Combined diffusion approximation-simulation model of AQM's transient behavior

Abstract The article introduces an approach combining diffusion approximation and simulation ones. Furthermore, it describes how it can be used to evaluate active queue management (AQM) mechanisms. Based on the obtained queue distributions, the simulation part of the model decides on package losses and modifies the flow intensity sent by the transmitter. The diffusion is used to estimate queue distributions and the goal of the simulation part of the model is to represent the AQM mechanism. On the one hand, the use of the diffusion part considerably accelerates the performance of the whole model. On the other hand, the simulation increases the accuracy of the diffusion part. We apply the model to compare the performance of fractional order P I η controller used in AQM with the performance of RED, a well known active queue management mechanism.

[1]  T. Czachorski,et al.  Priority disciplines - a diffusion approach , 2008, 2008 23rd International Symposium on Computer and Information Sciences.

[2]  Donald F. Towsley,et al.  On designing improved controllers for AQM routers supporting TCP flows , 2001, Proceedings IEEE INFOCOM 2001. Conference on Computer Communications. Twentieth Annual Joint Conference of the IEEE Computer and Communications Society (Cat. No.01CH37213).

[3]  G. F. Newell,et al.  Queues with Time-Dependent Arrival Rates : II . The Maximum Queue and the Return to Equilibrium , 2016 .

[4]  Mariusz Ciesielski,et al.  A Numerical Method for Solution of Ordinary Differential Equations of Fractional Order , 2001, PPAM.

[5]  Tadeusz Czachórski,et al.  Diffusion Model of RED Control Mechanism , 2001, ICN.

[6]  Hitay Özbay,et al.  On the design of AQM supporting TCP flows using robust control theory , 2004, IEEE Transactions on Automatic Control.

[7]  K. Miller,et al.  An Introduction to the Fractional Calculus and Fractional Differential Equations , 1993 .

[8]  The Fractional Order PID Control of the Forced Air Heating System , 2019, Pomiary Automatyka Robotyka.

[9]  Leszczynski Jacek,et al.  A numerical method for solution of ordinary differential equations of fractional order , 2002, math/0202276.

[10]  D. Iglehart,et al.  Multiple channel queues in heavy traffic. II: sequences, networks, and batches , 1970, Advances in Applied Probability.

[11]  Kwan Lawrence Yeung,et al.  Nonlinear RED: A simple yet efficient active queue management scheme , 2006, Comput. Networks.

[12]  S. Niculescu,et al.  Stability analysis of some classes of TCP/AQM networks , 2006 .

[13]  Igor Podlubny,et al.  Fractional-order systems and PI/sup /spl lambda//D/sup /spl mu//-controllers , 1999 .

[14]  S. Heppell,et al.  Application of diffusion approximation for risk assessments of sea turtle populations. , 2009, Ecological applications : a publication of the Ecological Society of America.

[15]  Gordon F. Newell,et al.  QUEUES WITH TIME-DEPENDENT ARRIVAL RATES II - THE MAXIMUM QUEUE AND THE RETURN TO EQUILIBRIUM , 1968 .

[16]  Joanna Domańska,et al.  Fluid Flow Analysis of RED Algorithm with Modified Weighted Moving Average , 2013 .

[17]  Hisashi Kobayashi,et al.  Modeling and analysis , 1978 .

[18]  Michele Pagano,et al.  The Fluid Flow Approximation of the TCP Vegas and Reno Congestion Control Mechanism , 2016, ISCIS.

[19]  Tadeusz Czachórski,et al.  Stability and Dynamics of TCP-NCR(DCR) Protocol in Presence of UDP Flows , 2006, EuroNGI Workshop.

[20]  Erol Gelenbe,et al.  On Approximate Computer System Models , 1975, JACM.

[21]  Tsuyoshi Murata,et al.  {m , 1934, ACML.

[22]  Daniel Alejandro Melchor-Aguilar,et al.  Stability Analysis of Proportional-Integral AQM Controllers Supporting TCP Flows , 2007, Computación y Sistemas.

[23]  Tadeusz Czachórski,et al.  GPU Accelerated Non-integer Order PI^α D^β P I α D β Controller Used as AQM Mechanism , 2018, CN.

[24]  Gordon F. Newell,et al.  Queues with time-dependent arrival rates. III — A mild rush hour , 1968 .

[25]  D. Iglehart,et al.  Multiple channel queues in heavy traffic. I , 1970, Advances in Applied Probability.

[26]  Daniel Melchor-Aguilar.  On the asymptotic stability of proportional-integral AQM controllers supporting TCP flows , 2007 .

[27]  Deniz Üstebay,et al.  Switching Resilient PI Controllers for Active Queue Management of TCP Flows , 2007, ICNSC.

[28]  Y. Mishura.  Diffusion approximation of recurrent schemes for financial markets, with application to the Ornstein-Uhlenbeck process , 2015 .

[29]  Martin May,et al.  Analytic evaluation of RED performance , 2000, Proceedings IEEE INFOCOM 2000. Conference on Computer Communications. Nineteenth Annual Joint Conference of the IEEE Computer and Communications Societies (Cat. No.00CH37064).

[30]  Allen J. Scott,et al.  Applications of Queueing Theory , 1972 .

[31]  Rosdiazli Ibrahim,et al.  Frequency Response Based Curve Fitting Approximation of Fractional–Order PID Controllers , 2019, Int. J. Appl. Math. Comput. Sci..

[32]  Hisashi Kobayashi,et al.  Application of the Diffusion Approximation to Queueing Networks I: Equilibrium Queue Distributions , 1974, JACM.

[33]  Mohammed Atiquzzaman,et al.  DSRED: improving performance of active queue management over heterogeneous networks , 2001, ICC 2001. IEEE International Conference on Communications. Conference Record (Cat. No.01CH37240).

[34]  Tomasz Nycz,et al.  Transient states analysis — diffusion approximation as an alternative to Markov models, fluid-flow approximation and simulation , 2009, 2009 IEEE Symposium on Computers and Communications.

[35]  T. Kaczorek,et al.  Fractional Differential Equations , 2015 .

[36]  Tadeusz Czachórski,et al.  Diffusion Approximation Model of TCP NewReno Congestion Control Mechanism , 2020, SN Comput. Sci..

[37]  Ward Whitt,et al.  Heavy-Traffic Limits for Queues with Many Exponential Servers , 1981, Oper. Res..

[38]  G. F. Newell.  Queues with time-dependent arrival rates I—the transition through saturation , 1968 .

[39]  Tadeusz Czachórski,et al.  Diffusion Approximation as a Modelling Tool , 2011, Network Performance Engineering.

[40]  Tadeusz Czachórski,et al.  The use of a non-integer order PI controller with an active queue management mechanism , 2016, Int. J. Appl. Math. Comput. Sci..

[41]  Dilip D. Kandlur,et al.  Adaptive packet marking for maintaining end-to-end throughput in a differentiated-services internet , 1999, TNET.

[42]  W. Whitt.  Multiple channel queues in heavy traffic. III: random server selection , 1970, Advances in Applied Probability.

[43]  Tadeusz Czachórski,et al.  Self-similarity Traffic and AQM Mechanism Based on Non-integer Order PI^α D^β Controller , 2017, CN.

[44]  I. Podlubny.  Fractional-order systems and PIλDμ-controllers , 1999, IEEE Trans. Autom. Control..

[45]  Tadeusz Kaczorek.  Reachability and controllability to zero of positive fractional discrete-time systems , 2007, 2007 European Control Conference (ECC).

[46]  Thabet Abdeljawad,et al.  On Riemann and Caputo fractional differences , 2011, Comput. Math. Appl..

[47]  Shinya Nogami,et al.  Evaluation of diffusion approximation for the G/G/1 queuing model , 2010, 8th Asia-Pacific Symposium on Information and Telecommunication Technologies.

[48]  Tadeusz Czachórski,et al.  The Influence of the Traffic Self-similarity on the Choice of the Non-integer Order PI ^\alpha Controller Parameters , 2018, ISCIS.

[49]  Mario Marchese,et al.  High performance TCP / IP networking - concepts, issues, and solutions , 2004 .

[50]  Yang Liu,et al.  Active Queue Management Algorithm for TCP Networks with Integral Backstepping and Minimax , 2019, International Journal of Control, Automation and Systems.

[51]  Abdon Atangana,et al.  Fractional derivatives with no-index law property: Application to chaos and statistics , 2018, Chaos, Solitons & Fractals.

[52]  Joanna Domańska,et al.  Fluid flow approximation of time-limited TCP/UDP/XCP streams , 2014 .

[53]  J. F. Gómez‐Aguilar,et al.  Decolonisation of fractional calculus rules: Breaking commutativity and associativity to capture more natural phenomena , 2018 .

[54]  Dariusz Rafal Augustyn,et al.  A RED modified weighted moving average for soft real-time application , 2014, Int. J. Appl. Math. Comput. Sci..

[55]  Yangquan Chen,et al.  Fractional order [proportional derivative] controller for a class of fractional order systems , 2009, Autom..

[56]  I. Podlubny.  Fractional-Order Systems and -Controllers , 1999 .

[57]  Van Jacobson,et al.  Random early detection gateways for congestion avoidance , 1993, TNET.

[58]  Silviu-Iulian Niculescu,et al.  Computing non-fragile PI controllers for delay models of TCP/AQM networks , 2009, Int. J. Control.

[59]  D. Iglehart.  Weak convergence in queueing theory , 1973, Advances in Applied Probability.

[60]  José Francisco Gómez-Aguilar,et al.  Series Solution for the Time-Fractional Coupled mKdV Equation Using the Homotopy Analysis Method , 2016 .

[61]  YangQuan Chen,et al.  Fractional order control - A tutorial , 2009, 2009 American Control Conference.

[62]  Vishal Misra,et al.  Fluid Methods for Modeling Large, Heterogeneous Networks , 2005 .

[63]  R. Agarwal,et al.  Fractional Sums and Differences with Binomial Coefficients , 2013 .

[64]  Zhang Hanqin,et al.  MULTIPLE CHANNEL QUEUES IN HEAVY TRAFFIC , 1990 .

[65]  Tadeusz Czachórski,et al.  AQM Mechanism with the Dropping Packet Function Based on the Answer of Several PI^α Controllers , 2019, CN.

[66]  Gordon F. Newell,et al.  QUEUES WITH TIME-DEPENDENT ARRIVAL RATES , 2016 .

[67]  Tadeusz Czachórski,et al.  Transient States of Priority Queues - A Diffusion Approximation Study , 2009, 2009 Fifth Advanced International Conference on Telecommunications.

[68]  H. D. Miller,et al.  The Theory Of Stochastic Processes , 1977, The Mathematical Gazette.

[69]  Tadeusz Czachórski,et al.  Comparison of AQM Control Systems with the Use of Fluid Flow Approximation , 2012, CN.