arXiv subjects

Ridha Nasri

Publications and source records attributed to Ridha Nasri.

At least 19 recordsLinked to original sources

arXiv2021

Asymptotic analysis of the sojourn time of a batch in an $M^{[X]}/M/1$ Processor Sharing Queue

In this paper, we exploit results obtained in an earlier study for the Laplace transform of the sojourn time $Ω$ of an entire batch in the $M^{[X]}/M/1$ Processor Sharing (PS) queue in order to derive the asymptotic behavior of the complementary probability distribution function of this random variable, namely the behavior of $P(Ω>x)$ when $x$ tends to infinity. We precisely show that up to a multiplying factor, the behavior of $P(Ω>x)$ for large $x$ is of the same order of magnitude as $P(ω>x)$, where $ω$ is the sojourn time of an arbitrary job is the system. From a practical point of view, this means that if a system has to be dimensioned to guarantee processing time for jobs then the system can also guarantee processing times for entire batches by introducing a marginal amount of processing capacity.

cs.PF↗

arXiv2021

Inversion of a Class of Singular Integral Operators on Entire Functions

Given constants $x, ν\in \mathbb{C}$ and the space $\mathscr{H}_0$ of entire functions in $\mathbb{C}$ vanishing at $0$, we consider the integro-differential operator $$ \mathfrak{L} = \left ( \frac{x \, ν(1-ν)}{1-x} \right ) \; δ\circ \mathfrak{M}\, , $$ with $δ= z \, \mathrm{d}/\mathrm{d}z$ and $\mathfrak{M}:\mathscr{H}_0 \rightarrow \mathscr{H}_0$ defined by $$ \mathfrak{M}f(z) = \int_0^1 e^{-z t^{-ν}(1-(1-x)t)} \, f \left (z \, t^{-ν}(1-t) \right ) \, \frac{\mathrm{d}t}{t}, \qquad z \in \mathbb{C}, $$ for any $f \in \mathscr{H}_0$. Operator $\mathfrak{L}$ originates from an inversion problem in Queuing Theory. Bringing the inversion of $\mathfrak{L}$ back to that of $\mathfrak{M}$ translates into a singular Volterra integral equation, but with no explicit kernel. In this paper, the inverse of operator $\mathfrak{L}$ is derived through a new inversion formula recently obtained for infinite matrices with entries involving Hypergeometric polynomials. For $x \notin \mathbb{R}^- \cup \{1\}$ and $\mathrm{Re}(ν) < 0$, we then show that the inverse $\mathfrak{L}^{-1}$ of $\mathfrak{L}$ on $\mathscr{H}_0$ has the integral representation $$ \mathfrak{L}^{-1}g(z) = \frac{1-x}{2iπx} \, e^{z} \int_1^{(0+)} \frac{e^{-xtz}}{t(t-1)} \, g \left (z \, (-t)^ν(1-t)^{1-ν} \right ) \, \mathrm{d}t, \qquad z \in \mathbb{C}, $$ for any $g \in \mathscr{H}_0$, where the bounded integration contour in the complex plane starts at point 1 and encircles the point 0 in the positive sense. Other related integral representations of $\mathfrak{L}^{-1}$ are also provided.

math.CA↗

arXiv2020

On the sojourn time of a batch in the $M^{[X]}/M/1$ Processor Sharing Queue

In this paper, we analyze the sojourn of an entire batch in a processor sharing $M^{[X]}/M/1$ processor queue, where geometrically distributed batches arrive according to a Poisson process and jobs require exponential service times. By conditioning on the number of jobs in the systems and the number of jobs in a tagged batch, we establish recurrence relations between conditional sojourn times, which subsequently allow us to derive a partial differential equation for an associated bivariate generating function. This equation involves an unknown generating function, whose coefficients can be computed by solving an infinite lower triangular linear system. Once this unknown function is determined, we compute the Laplace transform and the mean value of the sojourn time of a batch in the system.

math.PR↗

arXiv2020

A New Linear Inversion Formula for a class of Hypergeometric polynomials

Given complex parameters $x$, $ν$, $α$, $β$ and $γ\notin -\mathbb{N}$, consider the infinite lower triangular matrix $\mathbf{A}(x,ν;α, β,γ)$ with elements $$ A_{n,k}(x,ν;α,β,γ) = \displaystyle (-1)^k\binom{n+α}{k+α} \cdot F(k-n,-(β+n)ν;-(γ+n);x) $$ for $1 \leqslant k \leqslant n$, depending on the Hypergeometric polynomials $F(-n,\cdot;\cdot;x)$, $n \in \mathbb{N}^*$. After stating a general criterion for the inversion of infinite matrices in terms of associated generating functions, we prove that the inverse matrix $\mathbf{B}(x,ν;α, β,γ) = \mathbf{A}(x,ν;α, β,γ)^{-1}$ is given by \begin{align} B_{n,k}(x,ν;α, β,γ) = & \; \displaystyle (-1)^k\binom{n+α}{k+α} \; \cdot \nonumber \\ & \; \biggl [ \; \frac{γ+k}{β+k} \, F(k-n,(β+k)ν;γ+k;x) \; + \nonumber \\ & \; \; \; \frac{β-γ}{β+k} \, F(k-n,(β+k)ν;1+γ+k;x) \; \biggr ] \nonumber \end{align} for $1 \leqslant k \leqslant n$, thus providing a new class of linear inversion formulas. Functional relations for the generating functions of related sequences $S$ and $T$, that is, $T = \mathbf{A}(x,ν;α, β,γ) \, S \Longleftrightarrow S = \mathbf{B}(x,ν;α, β,γ) \, T$, are also provided.

math.CA↗

arXiv2020

Dynamic-TDD Interference Tractability Approaches and Performance Analysis in Macro-Cell and Small-Cell Deployments

Meeting the continued growth in data traffic volume, Dynamic Time Division Duplex (D-TDD) has been introduced as a solution to deal with the uplink (UL) and downlink (DL) traffic asymmetry, mainly observed for dense heterogeneous network deployments, since it is based on instantaneous traffic estimation and provide more flexibility in resource assignment. However, the use of this feature requires new interference mitigation schemes capable to handle two additional types of interference between cells in opposite transmission direction: DL to UL and UL to DL interference. The aim of this work is to provide a complete analytical approach to model inter-cell interference in macro-cell and dense small-cell networks. We derive the explicit expressions of Interference to Signal Ratio (ISR) at each position of the network, in both DL and UL, to quantify the impact of each type of interference on the system performance. Also, we provide the explicit expressions of the coverage probability as functions of different system parameters by covering different scenarios. Finally, through system level simulations, we analyze the feasibility of D-TDD implementation in both deployments and we compare its performance to the static-TDD (S-TDD) configuration.

cs.NI↗

arXiv2020

A 3D Beamforming Scheme Based on The Spatial Distribution of User Locations

Multi-antenna technologies such as massive Multiple-Input Multiple-Output (massive MIMO) and beamforming are key features to enhance performance, in terms of capacity and coverage, by using a large number of antennas intelligently. With the upcoming 5G New Radio (NR), FD-MIMO (Full Dimension MIMO) will play a major key role. FD-MIMO consists in arranging a large number of antennas in a 2D array, which enables to use 3D beamforming i.e., beamforming in both horizontal and vertical dimensions. The present paper provides a 3D beamforming model where beam steering depends on the random spatial distribution of users. We attempt to derive some analytical results regarding the probability distribution of antenna beamforming radiation pattern. Also, through system level simulations, we show how 3D beamforming can reduce interference impact, compared to the traditional 2D beamforming, and enhances system performance in terms of the coverage probability and users throughput.

cs.NI↗

arXiv2020

How To Dimension Radio Resources When Users Are Distributed on Roads Modeled by Poisson Line Process

Resources dimensioning aims at finding the number of radio resources required to carry a forecast data traffic at a target users Quality of Services (QoS). The present paper attempts to provide a new approach of radio resources dimensioning considering the congestion probability, qualified as a relevant metric for QoS evaluation. Users are assumed to be distributed according to a linear Poisson Point Process (PPP) in a random system of roads modeled by Poisson Line Process (PLP) instead of the widely-used spatial PPP. We derive the analytical expression of the congestion probability for analyzing its behavior as a function of network parameters. Finally we show how to dimension radio resources by setting a value of the congestion probability, often targeted by the operator, in order to find the relation between the necessary resources and the forecast data traffic expressed in terms of cell throughput. Different numerical results are presented to justify this dimensioning approach.

cs.NI↗

arXiv2020

3D Beamforming based Dynamic TDD Interference Mitigation Scheme

Dynamic Time Division Duplexing (D-TDD) allows cells to accommodate asymmetric traffic variations with high resource assignment flexibility. However, this feature is limited by two additional types of interference between cells in opposite transmission direction: downlink (DL) to uplink (UL) and UL to DL interference. Therefore, using this mode with macro-cell deployments requires interference mitigation techniques to reduce the strong DL to UL interference. 3D beamforming is an efficient technique that minimizes interference and enhances performance by exploiting a large 2D array of antennas intelligently. Combining D-TDD and 3D beamforming can make D-TDD feasible for macro-cells. The aim of this work is to provide a 3D beamforming analytical model in a D-TDD based macro-cells' deployment where beamforming horizontal and vertical radiation patterns depend on the spatial distribution of random users' locations. We evaluate interference in terms of Interference to Signal Ratio (ISR). We show that the cumulative ISR can be written in terms of convergent series and its expectation is an almost sure convergent series. Different numerical results are presented to justify the applicability of this scheme.

cs.NI↗

arXiv2019

Radio Resource Dimensioning with Cox Process Based User Location Distribution

The upcoming fifth generation (5G) New Radio (NR) interface inherits many concepts and techniques from 4G systems such as the Orthogonal Frequency Division Multiplex (OFDM) based waveform and multiple access. Dimensioning 5G NR interface will likely follow the same principles as in 4G networks. It aims at finding the number of radio resources required to carry a forecast data traffic at a target users Quality of Services (QoS). The present paper attempts to provide a new approach of radio resources dimensioning considering the congestion probability, qualified as a relevant metric for QoS evaluation. We distinguish between the spatial random distribution of indoor users, modeled by a spatial Poisson Point Process (spatial PPP) in a typical area covered by a 5G cell, and the distribution of outdoor users modeled by a linear PPP generated in a random system of roads modeled according to a Poisson Line Process (PLP). Moreover, we show that the total requested Physical Resource Blocks (PRBs) follows a compound Poisson distribution and we attempt to derive the explicit expression of the congestion probability by introducing a mathematical tool from combinatorial analysis called the exponential Bell polynomials. Finally we show how to dimension radio resources, for a given target congestion probability, by solving an implicit relation between the necessary resources and the forecast data traffic expressed in terms of cell throughput. Different numerical results are presented to justify this dimensioning approach.

cs.NI↗

arXiv2019

Inversion formula with hypergeometric polynomials and its application to an integral equation

For any complex parameters $x$ and $ν$, we provide a new class of linear inversion formulas $T = A(x,ν) \cdot S \Leftrightarrow S = B(x,ν) \cdot T$ between sequences $S = (S_n)_{n \in \mathbb{N}^*}$ and $T = (T_n)_{n \in \mathbb{N}^*}$, where the infinite lower-triangular matrix $A(x,ν)$ and its inverse $B(x,ν)$ involve Hypergeometric polynomials $F(\cdot)$, namely $$ \left\{ \begin{array}{ll} A_{n,k}(x,ν) = \displaystyle (-1)^k\binom{n}{k}F(k-n,-nν;-n;x), \\ B_{n,k}(x,ν) = \displaystyle (-1)^k\binom{n}{k}F(k-n,kν;k;x) \end{array} \right. $$ for $1 \leqslant k \leqslant n$. Functional relations between the ordinary (resp. exponential) generating functions of the related sequences $S$ and $T$ are also given. These new inversion formulas have been initially motivated by the resolution of an integral equation recently appeared in the field of Queuing Theory; we apply them to the full resolution of this integral equation. Finally, matrices involving generalized Laguerre polynomials polynomials are discussed as specific cases of our general inversion scheme.

math.CA↗

arXiv2018

Interference Analysis in Dynamic TDD System Combined or not With Cell Clustering Scheme

Dynamic Time Division Duplex (TDD) has been introduced as a solution to deal with the uplink and downlink traffic asymmetry, mainly observed for dense heterogeneous network deployments. However, the use of this feature requires new interference mitigation schemes capable to handle two additional types of interferences between cells in opposite transmission cycle: downlink to uplink and uplink to downlink interferences. Among them, Cell clustering has been proposed as an efficient solution to minimize inter-cell interferences in opposite transmission directions and somehow responds to the requirements of enhanced Interference Mitigation and Traffic Adaptation (eIMTA) problem. This work is devoted to provide a new analytical approach to model inter-cell interferences and quantify performances of Dynamic TDD system in terms of SINR (Signal to Interferences plus Noise Ratio) distribution. Analytical system performance investigation concerns two scenarios: i) basic Dynamic TDD without any other feature and ii) Dynamic TDD with interference mitigation schemes.

cs.NI↗

arXiv2016

Performance Analysis of Small Cells' Deployment under Imperfect Traffic Hotspot Localization

Heterogeneous Networks (HetNets), long been considered in operators' roadmaps for macrocells' network improvements, still continue to attract interest for 5G network deployments. Understanding the efficiency of small cell deployment in the presence of traffic hotspots can further draw operators' attention to this feature. In this context, we evaluate the impact of imperfect small cell positioning on the network performances. We show that the latter is mainly impacted by the position of the hotspot within the cell: in case the hotspot is near the macrocell, even a perfect positioning of the small cell will not yield improved performance due to the interference coming from the macrocell. In the case where the hotspot is located far enough from the macrocell, even a large error in small cell positioning would still be beneficial in offloading traffic from the congested macrocell.

cs.NI↗

arXiv2016

Offloading traffic hotspots using moving small cells

In this paper, the concept of moving small cells in mobile networks is presented and evaluated taking into account the dynamics of the system. We consider a small cell moving according to a Manhattan mobility model which is the case when the small cell is deployed on the top of a bus following a predefined trajectory in areas which are generally crowded. Taking into account the distribution of user locations, we study the dynamic level considering a queuing model composed of multi-class Processor Sharing queues. Macro and small cells are assumed to be operating in the same bandwidth. Consequently, they are coupled due to the mutual interferences generated by each cell to the other. Our results show that deploying moving small cells could be an efficient solution to offload traffic hotspots.

cs.NI↗

arXiv2016

System level analysis of heterogeneous networks under imperfect traffic hotspot localization

We study, in this paper, the impact of imperfect small cell positioning with respect to traffic hotspots in cellular networks. In order to derive the throughput distribution in macro and small cells, we firstly perform static level analysis of the system considering a non-uniform distribution of user locations. We secondly introduce the dynamics of the system, characterized by random arrivals and departures of users after a finite service duration, with the service rates and distribution of radio conditions outfitted from the first part of the work. When dealing with the dynamics of the system, macro and small cells are modeled by multi-class processor sharing queues. Macro and small cells are assumed to be operating in the same bandwidth. Consequently, they are coupled due to the mutual interferences generated by each cell to the other. We derive several performance metrics such as the mean flow throughput and the gain, if any, generated from deploying small cells to manage traffic hotspots. Our results show that in case the hotspot is near the macro BS (Base Station), even a perfect positioning of the small cell will not yield improved performance due to the high interference experienced at macro and small cell users. However, in case the hotspot is located far enough from the macro BS, performing errors in small cell positioning is tolerated (since related results show positive gains) and it is still beneficial in offloading traffic from the congested macrocell. The best performance metrics depend also on several other important factors such as the users' arrival intensity, the capacity of the cell and the size of the traffic hotspot.

cs.NI↗

arXiv2016

On the Analytical Tractability of Hexagonal Network Model with Random User Location

Explicit derivation of interferences in hexagonal wireless networks has been widely considered intractable and requires extensive computations with system level simulations. In this paper, we fundamentally tackle this problem and explicitly evaluate the downlink Interference-to-Signal Ratio (ISR) for any mobile location $m$ in a hexagonal wireless network, whether composed of omni-directional or tri-sectorized sites. The explicit formula of ISR is a very convergent series on $m$ and involves the use of Gauss hypergeometric and Hurwitz Riemann zeta functions. Besides, we establish simple identities that well approximate this convergent series and turn out quite useful compared to other approximations in literature. The derived expression of ISR is easily extended to any frequency reuse pattern. Moreover, it is also exploited in the derivation of an explicit form of SINR distribution for any arbitrary distribution of mobile user locations, reflecting the spatial traffic density in the network. Knowing explicitly about interferences and SINR distribution is very useful information in capacity and coverage planning of wireless cellular networks and particularly for macro-cells' layer that forms almost a regular point pattern.

cs.NI↗

arXiv2016

Product of parabolic cylinder functions involving Laplace transforms of confluent hypergeometric functions

In this paper, the product of parabolic cylinder functions $D_ν(\pm z)D_{ν+μ-1}(z)$, with different parameters $μ$ and $ν$, are established in terms of Laplace and Fourier transforms of Kummer's confluent hypergeometric functions. The provided integral representations are transformed to easily yield Nicholson-type integral forms and used to derive other series expansions for products of parabolic cylinder functions.

math.CA↗

arXiv2015

A New Alternative for Traffic Hotspot Localization in Wireless Networks Using O&M Metrics

In recent years, there has been an increasing awareness to traffic localization techniques driven by the problematic of hotspot offloading solutions, the emergence of heterogeneous networks (HetNet) with small cells' deployment and the green networks. The localization of traffic hotspots with a high accuracy is indeed of great interest to know how the congested zones can be offloaded, where small cells should be deployed and how they can be managed for sleep mode concept. We propose, in this paper, a new hotspot localization technique based on the direct exploitation of five Key Performance Indicators (KPIs) extracted from the Operation and Maintenance (O&M) database of the network. These KPIs are the Timing Advance (TA), the angle of arrival (AoA), the neighboring cell level, the load time and two mean throughputs: arithmetic (AMT) and harmonic (HMT). The combined use of these KPIs, projected over a coverage map, yields a promising localization precision and can be further optimized by exploiting commercial data on potential hotspots. This solution can be implemented in the network at an appreciable low cost when compared with widely used probing methods.

cs.NI↗

arXiv2015

Traffic Hotspot localization in 3G and 4G wireless networks using OMC metrics

In recent years, there has been an increasing awareness to traffic localization techniques driven by the emergence of heterogeneous networks (HetNet) with small cells deployment and the green networks. The localization of hotspot data traffic with a very high accuracy is indeed of great interest to know where the small cells should be deployed and how can be managed for sleep mode concept. In this paper, we propose a new traffic localization technique based on the combination of different key performance indicators (KPI) extracted from the operation and maintenance center (OMC). The proposed localization algorithm is composed with five main steps; each one corresponds to the determination of traffic weight per area using only one KPI. These KPIs are Timing Advance (TA), Angle of Arrival (AoA), Neighbor cell level, the load of each cell and the Harmonic mean throughput (HMT) versus the Arithmetic mean throughput (AMT). The five KPIs are finally combined by a function taking as variables the values computed from the five steps. By mixing such KPIs, we show that it is possible to lessen significantly the errors of localization in a high precision attaining small cell dimensions.

cs.NI↗