A Stochastic Approach for Resource Prediction Error and Bandwidth Wastage Evaluation in Advanced Dynamic Reservation Strategies

Many studies in literature have shown that the bandwidth of an ongoing flow can dynamically change during multimedia sessions and an efficient bandwidth prediction scheme must be employed in order to guarantee the needed Quality of Service. Many contributions in literature focused on some specific reservation schemes for cellular systems, 3G/4G/5G networks, WLAN and so on. However, few of them exploited the trade-off between resource prediction error and bandwidth wastage through a general prediction technique based on the analysis of Cell Stay Time, on the direction probabilities of hand-in and hand-out events of mobile nodes among wireless cells and on the resource over-provisioning due to the uncertain mobility patterns. Our proposed scheme is based on the stochastic computation and characterization of user movements and it is absolutely general. Two novel metrics such as prediction error e and resource wastage u are defined. Moreover, predictive reservation schemes, admission prediction and resource wastage probabilistically associated to the stochastic hand-in/hand-out movement directions are proposed, with an optimization problem for the optimal pre-reservation of bandwidth in wireless system. A threshold-based reservation scheme has been considered as a practical case and it has been evaluated in terms of reservation prediction errors and resource wastage, to prove the practical effectiveness and consistence of the defined theoretical bounds.


INTRODUCTION
I N recent years, the demand for wireless devices and applications has grown significantly, because of the manifold advantages offered by telecommunication systems. User mobility is the main reason for received Quality of Service (QoS) fluctuations because it has a notable impact on QoS parameters (like packet-delay, delay-jitter and packet-loss rate). When a mobile host changes its coverage cell (i.e., hand-off) with an active flow, the available bandwidth in the new base station (or access point) may be scarce and the congestion level may vary and the perceived service quality may fall below requested lower bounds. In the worst case, the connection may be dropped. Owing to these problems, deterministic service guarantees, used in wired networks, become unsuitable in a wireless scenario and a flexible service model, which allows variable QoS, is required [1], [2].
Mobile computing environments require flexible mechanisms for bandwidth reservations to face the adaptive nature of wireless networks [3], [4], [5], [6]. MRSVP and DRSVP are two well-known signaling protocols for resource reservation that can be applied in many wireless networks contexts. They allow more in advance cells the reservation with different bandwidth levels to be reserved mitigating the fading effects on the wireless channel or reducing the QoS parameters based on the traffic classes [3], [5].
However, the performance of these signaling protocols can change based both on the underlying mobility model and the adopted reservation policy. The mobility model has a heavy impact on the obtained results and they can be unrealistic if the model is not appropriate. In this contribution, we consider both synthetic models (such as the Smooth Random Mobility Model -SRMM [7]) and realistic ones (such as the Citymob for Roadmaps C4R [8], [9]), employed for a two-dimensional set of cell clusters. However, the proposed idea is for general application and is not dependent on the specific mobility model [7], [8] nor signaling protocol for resource reservation [3], [4], [5], [6]. It is, instead, based on the knowledge of two important statistics related to user movements: the Cell Stay Time (CST) distribution and the Hand-off Directions Probabilities values (HDP). After the definition of a Stochastic matrix for the movement characterization, two metrics, such as prediction error and resource wastage probabilities, are defined for the evaluation of the reservation scheme. It is one of the first contributions, to the best of our knowledge, that tries to consider both resource reservation error (due to mobility patterns) and resource wastage (related to the reservation policies) from a stochastic perspective. The contributions of the proposal are: 1) Definition of a local objective function given by the weighted sum of the novel defined metrics (resource reservation error and resource wastage) to trade-off the minimization or resource reservation error of the mobility patterns or minimize the resource wastage of the over-provisioning; 2) Formulation of an optimization problem to find the optimal threshold values associated to our considered multi-step reservation policy on the basis of the main parameters of mobility patters such as average and variance of direction probabilities; 3) Proposal of a multi-step prediction scheme combined with the CST evaluation to maintain QoS requirements during users' movements. The advanced reservation scheme can trade-off between resource utilization and error compensation through resource over-provisioning. 4) Theoretical bounds are verified to prove the practical effectiveness of the proposed math formulation. This paper is organized as follows: Section 2 gives an overview of the related work concerning the reservation techniques and studies proposed in the scientific literature in the wireless environment; Section 3 describes the problem statement, the definition of novel metrics (error prediction and resource wastage) for the reservation schemes evaluation, the characterization of the stochastic matrix of the handin and hand-out movement directions probabilities (HDP) and the optimization problem formulation. Multi-step prediction schemes and the threshold-based reservation are presented in Section 4; performance evaluation between single-step and multi-step reservation is presented in Section 5 and conclusions are summarized in the last section.

RELATED WORK
Many contributions have been proposed over the last decade on mobility prediction and resource reservation in wireless systems to improve system utilization, to reduce the signaling protocol overheads or reduce prediction errors. Some of these approaches used utility functions to better manage incoming calls and traffic under dynamic prediction and reservation schemes such as those proposed in [10], [11], [12], [13], [14]. Other approaches, instead, considered statistical methods to model user movements and to make prediction for in-advance reservation. Numerous contributions considered a "one-step" reservation (local reservation) without facing the issue related to "multi-step" reservation (more than one cell reserved in advance, under multiple handovers) [15].

User Mobility Profiling and Resource Pattern Prediction
In [16], [17], a hierarchical user mobility model based on an appropriate pattern matching and Kalman filtering is presented. In [18] two innovative and powerful resource prediction schemes are proposed to evaluate the required amount of bandwidth during hand-off events. In [19] it is proposed a resource reservation protocol and a call admission control based on the location estimation of the mobile user. However, these approaches are based on local reservation on the adjacent cells without accounting for QoS guarantees under different user mobility speeds.
In [20], [22] the authors propose a Bayesian approach with the use of Kalman filter (KM) or Extended Kalman Filter (EKM) to predict the position of a mobile node under the estimation of the posterior distribution of the node location after movement and observations. In [21], the authors use neural networks to predict node movement based on the aggregate behavior of users in the cell.
A spatial distribution of mobile users to predict the distribution of mobile users in cellular networks is presented in [23]. In [24] a Bayesian location prediction scheme with multiple restricting factors such as environment factors, movement factors and random factors is proposed. In [25] a cross-layer handover scheme based on movement prediction for mobile WiMAX networks is proposed. In [26] a pioneeristic work where the statistical behavior of mobile users is applied in the reservation process is proposed. The authors propose an integrated framework able to use the User Mobility Profile (UMP) to define the zone concept (zone where the user can move). This interesting contribution focuses on error prediction and overhead. However, the prediction resource wastage and a possible trade-off that can be obtained during a reservation process is not exploited. A user mobility patterns analysis in a macro-cellular wireless networks context is proposed in [27]. Also in [28], CST distribution and reservation policies are used in a WLAN cluster. In [29] a mobility profiler based on data mining techniques is proposed. The authors try to derive mobility user profiles based on the mobility pattern frequency.

Markovian and Space-Time Mobility Prediction
A semi-Markov model to predict future paths in a wireless system is proposed in [30]. In [31], the authors propose a reservation scheme and a mobility prediction algorithm based on the following key components: (1) the estimation of Expected Travel Distance (ETD); (2) the calculation of Cell Visiting Probability (CVP) for a Mobile Unit (MU) to visit the neighboring cells; (3) the formation of the shadow cluster comprising the neighboring cells having higher probability for a MU to visit. In [32], authors innovatively utilizes both incoming and outgoing handoff predictions such as those we proposed in [33]. In [34] the authors propose a mobility modelling based on a Bi-variate Gaussian process in order to modulate the different user mobility profiles. In [35], the authors propose a semi-Markov model to consider the user movement in a WLAN cluster where transient and steady state behaviors of mobility trajectory are considered. The limitations presented in [35], related to a low precision in a short time scale, have been overcome by another contribution proposed in [36], where a fine-grained mobility prediction based on a Hidden Markov model is applied. Authors consider higher mobility points of users in order to better model the space-time correlation of user movements and trajectories such as proposed also in [37], [38]. In [39], the authors propose a 2D reservation technique applied to the vehicular mobility, using in-advance reservations and bandwidth multiplexing on the basis of the mobility profile. All these mentioned approaches gave important contributions to the mobility prediction, especially in the WLAN context under a stochastic analysis perspective. In [40], an accurate cellular link bandwidth prediction in LTE networks has been proposed. The authors consider some parameters that can affect the link bandwidth proposing a machine learning based prediction framework, Link Forecast, to predict link bandwidth in real time. In this last case, prediction has been performed on the link bandwidth, but no correlation has been exploited considering mobility prediction and resource wastage in the allocation among more cells. The problem of location-dependent opportunistic bandwidth sharing between static and mobile downlink users in a cellular network is proposed in [41]. However, this last approach does not consider the error in the mobility prediction and the bandwidth wastage if the bandwidth cannot be shared with other users present in the predicted cells.

Work Motivations and Propositions
On this basis, we apply a promising stochastic approach with some novelty points. A stochastic approach is used to characterize some key design metrics such as reservation error and resource wastage probabilities in order to define some limits in the reservation policies, applied to some specific mobility patterns that can derive from deterministic, pseudo-random or completely random nature of user mobility. To the best of our knowledge, all previous works considered just the prediction error and no one modelled or related it with the resource wastage probability due to reservation policies, that can apply also resource over-provisioning to compensate the mobility error uncertainty related to variable mobility patterns (patterns with higher variance). The proposed approach has been modeled from a stochastic point of view. On the other hand, other approaches such as that presented in [42], consider the resource reservation and the analysis to determine the maximum consistent data rate that can be offered to a class of mobile users, both within a cell and within a region covered with multiple cells, given certain available resources. In the latter case, the mobility prediction of mobile users and the potential resource wastage is not considered but it is supposed that users can share unused reserved-bandwidth. They show that providing a consistent rate is rather expensive because a large percentage of the available resources remain unused most of the time. However, unused resources can be shared by the users in the group.
Moreover, the other cited proposals focus on integrated reservation techniques and mobility prediction models applied to a specific mobility model and network scenario. The work in [43], it is a recent paper where a data-driven approach based on deep neural networks is applied for mobility and traffic prediction. It is able to be independent by mobility model and traffic patterns. However, also this recent contribution did not consider the overprovisioning strategy to compensate possible errors and variance in the mobility patterns; this last one that can affect the in-advance reservation strategy. Moreover, it does not account about in-depth and in-advance reservation as a way to respect QoS constraints during the traveling. This contribution, instead, tries to focus mainly on the definition of two novel metrics such as prediction error and resource wastage, proposing a stochastic approach to find theoretical limits of the prediction error and system utilization, exploiting inadvance reservation policies on the basis of the reservation depth (number of handovers) and variance in the user's trajectories. The proposed approach is independent from the reservation policy. This means, that applying the same methodology, it is possible to associate a different reservation policy to the Stochastic Hand-off Direction Probability (HDP) matrix (defined later) to derive different theoretical bounds. As will be shown in the next paragraphs, we adopted a threshold-based prediction technique to inherit stochastic bounds related to the resource reservation policy. This contribution seeks to be the first attempt that makes use of stochastic modelling of the user behavior to obtain a closed formula of the prediction error probability and resource wastage prediction probability on the basis of the statistical distribution related to hand-in and hand-out directions of mobile users. The main contributions of this work are listed below: 1) Definition of two novel metrics such as resource prediction error and bandwidth wastage computed in a probabilistic way on the basis of the stochastic matrix associated to user behavior in the wireless system; 2) Definition and computation of an HDP Stochastic matrix, based on the main parameters of the probability density functions (p.d.f.) associated to each movement direction, such as average m and standard deviation s of user mobility pattern; 3) Proposal of a local objective function in the coverage area able to trade-off the opposite trend between error prediction and resource wastage; 4) Formulation of an optimization problem to find the optimal threshold values associated to our considered multi-step reservation policy; 5) Application of a Hand-off Direction based multi-step prediction scheme combined with the CST evaluation in order to respect QoS requirements during users' movements and to maintain a trade-off between resource utilization and error prediction. In the latter, theoretical bounds are verified in order to prove the practical effectiveness of the proposed math formulation.

STOCHASTIC APPROACH TO RESERVATION PREDICTION SCHEME (SARPS)
In this work, the considered system consists of a certain number of two-dimensional wireless clusters, as illustrated in Fig. 1. However, the proposed approach represents a methodology that can be applied to any context and it can be useful to any researcher or network service operator, who wants to evaluate the possibility to make in-advance reservation for mobile users, in order to improve the grade of service and maintain the QoS continuity without wasting bandwidth resources in future visited cells or committing high prediction errors. In our characterization of the problem, a mobility model is adopted (SRMM [7]), such as applied to [37], [38], but a Poisson call arrival time distribution with an exponentially distributed Call Holding Time (CHT) was considered (this hypothesis does not limit the proposed methodology that is based on statistical consideration). In order to obtain the predictive evaluation of the number of effective visited cells C e that the mobile host will cross during its call lifetime, the CST of mobile hosts was evaluated with a high number of monitor simulations (simulations dedicated only to the observation and monitoring of some interesting system parameters); their results showed that the CST distribution, under the SRMM with the chosen parameters, can be considered as Gaussian, for different values of v max [39]. Also in this case, if the CST distribution is different under other mobility models, this does not affect the methodology and just some other p.d.f. needs to be obtained for the CST. The assumptions for our model are the following ones: 1. Connections and calls are generated independently in cell clusters representing the network; 2. All cell coverage areas present the same area and hand-in and hand-out direction approximations; 3. There is a uniform distribution of mobile users in the simulation area; 4. The number of possible hand-off directions of mobile users per cell, which have to be taken into account for computing movement distributions, are fixed and equal for each cell. All these assumptions do not affect the generality of the methodology and mathematical modeling approach: they can be relaxed to extend the model such as presented in [38], where a coverage based on Voronoi tessellation is presented. An evaluation of Voronoi tessellation has been presented later also in our contribution.
The value of C e can be determined as in [39] but, unfortunately, without directional information about user mobility pattern, the predicted value of C e can only be used to make pre-reservations (the bandwidth is reserved in a cell before the QoS user arrived into) in a circular way, around the current cell (where the call has been admitted): following the same approach in [12], the value of C e can only be obtained a-priori, so the number of required passive reservations C r increases with polynomial trend, such as follows: For example, Fig. 1 shows how for C e ¼2, 3 or 4 a higher number of C r passive reservations must be made (Circular Reservation C.R. dotted lines), through a fixed circular cluster of C r ¼6, 18 or 36 cells with a radius of C e cells (including the active one).
This work introduces a novel algorithm, based on some additional information about user directional behavior, so the value of C r can be decreased (as shown in Fig. 1 with the Directional Reservation D.R. continuous line), making it nearer to C e , depending on the adopted reservation thresholds and policies.
In order to understand the proposed approach, it is necessary to consider the coverage area as a polygon, to collect statistics about user movements and to formulate a new mathematical modeling of the problem, for the efficient advanced n-steps reservation in a wireless system.

Coverage Area Under Regular and Heterogenous Shapes
A generic coverage area, generally with a circular shape, can be approximated with an n-edge regular polygon ( Fig. 1 shows the approximation for n¼6) and n can be considered as an input control parameter) or it can be represented more generally with a Voronoi tessellation, as shown in Fig. 2. It can be defined as the partition of a plane into regions (irregular convex polygons), each related to an internal seed (called also site or generator). For each seed there is a corresponding region consisting of all points of the plane closer to the seed than any other. In fact, when considering real coverage areas, we had to take into account that cellular shapes are affected by different factors, such as site availability, topography and traffic density. There are also several studies in literature which have shown that this problem can be well approached by Voronoi's theory [47], [48], [49]. In particular, in order to overcome the inaccuracy of the classical regular model, Voronoi diagrams are considered for tessellation and system optimization. For each cell c t 2C there will be an associated value n, indicated with n t , representing the number of sides (n cannot be defined a-priori, because it varies for each cell, depending on the particular shape). Let j j a-b j j denote the Euclidean distance between points a, b in a plane PL2R 2 and  A¼{a 1 , a 2 , ..., a s } a set of s points (seeds). Then, the Voronoi diagram is defined as a subdivision of PL into s corresponding polygonal regions PL v , v¼1,2,...,s, such that for each point b2PL, it is verified that: ka i À bk < ka j À bk, i,j¼1,2,. . .,s and i 6 ¼ j. This yields a plane tessellation, based on convex polygons around the points in the defining point set A, where any point within a polygon PL j is closer to the corresponding point a j than to any other points in A. Assuming that radio signal strength drops in relation to distance, A can represent site locations, considering the Voronoi regions as network cells. In our case, the analysis and modeling methodology can be applied in both hexagonal or Voronoi coverage areas.

Stochastic Matrix of the Direction Probabilities
A set S ho (hand-off directions set) of n possible movement directions (i.e., hand-off directions) can be then obtained: let us indicate movement directions with d 1 ; d 2 ; . . . ; d n , where d j ¼uÁ(2Áj-1)/2 rad., u¼2p/n rad. and j¼1..n, so S ho ¼ fd 1 ; d 2 ; . . . ; d n g and jS ho j ¼ n. Let us use, from now and in the following, the terms d in and d out to indicate hand-in and hand-out directions respectively, for a generic coverage cell ID p . The conditional probability that a mobile host will be handed-out to direction d out 2 S ho after CST (a normally distributed value) amount of time, if it was handed into current wireless cell from direction d in 2 S ho can be defined as p d in ;d out : p d in ;d out ¼ Probability to hand-out from a cell towards d out given the arrival from direction d in with d in ; d out 2 S ho .
Let C be a set containing all the cells of the considered system, C ¼ fID 1 ; ID 2 ; . . . ; ID W g with jCj ¼ W representing the number of cells in the system. Once n and S ho have been chosen (n ¼ jS ho j), a square nxn HDP matrix M can be defined with the following elements: Mðd in ; d out Þ ¼ p d in ;d out . In particular, M is a Stochastic Matrix where the single elements are represented by a p.d.f. to select a specific direction and they are characterized by the main parameters such as the average m and the standard deviation s through which it is possible to consider the related p.d.f.. Matrix M depends only on the adopted mobility model and network coverage cells subdivision. It is possible to consider different matrix for each cell because it is able to resume local stochastic mobility features. It has the hand-in directions on the rows and the hand-out ones on the columns and it can be filled out through a first addicted campaign of monitor simulations, while acquiring the CST distribution (as previously explained). As for the CST analysis, the Kolmogorov-Smirnov (KS) normality test was carried out on the n 2 elements of M: from different simulation runs, it resulted that p d in ;d out values are also distributed with a Gaussian trend, so each element of the n x n matrix M can be represented by synthetic values of the p.d.f. such as mean and standard deviation m d in ;d out and s d in ;d out respectively; in this sense, Mðd in ; d out Þ is a couple of values, the mean and the standard deviation of the obtained distribution, useful to extract the realization value of the probability p d in; d out , accounting for the specific p.d.f. obtained through monitor simulations. An example of M is shown in Fig. 3.
The KS test is based on the p-value concept [44]: remembering that a p-value for a statistical test is a measure of how much evidence there is against the null hypothesis; different p-values were obtained, showing the goodness of the Gaussian distribution hypothesis (for details about goodness-of-fit techniques to see [44].

Prediction Error and Resource Wastage
Probabilities Definition The number of predicted cells for the j-th hand-off event are chosen dynamically, with the minimization of prediction error as the main purpose. However, as it will be shown in the following, the prediction error minimization does not make sense if there is not a trade-off with resource reservation prediction error and bandwidth wastage. Also in this case C e is evaluated with the approach used in [10]. Let h ¼ C e À 1 indicate the number of hand-off events for a generic user. Let us assume, for now, that the elements of M are characterized by the main parameters of a generic distribution such as average and standard deviation. This approach allows us to make a general mathematical formulation that can also be adopted in other contexts and that can be specified on the basis of the specific distribution associated to the network contexts. The main aim of this manuscript is the formulation of an optimization problem for the prediction of future cells for each k-th hand-over, with k¼1,..,h. At this point, respecting the previous notation, let us define the value d in ðkÞ 2 S ho which represents the hand-in directions for a generic predicted cell ID p at the k-th hand-over (previous notation is used, but apices are added to discriminate the prediction step). Let us introduce the variable p ID i , i¼1,...,W as the probability that a cell ID i will be actually visited by the considered user and the term PrðID i Þ representing the general probability of considering the cell ID i as candidate in the predicted set, where resources will be reserved. For the sake of simplicity, let us refer to Fig. 4 where for a mobile user on the basis of the hand-in direction d in , next cells have to be selected reserving bandwidth in advance on these cells. It is assumed that the possible next cells are ID 1 , ID 2 , and ID 3 . Later, these hypotheses will be removed and the model will be generalized. The probability to make an error in the k-th reservation is given by the probability of not reserving in the ID i cell when a user moves there with p ID i ; it is possible to calculate the probability of not reserving in the rights cells in the following way: let us assume that mobile user can go on ID 1 or ID 2 or ID 3 .
It is to be noticed that term PrðID 1 Þ is also functions of the reservation depth k-th but we avoided introducing this explicit reference with the aim to simplify the notation.
Let Prð:ID i Þ be the probability of not reserving on the cell ID i . It is defined as: In this problem statement, PrðID i Þ depends on the adopted resource reservation policy. For example, if it is decided to reserve on the cell ID i on the basis of a specific policy consisting, for example, in the evaluation of the average value m d in ;d out of the distribution associated to hand-in direction d in and hand-out direction d out , it is possible to assign a channel (bandwidth) probabilistically for mobile user j that will go on ID 1 only if a random value p uniformly distributed in the range [0,1] is greater than m d in ;d out . Thus, many reservation policies can be defined based on reservation policy based on threshold value d in order to find a good trade-off between error and bandwidth wastage committed in the advanced reservation. Our criteria is absolutely general and it can be applied in many real contexts where specific mobility models or reservation policies are adopted.
The selected reservation policy can affect the resources associated to mobile users on the probably visited cells and, also, the resource reservation prediction error and bandwidth wastage. In order to understand the opposite trend associated to the error in the resource prediction and the wastage associated to the extra resource allocated in a wireless system, it is possible to observe Fig. 3 and to note how a reservation policy that involves all possible cells where mobile user will move is able to erase the effect of mobility prediction error on the reservation policy. However, the same reservation policy can also waste a significant resources because the bandwidth is allocated on all the possible local or far cells and it is not available for other potential users.
For this purpose, let us define two terms that allow to account for the contribution given to the prediction error and to the bandwidth wastage. We define two terms as the probability to make an error in the reservation e and the probability to reserve extra-bandwidth u in the cells where the user will not move. Obviously, these terms depend on the reservation policy P adopted and by the mobility model that can affect the stochastic matrix M. Considering Fig. 4a, let e(d) indicate the error committed in the resource reservation of a policy P. It is calculated as follows: This means that an error occurs if the mobile user goes in the cell ID 1 but no resource is reserved there or mobile user enters in the cell ID 2 but no bandwidth is reserved on that cell and so on. If all cells ðID 1 ; ID 2 ; ID 3 Þ are selected in the reservation phase, Prð:ID 1 Þ ¼ Prð:ID 2 Þ ¼ Prð:ID 3 Þ¼0 and e(d)¼0. On the other hand, if no cell is selected in the reservation policy, Prð:ID 1 Þ ¼ Prð:ID 2 Þ ¼ Prð:ID 3 Þ¼1 and e(d)¼1 (maximum error condition) as we expected. Thus, e(d) is a function depending on policy P and on a reservation threshold d with 0 eðdÞ 1. Let uðdÞ indicate the bandwidth wastage associated to the policy P. Considering  Fig. 4a, it is calculated as: This means that a bandwidth wastage occurs if a reservation is made on cells ID 2 and ID 3 when a mobile user moves in ID 1 or a wastage occurs if user goes in ID 2 and a reservation is made also on ID 1 and ID 3 and so on. Also in this case, it is possible to verify that if all cells are considered by the reservation policy P, This last condition leads to a maximum bandwidth wastage ( uðdÞ ¼ 1) and a minimum error condition (e(d)¼0). On the other hand, if PrðID 1 Þ ¼ PrðID 2 Þ ¼ PrðID 3 Þ ¼ 0 , the bandwidth wastage uðdÞ ¼ 0 but the reservation error is maximum (e(d)¼1). This means that e(d) and uðdÞ present two opposite trends such as shown in Fig. 4b and this suggests defining a novel objective function able to find the best trade-off between them.

Optimization Problem in Predictive Resource Reservation Strategies
We can define a local function as follows: With 0 eðdÞ 1,0 uðdÞ 1 and 0 p 1; p 2 1; p 1 and p 2 are weights defined in the local function to give more importance to the resource error prediction or to the resource wastage. They are defined so that p 1 þ p 2 ¼ 1.
Through the introduction of these weights the system operator or Telco companies can optimize the resource management in the system on the basis of the applications and user requests in the specific location areas. fðdÞ depends on the reservation step (k-th step) because the hand-in directions d in establish the hand-out probabilities (the hand-out p.d.f.) on the stochastic matrix M such as shown in Fig. 3. For this reason, we express the function f(d) in this way: On the basis of Eqs. (3) and (4), eðd; k; d in ðkÞÞ and uðd; k; d in ðkÞÞ can be expressed as follows: Eqs. (7) and (8) are also derived in Appendix I, which can be found on the Computer Society Digital Library at http:// doi.ieeecomputersociety.org/10.1109/TMC.2022.3176046. Because the objective function f(d) depends on the k-th reservation step, it can be expressed in a general way as below: where h is the number of hand-over events and n is the number of neighbor cells. This number can be estimated as proposed in [10] on the basis of the Cell Stay Time (CST) distribution. Eq. (9) is derived in Appendix II, available in the online supplemental material. At this point it is possible to extend the local objective function to all prediction steps. This approach can be useful to find the best trade-off between e(d) and uðdÞ if an inadvance reservation policy at h-steps is adopted. Thus, the total objective function is defined as: On the basis of Eq. (10), it is possible to formulate the following optimization problem that tries to find the best solution in terms of trade-off between prediction error and resource wastage, for each prediction step k, by choosing the appropriate d Ã (the optimal d).
Min s:t: f tot ðdÞ and . . . ; d Ã n is the optimal solution. In particular, the optimization problem presented in (11) needs to be solved. The proposed modelling strategy can be useful for both people and researchers working in the resource management area, applied to cellular coverage under 3G, 4G and 5G technologies. An example of optimal d Ã values found for one cluster is shown in Table 3.
The approach is practical because, after the collection of historical data about mobile user movements, it is possible to build a matrix in which mobility stochastic information is present and it is stored in the local base station (or access point) memory, which is covering a particular geographical area. Furthermore, they can be applied to establish how many cells (bandwidth) to reserve and with which depth in the reservation., 4G, 5G or next generation networks are applied [52].
This contribution could be applied, in conjunction with the work in [39] to define more advance reservation policies and getting advantage by bandwidth multiplexing and rate adaptation techniques. However, this integration can be considered as future work.

THRESHOLD-BASED DYNAMIC RESERVATION
Through a threshold-based comparison the algorithm must decide which the cells are those that a QoS user will visit with higher probability when handing-out the current considered cell, with a well-known hand-in direction d in ðkÞ 2 S ho , which specifies a unique row of M. The algorithm evaluates the probabilities of the n directions, considering all the possible hand-out direction d 2 S ho with the following inequality: If it is satisfied for any d, then the cell that is adjacent to the current one on direction d, must be considered as a possible future cell for the k-th step. Mðd i ; d j Þ consists of a couple of values that represent the main parameters associated to the direction p.d.f., that is to say Mðd i ; d j Þ ¼ Nðm d i ;d j ; s d i ;d j Þ then a statistical treatment is mandatory, in order to extrapolate a probability p d in ;d out characterized by the specific distribution.
When a constant value x is substituted by a stochastic variable X (with a Gaussian distribution in our case), a prediction error is implicitly introduced, especially if the standard deviation of X is not negligible. The main aim of the dynamic reservation scheme is the minimization of the prediction error without wasting a lot of resource through the appropriate definition of d. So the prediction error has to be defined according to general formulation in Eq. (10) and applied to the specific context. A realization of Mðd in ðkÞ; d out ðkÞÞ indicated with m d in ;d out (where d in ðkÞ is the known hand-in direction and d 2 S ho ) is obtained by the inversion of the p.d.f., such as explained in [41] and recalled in Appendix III, available in the online supplemental material. Once the realization m d in ðkÞ;d out ðkÞ is obtained, during the execution of the dynamic predictor algorithm, it can be used in Eq. (12) and it can be re-written as: Let us indicate The statistical prediction error at the k-th prediction step (i.e., a cell is not considered as probably-visited for the k-th hand-over) for a single cell prediction on direction d out ðkÞ with a known hand-in direction d in ðkÞ occurs if the realization of Mðd in ðkÞ; d out ðkÞÞ does not satisfy in Eq. (13), so the error probability p e at step k-th can be defined as Where Q(z) is the Q-function [44]. So, Eq. (17) represents the prediction error committed at the k-th step, with a known hand-in direction d in ðkÞ for a cell on the hand-out direction d out ðkÞ: Supposing that ID p is the cell which is adjacent to the current one on direction d out ðkÞ, then: At this point, the expression of eðd; k; d in ðkÞÞ and uðd; k; d in ðkÞÞ become: In Fig. 5 the dynamic predictor with a fixed reservation policy P is presented. The mathematical formulation of the problem is absolutely general and can be applied to any reservation scheme. This means that a specific reservation policy is applied in order to see the effects of a reservation scheme with the statistical parameters of the mobile user's movement and to verify that theoretical bounds of the reservation error and wastage are respected. The Data Flow Diagram in Fig. 5 shows the general behavior of a predictor, when a mobile host asks for a service in the considered network. First, the number of hand-over events for user i is determined (h i ), the next cell is predicted and vector vh i is initialized. Then, if more cells need to be predicted (k h i ), the probabilities of going out from the last predicted cell to other cells need to be evaluated. The index l is related to the number of cells to be predicted for the current hand-over event (index k), then the probability of going to an adjacent cell is evaluated as M(p in ,p) Ã p curr . If this value is lower than the considered threshold d, then the adjacent cell on direction p is considered as predicted, otherwise the test is made on the next adjacent cell on direction pþ1 (if p<n). This is just a general threshold-based predictor example, as a possible solution to evaluate a set of possibly visited cells.

PERFORMANCE EVALUATION
Numerous simulations were carried out to evaluate the performance of the proposed idea in terms of average prediction error, number of involved cells and system utilization.

Simulation Scenario
Our network consists of 7 clusters of 7 cells, the coverage radius is about 250 meters and users move toroidally (no physical topology borders are considered). Regarding mobility generation, C4R helped us to extract mobility patterns from real roadmaps, while the SRMM with the same mobility parameters used in [7] has been deployed for the generation of synthetic patterns (it makes users movements smoother and more realistic than other random models, because it relates speed and direction changes). C4R is a Java mobility pattern generator for vehicular networks, allowing simulation of vehicular traffic in different locations using real maps, extracted from OpenStreetMap [45]. On the other hand, the main concepts of the SRMM are two stochastic processes for direction ' and speed v: their values are correlated to the previous ones, in order to avoid unrealistic patterns and speed/direction changes.
Speed and direction changes follow two different Poisson processes and different typical patterns or environments can be modelled by setting some parameters, like the set of preferred speeds. This assumption, according with the one made by the authors in [7], allows to model the speed and the probability of a direction change obtaining more realistic movements.
This model is also based on a set of preferred speeds in the range ½v min ; v max and a mobile host moves with constant speed until a new target speed v Ã is chosen by the stochastic process, so it accelerates/decelerates in order to reach v Ã . The set of preferred speeds fv pref1 ; v pref2 ; . . . ; v prefn g is also defined in order to obtain a non-uniform speed distribution. More details on SRMM model can be found in [7]. The simulation parameters are presented in Table 1.
Regarding C4R, we used many urban maps of some European cities (about 10 Km 2 for each scenario), over which the set of coverage cells was considered (shown results are the average values evaluated on different maps). Dark and light lines refer to SRMM and C4R respectively. Moreover, simulations have been performed on a Java multithread simulator, where the main classes (mobile host, base station, etc.) have been carefully modelled.
The same simulator, with different statistics, has been used for obtaining the results of our previous works [38], [39]. Normal distribution of the CST with average value mCST and variance sCST M Stochastic matrix of hand-off direction probabilities M(x,y) matrix M element represented by two values (average mxy and standard deviation sxy h i number of hand-off events associated to matrix element M(x,y) p x,y value obtained by normal distribution associated to matrix element M(x,y) d in (k), d out (k) hand-in and hand-out direction at k-th hand-over c k current cell at k-th step (hand-over) P(k) probability to be in c k after (k-1)-th hand-over vector of thresholds associated to each direction function of three variables to compact formulas expressed in Eqs. (27) and (32).
function of three variables to compact formulas expressed in Eqs. (27) and (31).
dynamic thresholds applied for the prediction RD reservation depth. It presented the number of cells that is possible to reserve in advance in successive handover (hand-out) during the user movement. p IDi probability to move on the cell ID i Pr(ID i ) Probability to reserve on the cell ID i at a reservation dept k-th Q(z) Q-function [43] We provided to generate mobility traces by C4R and by the Java implementation of the Smooth Random Mobility Model (SRMM). An example of SRMM parameters used in our simulations are listed in Table 2. The obtained files have been, then, parsed in Java, in order to obtain the same format (tabular) with the following fields: [Mobile_ID, time, ID_x, ID_y, ID_z, ID_Vx, ID_Vy, ID_Vz]. Then the Voronoi tessellation (with the related number of generation points) and/or the regular hexagonal coverage have been implemented in Java, by considering the area extension of the different considered maps and creating the square area containing the coverage cells (with hexagonal/Voronoi shape). For each mobile user with the Mobile_ID identification, coordinates are chosen row by row from the parsed mobility file, having the possibility to identify the current covering cell and to store the sequence of the cells visited by each user (knowing the hand-in and hand-off sides of the coverage cell it is possible to build the matrix M, as illustrated in Fig. 3).

Other Prediction Techniques for Comparison
After an in-depth research in literature, it is possible to state that a very few works deal with a multi-step passive reservation approach; in order to compare our example of resource reservation with other strategies, we considered two of the algorithms summarised in [15], that are also respectively detailed in [26] and [46]. The first algorithm is able to perform a multi-step reservation (and to evaluate a set of probable cells for the next handovers), while the second one is designed to derive the most probable cell only for the next handover. In particular, the work in [26], called User Mobility Profile (UMP), does not evaluate the cells that will be traversed by the user but, given its mobility status (position, speed and direction), it derives the probabilities of each cell that the user can cross in a future moment. Several data structures are implemented and used for the prediction (most of them are matrices) and the main idea is to assign high location probability values to the cells taken in consideration for the prediction that has been previously traversed by users. For more details, please refer to [26]. The second algorithm [46], called Active Lempel-Ziv (ALeZi), is able to assign a particular symbol to each cell (like a unique cell identifier), obtaining a set of possible identifiers. The rationale behind ALeZi is the Prediction by Partial Matching (PPM), that observes the previous j-1 symbols, in order to predict the j-th one. More details can be found in [46].

Simulation Results
As illustrated in Fig. 6, the trend of the obtained prediction error (belonging to the interval [0,1]) can be observed. The horizontal axis represents the increasing threshold values. Dashed lines represent the trend of the theoretical error (evaluated as in Eq. 22), while continuous ones indicate the empirical values obtained by simulations. The curves are parameterized on the Reservation Depth (RD) of the predictions, that is to say RD¼2 means that we are considering the prediction error until the 2 nd cell (for RD¼3 we consider the prediction error until the 3 rd cell and so on). It can be seen how for lower threshold values (below 0.1) the error is completely negligible (under 10%): this is due to the high number of selected cells for the prediction (the table embedded in the figure illustrates the average  The curves obtained for C4R present a lower error: mobile hosts move following deterministic paths (on the roads existing in the considered map), so the obtained average variance in the evaluated matrix is lower than the one of the SRMM. Fig. 7 illustrates the trend of the average wastage (as introduced in Eq. (23)) belonging to the interval [0,1].
As for the previous figure, the horizontal axis represents the increasing threshold values and the meaning of RD, dashed lines and continuous lines is the same. As for the previous case, the number of predicted cells for each handover event belongs to the interval [2], [3] for lower threshold values (in the interval [0,0.08]): this leads the system to suffer an enormous wastage, because of the higher number of passive reservations.
For increasing values of RD, the wastage also increases, because more future cells are considered in the prediction phase (for the second and the third hand-over, if RD¼3). When the threshold becomes too high, the number of predicted cells decreases (as illustrated in the embedded table). The wastage decreases: it does not reach 0, because of the higher error percentage (as illustrated in Fig. 7), which leads to fail in the prediction, also for a single cell for each handover (only one cell is considered for a single prediction). In this case, the theoretical values represent an upper bound for the empirical wastage values.
From Figs. 6 and 7 it can be observed that prediction error and prediction wastage have opposite trends, as stated in Section 3, sub-paragraph B.1: this is always verified, independently on the considered parameters (coverage radius, reservation depth, etc.). We cannot show all the obtained results due to space limitation issues.
Figs. 8 and 9 depict the same variables of Figs. 6 and 7, but now the values of RD¼4 and RD¼5 are considered.     Also in this case, theoretical values represent lower and upper bounds for the obtained empirical values of prediction error and wastage respectively and the trend is still opposite (increasing for the error and decreasing for the wastage). The number of predicted cells for each hand-over does not change in a sensible way (as it can be seen from the embedded tables), but the prediction error and wastage are slightly higher than the previous ones, since we are considering a higher depth with RD¼4 and RD¼5.

Weights Evaluation on the Local Function f(d)
Average resource reservation error and resource utilization are respectively shown in Figs. 10 and 11. The reservation error increases for lower p 1 value. This is due to the lower weight given to the error e in the local function f(d), that allows the minimization of the overall average resource wastage improving the system utilization.
On the other hand, increasing the p 1 value determines reservation policy that tries to compensate the resource reservation error increasing the number of cells where to reserve bandwidth. This led to a higher resource wastage with lower system utilization and with a lower reservation error e such as it is possible to see in Fig. 11.
The best trade-off for our simulation, where a cell coverage of 200 m has been considered, is represented by a p 1 value of 0.5 because the system achieves an utilization around 80% with an error lower than 5%. However, the optimal p 1 value depends by the importance given to the metric to be optimized (resources or error prediction) and it can be established by the network operator. The study on the optimal p 1 selection under different constraints can be object of future works. Fig. 12 shows the trend of the average prediction error for different threshold values and different cell coverage radius (r 1 , r 2 and r 3 ), considering SRMM. The figure provides an opportunity to observe an interesting result, namely the error in the prediction phase increases for larger coverage areas. If the threshold is maintained below 0.1 the error can be considered acceptable, otherwise it increases till unacceptable values and theoretical values represent always a lower bound for the empirical ones. When a cell covers a larger geographical area, more user mobility behaviors have to be considered (longer roads, more crossroads, higher chance to observe different mobility decisions), so determinism in the prediction decreases: the variance in the distributions increases, as illustrated in Section 4 and the prediction error becomes higher. For space limitations the trend of the average wastage is not illustrated, but no relevant differences were observed for different coverage radius.

Reservation Thresholds and Cell Coverage Effect
In Fig. 13 the average variance ratio (evaluated as the variance value divided by the mean value) of the Cell Stay Time (CST) and a generic element of matrix M (in the figure the element M(1,1) has been considered).
The main purpose of this figure is the representation of the way the variance ratio has to increase for different cell coverage radius (the variance ratio of CST has been divided by 100 in order to respect the vertical axis proportions): we are not interested in the exact values but we want to underline that the variance ratio has an increasing trend for larger coverage; the relative variance value becomes more predominant if compared with the associated mean value. This verifies the descriptions given for the previous figure.

Comparison With Other Strategies
In Fig. 14, system utilization and the comparison among the three considered schemes are shown. We considered the system utilization for a single cell as the ratio among the number of used channels and the number of available channels (channels remain unused when there are no service requests or there are passive reservations, that is, in the second case, channels remain unused until the mobile host who made the passive request arrives). The results  illustrated here are averaged on the number of coverage cells. The one-step prediction scheme based on Lempel-Ziv compression (Active Le-Zi) outperforms the other two predictive schemes just because it reserves resources only on one next-cell, leaving the other hand-overs completely unmanaged (it is not guaranteed that a mobile host will find available channels in the future handovers).
The numbers 1, 2 and 3 for UMP represent the number of considered cells for each "next-handover" event. In addition, in the case of the ALeZi, it is not possible to select the number of cells over which the passive reservations can be made. Higher utilization leads to the lower bandwidth wastage, so it is clear that ALeZi performance is better than the one of the other schemes.
We considered the MRSVP as the reservation protocol [3], so also its overhead is considered in the curve trends. It can be seen that the system utilization for UMP goes decreasing for a higher number of reserved cells (more passive reservations), while SARPS maintains it almost constant, given its feature of dynamically selecting the proper number of future cells (over which a passive reservation is made) for each hand-over.
Our proposal offers much better results, especially for a larger coverage radius and the UMP predictive scheme is outperformed: in particular, when 2 or 3 cells are considered for the next hand-over event, for a coverage radius larger than 100m, the performance gap is higher than 35%. Fig. 15 illustrates the comparison among the schemes in terms of prediction error on the first hand-over. It can be seen that the worse results are obtained by ALeZi, especially for larger radius, because of the higher randomness of mobile nodes. In addition, no more than one cell is predicted each time, and for the consecutive hand-overs it cannot guarantee the availability of free channels. In fact, if the predictor fails to choose the right cell, there will be a lower chance to find available bandwidth (for higher coverages, more requests will be made inside a cell). As regards UMP, the prediction error for the first hand-over is lower than the one of ALeZi, and it decreases, for a given coverage area, when more than one cell is predicted.
The error increases for larger radiuses, given the higher variance in host movements. SARPS performs better than the other two schemes, given its capability of dynamically selecting the right number of future cells for each handover. This means that SARPS is more robust against movements or mobility patterns that present higher variance and where the only frequency of the visited cells or average frequency cannot guarantee a stable performance in the direction prediction under different coverage areas.
In Fig. 16 (CBP), the probability of obtaining a negative answer from the CAC module is shown (probability of a call to be blocked at the time of the request). In this case, ALeZi outperforms both SARPS and UMP: no passive reservations are requested, so the acceptance of a call depends only on the resource availability of the current cell (where the service has been requested). This is the reason for which the CBP is optimal for ALeZi. As regards UMP and SARPS, it can be seen that, also in this case, SARPS outperforms UMP (especially for UMP-2 and UMP-3): although SARPS has a "worse" performance than UMP-1, it should be reminded that SARPS is dynamic and the number of passive reservations is not fixed a-priori, but they are chosen dynamically on the basis of the coverage distribution.
In Fig. 17 (CDP), we can see the comparison of the ALeZi, UMP and SARPS performances in terms of Call Dropping Probability (the probability that a call is forced to be terminated during a hand-over event). It can be seen as ALeZi cannot guarantee a proper level of service-continuity (handover procedures are carried-on without service disruptions), due to the absence of passive reservations; the probability of success after a hand-over is related to the probability of having a good resource availability in the new cell. On the contrary, UMP and SARPS present acceptable values of CDP (the maximum for UMP is around 0.128, while the maximum for SARPS is 0.091, so below 0.1).

COMPLEXITY ANALYSIS
As regards the complexity of building a planar Voronoi tessellation, we can refer to [50] and [51], in which the  probability distribution of the number of sides of the Voronoi polygon is deeply analyzed, as well as the theory of the torus in which the Voronoi tessellation is built. In particular, we refer to a given regular planar graph G¼(V,E), where V is the vertex-set, E is the edge-set, and F denotes the set of faces of G. If we count the number of all pairs (v,e) belonging to the Cartesian product VÂE for which v is an endpoint of e (undirected edge), we have that the relation is always satisfied. In addition, 8 f 2 F, we use the notation e f to indicate the edges of f. If we repeat the operation of counting the number of pairs (e,f) in the Cartesian product EÂF for which e is an edge of f, we have: At this point, given that Euler characteristic of the torus is 0 [50], we have j V j À j E j þ j F j ¼0. So, considering this relation and the previous two equations, we can write that: X f2F e f ¼ 6 F j j Therefore, if we want to know which is the average number of sides of a random face chosen uniformly inside F, we have to divide by j F j , obtaining exactly 6. The result is also confirmed in [50], where an extensive set of simulations were carried out, showing that the average number of sides of a Voronoi cell follows a Gaussian distribution, with a mean value of 6, with a probability of having a pentagon or a heptagon of [0.2, 0.25].
On the basis of the considerations above, it is possible to evaluate the complexity of our in advance resource reservation and mobility prediction scheme based on stochastic matrix. In each cell base station needs to store 72 elements related to the matrix where 36 represent the average values and the other 36 the variance of the direction probabilities distribution. This means that the selection of next cells where the user will go need to scan maximum 6 elements per matrix if we know exactly the hand-in direction. Furthermore, in the intermediate steps where more cells can be involved the complexity will be limited by the reservation depth h. Thus, the overall asymptotic complexity of the proposed scheme in the worst case will be O(1)xO(h 2 ) where h is the number of cell C r in a circular reservation. In the best case, where a lower variance is present and only single cells are considered in the advance reservation, the complexity will be O(1)xO(h).

CONCLUSION
In this paper a novel mathematical characterization of user movements through a stochastic hand-in/hand-out direction probabilities (HDP) has been proposed. The p.d.f. associated to user movement allow us to define some network parameters such as error prediction and resource wastage useful to design good reservation schemes. The proposed approach is absolutely general and applicable to many contexts; an example of an optimization problem associated to the prereservation under stochastic matrixes is also proposed. This study aims to be a starting point for researchers working in the wireless systems field to propose efficient resource reservation techniques based on probabilistic user movements behaviour and to associate the network parameters to the probabilistic characterization. A dynamic threshold-based prereservation scheme has been applied to the advanced reservation in order to test the theoretical bounds of error prediction and resource wastage defined on the basis of the stochastic user movement characterization.