TDMA-based scheduling for multi-hop wireless sensor networks with 3-egress gateway linear topology
Linh Vu Nguyen, Nguyen Viet Ha, Masahiro Shibata, Masato Tsuru
GGraphical Abstract
TDMA-based scheduling for multi-hop wireless sensor networks with 3-egress gateway lineartopology
Linh Vu Nguyen,Nguyen Viet Ha,Masahiro Shibata,Masato Tsuru a r X i v : . [ c s . N I] F e b ighlights TDMA-based scheduling for multi-hop wireless sensor networks with 3-egress gateway lineartopology
Linh Vu Nguyen,Nguyen Viet Ha,Masahiro Shibata,Masato Tsuru• Modeling and investigating linear topologies with three gateways at the edges (Y-shaped topology) for multi-hopwireless sensor networks to resolve interferences among relay nodes and unreliable lossy links.• Designing a global static time-slot allocation to maximize the theoretical probability of successful delivery withinone cycle period with redundant transmissions.
DMA-based scheduling for multi-hop wireless sensor networkswith 3-egress gateway linear topology
Linh Vu Nguyen, Nguyen Viet Ha, Masahiro Shibata and Masato Tsuru
Computer Science and System Engineering, Kyushu Institute of Technology, Fukuoka, Japan
A R T I C L E I N F O
Keywords :Multi-hop wireless sensor networksTDMA-based transmission schedul-ingLinear topologyLagrangian multiplier method
A B S T R A C T
Packet transmission scheduling on multi-hop wireless sensor networks with 3-egress gatewaylinear topology is studied. Each node generates a data packet in every one cycle period and for-wards it bounded for either of gateways at edges. We focus on centrally-managed Time DivisionMultiple Access (TDMA)-based slot allocations and provide the design of a packet transmissionscheduling framework with static time-slot allocation and basic redundant transmission to re-duce and recover packet losses. We proposed three general path models, which would representall possible variations of this topology, and derived a global static time-slot allocation on eachof models to maximize the theoretical probability that all packets are successfully delivered toone of the gateways within one cycle period.
1. Introduction
Multi-hop wireless networks are in widespread use nowadays due to their cost-efficiency and flexibility in deploy-ment and operation. They can connect nodes in an extensive coverage area larger than a single hop radio range withproper transmission power. Consequently, multi-hop wireless networks are an excellent candidate for emerging IoTsystems, in which surveillance sensors are deployed along with a road, river, or electricity pylons network when acommercial communications infrastructure is unavailable or too costly. However, especially when the number of hopsis large, multi-hop wireless networks for field monitoring often suffer from frequent packet losses due to attenuationand fading on each link as well as radio interferences of simultaneous transmissions among nodes. Furthermore, intypical multi-hop sensor network scenarios, since each packet conveying sensing data should be forwarded toward oneof the sink nodes, the links near a sink are likely congested to forward all packets coming from upstream nodes. Ingeneral, to cope with frequent packet losses, there is a proactive approach, e.g., redundant transmissions of originalor coded packets with forwarding erasure correction (FEC); a reactive approach, e.g., retransmission of lost packetsby automatic repeat request (ARQ); and a combination of them, e.g., Hybrid ARQ. To avoid or reduce interferences(conflicts) of simultaneous packet transmissions, there is a centralized scheduling-based approach, e.g., Time DivisionMultiple Access (TDMA), and a decentralized contention-based approach, e.g., CSMA/CA.In this paper, our target is a stationary but lossy backbone network to forward the sensing data from sensors to mul-tiple egress “gateways” that are connected to a central data collection via an infrastructural reliable network. Therefore,a “node” does not represent each sensor but rather a low-cost relay node. Each node periodically gathers sensing datafrom nearby end sensors and forwards them in a hop-by-hop store-and-forward manner to one of the gateways withinone cycle time-period. Neighboring relay nodes communicate with each other by a simple omnidirectional antennausing the same single frequency. Note that there must be some way to gather sensing data from end sensors connectedto the relay node, e.g., a short-range wireless link different from the links between relay nodes in terms of types andfrequencies, which is out of the paper scope.In our study scope, to be more exact, we consider kinds of stationary “linear topologies” that are simple but typicalfor geographically elongated field monitoring, rather than mesh, complex, or dynamic topologies. In addition, sincetargeting a dedicated centrally-managed network (consisting of low-cost relay nodes, gateways, and a central server), weadopt a centrally-managed TDMA-based packet transmission scheduling with redundant transmissions assuming thatthe link layer does not provide any ARQ and transmission power adaptation mechanisms. A survey paper discussed thetypes of linear wireless sensor networks and their applications (Imad Jawhar (2011)). In linear topologies, the number [email protected] (L.V. Nguyen); [email protected] (N.V. Ha); [email protected] (M. Shibata); [email protected] (M. Tsuru)
ORCID (s):
Linh Vu Nguyen et al.:
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DMA-based scheduling with 3-egress gateway linear topology of neighboring nodes is limited as well as the distance between of neighboring nodes is generally long. Therefore,the number of potential interference patterns and the number of possible routing options are also limited, which mayallow us to pursue an optimality easier compared with general dense topologies. On the other hand, a small numberof possible routes will be a disadvantage in terms of robustness against a failure of nodes or links. It is worth notingthat we focus on cases in which gateways are placed outer-side rather than inner-side of a linear topology, e.g., at acenter, which is essentially different from “tree topologies”. Tree topologies are often used and can reduce the numberof necessary gateways but may suffer from heavy congestions around gateways. In our previous work, we focusedon tandemly-arranged topology networks with two gateways at both edges of a linear network (Agussalim (2016);R. Kimura (2020); R. Yoshida (2020)). Assuming that the topology, the data transmission rate (i.e., bandwidth) andthe time-averaged packet loss rate of each link, the packet size, and the packet generation rate of each node are known,we successfully derived an “optimal” static packet transmission time-slot allocation under a basic controlled redundanttransmission scheme.
Figure 1:
An example of Y-shaped network topology used in this research
Following our previous work, in this paper, we present the models and the performance investigation on the 3-egressgateway linear topology that is a linear topology with three gateways at the edges, called the Y-shaped topology. In anyY-shaped topology, there is a central node that potentially has three links but some of three links are not necessarily usedfor data transmission. An example is illustrated in Fig. 1. The Y-shaped topology represents not only a topologicalform as a graph but three gateways’ locations at the edges (outer-side) as mentioned above. This paper is a fully-extended version of our most recent publication in which we used the term T-shaped topology (L.V. Nguyen (2020)).Our proposed scheme aims at a global static time-slot allocation on Y-shaped topologies to maximize the theoreticalprobability that all packets are successfully delivered to one of the gateways within one cycle period with redundanttransmissions. A difficulty of transmission scheduling on Y-shaped topologies is the existence of patterns of potentialinterferences around the central node, in contrast to tandemly-arranged linear topologies with two gateways at the edges.On the other hand, such interferences may be efficiently avoidable because the data forwarding directions are opposite,in contrast to tree-like topologies with a gateway at the center. Note that, in our scheme, a central management serveris assumed to be able to know or estimate necessary information such as the packet loss rate of each link, compute aglobal time-slot allocation, and deliver the derived schedule to each node. Such system implementation issues will bediscussed later in the discussion section because it is not the main scope of this paper.The rest of this paper is organized as follows. The related work is reviewed in Sec. 2. The Y-shaped topologyand the path models are defined in Sec. 3. Section 4 explains how to derive optimal time-slot allocations by usingan example topology, which is evaluated through numerical simulations in Sec. 5. Discussions are provided in Sec. 6and finally, Sec. 7 concludes the paper. The appendix is given to explain an example of the detailed derivation of themathematical formula.
Linh Vu Nguyen et al.:
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2. Related work
For multi-hop wireless networks, there have been a variety of studies devoted to cope with the lossy unreliablewireless radio links and the conflicts (interferences) among simultaneous transmissions on adjacent links depending ona variety of requirements and restrictions. Even in TDMA-based transmission scheduling to resolve two fundamentalbottlenecks of multi-hop wireless networks, various methods have been developed (Sgora, Vergados and Vergados(2015)). In theory, it is conducted by defining a conflict-free TDMA for a given set of links, which is formulatedas a graph coloring. In addition, wireless conflicts are able to model with conflict graphs (K. Jain and Qiu (2005),Ramanathan (1999)). As a result, for example, K. Jain and Qiu (2005) obtained a graph coloring on the conflict graph,a conflict-free schedule formed from independent sets with appropriate cardinality. In specific adjacent links, somestudies addressed these issues by utilizing distributed implementation of RAND, a randomized time slot schedulingalgorithm (I. Rhee and Xu (2009)). Besides, S. Ergen (2010) introduced the shortest schedules based on two centralizedalgorithms, which was considered a more facile method to evaluate the performance of distributed algorithms. In asimilar concern about the shortest schedule, M. Sasaki (2016) introduced a min-max model and a min-sum modelas an efficient method for this point, but their large-scale application was hindered. Zeng and Dong (2014) utilizedscheduling algorithm, which was developed from the collaboration of nodes, to improve the packet receive ratio andenergy efficiency.However, almost all these studies concentrated on developing conflict graphs or heuristics to avoid interference ofsimultaneous data transmission on general network topology, and they do not deal with optimal redundant transmis-sions to recover lost packets.On the other hand, our previous work (Agussalim (2016); R. Kimura (2020); R. Yoshida(2020)) in this theme provided a packet transmission scheduling framework restricted to tandemly-connected topolo-gies onlys. L.V. Nguyen (2020) provided a preliminary result on the T-shaped topology with 3 gateways but not offereda comprehensive study. Herein, nodes located near the central point have the influence on interference following 3 di-rections of topology. Therefore, they designed a schedule of when and to which gateway each node sends a packet. Itis vital to control the efficiency of the system.
3. The Y-shaped topology model
Figure 2:
Y-shaped network topology model
In a Y-shaped topology network, the nodes and links are numbered separately (starting from 1) as shown in Fig. 1.The packet loss rate of link 𝑗 is denoted as 𝑞 𝑗 (0 < 𝑞 𝑗 < , and the packet generation rate of node 𝑖 is denoted aspositive integer 𝑟 𝑖 . As explained in Sec. 1, each node considered here is a relay node of the Y-shaped backbone ofa sensor network. Hence the packet generation rate of a node represents the number of packets to convey the totalamount of sensing data in one cycle period of 𝐷 gathered from end sensors managed by the node. In other words, node 𝑖 is assumed to generate 𝑟 𝑖 packets at (or before) the beginning of each 𝐷 and those packets are forwarded toward agateway either 𝑋 , 𝑌 , or 𝑍 . For concise formulations, all packets are assumed to have the same size and all links areto have the same data transmission rate. Therefore, let 𝑈 be the time duration of one time-slot, i.e., one packet can betransmitted on a link between adjacent two nodes in 𝑈 unit time, Then the total number 𝑇 of slots in one cycle periodis equal to 𝐷 ∕ 𝑈 . Linh Vu Nguyen et al.:
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We assume that each packet is forwarded along a single path (not a multi-path) toward only one of the threegateways. Based on this assumption, this study models the Y-shaped topology in three types as shown in Fig. 2. Aseparation link is a link on which no packet is transmitted because the two nodes at both sides of the link send packetsin opposite directions, i.e., the link is between two nodes each of which is at the most upstream of a path. Since theY-shaped topology has a single central node and three branches (called “segment”) terminated by gateways X, Y, andZ, there should be two separation links that separate the topology into three paths with different numbers of nodes ingeneral. We call it “the path model” representing packet routing paths; different path models can be considered bychoosing the locations of the separation links. To be more exact, we can assume one separation link is in the segmentto X and the other is in the segment to Y; no separation link in segment to Z without loss of generality. In this sense,the gateway name 𝑋 , 𝑌 , and 𝑍 should be given depending on the path model. The locations of separation links canclassify all nodes into three groups called 𝑆 𝑋 , 𝑆 𝑌 , and 𝑆 𝑍 which are the set of nodes whose packets are forwarded toX, Y, and Z, respectively. It is called the 𝑙 - 𝑟 - 𝑑 model where 𝑙 , 𝑟 , and 𝑑 represent the number of nodes in 𝑆 𝑋 , 𝑆 𝑌 , and 𝑆 𝑍 , respectively. Furthermore, based on the locations of separation links, three types of path models are considered.Type 1: no node on segment to X and Y is in 𝑆 𝑍 . Type 2: some nodes on only one of segments to X and Y are in 𝑆 𝑍 .Type 3: some nodes on segment to X and also some nodes in segment to Y are in 𝑆 𝑍 .Those types would represent all possible variations of Y-shaped topologies and affect the potential interferencepatterns among nodes nearby the central node that strongly impact the design of slot allocations, i.e., slot allocationpatterns. For example, as shown in Fig. 3, if the first separation link is set between nodes 3 and 4, and the second is setbetween nodes 4 and 5, then the path model is 3-2-3 model and this is Type 1. Group 𝑆 𝑋 has 3 nodes whose packetsare forwarded to gateway X ( 𝑆 𝑋 = {1 , , ), group 𝑆 𝑌 = {5 , , and group 𝑆 𝑍 = {4 , , . Similarly, Fig. ?? andFig. ?? illustrate the 2-2-4 model of Type 2 and 2-1-5 model of Type 3, respectively.
4. Path models and Time-slot allocation
In designing a global static time-slot allocation, there are two issues: how much each node can utilize a limitednumber 𝑇 of time-slots with redundant packet transmissions by considering the upstream-downstream relationshipamong nodes and the packet loss rate of each link; and how much it can avoid radio interferences in simultaneoustransmissions. To solve those issues, the following steps are performed. First, we list all possible path models ona given Y-shaped topology. Second, for each path model, by considering the potential interference patterns aroundthe central node, we list all possible “slot allocation patterns”. To prohibit nearby nodes from harmful simultaneoustransmissions, a static interference avoidance policy based on the distance between nodes is adopted. Although thepossible patterns are limited, we need to use geographical information such as distances and environmental conditionswhich are not represented by an abstract topology. For each pattern of each path model, we derive a static time-slotallocation to maximize the theoretical probability that all packets are successfully delivered to gateways within thetotal time-slots of 𝑇 . Lastly, by comparing all results in terms of the maximum theoretical probability of successfuldelivery, we can select the best path model with the best slot allocation.We explain an optimal static time-slot allocation for each allocation pattern of each path model on an exampletopology with nodes illustrated in Fig. 1. To express a slot allocation, in general, we denote 𝑠 𝑖,𝑗,𝑘 as the number ofslots allocated, i.e., available to use, for the 𝑘 -th packet generated by node 𝑖 on link 𝑗 in each cycle period. In otherwords, each node redundantly transmits a possessed packet (that is originally the 𝑘 -th packet generated by node 𝑖 ) ondownstream link 𝑗 in 𝑠 𝑖,𝑗,𝑘 times. If a packet is lost somewhere in upstream between that node and the node whichgenerated the packet, the slots allocated to the lost packet are used for the next packet. However, for concise explanation,packet generation rate 𝑟 𝑖 is assumed to be and thus 𝑠 𝑖,𝑗 is used instead of 𝑠 𝑖,𝑗,𝑘 . The extension to heterogeneous packetgeneration rates { 𝑟 𝑖 } is somewhat straight-forward as shown in Appendix. Note that we also introduce 𝑠 ′ 𝑖,𝑗 to indicatethe number of slots for an early stage transmission which happens before or at the same time of transmission of themost upstream node in the path, i.e., a concurrent use of the same time-slots for the same direction transmissions toincrease the efficiency.In general, the maximization problem for optimal slot allocations is defined as follows. The success probability ofdelivery of a packet generated by node 𝑖 is denoted by 𝑀 𝑖 ( 𝗌 𝗂 ) that can be calculated by a slot allocation 𝗌 𝗂 = { 𝑠 𝑖,𝑗 , 𝑠 ′ 𝑖,𝑗 | 𝑗 =…} for 𝑖 = 1 , , … , . Hence the problem to solve is max 𝑀 ( 𝗌 ) = ∏ 𝑖 =1 𝑀 𝑖 ( 𝗌 𝗂 ) subject to Linh Vu Nguyen et al.:
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Figure 3:
The 3-2-3 path model in the network topology 𝑇 = ( a linear function of 𝗌 in group X ) 𝑇 = ( a linear function of 𝗌 in group Y ) 𝑇 = ( a linear function of 𝗌 in group Z ) where 𝗌 = { 𝑠 𝑖,𝑗 , 𝑠 ′ 𝑖,𝑗 | 𝑖 = 1 , , … , 𝑗 = …} . Fig. ?? shows the 3-2-3 path model. In this example, since we assume nodes 3 and 7 are in the radio propagationdistance, 3 and 4 cannot send at the same time (to avoid interference at node 7). Since nodes 5 and 7 are in thepropagation distance, 5 and 4 cannot send at the same time (to avoid interference at node 7). On the other hand 3, 5,and 7 can send to their next node at the same time. We have two patterns for slot allocations. In pattern 1, we prioritizethe transmission in groups 𝑆 𝑋 (node 3-2-1) and 𝑆 𝑌 (node 5-6) first, then group 𝑆 𝑍 (node 4-7-8). In pattern 2, weprioritize group 𝑆 𝑍 first, then groups 𝑆 𝑋 and 𝑆 𝑌 . In other words, in pattern 2, the most upstream side node toward Z(i.e., node 4) can start its transmission earlier than the most upstream nodes toward X and Y (i.e., nodes 3 and 5).Pattern 1 is adopted when node group 𝑆 𝑋 or 𝑆 𝑌 is a bottleneck. We solve a sub-problem for group 𝑆 𝑋 and 𝑆 𝑌 first. Fig. 4 shows slot allocations pattern 1 in 3-2-3 model. In group 𝑆 𝑋 , 𝑀 = (1 − 𝑞 𝑠 , ) ,𝑀 = (1 − 𝑞 𝑠 , )(1 − 𝑞 𝑠 , ) 𝑀 = (1 − 𝑞 𝑠 , )(1 − 𝑞 𝑠 , )(1 − 𝑞 𝑠 , ) In group 𝑆 𝑌 , 𝑀 = (1 − 𝑞 𝑠 , )(1 − 𝑞 𝑠 , ) 𝑀 = (1 − 𝑞 𝑠 , ) In group 𝑆 𝑍 , 𝑠 , and 𝑠 ′7 , cannot be 0 at the same time, in any optimal schedule. 𝑀 = (1 − 𝑞 𝑠 , + 𝑠 ′8 , ) , Linh Vu Nguyen et al.:
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Page 5 of 16DMA-based scheduling with 3-egress gateway linear topology 𝑀 = ⎧⎪⎨⎪⎩ (1 − 𝑞 𝑠 ′7 , + 𝑠 , )(1 − 𝑞 𝑠 ′7 , ) if 𝑠 , = 0 , (1 − 𝑞 𝑠 , )(1 − 𝑞 𝑠 ′7 , + 𝑠 , ) if 𝑠 ′7 , = 0 ,𝑀 = (1 − 𝑞 𝑠 , )(1 − 𝑞 𝑠 , )(1 − 𝑞 𝑠 , ) To get a slot allocation in group 𝑆 𝑋 that maximizes 𝑀 𝑀 𝑀 subject to 𝑇 = 𝑠 , + 𝑠 , + 𝑠 , + 𝑠 , + 𝑠 , + 𝑠 , , (1)we apply the Lagrangian multiplier to a relaxation version of this problem to derive equations (2) where 𝑠 𝑖,𝑗 are notrestricted to natural numbers and 𝛼 is an unknown adjunct variable; please see Appendix. 𝑠 , = 𝑠 , = 𝑠 , = − log(1 − 𝛼 log( 𝑞 ))log( 𝑞 ) 𝑠 , = 𝑠 , = − log(1 − 𝛼 log( 𝑞 ))log( 𝑞 ) , 𝑠 , = − log(1 − 𝛼 log( 𝑞 ))log( 𝑞 ) (2)From Eqs.(1) and (2), 𝛼 can be numerically solved to get the real number solution { 𝑠 𝑖,𝑗 } of the relaxed problem. Thenwe should seek an appropriate natural number solution as the number of allocated slots near the derived real numbersolution. Let { 𝑠 ∗ 𝑖,𝑗 } be the natural number solution obtained for 𝑆 𝑋 ; let 𝐚 be 𝑠 ∗3 , .Independently and similarly to 𝑆 𝑋 , to get a slot allocation in group 𝑆 𝑌 that maximizes 𝑀 𝑀 subject to 𝑇 = 𝑠 , + 𝑠 , + 𝑠 , , (3)we have 𝑠 , = 𝑠 , = − log(1 − 𝛽 log( 𝑞 ))log( 𝑞 ) , 𝑠 , = − log(1 − 𝛽 log( 𝑞 ))log( 𝑞 ) (4)where 𝛽 is an unknown adjunct variable. From Eqs.(3) and (4), 𝛽 can be numerically solved to get the real numbersolution { 𝑠 𝑖,𝑗 } of the relaxed problem. Then we obtain the natural number solution { 𝑠 ∗ 𝑖,𝑗 } for 𝑆 𝑌 . Let 𝐚 be 𝑠 ∗5 , .Finally, to find a slot allocation in group 𝑆 𝑍 using 𝐚 = max( 𝐚 , 𝐚 ) , we tentatively maximize 𝑀 𝑀 𝑀 withoutconsidering 𝑠 ′8 , , 𝑠 ′7 , , 𝑠 ′7 , subject to 𝑇 = 𝑠 , + 𝑠 , + 𝑠 , + 𝑠 , + 𝑠 , + 𝑠 , (5)where its solution { 𝑠 ∗ 𝑖,𝑗 } can be solved in the same way. Let 𝐛 be 𝑠 ∗4 , , 𝐛 be 𝑠 ∗4 , , 𝐛 be 𝑠 ∗4 , . There are five cases:(c1) 𝐛 ≥ 𝐚 ; (c2) 𝐛 < 𝐚 and 𝐛 ≥ 𝐚 ; (c3) 𝐛 + 𝐛 ≥ 𝐚 ; (c4) 𝐛 + 2 𝐛 ≥ 𝐚 ; (c5) 𝐛 + 2 𝐛 < 𝐚 . Figure 4:
Transmission scheduling on 3-2-3 model (pattern 1, all 𝑟 𝑖 =1)Linh Vu Nguyen et al.: Preprint submitted to Elsevier
Page 6 of 16DMA-based scheduling with 3-egress gateway linear topology • In (c1), a final natural number solution is: 𝑠 , = 𝐛 , 𝑠 ′7 , = 𝐚 , 𝑠 , = 𝐛 − 𝐚 , 𝑠 , = 𝐛 𝑠 , = 𝑠 , = 𝑠 , = 𝐛 , 𝑠 ′8 , = 𝑠 ′7 , = 0 • In (c2), a final natural number solution is: 𝑠 , = 𝐛 , 𝑠 ′7 , = 0 , 𝑠 , = 𝑠 , = 𝐛 ,𝑠 , = 𝐛 − 𝐚 , 𝑠 , = 𝑠 , = 𝐛 , 𝑠 ′8 , = 𝐚 , 𝑠 ′7 , = 0 • In (c3), a final natural number solution is: 𝑠 , = 𝐛 , 𝑠 ′7 , = 𝐛 , 𝑠 , = 0 , 𝑠 , = 𝐛 , 𝑠 ′8 , = 𝐚 − 𝐛 ,𝑠 , = 𝐛 − ( 𝐚 − 𝐛 ) , 𝑠 , = 𝑠 , = 𝐛 , 𝑠 ′7 , = 0 • In (c4), a final natural number solution is: 𝑠 , = 𝐛 , 𝑠 ′7 , = 𝐛 , 𝑠 , = 0 , 𝑠 , = 𝐛 , 𝑠 ′8 , = 𝐛 , 𝑠 , = 0 ,𝑠 ′7 , = 𝐚 − ( 𝐛 + 𝐛 ) , 𝑠 , = 𝐛 , 𝑠 , = 𝐛 − ( 𝐚 − ( 𝐛 + 𝐛 )) • Case (c5) requires to solve other two equations independently:(i) For nodes and , the time-slot region is 𝐚 . 𝐚 = 𝑠 , + 𝑠 , + 𝑠 , By letting 𝑠 ∗ 𝑖,𝑗 be its solution, a final natural number solution is: 𝑠 ′7 , = 𝑠 ∗7 , , 𝑠 , = 0 , 𝑠 ′7 , = 𝑠 , = 𝑠 ∗8 , 𝑠 , = 𝑠 , = 0 (ii) For node 4, the time-slot region is 𝑇 − 𝐚 . 𝑇 − 𝐚 = 𝑠 , + 𝑠 , + 𝑠 , By letting 𝑠 ∗∗ 𝑖,𝑗 be its solution, a final natural number solution is: 𝑠 , = 𝑠 ∗∗4 , , 𝑠 , = 𝑠 ∗∗4 , , 𝑠 , = 𝑠 ∗∗4 , Figure 5:
Transmission scheduling on 3-2-3 model (pattern 2, all 𝑟 𝑖 =1)Linh Vu Nguyen et al.: Preprint submitted to Elsevier
Page 7 of 16DMA-based scheduling with 3-egress gateway linear topology
Pattern 2 is adopted when node group 𝑆 𝑍 is a bottleneck. We solve a sub-problem for group 𝑆 𝑍 first. Fig. 5 showsslot allocations pattern 2 in 3-2-3 model. In group 𝑆 𝑋 , 𝑠 , and 𝑠 ′2 , cannot be 0 at the same time, in any optimalschedule. 𝑀 = (1 − 𝑞 𝑠 , + 𝑠 ′1 , ) ,𝑀 = ⎧⎪⎨⎪⎩ (1 − 𝑞 𝑠 ′2 , + 𝑠 , )(1 − 𝑞 𝑠 ′2 , ) if 𝑠 , = 0 , (1 − 𝑞 𝑠 , )(1 − 𝑞 𝑠 ′2 , + 𝑠 , ) if 𝑠 ′2 , = 0 ,𝑀 = (1 − 𝑞 𝑠 , )(1 − 𝑞 𝑠 , )(1 − 𝑞 𝑠 , ) In group 𝑆 𝑌 , 𝑀 = (1 − 𝑞 𝑠 , )(1 − 𝑞 𝑠 , ) 𝑀 = (1 − 𝑞 𝑠 , + 𝑠 ′6 , ) In group 𝑆 𝑍 , 𝑀 = (1 − 𝑞 𝑠 , )(1 − 𝑞 𝑠 , )(1 − 𝑞 𝑠 , ) 𝑀 = (1 − 𝑞 𝑠 , )(1 − 𝑞 𝑠 , ) 𝑀 = (1 − 𝑞 𝑠 , ) The following process is almost the same approach as Pattern 1. To get a slot allocation in group 𝑆 𝑍 that maximizes 𝑀 𝑀 𝑀 subject to 𝑇 = 𝑠 , + 𝑠 , + 𝑠 , + 𝑠 , + 𝑠 , + 𝑠 , , we have 𝑠 , = 𝑠 , = 𝑠 , = − log(1 − 𝛾 log( 𝑞 ))log( 𝑞 ) 𝑠 , = 𝑠 , = − log(1 − 𝛾 log( 𝑞 ))log( 𝑞 ) , 𝑠 , = − log(1 − 𝛾 log( 𝑞 ))log( 𝑞 ) where 𝛾 can be numerically solved to get the real number solution of the relaxed problem, and then obtain the naturalnumber solution { 𝑠 ∗ 𝑖,𝑗 } for 𝑆 𝑍 .The obtained solution 𝑠 ∗4 , for 𝑆 𝑍 is used to solve group 𝑆 𝑌 . By starting from the tentative maximization of 𝑀 𝑀 without considering 𝑠 ′6 , subject to 𝑇 = 𝑠 , + 𝑠 , + 𝑠 , , a final natural number solution ( 𝑠 ∗5 , , 𝑠 ∗5 , , 𝑠 ∗6 , , ( 𝑠 ′6 , ) ∗ ) is obtained after checking a few conditions (cases) in a similarmanner as the pattern 1.The natural number solution 𝑠 ∗4 , for is also used to solve group 𝑆 𝑋 . By starting from the tentative maximizationof 𝑀 𝑀 𝑀 without considering 𝑠 ′1 , , 𝑠 ′2 , , 𝑠 ′2 , subject to 𝑇 = 𝑠 , + 𝑠 , + 𝑠 , + 𝑠 , + 𝑠 , + 𝑠 , , a final natural number solution ( 𝑠 ∗3 , , 𝑠 ∗3 , , 𝑠 ∗2 , , ( 𝑠 ′2 , ) ∗ , 𝑠 ∗3 , , 𝑠 ∗2 , , 𝑠 ∗1 , , ( 𝑠 ′2 , ) ∗ , ( 𝑠 , ) ∗ ) is obtained after checking a fewconditions. Linh Vu Nguyen et al.:
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Page 8 of 16DMA-based scheduling with 3-egress gateway linear topology
Fig. 7 shows the 2-2-4 path model. In this example, since nodes 4 and 5 are assumed to be in the radio propagationdistance, nodes 3 and 5 cannot send at the same time (to avoid an interference at node 4). Similarly, since nodes 5 and7 are in the propagation distance, nodes 4 and 5 cannot send at the same time (to avoid an interference at node 7). Onthe other hand, 5 and 7 can send to its next node at the same time. We have two patterns for slot allocations. In pattern1, we prioritize the transmission in group 𝑆 𝑍 (node 3-4-7-8) first, then group 𝑆 𝑌 (node 5-6). In pattern 2, we prioritizegroup 𝑆 𝑌 first, then group 𝑆 𝑍 . Note that group 𝑆 𝑋 is independent and solved separately.Pattern 1 is adopted when node group 𝑆 𝑍 is a bottleneck. We solve a sub-problem for group 𝑆 𝑍 first. Fig. 6 showsslot allocation pattern 1 in the 2-2-4 model. In group 𝑆 𝑋 , 𝑀 = (1 − 𝑞 𝑠 , ) , 𝑀 = (1 − 𝑞 𝑠 , )(1 − 𝑞 𝑠 , ) In group 𝑆 𝑍 , 𝑀 = (1 − 𝑞 𝑠 , )(1 − 𝑞 𝑠 , )(1 − 𝑞 𝑠 , )(1 − 𝑞 𝑠 , ) 𝑀 = (1 − 𝑞 𝑠 , )(1 − 𝑞 𝑠 , )(1 − 𝑞 𝑠 , ) 𝑀 = (1 − 𝑞 𝑠 , )(1 − 𝑞 𝑠 , ) , 𝑀 = (1 − 𝑞 𝑠 ′8 , ) In group 𝑆 𝑌 , 𝑀 = (1 − 𝑞 𝑠 , )(1 − 𝑞 𝑠 , ) , 𝑀 = (1 − 𝑞 𝑠 , + 𝑠 ′6 , ) The process to get a slot allocation is almost the same approach as the 3-2-3 model already explained. Thereforewe show only group 𝑆 𝑍 consisting of four nodes. To maximize 𝑀 𝑀 𝑀 𝑀 subject to 𝑇 = 𝑠 , + 𝑠 , + 𝑠 , + 𝑠 , + 𝑠 , + ⋯ + 𝑠 , , we need to solve two equations independently• Case 1: by ignoring 𝑠 , (if 𝑞 ≥ 𝑞 ), 𝑇 = 2 𝑠 , + 3 𝑠 , + 4 𝑠 , , 𝑠 ′8 , = 𝑠 , , 𝑠 , = 𝑠 ′8 , (6)• Case 2: by ignoring 𝑠 , (if 𝑞 < 𝑞 ), 𝑇 = 𝑠 , + 2 𝑠 , + 3 𝑠 , + 3 𝑠 , , 𝑠 ′8 , = 𝑠 , (7)By solving Eq. (6) or Eq. (7), a natural number solution for 𝑆 𝑍 is finally obtained. Based on the final naturalnumber solution, 𝑠 ∗3 , + 2 𝑠 ∗3 , is used to solve group 𝑆 𝑌 . Figure 6:
Transmission scheduling on 2-2-4 model (pattern 1, all 𝑟 𝑖 =1)Linh Vu Nguyen et al.: Preprint submitted to Elsevier
Page 9 of 16DMA-based scheduling with 3-egress gateway linear topology
Figure 7:
The 2-2-4 path model in the network topology
Figure 8:
The 2-1-5 path model in the network topology
Fig. 8 shows the 2-1-5 path model. In this example, since nodes 3, 4, and 5 are assumed to be in the radio prop-agation distance, nodes 3, 4 and 5 cannot send at the same time (to avoid an interference). On the other hand, nodes2 and 6 can send to its next node at the same time. Note that groups 𝑆 𝑋 and 𝑆 𝑌 are independent and can be solvedseparately. Therefore, we need only a single pattern.Fig. 9 shows a slot allocation on 2-1-5 model. In group 𝑆 𝑋 , 𝑀 = (1 − 𝑞 𝑠 , ) , 𝑀 = (1 − 𝑞 𝑠 , )(1 − 𝑞 𝑠 , ) In group 𝑆 𝑍 , 𝑀 = (1 − 𝑞 𝑠 , )(1 − 𝑞 𝑠 , )(1 − 𝑞 𝑠 , )(1 − 𝑞 𝑠 , ) 𝑀 = (1 − 𝑞 𝑠 , )(1 − 𝑞 𝑠 , )(1 − 𝑞 𝑠 , )(1 − 𝑞 𝑠 , ) Linh Vu Nguyen et al.:
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Page 10 of 16DMA-based scheduling with 3-egress gateway linear topology 𝑀 = (1 − 𝑞 𝑠 , )(1 − 𝑞 𝑠 , )(1 − 𝑞 𝑠 , ) 𝑀 = (1 − 𝑞 𝑠 , )(1 − 𝑞 𝑠 , ) , 𝑀 = (1 − 𝑞 𝑠 , + 𝑠 ′8 , ) In group 𝑆 𝑌 , 𝑀 = (1 − 𝑞 𝑠 , ) The process to get a slot allocation is almost the same approach as the previous models. We only mention group 𝑆 𝑍 . To maximize 𝑀 𝑀 𝑀 𝑀 𝑀 subject to 𝑇 = 𝑠 , + 𝑠 , + 𝑠 , + 𝑠 , + 𝑠 , + ⋯ + 𝑠 , , two cases should be examined and select best one. One case is to ignore 𝑠 , , i.e., node 8 does not generate ownpacket, and the other case is to ignore 𝑠 , and 𝑠 , , i.e., nodes 3 and 5 do not generate own packets, instead node 4generates three packets.
5. Numerical results
On our example of the Y-shaped topology network, we show a few numerical results for three different path modelsto evaluate the performance of derived time-slot allocations in three different cases in terms of the setting of link lossrates { 𝑞 𝑖,𝑗 } shown in Table 1; packet generation rates are uniform ( 𝑟 𝑖 = 1 ); the total number 𝑇 of time-slots is 𝑇 = 20 or 𝑇 = 30 . Highly lossy links (links with high loss rates) are located near gateway 𝑆 𝑍 in case 1; near gateway 𝑆 𝑋 in case 2; and near gateways 𝑆 𝑋 and 𝑆 𝑌 in case 3. Matlab is used to get the solutions of the maximization problemsfor the path model in the way described in Section 4. As a performance metric, the Theoretical Upper-Bound (TUB)value and the Model-based Computed (COM) value are used. TUB is the theoretical maximum value of the objectivefunction M(s) in the relaxed version of the maximization problem (i.e., any real number can be used). COM is thecomputed probability of delivering all packets using an optimal slot allocation according to a natural number solutionof the original integer-constraint maximization problem.In Fig. 10, the success delivery probability for all packets of TUB value performs better COM value in all cases.It is because TUB value is the theoretical maximum value of the objective function.In all conditions, two patterns in the 3-2-3 model represent the equal TUB value regardless of whether group 𝑆 𝑋 is prioritized or 𝑆 𝑍 is prioritized. It is attributed to the balanced locations of the separated links.In the case 1, the 3-2-3 mode illustrates the significantly higher performance than its counterparts due to the positiverecovery after loss link around gateway Z. Noticeably, the high resilience of this model possibly stem from the relativelyequal number of nodes in all groups, which relieve the effect of accumulated loss link from transmitting through manynodes. When T=30, for all cases, both patterns in the 3-2-3 model show more than 80 percent of the successful delivery. Figure 9:
Transmission scheduling on 2-1-5 model (all 𝑟 𝑖 =1)Linh Vu Nguyen et al.: Preprint submitted to Elsevier
Page 11 of 16DMA-based scheduling with 3-egress gateway linear topology
Figure 10:
Probability of success delivery for all nodes with T=20 (left) and T=30 (right)
Figure 11:
Probability of success delivery for each node with 𝑇 = 30 , pattern 1 in the 2-2-4 model-case 1 (left), 2-1-5model-case 1 (right) Table 1
Packet loss rate on each linkCase 𝑞 𝑞 𝑞 𝑞 𝑞 𝑞 𝑞 𝑞 𝑞 𝑞 Similarly, pattern 1 in the 2-2-4 model has demonstrated to be more effective in case that there is an impact of losslink to gateway X. It only attains the efficient probability of successful delivery of packets with no inference of loss linklocated at node Z. Besides, 2-2-4 model shows better performance than 2-1-5 one for both case 1 and 2. Especially, ithit the top result in case 2.The 2-1-5 model including 5 nodes in group 𝑆 𝑍 is always affected seriously by the accumulated loss link ofthe transmitting process towards gateway Z. In the case 3, the 2-1-5 model presents enhancement in this parametercompared with others. Our study indicates that this phenomenon only occurs in the condition that its links are notdamaged by interference but links near gateway X and Y are attacked seriously. The separation link is far away fromthe central node leads to the arrangement of packets become more complex and vulnerable.In the simulation model with a low time-slot value (T=20), we observe the probability of successful deliveryreduces significantly in all 3 cases. Herein, the effect on model 2-1-5 was the most significant because the number oftime-slots assigned to the link is low.Fig. 11 investigates the relationship between the probability of successful delivery for each node and their locationin the case of 2-2-4 model in case 1 and 2-1-5 model in case 2. For each node, the COM value is relatively equal tothe TUB value, excluding nodes under the considerable influence of the loss link. In specific, the COM value of group 𝑆 𝑍 in 2-1-5 model (case 2) is higher than its TUB value at nodes 7 and 4, but this edge is reversed at nodes 3 and 8. Linh Vu Nguyen et al.:
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Page 12 of 16DMA-based scheduling with 3-egress gateway linear topology
Table 2
Slot allocations for pattern 1 in 3-2-3 model with T=30 𝑠 , 𝑠 ′1 , 𝑠 , 𝑠 ′2 , 𝑠 , 𝑠 ′2 , 𝑠 , 𝑠 , 𝑠 , 𝑠 , 𝑠 , 𝑠 , TUB 5.5001 0 5.5001 0 3.9999 0 5.5001 3.9999 5.5001 3.4322 6.7617 4.3481COM 5 0 5 0 4 0 6 4 6 3 7 4 𝑠 , 𝑠 , 𝑠 , 𝑠 ′6 , 𝑠 , 𝑠 ′7 , 𝑠 , 𝑠 ′7 , 𝑠 , 𝑠 ′8 , TUB 11.8741 9.0630 9.0630 0 0 6.7617 3.5839 0.7642 0 4.3481COM 12 9 9 0 0 7 4 1 0 4
Table 3
Slot allocations for pattern 2 in 3-2-3 model with T=30 𝑠 , 𝑠 ′1 , 𝑠 , 𝑠 ′2 , 𝑠 , 𝑠 ′2 , 𝑠 , 𝑠 , 𝑠 , 𝑠 , 𝑠 , 𝑠 , TUB 5.5001 0 5.5001 0 0.5677 3.4322 5.5001 3.9999 5.5001 3.4322 6.7617 4.3481COM 5 0 5 0 1 4 5 4 6 4 7 4 𝑠 , 𝑠 , 𝑠 , 𝑠 ′6 , 𝑠 , 𝑠 ′7 , 𝑠 , 𝑠 ′7 , 𝑠 , 𝑠 ′8 , TUB 11.8741 9.0630 5.6307 3.4322 6.7617 0 4.3481 0 4.3481 0COM 12 9 5 4 7 0 4 0 4 0
In the same manner, we observe a similar phenomenon in the group 𝑆 𝑍 in 2-2-4 model (case 1) but the gap betweenTUB and COM values is negligible.Besides, the probabilities of successful delivery for the upstream nodes are generally lower than those of the down-stream nodes along a path (i.e., in a node group). The probability of successful delivery of each node is based on thenumber and location of nodes within a group.Table 2 and table 3 show slot allocations for pattern 1 and pattern 2 in the 3-2-3 model with T=30. In pattern1, we prioritize the transmission in groups 𝑆 𝑋 (node 3-2-1) and 𝑆 𝑌 (node 5-6) first, then group 𝑆 𝑍 (node 4-7-8). Inpattern 2, we prioritize group 𝑆 𝑍 first, then groups 𝑆 𝑋 and 𝑆 𝑌 .Due to a difference of pattern, we recognized the gap in the number of time-slots distributes to each node. Forinstance, at node 7 of pattern 2 in the 3-2-3 model, time-slots were assigned to 𝑠 , , whereas the counterparts in pattern1 in the 3-2-3 model were located at 𝑠 ′7 , .It is because node 4 cannot send at the same time as node 3 and node 5 in pattern 1 in the 3-2-3 model. Therefore,node 7 was prioritized to transmit simultaneously with node 3 and node 4. Otherwise, the number of time-slots allo-cated to 𝑠 ′7 , , 𝑠 ′7 , , 𝑠 ′8 , would depend on 𝑠 , or 𝑠 , . Hence, we utilized Model-based Comupted (COM) to obtained 𝑠 ′7 , , 𝑠 ′7 , , 𝑠 ′8 , with value at 7,1, 4, respectively, in condition that 𝑠 , is equal to 12.Otherwise, from results of Table 2 and table 3, the change of prioritized transmitting in each node group wouldfacilitate to avoid interference between different nodes in the same group. However, the performance was the same inpatterns 1 and 2, since the number of time-slots assigned to each node was large.
6. Discussion
We address a few issues that are not well mentioned in the main part of this paper but necessary to implement andextend our proposed scheme into practical systems. Firstly, our scheme is applicable to heterogeneous packet generationrates of nodes and heterogeneous data transmission rates of links, although only homogeneous cases were explainedfor concise formulations. In reality, each node may support different numbers of and/or types of sensors, and thus thenumber of packets necessary to convey them in one cycle period may differ. In addition, different types of backbonelinks may be mixed in the same network with the different data transmission rate to adjust some restrictions, e.g.,physical distance and cost. In our preliminary work on the tandemly-arranged topology networks with two gateways,we have shown the formulations and results on heterogeneous packet generation rates (R. Yoshida (2020)) and datatransmission rates (R. Kimura (2020)). Note that how the different packet generation rates ( 𝑟 𝑖 for node 𝑖 ) can bemanaged in our setting is shown in Appendix. Furthermore, the number of nodes can be extended, although only8-node example was shown. In general, even if the number of nodes, i.e., the length of a path, is increased, theinterference patterns around the central node are unchanged. Only the chances of a concurrent use of the same time- Linh Vu Nguyen et al.:
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Page 13 of 16DMA-based scheduling with 3-egress gateway linear topology slots for the same direction transmissions by two distanced nodes in the path is increased. However, an investigationon performance impacts and implications to scheduling design remains as future work.Secondly, in this paper, our scheme only adopts a basic redundant transmissions in which a node just redundantlytransmits each of its possessed packets in a specific times according to the given time-slot allocation. However it iswell-known that a packet-level coding as FEC increases the success probability of packet delivery, although it possiblyincreases the complexity of the system. Each node can combine multiple different packets its possessed by using somecoding and transmits possibly different coded packets within the allocated slots; those coded packets are decoded in thefinal receiver, e.g., a central data collection server. In our preliminary work, we have shown the benefits of XOR-basedsimple coding with consideration on fairness among nodes (R. Kimura (2020)). An detailed design and performanceinvestigation of packet-coding in Y-shaped topology setting remain as future work.Finally, our scheme is on a centrally-managed transmission scheduling for a network of relay nodes with gateways,and implicitly assumes a central management server that compute a global time-slot allocation. Therefore, the nextresearch will concentrate on the system architecture for real implementation. More specifically, a scheme to exchangeand share the involved information is necessary (i) for a server to know or estimate a network topology and relatedinformation such as data transmission rates (bandwidths) of links, distances between nodes, packet loss rates on links,and packet generation rates at nodes; and (ii) for each node to know a derived transmission schedule. In particular, theinformation exchange of (ii) requires an opposite direction communication from gateways to each node and is needednot only at the initial phase of the system but every time when environmental conditions change or periodically with arelatively long time interval.
7. Concluding Remarks
In this paper, we focus on the linear topology with three gateways at the edges to resolve existing problems, suchas interferences among relay nodes and unreliable lossy links. This goal would be achieved by utilizing a global statictime-slot allocation to maximize the theoretical probability that all packets are successfully delivered to one of thegateways within one cycle period with redundant transmissions.Our work presented three types of the path models on Y-shaped topologies. The optimization to the packet schedul-ing was built on completed formulas and calculations by Matlab. All models were evaluated in different cases to selectthe appropriate one for each case. In the future, as addressed in Discussion section, we will continue to improve boththe applicability and performance of the scheme, and to develop a system implement with performance evaluationbased on realistic wireless communication simulations.The research results have been achieved by the “Resilient Edge Cloud Designed Network (19304),” NICT, and byJSPS KAKENHI JP20K11770, Japan.
A. Appendix
We explain how to derive Eq.(2). For group 𝑆 𝑋 in transmission pattern 1 of 3-2-3 path model, let 𝑀 𝑖 be thetheoretical probability that a single packet generated by node 𝑖 is successfully delivered to gateway 𝑋 with the basicredundant transmission scheme. According to Fig. 4, we have 𝑀 = (1 − 𝑞 𝑠 , ) , 𝑀 = (1 − 𝑞 𝑠 , )(1 − 𝑞 𝑠 , ) , 𝑀 = (1 − 𝑞 𝑠 , )(1 − 𝑞 𝑠 , )(1 − 𝑞 𝑠 , ) , by letting 𝑠 𝑖,𝑗 be the number of allocated slots for one packet generated by node 𝑖 on link 𝑗 .Our final goal is to find a slot allocation maximizing the theoretical probability 𝑀 that all packets in one cycleperiod are successfully delivered to gateway 𝑋 with the basic redundant transmission scheme. The exact formulationof 𝑀 is somewhat complicated. By letting 𝑠 𝑖,𝑗,𝑘 be the number of allocated slots for the 𝑘 -th packet generated by node 𝑖 on link 𝑗 , this probability 𝑀 is 𝑟 ∏ 𝑗 =1 (1 − 𝑞 𝑠 , ,𝑗 ) 𝑟 ∏ 𝑗 =1 (1 − 𝑞 𝑠 , ,𝑗 )(1 − 𝑞 𝑠 , ,𝑗 ) 𝑟 ∏ 𝑗 =1 (1 − 𝑞 𝑠 , ,𝑗 )(1 − 𝑞 𝑠 , ,𝑗 )(1 − 𝑞 𝑠 , ,𝑗 ) . However, since we deal with a relaxation version of the maximization problem to apply the Lagrangian multipliermethod, the formulation of 𝑀 can be simpler in the relaxation version by considering 𝑠 𝑖,𝑗,𝑘 = 𝑠 𝑖,𝑗 for ∀ 𝑘 : 𝑀 = 𝑀 ( 𝐬 ) = 𝑀 𝑟 𝑀 𝑟 𝑀 𝑟 (8) Linh Vu Nguyen et al.:
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Page 14 of 16DMA-based scheduling with 3-egress gateway linear topology where 𝐬 = ( 𝑠 , , 𝑠 , , 𝑠 , , 𝑠 , , 𝑠 , , 𝑠 , ) and 𝑠 𝑖,𝑗 ( > are not restricted to natural numbers.Please note that “ 𝑠 𝑖,𝑗, ≠ 𝑠 𝑖,𝑗, ” may happen in the original maximization problem for slot allocation due to the totalslot number is restricted by a given 𝑇 . Therefore, as we showed in Sec. 4, we should find the exact optimal naturalnumbers { 𝑠 𝑖,𝑗,𝑘 | 𝑘 = 1 , , … , 𝑟 𝑖 } after obtaining a real number solution { 𝑠 𝑖,𝑗 } .The relaxation version problem can be solved as follows. max 𝑀 subject to 𝑇 = 𝑟 𝑠 , + 𝑟 ( 𝑠 , + 𝑠 , ) + 𝑟 ( 𝑠 , + 𝑠 , + 𝑠 , ) where 𝑀 is defined in Eq.(8).The corresponding Lagrangian function is: 𝐿 = 𝑀 − 𝜆 ( 𝑟 𝑠 , + 𝑟 ( 𝑠 , + 𝑠 , ) + 𝑟 ( 𝑠 , + 𝑠 , + 𝑠 , ) − 𝑇 ) . First we define two notations for conciseness. 𝐺 ( 𝑞, 𝑥 ) = − 𝑞 𝑥 log 𝑞 𝑞 𝑥 , 𝐹 ( 𝑞, 𝑦 ) = − log(1 − 𝑦 log 𝑞 )log 𝑞 (9)where 𝐺 ( 𝑞, 𝑥 ) = 1 𝑦 ⇔ 𝑥 = 𝐹 ( 𝑞, 𝑦 ) .If 𝐬 is a soluition within the internal region, 𝜕𝐿𝜕𝑠 𝑖,𝑗 = 0 should be held for every ( 𝑖, 𝑗 ) . Hence, by differentiating 𝐿 with respect to 𝑠 , , we have 𝜕𝑀𝜕𝑠 , = 𝜕𝑀 𝜕𝑠 , ( 𝑟 𝑀 𝑟 −11 ) 𝑀 𝑟 𝑀 𝑟 = 𝑟 (− 𝑞 𝑠 , log 𝑞 ) 𝑀 𝑟 −11 𝑀 𝑟 𝑀 𝑟 = 𝑟 𝐺 ( 𝑞 , 𝑠 , ) 𝑀,𝜕𝐿𝜕𝑠 , = 𝑟 𝐺 ( 𝑞 , 𝑠 , ) 𝑀 − 𝑟 𝜆 = 0 ,𝐺 ( 𝑞 , 𝑠 , ) = 𝜆𝑀 (10)where 𝐺 is defined in Eq.(9).In the same way, we have: 𝜕𝐿𝜕𝑠 𝑖,𝑗 = 𝑟 𝑖 𝐺 ( 𝑞 𝑗 , 𝑠 𝑖,𝑗 ) 𝑀 − 𝑟 𝑖 𝜆 = 0 , and thus, 𝐺 ( 𝑞 , 𝑠 , ) = 𝐺 ( 𝑞 , 𝑠 , ) = 𝐺 ( 𝑞 , 𝑠 , ) = 𝐺 ( 𝑞 , 𝑠 , ) = 𝐺 ( 𝑞 , 𝑠 , ) = 𝜆𝑀 (11)From Eqs.(10) and (11), by letting 𝛼 = 𝑀𝜆 as an adjunct variable, we have an explicit expression of each 𝑠 𝑖,𝑗 withan unknown positive variable 𝛼 : 𝑠 , = 𝑠 , = 𝑠 , = 𝐹 ( 𝑞 , 𝛼 ) , 𝑠 , = 𝑠 , = 𝐹 ( 𝑞 , 𝛼 ) , 𝑠 , = 𝐹 ( 𝑞 , 𝛼 ) (12)where 𝐹 is defined in Eq.(9). References
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L.V. Nguyen, M. Shibata, M.T., 2020. Message transmission scheduling for multi-hop wireless sensor network with t-shaped topology. Proceedingsof the 15-th International Conference on Broadband and Wireless Computing, Communication and Applications (BWCCA) , 120–130.M. Sasaki, T. Furuta, T.U.F.I., 2016. Tdma scheduling problem avoiding interference in multi-hop wireless sensor networks, journal of advancedmechanical design. Systems and Manufacturing 10.R. Kimura, M. Shibata, M.T., 2020. Scheduling for tandemly-connected sensor networks with heterogeneous link transmission rates. Proc. the 34thInternational Conference on Information Networking (ICOIN2020) , 590–595.R. Yoshida, M. Shibata, M.T., 2020. Transmission scheduling for tandemly-connected sensor networks with heterogeneous packet generation rates.Proc. the 12th International Conference on Intelligent Networking and Collaborative Systems (INCoS2020) , 437–446.Ramanathan, S., 1999. A unified framework and algorithm for channel assignment in wireless networks. Wireless Networks 5, 81–94.S. Ergen, P.V., 2010. Tdma scheduling algorithms for wireless sensor networks. Wireless Networks 16, 985–997.Sgora, A., Vergados, D.J., Vergados, D.D., 2015. A survey of tdma scheduling schemes in wireless multihop networks. ACM Computing Surveys47.Zeng, B., Dong, Y., 2014. A collaboration-based distributed tdma scheduling algorithm for data collection in wireless sensor networks. Journal ofNetworks 9, 2319–2327.Nguyen Vu Linh received the B.S. degree and M.S. degree in Electronics and Telecommunications, from University of Science, Ho Chi Minh City,Vietnam. He is currently pursuing a Ph.D. degree within the Graduate School of Computer Science and Systems Engineering, Kyushu Institute ofTechnology from Kyushu Institute of Technology, Japan. His research interests are network management and communication systems.Nguyen Viet Ha received the B.S. degree (2009), M.S. degree (2012) in Electronics and Telecommunications, from University of Science, HoChi Minh City, Vietnam and Ph.D. degree (2017) in Computer Science and System Engineering from Kyushu Institute of Technology, Japan. Hisresearch interests are transport layer protocols and network coding.Masahiro Shibata received B.E., M.E., and D.E. degrees in Computer Science from Osaka University, in 2012 2014, and 2017, respectively. Since2017, he has been an Assistant Professor in Kyushu Institute of Technology. His research interests include distributed algorithms and networkmanagement. He is a member of IPSJ and IEICE.Masato Tsuru received the B.E. and M.E. degrees from Kyoto University, Japan in 1983 and 1985, and then received his D.E. degree from KyushuInstitute of Technology, Japan in 2002. He has been a professor in the Department of Computer Science and Electronics, Kyushu Institute of Tech-nology since 2006. His research interests include performance measurement, modeling, and management of computer communication networks.He is a member of the ACM, IEEE, IEICE, IPSJ, and JSSST.
Linh Vu Nguyen et al.: