We introduce an original mathematical model to analyze the diffusion of posts within a generic online social platform. The main novelty is that each user is not simply considered as a node on the social graph, but is further equipped with his/her own Wall and Newsfeed, and has his/her own individual self-posting and re-posting activity. As a main result using our developed model, we derive in closed form the probabilities that posts originating from a given user are found on the Wall and Newsfeed of any other. These are the solution of a linear system of equations, which can be resolved iteratively. In fact, our model is very flexible with respect to the modeling assumptions. Using the probabilities derived from the solution, we define a new measure of per-user influence over the entire network, the$\Psi $-score, which combines the user position on the graph with user (re-)posting activity. In the homogeneous case where all users have the same activity rates, it is shown that a variant of the$\Psi $-score is equal to PageRank. Furthermore, we compare the new model and its$\Psi $-score against the empirical influence measured from very large data traces (Twitter, Weibo). The results illustrate that these new tools can accurately rank influencers with asymmetric (re-)posting activity for such real world applications.

We introduce an original mathematical model to analyze the diffusion of posts within a generic online social platform. Each user of such a platform has his own Wall and Newsfeed, as well as his own self-posting and re-posting activity. As a main result, using our developed model, we derive in closed form the probabilities that posts originating from a given user are found on the Wall and Newsfeed of any other. These probabilities are the solution of a linear system of equations. Conditions of existence of the solution are provided, and two ways of solving the system are proposed, one using matrix inversion and another using fixed-point iteration. Comparisons with simulations show the accuracy of our model and its robustness with respect to the modeling assumptions. Hence, this article introduces a novel measure which allows to rank users by their influence on the social platform, by taking into account not only the social graph structure, but also the platform design, user activity (self-and re-posting), as well as competition among posts.

Data plane development kit (DPDK) works as a specialized library that enables virtual switches to accelerate the processing of incoming packets by, among other things, balancing the incoming flow of packets over all the CPU cores and processing packets by batches to make a better use of the CPU cache. Although DPDK has become a de facto standard, the performance modeling of a DPDK-based vSwitch remains a challenging problem. In this paper, we present an analytical queueing model to evaluate the performance of a DPDK-based vSwitch. Such a virtual equipment is represented by a complex polling system in which packets are processed by batches, i.e., a given CPU core processes several packets of one of its attached input queues before switching to the next one. To reduce the complexity of the associated model, we develop a general framework that consists in decoupling the polling system into several queueing subsystems, each one corresponding to a given CPU core. We resort to servers with vacation to capture the interactions between subsystems. Our proposed solution is conceptually simple, easy to implement and computationally efficient. Tens of comparisons against a discrete-event simulator show that our models typically deliver accurate estimates of the performance parameters of interest (e.g., attained throughput, packet latency or loss rate). We illustrate how our models can help in determining an adequate setting of the vSwitch parameters using several real-life case studies.

In this paper, we analyze flow level performance of mobile users in 4G/pre-5G cellular networks such as LTE and LTE-A. To this end, we develop a two-levels model that provides users' performance in a cell and end-to-end performance in the network. At a cell-level, the model is a multi-class PS queue that captures mobility of users between zones of a cell, through a simple mobility model, that is decoupled from the cell model itself, enabling to directly apply the approach to more realistic mobility patterns. At a network-level, the model is a simple Discrete Time Markov Chain that reproduces the routing of mobile users between the different cells of the system. We first show that this model is consistent with known analytical bounds corresponding to a system with either static users or users having an infinite speed. The outcomes of our model confirm that mobility may improve both users and cells performance, and enable to quantify the gain. The model also shows that, while inter-cell mobility balances the load between cells, it does not lead to such improvement of throughput as intra-cell mobility.

In this paper, we investigate the influence of intra-and inter-cell mobility of users on performance of 4G/5G cellular networks, such as LTE and LTE-A. To this end, we develop a multi-class PS queue model that captures mobility of users between zones of a cell and between cells, through a simple mobility model, that is decoupled from the cell model itself, enabling to directly apply the approach to more realistic mobility patterns. We first show that this model is consistent with known analytical bounds corresponding to a system with either static users or users having an infinite speed. We then compare our model to simulations for more realistic speeds, and show that it provides user and cell performance with a very good accuracy. The outcomes of our model confirm that mobility may improve both users and cells performance, and enable to quantify the gain as a function of users speed.

Virtual switches are a key elements within the new paradigms of Software Defined Networking (SDN) and Network Function Virtualization (NFV). Unlike proprietary networking appliances, virtual switches come with a high level of flexibility in the management of their physical resources such as the number of CPU cores, their allocation to the switching function, and the capacities of the RX queues, which gives the opportunity for an efficient sizing of the system resources. We propose a model for the performance evaluation of a virtual switch. Our model resorts to servers with vacation to capture the involved interactions between queues resulting from the implemented polling strategies. The solution to the model is found using a simple fixed-point iteration and it provides estimates for customary performance metrics such as the attained throughput, the packet latency, the buffer occupancy and the packet loss rate. In the tens of explored examples, the predictions of the model were found to be accurate, thereby allowing their use for the purpose of sizing problems.

In this paper, we investigate the impact of mobility on user performance in the context of dense LTE-A networks. To this end, we propose simple analytical models that capture mobility through the distribution of the mobile users sojourn time, i.e., the time a mobile user is physically present in a given cell. We use these models to derive the throughput of users who remain spatially static during their whole data transmission and the amount of handovers generated by moving users. We analyze the impact on performance of some key parameters such as the size of the small cell, the speed and proportion of mobile users and the distribution of their sojourn time. Numerical evaluation and simulation results are provided to assess the accuracy of the latter model and gain insight into the global system performance.

In this paper, we follow up an existing modeling framework to analytically evaluate the performance of multi-hop flows along a wireless chain of four nodes. The proposed model accounts for a non-perfect physical layer, handles the hidden node problem, and is applicable under workload conditions ranging from flow(s) with low intensity to flow(s) causing the network to saturate. Its solution is easily and quickly obtained and delivers estimates for the expected throughput and for the datagram loss probability of the chain with a good accuracy.

IEEE 802.11 is implemented in many wireless networks, including multi-hop networks where communications between nodes are conveyed along a chain. We present a modeling framework to evaluate the performance of flows conveyed through such a chain. Our framework is based on a hierarchical modeling composed of two levels. The lower level is dedicated to the modeling of each node, while the upper level matches the actual topology of the chain. Our approach can handle different topologies, takes into account Bit Error Rate and can be applied to multi-hop flows with rates ranging from light to heavy workloads. We assess the ability of our model to evaluate loss rate, throughput, and end-to-end delay experienced by flows on a simple scenario, where the number of nodes is limited to three. Numerical results show that our model accurately approximates the performance of flows with a relative error typically less than 10%.

In this paper, we explore a way to find the right scheduling policy for WiMAX networks, that achieves the best compromise between an efficient use of the resource and a relative fairness among users. This problem is of primary importance as no scheduling policy has been recommended in the WiMAX standard. To do so, we develop an extension of our previous analytical model for WiMAX networks, that takes into account a more general scheduling policy than those previously studied (i.e., instantaneous throughput fairness, slot sharing fairness and opportunistic scheduling). We show that this general policy covers the two extreme cases, namely the instantaneous throughput fairness policy and the opportunistic policy, and offers intermediate policies that are good candidates for finding the right trade-off. In order to formulate the decision criterion, we introduce a new performance parameter, the mean throughput obtained by a user depending on its efficiency to use the resource. The model has a closed-form solution, and all performance parameters can be obtained instantaneously. This allows us to carry out dimensioning studies that require several thousands of evaluations, which would not be tractable with any simulation tool.

In this paper, we present a simple and accurate analytical model for performance evaluation of WiMAX networks with multiple traffic profiles. This very promising access technology has been designed to support numerous kinds of applications having different traffic characteristics. One of the QoS parameters considered by the standard for traffic classes is the maximum sustained traffic rate (MSTR), which is an upper bound for user throughput. Taking into account MSTR implies the implementation of a throttling scheduling policy that regulates the user peak rate. Our models take into account this policy and provides closed-form expressions giving all the required performance parameters for each traffic profile at a click speed. The model is compared with extensive simulations that show its accuracy and robustness.

This paper presents a simple and accurate analytical model to dimension mobile cellular networks with QoS differentiation. QoS per applicative flow is commonly defined in GPRS/EDGE and 3G systems where streaming applications with real time properties and elastic data applications have to share radio resources. The need for accurate and fast-computing tools is of primary importance to tackle complex and exhaustive dimensioning issues. In this paper, we present a generic QoS modeling methodology developed in the context of EDGE networks but that can be adapted easily to a different technology. We develop a ldquoconditional product-formrdquo Markovian model that takes into account the QoS differentiation between real time and non-real time classes and gives closed form expressions for all the required performance parameters at a click speed. We compare our model with simulation and show its accuracy. The speed of our analytical tool allows tackling dimensioning issues that require several thousands of evaluations, which is not tractable with any simulation tool.

Where distributed agents must share voluminous set membership information, Bloom filters provide a compact, though lossy, way for them to do so. Numerous recent networking papers have examined the trade-offs between the bandwidth consumed by the transmission of Bloom filters, and the error rate, which takes the form of false positives, and which rises the more the filters are compressed. In this paper, we introduce the retouched Bloom filter (RBF), an extension that makes the Bloom filter more flexible by permitting the removal of selected false positives at the expense of generating random false negatives. We analytically show that RBFs created through a random process maintain an overall error rate, expressed as a combination of the false positive rate and the false negative rate, that is equal to the false positive rate of the corresponding Bloom filters. We further provide some simple heuristics that decrease the false positive rate more than than the corresponding increase in the false negative rate, when creating RBFs. Finally, we demonstrate the advantages of an RBF over a Bloom filter in a distributed network topology measurement application, where information about large stop sets must be shared among route tracing monitors.

This paper is a contribution to the generic problem of having simple and accurate models to dimension radio cells with data traffic on a GPRS or EDGE network. It addresses the issue of capacity limitation in a given cell due to coupling with other cells because of a central equipment or transmission link of limited capacity. A mobile station can't access the requested resource although it is alone in a cell. The traffic on other coupled cells leads to reach the global capacity limit. Our objective is to avoid the derivation of any multi-dimensional Markovian (or semi-Markovian) model, where each dimension corresponds to a given cell of the system. Such direct extensions would be of non-manageable complexity. Instead we derive an analytical model that captures in an aggregate way the coupling between cells. We show that the performance parameters of the GPRS/EDGE network can be derived quickly and with a very good accuracy. Finally, as our modeling framework allows very fast computations, we show how to use it to perform complex iterative dimensioning studies.

In this paper, we derive a simple and accurate Erlang-like law for the dimensioning of radio resources in GSM/GPRS/EDGE networks taking into account the voice and data traffic mix in a cell. Resources are assumed to be shared according to the so-called partial partitioning scheme whereby some circuits are dedicated to voice, others to data, and the remaining are shared but voice traffic can preempt them.

A model for GPRS/EDGE network engineering is developed in order to obtain an Erlang-like law. It is based on a discrete-time Markov chain. It captures the main features of the GPRS/EDGE radio resource allocation and assumes an ON/OFF traffic (infinite sessions with geometric distributions of ON/OFF sizes) performed by a finite number of users over the cell. The Markov chain is simplified by Taylor series expansion and a simple and accurate Erlang-like law is obtained.

The authors consider closed queuing networks with general service times and subnetworks having population constraints. Each such subnetwork consist of a subset of stations of the original network. The population constraint of a subnetwork prescribes that the total number of customers than can be simultaneously present inside the subnetwork is limited to a fixed value, called the capacity of the subnetwork. A customer wanting to enter a subnetwork while it is full must wait in an external queue. As soon as a customer leaves the subnetwork, the first customer in line is allowed to enter the subnetwork. Such queuing networks with population constraints are of interest for modeling and analysis of manufacturing systems. Two approximate techniques for analyzing such networks, namely the aggregation technique and the method of R. Marie (1979, 1980), are compared.>