MULTIHOP wireless networks are a paradigm in wireless

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1 400 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 17, NO. 1, JANUARY 2018 Toward Optmal Dstrbuted Node Schedulng n a Multhop Wreless Network Through Local Votng Dmtros J. Vergados, Member, IEEE, Natala Amelna, Yumng Jang, Senor Member, IEEE, Katna Kralevska, Member, IEEE, and Oleg Granchn, Senor Member, IEEE Abstract In a multhop wreless network, t s crucal but challengng to schedule transmssons n an effcent and far manner. In ths paper, a novel dstrbuted node schedulng algorthm, called Local Votng, s proposed. Ths algorthm tres to sem-equalze the load (defned as the rato of the queue length over the number of allocated slots) through slot reallocaton based on local nformaton exchange. The algorthm stems from the fndng that the shortest delvery tme or delay s obtaned when the load s sem-equalzed throughout the network. In addton, we prove that, wth Local Votng, the network system converges asymptotcally toward the optmal schedulng. Moreover, through extensve smulatons, the performance of Local Votng s further nvestgated n comparson wth several representatve schedulng algorthms from the lterature. Smulaton results show that the proposed algorthm acheves better performance than the other dstrbuted algorthms n terms of average delay, maxmum delay, and farness. Despte beng dstrbuted, the performance of Local Votng s also found to be very close to a centralzed algorthm that s deemed to have the optmal performance. Index Terms Multhop wreless networks, node schedulng algorthm, wreless mesh networks, load balancng. I. INTRODUCTION MULTIHOP wreless networks are a paradgm n wreless connectvty whch has been used successfully n a varety of network settngs, ncludng ad-hoc networks [1], wreless sensor networks [2], and wreless mesh networks [3]. In such networks, the wreless devces may communcate wth each other n a peer-to-peer fashon and form a network, where ntermedate wreless nodes may act as routers and forward traffc to other nodes n the network [4]. Manuscrpt receved February 24, 2017; revsed July 31, 2017; accepted October 10, Date of publcaton October 31, 2017; date of current verson January 8, Ths work was supported n part by RFBR under Grant and Grant and n part by SPbSU under Grant The assocate edtor coordnatng the revew of ths paper and approvng t for publcaton was I. Guvenc. (Correspondng author: Dmtros J. Vergados.) D. J. Vergados s wth the School of Electrcal and Computer Engneerng, Natonal Techncal Unversty of Athens, GR Zografou, Greece (e-mal: djvergad@gmal.com). N. Amelna and O. Granchn are wth the Faculty of Mathematcs and Mechancs, Sant Petersburg State Unversty, St. Petersburg, Russa (e-mal: n.amelna@spbu.ru; o.granchn@spbu.ru). Y. Jang and K. Kralevska are wth the Department of Informaton Securty and Communcaton Technology, Norwegan Unversty of Scence and Technology, Trondhem, N-7491, Norway (e-mal: jang@ntnu.no; katnak@ntnu.no). Color versons of one or more of the fgures n ths paper are avalable onlne at Dgtal Object Identfer /TWC Due to ther many practcal advantages and ther wde use, there have been a lot of studes on the performance of multhop wreless networks. For example, the connectvty of a multhop wreless network has been studed under varous channel models n [4], [5]. Furthermore, ther capacty has been studed analytcally n [6] [9]. In addton, the stablty propertes of schedulng polces for maxmum throughput n multhop rado networks have been studed n [10], [11]. Also, a centralzed schedulng algorthm that emphaszes on farness has been proposed n [12]. In [13], the authors focused on the jont schedulng and routng problem wth load balancng n multrado, mult-channel and mult-hop wreless mesh networks. They also desgned a cross-layer algorthm by takng nto account throughput ncrease wth load balancng. Algorthms for jont power control, schedulng, and routng have been ntroduced n [14], [15]. In [16], the load balancng problem n a dense wreless multhop network s formulated where the authors presented a general framework for analyzng the traffc load resultng from a gven set of paths and traffc demands. Some more recent lterature works nclude [17] [25]. In [17], the authors present the state of the art n Tme Dvson Multple Access (TDMA) schedulng for wreless multhop network. Reference [18] proposes Genetc Algorthm for fndng Collson Free Set (GACFS) whch s a co-evolutonary genetc algorthm that solves the Broadcast Schedulng Problem (BSP) n order to optmze the slot assgnment algorthm n WMAX mesh networks. It s a centralzed approach and does not take nto consderaton the traffc requrements or the load n the network. Another schedulng soluton for wreless mesh networks based on a memetc algorthm that does not consder the traffc requrements s presented n [21]. An mproved memetc algorthm s appled for energy-effcent sensor schedulng n [26]. Reference [20] nvestgates the mn-slot schedulng problem n TDMA based wreless mesh networks, and t proposes a decentralzed algorthm for assgnng mn-slots to nodes accordng to ther traffc requrements. The authors n [19] propose a schedulng scheme for multcast communcatons where a conflct-free graph s created dynamcally based on each transmsson s destnatons. Reference [22] presents a probablstc topology transparent model for multcast and broadcast transmssons n moble ad-hoc networks. The novelty of the scheme s that nstead of guaranteeng that at least one conflct-free tme slot s assgned to each node, t only tres to brng the probablty of successful transmsson above a threshold. The authors have IEEE. Personal use s permtted, but republcaton/redstrbuton requres IEEE permsson. See for more nformaton.

2 VERGADOS et al.: TOWARD OPTIMAL DISTRIBUTED NODE SCHEDULING IN A MULTIHOP WIRELESS NETWORK 401 further presented performance mprovement for broadcastng n [27]. Another topology transparent schedulng algorthm s presented n [24]. The algorthm s not traffc dependent, and the acheved throughput s lower than the optmal manly due to the requrement for a guaranteed slot for each node. Reference [23] proposes a dstrbuted schedulng scheme for wreless sensor networks (WSNs). Fnally, the NP-hardness of the mnmum latency broadcast schedulng problem s proved n [25] under the Sgnal-to-Interference-plus-Nose- Rato (SINR) model. Two dstrbuted determnstc algorthms for global broadcastng based on the SINR model are presented n [28]. Effcent traffc load balancng and channel access are essental to harness the dense and ncreasngly heterogeneous deployment of next generaton 5G wreless nfrastructure [29]. Channel access n 5G networks faces nherent challenges assocated wth the current cellular networks [30], e.g. farness, adaptve rate control, resource reservaton, real-tme traffc support, scalablty, throughput, and delay. For nstance, beng able to do frequency and tme slot allocaton enables more adaptve and sophstcated mult-doman nterference management technques [31], [32]. In [32], TDMA s used to mtgate the co-ter nterference from tme doman perspectve n ultradense small cell networks. The modelng and the optmzaton of load balancng plays a crucal role n the resource allocaton n the next generaton cellular networks [33]. In ths paper, we focus on the problem of node schedulng n multhop wreless networks. In the node schedulng problem, each transmsson opportunty s assgned to a set of nodes n a such way whch ensures that there wll be no mutual nterference among any transmttng nodes. More specfcally, under node schedulng, two nodes can be assgned the same tme slot (and transmt smultaneously) f they do not have any common neghbors. We ntroduce the Local Votng algorthm. The dea behnd the algorthm was orgnated by the observaton that the total delvery tme n a network can be mnmzed, f the rato of the queue length over the number of allocated slots s sem-equalzed throughout the network. We call ths rato the load of each node. The proposed algorthm allows for neghborng nodes to exchange slots n a manner that eventually sem-equalzes the load n the network. The number of slots that are exchanged s determned by the relaton between the load of each node and ts neghbors, under the lmtaton that certan slot exchanges are not possble due to nterference wth other nodes. The prelmnary results were presented n [34]. Ths paper presents new algorthm and an analyss of ts performance, as well as new smulaton results. The smulaton results of the comparatve study between Local Votng and other representatve algorthms from the lterature show that Local Votng acheves the shortest end-to-end delvery tme and greatest farness compared to other dstrbuted algorthms for dfferent network denstes. We also show that ts performance s very close to a centralzed algorthm. The presented algorthm s a modfcaton of the Local Votng protocol wth non-vanshng to zero step-sze whch was suggested n [35]. It belongs to the more general class of stochastc approxmaton decentralzed algorthms whch have been studed early n [36], [37] Fg. 1. A multhop wreless network where the communcaton range and the nterference range of node are denoted by the crcle. The nodes wth whte background are two-hop neghbors of node, and the nodes wth gray background are outsde the two-hop neghborhood of node. wth decreasng to zero step-sze. However, changng the traffc parameters leads to an unsteady settng of the optmzaton problem. For smlar cases the stochastc approxmaton wth constant (or non-vanshng to zero stepsze) s useful [38], [39]. The paper s organzed as follows: Secton II descrbes thoroughly the network model. Secton III presents the proposed Local Votng algorthm where Secton III-B presents an analyss of the performance of the algorthm n terms of achevng consensus. The smulaton results n Secton IV compare the performance of the proposed algorthm wth other algorthms from the lterature. Fnally, Secton V concludes the paper. II. NETWORK MODEL AND LOAD BALANCING Consder a network that can be represented by a graph G = (N, E). N s the set of all wreless nodes that communcate over a shared wreless channel,.e. N ={1, 2,...,n}. E s the set of drectonal but symmetrc edges whch exst between two nodes f a broadcast from one node may cause nterference on the other node. We use the terms edges and lnks nterchangeably. Access on the channel s consdered to follow a paradgm of tme dvson multple access. There s no spatal movement of the nodes. The consdered schedulng algorthm s a node schedulng algorthm,.e. each slot s allocated to a node, nstead of a communcaton lnk. We study a smple protocol nterference model where two nodes are one-hop neghbors as long as ther dstance s less than the communcaton range. The nterference range s consdered to be equal to the communcaton range, and both values are consdered constant throughout the network. A multhop network s presented n Fg. 1 where the nodes wthn the crcle of node are one-hop neghbors of node, andtheone-hop neghborhood of node s denoted by N (1).WealsodefneN (2) as a two-hop neghborhood of node,.e. the set of all the nodes that are neghbors to node or that have a common neghbor wth node. Snce the ncluson N (1) N (2) holds, the nodes wth whte background n Fg. 1 are two-hop neghbors of node. The nodes presented wth gray background are outsde the two-hop neghborhood of node. Note that the nodes wthn the crcle of node are also

3 402 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 17, NO. 1, JANUARY 2018 wthn the nterference range of node because the nterference range and the communcaton range are equal. Two flows are depcted wth red and blue arrows, respectvely. Accordng to the protocol nterference model, two nodes can be assgned the same transmsson slot, wth no collson, as long as they do not have any common neghbors. Otherwse, a collson would happen, resultng n data loss. Node schedulng tres to guarantee that no such collson happens. Each node contans a queue wth packets to be transmtted, and the nternal schedulng on the queue s frst-come-frstserve. The maxmum length of each queue s consdered to be unbounded. Each node also has a set of slots that have been assgned to t, and neghborng nodes may exchange slots. Tme s dvded nto frames where each frame s denoted wth t and t = 0, 1,... In addton, each frame t s dvded nto tme slots. The number of tme slots n each frame s consdered to be fxed and equal to S where all tme slots have the same duraton. The number of slots n a frame S s consdered to be large enough for every node to be able to obtan at least one slot n each frame, f needed. Ths value can be determned by the chromatc number of the graph, where there s an edge between any two-hop neghbors n the orgnal graph G. The Greedy Colorng Theorem provdes an upper bound for ths chromatc number whch s equal to max N N (2) +1 [40]. The duraton of a tme slot s suffcent to transmt a sngle packet. The transmsson schedule of the network s defned as, { 1, f a slot s S s assgned to a node N; (1) 0, otherwse; X,s t = for t 0, wth X,s 0 = 0 by conventon. The transmsson schedule s conflct-free, fforanyt, Xt,s Xt j,s = 0, s S, N, j N (2), = j. (2) For each N, letñt denote a set of such nodes j that node can exchange slots wth node j and the produced schedule remans conflct-free and E t denote the correspondng subset of edges. The objectve of ths work s to desgn a load balancng node schedulng strategy to schedule nodes transmssons n such a way that the mnmum maxmal (mn-max) nodal delay s acheved. We wll study the followng scheme of slot assgnment and transmsson of packets (see Fg. 2). At the begnnng of frame t, the state of each node n the network s descrbed by three characterstcs: qt s the queue length, counted as the number of slots needed to transmt all packets at node at frame t; s the number of slots assgned to node at the p t 1 prevous frame t 1,.e. pt 1 S = s=1 X,s t 1 ; u t s the number of tme slots whch are assgned (u t > 0) or released (u t < 0) by node at the begnnng of frame t (u t s calculated by the schedulng polcy). For each node, the slot assgnment starts wth releasng tme slots accordng to the schedulng polcy when u t < 0, or otherwse wth assgnng slots to node from free tme slots Fg. 2. frame t. Procedure of slot assgnment and transmsson of packets durng or through redstrbuton of tme slots wth ts neghbors. After that, the transmsson of packets begns. Durng frame t new packets arrve. At the end of frame t, the schedulng polcy calculates {u t+1 } N locally based on the avalable data. So, the dynamcs of each node s descrbed by pt = pt 1 + n t + u t, N, t = 0, 1,..., qt+1 = max{0, q t p t }+z t, (3) where n t s the number of free slots that are allocated to node or the number of slots that are released due to an empty queue, and u t s the number of tme slots that node gans or loses at frame t due to the adopted slot schedulng strategy. These are slots that are exchanged between neghborng nodes, whle zt s the number of slots needed to transmt new packets receved by node at frame t, ether receved as new packets from the upper layers or from a neghborng node. If qt = 0, then no slot s allocated to the node,.e.weset pt = 0. For reader s convenence, we provde Table I wth the key notatons used n ths paper. A. Load Balancng The ultmate objectve of a schedulng algorthm n a multhop network s the packet flows to be delvered from the source to the destnaton n a short tme. Ths can be measured by the end-to-end delay per packet, the end-toend delvery tme of a packet burst, the throughput of each flow, and the farness n dstrbutng the resources among the competng flows. In general, the problem of optmal schedulng n terms of approxmatng the optmal throughput n a multhop wreless network s NP-hard as t s proven n [41]. A specfc challenge of havng such a schedulng algorthm s that t needs to examne per flow nformaton and use ths nformaton to schedule flows at every node whch we beleve s dffcult to mplement. For ths reason, we do not optmze the end-to-end delay for the whole wreless network, but nstead we focus on optmzng the nodal (per-node) delay n each transmtter. The proposed Local Votng algorthm may be consdered as a compromse, where we do node schedulng by usng the slots wthout nformaton about the ndvdual flows. Snce multhop end-to-end delay s the sum of nodal delays on the end-to-end path, we expect Local Votng to delver also good multhop end-to-end delay performance. To valdate ths, the evaluaton n Secton 4 has been focused on multhop end-to-end delay,

4 VERGADOS et al.: TOWARD OPTIMAL DISTRIBUTED NODE SCHEDULING IN A MULTIHOP WIRELESS NETWORK 403 TABLE I TABLE WITH NOTATIONS and the results ndcate that Local Votng does gve good or ndeed better multhop end-to-end performance than varous lterature algorthms. In the followng we show that the nodal delay may be optmzed (mn-max), f the load of each node n the network s balanced. The load of node at the begnnng of frame t s defned as zero when qt = 0, and otherwse t s defned as the rato of the queue length qt over the number of allocated slots pt (note that slots are not assgned to nodes that have nothng to transmt n an optmal schedulng strategy, so we have qt = 0fpt = 0),.e. [ ] q + 0.5, f q > 0, x = p (4) 0, f q (and consequently p ) = 0. where [ ] s the round functon (rounds a real number to the nearest nteger). Usng ths defnton we calculate the delay for each node (n tme slots) as x S. Defnton 1: Load balancng s the processes of equalzng the load between the nodes n the network by exchangng slots among them. Defnton 2: We defne a conflct-free schedule as nodally optmal or just optmal, f the maxmum delay per node n the network s smaller or equal than the maxmum per node delay for every other schedule (mn-max). Lemma 3 (Optmal Schedules Are Maxmal): An optmal schedule s a (or has an equvalent) maxmal schedule n the sense that 1 j N such that p j can be ncreased wthout reducng p k n at least one other node k N. Proof: Consder a schedule that s not maxmal. That means there exsts j N such that p j can be ncreased by one. For the new schedule, the delay for all the other nodes s unchanged (snce we dd not reduced slots [ for the other nodes). For node j, the new delay s x S = q (p +1) ] S x S. Thus, for every non-maxmal schedule, there exsts a maxmal schedule that has smaller or equal maxmum delay. Lemma 4 (Optmal Schedules Are Balanced): Assume that node k s the most loaded node n the network,.e k = argmax(x ), N. For all optmal schedules, t holds x k x j /(1 1/p j ) for the load of the most loaded node k and the load of every other node j where j Ñ k. Proof: Assume that an optmal schedule exsts where for the most loaded node k, x k > x j /(1 1/p j ) where j Ñ k. Snce k s the most loaded node, the maxmal delay for such a schedule s x k S. Snce node j Ñ k, t follows that a slot of node j can be reassgned to node k. After reassgnng, the new load for node k s [q k /(p k + 1) + 0.5], and the correspondng delay for node k s q k /(p k + 1) S < x k S. In addton, node j loses a slot so the new delay for node j s [q j /(p j 1) + 0.5] S =[(q j /p j )/(1 1/p j ) + 0.5] S =[(q j /p j ) + 0.5]/(1 1/p j ) S =x j /(1 1/p j ) S < x k S. Thus, the new allocaton has a maxmal delay that s smaller than or equal to the maxmal delay of the other allocaton, so the allocaton s not optmal. Based on the above reasonng, we desgn a load balancng strategy wth two goals: 1) The produced schedule should be maxmal, 2) The load n the schedule should be balanced n the sense of Lemma 4. For ths reason, we defne a slot exchange strategy that tres to equalze the load through load balancng, and n the next Secton III-B we prove that the Local Votng algorthm converges to a such soluton. It should be noted that, n general, a schedule could be both maxmal and balanced, but stll not optmal. Ths s because there could exst a reallocaton of the slots n the network that would produce a larger spectral effcency. Optmzng the schedule n ths sense would requre fndng a soluton for the NP-complete broadcast schedulng problem. Ths s not easy, so for the purposes of ths paper, we do not examne ways of escapng local optma and fndng the global optmum. However, we can see from the smulaton results 1 Symbol denotes the negaton of exstence

5 404 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 17, NO. 1, JANUARY 2018 that the performance of Local Votng s stll better than the performance of other dstrbuted algorthms that we compare wth, and also we see that optmzng the maxmal nodal delay also has a postve mpact on the end-to-end delay. Among all possble optons for load balancng, the mnmax nodal delay s acheved when all nonzero loads q t /p t are sem-equal. Ths comes as a result from the fndng that the mnmum expected nodal delay s acheved when the load n the network s equalzed on nodes (Lemma 1 and Corollary 1 from [35]). III. THE PROPOSED NODE SCHEDULING ALGORITHM: LOCAL VOTING In the prevous secton we have shown that an optmal schedule has three propertes: t s effcent, t s maxmal, and t s balanced. These are the propertes whch gude us n the desgn of the Local Votng algorthm. In order to be effcent, there should be no slots allocated to nodes that have an empty queue. For ths reason, before the begnnng of each frame, nodes wth an empty queue release all tme slots that they have reserved. In order to be maxmal, there should be no free tme slot n the neghborhood of any node, f that node has a postve queue, and assgnng the slot to the node would not cause a conflct wth other nodes. In order to meet ths objectve, after the frst step, free slots are allocated to the nodes that do not have an empty queue. Conflcts are resolved n a descendng order of the load. Fnally, the thrd objectve s to be balanced, whch can be formulated wth the followng control goal: to keep the rato qt /p t sem-equal throughout the network (as much as possble) for the nodes where the queue s not empty qt > 0. In other words, the number of slots assgned to each node should correspond to the amount of backlogged traffc. A consequent mplcaton s that, n order to acheve ths optmal strategy, we should be able to freely exchange slots among any two nodes n the network. However, n realty, t s not always possble due to the potental nterference wth other nodes n network. That s expressed through Eq. (2). In the followng, we propose a novel algorthm that adopts the local votng control strategy. For the proposed Local Votng algorthm, ts sem-consensus propertes wth respect to the local balancng are proved n Secton III-B. A. The Proposed Algorthm: Local Votng At the end of frame t, each node computes a schedulng polcy. The u t+1 value s calculated as follows. Each node uses the characterstcs of ts own state qt+1, p t and ts neghbors states q j t+1, p t j f Ñt =. Let us for tme frame t and for each node, N : q t+1 > 0, defne sem-nverse load x t : x t = p t/qt+1,and consder the followng modfcaton n the already known Local Votng (LV) protocol [35]: u t+1 = γ at, j ( x t j x t ) (5) j Ñt where γ > 0 s a LV protocol step-sze, and LV protocol matrx coeffcents a, j t : a, j t = q j t k Ñ t Note, t s not so hard to see that u t+1 = γ q k t+1 q t+1 j Ñt qt+1 p t j q j t+1 p t q t+1 + j Ñ t. q j t+1. For all other case we defne u t+1 = 0and x t = p t.we set at, j = 0 for other pars, j and denote the matrx of the protocol as A t =[at, j ]. The elements at, j n adjacency matrx A t are at, j > 0 f node can exchange slots wth node j and the produced schedule remans conflct-free; and at, j = 0otherwse. When γ = 1, Eq. (5) has a form: u t+1 = qt+1 p t + j Ñt pt j qt+1 + j Ñt q j pt. t+1 Example. Let s consder the network wth S =50, three nodes, all neghbors wth each other (sngle hop), wth the followng ntal queue lengths of q0 1 = 400, q2 0 = 100, q0 3 = 310, and p1 0 = 20, p2 0 = 20, p3 0 = 10. The ntal values for the loads are the followng: x0 1 = 20, x 0 2 = 5, x 0 3 = 32. The queue lengths at the end of frame t = 0 wll be q1 1 = = 380, q2 1 = = 80, q3 1 = = 300. Usng Eq. (5) we get u 1 1 u 2 1 u 3 1 = [380 ( )/( )] 20 = 5, = [80 ( )/( )] 20 = 15, = [300 ( )/( )] 10 = 10, and we have three sem-equal loads x1 1 = 380/25 = 15.2, x 1 2 = 80/5 = 16, x 1 3 = 300/20 = 15 Eventually, node gans a slot n the followng scenaros: Its queue length s postve and there exsts an avalable tme slot that s not allocated to one-hop or two-hop neghbors of node ; Its queue length s postve and there exsts a neghbor j Ñt that has a value ut j lower than zero. It s mportant to note that the quanttes n protocol (5) are dscrete-values,.e. the state and other relevant quanttes may only take a countable set of values. In that case, t makes sense to consder a quantsed consensus problem [42], [43]. The proposed Local Votng algorthm conssts of two functons: requestng and releasng free tme slots, and load balancng. For the frst functon (Fg. 3) nodes are examned sequentally at the begnnng of each frame. If a node has an empty queue, then t releases all ts tme slots. If a node has a postve backlog (.e. ts queue s not empty), then t s gven tme slots. All tme slots are examned sequentally, and the frst avalable tme slots that are found, whch are not reserved by one-hop

6 VERGADOS et al.: TOWARD OPTIMAL DISTRIBUTED NODE SCHEDULING IN A MULTIHOP WIRELESS NETWORK 405 balancng functon. If a node has a postve u t value, then t checks f any of ts neghbors has a load lower that ts own and may gve a slot to t wthout causng a conflct. Note that ths s not always the case, because the requestng node may not be able to obtan a slot f one of ts other one-hop or twohop neghbors has also allocated the same slot. The neghbor wth the smallest ut j value gves slots to node. Afterthe exchange u t s reduced by r = mn{u t, u t u t j, p j t 1 },and u j s ncreased by r. Ths procedure s repeated untl u t s t postve, or untl none of the neghbors of node can gve any slots to node wthout causng a conflct. In ths way, n general, slots are removed from nodes wth lower load and are offered to nodes wth hgher load, and eventually the load between nodes wll reach a common value,.e. sem-consensus wll be acheved. Fg. 3. Requestng and releasng tme slots functon for the Local Votng algorthm. Fg. 4. Load balancng functon for the Local Votng algorthm. or two-hop neghbors for transmsson, are allocated to the node. The message exchanges for requestng and releasng slots are consdered equvalent to message exchanges n the DRAND algorthm [44]. If no avalable slot s found (all slots have been allocated to one-hop or two-hop neghbors of the examned node), then no new slot s allocated to the node. On the contrary, f the queue of the node s found to be empty and the node has allocated slots, then all slots are released. The load balancng functon (Fg. 4) s nvoked n order to acheve the objectve of keepng the load balanced. Every node N has a value u t (from the schedulng polcy calculated at the end of the prevous frame) whch determnes how many slots the node should deally gan or lose by the load B. Consensus Propertes of Local Votng 1) Notaton: For the consdered network, N (1) and N (2) do not change over tme snce there s no spatal movement of the nodes. However, the network changes over tme due to the slot allocaton whch s dynamc. Takng ths nto consderaton, we descrbe the structure of the dynamc network (network topology) usng a sequence of drected graphs G At ={(N, E t )} t 0, where E t E. In the consdered case, E t defnes a subset whch conssts of lnks between the nodes that can exchange slots at tme t. Note that these drected graphs G At are not the same as the communcaton graph G. Instead, they defne to whch of the other nodes a node can offer a slot. More specfcally, f there s an edge from node to node j n G At, t means that node has a slot to offer to node j, and after the exchange the produced schedule wll stll reman conflct-free wth respect to Eq. (2). A t =[at, j ] s the correspondng { adjacency } matrx. As defned earler, Ñt = j : at, j > 0 denotes the set of neghbors of node N at tme t,.e. the set of neghbors that can exchange slots wth node. Generally, Ñt = f s S : Xt,s = 1and k N (2), Xt,s Xt k,s = 0. Note that n contrast to N (1) and N (2),thesetÑt N (1) changes n tme. Let E max ={( j, ) : sup t 0 at, j > 0} stand for the maxmal set of communcaton lnks (a set of edges that appear wth non-zero probablty n Ñt ). For any matrx A we defne the weghted n-degree of node as a sum of -th row of the matrx A: d (A) = n j=1 a, j,andd(a) = dag{d (A)} as the correspondng dagonal matrx. Let L(A) = D(A) A denote the Laplacan matrx of the graph G A,andλ 1,...,λ n stand for the egenvalues of the matrx L( A) ordered by ncreasng absolute magntudes. The symbol d max (A) accounts for a maxmum n-degree of the graph G A. 2) Assumptons: Let (, F, P) be the underlyng probablty space correspondng to the sample space, the collecton of all events, and the probablty measure, respectvely, and {F t } be a sequence of σ -algebras whch are generated by {qk, p k } =1,...,n,k=1,...,t. The symbol E accounts for the mathematcal expectaton, E Ft s a condtonal mathematcal expectaton wth respect to the σ -algebra F t, and the followng assumptons are satsfed:

7 406 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 17, NO. 1, JANUARY 2018 A1. a) For all N, j Nmax an appearance of varable edges ( j, ) n the graph G At s an ndependent random event. Nmax s defned by the topology E max. Denote by aav, j the average value of at, j.leta av stand for the adjacency matrx of averaged values aav, j. b) For all N, t = 0, 1,..., the number of slots zt requred to transmt new packets receved by node at frame t n Eq. (3) are random varables do not depend on F t. Note that new packets refer to new ncomng packets from new connectons and new packets arrved from neghbors. c) For all N, j Ñ t and b, j t = q j t+1 qt+1 + k Ñt qt+1 k there exst condtonal average values bav, j = E Ft 1 (bt, j ), whch do not depend on t. Note that b, j t = a, j t /qt+1 and B t = A t Q 1 t+1 where Q t+1 = dag{max{1, qt+1 }}. There exsts a postve defnte matrx Q av > 0suchthat av 2 σq 2. A av = B av Q av,ande Q 1 t+1 Q 1 d) For matrces B t =[b, j t ] and B av =[bav, j ] there exsts a matrx R such that E(L(B av ) L(B t )) T (L(B av ) L(B t )) R, and ts maxmum on the absolute magntude egenvalue: λ max (R) <. e) For all N, t = 0, 1,..., the errors of roundng n LV protocol (5) w t = γ j Ñ t at, j ( x t j x t ) [γ j Ñt at, j ( x t j x t )] (6) are centered, ndependent, and they have a bounded varance E(wt )2 = σw 2 and ndependent of F t. f) For all N, t = 0, 1,...,thevarableset+1 are random, ndependent and dentcally dstrbuted wth mean values ē and varance σe 2, and they do not depend on F t. All varables zt, e t+1,w t are mutually ndependent. We assume that the followng assumpton for the average matrx of the network topology s satsfed: A2: Graph G Aav has a spannng tree, and for any edge ( j, ) E max t holds aav, j > 0. 3) Mean Square ɛ-consensus: Consder the state vectors x t R n, t = 0, 1,..., whch consst of the elements x t 1, x t 2,..., x t n. Note that f state values x t, N, are semequal then the nverse values qt+1 /p t, N for q t+1 > 0, pt > 0 are sem-equal. The followng theorem gves the condtons when the sequence {x t } converges asymptotcally n the mean squared sense to some bounded set around a trajectory x t of the correspondng averaged model x t+1 = x t γ L(B av ) x t + Q 1 av ēt+1, x 0 = 0(= x 0 ). (7) If ē t 0 then x t x as t, and x s a left egenvector of the matrx A av correspondng to ts zero egenvalue. Note that f A av s a symmetrc matrx, then x s equal to x 1 n where 1 n s n-vector of ones,.e. we wll get the asymptotcal consensus for the state vectors { x t }. Theorem 1: If Assumptons A1 A2 are satsfed and 0 <γ < 1 d max (B av ), (8) then TABLE II SIMULATION SETUP ρ = (1 λ 2 (B av )) 2 < 1 (9) and the trajectory { x t } of the system (7) converges to the vector x whch s a left egenvector of the matrx A max correspondng to ts zero egenvalues, and the followng nequalty holds: E x t+1 x t+1 2 2( 1 ρ +ρt E p 0 2 +σq 2 Q av x t+1 2 ), (10) where = n(λ max (R) S +σ 2 e + σ 2 w ). If t, then the asymptotc mean square ε-consensus s acheved wth ε 2 1 ρ + 2σ q 2 Q av x t+1 2. Proof s n the Appendx. Theorem 1 shows that our protocol (5) provdes an approxmate consensus,.e. gves an almost optmal behavor of the system. IV. EVALUATION We have performed a set of smulatons n order to evaluate the performance of dfferent schedulng algorthms. These smulatons are carred out by usng a custom bult, eventdrven smulaton tool developed n Java. The smulaton setup s summarzed n Table II. Although several routng algorthms for load balancng n multhop networks exst, e.g. [45], n ths paper we focus on the nteracton of schedulng and load balancng algorthms. The routng n the network s consdered to follow a smple shortest path routng algorthm. A. The Smulaton Tool The source code that was developed for evaluatng dfferent schedulng algorthms has been made open source and s avalable. 2 The scrpts for runnng the smulatons and producng the results have also been made avalable. 3 The smulaton tool focuses on the evaluaton of the schedulng algorthms. There are two types of scenaros that

8 VERGADOS et al.: TOWARD OPTIMAL DISTRIBUTED NODE SCHEDULING IN A MULTIHOP WIRELESS NETWORK 407 were evaluated. In the frst class of scenaros, a varable number of connectons s consdered, each connecton starts wth a fxed number of packets. Ths represents the response to a sudden burst of traffc. Dfferent load n the network s calbrated by changng the number of connectons. The smulaton s executed untl all packets have reached ther destnatons. In the second class of scenaros, connectons are added constantly, followng a Posson process. The load s calbrated by changng the connecton arrval rate. Ths scenaro s executed for a fxed tme duraton. The measured metrcs for each connecton are: the delvery tme, whch s the tme needed for all packets of a connecton to reach ther fnal destnaton; The delay, whch s tme from the moment each packet s generated untl t has been receved by ts fnal destnaton; The throughput, whch s the number of packets n the connecton, dvded over the tme dfference (n slots) between the start and the completon of the connecton. For each smulaton we used the per connecton metrcs n order to take the average value between the connectons per smulaton, the maxmum and mnmum values for each connecton, and the farness, whch was calculated usng Jan s farness ndex [46]. The smulaton software s organzed nto four packages: the network package contans the mplementaton of the network elements and algorthms, the smulator package whch contans the objects for mplementng the dscrete event smulator, the applcaton package whch mplements the network connectons and the statstcs gatherng functonalty, and the stablty package whch contans the dfferent scenaros to be executed. Some of the network functons that were mplemented n the smulaton tool nclude the followng: a Connecton object represents the applcaton layer. For the purposes of ths smulaton, each connecton has a random source and destnaton. It s ntalzed wth a number of packets that are transmtted. For the frst scenaro (traffc bursts), each connecton has 100 packets. For the steady state scenaro, the number of packets are calculated based on an exponental dstrbuton. The Node object represents each wreless staton n the network. It contans an nfnte FIFO queue that s common for all outgong transmssons. It also has a routng table that s created usng a shortest path algorthm. It contans a set of slot reservatons, as well as X-Y coordnates. A Reservaton object represents the slot reservaton. It contans felds for the transmttng node, as well as the nodes that are blocked due to ths reservaton (all nodes n the two-hop neghborhood, except for the lnk-schedulng case). The Network object mplements network functons, such as routng. The Scenaro object contans the scenaro to be executed, and defnes the scheduler type, the transmsson range, the number of tme slots n each frame, the number of nodes n the network, and the sze of the topology. Each Scheduler also has a dfferent class whch nherts from the TDMAScheduler class. The wreless channel s lossless (unless otherwse specfed). Two nodes are one-hop neghbors f ther dstance s smaller than the transmsson/nterference range. All schedulng algorthms are conflct-free usng the protocol nterference model where two nodes are not scheduled to transmt as long as they are two-hop neghbors. We also consder a scenaro wth a lnkschedulng algorthm where two transmssons are allowed to be concurrent, f each recever receves at most one packet at a tme. B. Implemented Algorthms In ths subsecton we brefly descrbe the operaton of some algorthms for node schedulng from the lterature. We have mplemented these algorthms n our smulaton platform, and compared ther performance wth the performance of Local votng algorthm. A typcal example of a dstrbuted, traffc ndependent, topology dependent node schedulng algorthm s DRAND [44]. DRAND defnes a communcaton protocol for obtanng a conflct-free schedule, usng nformaton from the two-hop neghborhood. The protocol assgns a sngle tme slot to each node. The frame length s constant throughout the network, and t s determned by the maxmum densty of the nodes. Another example of a dstrbuted, traffc ndependent, topology dependent node schedulng algorthm s Lyu s algorthm [47], [48]. The algorthm frst assgns a color to each node, usng exstng graph colorng technques, wth the lmtaton that two nodes are not assgned the same color f they are n the same two-hop neghborhood. Dependng on the color that s assgned to a node, t s a canddate to transmt n any tme slot for whch t mod p(c k ) = c k mod p(c k ), where t s the tme slot, c k s the color assgned to node k, and p(c k ) s the smallest power of 2 greater than or equal to c k. Among these canddate nodes, n each twohop neghborhood, the node wth the largest color transmts. Therefore, n Lyu s algorthm, the nodes have more than one transmsson opportunty n each frame, and there s no common frame length for the entre network. Ths makes slot assgnment easer than n DRAND where the frame length must be known n advance. Lyu s algorthm also has better performance snce the nodes can transmt more frequently, and the performance n sparse areas s not affected by larger node densty elsewhere. The Load-Based Transmsson Schedulng (LoBaTS) [49] protocol s an example of a dstrbuted, traffc dependent, topology dependent node schedulng scheme. It schedules the transmssons usng Lyu s algorthm, but now nstead of each node havng a sngle color, addtonal colors can be assgned to nodes that experence hgh load. Each node mantans an estmate of the utlzaton of every node n ts two-hop neghborhood. If the queue length exceeds a threshold, then the node tres to fnd an addtonal color that: a) s not assgned to any other node n the two-hop neghborhood, and b) does not cause the utlzaton of any other node n the neghborhood to exceed one. If such a color s found, then the node nforms ts neghbors about the new assgnment, and t uses Lyu s algorthm to calculate the new transmsson schedule. A centralzed, traffc dependent, topology dependent node schedulng algorthm was proposed n [50], called Longest Queue Frst (LQF) schedulng. Accordng to ths schedulng

9 408 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 17, NO. 1, JANUARY 2018 Fg. 5. Smulaton results of the dfferent schedulng algorthms for dfferent traffc load n the delvery tme scenaro. algorthm, nodes that have a packet to transmt are ordered accordng to ther queue length n a descendng order. The node wth the longest queue s assgned to transmt n the current tme slot. The remanng nodes are examned one by one, and any node that can transmt n the same tme slot wthout causng a conflct s also assgned to transmt. The LQF polcy s a smple heurstc for slot assgnment, but t s not really practcal, snce t s centralzed and the scheduler requres nformaton about the queue lengths of all nodes n the network. Nevertheless, due to ts smplcty and good performance, ths algorthm has been often used for obtanng theoretcal results and as a benchmark for comparng the performance of schedulng schemes. Ths algorthm s also known as the Greedy Maxmal Schedulng algorthm, and ts performance n terms of capacty has been analyzed n [51]. For the fnal scenaro we used a lnk-schedulng varant of the LQF algorthm. In ths verson of the algorthm, agan the nodes are examned n decreasng queue sze. Ths tme, however, whether the packet wll conflct wth other transmssons depends on the destnaton of the packet (snce we have lnk schedulng). For ths reason, we examne the packets from the start of the queue untl we fnd the frst packet that has a destnaton that doesn t cause a convct wth the already scheduled transmssons n ths slot. Ths packet s added to the slot, and the algorthm contnues wth the next node. C. Delvery Tme Scenaro In ths experment we nvestgate the delvery tme of fxed szed messages, all ntalzed at the same tme. The scenaro has been repeated 500 tmes for each number of connectons and for each of the algorthms. The total number of experments s connectons 5 protocols = experments. At the begnnng of each smulaton a varyng number from 1 to 30 concurrent connectons s generated wth random sources and destnaton nodes. Each connecton generates 1 packet every 5 tme unts untl a total of 100 packets per connecton s generated. The results of the smulaton are depcted n Fg. 5. For each number of concurrent connectons and each algorthm, the above metrcs are averaged over the 500 dfferent smulaton runs. Fg. 5(a) depcts the average end-to-end delvery tmes among all the concurrent connectons. The LQF and the

10 VERGADOS et al.: TOWARD OPTIMAL DISTRIBUTED NODE SCHEDULING IN A MULTIHOP WIRELESS NETWORK 409 Fg. 6. Smulaton results for varyng network densty. Local Votng algorthms acheve the shortest delvery tmes, followed by LoBaTS.TheDRAND and Lyu algorthms exhbt the worst performance, that s expected, snce these two algorthms assgn a fxed number of slots to each node wthout consderng the traffc condtons. Fg. 5(b) presents the farness n terms of the end-to-end delvery tme among connectons that s calculated usng Jan s farness ndex. The LQF and Local Votng algorthms clearly acheve superor farness than other algorthms, regardless of the number of concurrent connectons. Ths llustrates the sgnfcance of load balancng when consderng farness. The LoBaTS algorthm comes thrd (for most traffc loads) snce t s also traffc dependent, whle the DRAND follows t. Lyu s algorthm has the worst farness, and ths valdates what s expected, snce t assgns a dfferent number of tme slots accordng to the nodes color, wthout consderng the traffc condtons. The lack of farness s notceable for all algorthms except LQF and Local Votng, even when the number of connectons s lmted. As the number of connectons ncreases, farness deterorates for all algorthms, but the dfference n performance among the Local Votng and LQF algorthms and the remanng algorthms ncreases as the traffc load ncreases. It should be noted that even the LQF algorthm cannot acheve perfect farness, and ths s due to the dfferent levels of congeston n varous parts of the network. Namely, flows that encounter no (or only lmted) congeston on ther path have shorter delvery tmes than flows that encounter congeston, and ths effect cannot be mtgated by schedulng polces alone. Fg. 5(c) demonstrates the maxmum end-to-end delvery tme, whch s the completon tme of the connecton that ends the latest. Ths s an mportant metrc because t shows after how much tme the system has delvered all packets to ther destnaton, thus, t s related to the capacty of the network. The results confrm our expectatons that the LQF algorthms acheves the best performance. However, the performance of the Local Votng algorthm s very close to optmal. Ths valdates the results of Secton II that load balancng can decrease the overall delvery tme. The slght dfference among these two algorthms can be explaned by two facts: 1) the Local Votng algorthm s dstrbuted, therefore, the delays n propagatng the state affect ts effcency, and 2) slot exchange between two nodes s not always possble n real systems snce allocatons by other neghbors may cause a conflct, thus, t lmts the amount of load balancng that s feasble. The LoBaTS algorthm exhbts worse performance than the frst two algorthms, possbly because t assgns at least one slot to each node, even f the node does not have traffc. DRAND and Lyu s algorthms perform equally badly,.e. several orders of magntude behnd the rest of the algorthms. Ths s expected snce both algorthms do not adapt the schedulng to traffc requrements. Fg. 5(d) depcts the end-to-end delvery tme for the connectons wth the shortest delvery tme. In general, the Local Votng algorthm has slghtly better performance n terms of the mnmum delay compared to the other algorthms. D. The Effect of the Network Densty In ths scenaro we have repeated the experments of secton IV-C, but ths tme we have changed the sze of the topology, whle the number of nodes s kept constant. Ths allows us to nvestgate how the network densty affects the performance of the algorthms. We vary the sze of the network from 10 unts to 200 unts, whle the number of nodes s stll equal to 100, and the transmsson and the nterference ranges are equal to 10 unts. The results are depcted n Fg. 6 for 10 and 30 concurrent connectons, respectvely. In all cases the Local Votng and LQF algorthms have the best performance. Addtonally, the performance of the proposed Local Votng algorthm s very close to the performance of the centralzed LQF scheme n terms of maxmum delvery tme. E. Steady State Scenaro In ths subsecton we evaluate the steady state performance of the load balancng algorthm. Ths scenaro s set up on

11 410 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 17, NO. 1, JANUARY 2018 Fg. 7. Smulaton results for the steady state scenaro. the same network as the prevous one. However, nstead of startng all connectons at the begnnng of the smulaton, the connectons start followng a Posson process where the arrval rate λ s n the range of [ 10 4, 10 1] slots 1, the duraton of each connecton s dstrbuted exponentally wth a parameter of 1/μ = 10 3 slots 1, and the packet nter-arrval tme wthn a connecton s 1 packet every 5 tme slots. The source and the destnaton of the connecton are chosen randomly, followng a unform dstrbuton. The duraton of the smulaton s tme slots. The packets that are receved before slots have elapsed snce the begnnng of the smulaton are gnored. We measure the average end-to-end delvery tme, the average end-to-end delay, the average throughput, and the farness n terms of throughput. Fg. 7(a) presents the average end-toend delvery tme, from the transmsson of the frst packet to the recepton of the last packet of all connectons. The Local Votng algorthm acheves the best performance that s very close to the LQF algorthm. The performance of the LoBaTS algorthm s a bt behnd the frst two algorthms, and the traffc ndependent algorthms acheve the worst performance. In Fg. 7(b) we can see the average end-to-end delay, from the moment a packet was generated untl t was receved by the fnal destnaton. For low arrval rates, the LQF algorthm has the smallest end-to-end delay, followed by the Local Votng, LoBaTS, Lyu s and DRAND algorthms. On the contrary, the average throughput for the LQF, Local Votng, andlobats algorthms has a smlar value, but Lyu s and DRAND acheve lower average throughput (Fg. 7(c)). Fnally, n terms of farness, the Local Votng algorthm s superor for medum arrval rates, but LQF has a superor performance for hgh and low arrval rates. Fg. 8(a) shows the evoluton of the delay per packet per node, for the dfferent algorthms for an arrval rate of 10 3 new connectons per tme slot. The LQF algorthm has the hgher percentage of packets wth very low delay, and ths s expected because there s no frame length, so packets are elgble to be transmtted at the next tme slot. On the contrary, the Local Votng algorthm has a peak n the delay dstrbuton that s close to the frame length of 10. The LoBaTS algorthm has hgher delay, followed by DRAND and Lyu. In Fg. 8(b) we plot the dstrbuton of the end-to-end delay per packet. We can see that the rankng of the algorthms s smlar to the per hop rankng. Ths result valdates that

12 VERGADOS et al.: TOWARD OPTIMAL DISTRIBUTED NODE SCHEDULING IN A MULTIHOP WIRELESS NETWORK 411 Fg. 8. The dstrbuton of delay per packet (nodal and end-to-end) for the steady state scenaro. Fg. 9. Smulaton results for dfferent values of the packet loss, for an arrval rate of 10 4 connectons/slot. Fg. 10. Smulaton results for dfferent values of the packet loss for an arrval rate of 10 3 connectons/slot. optmzng per-node delay through load balancng has a postve effect on end to end delay n a multhop network. F. The Effect of Packet Loss In ths scenaro, we evaluate the performanceof the schedulng algorthms when errors can occur durng the transmsson between nodes. We kept the same parameters as the prevous scenaro, but ths tme we consdered a packet loss probablty n a range from zero (.e. no packet loss) to 0.9. We measure the average delvery tme, the average end-to-end delay, and the average throughput for arrval rates of 10 4 and 10 3 connectons per tme slot. Fg. 9 shows the results for an arrval rate equal to 10 4, and Fg. 10 presents the results for an arrval rate equal to 10 3 connectons per tme slot. In both cases, when the packet loss ncreases, the end-to-end delay also ncreases. Ths s expected, because an ncreased packet loss causes the packets to be re-transmtted, thus, an addtonal delay s experenced. Smlar results may be seen for the delvery tme and the throughput, but are omtted due to space page lmtaton. Fg. 11. Smulaton results for local votng protocol, on the steady state scenaro, wth γ rangng from 10 3 to G. The Effect of the γ Value In ths scenaro we nvestgate the effect of the γ value on the performance of the network. We execute the steady state scenaro for the Local Votng algorthm, but ths tme, we set the γ parameter to dfferent values, from 10 3 to 10 3.

13 412 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 17, NO. 1, JANUARY 2018 the centralzed LQF algorthm whch s consdered to have the best performance. To summarze, we showed the advantage of load balancng when performng schedulng n wreless multhop networks, proposed Local Votng algorthm for load balancng/schedulng, found theoretcal condtons for convergence (reachng consensus), and demonstrated by smulatons that the Local Votng algorthm shows good performance n comparson wth other schedulng algorthms. Fg. 12. Node vs. Lnk schedulng. The results are depcted n Fg. 11. There are sgnfcant dfferences n terms of the end-to-end delay. For the network settngs tested, we observed the best performance wth n terms of delay for γ = 1. H. Node Schedulng vs. Lnk Schedulng All the algorthms studed n ths paper are node-schedulng algorthms. Ths means that the destnaton of each transmsson s not consdered, so the nterference model that s used under node-schedulng s more conservatve than lnkschedulng. On the other hand, node schedulng has a multplexng advantage under ntermttent load. Fg. 12 depcts the results of the frst scenaro, ncludng a lnk-schedulng varant of the LQF algorthm. V. CONCLUSION The problem of schedulng s one of the bg challenges n wreless networks. In ths paper we studed the nteracton of schedulng and load balancng. We showed that the problem of mnmzng the overall delvery tme through a multhop network can be modeled as a consensus problem, where the goal s to sem-equalze the fracton of the number of slots allocated to each node over the queue length of the node. We ntroduced the schedule exchange graph, that s a drected, tme-varyng graph, whch represents whether a node can gve a slot to another node. The problem of wreless schedulng was modeled as a load balancng problem. Takng nto consderaton the dynamcally changng network topology, we ntroduced Local Votng protocol (consensus protocol) to solve the schedulng/load balancng problem. Fnally, we found the condtons that should be met n order for the Local Votng protocol to acheve approxmate consensus, and therefore optmze the delvery tme throughout the network. We compared the performance of the Local Votng algorthm wth other schedulng algorthms from the lterature. Smulaton results valdated the theoretcal analyss and showed that the delvery tmes are mnmzed wth the use of the Local Votng algorthm. The proposed algorthm acheves better performance than the other known dstrbuted algorthms from the lterature n terms of the average delay, the maxmum delay, and the farness. Despte beng dstrbuted, the performance of the Local Votng algorthm s very close to the performance of APPENDIX PROOF OF THEOREM 1 Proof: The result of ths Theorem and ts proof are dfferent from correspondng parts n [35]. The dfference s caused by the dfferent ways of achevng consensus. Whle n [35], consensus s acheved through re-dstrbutng the load or qt, n ths paper consensus s reached through re-dstrbutng slots n a frame,.e. pt. The dea of the proof follows the paper [52]. By vrtue the Eqs. (3) and (6), the dynamcs pt of the closed loop system wth protocol (5) are as follows p t+1 = p t + e t+1 +[γ Ñ t = p t + γ Ñ t a, j t ( x j t x t )] at, j ( x t j x t ) + e t+1 + w t. (11) Denote by p t R n a vector whch conssts of pt, e t+1 R n a vector whch conssts of et+1, and by w t R n a vector of the errors wt,wheret = 0, 1,... Due to the vew of the Laplacan matrx L(A t ) and defnton of Q t+1, we can rewrte Eq. (11) n a vector-matrx form as: p t+1 = p t γ L(A t )Q 1 t+1 p t + e t+1 + w t. (12) We consder that p t = Q av x t. If we multply both sdes of Eq. (7) by Q av, we get that the sequence { p t } s a trajectory of the average system p t+1 = p t γ L(B av ) p t +ē t+1. (13) The vector 1 n s the rght egenvector of the Laplacan-type matrces L t = γ L(A t )Q 1 t+1 = γ L(B t) and L B = γ L(B av ) correspondng to the zero egenvalue: L t 1 n 1 n = L B 1 n = 0. Sums of all elements n the rows of the matrces L t or L B are equal to zero and, moreover, all the dagonal elements are postve and equal to the absolute value of the sum of all other elements n the row. The next Lemma from [53] s useful. Lemma [53]: Laplacan matrx L(B) of graph G B has an algebrac multplcty equal to one for ts egenvalue λ 1 = 0 f and only f graph G B has a spannng tree. Note that graph G Bav has a spannng tree when condtons A1.c and A2 hold. Due to the defntons of the matrces L t and L A,wederve from (12),(13) for the dfference r t+1 = p t+1 p t+1 r t+1 = r t L t p t + L B p t + e t+1 ē t+1 + w t = = (I L B )r t ( L t L B )p t + (e t+1 ē t+1 ) + w t, where I s the dentty matrx.

14 VERGADOS et al.: TOWARD OPTIMAL DISTRIBUTED NODE SCHEDULING IN A MULTIHOP WIRELESS NETWORK 413 Consder the condtonal mathematcal expectaton of the squared norm r t+1 accordng to σ -algebra F t. By vrtue of Assumptons A1.d f we derve E Ft r t+1 2 (I L B )r t 2 + p T t Rp t + nσ 2 e + nσ 2 w. Further, by takng uncondtonal expectaton we get: E r t+1 2 ρe r t 2 +. By Lemma 1 from Chapter 2 of [54] t follows that E r t ρ + ρt E p 0 2. (14) Due to the defntons we have E x t+1 x t+1 2 = E Q 1 t+1 (p t+1 p t+1 ) + (Q 1 t+1 Q av I) x t+1 2 2E Qt+1 1 r t E (Q 1 t+1 Q 1 av )Q av x t+1 2 2( 1 ρ + ρt E p 0 2 ) + 2σq 2 Q av x t+1 2. The proof of the frst part of Theorem 1 s completed. The second concluson about the asymptotc mean square ε-consensus follows from nequalty (10) f t. Snce(9) s satsfed, then the thrd term of (10) exponentally tends to zero. ACKNOWLEDGMENT The authors would lke to thank the anonymous revewers for ther very valuable comments. REFERENCES [1] W. Kess and M. 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Beard, Consensus seekng n multagent systems under dynamcally changng nteracton topologes, IEEE Trans. Autom. Control, vol. 50, no. 5, pp , May [54] B. T. Polyak, Introducton to Optmzaton. New York, NY, USA: Optmzaton Software, Inc., Dmtros J. Vergados (M 04) receved the Ph.D. degree from the School of Electrcal and Computer Engneerng, Natonal Techncal Unversty of Athens, n Snce 2009, he has been nvolved n research projects and as a Vstng Lecturer n several unverstes n Greece and Norway. He s currently a Researcher wth the School of Electrcal and Computer Engneerng, Natonal Techncal Unversty of Athens. He has authored several publcatons n these areas. Hs research nterests nclude wreless networks, smulaton modelng, schedulng algorthms, and multhop networks. Natala Amelna receved the degree from the Department of Informatcs, Sant Petersburg State Unversty, Russa, n 2009, and the Ph.D. degree n mathematcal cybernetcs from the Department of Theoretcal Cybernetcs, Sant Petersburg State Unversty, n She was an ERCIM Post- Doctoral Research Scholar wth the Department of Telematcs, Norwegan Unversty of Scence and Technology, Trondhem, Norway, from 2012 to 2013, and then as a Post-Doctoral Researcher wth Sant Petersburg State Unversty. She s currently a Researcher wth the Faculty of Mathematcs and Mechancs, Sant Petersburg State Unversty. Her research nterests nclude the dstrbuted control of multagent systems, consensus problem, load balancng, and the cooperatve control of UAVs, sensor, and wreless networks. Yumng Jang (SM 14) receved the Ph.D. degree from the Natonal Unversty of Sngapore n From 1996 to 1997, he was wth Motorola, Bejng, Chna, and from 2001 to 2003, wth the Insttute for Infocomm Research, Sngapore. He has been a Professor wth the Norwegan Unversty of Scence and Technology, Trondhem, Norway, snce Hs research nterests are the provson, analyss, and management of qualty of servce guarantees n communcaton networks, wth a partcular focus on stochastc network calculus. Katna Kralevska (M 13) receved the B.Sc. and M.Sc. degrees n telecommuncatons from Ss. Cyrl and Methodus Unversty-Skopje, Macedona, n 2010 and 2012, respectvely, and the Ph.D. degree from NTNU n She s currently a Post- Doctoral Researcher wth the Department of Informaton Securty and Communcaton Technology, Norwegan Unversty of Scence and Technology. Her research nterests nclude appled erasure codng n dstrbuted storage systems and mult-hop wreless communcatons. Oleg Granchn (SM 14) receved the Ph.D. degree wth Lenngrad State Unversty n Hs doctoral thess was Randomzed Algorthms of an Estmaton and Optmzaton Under Arbtrary Noses n 2001 at the Insttute of Control Scences, Russan Academy of Scences (RAS), Moscow. He was awarded the ttle of Professor n He s currently a Professor wth the Computer Scence Department, Faculty of Mathematcs and Mechancs, Sant Petersburg State Unversty, and he s also wth the Insttute of Problems n Mechancal Engneerng, RAS. Hs research actvtes are focused on the multagent adaptve control, compressve sensng, clusterng (data mnng), randomzed algorthms, analyss, and the desgn of complex systems wth uncertanty, and varous applcatons wthn nformaton technology.

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