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# A positive systems model of TCP-like congestion control: Asymptotic results

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## Abstract

We study communication networks that employ drop-tail queueing and Additive-Increase Multiplicative-Decrease (AIMD) congestion control algorithms. It is shown that the theory of nonnegative matrices may be employed to model such networks. In particular, important network properties such as: (i) fairness; (ii) rate of convergence; and (iii) throughput; can be characterised by certain non-negative matrices. We demonstrate that these results can be used to develop tools for analysing the behaviour of AIMD communication networks. The accuracy of the models is demonstrated by several NS-studies.

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... These systems, primarily motivated from their broad applications, have attracted wide attention in recent times. For instance, switched positive systems have been employed to devise optimal drug treatments to prevent HIV mutation [10], to model communication network [11] and traffic control of crossway [12], to design l 1 filtering of the Foschini-Miljanic power regulation algorithm [13]. Stimulated by these applications, recently, researchers have been drawn to study the switched positive systems. ...
... Hence, exploiting (11), one can know that (18) also holds for t = t k + 1 . And recalling (13), we immediately attain ...
... As a sequence, the simulation results exhibit the usefulness of the developed control scheme. Example 2: An important application of positive systems is communication networks model [11]. Due to the limit of network throughput and bandwidth allocation amongst flows, network congestion often arises in the practical running process. ...
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This study handles the issues of stability and guaranteed cost for switched positive systems based on a novel multiple linear copositive Lyapunov functions approach. Two different switching policies are devised. The first one relies on continuous time strategy and the other depends on the sampled‐data strategy. The designed switching policies not only render the considered systems globally exponentially stable with a pre‐specified adequate level of performance, but also decrease the switching frequency to rule out the Zeno behaviour. New stability conditions are presented in form of linear vector equality as well as linear vector inequalities, which can be readily solved by employing Linprog function. Moreover, based on the attained stability results, the authors further research an infinite time cost function. The effectiveness of the proposed methods is validated and illustrated via simulation results.
... In the real world, there is a special class of switched systems, namely, positive switched systems [1][2][3]. Such a class of systems has been widely applied in chemical engineering [4], ecology [5], network employing [6], and so on. Some important results on positive switched systems [7][8][9][10][11] have been reported. ...
... Communication networks follow the switching rules between the idle and the busy time models in operation. The literature [6,47] established a communication network consisting of three nodes through positive switched systems (see Fig. 1). Large data packets were sent from control center to network terminal. ...
Article
This paper proposes a reliable control of positive switched systems with random nonlinearities which may induce the security problem of the systems. The random nonlinearities are governed by stochastic variables obeying the Bernoulli distribution. A switched linear copositive Lyapunov function is employed for the systems. Using a matrix decomposition approach, the gain matrix of controller is formulated by the sum of nonnegative and non-positive components. A reliable controller is designed for positive switched systems with actuator faults by virtue of linear programming. Under the designed reliable controller, the systems can resist some possible security risks triggered by random nonlinearities and actuator faults. The obtained approach is developed for systems subject to exogenous disturbances. Finally, two examples are provided to verify the validity of the obtained results.
... Noting that each subsystem is positive, the trajectory of the positive switched system under arbitrary switching signal will remain nonnegative if the initial state is nonnegative. It is well known that positive switched system has wide applications in practice, such as consensus of multiagent systems [2], congestion control in communication networks [3], chemical reaction networks [4], drug delivery control in anesthesia [5] and Markovian jump systems [6], [7]. ...
... Therefore, system (1) is asymptotically stable under the statedependent switching signal (3). This completes the proof of Theorem 1. ...
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In this paper, we focus on the stabilization of positive switched linear time-varying systems with delay. By constructing Lyapunov-Metzler inequalities, state-dependent switching signals have been designed such that the system is asymptotically stable, even if each subsystem is not asymptotically stable. Both the continuous-time and the discrete-time positive switched linear time-varying systems have been taken into consideration. Numerical examples are also given to verify the validity of our results.
... ey have the property that all descriptive variables can only take positive values, or at least nonnegative values. ese systems can be found in economics [15], biology, stochastic processes (Markov chains or hidden Markov models) [16], chemical processing [17], communication science and information [18], etc. e theory of positive systems is more complicated than the one of the standard systems because positive linear systems are defined on cones and not on linear spaces [19]. As a result, some known properties of linear systems cannot be applied for positive systems (for more details, see [20]). ...
... As an example, in the field of biology and pharmacokinetics, they are used to describe the dynamics of the viral mutation under drug treatment [22]. It is also applied in HIV treatment modelling [21], formation flying [23], and communication systems [18]. Many problems have been examined concerning positive switched systems, such as stability and stabilizability [24] as well as structural properties, like reachability, controllability, and observability [25,26]. ...
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In this paper, we present a sufficient condition for the output reachability of discrete-time positive switched systems. Besides, necessary and sufficient conditions for output monomial reachability and zero output controllability are provided. Further, some examples are shown. 1. Introduction In recent years, engineers and applied mathematicians have been interested in the study and analysis of switched systems, which represent an important class of hybrid dynamical systems. A switched system is the association of a finite set of differential or difference subsystems and a switching law that indicates at each instant the active system. Switched systems are of a great interest, since they are very convenient in the mathematical modelling of several systems such as network control systems, near-space vehicle control systems [1], biological systems [2], dc/dc convertors, oscillators [3], chaos generators [4], and so on. Based on previous research, many mathematical problems have been posed and investigated such as stability and stabilizability properties [5, 6]. Recent studies examined other issues such as reachability and controllability [7–13]. It is important to note that Babiarz [14] provided important results on the output controllability for standard switched systems. It should be noted that positive systems are of great importance in practice as they appear naturally in various fields of science and technology. They have the property that all descriptive variables can only take positive values, or at least nonnegative values. These systems can be found in economics [15], biology, stochastic processes (Markov chains or hidden Markov models) [16], chemical processing [17], communication science and information [18], etc. The theory of positive systems is more complicated than the one of the standard systems because positive linear systems are defined on cones and not on linear spaces [19]. As a result, some known properties of linear systems cannot be applied for positive systems (for more details, see [20]). Combing the characteristics of general switched systems and positive systems, results are obtained on positive switched systems [21]. The strong interest that this type of system has recently raised is due to its strong presence in the most important areas. As an example, in the field of biology and pharmacokinetics, they are used to describe the dynamics of the viral mutation under drug treatment [22]. It is also applied in HIV treatment modelling [21], formation flying [23], and communication systems [18]. Many problems have been examined concerning positive switched systems, such as stability and stabilizability [24] as well as structural properties, like reachability, controllability, and observability [25, 26]. This paper which deals with the output reachability and controllability problem of discrete-time positive switched systems is organized as follows. After some preliminaries in the following section, we provide necessary conditions for the output reachability in Section 3. In Section 4, necessary and sufficient conditions for the output monomial reachability are provided. The zero output controllability problem is explored in Section 5. 2. Preliminaries The symbols and denote the sets of nonnegative integers and nonnegative real numbers, respectively. is the n-dimensional Euclidean space and is the set of all n-dimensional nonnegative real vectors. In addition, represents the space of matrices with real entries and represents the set of all matrices with nonnegative entries. If , we say that is nonnegative and write . We write for the transpose of the matrix and the identity matrix. For any , with , we set = . Let , be an alphabet whose elements are called letters. A word over the alphabet is a finite sequence of elements of ; it will be denoted by where , . The length of the word is the number of letters it is composed of, written as . The set of all words over the alphabet is a free monoid for concatenation, whose neutral element is the empty word denoted by . Clearly, for any word , and . Let be a set of matrices and . If is a word in , we set Next, we introduce a class of nonnegative matrices, namely, the monomial matrices. Definition 1. A nonnegative vector is said to be monomial if it contains precisely one nonzero entry. We will call it an -monomial vector if the nonzero component is in the position. Definition 2. A nonnegative matrix is a monomial matrix if it has only one nonzero entry in every row and every column. In this paper, we consider a discrete-time switched system described by the difference state equationwhere is the state vector, is the control input, is the output vector, and is a switching sequence. Given a control , , and a switching sequence , , the solution of system (2), with the initial condition , at time , can be expressed as [25]where Definition 3. The discrete system (2) is called positive if for any switching sequence , any initial condition , and for any input , , the state and the output for all . Proposition 1. The discrete system (2) is positive if and only if, for each , , , and . Proof. If , , and for all , then equations (3) and (4) imply that for all and we have and for all . Conversely, assume that the positive switched system (1) is positive. Let , , and for all . Then, from (2), for , we obtain and . Thus, and , since may be arbitrary. Now, assuming that , then from (2), for , we have . It follows that , since may be arbitrary. 3. Output Reachability of Switched Positive Systems In the main result of this section, we provide a sufficient condition for the output reachability of system (2). Before giving our result, some definitions concerning the output reachability of positive switched systems should be cited. Definition 4. An output is said to be reachable in steps if there exists a switching sequence , , and inputs for that steer the output of system (2) from to , namely, . System (2) is called output reachable if every nonnegative output is reachable in some step . It is clearly seen that when , the output can be written aswhere is called the output reachability matrix associated to the switching sequence . Definition 5. The set of all nonnegative linear combinations of the columns of a matrix is called polyhedral convex cone, namely,Polyhedral convex cones play an important role in the output reachability of positive systems since the set of all reachable outputs in steps is a polyhedral cone belonging to the nonnegative orthant. is a polyhedral cone generated by the columns of the output reachability matrix associated to the switching sequence of length . The length of the switching sequence is the cardinality of the discrete-time interval and it is denoted, for short, by means of the notation . Clearly, the positive switched system (2) is output reachable if there exist switching sequences of lengths , respectively, , such that Example 1. Consider positive switched system (2) with and the following matrices:DefineWe getTherefore, the system is output reachable. 4. Output Monomial Reachability of Switched Positive Systems We study in this section the concept of output monomial reachability and provide necessary and sufficient conditions for this property. First, we recall the following definition and give some preliminary results. Definition 6. The positive switched system (2) with is said to be output monomially reachable if, for all , there exist , a switching sequence , and nonnegative control inputs such thatwith being the ith canonical vector of . Lemma 1. If and are such that is an i-monomial vector, then includes an i-monomial column. Proof. Let , with being the vector columns of and . If is i-monomial, thenwhich implies that for all , we have and there exists some such that . Therefore, there exists such that . Hence, includes an i-monomial column. Corollary 1. Let and . If includes an i-monomial column, then has an i-monomial column. Proof. Let be the vector columns of ; then,Since contains an i-monomial column, then there exists such that is an i-monomial vector. Applying Lemma 1, it yields that the matrix has an i-monomial vector. The proposition below contains a necessary and sufficient condition for output monomial reachability using the output reachable matrix associated with all possible switching sequences. Proposition 2. The positive switched system (2) is output monomially reachable if and only if there exists some positive integer N such that the output reachability matrix in N stepsincludes an monomial submatrix. Proof. Assume that for all , there exist , a switching sequence , and nonnegative control inputs such that . This implies that the following equality Then, there exists such that is an -monomial vector. By Lemma 1, includes an i-monomial column. Letand pose and . Then, includes an i-monomial column. For , we have includes an monomial submatrix. Conversely, let ; then, includes an i-monomial column, which implies that there exist , , and such that contains an i-monomial column. Let and poseLet satisfying , and , where . Then,Set . Then, there exists such that and , for . Let and , for all . We get from (5) thatTherefore, the system is output monomially reachable. Remark 1. In the case of single output systems , the Proposition 2 gives in fact a characterization of the output reachability of system (2). Let us now consider some examples. Example 2. Consider positive switched system (2) consisting of two subsystems with the following matrices:For the two subsystems, we have , , and . So, neither one is output reachable in one step. But , and hence the positive system (2) is output reachable in one step. Indeed, let and ; then, for all , for we get . Example 3. Consider the positive system switching among the following subsystems:We haveSo, the two subsystems are not output monomially reachable in one step. ButHence, the positive system (2) is output monomially reachable in one step. Indeed, for any , let and . Then, . Also, for any , let and . Then, . On the other hand, it is clearly seen that this system is not reachable in one step because the vector can never be reached in one step. Corollary 2. If the positive switched system (2) is output monomially reachable, then the matrix has an monomial submatrix. Proof. Suppose the system is output monomially reachable. Thus, for all , there exist , such that has an i-monomial column. Applying Corollary 1, it yields that the matrix has an i-monomial column. Hence, the matrix has an monomial submatrix. 5. Zero Output Controllability To present our main results for zero output controllability, we introduce the following definition. Definition 7. The positive switched system (2) is said to be zero output controllable if, for all , there exist , a switching sequence , and nonnegative control input , such that Proposition 3. The positive switched system (2) is zero output controllable if and only if there exist and such that Proof. If the system is zero output controllable, then in particular, for , there exist , a switching sequence , and , such that . It follows thatThen,Let and ; then, . Conversely, let with , , , , and . Then, for each , we haveIf , then , and for and , we obtain , for all , which completes the proof. Corollary 3. The positive switched system (2) is zero output controllable if there exists such that is nilpotent. Proof. Assume that there exists such that Let . Then, . According to Proposition 3, system (2) is zero output controllable. Example 4. Consider the positive switched system composed of two subsystems with the following matrices:By choosing and , we get . Therefore, positive switched system (2) is zero output controllable. Also, the positive switched system (2) is zero output controllable, since there exists a word such that , that is, is nilpotent. 6. Conclusions In this paper, we have addressed a number of issues related to the output reachability, output monomial reachability, and the zero output controllability properties of discrete-time positive switched systems. By means of certain concepts borrowed from the algebra of noncommutative polynomials, we have been able to establish the necessary and sufficient conditions guaranteeing the output monomial reachability (Proposition 2) and the zero output controllability of discrete-time positive switched systems (Proposition 3). These conditions were then applied to numerical examples to illustrate their application and to support the theoretical results. The results discussed here will be of great value for our future work that will treat another class of positive systems. Data Availability No data were used to support this study. Conflicts of Interest The authors declare that they have no conflicts of interest regarding the publication of this paper.
... The non-negative variables, for example, the number of biological populations and the concentration of chemical reactants, are contained in many practical engineering. The systems with non-negative variables are generally described as positive systems (see Farina & Rinaldi, 2000;Kaczorek, 2002), which often appear in biomedicine, chemistry, communication engineering and some other real fields (see Sandberg, 1978;Shorten et al., 2006;Xiao et al., 2019). For positive systems, the special positive property may generate not only interesting results but also difficulties and challenges in system analysis and design. ...
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For non-linear positive switched systems (NPSSs) with time-varying delay, the $H_{\infty }$ estimation problem is addressed in this paper. Firstly, in light of T-S fuzzy modeling method, the considered NPSS is equivalently transformed into a switched positive T-S fuzzy system. Then, a less conservative $H_{\infty }$ performance co1ndition is provided for the error system via presenting a new piecewise quadratic copositive Lyapunov–Krasovskii functional. Furthermore, an observer is constructed to make the underlying system exponentially stable with a preset $H_{\infty }$ performance and an iterative optimization algorithm is proposed to obtain the observer parameters. To verify the effectiveness of the theoretical results, two examples are finally presented.
... Positive systems [25] are an important class of dynamical systems whose state is confined in the nonnegative orthant. Such systems have been considered for the modeling of a wide variety of real world processes such as the modeling of populations [38], physiological systems [33], biochemical systems [7,9,10,14,30], communication networks [13,52], etc. The control/controllability/reachability of systems with positive inputs have been well studied; see e.g. ...
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The integral control of positive systems using nonnegative control input is an important problem arising, among others, in biochemistry, epidemiology and ecology. An immediate solution is to use an ON-OFF nonlinearity between the controller and the system. However, this solution is only available when controllers are implemented in computer systems. When this is not the case, like in biology, alternative approaches need to be explored. Based on recent research in the control of biological systems, we propose to develop a theory for the integral control of positive systems using nonnegative controls based on the so-called \emph{antithetic integral controller} and two \emph{positively regularized integral controllers}, the so-called \emph{exponential integral controller} and \emph{logistic integral controller}. For all these controllers, we establish several qualitative results, which we connect to standard results on integral control. We also obtain additional results which are specific to the type of controllers. For instance, we show an interesting result stipulating that if the gain of the antithetic integral controller is suitably chosen, then the local stability of the equilibrium point of the closed-loop system does not depend on the choice for the coupling parameter, an additional parameter specific to this controller. Conversely, we also show that if the coupling parameter is suitably chosen, then the equilibrium point of the closed-loop system is locally stable regardless the value of the gain. For the exponential integral controller, we can show that the local stability of the equilibrium point of the closed-loop system is independent of the gain of the controller and the gain of the system. The stability only depends on the exponential rate of the controller, again a parameter that is specific to this type of controllers. Several examples are given for illustration.
... Positive systems [25] are an important class of dynamical systems whose state is confined in the nonnegative orthant. Such systems have been considered for the modeling of a wide variety of real world processes such as the modeling of populations [38], physiological systems [33], biochemical systems [7,9,10,14,30], communication networks [13,52], etc. The control/controllability/reachability of systems with positive inputs have been well studied; see e.g. ...
Article
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The integral control of positive systems using nonnegative control input is an important problem arising, among others, in biochemistry, epidemiology and ecology. An immediate solution is to use an ON-OFF nonlinearity between the controller and the system. However, this solution is only available when controllers are implemented in computer systems. When this is not the case, like in biology, alternative approaches need to be explored. Based on recent research in the control of biological systems, we propose to develop a theory for the integral control of positive systems using nonnegative controls based on the so-called \emph{antithetic integral controller} and two \emph{positively regularized integral controllers}, the so-called \emph{exponential integral controller} and \emph{logistic integral controller}. For all these controllers, we establish several qualitative results, which we connect to standard results on integral control. We also obtain additional results which are specific to the type of controllers. For instance, we show an interesting result stipulating that if the gain of the antithetic integral controller is suitably chosen, then the local stability of the equilibrium point of the closed-loop system does not depend on the choice for the coupling parameter, an additional parameter specific to this controller. Conversely, we also show that if the coupling parameter is suitably chosen, then the equilibrium point of the closed-loop system is locally stable regardless the value of the gain. For the exponential integral controller, we can show that the local stability of the equilibrium point of the closed-loop system is independent of the gain of the controller and the gain of the system. The stability only depends on the exponential rate of the controller, again a parameter that is specific to this type of controllers.
... S WITCHED systems are an important class of control systems in the field of control theory and applications [1]- [4]. In practice, many control systems contain quantities taking nonnegative values such as communication networks, biological systems, industrial processes, etc. Positive systems can describe the systems consisting of nonnegative quantities [5]- [7]. Consequently, a new class of switched systems, named positive switched systems, was introduced [8]- [10]. ...
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This paper investigates the double switchings reliable control of positive switched systems with actuator faults. Different from existing fixed actuator faults, a class of unfixed actuator faults is introduced, where each subsystem can have several faults rather than only a fault. First, a periodic occurring fault is considered for the subsystems. Using linear Lyapunov functions with linear programming, a set of reliable controllers is designed for positive switched systems. Then, an average dwell time based switching law between the normal and faulty modes of subsystems is established. In detail, the considered systems consist of several switched subsystems, where the subsystems have normal and faulty modes. In such a case, two classes of multiple linear Lyapunov functions are constructed for the subsystems and the whole switched systems, respectively. Accordingly, two classes of reliable controllers are designed for the normal and faulty modes, respectively. Meanwhile, two classes of average dwell time switching laws, named double switchings laws, are designed for the subsystems and the whole switched systems, respectively. Finally, two examples are provided to illustrate the effectiveness of the proposed design.
... Recently, finite-time control problems have been actively investigated in the context of positive systems [10], which are dynamical systems whose state variables are confined to be within the positive orthant and naturally arise in various application areas including pharmacology [11], epidemiology [12,13], and communication networks [14]. For example, the authors in [15] derived a necessary and sufficient condition for the finite-time stability of switched positive linear systems by using the co-positive Lyapunov approach. ...
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In this paper, we study a class of finite-time control problems for discrete-time positive linear systems with time-varying state parameters. Although several interesting control problems appearing in population biology, economics, and network epidemiology can be described as the class of finite-time control problems, an efficient solution to the control problem has not been yet found in the literature. In this paper, we propose an optimization framework for solving the class of finite-time control problems via convex optimization. We illustrate the effectiveness of the proposed method by numerical simulation in the context of dynamical product development processes.
... For example, such models are common when the state variables represent concentrations, absolute temperatures, prices, production quantities, population levels, buffer sizes, queue lengths, charge levels, light intensity levels, and so on. Positive system models have therefore flourished in several research areas, including biology, ecology, physiology, and pharmacology (26)(27)(28)(29)(30); biomolecular and biochemical modeling (31)(32)(33); thermodynamics (28,31); epidemiology (34)(35)(36); traffic and congestion modeling (31,37); power systems (38); filtering and charge routing networks (39)(40)(41); and econometrics (42). In addition, since probabilities are nonnegative quantities, Markov chains (43), hidden Markov chains, and other probabilistic models represent special cases of positive systems. ...
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In this article, we first present some foundational results about the stability and positive stabilization of continuous-time positive systems. Necessary and sufficient conditions for achieving stability are provided, together with some desired performance in terms of disturbance attenuation. These conditions are expressed in terms of linear programming and scale well with the system size. We then discuss the interconnection of positive subsystems by means of a static output feedback that preserves positivity, and propose conditions to achieve both stability and the asymptotic alignment of the closed-loop output to a desired vector. Finally, we describe some results for a class of parameterized positive systems. The second part of the article presents some interesting applications of the results presented in the first part. Specifically, control problems for heating networks, formation control, power control in wireless communication, and the evolutionary dynamics of cancer and HIV are formalized and solved as optimal control problems for positive systems. Expected final online publication date for the Annual Review of Control, Robotics, and Autonomous Systems, Volume 4 is May 3, 2021. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.
... In last two decades, many models have been presented to understand the dynamics of TCP/AQM system. The fluid flow model MGT is initially presented in [8] and reviewed later without considering any AQM controller where a nonnegative matrix method is proposed to model the dynamics of TCP system [33]. The analytical model for TCP/AQM system is presented in [2]. ...
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... Positive systems can be used to accurately describe the systems consisting of nonnegative quantities. Such kind of systems has drawn many interests in the control field owing to their effectiveness in medical treatment [1,2], communication [3,4], water systems [5] and challenges in theory [6,7,8,9]. Over past two decades, much effort has been devoted to stability [10,11], control synthesis [12,13], observation [14], filter [15], etc. ...
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This paper investigates the event-triggered model predictive control of positive systems with actuator saturation. Interval and polytopic uncertainties are imposed on the systems, respectively. First, a new model with actuator saturation obeying Bernoulli distribution is established, which is more general and powerful for describing the saturation phenomenon than the saturation in a certain way. Then, a linear event-triggering condition is constructed based on the state and error signal. An interval estimate approach is presented to reach the positivity and stability of the systems. The saturation part in the controller is technically transformed into a non-saturation part. Thus, a linear programming approach is proposed to compute the event-triggered controller gain and the corresponding domain of attraction gain. A predictive algorithm is introduced for the computation of the event-triggered controller parameters. Finally, an example is provided to illustrate the validity of the design.
... ey have extensive applications in the fields of biology systems and pharmacokinetics [3,4]. In practice, there have been many systems that can be modeled as positive switched systems such as formation flying [5] and network employing TCP [6,7]. In the literature [8], the stability of positive switched linear systems with average dwell time (ADT) was studied using multiple linear copositive Lyapunov functions (MLCLFs). ...
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... Definition 3 [12]: For a switching signal ( ) t  and each 2 ...
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... The research on positive linear systems traces back to David G. Luenberger, who systematically introduced the concept of such class of systems in a fundamental book [23]. Since then, positive systems theory has seen broad applications in many industrial problems, such as biochemical engineering and traffic control [7,36]. For many real-world physical systems, the descriptor variables usually have intrinsically positive or non-negative features [11]. ...
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... States of several dynamical systems are constraint in the first orthant; these systems belong to the class of positive systems [1]. Communication networks [2], biological networks [3], pharmacology [4], population growth, and diseases (See refs. [5][6][7]) are some examples of these systems. ...
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... [15], [16]) involving the eigenproblem of rank-one perturbations of symmetric matrices (Theorems 1 and 2). This formulation leads to a new admission control algorithm with AIMD structure which is stable irrespective of the AIMD tuning and the number of nodes, and inherits attractive features of the standard AIMD algorithm (e.g., fairness among nodes, tunable convergence rate) [17]. To the best of our knowledge, this paper presents a new admission control policy with AIMD dynamics for scheduling tasks. ...
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... An event-triggering state feedback law for positive systems was constructed in terms of linear matrix inequalities in [38]. Positive switched systems were used for dealing with the networks congestion problem of communication networks in [5]. Indeed, in the idle time of communication networks, the communication control centre does not need to change the network speed or restrict the number of data packages transited in the network. ...
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... Positive systems can be used to accurately describe the systems consisting of nonnegative quantities. Such kind of systems has drawn many interests in the control field owing to their effectiveness in medical treatment [1,2], communication [3,4], water systems [5] and challenges in theory [6][7][8][9]. Over past two decades, much effort has been devoted to stability [10,11], control synthesis [12,13], observation [14], filter [15], etc. ...
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Virtual queue-based active queue management schemes have been proposed to provide low-loss, low-delay service in the Internet. In an earlier work, we had proposed a particular scheme called the adaptive virtual queue (AVQ) algorithm where the capacity of the virtual queue is adapted to the traffic conditions to achieve a desired level of utilization in the network. Here, we study the choice of the parameters of the congestion-controllers at the sources and the AVQ scheme at the links that is required to ensure stability. In particular, we consider a system in which users with diverse round-trip delays and fairness requirements access a general topology network. For this system, we show that, by choosing the speed of adaptation at the sources and the links appropriately, one can guarantee the stability of the network.
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We propose a natural and simple model for the joint throughput evolution of a set of TCP sessions sharing a common tail drop bottleneck router, via products of random matrices. This model allows one to predict the fluctuations of the throughput of each session, as a function of the synchronization rate in the bottleneck router; several other and more refined properties of the protocol are analyzed such as the instantaneous imbalance between sessions, the autocorrelation function or the performance degradation due to synchronization of losses. When aggregating traffic obtained from this model, one obtains, for certain ranges of the parameters, short time scale statistical properties that are consistent with a fractal scaling similar to what was identified on real traces using wavelets.
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We study stability properties of a finite set Sigma of n×n-matrices such as paracontractivity, BV- and LCP-stability, and their relations to each other. The conjecture on equivalence of paracontractivity and LCP-stability is proved. Moreover, we prove the equivalence of the uniform BV-stability and the property of vanishing length of steps of any trajectory of Sigma.
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Introduction. In the investigation of chaotic iteration procedures for linear consistent systems matrices which are paracontracting with respect to some vector norm play an important role. It was shown in [EKN], that if A 1 ; : : : ; Am are finitely many k--by--k complex matrices which are paracontracting with respect to the same norm, then for any sequence d i ; 1 d i m; i = 1; 2; : : : and any x 0 the sequence x i+1 = A d i x i i = 1; 2; : : : is convergent. In particular A (d) = lim i!1 A d i : : : A d1 exists for all sequences fd i g
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In this paper we develop a simple analytic characterization of the steady state throughput, as a function of loss rate and round trip time for a bulk transfer TCP flow, i.e., a flow with an unlimited amount of data to send. Unlike the models in [6, 7, 10], our model captures not only the behavior of TCP's fast retransmit mechanism (which is also considered in [6, 7, 10]) but also the effect of TCP's timeout mechanism on throughput. Our measurements suggest that this latter behavior is important from a modeling perspective, as almost all of our TCP traces contained more timeout events than fast retransmit events. Our measurements demonstrate that our model is able to more accurately predict TCP throughput and is accurate over a wider range of loss rates. This material is based upon work supported by the National Science Foundation under grants NCR-95-08274, NCR-95-23807 and CDA-95-02639. Any opinions, findings, and conclusions or recommendations expressed in this material are those of...
Throughput analysis of TCP networks
• D Leith
• R Shorten
D. Leith and R. Shorten, " Throughput analysis of TCP networks, " 2004, to appear in the proceedings of Networking 2004.
Traffic phase effects in packet-switched gateways Journal of Internetworking: Practice and Experience
• S Floyd
• V Jacobson
S. Floyd and V. Jacobson, " Traffic phase effects in packet-switched gateways, " Journal of Internetworking: Practice and Experience, vol. 3, no. 3, pp. 115–156, September, 1992. [Online]. Available: citeseer.ist.psu.edu/floyd92traffic.html