Achieving Transparent Coexistence in a Multi-hop Secondary Network Through Distributed Computation

Transcription

1 Achevng Transparent Coexstence n a Mult-hop econdary Network Through Dstrbuted Computaton Xu Yuan Y h Y. Thomas Hou Wenng Lou cott F. Mdkff astry Kompella Vrgna olytechnc Insttute and tate Unversty, UA U.. Naval Research Laboratory, Washngton, DC, UA Abstract Transparent coexstence, also known as underlay, offers much more effcent spectrum sharng than tradtonal nterweave coexstence paradgm. In a prevous work, the transparent coexstence for a mult-hop secondary networks s studed. In ths paper, we desgn a dstrbuted soluton to acheve ths paradgm. In our desgn, we show how to ncrease the number of data streams teratvely whle meetng constrants n the MIMO nterference cancelaton (IC) model and achevng transparent coexstence. All steps n our dstrbuted algorthm can be accomplshed based on local nformaton exchange among the neghborng nodes. Our smulaton results show that the performance of our dstrbuted algorthm s hghly compettve when compared to an upper bound soluton for the centralzed problem. I. INTRODUCTION Coexstence of the secondary network wth a prmary network s the key approach to mprove rado spectrum utlzaton. In [7], a novel coexstence paradgm under the name of transparent coexstence s explored for a secondary mult-hop network. Under ths paradgm, there s no change on the prmary network. A prmary network consders tself as the only wreless network that s usng the spectrum and ts spectrum access behavor s not affected by the secondary network. In ths regard, the prmary network s behavor s smlar to that n the nterweave paradgm [2]. The dfference s n the behavor for the secondary network. Instead of accessng the spectrum only through spectrum holes n tme, frequency, or space, a secondary network under the transparent coexstence paradgm s allowed to access the spectrum n the same tme, frequency, and locaton wth the prmary network, as long as ts actvtes are nvsble to the prmary network. uch transparency s acheved by havng the secondary network proactvely cancel ts nterference to the prmary network. ror to [7], there has been some efforts on underlay coexstence paradgm (see, e.g., [1], [4], [8], [9]). But these efforts have been lmted to very smple network settngs, e.g., several nodes or lnk pars, all for sngle-hop communcatons. The transparent coexstence that was studed n [7] focused on mult-hop communcatons, both for the prmary and secondary networks. The results n [7] showed the concept of achevng transparent coexstence for mult-hop prmary and secondary networks through a centralzed soluton. But t s desrable to have a dstrbuted soluton to solve ths problem. Ths s the goal of ths paper. The man contrbuton of ths paper s the development of a dstrbuted schedulng For correspondence, please contact rof. Tom Hou (thou@vt.edu). rmary nodes econdary nodes Fg. 1. A mult-hop secondary network overlad on a mult-hop prmary network. algorthm to allocate secondary network resource to acheve the transparent coexstence paradgm. For IC, we assume each secondary network s equpped wth MIMO, whle there s no requrement on the prmary nodes. We employ a MIMO nterference cancelaton (IC) model that was developed n et al. [5] to keep track of DoF allocaton for IC. It was shown n [5] that ths IC model s effcent n DoF allocaton whle guaranteeng feasblty n the fnal soluton. By feasblty, we mean there exsts a feasble precodng and decodng vector for each data stream at the physcal layer. However, ths model requres to mantan an orderng among the secondary nodes n the network, whch poses a challenge n a dstrbuted network envronment. We show how such orderng relatonshp among the neghborng nodes can be establshed and mantaned at each node n a dstrbuted envronment. We also show how to adust node orderng n our dstrbuted algorthm should t become necessary (e.g., remove bottleneck n DoF resource allocaton). Through numercal results, we show that the teratve dstrbuted algorthm that we propose offers compettve performance when compared wth an upper bound result. The remander of ths paper s organzed as follows. In ecton II, we gve a centralzed problem formulaton under transparent coexstence for a mult-hop prmary and secondary networks. ecton III presents our desgn of an teratve dstrbuted algorthm to acheve the transparent coexstence for a secondary mult-hop network. ecton IV presents smulatons results and demonstrates the compettve performance of the proposed dstrbuted algorthm. ecton V concludes ths paper.

2 II. A CENTRALIZED ROBLEM FORMULATION OF TRANARENT COEXITENCE We consder a mult-hop prmary network, where each node s only equpped wth a sngle antenna. Ths prmary network s co-located wth a mult-hop secondary network n the same geographcal regon, as shown n Fg. 1. For the secondary network, we assume that each node s equpped wth MIMO, whch has the IC capablty that s requred to acheve transparent coexstence. uppose that the prmary and secondary networks share the same spectrum channel n the tme doman, wth T tme slots n a frame. For the prmary network, each node can access any tme slot wthout any consderaton of the secondary network. A secondary node, however, s allowed to use a tme slot only f ts actvty s transparent to the prmary network. Havng accurate channel state nformaton (CI) s crtcal for the secondary nodes to cancel nterference to/from prmary nodes. The problem here s: how can a secondary node obtan the CI between tself and ts neghborng prmary nodes whle remanng transparent to the prmary nodes? We propose the followng soluton to resolve ths problem. For each prmary node, t typcally sends out a plot sequence (tranng sequence) to ts neghborng prmary nodes such that they can estmate the CI between them for communcaton. Ths s the practce for current cellular networks and we assume such a mechansm s avalable for a prmary network. We wll explot ths feature for the secondary nodes to estmate CI. pecfcally, the secondary nodes can overhear the plot sequence sgnal from the prmary node whle stayng transparent. uppose the plot sequence from the prmary nodes s publcly avalable (as n cellular networks) and s known to the secondary nodes. Then the secondary nodes can use ths nformaton and the actual receved plot sequence sgnal from the prmary node for channel estmaton. Based on the recprocty property of a wreless channel [6], the estmated CI can also be used as CIT (channel state nformaton at transmtter sde). mlarly, for each secondary node, t can send out plot sequence to ts neghborng secondary nodes such that those nodes can estmate the CI. Therefore, a secondary node can obtan complete CI between tself and ts neghborng prmary and secondary neghborng nodes. uppose that there s a set of sessons F n the prmary network. The routng for each prmary sesson can be found by standard routng protocol, e.g., OLR or AODV. Denote L as the set of actve lnks assocated wth sessons F. Denote z ( l) (t) as the number of data streams on lnk l L n tme slot t. nce each prmary node has only a sngle antenna, z ( l) (t) = 1 f lnk l s actve n tme slot t and 0 otherwse. For the secondary network, suppose that there s a set of sessons F n. Agan, the routng of each secondary sesson can be found by standard routng protocol. Denote L as the set of lnks assocated wth sessons F. Denote A as the number of antennas on a node. Denote z (l) (t) as the number of data streams on lnk l L n tme slot t. For each secondary node, t needs to know the neghborng prmary lnk schedulng, whch can be obtaned by montorng/sensng ts neghborng prmary nodes actvtes. Tme lot chedulng. Denote x (t) as a bnary varable ndcatng whether or not secondary node s a transmtter n tme slot t. That s, x (t) = 1 f node s transmttng n t and 0 otherwse. mlarly, denote y (t) as a bnary varable ndcatng whether or not secondary node s a recever n tme slot t. We assume that the secondary transcever s half-duplex. nce a node cannot transmt and receve n the same tme slot, we have: x (t) + y (t) 1 (, 1 t T ). (1) M and IC at a econdary Transmtter. If a secondary node s a transmtter n a tme slot, t should allocate DoF for M and IC. For M, denote L Out as the set of outgong lnks from node. Then, the number of DoFs at secondary node that s used for M s z (l) (t). For IC, the secondary transmtter should perform both nter-network IC (.e., cancelng ts nterference to ts neghborng prmary recevers) and ntra-network IC (.e., cancelng ts nterference to a subset of ts neghborng secondary recevers). () For IC to ts neghborng prmary recevers, denote Ĩ as the set of neghborng prmary nodes that are located wthn In the nterference range of node. Denote L p as the set of ncomng prmary lnks to node p Ĩ. Then, the number of DoFs requred for IC to node s neghborng prmary recevers s z ( l) (t). p Ĩ l LIn p () For IC to node s neghborng secondary recevers, we employ the node-orderng scheme proposed n [5]. The dea s to put all the nodes n the network nto an ordered node lst. An optmal orderng of such a node lst s part of the optmzaton problem. Denote π(t) as an ordered lst of the secondary nodes n tme slot t (1 t T ) and π (t) as the poston of node n π(t). Therefore, 1 π (t), (, 1 t T, = ). (2) To model the relatve orderng between any two secondary nodes and n π(t), we denote a bnary varable θ, (t) as the relatve poston of node and node n π(t) as follows: θ, (t) = 1 f node s before node and 0 otherwse. Then, we have π (t) θ, (t)+1 π (t) π (t) θ, (t)+ 1, (3) where (,, 1 t T ). Denote I as the set of neghborng secondary nodes that are located wthn the nterference range of node, and L In as the set of ncomng secondary lnks to node. It was shown n [5] that node only needs to cancel ts nterference to these secondary recevers that are before tself n the ordered node lst π(t). Node does not need to be concerned wth ts nterference to those secondary recevers that are after tself n π(t) as such nterference wll be canceled by those nodes later when we consder them n the ordered node lst. Therefore, the

3 number of DoFs requred for IC to secondary recevers s Tx(k) θ, (t) z (k) (t). I Combnng DoF consumpton for M and IC, we have: z (l) (t) + z ( l) (t) + I θ,(t) p Ĩ Tx(k) l LIn p z (k) (t) A, (4) whch means the total number of DoFs consumed at ths transmtter for M and IC cannot exceed the total number of ts DoFs. M and IC at a econdary Recever. If a secondary node s a recever n a tme slot, t should allocate DoFs both for M and IC. For M, The number of DoFs used for M at a secondary recever n tme slot t s z (k) (t). For IC, the secondary recever should perform both nter-network IC (.e., cancelng the nterference from ts neghborng prmary transmtters) and ntra-network IC (.e., cancelng the nterference from a subset of ts neghborng secondary L Out transmtters). Denote p as the set of outgong prmary lnks from prmary node p. Then at secondary recever, the number of DoFs used for IC from prmary transmtters s z ( l) (t). To cancel the nterference p Ĩ l LOut p from ts neghborng secondary transmtters, we agan employ the node-orderng scheme. mlar to that n the secondary transmtter case that was dscussed earler, node only needs to cancel the nterference from these secondary transmtters that are before tself n the ordered node lst π(t). Node does not need to be concerned wth the nterference from those secondary transmtters that are after tself n π(t) as such nterference wll be canceled by those nodes later when we consder them n the ordered node lst. At receve node, the number of DoFs used to cancel nterference from other secondary Rx(l) transmtters s θ, (t) z (l) (t). ummng up I DoF allocatons for M and IC at a secondary recever, we have: z (k) (t) + z ( l) (t) + I θ, (t) p Ĩ Rx(l) l LOut p z (l) (t) A. (5) Lnk Capacty Constrant. For smplcty, we assume that fxed modulaton and codng scheme (MC) s used for each data stream and that each data stream corresponds to one unt data rate. Denote r(f) as the rate of sesson f F. Then, for each lnk l L, we have the followng lnk capacty constrant: f traversng l f F r(f) 1 T T z (l) (t) (l L). (6) A. roblem Formulaton For the centralzed problem formulaton, we consder a throughput optmzaton problem, wth the obectve of maxmzng the mnmum sesson rate r mn among all secondary sessons. The optmzaton problem can be wrtten as follows: t=1 max r mn s.t r mn r(f) (f F); Half duplex constrants: (1); Node orderng constrants: (2), (3); M and IC at secondary transmtters: (4); M and IC at secondary recevers: (5); Lnk capacty constrants: (6). The above formulaton s n the form of mxed nteger nonlnear programmng (MINL), but can be reformulated nto a mxed nteger lnear programmng (MIL) through Reformulaton-Lnearzaton Technque(RLT) [3, Chapter 6]. However, a MIL problem s stll N-hard n general. The goal of ths paper s to desgn a dstrbuted algorthm that can offer compettve results to ths problem. III. A DITRIBUTED OLUTION We propose an effcent dstrbuted schedulng algorthm to acheve transparent coexstence of the mult-hop secondary network wth the prmary network. The man dea s to ncrease the number of data streams for each lnk by 1 (by consderng all actve lnks) durng an teraton. For each lnk, we try to ncrease the number of DoFs for M on ths lnk by 1 at both the transmtter and the recever. Ths ncrement s successful only f the neghborng nterference constrants are satsfed at both transmt and receve nodes of ths lnk. If ths ncrement s not successful for any lnk durng an teraton, we wll then try to make a local adustment of node orderng, wth the goal of freeng up some DoFs for M/IC after the adustment. Agan, a local node orderng adustment s successful only f the DoF constrants at each node are satsfed after the adustment. At any teraton when a local node orderng adustment s not successful (and thus the number of data streams on the assocated lnk cannot be further ncreased), our algorthm termnates. A. tep 1: Choosng a Lnk The goal of ths step s to choose each actve lnk durng an teraton (for data stream ncrement). If a lnk s traversed by multple sessons, t s necessary to represent the lnk multple tmes to ensure that each sesson s consdered for data stream ncrement. There are many ways to acheve ths. In our soluton, we propose to employ the so-called dstrbuted rankng algorthm by Zaks [10]. Ths algorthm was desgned to solve the problem of sortng and rankng n processors n a dstrbuted system. The nput s an ntal value for each processor (not necessarly dstnct). The output s a rankng of all n processors.

4 To apply the dstrbuted rankng algorthm, we need to assgn an ntal value for each lnk. Ths value wll be randomly generated and mantaned by the transmtter of the lnk. After applyng the dstrbuted rankng algorthm, the transmtter of each lnk wll mantan ths lnk s rank. After each lnk obtans ts rank, t wll know precsely whch tme slot t wll be chosen for data stream ncrement. pecfcally, for lnk l wth rank(l), t wll be chosen n the [kl+rank(l)]-th tme slot, 1 where k = 0, 1, 2,, and L s the total number of actve secondary lnks. B. tep 2: Data tream Increment Once a lnk s chosen (n tep 1), we attempt to ncrease one data stream (for M) on the selected lnk, whle satsfyng IC and transparency to the prmary network. We frst present the data structure that s mantaned at each secondary node. Then we present the necessary condtons under whch one more data stream can be added on the lnk n a tme slot. Fnally, we descrbe how to update state nformaton on the nodes that are nvolved n ths ncrement. tate Informaton In our algorthm, a secondary node mantans the followng state nformaton: s (t): The status of node n tme slot t,.e., s (t) = Tx, Rx or Idle represents that node s a transmtter, recever or dle n tme slot t. ω (t): A lst that represents a local orderng of nodes wthn node s nterference range (ncludng node ). B (t): The set of nodes that are before node n ω (t) n tme slot t. Y (t): The set of nodes that are after node n ω (t) n tme slot t. (t): The number of DoFs that node has allocated for M n tme slot t. λ IC (t): The number of DoFs that node has allocated for IC n tme slot t. (t): The number of remanng DoFs at node n tme slot t,.e., (t) = A λ M (t) λ IC (t). α (t): The total number of data streams transmtted by those prmary transmtters that may nterfere wth node s recepton n tme slot t,.e., α (t) = z ( l) (t) λ M. p Ĩ l LOut p β (t): The total number of data streams receved by those prmary recevers that are wthn node s nterference range n tme slot t,.e., β (t) = z ( l) (t). p Ĩ l LIn p α (t): The total number of data streams transmtted by those secondary transmtters that may nterfere wth node s recepton n tme slot t,.e., α (t) = I Rx(k) k L Out z (k) (t). β (t): The total number of data streams receved by those secondary recevers that are wthn node s nterference 1 Ths tme slot refers to schedulng n a control channel, whch s dfferent from schedulng for data transmsson n the data channel. Fg. 2. Four cases of lnk status. range n tme slot t,.e., β (t) = I Tx(k) z (k) (t). z, (t): The number of data streams transmtted from node to node. Durng the ntalzaton stage, each node s set to Idle,.e., s (t) =Idle for, t = 1, 2,, T. The ntal lst ω (t) s set to for, ndcatng that a local orderng of nodes s null. As a result, B (t) = and Y (t) = for. nce the ntal DoF allocaton for M and IC at each node s 0, we have λ M (t) = λ IC (t) = 0, (t) = A and z, (t) = 0 for,, t = 1, 2,, T. α (t) and β (t) are constants and are calculated based on actve sessons n the prmary network. On the other hand, the ntal values for α (t) and β (t) are 0. Except that α (t) and β (t) are constants, the values for s (t), ω (t), B (t), Y (t), λ M (t), λ IC (t), (t), z, (t), α (t) and β (t) are varables and wll be updated durng each teraton of the algorthm. uffcent Condtons for Data tream Increment. We now dscuss when the number of data streams on a chosen lnk can be ncremented by 1 n a gven tme slot. uppose lnk (, ) s the lnk. Then both nodes and frst check ther current status ( Tx, Rx, or Idle ). ome cases can be clearly ruled out for consderaton,.e., s (t) = Rx or s (t) = Tx. In ths case, lnk (, ) cannot be consdered for data stream ncrement n t and we move to the next tme slot (t + 1) mmedately. Otherwse, f lnk (, ) s sutable for stream ncrement, there are four possble states as shown n Fgure 2. The suffcent condtons for data stream ncrement on lnk (, ) are as follows: Case (a): s (t) = Idle and s (t) = Idle. s (t) = Idle: nce node s dle, ts local orderng lst ω (t) s empty. We need to establsh a new ω (t). But the local orderng of node s actve neghborng nodes are not yet establshed. To get around ths ssue, we can consder puttng node ether as the frst or the last node n ω (t), by treatng the relatve orderng of the other nodes n ω (t) as a blackbox (.e., wthout explct knowledge of ts detals). If node s at the begnnng of ω (t), then the followng two condtons must be satsfed: () the total number of DoFs at node should be greater than the total number of data streams receved by ts neghborng prmary recevers,.e., A > β (t), () all secondary recevers that are after node n ω (t) must have at least one remanng DoF to cancel one more nterference stream from node. If node s at the end of ω (t), the followng condton must be satsfed: the total number of DoFs

5 at node s more than the sum of data streams receved by both neghborng prmary and secondary recevers,.e., A > β (t) + β (t). s (t) = Idle: mlar to node, we consder puttng receve node ether as the frst or the last node n ω (t) by treatng the relatve orderng of the other nodes n ω (t) as a blackbox. The suffcent condtons for as the frst or the last node are smlar as, we omt ts dscusson here. If the condtons for s (t) = Idle and s (t) = Idle are both satsfed, we proceed wth ths ncrement and update state nformaton at nodes and and ther neghborng nodes accordng to Fgure 3 and Fgure 4. tate update at dle node and neghborng recever k 1. If s (t) = Idle: 2. Update s (t) = Tx; λ M (t) λ M (t) + 1; (t) (t) 1; z,(t) z,(t) If node s put at the begnnng of ω (t): 4. Y (t) {Neghborng actve secondary recevers.} 5. Update λ IC (t) β (t); (t) (t) λ IC (t); 6. For each receve node k Y (t): 7. B k (t) B k (t) {}; λ IC k (t) λ IC k (t) + 1; k (t) 8. Else f node s put at the end of ω (t): 9. B (t) {Neghborng actve secondary recevers.} 10. Update λ IC (t) β (t) + β (t); (t) (t) λ IC (t). 11. For each node k B (t): 12. Y k (t) = Y k (t) {}. Fg. 3. seudocode to update state nformaton when s (t) = Idle. tate update at dle node and neghborng transmtter k 1. If s (t) = Idle: 2. Update s (t) = Rx; λ M (t) λ M (t) + 1; (t) (t) 1; z, (t) z, (t) If node s put at the begnnng of ω (t): 4. Y (t) {Neghborng actve secondary transmtters.} 5. Update λ IC (t) β (t); (t) (t) λ IC (t) 6. For each transmt node k Y (t): 7. B k (t) B k (t) {}; λ IC k (t) = λ IC k (t) + 1; k (t) = 8. Else f node s put at the end of ω (t): 9. B (t) {Neghborng actve secondary transmtters.} 10. Update λ IC (t) α (t) + α (t); (t) (t) λ IC (t). 11. For each k B (t): 12. Y k (t) Y k (t) {}. Fg. 4. seudocode to update state nformaton when s (t) = Idle. Case (b): s (t) = Tx and s (t) = Idle. s (t) = Tx : In ths case, the followng condtons must be satsfed f node wants to ncrease one more data stream on lnk (, ): () node has at least one remanng DoF for M,.e., (t) 1; () each receve node k n ω (t) that s after node,.e., k Y (t), has at least one remanng DoF to cancel the new nterference from node. s (t) = Idle : Ths case has been dscussed n Case (a). If the condtons for s (t) = Tx and s (t) = Idle are both satsfed, we proceed wth ths ncrement and update state nformaton at nodes, and ther neghborng nodes accordng to Fgure 5 and Fgure 4. tate update at transmt node and neghborng recever k 1. If s (t) = Tx: 2. Update λ M (t) λ M (t) + 1; (t) (t) 1; z, (t) = z, (t) For each receve node k Y (t): 4. Update λ IC k (t) λ IC k (t) + 1; k (t) Fg. 5. seudocode to update state nformaton when s (t) = Tx. Case (c): s (t) = Idle and s (t) = Rx. s (t) = Idle: Ths case has been dscussed n Case (a). s (t) = Rx: In ths case, the followng condtons must be satsfed f node wants to ncrease one more data stream on lnk (, ): () node has at least one remanng DoF for M,.e., (t) 1; () each transmt node k n ω (t) that s after node,.e., k Y (t) has at least one remanng DoF to cancel ts nterference to node. If the condtons for s (t) = Idle and s (t) = Rx are both satsfed, we proceed wth ths ncrement and update state nformaton at nodes, and ther neghborng nodes accordng to Fgure 3 and Fgure 6. tate update at receve node and neghborng transmtter k 1. If s (t) = Rx: 2. Update λ M (t) λ M (t) + 1; (t) (t) 1; z, (t) = z, (t) For each transmt node k Y (t): 4. Update λ IC k (t) λ IC k (t) + 1; k (t) Fg. 6. seudocode to update state nformaton when s (t) = Rx. Case (d): s (t) = Tx and s (t) = Rx. The case for s (t) = Tx has been dscussed n Case (b) and s (t) = Rx has been dscussed n Case (c). If the condtons for s (t) = Tx and s (t) = Rx are both satsfed, we proceed wth ths ncrement and update state nformaton at nodes, and ther neghborng nodes accordng to Fgure 5 and Fgure 6. Recall that there are T tme slots n a tme frame. If the data stream ncrement operaton descrbed above fals n the frst tme slot, we try t agan n the second tme slot and so forth, untl a data stream ncrement s successful n a tme slot or fals after all T tme slots. C. tep 3: Adustng Local Node Orderng If the suffcent condtons at ether node or node cannot be satsfed, we move on to ths step. The only reason why lnk (, ) fals to ncrease one data stream n step 2 s the lack of DoF resources at some nodes. nce the local orderng of a node drectly affects the node s DoF consumpton for IC (see (4) and (5)), we wll try to adust a node s local orderng and thus change ts DoF consumpton for IC. Ths can be done by swappng the poston of the bottleneck node wth

6 some other node ahead of tself n the local node orderng, thereby transferrng the IC responsblty from tself to the other node. nce some new DoF resources for the bottleneck become avalable, a new data stream ncrement on lnk (, ) may be possble. The man dea of ths step s as follows. For each tme slot t, we dentfy the set of bottleneck nodes (denoted as D (,) (t)), whch do not have enough remanng DoF resources should one more data stream s added onto lnk (, ). For each node k D (,) (t), we perform local orderng adustment for k by swappng ts poston wth some other node before k n ts local node orderng. To ensure feasblty, only a subset of nodes (denoted as B k (t)), Bk (t) B k (t), s elgble for swappng wth k. After dentfyng B k (t) for k, we consder nodes n B k (t) n the order of non-ncreasng remanng DoFs,.e., startng wth the one that has the maxmum remanng DoF (denoted as node a) after t s swapped wth node k. If ths swap s nfeasble, then local node orderng adustment fals n ths tme slot and we move on to the next tme slot. Otherwse, we swap k and a and update ther state nformaton. In ts new poston, f a new data stream can be added on lnk (, ), we are done. Otherwse, we contnue by consderng swappng k wth the next node n B k (t) that has the maxmum remanng DoF (denoted as node b) followng the same process. The algorthm termnates upon a new data stream can be successfully added on lnk (, ) or all nodes n D (,) (t) are consdered for all tme slots n a frame. IV. IMULATION REULT We present smulaton results to demonstrate the performance of the proposed dstrbuted algorthm. nce the centralzed problem formulaton s MIL, whch s N-hard n general, we cannot obtan the optmal soluton for comparson. Instead, we wll compare the performance of our algorthm to an upper bound of the obectve for the centralzed problem. uch an upper bound can be obtaned by runnng CLEX for a gven termnaton tme (e.g., 8 hours n our study). uch a comparson approach s very conservaton. Ths s because the optmal obectve value (not obtanable) to the centralzed problem les between the upper bound and the feasble soluton obtaned by our dstrbuted algorthm. Therefore, f the feasble soluton from our dstrbuted algorthm s close to the upper bound by CLEX, then we can clam that our soluton (obectve) s even closer to the optmal obectve and thus s compettve. We consder a secondary CR network co-locates wth a prmary network wthn an area of For generalty, we normalze the unts for dstance, bandwdth, and data rate wth approprate dmensons. Each node (both prmary and secondary) s randomly deployed nsde the area. The prmary nodes are tradtonal sngle-antenna node whle the secondary nodes are equpped wth MIMO, wth four antennas on each node. We assume that each node s transmsson range and nterference range are 30 and 50, respectvely. We assume a tme frame s dvded nto T = 10 tme slots. We show results for one network nstance, wth 20 prmary nodes and 30 secondary nodes. The locatons of each node are shown n Fgure 7.We assume there are three prmary sessons and four secondary sessons, wth each sesson s source and destnaton nodes shown n ths fgure. We assume that mnmum-hop routng s used for each prmary and secondary sesson. Fgure 7 shows the routng n prmary and secondary network (where the sold arrows represent prmary lnks and the dashed arrows represent the secondary lnks). chedulng for the prmary and secondary lnks s also gven n ths fgure, where numbers n the box represents the tme slots used by ether a prmary or secondary lnk. Note that schedulng for the prmary lnks s decded by the prmary network, whle schedulng for each secondary lnk s found by our dstrbuted algorthm Fg. 7. Routng for each sesson and schedulng on each lnk for both prmary and secondary networks. The numbers n the box next to a lnk show the tme slots when the lnk s actve. The obectve value obtaned from our dstrbuted algorthm s 0.6 (n less than a second computaton tme). On the other hand, the upper bound obtaned by CLEX s 0.7 (wth a cutoff tme of 8 hours). As dscussed, snce the optmal soluton les between 0.6 and 0.7, our obectve value (0.6) should be very close to the unknown optmal. To show the transparent coexstence between prmary and secondary networks, we focus on one tme slot, say 6. Fgure 8 shows the set of actve lnks n tme slot 6 for both networks. In ths tme slot, transparent coexstence s acheved on secondary lnks 28 17, 13 24, 30 12, 3 1, 4 11 and 4 5 We frst consder nter-network IC: For secondary lnk 28 17, ts nterference to 9 on prmary lnk 4 9 s canceled by 28 wth 1 DoF, whle the nterference from 4 and 1 to 17 s canceled by 17, each wth For secondary lnks 3 1, 30 12, 13 24, 4 11 and 4 5, the nterference from ther transmtters ( 3, 30, 13, 4 ) to recever 8 on prmary lnk 1 8 s canceled by 3, 30, 13 and 4, each wth The nterference from 1 to 12 and 24 s

7 canceled by 12 and 24 wth 1 DoF, respectvely, and the nterference from 4 to 11 s canceled by 11 wth For ntra-network IC wthn the secondary network, our soluton shows that: 11 s cancelng nterference from 3 and 4, each wth 5 s cancelng nterference from 3 and 4, each wth The nterference from 4 to 1 s canceled by 1 wth 1 DoF. The nterference from 3 to 12 s canceled by 12 wth The nterference from 13 to 12 s canceled by 13 wth 2 DoFs. The nterference from 30 to 1 and 11 s canceled by 30, each wth The detals of DoF allocaton for M and IC at each actve secondary node n tme slot 6 are shown n Table I. In ths table, the second and thrd columns represent the set of secondary nodes that are before and after ths node n ts local ordered lst, respectvely. The fourth column represents the number of DoFs allocated for M. The ffth column represents the number of DoFs that are allocated for IC to/from prmary network. The last column represents the number of DoFs allocated for IC for the set of secondary nodes before tself n the local ordered lst wthn the secondary network. Fg rmary lnk econdary lnk Interference Actve lnks n tme slot 6 n both prmary and secondary networks. V. CONCLUION Transparent coexstence paradgm s attractng attenton as t offers much mproved performance than tradtonal nterweave paradgm. In ths paper, we studed how to desgn a dstrbuted optmzaton algorthm to acheve transparent coexstence for mult-hop prmary and secondary networks. By employng MIMO at each secondary node for IC, we showed that a dstrbuted algorthm can be desgned to ensure nter-network nterference s canceled by the secondary nodes whle undesrable ntra-network nterference can be canceled wthn the TABLE I DOF ALLOCATION FOR M AND IC AT EACH ACTIVE ECONDARY NODE IN TIME LOT 6. Node B (t) Y (t) DoF IC to/from DoF for IC wthn for M prmary secondary network 1 { 4} { 30} { 5, 11, 12} { 1, 11} { 3, 4 } { 3, 4 } { 30 } { 3 } { 13 } { 12 } { 1, 11 } secondary network. We showed that each step n the dstrbuted algorthm can be accomplshed through local computaton based on nformaton exchange among the neghborng nodes. Through smulaton study and comparng our results to an upper bound result, we conclude that the dstrbuted algorthm offers compettve throughput performance when achevng transparent coexstence. ACKNOWLEDGMENT Ths work was supported n part by the NF, ONR, and Vrgna Tech Insttute for Crtcal Technology and Appled cence. The work of Dr.. Kompella was supported n part by the ONR. art of rof. W. Lou s work was completed whle she was servng as a rogram Drector at the NF. Any opnon, fndngs, and conclusons or recommendatons expressed n ths paper are those of the authors and do not reflect the vews of the U.. government. REFERENCE [1] F. Gao, R. Zhang, Y.-C. Lang, and X. Wang, Desgn of learnngbased MIMO cogntve rado systems, IEEE Transactons on Vehcular Technology, vol. 59, no. 4, pp , May [2] A. Goldsmth,.A. Jafar, I. Marc, and. rnvasa, Breakng spectrum grdlock wth cogntve rados: An nformaton theoretc perspectve, roceedngs of the IEEE, vol. 97, no. 5, pp , May [3] Y.T. Hou, Y. h, and H.D. heral, Appled Optmzaton Methods for Wreless Networks, Cambrdge Unversty ress, 2014, IBN-13: [4].-J. Km and G.B. Gannaks, Optmal resource allocaton for MIMO ad hoc cogntve rado networks, IEEE Transactons on Informaton Theory, vol. 57, no. 5, pp , May [5] Y. h, J. Lu, C. Jang, C. Gao, and Y.T. Hou, A DoF-based lnk layer model for mult-hop mno networks, IEEE Trans. on Moble Computng, vol. 12, ssue 7, pp , July [6] G.. mth, A drect dervaton of a sngle-antenna recprocty relaton for the tme doman. n IEEE Trans. on Antennas and ropagaton, vol. 52, no. 6, pp , June [7] X. Yuan, C. Jang, Y. h, Y.T. Hou, W. Lou, and. Kompella, Beyond nterference avodance: On transparent coexstence for multhop secondary CR networks, roc. IEEE ECON, pp , New Orleans, LA, June 24 27, [8] R. Zhang and Y.-C. Lang, Explotng mult-antennas for opportunstc spectrum sharng n cogntve rado networks, IEEE Journal of elected Topcs n gnal rocessng, vol. 2, no. 1, pp , February [9] Y.J. Zhang and A.M.-C. o, Optmal spectrum sharng n MIMO cogntve rado networks va semdefnte programmng, IEEE Journal on elected Areas n Communcatons, vol. 29, no. 2, pp , February [10]. Zaks, Optmal dstrbuted algorthms for sortng and rankng, IEEE Trans. on Computers, vol. 5, no. 1, pp , Aprl 1985.