modeling interference (contd.)

Physical model of interference Pv d(sv ,rv )α Pu d(su ,rv )α

(Good) Signal at rv = Interference at rv = Ambient noise N


P

Pv d(sv ,rv )α Pu u∈S d(su ,rv )α +N

A link set S is feasible if SINR ≥ β for all links in S. Halld´ orsson and Mitra ()


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The Scheduling Problem Given a link set L = {`1 , `2 . . . `n }, partition L in to a minimum number of feasible sub-sets. Equivalently, find the minimum number of slots required to schedule each link successfully.



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The Scheduling Problem Given a link set L = {`1 , `2 . . . `n }, partition L in to a minimum number of feasible sub-sets. Equivalently, find the minimum number of slots required to schedule each link successfully. Assumptions: Powers are given: “fixed power” Powers are oblivious, ie, Pv is a function of d(sv , rv ). Also, the powers are 1 2

Length monotone. Sub-linear.

Pv = constant (uniform power) Pv = p d(sv , rv )α (linear power) Pv = d(sv , rv )α (mean power)



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The Scheduling Problem (contd.) State of the art: 1 2

Centralized: O(log n) approximation (Halld´ orsson and Mitra, 2011) Distributed: O(log2 n) approximation. (Kesselheim and Vöcking, 2010)

“Distributed”: links know their power assignments, an estimate of n and have synchronized time slots.

Thm. Let the optimal solution be T . There is randomized distributed algorithm that schedules the links in O(T × log n) slots, with high probability. In general, no distributed algorithm can do better.



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Algorithm for a link (KV10)

Choose probability q = “the right value” 2: for lnqn slots do 3: transmit with i.i.d. probability q 4: end for 1:



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P

Pv d(sv ,rv )α Pu u∈S d(su ,rv )α

S is feasible if SINR ≥ β for all links in S.

Affectance 1 = SINR

Pu u∈S d(su ,rv )α Pv d(sv ,rv )α

P

SINR ≥ β ≡ au (v ) = β `v succeeds iff

P

u∈S


X Pu · d(sv , rv )α β ≤1 Pv · d(su , rv )α u

Pu · d(sv , rv )α Pv · d(su , rv )α

(1) (2)

au (v ) ≤ 1


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P

Pv d(sv ,rv )α Pu u∈S d(su ,rv )α

S is feasible if SINR ≥ β for all links in S.

Affectance au (v ) = β

Pu · d(sv , rv )α Pv · d(su , rv )α

Define P Incoming affectance aL (v ) = Pu∈L au (v ) Outgoing affectance av (L) = u∈L av (u)



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Analysis (KV10) ¯= Let, average affectance, A

1 n

P

v ∈L aL (v ).

¯ Call Now, there must be at least n/2 links `v for which aL (v ) ≤ 2A. ¯ these links L Set q =

1 ¯. 4A

¯ E(aS (v )) = qaL (v ) ≤ Then for `v ∈ L, of active links.

1 ¯ ¯ 2A 4A

= 21 , where S is the set

Upshot In expectation, Ω( A1¯ ) fraction of the links succeed in the first slot. Thus, ¯ log n slots. we need A

Question ¯ relate to the optimal scheduling length T ? How does A



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Analysis (KV10) – contd. Question ¯ relate to the optimal scheduling length T ? How does A

Answer ¯ = O(T log n) →→ log2 n approximation A

Proof Pattern 1 Find measure M, show that with transmission probability Θ( M ), algorithm takes O(aM) steps.

Show that M = O(bT ). Derive O(a × b) approximation factor.



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Our Analysis Proof Pattern 1 Find measure M, show that with transmission probability Θ( M ), algorithm takes O(aM) steps.

Show that M = O(bT ). Derive O(a × b) approximation factor.

New measure Λ = arg mint |{ù ∈ L : aL (u) ≤ 4t}| ≥ |L|/4 . We show: a = log n, b = O(1).



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The crux

¯ = O(T log n). A Λ = O(T ).



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The crux Lemma (KV10) Assume S is feasible, and ù ≤ S. Then, aS (u) = O(1) and au (S) = O(log n).



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The solution Λ = O(T ).

Lemma Assume S is feasible. Then there exists S 0 ≥ S/2 such that au (S) ≤ 4 (we call S 0 anti-feasible).

Lemma Assume S is feasible and anti-feasible and ù ≤ S. Then, aS (`) = O(1) and au (S) = O(1).



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The solution Only feasible = bad. Feasible+Anti-feasible = good. Every feasible set has a large subset that is Feasible+Anti-feasible. Λ = arg mint |{ù ∈ L : aL (u) ≤ 4t}| ≥ |L|/4 .



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The full algorithm Algorithm 1 Distributed 1: k ← 0 2: while transmission not successful do 3: q = 21k 4: for lnqn slots do 5: transmit with i.i.d. probability q 6: end for 7: k ←k +1 8: end while

Issues Scaling down to the right probabiliy Acknowledgements



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Lower bound

Point: At any time t, the probability of only one links in a pair transmitting is ≤ 1/2 Thus, in expectation, no more than 1/2 of the pairs will succeed in each time slot. Hence we will need log n slots in expectation. Halld´ orsson and Mitra ()


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Discussion

Can we get real multiplicative lower bounds? Use carrier sense for better bounds? Utilize and transcend anti-feasibility.



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Thanks!



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