TY - JOUR
AU - Sanchez-Perez, Juan, M
AB - Abstract Mobility management is one of the most important operations in any public land mobile network. For idle terminals, most of mobility management strategies consist of two main procedures: subscriber’s location update and paging. These two procedures define a multiobjective optimization problem in which the challenge is to find the network configurations that minimize the number of location updates and the number of paging messages. In this optimization process, the use of accurate information is very important to find the network configurations that best match the subscribers’ activity in a mobile network. A mobile activity trace is a very accurate way for representing the subscribers’ activity in a mobile network in form of a chronological list of events related to the subscribers’ movements and the incoming and outgoing calls. This kind of representation can be used to study the performance of any mobility management strategy at the expense of considerably increasing the time needed to evaluate the objective functions. In fact, we show that more than 99% of the optimizer runtime is used to evaluate the objective functions. With the aim of facing this problem, we propose a fine-grained parallel approach focused on parallelizing the objective functions exclusively. Results show that the efficiency of our approach is always higher than 50% and that the optimizer runtime is drastically reduced. 1 Introduction Mobile telecommunication networks are widely extended throughout the world and are one of the most influential technologies in today’s society. In fact, as reported by the GSM (Global System for Mobile Communication) Association, approximately the half of the world population will use mobile communications in 2017 [19]. This market penetration is even higher in developed countries wherein near 80% of the population make use of mobile phones in its daily life. This huge service demand is faced by arranging the coverage area in smaller zones among which the scarce radio-electric resources are distributed and reused. These smaller zones are commonly referred as cells and can be of different sizes to cope with different traffic demands, being their size smaller with the higher traffic densities. Due to the fact that a subscriber can be in any cell and at any time, a direct consequence of a cell-based access network is that the network must now control the subscribers’ movement across the different cells to redirect the incoming calls to the mobile stations. It is important to differentiate between mobility management for active and idle mobile stations. Mobility management of active mobile stations (i.e. mobile stations that has got an active data traffic session) are mainly focused on minimizing the signalling overload due to handover processes. These management strategies have been widely analysed in prior art papers [4, 17, 25, 33], and are out of the scope of this article. On the other hand, mobility management of idle mobile stations consist in partially tracking the subscribers’ movement along the coverage area with the aim of minimizing their location uncertainty. With regard to strategies to manage the mobility of idle mobile stations, there are several mobility management strategies proposed in the literature [26], all of them composed by two main procedures: location update and paging [22]. Location update is the procedure by means of which the network maintains updated the location of all its subscribers in the core network databases, and paging is the procedure with which the network finds the exact cell in which the callee’s terminal is located prior to redirect the incoming call. In this work, we study the mobility management strategy based on registration areas because it is commonly used in current mobile networks [23]. In an area-based mobility management strategy, every mobile station initiates a location update whenever it crosses the boundary between registration areas, being a registration area a continuous group of network cells. On the other hand and due to the fact that the network knows the location of its subscribers at a registration area level, the paging procedure should only be performed within the cells of the last updated registration area for the callee’ mobile station. Both location update and paging are two conflicting procedures in terms of signalling traffic. Suppose, e.g., that the network knows the exact location of every mobile station at any time, i.e. the case when a mobile station initiates a location update whenever it moves from one cell to another. In this extreme case, the number of location updates reaches its maximum and the number of paging messages is minimum because only the last updated cell should be paged. If on the other hand we suppose that a mobile station never initiates a location update (i.e. the case wherein the uncertainty about the location of a mobile station is at its maximum), there are no location updates but the number of paging messages reaches its maximum because all the network cells could be paged per each incoming call. That is, every mobility management strategy can be described as a multiobjective optimization problem with two conflicting objective functions: find the network configurations (i.e. registration areas configurations) that (i) minimize the number of location updates and (ii) minimize the number of paging messages. These are the desired solutions for the problem we are tackling. Furthermore, it is stated in the literature that the Registration Areas Planning Problem (RAPP) is a multiobjective NP-hard optimization problem [18, 22]. In the literature, RAPP was commonly tackled considering only few arguments per cell, e.g. number of incoming calls and number of subscribers’ movements among neighbouring cells [1, 2, 15, 16, 18, 35–38]. This allows reducing the optimizer runtime at the expense of also reducing the input data accuracy, which leads to less fitted solutions. This is the reason why we proposed in our latest works the use of mobile activity traces [9, 10]. Mobile activity traces are a very accurate way for representing the subscribers’ activity in a mobile network in form of a chronological list of events related to the subscribers’ movements and the incoming and outgoing calls. Furthermore, this kind of representation can be used to analyse the performance of any mobility management strategy. However, the mobile activity trace of a typical mobile network might have millions of events, and therefore, the assessment of these activity traces increases the optimizer runtime considerably. Nonetheless, this should not be a limitation because most of today’s devices have multi-core processors, e.g. personal computers, laptops and even mobile phones. Therefore, research different parallel techniques with the aim of decreasing the optimizers runtime when solving RAPP can be considered a step forward compared to the state of the art. Our first attempt was presented at [9], where we proposed a preliminary parallel version of our best optimizer [6–8], which is the standard NSGAII [14] with our specific evolutionary operators to solve RAPP. This preliminary approach was developed to be run in a multi-core processor with shared-memory architecture. For this purpose, we parallelized those operations inside NSGAII that can be done per each individual independently, i.e. initialization of the population, crossover or recombination of parents, and mutation. Thanks to this preliminary approach, the optimizer runtime was reduced considerably but we realized that there is still much room for improvement. In this work, we propose a fine-grained parallel approach that outperforms considerably the results published in [9]. Furthermore, this previous work has been extended and improved considerably by analysing the performance of both proposals when solving a realistic mobile network based on data gathered in a mobile network of Rome (Italy). The rest of the article is organized as follows. The related work is discussed in Section 2. Section 3 defines the operation mode of a mobility management strategy based on registration areas. Section 4 presents some basic concepts of multiobjective optimization as well as the optimization framework used in this work. The reason why it is interesting the use of parallelism in the optimization of registration areas is discussed in Section 5. Results and comparison between parallel approaches are shown in Section 6. Section 7 presents a comparison with other optimizers by other authors. Finally, conclusion and future work are discussed in Section 8. 2 Related work There are many different papers in the literature wherein the goal is to find the best possible configuration of registration areas. P. R. L. Gondim was one of the first authors in studying this optimization problem. In his work [18], P. R. L. Gondim realized that RAPP is an NP-hard optimization problem due to the huge size of the objective space and proposed a genetic algorithm with the aim of finding near-optimal configurations of registration areas. Afterwards, the search of the best optimizer to solve RAPP became an important research topic. P. Demestichas et al. proposed three different metaheuristics to tackle this optimization problem considering macro and micro-cellular networks. They implemented their adaptation of the following optimizers: simulated annealing, taboo search and genetic algorithm [15]. R. Subrata and A. Y. Zomaya proposed a dynamic location update strategy based on registration areas wherein a central controller was in charge of assigning individualized configurations of registration areas to each mobile station [34]. Though this dynamic strategy performed better than other location update strategies such as the distance-based strategy, it might not be implemented in current mobile networks due to the use of a different communication protocol. I. Demirkol et al. proposed another adaptation of the simulated annealing algorithm in [16]. The main difference with respect to their predecessors is that they considered one of the objective functions as a constraint, concretely the paging load. The drawback of such assumption is that the optimizer is focused on minimizing only one objective function and not the global signalling traffic, which is composed by the load of both location updates and paging. Furthermore, their proposal was only compared with non-metaheuristic methods. This assumption was also made in [12, 21]. In [21], T. James et al. developed a grouping genetic algorithm hybridized with a taboo search-based operator. This approach performed better than traditional grouping genetic algorithms; and S. N. Chaurasia and A. Singh proposed in [12] a steady-state grouping genetic algorithm that outperformed the method previously published in [21]. J. Taheri and A. Y. Zomaya proposed many different kinds of optimization techniques to tackle RAPP. They developed their adaptation of the following metaheuristics: Hopfield neural network [35], simulated annealing [36], genetic algorithm [37] and combinations between their Hopfield neural network with their genetic algorithm [38]. The feasibility of these metaheuristics was assessed in different test networks of different complexity whose sizes varied from 25 to 63 cells. They concluded that all registration areas must be configured in size and shape to fit the mobile activity in a network. S. M. Almeida-Luz et al. developed another two metaheuristics to improve the results published in [35–38]. In [2], they proposed an adaptation of the differential evolution; and in [1] an adaptation of the scatter search. M. Toril et al. researched a method for the automatic replanning of registration areas by using an evolutionary graph partitioning algorithm [40]. Their work showed that the configuration of registration areas should be replanned twice a week due to changes in the subscribers’ activity at weekends. It is important to mention that all of these works proposed a single-objective approach to tackle RAPP. This was done either by linearly combining the two objective functions into a weighted objective function or by considering one of the two objective functions as a constraint. A single-objective optimization of RAPP allows reducing the complexity of the optimizer but has several drawbacks. On the one hand, the optimization problem is not fully studied because the single-objective optimizers only provide one solution, the one that best fits their objective function. On the other hand, it is necessary a very accurate knowledge of the problem prior to set the weight coefficients. Furthermore, the proper value of these weight coefficients might be different for different problem instances. These are the reasons why we proposed different multiobjective optimizers in our previous works [6–8]. Furthermore, most of this related work addressed RAPP considering only few arguments per network cell, e.g. number of incoming calls and number of subscribers’ movements among neighbouring cells. This assumption allows reducing the optimizers runtime at the expense of decreasing the data input accuracy. As a result, this lack in the input data accuracy leads to less fitted solutions. In our latest work [9, 10], we avoid this problem by using mobile activity traces. A mobile activity trace is composed by a chronological list of events related to the incoming and outgoing calls and the subscribers’ movement at a cell level. Therefore, a mobile activity trace is a very accurate way to represent the subscribers’ activity in a mobile network. However, the mobile activity trace of a typical mobile network might have millions of events, and hence, the assessment of these activity traces increases the optimizer runtime considerably. Thus, research different techniques of parallel computing to tackle this issue becomes an essential step to study the throughput of different mobility management strategies quickly and accurately. Our first attempt was presented at [9], where we proposed an individual-based parallelization of NSGAII. In contrast to our previous work [9], this work presents a fine-grained parallel approach that is able to obtain much better results. Furthermore, this previous work has been extended and improved considerably by analyzing the performance of our proposal when solving a realistic mobile network based on data gathered in a mobile network of Rome (Italy). 3 Mobility management based on registration areas The most popular strategy for managing the mobility of idle terminals in a public land mobile network is based on registration areas (RAs). It is important to mention that the name of the RA is dependent on the underlying mobile technology. For example, a RA is called location area in GSM networks, routing area in GPRS networks, UTRAN registration area in UMTS and tracking area in LTE networks. In an area-based strategy, the network cells are arranged in continuous sets to partially track the subscribers’ movement, see Figure 1. For this purpose, a mobile station (i.e. the subscriber’s terminal) initiates a location update procedure whenever it crosses the boundary between RAs. As a consequence, the mobile network knows the location of its subscribers at a RA level, and therefore, the callee’s terminal must only be searched in the cells within its last updated RA. Fig. 1. Open in new tabDownload slide Registration areas strategy. Fig. 1. Open in new tabDownload slide Registration areas strategy. It should be noted that the signalling traffic due to mobility management is directly dependent on the number and size of the RAs. For a given mobile activity trace, the signalling traffic due to location updates will be higher for smaller RAs; and the signalling traffic due to paging messages will be higher for higher RAs. Thus, there is an inherent trade-off between the signalling loads due to location updates and paging messages and the RAPP can be described as a multiobjective optimization problem wherein the main challenge consists in finding the configurations of RAs that simultaneously minimize the number of location updates and the number of paging messages. These two objective functions are described in Equation (3.1) and Equation (3.2) respectively: \begin{equation} \textbf{f}_{\text{1}}=\text{min}\left\{\text{LU}=\sum_{\text{t=T}_{\text{ini}}}^{\text{T}_{\text{fin}}}\sum_{\text{i=1}}^{\text{N}_{\text{user}}}\gamma_{\text{t,i}}\right\}, \end{equation}(3.1) \begin{equation} \textbf{f}_{\text{2}}=\text{min}\left\{\text{PA}=\sum_{\text{t=T}_{\text{ini}}}^{\text{T}_{\text{fin}}}\sum_{\text{i=1}}^{\text{N}_{\text{user}}}\rho_{\text{t,i}}\cdot \mid \text{RA}_{\text{t,i}}\mid \right\}, \end{equation}(3.2) where |$\left[\text{T}_{\text{ini}}, \text{T}_{\text{fin}}\right]$| is the time interval during which the RAs strategy is evaluated. |$\text{N}_{\text{user}}$| is the number of subscribers. |$\gamma_{\text{t,i}}$| is a binary variable which is equal to 1 when the mobile station i crosses the boundary between RAs in the time t. |$\rho_{\text{t,i}}$| is a binary variable which is equal to 1 when the mobile station i has an incoming call in the time t. |$\mid \text{RA}_{\text{t,i}}\mid$| is the number of cells inside the last updated RA for the mobile station i. Though our parallel approach can be used for any paging procedure, for simplicity, we use the simultaneous paging in this work. As its name suggests, in a simultaneous paging procedure, all the network cells inside the last updated RA are polled simultaneously. 4 Multiobjective optimization for the RAPP As stated above, the RAPP is a multiobjective optimization problem with two objective functions (see Equation (3.1) and Equation (3.2)). A multiobjective optimization problem can be described as the problem in which two or more conflicting objective functions must be optimized simultaneously under certain constraints [13]. In contrast to single-objective optimization where the goal consists in finding only the solution that best fits the objective function, the main challenge in every multiobjective optimization problem is to find the best possible set of non-dominated solutions. Assuming a bi-objective optimization problem (such as RAPP), the solution x|$^{\text{i}}$| is said to dominate the solution x|$^{\text{j}}$| (expressed as x|$^{\text{i}} \prec$|x|$^{\text{j}}$|⁠) if and only if satisfies that |$\forall \text{k}\in \left[1,2\right],\textbf{z}^{\text{i}}_{\text{k}}={\boldsymbol{f}}_{\text{k}}\left(\textbf{x}^{\text{i}}\right) \leq \textbf{z}^{\text{j}}_{\text{k}}={\boldsymbol{f}}_{\text{k}}\left(\textbf{x}^{\text{j}}\right) \wedge \exists \text{k}\in \left[1,2\right]: \textbf{z}^{\text{i}}_{\text{k}} < \textbf{z}^{\text{j}}_{\text{k}}.$| As can be deduced from this definition, every non-dominated solution corresponds to a specific trade-off between the considered objectives. In the literature, a set of non-dominated solutions is commonly referred as Pareto set, and its image in the objective space is called Pareto front. There are many different optimizers specifically defined to tackle this kind of problems, each one with its own particularities. Nonetheless, most of them share more or less the main features of the Non-dominated Sorting Genetic Algorithm II (NSGAII) [42]. Furthermore, NSGAII has been satisfactorily applied to a wide range of optimization problems in fields such as telecommunications [11, 20, 41], power systems [32], natural resources management [31], magnetics [5], biomedical engineering [27], industrial informatics [3], etc. These are the reasons why NSGAII could be considered as a de-facto standard in the multiobjective optimization field. Besides, NSGAII is the optimizer by means of which we have obtained our best results [6–8]. A brief description of our adaptation of NSGAII to solve RAPP is shown below in Section 4.1. Please, take into account that when we mention our adaptation of NSGAII we refer to the standard NSGAII with our specific evolutionary operators to solve RAPP. 4.1 Optimization framework The Non-dominated Sorting Genetic Algorithm II is a population-based multiobjective optimizer that uses the evolutionary operators of biological systems to iteratively improve a set of randomly initialized solutions (Ind) [14]. For definition, population-based algorithms are those that manage one or more populations of individuals, being every individual an encoded solution of the problem; and evolutionary operators are used for creating and/or modifying new individuals. In this article, each individual is encoded as a vector of |$\text{N}_{\text{cell}}$| (i.e. number of cells) elements in which each vector position represents the RA assigned to the corresponding network cell. The first population of individuals is generated by using the following procedure. Firstly, each individual (i.e. each vector) is filled with a randomly uniform pattern of 0s and 1s. Afterwards, the resulting RA configuration is determined by grouping identical symbols into contiguous sets. An example of this procedure is shown in Figure 2. On the other hand, the evolutionary operators used by NSGAII are: crossover or recombination of parents, mutation of the offspring and natural selection of the fittest individuals. Fig. 2. Open in new tabDownload slide Initialization procedure. Fig. 2. Open in new tabDownload slide Initialization procedure. Algorithm 4.1. (Pseudo-code of NSGAII) The pseudo-code of NSGAII is shown in Algorithm 4.1. In this pseudo-code, after initializing and evaluating the first population of |$\text{N}_{\text{pop}}$| parents according to the procedure shown in Figure 2, the crossover operator is used with probability |$\text{P}_{\text{C}}$| to create a new population of |$\text{N}_{\text{pop}}$| individuals (the offspring) by recombining the solutions stored in the population of parents. Our optimizer uses an elitist crossover based on the binary tournament and a random number of crossover points between 1 and 4. Afterwards, the mutation operator is used with probability |$\text{P}_{\text{M}}$| to slightly change the offspring. We have implemented two specific mutation operators to solve RAPP. The first one merges the smallest RA with its smallest neighbouring registration area (Figure 3); and the second assigns a randomly selected cell with its smallest neighbouring RA (Figure 4). This randomly selected cell must be a border cell between RA. It should be noted that the objective functions must be assessed for each new individual to know its objective vector. Finally, natural selection is used to select the best individuals as the parent population in the next generation. For this purpose, NSGAII has its own fitness function to determine the quality of a solution in the multiobjective context (a detailed description of this fitness function can be found in [14]). This iterative method is done until satisfying the stopping condition. In this work, as in other previously published works, the stopping condition is the number of generations (⁠|$\text{N}_{\text{G}}$|⁠). This operation mode is also described in Algorithm 4.1, where we present the pseudo-code of NSGAII. Fig. 3. Open in new tabDownload slide RA-based mutation. Fig. 3. Open in new tabDownload slide RA-based mutation. Fig. 4. Open in new tabDownload slide Cell-based mutation. Fig. 4. Open in new tabDownload slide Cell-based mutation. 5 The need for parallelism As stated above, a mobile activity trace is a very accurate way for representing the subscribers’ activity in a mobile network in terms of incoming and outgoing calls and subscribers’ movements at a cell level. Besides, this accurate representation can be used for analysing the performance of any mobility management strategy. As a counterpart, the mobile activity trace of a current mobile network (with thousands of mobile subscribers) might be composed by several millions of activity events. This means that the time needed to evaluate the objective functions, and therefore the optimizer runtime, is increased considerably. Let us consider a mobile activity trace of a mobile network deployed in Rome (Italy) [8]. In this scenario with 21,800 mobile subscribers, the mobile activity trace is composed by 1,633,987 activity events and the optimizer runtime (Section 4.1) rises up to 1.5 days. Table 1 shows the mean runtime (of 31 independent runs) needed to evaluate each main operation of the optimizer, i.e. initialization of the population, natural selection, mutation, crossover, evaluation of the fitness function, and evaluation of the objective functions. This table highlights that most of time (⁠|${>}99\%$|⁠) is spent in evaluating the objective functions, i.e. most of time is spent in assessing the mobile activity trace for each new individual. According to the work described in [40], subscribers’ mobile activity can be decomposed into two main patterns, one during the working days and the other during the weekend. This implies that the optimization of the RAs for the following week should be performed in less than two days (the weekend), which corroborates the need of efficient techniques of parallel computing for reducing the runtime associated with the assessment of the mobile activity trace during the optimization process. Furthermore, it should be taken into account that mobile technology is continuously evolving, so it is essential to use tools to quickly evaluate the performance of different mobility management strategies. For these reasons, research on parallelism techniques to reduce this huge runtime can be considered as an interesting research topic. Table 1. Mean runtime per operation Initialization Natural selection Mutation Crossover Fitness ev. Objective functions ev. . |$\bar{\text{T}}$|(s) 0.02 4.98 69.44 162.72 57.55 133,438.71 Initialization Natural selection Mutation Crossover Fitness ev. Objective functions ev. . |$\bar{\text{T}}$|(s) 0.02 4.98 69.44 162.72 57.55 133,438.71 Table 1. Mean runtime per operation Initialization Natural selection Mutation Crossover Fitness ev. Objective functions ev. . |$\bar{\text{T}}$|(s) 0.02 4.98 69.44 162.72 57.55 133,438.71 Initialization Natural selection Mutation Crossover Fitness ev. Objective functions ev. . |$\bar{\text{T}}$|(s) 0.02 4.98 69.44 162.72 57.55 133,438.71 5.1 Mobile activity trace The mobile activity trace used in this work is based on data gathered in a mobile network deployed in Rome (Italy). This mobile activity trace has been generated by following the guidelines stated in [22] to mimic the behaviour of real mobile subscribers in a city. For this purpose, some considerations have been taken into account. First, it is assumed that different moves take place depending on the time of the day, e.g. movements towards workplace or movements towards home. Secondly, we also consider that subscribers move in a city according to their socio-economic classification, e.g. employed, pensioners, students, etc. Finally, we take into account the fact that call intensity during the day is generally higher than during the night. This is done by using a Poisson distribution (widely used in prior related work [24, 29, 30, 43]) with different values of |$\lambda$|⁠: |$\lambda_{\text{day}}=0.5$| calls/hour and |$\lambda_{\text{night}}=0.3$| calls/hour. These guidelines were also followed in [39]. The transportation network of this mobile activity trace is based on the topology of a real network deployed in Rome (Italy), a mobile network with 218 cells. A comparison between the mobile activity trace generated in this work and the real mobile activity measured in a network of Rome [28] is shown in Figure 5. As can be drawn from this figure, the mobile activity trace generated in this work follows the same behaviour than the mobile activity measured in real networks, so we can ensure that our proposal is evaluated under realistic conditions. This section presents a brief description of the mobile activity trace used in this work. For more information, please consult [8]. Fig. 5. Open in new tabDownload slide Hourly mobile activity trace. Fig. 5. Open in new tabDownload slide Hourly mobile activity trace. 6 Fine-grained parallelization The work presented at [9] was our first attempt to reduce the optimizer runtime by using parallel techniques in a multi-core processor system. In [9], we thought that due to the fact that the objective functions are evaluated per each new solution (i.e. per each new individual) and that there are some operations that are done per each individual in a population-based optimizer, we could reduce the optimizer runtime by parallelizing the code between lines L2-L5 and L11-L21 in the pseudocode shown in Algorithm 4.1. This preliminary approach allowed us to reduce the optimizer runtime considerably, but we realized that there is still much room for improvement. In this work, we focus our efforts on parallelizing the objective functions exclusively. If we pay attention to Equation (3.1) and Equation (3.2), we can realize that the total location management cost is the sum of the costs obtained per each subscriber. Therefore, these two objective functions can be parallelized by assigning the activity events of a block of subscribers to each core (or equivalently to each computation thread). For this purpose, the mobile activity trace must be sorted according to the subscriber ID. Besides, all the activity events for a given subscriber must be sorted in a chronological order. Afterwards, activity events of blocks of subscribers are assigned to the computation threads dynamically. An example of this method is shown in Figure 6, where we represent the mobile activity trace of 4 subscribers (Figure 6a), this mobile activity trace chronologically sorted by subscribers IDs (Figure 6b) and a possible block-to-thread assignation. In this figure, we assume a block size (or equivalently a chunk size) equal to 1 (i.e. a block of activity events has only events of 1 subscriber) and 2 computation threads which dynamically process blocks of activity events. Fig. 6. Open in new tabDownload slide Mobile activity trace arranged per subscriber ID Fig. 6. Open in new tabDownload slide Mobile activity trace arranged per subscriber ID As can be seen in Table 2 and Figure 7, this new approach is much more efficient than the method proposed in [9]. Concretely and considering 16 cores, the method proposed in this work presents an efficiency 22.55% higher than the method presented at [9]. Therefore, if we compare the results showed in Table 1 and Table 2, the total runtime has been drastically reduced from 1.55 days to 3.98 hours. Furthermore, it is important to note that the differences between both approaches increase with the number of cores, which means that this new approach scales better than the one proposed in [9]. Moreover, it is noteworthy that the efficiency of our proposal is always higher than 50%. All these experiments consists of 10 independent runs and have been performed up to 16 cores (sufficient to evaluate the quality of our proposal in current personal computers) in a machine with: 2 processors AMD Opteron(tm) 6174 @ 2.2 GHz, 64-GB RAM and Scientific Linux 6.1. Furthermore, we have used the OpenMP Application Program Interface because it is considered as a de-facto standard for parallel computing in shared-memory systems. After a previous study, we concluded that the dynamic scheduler with a chunk size equal to 100 provides good results. Table 2. Scalability study . . Fine-grained . Individual-based [9] . 2 cores Speedup 1.86 1.81 Efficiency(%) 92.85 90.45 |$\bar{\text{T}}$|(s) 72,019.73 73,929.83 4 cores Speedup 3.32 3.10 Efficiency(%) 82.94 77.45 |$\bar{\text{T}}$|(s) 40,311.60 43,168.58 8 cores Speedup 6.07 4.76 Efficiency(%) 75.86 59.53 |$\bar{\text{T}}$|(s) 22,038.29 28,083.00 16 cores Speedup 9.33 5.72 Efficiency(%) 58.32 35.77 |$\bar{\text{T}}$|(s) 14,332.81 23,365.76 . . Fine-grained . Individual-based [9] . 2 cores Speedup 1.86 1.81 Efficiency(%) 92.85 90.45 |$\bar{\text{T}}$|(s) 72,019.73 73,929.83 4 cores Speedup 3.32 3.10 Efficiency(%) 82.94 77.45 |$\bar{\text{T}}$|(s) 40,311.60 43,168.58 8 cores Speedup 6.07 4.76 Efficiency(%) 75.86 59.53 |$\bar{\text{T}}$|(s) 22,038.29 28,083.00 16 cores Speedup 9.33 5.72 Efficiency(%) 58.32 35.77 |$\bar{\text{T}}$|(s) 14,332.81 23,365.76 Table 2. Scalability study . . Fine-grained . Individual-based [9] . 2 cores Speedup 1.86 1.81 Efficiency(%) 92.85 90.45 |$\bar{\text{T}}$|(s) 72,019.73 73,929.83 4 cores Speedup 3.32 3.10 Efficiency(%) 82.94 77.45 |$\bar{\text{T}}$|(s) 40,311.60 43,168.58 8 cores Speedup 6.07 4.76 Efficiency(%) 75.86 59.53 |$\bar{\text{T}}$|(s) 22,038.29 28,083.00 16 cores Speedup 9.33 5.72 Efficiency(%) 58.32 35.77 |$\bar{\text{T}}$|(s) 14,332.81 23,365.76 . . Fine-grained . Individual-based [9] . 2 cores Speedup 1.86 1.81 Efficiency(%) 92.85 90.45 |$\bar{\text{T}}$|(s) 72,019.73 73,929.83 4 cores Speedup 3.32 3.10 Efficiency(%) 82.94 77.45 |$\bar{\text{T}}$|(s) 40,311.60 43,168.58 8 cores Speedup 6.07 4.76 Efficiency(%) 75.86 59.53 |$\bar{\text{T}}$|(s) 22,038.29 28,083.00 16 cores Speedup 9.33 5.72 Efficiency(%) 58.32 35.77 |$\bar{\text{T}}$|(s) 14,332.81 23,365.76 Fig. 7. Open in new tabDownload slide Comparison between approaches Fig. 7. Open in new tabDownload slide Comparison between approaches 7 Comparison with other optimizers The quality of our optimizer has been already checked in our previous work [6–8]. However and with the aim of making the article self-contained, this section presents a comparison with optimizers developed by other authors [1, 2, 35–38]. To perform a fair comparison with these other metaheuristics, our optimizer has been configured with the same population size (⁠|$\text{N}_{\text{pop}}=250$|⁠) and the same number of generations (⁠|$\text{N}_{\text{G}}=5,000$|⁠). The other parameters of NSGAII (i.e. |$\text{P}_{\text{C}}$| and |$\text{P}_{\text{M}}$|⁠) have been configured by means of a parametric study of 31 independent runs per experiment. We have chosen the configuration that maximizes (on average) the Hypervolume value, one of the most popular indicators used to estimate the quality of a set of non-dominated solutions [13]. The configuration with the best Hypervolume value was: |$\text{P}_{\text{C}}=0.90$| and |$\text{P}_{\text{M}}=0.25$|⁠. This comparative study can be shown in Table 3, wherein we present the results obtained in the set of test networks used in [1, 2, 35–38]: LA25 (a test network with 25 cells), LA35 (a test network with 35 cells), LA49 (a test network with 49 cells) and LA63 (a test network with 63 cells). These test networks, although provide only two arguments per network cell (number of incoming calls and number of subscribers’ movements among neighbouring cells), allow us to compare our optimizer with other prior art techniques in terms of solutions quality. It is important to note that the metaheuristics proposed in [1, 2, 35–38] are single-objective optimizers. Therefore, to perform a fair comparison with these single-objective metaheuristics, we have searched in our Pareto fronts the non-dominated solutions that minimize the objective function defined in [1, 2, 35–38]. This table reveals that our optimizer is very competitive because it is able to obtain the best solution found to date with single-objective optimizers in all the test networks. For further information about the quality of the Pareto fronts provided by our optimizer, please consult our previous work [6–8, 10]. Table 3. Comparison with other optimizers by other authors . NSGAII . HNN [35] . SA [36] . GA [37] . GA-HNN1 [38] . GA-HNN2 [38] . GA-HNN3 [38] . DE [2] . SS [1] . LA25 26,990 27,249 26,990 28,299 26,990 26,990 26,990 26,990 26,990 LA35 39,832 39,832 42,750 40,085 40,117 39,832 39,832 39,859 39,832 LA49 60,685 63,516 60,694 61,938 62,916 62,253 60,696 61,037 60,685 LA63 89,085 92,493 90,506 90,318 92,659 91,916 91,819 89,973 89,085 . NSGAII . HNN [35] . SA [36] . GA [37] . GA-HNN1 [38] . GA-HNN2 [38] . GA-HNN3 [38] . DE [2] . SS [1] . LA25 26,990 27,249 26,990 28,299 26,990 26,990 26,990 26,990 26,990 LA35 39,832 39,832 42,750 40,085 40,117 39,832 39,832 39,859 39,832 LA49 60,685 63,516 60,694 61,938 62,916 62,253 60,696 61,037 60,685 LA63 89,085 92,493 90,506 90,318 92,659 91,916 91,819 89,973 89,085 Table 3. Comparison with other optimizers by other authors . NSGAII . HNN [35] . SA [36] . GA [37] . GA-HNN1 [38] . GA-HNN2 [38] . GA-HNN3 [38] . DE [2] . SS [1] . LA25 26,990 27,249 26,990 28,299 26,990 26,990 26,990 26,990 26,990 LA35 39,832 39,832 42,750 40,085 40,117 39,832 39,832 39,859 39,832 LA49 60,685 63,516 60,694 61,938 62,916 62,253 60,696 61,037 60,685 LA63 89,085 92,493 90,506 90,318 92,659 91,916 91,819 89,973 89,085 . NSGAII . HNN [35] . SA [36] . GA [37] . GA-HNN1 [38] . GA-HNN2 [38] . GA-HNN3 [38] . DE [2] . SS [1] . LA25 26,990 27,249 26,990 28,299 26,990 26,990 26,990 26,990 26,990 LA35 39,832 39,832 42,750 40,085 40,117 39,832 39,832 39,859 39,832 LA49 60,685 63,516 60,694 61,938 62,916 62,253 60,696 61,037 60,685 LA63 89,085 92,493 90,506 90,318 92,659 91,916 91,819 89,973 89,085 8 Conclusion and future work This work presents a fine-grained parallel approach developed with the aim of reducing the runtime needed to assess mobile activity traces in the RAPP. Mobile activity traces are a very accurate way for representing the subscribers’ activity in a mobile network in form of a chronological list of events related to the subscribers’ movements and the incoming and outgoing calls. This kind of representation can be used to study the performance of any mobility management strategy at the expense of increasing the optimizer runtime considerably, and concretely the time needed to evaluate the objective functions. In fact, we have shown in Section 5 that more than 99% of the total runtime is used to evaluate the objective functions. In contrast to previously published work, the parallel approach proposed in this work is focused on parallelizing the objective functions exclusively. Results show the good efficiency of our fine-grained approach, always higher than 50% and even 22.55% higher than the method proposed in [9]. As a consequence, the optimizer runtime is drastically reduced from 1.55 days to 3.98 hours. The benefits of a fine-grained approach are shown in this work. Nonetheless, as a future work, it could be interesting to explore the efficiency of other parallel approaches, e.g. considering different parallel schedulers and/or other arrangements of mobile activity events. Acknowledgements This work was partially funded by the Spanish Ministry of Economy and Competitiveness and the ERDF (European Regional Development Fund), under the contract TIN2016-76259-P (PROTEIN project). 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TI - A fine-grained parallel approach for the registration areas optimization
JF - Logic Journal of the IGPL
DO - 10.1093/jigpal/jzx032
DA - 2017-12-01
UR - https://www.deepdyve.com/lp/oxford-university-press/a-fine-grained-parallel-approach-for-the-registration-areas-rIx76GXVFb
SP - 862
VL - 25
IS - 6
DP - DeepDyve
ER -