On the minimum frame length problem in wireless mesh networks with - - PowerPoint PPT Presentation

network setting

• ptimization model

numerical results concluding remarks

On the minimum frame length problem in wireless mesh networks with multicast periodic packet traffic

Michał Pióro

Warsaw University of Technology (Poland) and Lund University (Sweden)

Centre d’Enseignement et de Recherche en Informatique, Université d’Avignon, September 8, 2016

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• utline

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concluding remarks

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area of applications – wireless networks

wireless mesh networks (WMN) composed of mesh routers and gateways that employ multi-hop routing most common transmission technology: Wi-Fi radio using the IEEE 802.11-family standards

• ther standards (WiMAX IEEE 802.16a, Bluetooth, Zigbee, ANT)

also support WMN WMN deployments: from residential area networks providing Internet access, to sensor networks in industrial, environmental, smart city and intelligent home applications the optimization model assumes TDMA (time division multiple access)

provides performance bounds for the more common CSMA (carrier sense multiple access) case

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network graph and traffic demands

network graph G = (V, A) – directed (usually bi-directed) composed of links (arcs) with the end nodes in the transmission range δ+(v), δ−(v) – outgoing/incoming star of arcs outgoing from/incoming to node v set of traffic demands (multicast packet streams) D each stream d ∈ D sends packets from its originating node

• (d) to the set of destinations nodes D(d)

(D(d) ⊆ V \ {o(d)}) packets from stream d follow a multicast tree A(d) (a directed Steiner tree) – an arborescence rooted at o(d) and spanning all nodes in D(d); the trees are given and fixed

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4 × 4 network multicast tree

5 6 7 8 9 1 2 10 11 3 4 12 13 14 15 16 5 6 7 8 9 1 2 10 11 3 4 12 13 14 15 16

sources v = 1, 2, 3, 4 – dark grey destinations (gateways) v = 5, 6, . . . , 16 – light gray undirected lines represent two oppositely directed links multicast tree A(d) for s = 1 with o(d) = 1 and D(d) = {5, 6, . . . , 16}

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TDMA transmission of packets

a common TDMA frame T , composed of T time slots t (t ∈ T ) of equal length, is periodically repeated (a train of identical frames) the transmission pattern in each slot of the frame is specified in the form

• f a compatible set (c-set in short)

a c-set c ∈ ˆ C is a subset of nodes that transmit simultaneously; ˆ C – the family of all (exponentially many) c-sets each node transmits to a dedicated set of receiving nodes (the subsets

• f the receiving nodes are mutually disjoint) and the transmissions do

not interfere with each other in essence, c-sets are composed of disjoint stars the packets are of equal length; it takes exactly one slot to send one packet along a link each stream d ∈ D generates a packet at the beginning of each consecutive frame

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examples of c-sets

5 6 7 8 9 1 2 10 11 3 4 12 13 14 15 16 5 6 9 1 4 12 15 16

• n the left:
• ne transmitting node (a star)
• n the right:

two transmitting nodes (two stars)

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scheduling of packets in the frame

t 1 2 3 4 5 6 7 8 9 10 1 2 3 4 1,4 1,4 1,4 2,3 2,3 2,3 v = 1 1 × × × 4 2 3 × × × v = 2 × 2 × × × × × 1 3 4 v = 3 × × 3 × × × × 1 2 4 v = 4 × × × 4 1 2 3 × × ×

the frame is composed of 10 time slots (T = 10) the columns correspond to the c-sets (transmitting nodes – indicated) the rows correspond to the transmitting nodes the entries specify the number of the stream of the transmitted packet (× means that the node does not transmit in the c-set)

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the considered problem

find the minimal frame length T and the corresponding frame composition ct, t = 1, 2, . . . , T (i.e., the c-sets to be used in the consecutive time slots of the frame) so that by means of this frame the network is capable of delivering the packets from their sources to destinations with a finite delay equivalent to maximization of carried traffic: min T ⇔ max |S|

T

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c-sets – description

a c-set c is characterized by the set W(c) of the transmitting nodes, and the family of the mutually disjoint and non-empty sets of the receiving nodes U(c, w), w ∈ W(c) W(c) ⊆ V U(c, w) ⊆ V \ W(c), U(c, w) = ∅, w ∈ W(c) U(c, w) ∩ U(c, w′) = ∅, w, w′ ∈ W(c), w = w′ transmissions do not interfere each other: U(c, w) ⊆ {u ∈ V \ W(c) : p(w, u) η +

v∈W\{w} p(v, u) ≥ γ}

c can be specified by means of binary variables

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c-sets – characterization in the IP form

Xw ≥ Ywu, w ∈ V, u ∈ δ+(v) (1a) Xw ≤

u∈δ+(w) Ywu,

w ∈ V (1b) Xw +

u∈δ−(w) Yuw ≤ 1,

w ∈ V (1c) p(w, u) + M(w, u)(1 − Ywu) ≥ ≥ γ(η +

v∈V\{w,u} p(v, u)Xv),

(w, u) ∈ A (1d) Ywu ∈ B, w ∈ V, u ∈ δ+(w); Xw ∈ B, w ∈ V (1e)

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• ptimization problem – parameters and variables

C – given list of (allowable) c-sets (C ⊆ ˆ C) a(e), b(e) – the originating and terminating node of link e ∈ E C(e) – family of allowable c-sets with a(e) ∈ W(c) and b(e) ∈ U(c, a(e)) (transmission on e) Tc, c ∈ C – the number of times c is used in the frame (T – frame length) (variables) hdwc ∈ {0, 1}, d ∈ D, c ∈ C, w ∈ W(c) – the number of slots (0 or 1) in the frame that use c-set c to broadcast a packet of stream d from node w (variables) yde ∈ {0, 1}, d ∈ D, e ∈ E – specify the tree A(d) (variables) zdwe ≥ 0, d ∈ D, w ∈ D(d), e ∈ E – link flows that assure connectivity of A(d) (variables)

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the problem

frame length minimization – primal MIP min T =

c∈C Tc

(2a) [ϕdwv]

• e∈δ−(v) zdwe + I(d, w, v) =

e∈δ+(v) zdwe,

d ∈ D, w ∈ D(d), v ∈ V (2b) [σdwe] zdwe ≤ yde, d ∈ D, w ∈ D(d), e ∈ E (2c) [λde]

• c∈C(e) hda(e)c ≥ yde,

d ∈ D, e ∈ E (2d) [πcw]

• d∈D hdwc ≤ Tc,

c ∈ C, w ∈ W(c) (2e) yde ∈ {0, 1}, d ∈ D, e ∈ E (2f) zdwe ∈ R+, d ∈ D, w ∈ D(d), e ∈ E (2g) hdwc ∈ {0, 1}, d ∈ D, c ∈ C, w ∈ W(c) (2h) Tc ∈ R, c ∈ C (2i)

I(d, w, o(d)) = 1, I(d, w, w) = −1, and I(d, w, v) = 0 for v ∈ V \ {o(d), w}

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dual problem and constraint violation

dual (λ∗: optimal solution) max H =

d∈D

• w∈D(d)(ϕdwo(d) − ϕdww)

(3a)

• w∈D(d) σdwe ≤ λde,

d ∈ D, e ∈ E (3b) ϕdwa(e) − ϕdwb(e) ≤ σdwe, d ∈ D, w ∈ D(d), e ∈ E (3c)

• w∈W(c) πcw = 1,

c ∈ C (3d)

• v∈U(c,w) λd(w,v) ≤ πwc,

d ∈ D, c ∈ C, w ∈ W(c) (3e) σdwe ∈ R+, d ∈ D, w ∈ D(d), e ∈ E (3f) λde ∈ R+, d ∈ D, e ∈ E (3g) πwc ∈ R+, c ∈ C, w ∈ W(c). (3h) introducing a new c-set to the problem: can result in violation of red constraints by the optimal λ∗

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pricing problem

constraint violation for given c / ∈ C and λ∗ the sum of violations of constraints (3e) corresponding to c violated by λ∗ is computed as follows P(c) := min π ≥ 0 :

w∈W(c) πw = 1Q(π; c)

where Q(π; c) :=

w∈W(c)

• d∈D(max {0,

v∈U(c,w) λ∗ d(w,v) − πw})

maximization of P(c) over c: by means of an appropriate MIP

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resolution algorithm

price and branch step 1: assume an initial family C ⊆ ˆ C of c-sets (the family of all maximal one-node c-sets) step 2: solve the dual problem (3) to obtain λ∗ step 3: solve the pricing problem to generate an extra c-set for which the constraints (3e) are most violated by λ∗ step 4: if such a c-set exists, add it to C; go to step 2 step 5: solve the primal MIP (2) for the so obtained C

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N × N grid network

for each N the horizontal/vertical distance between adjacent nodes: 40 meters (diagonal distance: 56.6 meters) noise power: η = 101 dBm, node transm. power: P = 20 dBm (i.e., 100 mW) power gain between nodes v and w : g(v, w) = d(v, w)−4 where d(v, w) is the distance (so that p(v, w) = P · d(v, w)−4) transmission range: L(γ) = ( P

γ·η )

1 4 ; γ = 8 dB:

L(8) = 66.8 meters sources: dark grey (inner) nodes — destinations: light gray (outer) nodes

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9 × 9 network and computing environment

N = 9 node network V = 81 nodes; A = 800 arcs 49 demands, each sending a packet stream from one of the 49 inner nodes to all of the 32 outer nodes 49 one-node c-sets used as the initial list of allowable c-sets computational resources

the optimization model/optimization procedure implemented in AMPL the resulting AMPL program executed using the AMPL 12.6 interpreter, the LP and MIP subproblems solved with the 64-bit version of the CPLEX 12.6.2 solver computations run on a dedicated Windows Server 2012 R2 x64 virtual machine assigned 20 logical processors and configured to use up to 80GB of RAM

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results I (OT – trees optimized, FT – trees fixed)

N = 9, G = 32: optimality T 0 T LR T MIP ∆ |C0| |CLR| |CMIP| iter OT 606 119.5 232 48% 49 103 48 54 FT 2401 481.5 482 0% 49 272 122 223 N = 9, G = 32: computation times time/iter time/LR time/PP time/MIP total time OT 15m26s 15m16s 10s 2h 46m 40s (*) 16h 39m 49s FT 8.7s 1.5s 7.2s 3m 28s 35m 42s

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results II (OT – trees optimized, FT – trees fixed)

scalability N T 0 T LR T MIP ∆ |C0| |CMIP| iter total time OT 9 606 119.5 232 48% 49 48 54 16h 39m 49s FT 2401 481.5 482 0% 49 122 223 35m 42s OT 8 381.4 87 153 36.5% 36 26 35 3h 8m 49s FT 1296 336 336 0% 36 91 120 10m 1s OT 7 223.4 59.5 91 24.7% 25 30 46 1h 32m 30s FT 625 215 216 0% 25 59 70 57s OT 6 124.8 37.5 46 11.5% 16 20 33 20m 9s FT 256 120 120 0% 16 41 37 14s OT 5 46 20.8 25 0% 9 14 18 12m 39s FT 81 53 53 0% 9 18 9 3s OT 4 16 10 10 0% 4 6 2 ∼ 1s FT 16 15 15 0% 4 5 1 ∼ 0s

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general conclusion

both OT and FT are (most likely) NP-hard (PP and MIP) for the FT case (fixed trees) the presented algorithm resolves the frame minimization problem effectively (for up to N = 11) in terms of the solution quality and the computation time improvement of the initial objective function value (i.e., the frame length) is substantial (more than 5 times for N = 9), showing that using multi-node c-sets is decisive for the frame length minimization adding multicast tree optimization (FT → OT) makes the problem considerably harder to solve yet, even suboptimal frames achieved with our algorithm for OT are significantly shorter than for FT (roughly 2 times)

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extensions

variable packet length variable MCS (modulation and coding scheme) to solve the tradeoff between transmission ranges, rates, and interference the multichannel case (with OFDMA – orthogonal frequency division multiple access) minimizing packet delays (bicriteria optimization, requires modeling of packet scheduling) minimizing node energy consumption (for example, minimizing the maximum number of receptions at the network node) traffic flows instead of packet streams (traditional approach) simplify the problem

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acknowledgement

Artur Tomaszewski (Warsaw University of Technology) developed the

• ptimization software, performed all the computations, and

substantially contributed to adding multicast tree optimization to the presented model.

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