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

network setting optimization 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 1 / 23

network setting optimization model numerical results concluding remarks outline 1 network setting 2 optimization model numerical results 3 concluding remarks 4 2 / 23

network setting optimization model numerical results concluding remarks 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 other 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 3 / 23

network setting optimization model numerical results concluding remarks 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 o ( 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 4 / 23

network setting optimization model numerical results concluding remarks 4 × 4 network multicast tree 13 14 15 16 13 14 15 16 11 3 4 12 11 3 4 12 9 1 2 10 9 1 2 10 5 5 6 7 8 6 7 8 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 } 5 / 23

network setting optimization model numerical results concluding remarks 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 of 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 of 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 6 / 23

network setting optimization model numerical results concluding remarks examples of c-sets 13 14 15 16 15 16 11 3 4 12 4 12 9 1 2 10 9 1 5 6 7 8 5 6 on the left: one transmitting node (a star) on the right: two transmitting nodes (two stars) 7 / 23

network setting optimization model numerical results concluding remarks scheduling of packets in the frame 1,4 1,4 1,4 2,3 2,3 2,3 1 2 3 4 × × × × × × v = 4 4 1 2 3 × × × × × × v = 3 3 1 2 4 × × × × × × v = 2 2 1 3 4 × × × × × × v = 1 1 4 2 3 t 5 1 2 3 4 6 7 8 9 10 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) 8 / 23

network setting optimization model numerical results concluding remarks the considered problem find the minimal frame length T and the corresponding frame composition c t , 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 9 / 23

network setting optimization model numerical results concluding remarks 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: p ( w , u ) U ( c , w ) ⊆ { u ∈ V \ W ( c ) : v ∈W\{ w } p ( v , u ) ≥ γ } η + � c can be specified by means of binary variables 10 / 23

network setting optimization model numerical results concluding remarks c-sets – characterization in the IP form w ∈ V , u ∈ δ + ( v ) X w ≥ Y wu , (1a) X w ≤ � w ∈ V u ∈ δ + ( w ) Y wu , (1b) X w + � u ∈ δ − ( w ) Y uw ≤ 1 , w ∈ V (1c) p ( w , u ) + M ( w , u )( 1 − Y wu ) ≥ ≥ γ ( η + � v ∈V\{ w , u } p ( v , u ) X v ) , ( w , u ) ∈ A (1d) Y wu ∈ B , w ∈ V , u ∈ δ + ( w ); X w ∈ B , w ∈ V (1e) 11 / 23

network setting optimization model numerical results concluding remarks optimization 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 ) T c , c ∈ C – the number of times c is used in the frame ( T – frame length) (variables) h dwc ∈ { 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) y de ∈ { 0 , 1 } , d ∈ D , e ∈ E – specify the tree A ( d ) (variables) z dwe ≥ 0 , d ∈ D , w ∈ D ( d ) , e ∈ E – link flows that assure connectivity of A ( d ) (variables) 12 / 23

network setting optimization model numerical results concluding remarks the problem frame length minimization – primal MIP min T = � c ∈C T c (2a) [ ϕ dwv ] � e ∈ δ − ( v ) z dwe + I ( d , w , v ) = � e ∈ δ + ( v ) z dwe , d ∈ D , w ∈ D ( d ) , v ∈ V (2b) [ σ dwe ] z dwe ≤ y de , d ∈ D , w ∈ D ( d ) , e ∈ E (2c) [ λ de ] � c ∈C ( e ) h da ( e ) c ≥ y de , d ∈ D , e ∈ E (2d) [ π cw ] � d ∈D h dwc ≤ T c , c ∈ C , w ∈ W ( c ) (2e) y de ∈ { 0 , 1 } , d ∈ D , e ∈ E (2f) z dwe ∈ R + , d ∈ D , w ∈ D ( d ) , e ∈ E (2g) h dwc ∈ { 0 , 1 } , d ∈ D , c ∈ C , w ∈ W ( c ) (2h) T c ∈ 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 } 13 / 23

network setting optimization model numerical results concluding remarks dual problem and constraint violation ( λ ∗ : optimal solution) dual max H = � � w ∈ D ( d ) ( ϕ dwo ( d ) − ϕ dww ) (3a) d ∈D � w ∈ D ( d ) σ dwe ≤ λ de , d ∈ D , e ∈ E (3b) ϕ dwa ( e ) − ϕ dwb ( e ) ≤ σ dwe , d ∈ D , w ∈ D ( d ) , e ∈ E (3c) � w ∈W ( c ) π cw = 1 , c ∈ C (3d) � v ∈U ( c , w ) λ d ( w , v ) ≤ π wc , d ∈ D , c ∈ C , w ∈ W ( c ) (3e) σ dwe ∈ R + , d ∈ D , w ∈ D ( d ) , e ∈ E (3f) λ de ∈ R + , d ∈ D , e ∈ E (3g) π wc ∈ 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 ∗ 14 / 23

network setting optimization model numerical results concluding remarks pricing problem ∈ C and λ ∗ constraint violation for given c / 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 ) π w = 1 Q ( π ; c ) where v ∈U ( c , w ) λ ∗ d ∈D ( max { 0 , � d ( w , v ) − π w } ) Q ( π ; c ) := � � w ∈W ( c ) maximization of P ( c ) over c : by means of an appropriate MIP 15 / 23

network setting optimization model numerical results concluding remarks 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 16 / 23