A novel predictive approach for mobility activeness in mobile wireless networks

Computer Networks 226 (2023) 109689 Available online 9 March 2023 1389-1286/© 2023 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/bync-nd/4.0/). Contents lists available at ScienceDirect Computer Networks journal homepage: www.elsevier.com/locate/comnet A novel predictive approach for mobility activeness in mobile wireless networks Peppino Fazio a,b,∗,Miralem Mehic b,c,Miroslav Voznak b,Floriano De Rango d,Mauro Tropea d aDepartment of Molecular Sciences and Nanosystems, Ca’ Foscari University, Via Torino 155, Mestre (VE), 30172, Italy bVSB – Technical University of Ostrava, 17. listopadu 2172/15, Ostrava, 70833, Czechia, cDepartment of Telecommunications, Faculty of Electrical Engineering, University of Sarajevo, Zmaja od Bosne bb, Sarajevo, 71000, Bosnia and Herzegovina dDIMES, University of Calabria, via P. Bucci 39/C, Arcavacata di Rende (CS), 87036, Italy ARTICLE INFO MSC: 0000 1111 Keywords: Mobile networks Mobility Routing Networking Metric Stability ABSTRACT Nowadays, mobile computing has become a key component of telecommunication systems, and the Open Systems Interconnection (OSI) layer operations are affected by the effects of node movements along the roads, from the physical to the routing/transport layers. In particular, routing approaches have been investigated from many years, trying to optimize the performance of the whole considered system, under different points of view. In this paper we are focusing the attention on the analysis of the mobility grade trend for a mobile ad-hoc network environment, as well as on the way it can be a-priori known, in order to have the possibility to study how the dynamics of mobile nodes can be described and in-advance known, with a predicted knowledge of nodes stability (in terms of mobility). Our simulations considered mobility in real geographical maps, and the obtained results confirmed the goodness of our proposed study. 1. Introduction The emergence of the Internet of Things (IoT), autonomous and aerial vehicles has led to an increased demand for network connectivity and, at the same time, an increasing of mobility level (more devices are related to vehicles which move with less physical constraints, such as drones, aerial devices and mobile sensors). In addition, peer to peer communications with no infrastructure pose special problems and some parameters to be taken into account, such as routing performance, energy consumption, scalability and security. These issues have already been addressed within ad hoc networks, where routes between two hosts may consist of hops through other network hosts which, due to dynamic nature of network nodes, can cause frequent and unpredictable topology changes [1]. As the number of nodes increases, there is a growing need for effective network management and organization, where the questions of scalability and robustness become vital. In this paper we analyze what happens to mobile nodes in terms of Mobility Activeness (MA), or mobility grade, considered as a parameter referring to the way the entire mobile system evolves in time, with some well-known consequences, such as high call dropping probability, huge signal evanescence and, above all, peer-to-peer link intermittence and the consequent network unstability [2], [3]. First of all, a detailed analysis of the MA in mobile scenario is given and, then, a possible approach to the prediction of its trend is described. The advantages of ∗Corresponding author at: Department of Molecular Sciences and Nanosystems, Ca’ Foscari University, Via Torino 155, Mestre (VE), 30172, Italy. E-mail address: [email protected] (P. Fazio). this kind of predictive study will be clearly underlined, also by the help of a deep simulation campaign, able to show some interesting results about the MA evolution. In dynamic networks (where the adjective dynamic refers to some aspects of the network, such as mobility, topology, energy, transmission power, etc.), having the possibility to apply predictive approaches for the enhancement of the overall network performance is always desirable, especially if we refer to the next frontiers of mobile communications, i.e. 6G [4]. Let us think, for example, to a routing protocol, where multi-objective metrics can be used to react to frequent changes in network topology [5], or where (typically) metrics are defined and evaluated at the moment at which the decision should be taken (e.g. packet forwarding, routing table building, best path evaluation, traffic requests, etc.): if the network condition in the immediate future could be known in-advance, routing decisions could anticipate a future network configuration, avoiding undesirable performance [6]. If we refer to nodes mobility, as illustrated in [7], many ideas have been introduced by the researchers and all of them may help network administrators, protocols and algorithms to behave differently, on the basis of the knowledge future conditions. In this work, we focused our attention on Mobile Ad-hoc NETworks (MANETs) environments: the works in [8,9] considered the way adhoc nodes move into the considered network and the authors take https://doi.org/10.1016/j.comnet.2023.109689 Received 8 June 2022; Received in revised form 12 February 2023; Accepted 6 March 2023 Computer Networks 226 (2023) 109689 2 P. Fazio et al. advantage from nodes mobility behavior to optimize routing operations through predictions and guarantee a given level of Quality of Service (QoS). In addition, in [9], the concept of energy in ad-hoc networks is also described. Firstly, the authors consider the ‘‘information amount’’ which is exchanged through packet signaling; secondly, the authors argue about the mobile energy, defining it as a function of node distribution and the propagation power law exponent. In [10–13] the authors propose some deep analysis of the way the stability of a pointto-point connection can be predicted, in ad-hoc environments. The main contributions of this proposal can be summarized as follows: •A deep analysis and definition of the mobility grade concept for distributed networks, aimed to define a new parameter which can help the system to enhance the overhead and the overall performance; •A new prediction approach for considering the future values of mobility grade for mobile nodes; •A deep analysis of the numerical results, in order to establish the stochastic properties of the proposed model. Before concluding the introduction, we would like to underline that we defined the MA by relating it to the Shannon’s entropy definition. So in the following, we refer to the concept of entropy referring to MA (in the case of mobile nodes). We can affirm that, the first main difference with the existing works is that most of them are related to the entropy contained into the exchanged information (informationentropy), while we focused on the mobility grade of the nodes which compose the network topology. In addition, most of the existing works take into account only networks in which nodes are completely mobile, disregarding the fact that the routing table of a static node (with no mobility) is affected by the entropy generated from neighbors, as they move and influence the complete network topology (Reciprocal Mobility Activeness, RMA, defined in next sections). Moreover, there are already some predictive approaches for MANET routing, but our proposal offers the lowest computational complexity, because it is based on an order-1 Auto-Regressive (AR) model which is, from our point of view, the simplest analytical method from predicting data of a time-series (we discovered that mobility activeness in a real mobile network, based on real paths, can be modeled as an order-1 process). The last consideration about the novelty and enhancement introduced by our contribution regards the type of mobility considered for simulations: most of the existing works consider a synthetic mobility model, which models mobility by stochastic and analytical equations. In this way, node movements may be unnatural (e.g. high steering degrees with high speed). We created mobility, instead, by considering real maps with City4Roadmaps (C4R) [14,15], which is based on the OpenStreetMap core (for extracting road-maps from the real world) and SUMO core (for creating real nodes movements in function of the extracted map). In this way, we are sure that our simulations consider real natural mobility. In Table 1 the main abbreviations used in the paper are illustrated. As regards the structure of the paper, the next section gives a detailed overview of the main scientific works existing in literature, Section 3 introduces the mobility grade concept for static and moving nodes. Section 4describes the deployment of adaptive filtering for temporal prediction of nodes evolution, while Section 5provide details about the main obtained results, in terms of mobility grade values in function of different system parameters and prediction possibilities, discussing the broader aspects of our approach. At the end, a comparison with the AODV protocol with and without our proposed metric is illustrated. Section 6concludes the paper. 2. State of the Art Predictive approaches have been often considered in mobile networks, in order to enhance the overall performance of the considered system. Clearly, they depend on the accuracy of the proposed idea, as Table 1 List of acronyms. Acronym Description ACF AutoCorrelation Function AF Adaptive Filtering AF-LMS Adaptive LMS filter AR Auto Regressive FOA Filter Optimization Algorithm IoT Internet of Things LMS Least Mean Square MA Mobility Activeness MANET Mobile Ad-hoc NETwork OSI Open Systems Interconnection PACF Partial ACF QoS Quality of Service RA Reciprocal Activeness RMA Reciprocal and Mobility Activeness SMA Simple Moving Average SNR Signal-to-Noise Ratio UKF Unscented Kalman Filter VANET Vehicular Ad-hoc NETworks WSN Wireless Sensors Network well as on the intrinsic traffic/mobility dynamics. Integrating a routing protocol with a predictive approach leads always to the enhancement of the overall performance [6]. In fact, as illustrated in [7], many ideas have been introduced by the researchers and all of them may behave differently. In particular, when referring to ad-hoc networks, node mobility is one of the key aspects that have been investigated and predicted, given that it is crucial for MANETs. 2.1. The concept of entropy in dynamic networks In the works [8,9], the main focus is targeted on the way mobile nodes move into the considered ad-hoc network. The authors base their proposal on the ‘‘entropy’’ concept to improve routing operations by predicting users movements, reflecting several enhancements on the QoS. In addition, in [9], the concept of information energy in ad-hoc networks is also described: first of all, the authors refer to Shannon’s information theory, considering the ‘‘amount of information’’ exchanged through packet exchanges, then the authors consider node communications from the energy point of view, modeling them as functions of nodes distribution and the propagation power law exponent. The article in [16] argues about the concept of topology changes measurements for MANETs, considered as the uncertainty of changes in network topology. It is strictly related to the minimum overhead required by nodes, during routing operations, to reach the ‘‘converged’’ status (that is to say the complete topology is known by all nodes). The article in [17] argues about the conditional entropy in wireless networks, by characterizing the topological uncertainty using the formalism of graph entropy, while the authors of [18] take into account the trustiness between terrestrial and satellite nodes, by analyzing the packets entropy of the exchanged information. 2.2. The concept of link stability/lifetime in dynamic networks Another parameter that can be optimized in ad-hoc networks is the link lifetime (or link stability), which is heavily affected by mobility or residual energy. In [10–13,19] the authors propose some deep analysis of the way the stability of a point-2-point connection can be predicted, in ad-hoc environments. In [10,13] the authors make use of the interpolation concept to predict the time interval over which a considered node can be considered as trusted, defining a new metric for routing table construction based on the best Signal-to-Noise Ratio (SNR) value. Given that routing protocols are responsible for searching and maintaining the best routes from a generic source to a generic destination, in [11] a novel forwarding approach is proposed, based on path stability. Also in this case, the authors based the choice of the Computer Networks 226 (2023) 109689 3 P. Fazio et al. Table 2 Symbols used in the proposal. Symbols Description 𝐺Geographical area 𝑔𝑖𝑗 Sub-areas 𝑁, 𝑂 Dimension of G in meters 𝑙𝑥, 𝑙𝑦Side size of a generic 𝑔𝑖𝑗 ∈𝐺 𝑛 𝑛 =⌈𝑁∕𝑙𝑦⌉ 𝑚 𝑚 =⌈𝑂∕𝑙𝑥⌉ 𝑀𝑂𝐵 Set of mobile nodes with size 𝑀 𝑀Size of set 𝑀𝑂𝐵 𝑣𝑘𝑘-th mobile nodes 𝑊Observation window size 𝑝𝑊 𝑘(𝑔𝑘 𝑖𝑗 )Probability of visiting 𝑔𝑘 𝑖𝑗 𝑉∗ 𝑘Number of distinct 𝑔𝑖𝑗 visited by 𝑣𝑘 𝐼𝐷𝑘Identifier of 𝑘-th node 𝑛𝑔𝑘Number of one-hop neighbor nodes of 𝑣𝑘 𝑅𝐴𝑠Reciprocal Activeness contribution 𝑅𝑀𝐴 Reciprocal and Mobility Activeness 𝛾smoothing factor ([0..1]) 𝑊𝑗𝑗-th observation window 𝐼𝑅𝑊𝑗Impulse response at 𝑊𝑗 𝑃 𝑅𝐸𝑅𝑀𝐴 Predicted output 𝐷𝐸𝑆𝑅𝑀𝐴 Desired output 𝑐𝑓 Convergence factor 𝜖𝑗Difference between the predicted RMA and the desired RMA at step 𝑗 𝜇Mean of the process next hop on the signal strength prediction, which takes into account link stability by a distance and time based theoretical formulation, able to predict how long a link becomes stable for use by the help of mobility. In [12], link lifetime is predicted through the deployment of the Unscented Kalman Filter (UKF), used to model a nonlinear system and to compute the estimates of the remaining link lifetime. Authors suggest to apply the UKF recursively, in order to compute system’s states, using as inputs periodical measurements of the distance between the two link’s nodes. The work in [20] takes into account the residual energy concept for predicting the lifetime of a link among a couple of nodes (power aware routing). In few words, the authors make use of an optimization problem, defining an objective function (maximizing the lifetime of a chosen path) and the associated constraints, integrated into the RREQ/RREP mechanism. The core of the idea is based on the individual battery lifetime prediction made by each single node, based on its past activity (using a Simple Moving Average (SMA) predictor). In the next section, our proposal is deeply introduced and described. 3. Mobility Activeness in mobile networks We start our proposal by defining the concept of Mobility Activeness (MA) as a measure of the uncertainty in a generic statistical model [21]. This definition is based on the amount of node mobility and higher MA leads to harder prediction operations, with lower accuracy. Table 2 shows the main symbols used in the mathematical formulation for explaining our proposal. 3.1. Mobility Activeness of mobile nodes: the definition We associate a certain level of MA to a node by considering its geographical position, its way to move among different areas or how it communicates with its neighbors. So, first of all, let us assume that all nodes into the system are mobile (static nodes can be considered as a particular case of mobile nodes, with MA equals to zero). A generic 2D geographical area G can be considered as the result of a partitioning operation, able to subdivide G(where mobile nodes are moving) into a finite set of 𝑛𝑥𝑚 square/rectangular sub-areas 𝑔𝑖𝑗 , such as: 𝐺=𝑔11 ∪𝑔12 ∪⋯∪𝑔1𝑚∪𝑔21 ∪𝑔22 ∪⋯∪𝑔𝑛(𝑚−1) ∪𝑔𝑛𝑚 𝑔11 ∩𝑔12 ∩... ∩𝑔1𝑚∩𝑔21 ∩𝑔22 ∩... ∩𝑔𝑛(𝑚−1) ∩𝑔𝑛𝑚 = ∅.(1) Fig. 1. An example of partition applied to a geographical area 𝐺𝑁𝑒𝑤𝑌 𝑜𝑟𝑘 with N=12 km, O=22 km, and an area of O*N=264 km2; the values of 𝑚and 𝑛are 4 and 6 respectively, with 𝑙𝑥= 3 km and 𝑙𝑦≈ 3.67 km. which can be rewritten, in compact form, as: 𝐺= 𝑛 ⋃ 𝑖=1 𝑚 ⋃ 𝑗=1 𝑔𝑖𝑗 𝑤𝑖𝑡ℎ 𝑛 ⋂ 𝑖=1 𝑚 ⋂ 𝑗=1 𝑔𝑖𝑗 = ∅.(2) and 𝑛, 𝑚 ∈N+. The values of 𝑛and 𝑚can be set or derived from the dimensions of 𝐺, assumed to be 𝑁and 𝑂(in meters), so for each 𝑔𝑖𝑗 ∈𝐺the relations 𝑛=⌈𝑁∕𝑙𝑦⌉and 𝑚=⌈𝑂∕𝑙𝑥⌉are always valid, with 𝑙𝑥and 𝑙𝑦representing the side sizes of the generic 𝑔𝑖𝑗 ∈𝐺. We assume that each sub-area has the same dimensions of the other ones, as depicted in the example of Fig. 1. For simplicity of notation, 𝐺can be represented by its partition set 𝑔𝑖𝑔 ∈𝐺of sub-areas as a matrix (𝑛x𝑚): 𝐺=⎡ ⎢ ⎢ ⎣ 𝑔11 𝑔12 ... 𝑔1𝑚 ... ... ... ... 𝑔𝑛1𝑔𝑛2... 𝑔𝑛𝑚 ⎤ ⎥ ⎥ ⎦ (3) At this time, the MA value for each mobile node should be defined: we need to introduce also an observation time Window W, during which mobile hosts move and define their current MA. The idea is to consider the number of visited 𝑔𝑖𝑗 ∈𝐺during 𝑊and relating it to the definition of MA. To this aim, given the set of mobile nodes 𝑀𝑂𝐵 = {𝑣1,…, 𝑣𝑀}, with ‖𝑀𝑂𝐵‖=𝑀, then for the 𝑘th mobile node 𝑣𝑘, we can define the set of areas visited by 𝑣𝑘∈𝑀𝑂𝐵 during W: 𝑣𝑊 𝑘= {𝑔𝑘 𝑖𝑗1...𝑔𝑘 𝑖𝑗𝑉𝑘|𝑔𝑘 𝑖𝑗𝑙∈𝐺, 𝑙 = 1..𝑉𝑘},(4) with ‖𝑣𝑊 𝑘‖=𝑉𝑘. At this point, the probability of visiting 𝑔𝑘 𝑖𝑗 by 𝑣𝑘in the current W can be defined as: 𝑝𝑊 𝑘(𝑔𝑘 𝑖𝑗 ) = ∑𝑉𝑘 𝑙=1 𝑐𝑜𝑢𝑛𝑡(𝑙, 𝑔𝑘 𝑖𝑗𝑙, 𝑣𝑊 𝑘) 𝑉𝑘 ,(5) where the argument of the summation 𝑐𝑜𝑢𝑛𝑡(𝑙, 𝑔𝑘 𝑖𝑗𝑙, 𝑣𝑊 𝑘)counts how many times node 𝑣𝑘visited 𝑔𝑖𝑗 . At this point, we apply the fundamental entropy definition given by Shannon in [21], based on a set of symbols and the probabilities of those symbols to appear in the sequence; so it is easy to see that: 𝑀𝐴(𝑣𝑊 𝑘)=− 𝑉∗ 𝑘 ∑ 𝑙=1 𝑝𝑊 𝑘(𝑔𝑘 𝑖𝑗𝑙)⋅𝑙𝑛[𝑝𝑊 𝑘(𝑔𝑘 𝑖𝑗𝑙)] (6) where 𝑉∗ 𝑘is the number of distinct 𝑔𝑖𝑗 visited by 𝑣𝑘. Then, we apply the definition of MA given in (6) to extract knowledge about the evolution of a network with a dynamic topology. 3.2. The Reciprocal Activeness: how nodes are influenced by each other In Wireless Sensors Networks (WSNs), Ad-hoc Networks, MANETs or Vehicular Ad-hoc NETworkss (VANETs), during routing operations some links may break, then route recovery and maintenance procedures Computer Networks 226 (2023) 109689 4 P. Fazio et al. Fig. 2. The influence of nodes 𝑣𝑖,𝑣𝑗,𝑣𝑙on 𝑣𝑘’s RA (curved arrows), where 𝑟𝑖,𝑟𝑗,𝑟𝑘 and 𝑟𝑙are the coverage radius. need to be executed. But, such procedures consume various resources such as the energy which needs to be preserved. To minimize route interruptions, it is of crucial importance to find a route that endures longer time. Many works in literature, such as [22–24], emphasize the importance of the link stability parameter. In the case of distributed wireless networks, relaying operations affect the performance of the whole system, so the routing metric should be chosen carefully. In this sense, the MA can provide precious information to characterize nodes behavior and their uncertainty in terms of reliability over time. Another key aspect could be represented, for example, by the reflection of the MA into a routing table [25]: on the basis of the way the entries are stored, it is possible to analyze nodes routing stability and, consequently, it is possible to introduce an ageing mechanism to set the periodic signaling interval (such as Hello messages). In addition, it is easy to see that, if the network is dealing with static (or almost static, with low mobility grade) nodes, the mobility contribution to MA is null (MA is equal to zero for each node), or it could be evaluated for a very large 𝑊, given that each node is visiting only one area in 𝑊, with probability equal to 1. So, in this case, the way for calculating MA should be different. In particular, we take into account the number of one-hop neighbor nodes for node 𝑣𝑘, as an index of the local influence of 𝑣𝑘’s neighbors on 𝑣𝑘(local world). As regards the implementation of this approach, let us imagine that each node has an associated 𝐼𝐷𝑘and it can manage a shared structure (a list) in which for each 𝐼𝐷𝑘the number of its onehop neighbor can be inserted. Neighboring information can be derived, for example, by the routing operations (Hello messages, RREQ/RREP mechanism, etc.), so we are considering the general case. After the convergence time, each node will know the exact number of one-hop neighbors for the other nodes. So, if 𝑣1, ..., 𝑣𝑀are the considered mobile nodes (of the whole network) and 𝑛𝑔𝑘is the number of one-hop neighbor nodes of 𝑣𝑘, then the Reciprocal Activeness (RA) contribution 𝑅𝐴𝑠of the 𝑛𝑔𝑘nodes on 𝑣𝑘in the 𝑊period can be expressed after the re-definition of: 𝑝𝑊 𝑘(𝑡) = 𝑛𝑔𝑊 𝑘(𝑡) ∑𝑀 𝑙=1 𝑛𝑔𝑊 𝑙(𝑡) , 𝑡 ∈𝑊(7) where 𝑡is a time instant inside the range 𝑊(the addition of the time dependence is needed because in the time window 𝑊the number of neighbors for node 𝑣𝑘can change over the time). Then, as from Eq. (6): 𝑅𝐴𝑠(𝑛𝑔𝑊 𝑘)=−∑ 𝑡∈𝑊 𝑝𝑊 𝑘(𝑡)⋅𝑙𝑛 [𝑝𝑊 𝑘(𝑡)](8) that is to say the influence of 𝑣𝑘’s neighbors on 𝑣𝑘in 𝑊(as illustrated in Fig. 2). When nodes move, assuming an ON-OFF behavior (with failures/retrievals), through a beaconing or routing signaling each 𝑣𝑘 can ’’sense’’ the absence/presence of a neighbor. In this case, the related entry of the shared list is updated. If a new node enters the network it will start the update procedure from the beginning. In general, since our approach does not consider a specific situation (there could be fixed nodes with a high number of neighbors, or moving Fig. 3. The general scheme of an AF applied to RMA process. Fig. 4. The 𝐺map considered for simulations and its 10 x 10 partition. Fig. 5. An example of the trend of the activeness associated to a mobile host, with 𝑎𝑣𝑔_𝑠𝑝𝑒𝑒𝑑 of 14 m/s, 𝐿= 50 m and an observation time window size of 𝑊= 10 s. The obtained trends for 𝑀𝐴(𝑣𝑊 𝑘)and 𝑅𝐴(𝑛𝑔𝑊 𝑘)are shown in two lines for better readability. nodes with a low number of neighbors for example), the activeness index should be composed by both terms (mobility and reciprocal activenesses); this is the reason why in our approach, considering (6) and (7) we define the Reciprocal and Mobility Activeness (RMA): 𝑅𝑀𝐴(𝑣𝑊 𝑘) = 𝛾⋅[𝑀𝐴(𝑣𝑊 𝑘)] + (1 − 𝛾)⋅[𝑅𝐴(𝑛𝑔𝑊 𝑘)],(9) with 𝛾∈ [0..1] as smoothing factor. 4. Activeness Prediction via Adaptive Filtering The second idea of this paper relies on the utilization of a predictive approach, to know in advance what the trend of nodes RMA will be. In the next subsections we describe the made assumptions for the Computer Networks 226 (2023) 109689 5 P. Fazio et al. Fig. 6. The trend of the RMA (60 samples) associated to a mobile node (Eq. (9)), with 𝑎𝑣𝑔_𝑠𝑝𝑒𝑒𝑑 of 11 m/s, area side L =50 m and an observation time window W =10 s. Different values of 𝛾have been considered. Fig. 7. Average RMA associated to mobile nodes for different values of 𝑊and 𝐿. predictive approach and how the RMA dynamics during nodes mobility can be captured, analyzed and predicted. 4.1. Adaptive Filtering for Dynamic Processes Given the periodical nature of the defined parameters (MA, RA and RMA) and routing protocols (given their update interval), we decided to base our proposal on the Adaptive Filtering (AF) [26] theory, since an AF is able to adapt the coefficients of its impulse response in function of a given optimization algorithm. In fact, in our case, the particular activeness (MA, RA or RMA) changes in function of 𝑊and the closed loop (of the filter) treats the feedback as an error signal, to re-define its transfer function parameters. Fig. 3 shows the general scheme of an AF: the 𝐼𝑅𝑊𝑗is the impulse response at the 𝑗th observation window 𝑊𝑗,𝑅𝑀𝐴(𝑣𝑊𝑗 𝑘)is the input RMA at 𝑊𝑗; the filter evaluates its output as the convolution if its current impulse response and the most recent values of RMA, then the predicted output 𝑃 𝑅𝐸𝑅𝑀𝐴 is compared with the desired 𝐷𝐸𝑆𝑅𝑀𝐴 and the difference together with the input are given as inputs to the Filter Optimization Algorithm (FOA), able to recalculate and optimize the impulse response weights. We decided to use the Least Mean Square (LMS) [27,28] as FOA, able to update filter weights in the following way: 𝑐𝑙,𝑗+1 =𝑐𝑙,𝑗 + 2 ⋅𝑐𝑓 ⋅𝜖𝑗⋅𝑅𝑀𝐴(𝑣𝑊𝑗,𝑙 𝑘),(10) where we considered the 𝐾−𝑡ℎ order Adaptive LMS filter (AF-LMS) (𝑙= 1..𝐾), 𝑗is the previous observation step and 𝑊𝑗is its related observation window, 𝑐𝑓 is called convergence factor and 𝜖𝑗is the difference between the predicted RMA (𝑃 𝑅𝐸𝑅𝑀𝐴) and desired RMA (𝐷𝐸𝑆𝑅𝑀𝐴) at step 𝑗. As regards the parameter 𝑐𝑓 , it controls the speed and accuracy of the algorithm convergence: generally it is larger at the beginning for a rapid convergence and decreased to minimize overshooting actions (0< 𝑐𝑓 < 1). 4.2. Adaptive Filtering as an Auto Regressive Process The AF-LMS approach fits perfectly with our scope, since the activeness is evaluated periodically (let us think, for example, to the periodic beaconing, or periodic routing updates, etc.), giving us the possibility to assume and consider it as a sequence of time samples and, in particular, as an Auto Regressive (AR) process, where the last observed value depends linearly on the previous K ones. The only remaining concern for the proposed analysis is the determination of the value of 𝐾, that is the order of the AF-LMS and, hence, of the underlying AR process. To this aim, we consider the AutoCorrelation Function (ACF) and the Partial ACF (PACF) [29]. In fact, an index of the correlation between two values of an 𝐴𝑅(𝐾)process is the ACF. For a generic process 𝑋𝑡, 𝑡 = 0,1,2,…the autocovariance [30,31] at lag 𝐾is defined as: 𝛾𝑋 𝑘=𝐶𝑜𝑣(𝑋𝑡, 𝑋𝑡−𝐾) = 𝐸[(𝑋𝑡−𝜇)⋅(𝑋𝑡−𝐾−𝜇)] (11) where 𝜇is the mean of the process, i.e. 𝜇=𝐸[𝑋(𝑡)], and the autocorrelation coefficient at lag 𝐾is: 𝜌𝑋 𝐾=𝛾𝑋 𝐾 𝛾𝑋 0 (12) where the autocovariance at lag zero 𝛾𝑋 0is the variance of the process. It is clear that, from the definition, the autocorrelation coefficient 𝜌𝑋 𝐾 is dimensionless, so independent on the measurement scale, and it belongs to the interval [−1,1]. From [32], it is known that the term in Eq. (12) is the theoretical ACF. A lag 𝐾autocorrelation represents, in our case, the relation between activeness values that are 𝐾time periods apart. So, the ACF is a way to consider the linear relationship between a time instant 𝑡and all the process observations at previous times. In our work, we assume that the activeness dynamics can be modeled as an 𝐴𝑅(𝐾)process, but we want to know which is the relation among 𝑅𝑀𝐴(𝑣𝑊𝑗 𝑘)and 𝑅𝑀𝐴(𝑣𝑊𝑗−𝐾 𝑘), without considering the contributions of 𝑅𝑀𝐴(𝑣𝑊𝑗−1 𝑘), ..., 𝑅𝑀𝐴(𝑣𝑊𝑗−𝐾+1 𝑘). Clearly, at lag 1, PACF(1) is the same as ACF(1). Following the theory in [33], in order to describe the expression for the PACF, we have to consider 𝐾Yule– Walker equations [33] written for the 𝐴𝑅(𝐾)process, and solve them for the 𝐾variables 𝜙𝐾1,…, 𝜙𝐾𝐾 . Typically they are written in a matrix form as follows (we used the notation 𝑅for the 𝑅𝑀𝐴 process in order to obtain a compact notation): ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1𝑅(𝑣𝑊1 𝑘)... 𝑅(𝑣𝑊𝐾−1 𝑘) 𝑅(𝑣𝑊1 𝑘) 1 ... 𝑅(𝑣𝑊𝐾−2 𝑘) 𝑅(𝑣𝑊2 𝑘)𝑅(𝑣𝑊3 𝑘)... 𝑅(𝑣𝑊𝐾−3 𝑘) ... ... ... ... 𝑅(𝑣𝑊𝐾−1 𝑘)𝑅(𝑣𝑊𝐾−2 𝑘)... 1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⋅ ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ 𝜙𝐾1 𝜙𝐾2 𝜙𝐾2 ... 𝜙𝐾𝐾 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 𝑅(𝑣𝑊1 𝑘) 𝑅(𝑣𝑊2 𝑘) 𝑅(𝑣𝑊3 𝑘) ... 𝑅(𝑣𝑊𝐾 𝑘) ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (13) and the PACF will be represented by the 𝐾−𝑡ℎ solution 𝜙𝐾𝐾 , a function of lag 𝐾. Based on the discussion above, as shown in next sections, we can conclude that using the PACF instead of the ACF will lead us to obtain a meaningful value of 𝐾and, as stated in [29,34], the PACF represents the most useful ‘‘tool’’ for determining the order of an AR model. So, the ACF and PACF are statistical measures that reflect how the observations of a process evolution are related to each other. In addition, as stated in [32], it is often useful to plot these functions against consecutive time lags. All the graphical approaches, for assessing the lag/order of an AR model, include looking at the ACF/PACF values versus the lag (correlogram). In an ACF correlogram, as the ones shown in the Computer Networks 226 (2023) 109689 6 P. Fazio et al. Fig. 8. RMA trend fitting by using linear filtering. Fig. 9. PACF trend for different lags 𝐾and 𝛾values (𝑊= 10 s, 𝐿= 50 m, 𝑎𝑣𝑔_𝑠𝑝𝑒𝑒𝑑 = 14 m/s). Fig. 10. Correlogram of the PACF for different lags 𝐾and 𝛾values (on the X axis), with 𝑊= 10 s, 𝐿= 50 m, 𝑎𝑣𝑔_𝑠𝑝𝑒𝑒𝑑 = 13.9m/s. next section, if there are large values with a non-random pattern, then there is a high probability that the values are correlated. In a PACF correlogram, instead, the pattern is usually randomly defined, but large values for a given 𝐾indicate that it is a possible choice for the lag/order of the whole process [32]. In the next section a full and deep analysis of these concepts is carried out, giving to the reader the possibility to well understand how the activeness parameter can be analyzed, predicted and applied in mobile networking. 5. Numerical analysis and results This section is dedicated to show the main numerical results reachable by the proposed AF-LMS approach. In Table 3 the values used in Table 3 The main values used in numerical analysis. Parameter Value 𝑁=𝑂2000 m 𝑁x𝑂4 km2 𝑙𝑥=𝑙𝑦50–200 m 𝑛=𝑚[10, ...,40] 𝑔𝑖𝑗 ∈𝐺100–1600 𝑟𝑘=𝑟50 m 𝛾0.6 𝑎𝑣𝑔_𝑠𝑝𝑒𝑒𝑑 11, 14, 20 m/s the numerical analysis are listed. Mobility has been generated on the basis of the OpenStreetMap core [15] (which gives the opportunity to select and export a desired geographical map 𝐺) and City4Roadmaps (C4R) [14] (able to generate mobility patterns by following real movements). A square Gwith N=O=2000 meters and an area of about 4 km2has been considered, extracted from the center and peripheral of Rome city (refer to Fig. 4). Mobility traces have been generated as urban mobility, with variable average speeds (𝑎𝑣𝑔_𝑠𝑝𝑒𝑒𝑑) of 11, 14 and 20 m/s, while the partitioning sub-areas have been considered to be square, with a side 𝑙𝑥=𝑙𝑦=𝐿ranging from 50 m to 200 m (so values of 𝑛=𝑚∈ [10,…,40] and a total number of sub-areas 𝑔𝑖𝑗 ∈𝐺going from 100 to 1600). Vehicles arrival times belong to a Poisson distribution and the coverage radius for each node has been considered to be 𝑟𝑘=𝑟= 50 m ∀ 𝑣𝑘∈𝑀𝑂𝐵, 𝑘 = 1..𝑀. An object-oriented Python application has been designed, in order to create the map, its partition, mobility traces and the evaluation of Eqs. (6),(8) and (9), taking 𝐺,𝑁,𝑂, and 𝐿as input parameters. Fig. 5 shows the trend of 𝑀𝐴(𝑣𝑊 𝑘)and 𝑅𝐴(𝑛𝑔𝑊 𝑘)for a generic mobile node. The shown trend is general and we verified that it is valid for all the involved nodes during their active sessions. In order to give an idea of the generic trend of the values of 𝑅𝑀𝐴 as defined in Eq. (9),Fig. 6 is also shown. In Fig. 6, the total number of samples has been reduced in order to make the figure more readable. If we refer to the average trend of RMA (𝛾= 0.6) in function of 𝑊and 𝐿,Fig. 7 can be considered. The first interesting result of our analysis shows how the average MA value is influenced by the selected parameters. In fact, Fig. 7 illustrates the trend of 𝑅𝑀𝐴(𝑣𝑊 𝑘)by varying the 𝑊length and the value of 𝐿(with an 𝑎𝑣𝑔_𝑠𝑝𝑒𝑒𝑑 of 11 m/s and 𝛾=0.6). For bigger subareas, the activeness goes decreasing given that each mobile host takes more time to go outside each area (the current one), while for higher values of 𝑊RMA increases, given that each mobile node is able to move among a higher number of locations. At this point, we have to verify that an 𝐴𝑅(𝐾)approach can help to analyze the dynamics of mobile nodes, giving to them an a-priori knowledge of future RMA behaviors. As illustrated in the following, we provided to implement the LMS adaptive filter in python, through the Python Adaptive Signal Processing library v1.1.1 [35]. Computer Networks 226 (2023) 109689 7 P. Fazio et al. Fig. 11. Activeness samples prediction with LMS for 𝐾=1, 𝑐𝑓 =0.1, 0.3, 0.5, 0.7. In particular, we integrated the previously implemented parser (in order to make the mobility generated by C4R readable) with the library in [35], in order to evaluate the ACF and the PACF functions related to the collected samples, after the 𝐺partitioning operation, and assuming that 𝑅𝑀𝐴(𝑣𝑊 𝑘)is an 𝐴𝑅(𝐾)process. This approach fits perfectly with the main aim of this paper, that is the possibility to predict future samples in real-time, such as very important system parameters (weights in network routing, the overall cost on a path from a source to a destination, links stability, etc.). In order to obtain some suitable results in this direction, we provide to apply the ACF/PACF definitions to find the order of the RMA process. First of all, the order of the adaptive filter needs to be decided. To this aim, we provided to use the 𝑠𝑐𝑖𝑝𝑦.𝑠𝑖𝑔𝑛𝑎𝑙 and 𝑠𝑝𝑒𝑐𝑡𝑟𝑢𝑚 libraries in Python, where the pyule function is available [33], in order to obtain the PACF related to the lag 𝐾.Fig. 8, for example, shows two representations of the trend of the original sequence of 20 RMA samples (cross points) and their estimation by a linear AR filter with LMS optimization (𝐾= 10). On the left side a more evident trend of the committed error is underlined, while on the right it can be observed how a linear filter is able to follow the right trend, with a prediction error (variance) 𝜎2=0.18. So, in order to discover and choose an adequate value for the filter order 𝐾, we provided to evaluate the PACF, considered, as defined earlier, as the autocorrelation between 𝑅𝑀𝐴(𝑣𝑊𝑗 𝑘)and 𝑅𝑀𝐴(𝑣𝑊𝑗−𝐾 𝑘), without the linear dependence of 𝑅𝑀𝐴(𝑣𝑊 𝑘)on 𝑅𝑀𝐴(𝑣𝑊𝑗−1 𝑘)through 𝑅𝑀𝐴(𝑣𝑊𝑗−𝐾+1 𝑘)[36]. We provided to carry out the analysis of several activeness samples related to different nodes into 𝐺and, consequently, the PACF analysis has been carried out, varying 𝑊,𝐿,𝛾,𝑎𝑣𝑔_𝑠𝑝𝑒𝑒𝑑: for space issues we cannot show all the obtained results, but we summarize them with the following figures. Fig. 9 shows the trend of the PACF, taking into account the possible values of lags from 𝐾=1 to 𝐾=20. We can observe and conclude that the memory effect is noticeable for low values of 𝐾. In particular the activeness evolution can be considered as an 𝐴𝑅(1) process, since the function spike occurs for 𝐾=1, independently of the chosen 𝛾value. For 𝐾=2 or 𝐾= 3 the absolute value of PACF goes drastically decreasing, while for higher values it approaches to 0. Clearly, Fig. 9 is obtained for a particular combination of simulation parameters, but it reflects the general PACF trend. Fig. 10 shows the same values in a different form (a correlogram): independently from 𝛾,𝐾=1 (biggest and darkest marker) leads to the highest absolute value of PACF, while for decreasing lag values, PACF is negligible, showing that no correlation exists among samples after larger time periods. Table 4 Parameter values of the simulation. Parameter Value Simulation area 1000 ×1000 m2 Number of Nodes 10, 30, 50 Number of NetDevices per node 1 Wifi Phy mode DsssRate11Mbps Wifi Propagation Delay Constant Speed Propagation Model Data Traffic Type UDP Constant Bit Rate Data Traffic Rate 512 kbps Data Traffic Application OnOffApplication Mobility Model Random WayPoint Mobility Model Pause Interval Constant (0.5 s) Node speed in mobility model 10, 30 m/s Total Simulation time 100 s For that value of 𝐾the prediction error could be minimized, given that the activeness evolution is not completely random, but it is regulated by a heavy correlation between samples which are 𝐾steps away. So, for the next results, we set 𝐾=1 and applied the LMS algorithm to the RMA process in order to obtain the optimal coefficient. Fig. 11 shows the results obtained by considering 120 RMA samples, with 𝑊=10s, 𝐿=30 m, 𝑎𝑣𝑔_𝑠𝑝𝑒𝑒𝑑=14 m/s. It can be seen how, in general, the LMS is able to evaluate future samples with high accuracy and the 𝑐𝑓 parameter does not heavily affect the overall error. 5.1. Metric application simulation analysis To analyze the usage of RA and MA in routing protocols, we simulated networks with random topologies consisting of 10, 30 and 50 nodes. We considered the impact of node mobility and the geographical size of sub-areas that are used to calculate MA value (𝑔𝑖𝑗 in Eq. (3)). The simulations were performed using the NS-3 Simulator of version 3.37 [37]. For the same parameters of the number of nodes, the speed of movement, and the number of traffic-generating applications, 10 different random scenarios were generated, which resulted in total 1712 simulations. The BRITE topology generator to generate random topologies since it is supported under NS-3 and the source code is freely available [38]. Table 4 lists the simulation parameters including parameters of WiFi NetDevices which were set to provide a maximal coverage area of 150 m2and enable multi-hop communication. Parameters not given here are default parameters of the NS-3 v3.37 simulator. Each simulation included two traffic-generated applications. For each of the applications, a node is randomly selected from the (0,(𝑛∕2)− 1) range of nodes and the source traffic application is installed on the selected node. For each of the applications, a node is randomly selected Computer Networks 226 (2023) 109689 8 P. Fazio et al. from the (𝑛∕2, 𝑛 − 1) range of nodes and the destination traffic application is installed on the selected node. Location-based monitoring is implemented as dedicated module in the NS-3 simulator, which periodically analyzes the movement of nodes every three seconds. Based on the measured values, the MA value for each node is periodically calculated. Also, location-based monitoring was extend to provide RA value by analyzing the routing tables of each node in the network. In particular, we considered the application of the RA and MA in AODV routing protocol. AODV is known as a reactive routing protocol where routing paths are searched only when needed by flooding the network [39,40]. The discovery procedure terminates when either a route has been found, or no route is available after all route permutations have been checked. Due to flooding, an intermediary node may receive multiple RREQ queries to find a path to a remote destination. By default, AODV stores the first RREQ while discarding all subsequent requests as they are considered duplicates. In our variant, we considered the application of the RA and MA when analyzing that broadcasted RREQ queries. Each time node receives RREQ request from its neighbor and there is already previously processed RERQ and stored in cache memory with the same origin and destination, it will calculate RA and MA values using Eq. (9) for itself and the neighboring node which forwarded RREQ request. Suppose that the calculated RMA value of neighbor is lower then the calculated RMA value of itself. Then the received RREQ request will be ignored. However, in opposite case, it will be processed and AODV route will be updated over the neighboring node which forwarded RREQ request. Figs. 12 and 13 shows the obtained results. One can note that when parameter gamma (𝛾from Eq. (9)) is set to 0, the value RMA is based on RA, and thus, there is no influence of the geographical size of sub-areas that are used to calculate MA value (𝑔𝑖𝑗 in Eq. (3)). This case is denoted with blue box-plots on graphs that are identical in subfigures. However, as value 𝛾increases, the RAM value considers RA and MA values. In the case of a network with a smaller number of nodes (i.e. 10), there are no significant changes in obtained results. The reason is that a small number of nodes do not lead to rapid changes in the network from the aspect of routing table entries and overall network dynamics. However, when the network is formed with a larger number of nodes, more dynamics lead to significant changes in routing tables and MA values. As the number of network nodes increases, more mobile nodes are marked as candidates as messenger nodes between different mobility regions. Thus, there are more chances to find a better AODV route. Simulations were performed with identical network topologies and random seeds, guaranteeing the simulation’s repeatability. While comparing results from Figs. 12 and 13, one can note that the obtained PDR values are significantly lower. The reason is that as the mobility of nodes increases, there are more interruptions of established AODV routes. There is also an increased number of chances to find alternative AODV routes, but due to high mobility, these alternative routes last only for a short period of time. The impact of geographical sizes of sub-areas is more significant, and AODVM can for different values of gamma (𝛾) outperform pure AODV. In some cases, our modification of AODV resulted in equal or better routing (best expressed with purple box-plots when 𝛾= 1), while in others, it resulted in degradation. It depends on values of 𝛾and sizes of sub-areas. However, the simulated scenarios denote only one example of using the RMA approach. It is possible to find better scenarios in which RMA values will have a more significant impact. In our example with AODV RREQ records, RMA is considered only when the route is interrupted and needs to be refreshed by processing new RREQ records (the processing of the first RREQ records to establish the initial route is identical for AODV and AODVM protocols). Such cases are not frequent (especially for networks with low mobility and dynamics), and a more significant influence of RMA records is expected in proactive routing protocols, e.g., when processing periodic hello messages to consider the network state (i.e., DSDV routing protocol [41]). However, the described example shows that the RMA value can significantly impact network performance, even considered through application to AODV RREQ requests. Fig. 12. Simulation results comparing AODV and AODVM routing protocols for different sizes of network (number of nodes). The mobility speed of nodes was set to 10 m/s. 6. Conclusion and future works In this work, we presented a stochastic analysis of the concept of mobility activeness in mobile networks, given its capability to influence network dynamics (routing, physical channel, etc.). We provided to define it and give emphasis to the main features which are able to influence its value when mobile nodes move inside a geographical area. We underlined the importance of considering mobility activeness (direct or reciprocal), as well as the possibility to predict it, by the use of an adaptive filter, optimized by the LMS algorithm for the weights update. In addition, we discovered that the activeness process can be considered to be an order-1 auto regressive process. The obtained results have shown that the trend of mobile activeness can be predicted with a very negligible error and this feature can give to the network a very important feedback on the future behavior of mobile nodes, Computer Networks 226 (2023) 109689 9 P. Fazio et al. Fig. 13. Simulation results comparing AODV and AODVM routing protocols for different sizes of network (number of nodes). The mobility speed of nodes was set to 30 m/s. especially on their stability in the near future. We provided, also, to carry out a performance comparison between the classical AODV protocol (whose metric is based on the hop-count), and the AODVM (with the RA and MA metrics), in order to show the benefits of our proposal. As future extensions of the proposed idea, we plan to consider also Artificial Intelligence (AI)-based predictive schemes for routing optimization, in order to consider and compare the possible obtainable enhancements, at the cost of a higher computational complexity. CRediT authorship contribution statement Peppino Fazio: Conceptualization, Investigation, Writing – original draft, Methodology, Software. Miralem Mehic: Conceptualization, Writing – review & editing, Software. Miroslav Voznak: Visualization, Supervision, Funding acquisition. Floriano De Rango: Resources, Investigation, Supervision. Mauro Tropea: Software, Data curation, Validation. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Data availability Data will be made available on request. Acknowledgment The research received a financial support from the Student Grant System (SGS) No. SP2018/59 ‘‘Networks and Communication Technologies for Smart Cities’’, VSB - Technical University of Ostrava, Czech Republic. References [1] S. Basagni, et al., in: S. Basagni, et al. (Eds.), Mobile Ad Hoc Networking, John Wiley & Sons, Inc., Hoboken, NJ, USA, 2013. [2] F. De Rango, et al., Utility-based predictive services for adaptive wireless networks with mobile hosts, IEEE Trans. Veh. Technol. 58 (3) (2009) 1415–1428. 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