The principle limitation of the current technique is that the number of semidefinite constraints it generates is pro- portional to the number of paths. This path dependent formulation may be too huge for large networks. Though the state of art semidefinite programming software is rel- atively advanced we need techniques to effectively handle this path dependency in the formulation. One such idea may be to evaluate heuristics to determine paths that are most likely of becoming the critical path. This could be then used to reduce the number of semidefinite constraints , . Another promising technique could be one that uses cutting plane techniques to solve the semidefinite pro- gram with huge number of semidefinite constraints effec- tively . It is necessary to evaluate and develop these techniques and then test them effectively on large projects with large number of paths.
5 Evaluation of the flexibility of W ave
5.1 Simulation parameters
In this section, we conduct a comparative performance evaluation of W ave with a well-known centralized scheduling algorithm T MCP  and DeT AS  a distributed scheduling algorithm. This evaluation is qualitative for DeTAS and is quantitative for TMCP. For the quantitative evaluation, we use our simulation tool based on GNU Octave  to evaluate the number of slots required by these conflict-free scheduling algorithms. The number of nodes varies from 10 to 100. To generate routing graphs, we use the Galton-Watson process as a branching stochastic process: the maximum number of children per node is 3. We suppose that all the nodes except the sink have a single radio interface and we vary the number of sink radio interfaces from 1 to 3. The number of available channels varies from 2 to 3. We consider both cases: 1) homogeneous traffic demands, where each node different from the sink generates one packet and 2) heterogeneous traffic demands where the number of packets locally generated on a node is randomly chosen between 1 and 5. In the following, each result depicted in a curve is the average of 20 simulation runs for topologies with a number of nodes ≤ 30 and 100 runs for larger topologies.
Wireless networks performances study motivates various research works. WMNs deployment in operational situations such as urban areas needs QoS guarantees because of capacity constraints. Indeed it has been prooved under some hypotesis that random network performances degrade with a factor at least O(1/ √ n) when its size, n, grows [5, 6, 7]. Some capacity evaluation frameworks have been developped to get a network behaviour estimation and a stochastic analysis confirmation [8, 9]. In WMNs, it is known that routers far from the gateways may be starved by routers close to the gateways. Therefore, we consider a max-min model to achieve high throughput but with a good fairness. In this work, we focus on the problem of providing fair throughput guarantees for multi-hop transmissions considering interferences among multiple simultaneous transmissions.
This work is a first attempt towards utilizing a combination of controlled mobility and wireless transmission for data collection instochastic and dynamic wireless networks. Therefore, there are many related open problems. In this paper we have utilized a simple wireless communication model based on a communication range. In the future we intend to study more advanced wireless communi- cation models such as modeling the transmission rate as a function of the transmission distance. For the case of multiple-collectors whose transmissions are subject to interference constraints, we intend to study interference models that do not restrict the collectors’ motion to a grid. Note that such a joint server routing and scheduling problem is significantly more involved. For instance, the stability region of such a system depends on the interference constraints, and it is unknown since there are uncountably many possible activation vectors.
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In this paper, we propose a new stochastic approach for the automatic detection of network structures in raster data. We represent a network as a set of trees with acyclic planar graphs. We embed this model in the probabilistic frame- work of spatial point processes and determine the most probable configuration of trees by stochastic sampling. That is, different configurations are constructed randomly by modifying the graph parameters and by adding or removing nodes and edges to/ from the current trees. Each configuration is evaluated based on the probabilities for these changes and an energy function describing the con- formity with a predefined model. By using the Reversible jump Markov chain Monte Carlo sampler, an approximation of the global optimum of the energy function is iteratively reached. Although our main target application is the ex- traction of rivers and tidal channels in digital terrain models, experiments with other types of networksin images show the transferability to further applica- tions. Qualitative and quantitative evaluations demonstrate the competitiveness of our approach with respect to existing algorithms.
Index Terms—Maximum weight scheduling, backlog/delay bounds, capacity region, order optimal delay
I. I NTRODUCTION
Wireless scheduling has been known to be a key problem for throughput/capacity optimization in wireless networks. The well-known maximum weight scheduling algorithm has been proposed by Tassiulas in his seminal paper  where he proved its throughput optimality. Latter developments in this area include extension of this maximum weight scheduling algorithm to wireless networks with rate/power control , , network control when offered trafﬁc is outside the ca- pacity region , and other scheduling policies with lower- complexity -. While most existing works in the area of stochastic network control focused on throughput perfor- mance of optimal and suboptimal scheduling policies, delay properties of most scheduling policies proposed for wireless ad hoc networks remain unknown. In this paper, we study backlog/delay properties of the maximum weight scheduling algorithm in wireless ad hoc networks.
I. I NTRODUCTION
Weather-related outages in electricity distribution networks (DNs) continue to show an upward trend as utilities face the dual problems of deteriorating power grid infrastructure and higher frequency of natural disasters such as hurricanes , . Prolonged delays in restoring the power supply for Puerto Rico in the aftermath of Hurricane Maria highlight the importance of strategic planning and efficient response to extreme events. This paper is motivated by the need for developing a modeling framework that (i) accounts for the likely locations of component failures for damage as- sessment; and (ii) enables the design of pre-storm resource allocation strategies as well as post-storm repair operations. To address these issues, we formulate a two-stage stochastic optimization problem based on an uncertainty model of storm-induced failures.
Rule-based: These types of methods use more or less complex rule-based systems to solve the power system scheduling problem. These rules may be static (i.e.: predefined and fixed through time) or they may evolve in time whenever machine learning techniques are employed. In addition, these rules may be simple rule-of-thumb ones (e.g.: successively commit generators having the lowest average incremental cost until the requirements of the problem are met) or they may be composed of more or less complex inference systems. These systems try to mimic the actions of an expert  based on actual human expert inputs (in which case they are commonly called expert systems) or they may be created from historical data. In this last case, inference systems based on artificial neural networks are widely used in the literature . These inference systems may be static or they may evolve in time “learning” from experience . Finally, the rules incorporated by these rule-based systems may take the form of single values (i.e.: crisp values) or by fuzzy numbers. In the first case, the single values may define, for instance, thresholds to respect. It the second case it is more or less the same with the difference that these thresholds are no longer represented by crisp values, but by fuzzy numbers. These fuzzy numbers may model the uncertainty around a given numeric value (e.g.: the system load will be between 200 MW and 250 MW), or they may translate some qualitative measure (e.g.: the system load will be average). Consequently, when fuzzy numbers are used, the scheduling process has to incorporate fuzzy logic (provided by fuzzy set theory) for scheduling the power system .
d = 1 in Figure 3(b)).
The traffic is single-hop and the arrival process to each link of the network is assumed to be stochastic, with characteristics not necessarily known by the network designers. The goal is to schedule active links at each step in order to insure the stability of the system and, in particular, to activate links which are the most loaded. In the primary node interference model this corresponds to finding a maximum matching or a large matching. Centralized algorithms have been proposed to solve this problem both for random arrivals in [23, 24] and deterministic arrivals in . As example, if the network is a square grid of 4 nodes (Figure 1) with the primary node interference model (d = 0), we can activate at one step either vertical links (Figure 1(a)) or horizontal links (Figure 1(b)). It is also possible to have a single active link in the network but we consider only the two previous sets of active links (maximal sets). A good scheduling algorithm has to insure the stability of the system (stability of the four queues associating to the four links in Figure 1). For example if the capacity of each link is 1 (if a link is active during a step, it sends 1 message), and if
A nonconforming discretization of DFN allows to reduce the number of un- knowns and facilitate mesh refinement. Sharp angles are managed by a staircase- like discretizations of the fractures’ contours . The non-matching feature at the fractures’ intersections is handled via a Mortar method [4, 5, 1] developed for DFN in [33, 34] for a mixed hybrid finite element formulation. It consists in defining, for each intersection between fractures, master and slave sides. Due to the staircase- like discretizations, a shared edge may be labeled several times with master and/or slave properties, it is called in the paper a multi-labeled edge. Continuity conditions are enforced between the unknowns on both sides. The derived linear system has only inner and master traces of hydraulic head as unknowns. The matrix A of this system is a symmetric definite positive (SPD) arrow matrix in presence of Dirichlet boundary conditions .
Figure 2 Measurements of the dimensions at a station i of the net- work. The elliptic approximation appears in red.
Figure 3 Schematisation of the width-height ratio.
The data used for this paper consists in 49 different net- works, shared with us by various speleologists during two pre- vious studies ( Collon et al. , 2017 ; Jouves et al. , 2017 ) and presented in Appendix A. The extent of these networks can be quite different, the widest one, Sieben Hengste (Switzerland), extending itself over 80 kilometers with 15340 data points, while the smallest one, Baume Galinière (France), has less than 200 meters of conduits with 50 data points. Most of them are rather small, their median length being 2135 meters long, and are sampled with a median number of 269 points. The av- erage sampling distance is every 7.5 meters. It has to be noted that the sampling is not homogeneous and some network parts lack geometrical information. There is also a great uncertainty about the sampling itself.
We formulated the pump scheduling problem in water dis- tribution network as a new generic non-compact linear program, based on the approximation of the head at the water tanks and on the relaxation of the pump aging con- straints. This approximation turned out to be both tight and easy to solve when experimented on two networks with different characteristics. We were then able to quickly find low cost feasible solutions by searching in a neighborhood of the approximated solutions. These results lead us to believe that this method could deal with networks larger than with the currently known approaches. Failing to dispose of such study cases, we envisage to build new realistic instances to confirm our claim. Perspectives to extend our method are, first, to exploit the new LP approximation in a global optimization approach, and, second, to exploit historical data of network operations to build the configuration set. REFERENCES
In future works, we aim at identifying more Process Hitting patterns leading to the emergence of particular behaviours (e.g. oscillations) and especially hybrid patterns coupling both discrete structure and continuous temporal and stochas- tic parameters. The verification of Process Hittings could be performed by using a translation into Petri Nets or into PRISM. Translating Process Hittings into more sophisticated process aglebras (Beta Workbench, Bio-PEPA, etc.) may also be of interest. Finally, techniques have to be developed to infer automatically temporal and stochastic parameters of Process Hittings modelling GRNs. Supplementary Material
We notice that the colors (time slots) used in the weighted min-max cliques around the gateways is enough to color all routes. The weighted min-max cliques around the gateway can be determined by a distribution of the total flow comming from the nodes over its links. The cliques are generated in a way to minimize the maximal one, consequently it will minimize the total number of colors needed (total routing time W ). The figure 6.a shows an example of a bad traffic distribution giving a worse result and the best configuration for grid networks with the gateway in the center. The best configuration shows that there is 4 nodes close to the gateway that have to go far from it and come back after through the central cliques. This routing minimizes the maximal cliques. The Grid result in figure 6.a has been proved in . Another result can be seen in figure 6.b, we show a larger network and the maximum clique with weight equals the optimal value.
These multi-hop networks are expected to carry high throughput. The ca- pacity of WMNs, that is, the throughput offered to each flow, is however affected by many factors such as network topology, traffic pattern, resource sharing and radio interferences [2, 3]. Several analytical studies on the capacity of wireless ad-hoc networks have all shown that the capacity decreases as the network size increases [4, 5]. Unlike ad-hoc networks, WMNs are stationary networksin which traffic is mainly router-to-gateway (respectively gateway-to-router) ori- ented. This special feature makes a bottleneck appear around the gateways, lead- ing to a more constrained available capacity per node . Optimization-based approaches have been investigated trying to maximize the network capacity . A key issue in wireless networking is to cope with the interferences produced by concurrent transmissions. If many concurrent transmissions are successful, they have to be pairwise non interfering. Consequently, MAC protocols achieving conflict-free link scheduling have been developed to avoid interferences [2, 8]. The evolution of a network can thus be seen as the sequential activation of conflict- free sets of links, called rounds in the following.
I. I NTRODUCTION
In wireless networks, the communication channels are shared among the terminals. Thus, one of the major problems faced is the reduction of capacity due to interferences caused by simultaneous transmissions . In this work, we call a round a collection of links that can be simultaneously activated in the network. We address the problem called Round Weight- ing Problem (RWP)  that consider joint routing and schedul- ing. We present a cross-layer formulation of the problem. We have to find a minimum-length schedule of selected links in a TDMA (Time Division Multiple Access) based wireless network. As we deal with multi-hop networks, these selected links represent a routing solution (paths) providing enough capacity to achieve the routers requirements of bandwidth. Scheduling methods such TDMA can guarantee achieving higher capacities by allowing time slots to be shared by simultaneous transmissions.
The Shu ffledJoinRDD implementation is very similar to ShuffledRDD. Instead of fetch- ing map output partitions from just one dependency, it fetches the corresponding map out- put partitions from both dependencies. The user specificies the number of Shu ffledJoin- RDD partitions and each paritition requests a corresponding fraction of the map output partitions. For instance, Shu ffledJoinRDD partition 1 will fetch Dataset1 Partition 1 and Dataset2 Partition 1 from all of the workers. Once these partitions are fetched, it creates a map with the key-value pairs of the smaller partition. Subsequently, it iterates through the keys of the bigger partition, seeing if they are present in this map, and if so, adding the intersection to the output.
Abstract: We consider a multi-hop wireless mesh network composed by routers which route traffic to the Internet through several gateway. In such network, a bottleneck phenomenon limits the performances around the gateways, the net- work capacity does not scale with its size. In this work, we propose a traffic scheduling strategy around the gateways in a 802.11-based wireless mesh net- work. We distinguish two kinds of nodes according to their location in the network and the medium sharing strategy used: those located within k-hop of the gateway run a TDMA medium access protocol while the nodes further run a CSMA/CA MAC layer. We investigate on the impact of the size of the TDMA area on the network capacity when an optimal scheduling is im- plemented. Through extensive simulations, it is shown that network capacity, fairness and packet loss rate are improved by our approach.