Article

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

Authors:
To read the full-text of this research, you can request a copy directly from the authors.

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.

No full-text available

Request Full-text Paper PDF

To read the full-text of this research,
you can request a copy directly from the authors.

... 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. ...
Article
Full-text available
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. ...
Article
Full-text available
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]. ...
Article
Full-text available
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. ...
Article
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. ...
Preprint
Full-text available
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
Full-text available
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]. ...
Article
Full-text available
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. ...
Preprint
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. ...
Article
Full-text available
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]. ...
Article
Full-text available
In this work a self-tuning rate and queue based proportional and integral controller called SRQ-PI is proposed to efficiently control the queue length with small overshoot and faster settling time. SRQ-PI proposes a new control tracking function that maps level of congestion to the packet drop probability dynamically. In SRQ-PI, the incoming traffic rate is estimated and used with the proportional and integral controller. The SRQ-PI tunes itself and stabilizes the system with internal feedback without requiring any external feedback. Furthermore, the stability of the SRQ-PI is analyzed using control theory and presents systematic guidelines to select the control gain parameters. NS2 is used to carry out the simulation work. The simulation result demonstrates that SRQ-PI is stable and gets faster transient response due to lower average delay jitter and robust against dynamic network parameters. The SRQ-PI outperforms proportional integral (PI), Intelligent adaptive PI (IAPI) and Random exponential marking (REM) algorithm.
... 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. ...
Preprint
Full-text available
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). ...
Article
Full-text available
This paper investigates the event-triggered asynchronous filter of positive switched systems with state saturation using linear programming and multiple Lyapunov functions. First, a filter is constructed for continuous-time positive switched systems. Under the asynchronous switching law, an error system is proposed with respect to positive switched systems and their filters, where the state saturation term is described in a polytopic form by virtue of the saturation property. A novel event-triggering condition is addressed based on a 1-norm inequality. Under the event-triggering condition, the error system is transformed into an interval system with lower and upper bounds. By using multiple Lyapunov functions and linear programming, the positivity and stability of the error system are achieved by considering the corresponding properties of the lower and upper bounds, respectively. Then, the event-triggered l1-gain filter and nonfragile filter are also proposed for the systems with disturbances. Moreover, the presented filter framework is extended to the discrete-time case. Finally, two examples are given to verify the effectiveness of the proposed filters.
... Definition 3 [12]: For a switching signal ( ) t  and each 2 ...
Article
Full-text available
In this paper, the finite-time stabilization for a class of switched positive systems with time-varying delays and actuator saturation under average dwell time switching is investigated. Through multiple copositive Lyapunov-Krasovskii functionals method, the sufficient conditions for finite-time bounded of continuous-time case are given, and the appropriate switching rules and state feedback controllers are presented together. Moreover, the convex hull technique is employed to deal with actuator saturation. Finally, an illustrative example is given to show the effectiveness of the proposed method.
... 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]. ...
Article
This article aims to design proportional-derivative (PD) controllers for interval positive linear systems in the continuous-time domain, which still remains a widely-discussed open problem in positive systems theory. The specific objective is to design a PD controller for the system with interval uncertain parameters and time-varying delay, which simultaneously ensures closed-loop system stability and preserves positivity. The work proposes a systematic framework, with the aim of finding PD controller gains for positive robust stabilization. The methodology and algorithm are presented first in the study, and the performance of such methods is instantiated by numerical examples.
... 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. ...
Article
Full-text available
Abstract This paper investigates a bumpless transfer control problem for uncertain switched positive linear time‐delay systems (SPLTDSs) with discrete and distributed delays. The innovation of this study is to develop a linear feedback controller and design a switching law that ensures lower controller signal bumps due to the switches of the SPLTDSs and also guarantee the L1–gain performance of the system. To this end, a bumpless transfer performance is proposed, and stabilisation constraints for SPLTDSs with synchronous switching are developed. Dwell time criteria are utilised in the stability analysis. Also, the results are improved to cover the uncertain SPLTDSs with interval or polytopic uncertainties. Then, asynchronous switching is considered, and the proposed method is extended to deals with asynchronous switching mode. Asynchronous switching means switches of the controller have lagged behind the subsystem switches. All the stability conditions are derived by multiple co‐positive Lyapunov–Krasovskii functional for synchronous and asynchronous switching conditions. At last, two numerical examples are presented to show the effectiveness of the proposed method compared with the leading existing solutions.
... [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. ...
Preprint
Full-text available
We consider the problem of simultaneous scheduling and resource allocation of an incoming flow of requests to a set of computing units. By representing each computing unit as a node, we model the overall system as a multi-queue scheme. Inspired by congestion control approaches in communication networks, we propose an AIMD-like (additive increase multiplicative decrease) admission control policy that is stable irrespective of the total number of nodes and AIMD parameters. The admission policy allows us to establish an event-driven discrete model, triggered by a locally identifiable enabling condition. Subsequently, we propose a decentralized resource allocation strategy via a simple nonlinear state feedback controller, guaranteeing global convergence to a bounded set in finite time. Last, we reveal the connection of these properties with Quality of Service specifications, by calculating local queuing time via a simple formula consistent with Little's Law.
... 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. ...
Article
Full-text available
This study focuses on the problem of event‐triggered control for positive switched systems without/with input saturation in both continuous‐time and discrete‐time contexts. First, a 1‐norm based event‐triggering mechanism is established for continuous‐time positive switched systems. By means of the matrix decomposition technique, an event‐triggered controller for the systems is designed by decomposing controller gain matrix into non‐positive and non‐negative components. Under the designed controller, the resulting closed‐loop systems are positive and stable. For the systems with input saturation, the saturation term is transformed into interval form under the event‐triggering condition. Then, an event‐triggered controller gain matrix and a cone domain of attraction gain matrix are designed for the corresponding interval systems in terms of linear programming, respectively. Furthermore, the presented approaches are extend to discrete‐time positive switched systems without/with input saturation. Compared with existing results on positive switched systems, the designed event‐triggered controller can reduce sampling frequency and save resources. Finally, two numerical examples are provided to verify the effectiveness of the obtained results.
... 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. ...
Article
Full-text available
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 deterministic way. Then, a linear event-triggering condition is constructed based on the state and error signal. Under the event-triggering condition, 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 gain of attraction domain. 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.
Article
Full-text available
In this article, the issue of positive L₁ filter design is investigated for positive nonlinear stochastic switching systems subject to the phase-type semi-Markov jump process. Many complicated factors, such as semi-Markov jump parameters, positivity, T-S fuzzy strategy, and external disturbance, are taken into consideration. Practical systems under positivity constraint conditions and unpredictable structural changes are characterized by positive semi-Markov jump systems (S-MJSs). First, by the key properties of the supplementary variable and the plant transformation technique, phase-type S-MJSs are transformed into Markov jump systems (MJSs), which means that, to an extent, these two kinds of stochastic switching systems are mutually represented. Second, with the help of the normalized membership function, the associated nonlinear MJSs are transformed into the local linear MJSs with specific T-S fuzzy rules. Third, by choosing the linear copositive Lyapunov function (LCLF), stochastic stability (SSY) criteria are given for the corresponding system with L₁ performance. Some solvability conditions for positive L₁ filter are constructed under a linear programming framework. Finally, an epidemiological model illustrates the effectiveness of the theoretical findings.
Article
This article focuses on the design of proportional-derivative (PD) controllers for positive linear systems in the continuous-time domain, which is a well-known open problem in positive systems theory. The main objective is to design a PD controller subject to interval gain variations, which simultaneously preserves the closed-loop system stability and positivity. The provided results fill the literature gap by considering both the PD controller structure and positivity. Although various feedback control strategies of positive systems have been developed recently, the work provides, for the first time, a systematic and tractable framework for finding nonfragile PD controller gains for positive stabilization. Finally, the theory and algorithm performance are evaluated and illustrated by numerical examples.
Preprint
Full-text available
The design and implementation of regulation motifs ensuring robust perfect adaptation are challenging problems in synthetic biology. Indeed, the design of high-yield robust metabolic pathways producing, for instance, drug precursors and biofuels, could be easily imagined to rely on such a control strategy in order to optimize production levels and reduce production costs, despite the presence of environmental disturbance and model uncertainty. We propose here a motif that ensures tracking and robust perfect adaptation for the controlled reaction network through integral feedback. Its metabolic load on the host is fully tunable and can be made arbitrarily close to the constitutive limit, the universal minimal metabolic load of all possible controllers. A DNA implementation of the controller network is finally provided. Computer simulations using realistic parameters demonstrate the good agreement between the DNA implementation and the ideal controller dynamics.
Article
This paper studies the parameter tuning problem of positive linear systems for optimizing their stability properties. We specifically show that, under certain regularity assumptions on the parametrization, the problem of finding the minimum-cost parameters that achieve a given requirement on a system norm reduces to a geometric program, which in turn can be exactly and efficiently solved by convex optimization. The flexibility of geometric programming allows the state, input, and output matrices of the system to simultaneously depend on the parameters to be tuned. The class of system norms under consideration includes the $H^2$ norm, $H^{\infty}$ norm, Hankel norm, and Schatten p-norm. Also, the parameter tuning problem for ensuring the robust stability of the system under structural uncertainties is shown to be solved by geometric programming. The proposed optimization framework is further extended to delayed positive linear systems, where it is shown that the parameter tunning problem jointly constrained by the exponential decay rate, the $\mathcal{L}^1$-gain, and the $\mathcal{L}^\infty$ -gain can be solved by convex optimization. The assumption on the system parametrization is stated in terms of posynomi
Preprint
Full-text available
In this paper, we present an impulse response identification scheme that incorporates the internal positivity side-information of the system. The realization theory of positive systems establishes specific criteria for the existence of a positive realization for a given transfer function. These transfer function criteria are translated to a set of suitable conditions on the shape and structure of the impulse responses of positive systems. Utilizing these conditions, the impulse response estimation problem is formulated as a constrained optimization in a reproducing kernel Hilbert space equipped with a stable kernel, and suitable constraints are imposed to encode the internal positivity side-information. The optimization problem is infinite-dimensional with an infinite number of constraints. An equivalent finite-dimensional convex optimization in the form of a convex quadratic program is derived. The resulting equivalent reformulation makes the proposed approach suitable for numerical simulation and practical implementation. A Monte Carlo numerical experiment evaluates the impact of incorporating the internal positivity side-information in the proposed identification scheme. The effectiveness of the proposed method is demonstrated using data from a heating system experiment.
Article
This article studies the exponential stability of continuous‐time switched positive systems consisting of unstable subsystems. Different from the existing results, both stabilizing and destabilizing switching behaviors act in the switching sequences. By employing multiple composite copositive Lyapunov functions, sufficient condition is derived to ensure the exponential stability of the system, which evaluates the ratio of stabilizing switching behaviors to compensate the state divergence caused by either unstable subsystems or destabilizing switching behaviors. Simulations demonstrate the effectiveness of the result.
Article
This paper is concerned with the event-triggered L1-gain control of a class of nonlinear positive switched systems. First, an event-triggering condition in the form of 1-norm is presented for the systems. By virtue of the event-triggering strategy, the original system is transformed into an interval uncertain system. An event-triggered L1-gain controller is designed by decomposing the controller gain matrix into the sum of nonnegative and non-positive components. Under the design controller, the resulting closed-loop systems are positive and L1-gain stable. The obtained approach is developed for the systems subject to input saturation. All presented conditions are solvable in terms of linear programming. Finally, two examples are provided to verify the effectiveness of the design.
Article
This article investigates the stability analysis and control design of a class of nonlinear positive Markovian jump systems with randomly occurring actuator faults and saturation. It is assumed that the actuator faults of each subsystem are varying and governed by a Markovian process. The nonlinear term is located in a sector. First, sufficient conditions for stochastic stability of the underlying systems are established using a stochastic copositive Lyapunov function. Then, a family of reliable L 1‐gain controller is proposed for nonlinear positive Markovian jump systems with actuator faults and saturation in terms of a matrix decomposition technique. Under the designed controllers, the closed‐loop systems are positive and stochastically stable with an L 1‐gain performance. An optimization method is presented to estimate the maximum domain of attraction. Furthermore, the obtained results are developed for general Markovian jump systems. Finally, numerical examples are given to illustrate the effectiveness of the proposed techniques.
Article
In this paper, the stability and estimator design issues are addressed for switched positive linear systems (SPLSs) with average dwell time (ADT). Based on a reverse timer which starts timing at the end of the activated time of a subsystem, a reverse-timer-dependent linear co-positive Lyapunov function (RLCLF) is constructed. With the help of the RLCLF approach, a new stability criteria is derived. Then, based on the stability criteria, a sufficient condition checking the existence of state estimator is derived for SPLSs with ADT. All the results can be relaxed into sum of square (SOS) program by using a SOS approximation approach. It’s shown that the results obtained by RLCLF approach are less conservative compared with those of the literature. Finally, the advantages of the results are illustrated within two examples.
Article
This paper proposes the event-triggered control of positive systems with state saturation both in discrete-time and continuous-time cases. A 1-norm based event-triggered mechanism is established for positive systems. Under the event-triggered mechanism, the error term between actual and sample states is transformed into interval uncertain form. Together with the properties of saturation, the systems with state saturation are transformed into interval uncertain systems and the corresponding lower and upper bounds of the system matrices are obtained, respectively. Using a linear co-positive Lyapunov function, the event-triggered controller and the auxiliary gain matrix of the domain of attraction are designed in terms of linear programming, respectively. Then, the systems with state and input saturation are also described via interval uncertain systems. An event-triggered controller is designed and thus the closed-loop systems are positive and stable under the designed controller. Furthermore, the presented event-triggered control approach is extended to the continuous-time case. Compared to existing control approaches, the event-triggered control can reduce energy consumption and increase the practicability. Finally, several numerical examples are given to illustrate the effectiveness of the proposed design.
Preprint
Full-text available
An outstanding problem in the design of distributed ledgers concerns policies that govern the manner in which users interact with the network. Network usability is crucial to the mainstream adoption of distributed ledgers, particularly for enterprise applications in which most users do not wish to operate full node. For DAG-based ledgers such as IOTA, we propose a user-node interaction mechanism that is designed to ensure the risk of a user experiencing a poor quality of service is low. Our mechanism involves users selecting nodes to issue their transactions to the ledger based on quality of service indicators advertised by the nodes. Simulation results are presented to illustrate the efficacy of the proposed policies.
Article
This paper investigates the event-triggered control of positive switched systems with randomly occurring actuator saturation and time-delay, where the actuator saturation and time-delay obey different Bernoulli distributions. First, an event-triggering condition is constructed based on a 1-norm inequality. Under the presented event-triggering scheme, an interval estimation method is utilized to deal with the error term of the systems. Using a co-positive Lyapunov functional, the event-triggered controller and the cone attraction domain gain matrices are designed via matrix decomposition techniques. The positivity and stability of the resulting closed-loop systems are reached by guaranteeing the positivity of the lower bound of the systems and the stability of the upper bound of the systems, respectively. The proposed approach is developed for interval and polytopic uncertain systems, respectively. Finally, two examples are provided to illustrate the effectiveness of the theoretical findings.
Article
Positive systems can be used as mathematical models for many practical systems, such as biological systems, communication networks, and interconnected systems. In this letter, we propose proximal alternating linearized minimization (PALM) and PALM-like algorithms to determine the nearest discrete-time linear positive system to a given system, with the same order as that of the considered system. Global convergence of the PALM algorithm to a critical point of the considered objective function is ensured by using the Kurdyka-Łojasiewicz and semi-algebraic properties. Numerical experiments are performed to compare the PALM and PALM-like algorithms.
Article
The article addresses the problems of robust stability and stabilization in exponential sense for uncertain switched positive systems (SPSs) under mode‐dependent dwell‐time constraints including mode‐dependent constant dwell time (MDCDT) constraint, mode‐dependent minimum dwell time (MDMDT) constraint, and mode‐dependent ranged dwell time (MDRDT) constraint. At first, the mixed time‐varying delay case is studied, and sufficient conditions of robust stability and stabilization in exponential sense are provided for uncertain SPSs with MDMDT, MDCDT, and MDRDT constraints, respectively. Next, the robust stability and stabilization issues are also studied for SPSs in delay‐free case. With mode‐dependent dwell‐time constraints, a new discretized linear copositive Lyapunov–Krasovskii functional technique is introduced in this article. Meanwhile, for some specific situations, the proposed conditions of robust stability and stabilization can be degenerated into the ones for SPSs in mode‐independent dwell‐time constraint case and in non‐uncertainty case, respectively. At last, we give five examples to explain the importance and effectiveness of the results.
Article
Full-text available
We present a model of a network of synchronised sources operating additive increase multiplicative decrease (AIMD) congestion control algorithms. We show: (i) that networks of such devices in the presence of a drop-tail bottleneck buffer may be modelled as a positive linear system; (ii) that such networks possess a unique stationary point; and (iii) that this stationary point is globally exponentially stable. We use these results to establish conditions for the fair co-existence of traffic in networks employing heterogeneous AIMD algorithms and to design a new protocol for operation over high-speed and long-distance links.
Conference Paper
Full-text available
High-speed networks with large delays present a unique environment where TCP may have a problem utilizing the full bandwidth. Several congestion control proposals have been suggested to remedy this problem. The existing protocols consider mainly two properties: TCP friendliness and bandwidth scalability. That is, a protocol should not take away too much bandwidth from standard TCP flows while utilizing the full bandwidth of high-speed networks. This work presents another important constraint, namely, RTT (round trip time) unfairness where competing flows with different RTTs may consume vastly unfair bandwidth shares. Existing schemes have a severe RTT unfairness problem because the congestion window increase rate gets larger as the window grows ironically the very reason that makes them more scalable. RTT unfairness for high-speed networks occurs distinctly with drop tail routers for flows with large congestion windows where packet loss can be highly synchronized. After identifying the RTT unfairness problem of existing protocols, This work presents a new congestion control scheme that alleviates RTT unfairness while supporting TCP friendliness and bandwidth scalability. The proposed congestion control algorithm uses two window size control policies called additive increase and binary search increase. When the congestion window is large, additive increase with a large increment ensures square RTT unfairness as well as good scalability. Under small congestion windows, binary search increase supports TCP friendliness. The simulation results confirm these properties of the protocol.
Conference Paper
Full-text available
The objective of this paper is to study the small-signal stability of congested TCP networks. The starting point for this analysis is a fluid model describing the interaction of TCP-controlled traffic with congested routers. Assuming integral action in the active queue management scheme, we compute the network's equilibrium point about which we determine the network's small-signal dynamics. Specializing to networks comprised of m links and m+1 sources, we study closed-loop stability, study network scaling, and evaluate robustness to variations in network parameters.
Conference Paper
Full-text available
Recent work has shown the benefit of using proportional feedback in TCP/AQM (transmission control protocol/active queue management) networks. By proportional feedback we mean the marking probability is proportional to the instantaneous queue length. Our earlier work (2001) relied on linearization of nonlinear fluid-flow models of TCP. In this work we address these nonlinearities directly and establish some stability results when the marking is proportional. In the case of delay-free marking, we show the system's equilibrium point to be asymptotically stable for all proportional gains. In the more realistic case of delayed feedback, we establish local asymptotic stability and quantify a region of attraction
Conference Paper
Full-text available
We use a previously developed nonlinear dynamic model of TCP to analyze and design active queue management (AQM) control systems using random early detection (RED). First, we linearize the interconnection of TCP and a bottlenecked queue and discuss its feedback properties in terms of network parameters such as link capacity, load and round-trip time. Using this model, we next design an AQM control system using the RED scheme by relating its free parameters such as the low-pass filter break point and loss probability profile to the network parameters. We present guidelines for designing linearly stable systems subject to network parameters like propagation delay and load level. Robustness to variations in system loads is a prime objective. We present no simulations to support our analysis
Article
Full-text available
In active queue management (AQM), core routers signal transmission control protocol (TCP) sources with the objective of managing queue utilization and delay. It is essentially a feedback control problem. Based on a recently developed dynamic model of TCP congestion-avoidance mode, this paper does three things: 1) it relates key network parameters such as the number of TCP sessions, link capacity and round-trip time to the underlying feedback control problem; 2) it analyzes the present de facto AQM standard: random early detection (RED) and determines that REDs queue-averaging is not beneficial; and 3) it recommends alternative AQM schemes which amount to classical proportional and proportional-integral control. We illustrate our results using ns simulations and demonstrate the practical impact of proportional-integral control on managing queue utilization and delay
Article
Full-text available
We investigate how congestion control can achieve efficient usage of network resources in the presence of heterogeneous communication delays between network users and resources. To this end, we consider a fluid flow model of network behavior. We study the stability of the system's behavior under small perturbations around the target equilibrium point (local stability). We establish several criteria for stability of certain linear delay-differential equations, via a technique which essentially reduces the question to studying stability of ordinary differential equations. These results are then used to derive sufficient conditions for local stability of the network congestion control problem. The same issue has been studied by Johari et al. (2001), where the authors propose a conjecture according to which local stability can be ensured in a distributed way. The correctness of the conjecture was established by Johari et al., only in degenerate cases where feedback delays coincide. Our results show that a modified form of the conjecture holds true for arbitrary feedback delays
Article
Full-text available
this paper we define the notion of traffic phase in a packet-switched network and describe how phase differences between competing traffic streams can be the dominant factor in relative throughput. Drop Tail gateways in a TCP/IP network with strongly periodic traffic can result in systematic discrimination against some connections. We demonstrate this
Article
Full-text available
Modern communication networks are able to respond to randomly fluctuating demands and failures by adapting rates, by rerouting traffic and by reallocating resources. They are able to do this so well that, in many respects, large-scale networks appear as coherent, almost intelligent, organisms. The design and control of such networks present challenges of a mathematical, engineering and economic nature. This paper outlines how mathematical models are being used to address current issues concerning the stability and fairness of rate control algorithms for the Internet.
Article
This article reviews the current transmission control protocol (TCP) congestion control protocols and overviews recent advances that have brought analytical tools to this problem. We describe an optimization-based framework that provides an interpretation of various flow control mechanisms, in particular, the utility being optimized by the protocol's equilibrium structure. We also look at the dynamics of TCP and employ linear models to exhibit stability limitations in the predominant TCP versions, despite certain built-in compensations for delay. Finally, we present a new protocol that overcomes these limitations and provides stability in a way that is scalable to arbitrary networks, link capacities, and delays.
Article
We study the spectrum of a positive matrix that arises in the study of certaincommunication networks. Bounds are given on the rate of convergence of thenetworks to the equilibrium condition.
Article
We derive decentralized and scalable stability conditions for a fluid approximation of a class of Internet-like communications networks operating a modified form of TCP-like congestion control. The network consists of an arbitrary interconnection of sources and links with heterogeneous propagation delays. The model here allows for arbitrary concave utility functions and the presence of dynamics at both the sources and the links.
Chapter
Most of today’s Internet traffic uses the Transmission Control Protocol (TCP) and it is therefore not surprising that modeling TCP’s behavior has been attracting the attention of academia and industry ever since this protocol was proposed.
Article
High-speed communication networks are characterized by large bandwidth-delay products. This may have an adverse impact on the stability of closed-loop congestion control algorithms. In this paper, classical control theory and Smith’s principle are proposed as key tools for designing an effective and simple congestion control law for high-speed data networks. Mathematical analysis shows that the proposed control law guarantees stability of network queues and full utilization of network links in a general network topology and traffic scenario during both transient and steady-state condition. In particular, no data loss is guaranteed using buffers with any capacity, whereas full utilization of links is ensured using buffers with capacity at least equal to the bandwidth-delay product. The control law is transformed to a discrete-time form and is applied to ATM networks. Moreover a comparison with the ERICA algorithm is carried out. Finally, the control law is transformed to a window form and is applied to Internet. The resulting control law surprisingly reveals that today's Transmission Control Protocol/Internet Protocol implements a Smith predictor for congestion control. This provides a theoretical insight into the congestion control mechanism of TCP/IP along with a method to modify and improve this mechanism in a way that is backward compatible.
Conference Paper
We propose a model for Stochastic Hybrid Systems (SHSs) where transitions between discrete modes are triggered by stochastic events much like transitions between states of a continuous-time Markov chains. However, the rate at which transitions occur is allowed to depend both on the continuous and the discrete states of the SHS. Based on results available for Piecewise-Deterministic Markov Process (PDPs), we provide a formula for the extended generator of the SHS, which can be used to compute expectations and the overall distribution of the state. As an application, we construct a stochastic model for on-off TCP flows that considers both the congestion-avoidance and slow-start modes and takes directly into account the distribution of the number of bytes transmitted. Using the tools derived for SHSs, we model the dynamics of the moments of the sending rate by an infinite system of ODEs, which can be truncated to obtain an approximate finite-dimensional model.
Conference Paper
Since TCP traffic is elastic, a fundamental issue is the behaviour of multiple TCP flows competing for bandwidth on a shared link. Motivated by the ubiquity of drop-tail queueing in current networks, our focus in this paper is on developing analytic models suited to characterising the throughput and fairness of competing TCP flows in drop-tail environments. Building on recent ideas from the theory of positive linear systems, we obtain simple, insightful closed-form expressions for throughput and fairness. The accuracy of these expressions is confirmed in extensive simulations across a range of network conditions. In particular, they are found to provide accurate estimate of mean fairness and throughput even when flows are not synchronised.
Article
In this paper tools are developed to analyse a recently proposed random matrix model of communication networks that employ additive-increase multiplicative-decrease (AIMD) congestion control algorithms. We investigate properties of the Markov process describing the evolution of the window sizes of network users. Using paracontractivity properties of the matrices involved in the model, it is shown that the process has a unique invariant probability, and the support of this probability is characterized. Based on these results we obtain a weak law of large numbers for the average distribution of resources between the users of a network. This shows that under reasonable assumptions such networks have a well-defined stochastic equilibrium. ns2 simulation results are discussed to validate the obtained formulae. (The simulation program ns2, or network simulator ,i s an industry standard for the simulation of Internet dynamics.)
Conference Paper
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.
Article
Under the assumption that queueing delays will eventually become small relative to propagation delays, we derive stability results for a fluid flow model of end-to-end Internet congestion control. The theoretical results of the paper are intended to be decentralized and locally implemented: each end system needs knowledge only of its own round-trip delay. Criteria for local stability and rate of convergence are completely characterized for a single resource, single user system. Stability criteria are also described for networks where all users share the same round-trip delay. Numerical experiments investigate extensions to more general networks. Through simulations, we are able to evaluate the relative importance of queueing delays and propagation delays on network stability. Finally, we suggest how these results may be used to design network resources
Article
The steady-state performance of a bulk transfer TCP flow (i.e., a flow with a large amount of data to send, such as FTP transfers) may be characterized by the send rate, which is the amount of data sent by the sender in unit time. In this paper we develop a simple analytic characterization of the steady-state send rate as a function of loss rate and round trip time (RTT) for a bulk transfer TCP flow. Unlike the models of Lakshman and Madhow (see IEE/ACM Trans. Networking, vol.5, p.336-50, 1997), Mahdavi and Floyd (1997), Mathis, Semke, Mahdavi and Ott (see Comput. Commun. Rev., vol.27, no.3, 1997) and by by Ott et al., our model captures not only the behavior of the fast retransmit mechanism but also the effect of the time-out mechanism. Our measurements suggest that this latter behavior is important from a modeling perspective, as almost all of our TCP traces contained more time-out events than fast retransmit events. Our measurements demonstrate that our model is able to more accurately predict TCP send rate and is accurate over a wider range of loss rates. We also present a simple extension of our model to compute the throughput of a bulk transfer TCP flow, which is defined as the amount of data received by the receiver in unit time
Article
A number of empirical studies of traffic measurements from a variety of working packet networks have demonstrated that actual network traffic is self-similar or long-range dependent in nature-in sharp contrast to commonly made traffic modeling assumptions. We provide a plausible physical explanation for the occurrence of self-similarity in local-area network (LAN) traffic. Our explanation is based on convergence results for processes that exhibit high variability and is supported by detailed statistical analyzes of real-time traffic measurements from Ethernet LANs at the level of individual sources. This paper is an extended version of Willinger et al. (1995). We develop here the mathematical results concerning the superposition of strictly alternating ON/OFF sources. Our key mathematical result states that the superposition of many ON/OFF sources (also known as packet-trains) with strictly alternating ON- and OFF-periods and whose ON-periods or OFF-periods exhibit the Noah effect produces aggregate network traffic that exhibits the Joseph effect. There is, moreover, a simple relation between the parameters describing the intensities of the Noah effect (high variability) and the Joseph effect (self-similarity). An extensive statistical analysis of high time-resolution Ethernet LAN traffic traces confirms that the data at the level of individual sources or source-destination pairs are consistent with the Noah effect. We also discuss implications of this simple physical explanation for the presence of self-similar traffic patterns in modern high-speed network traffic
Article
This article reviews the current transmission control protocol (TCP) congestion control protocols and overviews recent advances that have brought analytical tools to this problem. We describe an optimization-based framework that provides an interpretation of various flow control mechanisms, in particular, the utility being optimized by the protocol's equilibrium structure. We also look at the dynamics of TCP and employ linear models to exhibit stability limitations in the predominant TCP versions, despite certain built-in compensations for delay. Finally, we present a new protocol that overcomes these limitations and provides stability in a way that is scalable to arbitrary networks, link capacities, and delays
Article
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.
Article
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.
Article
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.
Article
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
Article
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