EP1264439A1

EP1264439A1 - Improved monitoring and simulating of complex systems, in particular of flow and congestion mechanisms and controls in communication networks

Info

Publication number: EP1264439A1
Application number: EP01909920A
Authority: EP
Inventors: François BACCELLI; Dohy Hong
Original assignee: Institut National de Recherche en Informatique et en Automatique INRIA
Current assignee: Institut National de Recherche en Informatique et en Automatique INRIA
Priority date: 2000-03-01
Filing date: 2001-02-28
Publication date: 2002-12-11
Also published as: FR2805945A1; JP2003526262A; FR2805945B1; WO2001065772A1; CA2401312A1; US20030161266A1

Abstract

The invention concerns a device and a method assisting in monitoring and/or simulating a complex system, in particular a telecommunication network. It consists in storing data in matrix form, of dynamically variable structure, with iterative multiplication in max-plus algebra of a current matrix of network data and an instantaneous matrix of the network parameters.

Description

WO 01/65772 PCT / FR01 / OOS79

Advanced monitoring and simulation of complex systems, including mechanisms and flow and congestion controls in communication networks

The invention relates to the monitoring and simulation of complex systems.

In the context of flow control and congestion in communication networks, especially internet type, a fine analysis of the speed offered is desired to estimate the respective influences of the network parameters.

With the development of online communication techniques and the problems encountered, in particular congestion, various flow control and congestion protocols have emerged, notably TCP control (from the English "transmission control protocol").

Methods of analyzing these protocols are known. Among these known methods, a method based on a mathematical expression adapted to the throughput in a TCP type protocol has enabled an analytical approach to control. The principle on which this process is based is described in particular in:

M. Mathis, J. Semske, J. Mahdavi and T. Ott, "The Macroscopic Behavior of the TCP Congestion Avoidance Algorithm", Computer Communication Review, 27 (3), July (1997).

^This process, although promising, has shown its limits in practical applications, in particular the fact that the random nature of the traffic is only partially taken into account, or that it requires an approximation of all the nodes of the network to a single equivalent, virtual node.

Another known process, taking better account of the stochastic approach, made it possible to grasp the randomness traffic. This more recent process stems from the principle described in the following works:

Padhye, J., Firiou, V., Towsley, D. and Kurose, J. "Modeling TCP throughput: a simple model and its empirical validation", Proc. of ACM SIGCOMM (1998)

Padhye J., Firiou V., Towsley D., "A Stochastic Model of TCP Reno Congestion Avoidance and Control", Technical Report, 99-02, CMPSCI, Univ. of Massachusetts, Amherst (1999).

However, this method has also shown its limits, in particular the fact that it still requires an approximation of all the nodes of the network to an equivalent node.

The present invention improves the situation.

According to a different approach, the invention proposes to use a representation in so-called "max-plus" algebra of complex systems, such as communication networks and in particular flow and congestion control.

To obtain details of the mathematical principles on which such a representation is based, reference may be made to the following work:

F. Baccelli, G. Cohen, G.J. Olsder, and J.P. Quadrat, Synchronization and Linearity, iley (1992).

Globally, the scalar max-plus algebra is a semi-ring on the real line where the addition becomes the "max" function (largest value among a set of values) and the multiplication, the "plus" function (sum ). The use of max-plus algebra allows the calculations of a complicated system to be reduced to a simple matrix representation.

The Applicant has shown and verified in practice that the use of max-plus algebra adapts very satisfactorily. monitoring and simulating systems such as a communications network, whether controlled or not. It makes it possible in particular to overcome the random nature of the parameters of the network, while considering a plurality of nodes. In addition, the Applicant has shown that the representation of a network using a TCP protocol is linear in max-plus algebra, which makes it possible, in practice, to apply simple data processing.

The present invention therefore relates to a device for assisting in monitoring and / or simulating a complex system, in particular a communication network.

According to an important first of the invention, the device comprises:

a memory for storing first data representative of network parameters, as well as for receiving at least second data representative of events in the network, a portion of said memory being reserved for storing data in matrix form,

a calculation module, capable of performing, on at least two matrices of dynamically variable structure, an operation forming a product according to the algebra known as MAX-PLUS,

a modeling module for constructing at least a first matrix and a current matrix respectively as a function of the first data and of the second data, according to a chosen model, and

a pilot module for repeatedly applying the first matrix and the current matrix to the calculation module, the product matrix obtained becoming a new first matrix.

Preferably, the first data includes information on the network topology, such as the number of routers crossed by the connection to be monitored or simulated, the properties of these routers (memory sizes called "buffers", or other), statistical properties. traffic offered in the network, etc. According to another advantageous characteristic of the invention, the modeling module comprises:

- a static modeling sub-module, to build the first matrix as a function of the first data, and - a dynamic modeling sub-module, to build at least one current matrix as a function of the second data.

Advantageously, the device of the invention is capable of processing matrices comprising dynamically variable coefficients, including at least the above-mentioned current matrix. In the processing carried out by the device according to the invention, the matrices constructed are advantageously of the same dimension.

These second data preferably include information relating to losses in the network, to transverse flows in the network, with respect to a controlled connection to be monitored or simulated, to congestion in the network, or even to timeouts in the so-called "time-out" network.

According to another advantageous preferential characteristic, the product matrix obtained is a vector represented by a matrix with a single column, which makes it possible to limit the treatments and their duration. The first matrix, representative of the parameters of the network, is advantageously structured at the outset as a vector.

According to a second important characteristic of the invention, the product matrix obtained is representative of a throughput in the network associated with the connection to be monitored or simulated, of an average throughput in the network, or even of fluctuations in a throughput snapshot in the network.

According to a third important characteristic of the invention, the modeling module is arranged to successively build a plurality of matrices, in number corresponding substantially to the number of packets in the network. The model chosen preferably includes consideration of the variable size of a window used to control the number of packets in the network.

It can be a network controlled by a TCP type protocol, typically comprising discipline routers of the "first come first served" type, or alternatively routers with discipline of the FQ type (from the English "weighted fair queu - ing "). The TCP protocol controlling the network can also be based on a Reno model or a Tahoe model, as we will see below.

The network service can be deterministic, or even random, as we will see in detail below.

The present invention also relates to a method for assisting in monitoring a complex system, in particular a communication network. Such a method generally comprises the following steps: a) obtaining first data representative of parameters of the network, b) constructing a first matrix, according to a chosen model, as a function of said first data, c) receiving, at a chosen time, at least second data representative of events in the network, d) construct at least one second matrix of dynamically variable structure, according to the chosen model, as a function of the second data, and e) perform on said matrices a product forming operation according to algebra called MAX-PLUS, the product matrix obtained being representative of the state of the network at said selected time.

If it is desired to follow a temporal evolution of the state of the network at selected times, the method advantageously includes the following additional step: f) repeating, at selected times, steps c), d) and e) , while the product matrix obtained becomes the first matrix after step e). The present invention also relates to a method for simulating a complex system, in particular mechanisms and controls of flow and congestion in a communication network. This process generally comprises the following steps: a) obtaining first data representative of parameters specific to the network, b) constructing a first matrix, according to a chosen model, as a function of said first data, c) simulating events in the network and predicting at less second data representative of said events, d) construct at least a second matrix according to the chosen model, as a function of said second data, and e) perform on said matrices a product forming operation according to the so-called MAX-PLUS algebra, the product matrix obtained being representative of a state of the network undergoing said events.

To predict an evolution of the state of the network as a function of the events it undergoes, this method advantageously comprises the following additional step: f) repeating, for successive events, steps c), d) and e) , while the product matrix obtained becomes the first matrix after step e).

Other characteristics and advantages of the invention will appear on examining the detailed description below, and the attached drawings in which:

FIG. 1A schematically represents a device within the meaning of the present invention,

FIG. 1 schematically represents a number K of tandem queues in a network, with flow control,

FIG. 2A represents an example of a step-by-step evolution of daters and of the size of a window used to control the number of packets in the network, FIG. 2 represents interactions between several packets in the network,

FIG. 3 represents a variation of the throughput (curve in solid lines), obtained by simulation, in a TCP protocol network based on the Tahoe model without the exponential phase,

FIG. 4 illustrates a graphic interpretation of the asymptotic rates in a TCP protocol network based on the Reno model with deterministic delays,

FIG. 5 represents a variation in the throughput, obtained by simulation and showing a decrease in the throughput in the event of random losses in the network,

FIG. 6 represents a variation of the throughput (curve in solid lines), obtained by simulation, in a TCP protocol network based on a Reno Markovian model,

FIG. 7 represents comparative variations in throughput, obtained by simulation, in TCP protocol networks based respectively on a Reno deterministic RD (solid lines), Reno markovian RM (long dashed lines), Tahoe deterministic TD ( medium dotted lines) and Tahoe markovian TM (short dotted lines),

FIG. 8 represents a variation of the throughput (curve in solid lines), obtained by simulation, in a TCP protocol network based on a Tahoe model with exponential phase,

FIG. 9 represents comparative variations in bit rates, obtained by simulation, in a TCP protocol network based on a Tahoe model with services s ₃ and s ₈ respectively constant and equal to 1, and - Figure 10 schematically shows a network with its queues and routers.

Annex I includes the formulas and equations E1 to E28 to which the detailed description below refers.

Annex II includes the bibliographic references [1] to [13] indexed in square brackets in the description below.

The drawings, appendices and description below essentially contain elements of a certain nature. They can therefore not only serve to better understand the present invention, but also contribute to its definition, if necessary.

Referring to Figure 1, the device is in the form of a computer comprising a central unit UC provided with a microprocessor μP which cooperates with a motherboard CM. This motherboard is connected to various equipment, such as a COM communication interface (of Modem type or other), a ROM read only memory and a RAM working memory (random access memory). The motherboard CM is also connected to a graphical interface IG, which controls the display of data on an ECR screen that the device includes. There are also provided input means, such as a CLA keyboard and / or an input device called "mouse" SOU, connected to the central unit UC and allowing a user interactivity with the device.

The ROM memory, or even the RAM memory stores the first aforementioned data, representative of the parameters of the network (topology, properties of the routers, etc.). In the example, the RAM memory receives the aforementioned second data, representative of events in the network (transverse flows, congestion, losses, etc.). As part of a network monitoring aid, these second data can be received by the COM communication interface. In the context of a simulation, the acquisition of these second data can be performed by a calculation based on a simulation model, as will be seen below.

The RAM memory at least can be addressable as a function of rows and columns of matrices and thus allow storage of data in matrix form.

The ROM memory includes a MOD modeling module which, in cooperation with the microprocessor μP, makes it possible to construct the aforementioned first matrix and a current matrix, respectively as a function of the first data and of the second data, according to a chosen model which will be seen further.

The ROM memory comprises a module CAL which, in cooperation with the microprocessor μP, makes it possible to carry out, on at least two matrices of dynamically variable structure, a product forming operation according to the MAX-PLUS algebra. In the example, the term "dynamically variable structure matrices" means matrices whose coefficients at least are dynamically variable. Advantageously, the models which will be described below make it possible to reduce the constructed matrices (and more particularly the current matrices) to matrices of which only the coefficients are dynamically variable.

The MOD modeling module then comprises: - a static modeling sub-module ST, to build the first matrix as a function of the first data, and - a dynamic modeling sub-module DYN, to build, as a function of the second data, at minus a current matrix whose coefficients are dynamically variable.

The ROM memory furthermore comprises a PIL module which, in cooperation with the microprocessor μP, makes it possible to apply repeatedly the first aforementioned matrix (comprising the first data) and the current matrix (comprising the second data) to the calculation module CAL. The product matrix obtained is stored βn memory and becomes a new first matrix. It can then be multiplied (in the MAX-PLUS algebra) to another current matrix, comprising new second data representative of new events in the network.

Various approaches have been proposed to apprehend the key properties of the TCP window type flow control mechanism based in particular on heuristic considerations, simulations, fluid approximations or even reference arkovian analyzes [10,11,1,12 , 13,14]. All analytical models are based on the reduction of the network to a single node representing the bottleneck [9].

Furthermore, it has recently been demonstrated that window flow control of a multidimensional network admits a max-plus linear representation when the window size is constant (reference [5]). Here, the Applicant is mainly interested in models which combine the adaptive control mechanism of TCP and a multidimensional network made up of several routers in series. The dynamics of such a controlled xnet are advantageously described at the "packet" level via iterations of matrix products in max-plus algebra. We consider here both the case where the packet transmission times are deterministic and the various stochastic models which have been used in the prior art, in particular the cases where there are random losses in addition to the losses due to exceeding the capacity of the buffer memories (or "buffers"), and advantageously the case where the packet transmission times are randomly disturbed by other traffic.

Many key aspects of the protocol can be represented: congestion losses, random losses, propagation delays (or "time-outs") or delays due to expectations or the flow control mechanism, etc. As will be seen below, this approach makes it possible to obtain explicit formulas for the maximum allocated speed, when the disturbances are deterministic or random. These formulas are asymptotically compatible with the known formulas when the maximum size of the window tends towards infinity.

In addition, the present invention makes it possible to analyze instantaneous and random fluctuations in bit rate, which can be useful for estimating the quality of service offered to a connection. It also adapts well for efficient simulations of the dynamics of a TCP session operating under control, end-to-end, on a large network.

Hereinafter, a max-more general representation is given of the basic model used by the present invention. Next, deterministic services with a deterministic and then Markovian evolution of the window size are described. The Applicant has shown that for these deterministic models, the speed depends only on the round trip time or RTT time and the bandwidth. Variants and extensions of the deterministic model have also been implemented, in particular:

- the case of random losses in addition to loss of leave;

- detailed monitoring of the buffer size and congestion losses;

- the case of random services;

- the case in the presence of time-outs.

The extensions of the deterministic model all lead to analytical formulas or to new principles of simulations based on the calculation of the product of a large number of matrices. In particular, the Applicant has shown that the cost of the simulation by this approach to the transmission of packets on K n routers is advantageously 2n (KW _max) ^2, where ^w _a ^is ^a maximum window size.

The scalar max-plus "algebra" is a semi-ring on the real line where the addition is replaced by max (denoted Φ) and multiplication by plus (denoted ®). The law ® is distributive with respect to the law θ, which allows us to extend the usual concepts of linear algebra to this framework, and in particular the theory of matrices. This semi-ring is noted (R _ma x _f Φ _f ®) _ι where R _j - ^ x is the real line completed with minus infinity, which is the neutral element of θ. Thereafter, the set of square matrices of dimension d in this algebra is noted (R _m ' _aχ , θ, ®), where the two operations θ and, when applied to matrices, are linked by the relation El of Annex I.

For more details on this algebra, which is also used for QoS guarantees in networks, one can refer to [2] or [6] in the references given in appendix II.

In a PAPS network, of the “first come, first served” type, with K queues in tandem, the n-th client arriving at station i receives a service σ (n). In the context of TCP, this network models a single source sending packets to a single destination, through a path composed of K routers. The variable σ ^ n) is the random delay caused by the transverse traffic (the other users) at the level of router i on the n-th packet. This delay does not include wait times, but only the slowdown in server speed due to the presence of cross traffic. The propagation delay of the n-th packet between the routers i to j is noted below d ^ _j (n).

The flow of the input stream is controlled by a dynamic window (n), the size of which is equal to the total number of packets sent by the source at a given instant and which have not reached the destination (or more precisely the packets which have not yet been "acquitted").

The size of the window has a general evolution defined by the recurrence E2 given in the appendix. _In the equations E2, ACK (n) is the flow / congestion control signal giving information on the state of the system at time n and where f is a function specified below.

For example, ACK (n) = l if no congestion (or no packet loss) is observed by the n-th packet, otherwise ACK (n) = 0. Thereafter and in certain cases, f is supposed to also depend on s, which is, for the TCP models, the threshold which separates the phase of exponential growth (called "slow-start phase") from the phase of linear growth (called "congestion-avoidance phase").

The recurrence relation (given in appendix by E3) defines a reference size of the window. The effective size is then defined as the whole part of the reference size, according to equation E3 in the appendix.

The maximum size of the window is assumed to be finished and noted w ^* (see equation E4 in the appendix).

In addition, the change in window size can be broken down into two phases which depend on ACK (n) as follows: a growth phase defined by E51 and a decrease phase defined by E52.

In the model described here, ACK (n) (worth 0 or 1) is the acknowledgment signal of the n-th packet which detects a congestion state or a packet loss. The usual examples of the ideal window evolution policy are given by equations E61 to E67 in the appendix, in which the real α is between 0 and 1, strictly. The relations E61 and E62 are given respectively for exponential and linear phases.

In the following examples, the value of α is set to 1/2.

The queue at the entrance is considered below as saturated (even if the unsaturated case can be easily integrated, as will be seen below). The network then behaves as a closed network and its speed gives the maximum rate at which the source can send packets while keeping a stable input buffer. We can refer to article [5] given in the appendix of references for more details on this case.

We refer to Figure 1 showing K files in tandem, with flow control. x ^ (n) is the date on which the n-th packet arriving at router i begins its service on this router.

y _i (n) is the date on which the n-th client leaves router i. By the Formulas E7 of the appendix, we deduce the vector Z (n), called "daters vector" thereafter.

The variables Mi, i belonging to {1, ..., w ^* }, are given matrices of (lζ _ιax ). e is the matrix of (^ _a x) of which all the elements are equal to minus infinity. Below, we note (M _x | M ₂ ! .. • | M _W ^* ) the matrix of (R ^ _a ') defined by blocks of size kx k. The Applicant has shown that all the blocks are equal to the matrix e of (R ^ _a x) 'except for the first row of blocks which is equal to M _{1 #} M ₂ , ..., M _W ^* . The Applicant has also shown that if initially the system is empty, the evolution of the date vector Z (n) is given by the max-plus linear recurrence E8.

In the formula E8, D is the matrix of dimension Kw ^{* of} which all the elements are equal to minus infinity except those of the form D _{κ + ii} (with i belonging to {1, ..., k (w ^* - l)}) which are all equal to 0.

We can define the Lyapunov exponent of the station k by the limit of the expression y _k (n) / n when n tends towards more infinity. Because of the monotony, it is clear that this limit is independent of k. This property is more generally true under the irreducibility hypothesis defined in reference [8]. At the level of the representation described here, the error control is not taken into account and no difference is marked between the original packets and the retransmitted packets. In particular, no distinction has yet been made between "send rate", throughput "or" goodput "within the meaning of reference [13].

The equation E8 is, in this realization, the basis of the algebraic simulation scheme to which it was alluded above. As the matrices A _Wn _ ₁ (n) are of dimension Kw ^* , and as only the matrix-vector products are necessary, we can simulate the controlled transmission of n packets through the network in 2n (Kw ^* ) ² operations on a single processor.

FIG. 2A represents an example of an explicit step-by-step evolution of the daters and of the size of the window (K = 5 with (σ _λ , ..., σ ₅ ) = (1, 1, 2, 1, 1) and w ^* = 4). The evolution of the window size is deterministic (TCP Tahoe without the "slow start" phase, ie (w _χ , w ₂ , w ₃ , ...) = (1, 2, 2, 3, 3, 3, 4, 1, 2, 2, ...)).

For example, package # 5 corresponds to the size of window 3, - which is noted 5 (3) in Figure 2A. Thus, just after the transmission of packet # 5, three packets in the network have not yet been "acknowledged". As Figure 2A shows, the delays imposed on packets (see for example packet # 6) in internal routers can be quite complicated.

For the simplicity of the presentation, we first consider the case where all the propagation delays d _{i? J} (n) are worth 0, and then an adaptation to cover the case of non-zero propagation delays.

In a first approximation, the services σ _± (n) are all deterministic and constant (σ ^). Congestion is normally detected by the formula E9 policy. PR

16

This detection can be interpreted as giving the service rate 1 / σ ^* of the network bottleneck and S, a round trip time RTT. Congestion is therefore detected when the average emission rate (w _n / RTT) reaches the rate of the bottleneck. For the static window model, w ^* is the optimal window size given by the product: bandwidth x delay (reference [9]).

The Applicant has shown that, under these conditions, w ^* is given by the formula ElO and that w _n becomes periodic with a period T which can be decomposed into t ₁ + t ₂ + ... + t _{w *} , where ti is the number of occurrences of w _n = i during a period.

By first considering the TCP Tahoe model without the exponential phase, under the E9 policy, the Eli formulas can be deduced from the appendix. The matrix H (formula E12) is irreducible and its eigenvalue is equal to nS + σ ^* . The Applicant has also shown that if all the delays are deterministic, this eigenvalue is given by the formula E13.

Consequently, the flow depends on the delays (σ ₁ , ... σ _k ) only through S and σ ^* and this flow is given by the formula E14. Thus, the asymptotic flow becomes l / (2σ ^* ) when w ^* tends towards infinity.

In Figure 2 (where we assume that w ^* = 6) the step-by-step evolution of the entry date y ₀ (n) and the departure date y _k (n) (of package # n) are represented. The above algebraic properties are interpreted as follows.

Before congestion detection, the packets sent behave as if there was no interaction between them, except for the couple of packets sent at the same time when the size of the window increases by one. For these latter packets, the second packet always leaves station K with a delay of σ ^* relative to the first. One can thus read directly the eigenvalue 5S + 6σ ^* on the graphic evolution. The saturation rate for TCP Tahoe with the exponential phase is given by the formula E15 and the asymptotic rate (w ^* infinite) is given by E16.

Figure 3 shows the appearance of the ratio n / y ₄ (n) (solid lines) and n (dotted lines) under the following conditions: TCP Tahoe without the exponential phase with four tandem queues (σ ₁ = 3.2 ; σ ₂ = 4.61; σ ₃ = 2.7; σ ₄ = 4.61 and w ^* = 4). The saturation flow rate is equal to 0.140084 using the formula E14.

A deterministic periodic evolution of TCP Reno was considered in reference [11] to obtain a heuristic value for the throughput. The max-plus representation above leads to a new Formula which refines that of [11].

If all the delays are deterministic, the saturation rate depends only on S and σ ^* and it is given by the formulas E17 in the appendix.

Referring to FIG. 4, the asymptotic flows (case 1, case 2, case 3) are obtained in a fairly intuitive manner from the fluid approximation of the evolution of the size of the window.

We call d ₀ = l / σ ^* the flow corresponding to w _n = w ^* (case 0); when w _n increases linearly from 1, the volume of the flow, which is proportional to the integral of W (t) over a period, is indeed 1 / 2.d _Q (case 1); when w _n increases linearly from w ^* / 2, the volume of the flow decreases by a factor of 3/4 (case 2).

The well-known formula for the throughput of a TCP connection as a function of the loss rate p _loss and the RTT round trip time is of the form given by the reference [11] and transcribed by:

flow = c ₀ /(RTT.(p _loss ) ^{1 2} ) where c ₀ is a real constant. For the deterministic model above, with RTT = S, the values of P _loss ^{according to the} different cases (Cl, C2, C3) are expressed according to the formulas E18.

Therefore, for large values of w ^* (or small values of p _loss) ^ ^has asymptotic formula coincides well with that of the reference [11].

The Applicant has verified that all the above results are true for propagation delays of constant ± _j provided to replace the value of S by one of the formula 1 E19 Schedule.

For deterministic models, the speed obtained can be compared to that simulated by a simulator (for example of the NS type) by choosing an arbitrary packet size and taking a service speed of the router i corresponding to σ _i . The differences between the flow rates obtained by the NS simulation and by the above formulas can only come from differences in the mechanisms of detection of the loss by congestion, or from the fact that the integer part of W (n) in f ( as in reference [9]) is the only one considered here, while NS uses (n). Indeed, for all deterministic models with the same periodic evolution of W (n), the evolutions are exactly the same. For example, a simulated TCP protocol on NS with an ftp source gives, for K = 10, a packet size equal to 1250 (40 for acknowledgments), and a buffer size equal to 2. All the deadlines of _i? are equal to 0.1ms except d _κ0 which is equal to 1ms. The service speed is (10,5,4,2,5,4,5,5,4,5,5) Mb / s for the links 0-1, .., 9-10, 10-0. At t = 100s, the NS simulator gives 152.27 packets / s. For this example, S = 25.5 ms and σ ^* = 5 ms; therefore, by the formula E15 we would have 134 packets / s. However, w ^* is actually equal to 7 in the NS simulation instead of 6 in the model described here. An RTT is necessary to detect a so-called "triple-acks" delay on this NS model. By evaluating the flow with the formula E15 and w ^* = 7, we have 152.55 packets / s. Finally, for all deterministic models, the sequence {w _n } is deterministic, and we can therefore calculate the evolution of the emission rate exactly from the products of corresponding max-plus matrices.

We describe below a Markov model with deterministic services.

Services are all considered deterministic. Now, W (n) is supposed to evolve independently of the other elements of the network by the Markov transition matrix P given by the following formulas:

- for all n integers,

. if w (n) <w _maχ , w (n + l) = w (n) with probability p ₀ h (w (n)) with probability p ₊ g (w (n)) with probability p_

. if (n) = _max , (n + l) = g (w (n)) with probability 1,

where h (W (n))> (n), g ((n)) <(n) and where _maχ is arbitrary.

If all the services are deterministic, for any Markovian evolution of (n) with the above structure, the flow depends on the delays only through S and σ ^* . In addition, the Lyapunov exponent γ (the inverse of the flow) is of the form given by the formula E20 in which π is the stationary probability of the set A.

If both transitions are possible, w ^* can be defined by policy E9 and ^w _max = w ^* , with policy E21 in the appendix. This model can be compared to that of reference [12], where an overall probability of loss is used to capture at the same time the time-outs (TO) due to packet losses and the triple-duplication of ACKs (TD) due to congestion. In the model described here, these two mechanisms are on the contrary described separately; the loss of packets which generate the TO constitute an iid sequence (independent and identically distributed), independent of all the other elements of the network and are captured by the parameter p_; losses due to congestion are captured by the w ^* parameter.

We show that for TCP Tahoe with a Markovian evolution as above (that is to say with E21 and g ((n)) = l), the bit rate is given by the formulas E22.

The Applicant has verified that p_ = 0 corresponds to the case of deterministic TCP Tahoe (without the exponential phase). Furthermore, if the packet losses are linked to congestion in a way that is on average similar to that of the case of deterministic TCP Tahoe, the parameter C _{0 is} worth 60%. Therefore, the performance degradation can be significant (approximately 15%) when we pass from the deterministic model (where C _o ≈0.71) to the Markovian model. The impact of TOs on performance has already been noted, for example in references [13,10]. The predominant influence of p- on the overall probability of loss can be quantified from the above analytical model as follows.

The overall probability of loss for this model is given by:

μ (w ^* ) + p_ (l-μ (w ^* )) = μ (l) where μ (w ^* ) is the loss due to congestion and p_ (l-μ (w ^* ) is the loss due to TO .

For μ (l) fixed, Figure 5 shows the decrease in flow in p_ (σ ^* = l and S = w ^* -1).

We see here to what extent the impact of losses by TO is preponderant compared to losses due to congestion.

By applying the above models to the buffers (or "buffers") of a network, similar results are obtained for models of the TCP Reno type and also for models based on a Markovian change in window size with three (or more) transitions.

For example, let K = 4 with σ _χ = 3.2; σ ₂ = 4.61; σ ₃ = 2.7; σ ₄ = 4.61, and w _* = 4. Figure 6 shows the evolution of n / y ₄ (n) and w _n for TCP Reno markovien with (p ₊ / PorP-) ⁼ (0.8,0.1,0.1).

Figure 7 shows the bit rates of TCP Reno and TCP Tahoe in the deterministic and Markovian cases ((p ₊ , p ₀ , p_) = (0.8, 0.1, 0.1)). In the deterministic case, the flows are equal to 0.140084 (Tahoe) and 0.172166 (Reno). The flows of the deterministic model seem to be higher than those of the Markov models.

For finite buffers, tandem queues with deterministic services and with a fixed buffer size b for all stations are first considered. In this case, according to Little's formula, it is natural to define the detection of congestion by the formula E23.

This model also leads to a deterministic and periodic behavior of _n and to an analysis similar to that described above. Note that the policy model E9 is a special case of E23 with the size of the buffer b = l.

For random services, tandem queues with random services and with a fixed buffer size b for all stations are now considered. The service time supports are assumed to be limited.

We then consider the relationships E24 expressed in the appendix. (n) has a deterministic evolution when (n) belongs to {l, ..., b} and a random evolution when it belongs to {b + 1, ..., b + k}. The interpretation of this model is as follows: for an evolution of the size of the TCP type window, the congestion start date is the time when w (n) = S (n) / σ ^* (n) Then, a packet loss occurs when the number of packets in the system exceeds the size of the buffer, which corresponds substantially to the event W (n) = b + r (n).

Another interpretation is as follows: b is an arbitrary size (for example, half the actual size of the buffer) such that if the number of pending packets exceeds this threshold, congestion is detected.

Under the policy E24, if the service time vectors σ _i , ... σ _k are iid at n, the throughput of TCP Tahoe without exponential phase is given by the expression E25 where 0 is the vector (0, -, 0) ^fc of R * ' ¹ .

Time-out mechanisms can be taken into account, for example, by the condition S (n)> TO or by the condition: vk ⁽ⁿ⁾ - yκ ⁽ⁿ -n-ι ⁾ > ^τo -

For example, as soon as Z (n) is Markovian and that the support D of absolute values {y _κ (n) -y _κ (nw _n _ ₁ )} is finished, we find the same regenerative structure as above. The law of a regeneration cycle T1 can be explicitly calculated by the recurrence E26 of the appendix.

This recurrence formula is valid for n < _max . Thus, one can obtain a formula for flow using the ergodic theorem for regenerative processes as above.

Those skilled in the art can obtain a similar formula for TCP Tahoe with exponential phase or TCP Reno, and also for various extensions of the above model by including packet losses as according to formula E21 above.

In another example, where we consider four queues in tandem with a buffer size b = 50 and with an independent multinomial distribution for the services, Figure 8 shows the evolution of n / y ₄ (n) and (n ) for the TCP model Tahoe with the exponential phase and with b = 50. The σ ^ n) are iid and mutually independent, with values in {1,5,10} with respective probabilities of 0.3, 0.4, 0.4.

Figure 9 shows a comparison of the bit rates of two Tahoe TCPs with the exponential phase: b = 10 and K = 10. In the first case, σ ₃ (n) = l and {σ _j (n) (with j ≠ 3)} is an iid sequence, with values in {1, 10, 20} with the respective probabilities 0.3, 0.2, 0.5 . In the second case σ ₈ (n), is constant equal to 1 and the other services are as above.

Calculating throughputs using the formula E25 shows that switching the characteristics of two routers can change the value of the throughput. So, already in this case, the model cannot be reduced to a model with a single server, that of the bottleneck. It should also be noted that knowledge of the average values is insufficient to predict the value of the average flow.

The model described above finds many applications, in particular:

- relative to the law of instantaneous debits which, within the framework of the formula E20, is equal to the expression E27 of the appendix: it is an important value which can define a natural indicator of QoS in addition to the average value ;

- the law of end-to-end deadlines (formula E28);

- the law of time T necessary for the transmission of a file of size F; as a first approximation, this law is given by the relation: P (T> t) = P (y _κ (F)> t), but a more precise formula can be given by taking into account retransmissions of lost packets.

In this context, the following problems can be addressed at least in the context of algebraic simulation: - open models (where the source is not saturated: HTTP instead of FTP) where the process of arrivals is described by its statistical characteristics;

- multiple connections and interactions between multiple users;

- multipoint connections through a network with a tree structure instead of a linear structure of routers in series (see reference [7] for the case with constant window).

Some details are given below on the structure of an algebraic simulator which would make it possible to predict the performance of TCP or other flow control mechanisms originating from TCP in IP networks already existing or under design.

The basic principle of this simulator is the analysis of a controlled connection. The simulation of this connection begins with the acquisition of data concerning the network:

number of routers crossed by the connection, characteristics of each of the routers: capacity in Mb / s, size of bi buffers, forecasts, etc .;

propagation delays on each link (d _{± i + 1} );

statistical characteristics of transverse flows F _L when these are known (as for example in the case of Internet backbone routers);

statistical characteristics of the traffic offered by the controlled connection;

statistical characteristics of the traffic offered by the other flows in competition with the controlled connection on the access router. These characteristics appear in Figure 10.

One then builds according to these data and the characteristics of the version of TCP chosen (or of the algorithm of control of flow retained) an algebraic simulator which consists in the construction step by step of the matrices A _wn above and the computation by recurrence of the system daters as indicated.

We deduce from the mathematical analysis that was presented above that we can calculate from the simulation the main characteristics of the effect of TCP on the controlled connection (average speed, average speed over a given period of time, instantaneous flow fluctuations etc).

To complete this description of the simulator, some indications on how to take certain fine mechanisms into account are given below.

For a router i, let bi be the size of its buffer per stream; the detection of the exceeding of this capacity is done by the following condition on the daters: x _i (n + b _i ) ≤y _i (n)

As already indicated, the precise mechanisms of time-outs can be taken into account by the condition:

This formulation also applies when there is evolution of TO itself (in some versions, the variable T0 is updated at each timeout: TO is doubled in case of timeout, until we reach 64 times the initialization value). The present invention can be applied to the estimation of the variations and to the adaptations of the window in the max-plus linear framework. The variations and adaptations of the variable TO are of a similar nature and are therefore also representable in this context. In some versions of TCP, it is recommended to group the acknowledgments. We wait for example until three packets have arrived at the destination to return a single acknowledgment for the three packets. This can also be represented by means of a chain of two variable window mechanisms, and fits quite naturally in the context of a max-more linear description.

A scheduling mechanism in a router (for example first come first out "FIFO" (or Weighed Fair Queueing) with possibly differentiations of services) can be represented in this simulation by taking into account the influence of this mechanism on the service lives σ ^ (); once calculated as a function of the transverse traffic, the traffic of the controlled connection and the scheduling mechanism, these random durations can be injected into the simulator.

Obviously, for access network infrastructures being designed, it is not possible to measure the statistical characteristics of transverse traffic on access routers. Assuming that we have a statistical description of the traffic offered planned on the access routers of this network (for example N HTTP traffic), we can then use the simulator to calculate the unknown transverse traffic as a fixed point of the system : either t the traffic law of the controlled connection observed at the access router; t must be such that if we take as offered traffic on the controlled connection that of an HTTP session, - the traffic of the controlled connection observed on the access router has the law t; and the transverse traffic observed at this point is the sum of N traffics of law t.

Thus, in saturated or unsaturated networks, the control mechanism in adaptive feedback of TCP is a linear feedback in max-plus algebra. This leads to a simple representation of the effect of this protocol on any network which itself admits a max-plus representation in the absence of control, as is the case for example for tandem queues or fork-join networks found in multipoint trees.

The formulas above confirm that in this case, the speed depends only on the RTT and the rate of the bottleneck. New formulas have also been obtained when the services are random. The hazards here represent the effect of the rest of the traffic on the controlled connection. In this case, throughput cannot be obtained only from average considerations, and the order and fine statistical behavior of the routers cannot be ignored.

The set of models falling within this framework is very rich: - one can indeed choose deterministic or random services, flow control based on loss or leave;

- losses can come from congestion or ti e-outs, or be random, or even be a combination of these three possibilities;

- a version of Reno or Tahoe can be chosen, with or without the exponential phase etc.

The present invention finds interesting applications in the analysis of these few combinations, since all combinations can in principle be analyzed in this context.

More generally, the invention defines a generic framework for the simulation of protocols of the TCP type on networks which may be large. The simulation is based on a simple processing which exploits linearity and which has a controlled complexity.

Of course, the invention is not limited to the embodiment described above by way of example; it extends to other variants which nevertheless remain defined in the context of the claims below. E '. (A® B) _tj = A _ik θ B _kj = maxiAa.Bkj), (A ® B) _tj = φ A _ik ® B _kj = max A _ik + B _kj l≤k <d * ≤ ^k ≤ ^Λ

£ 2: W (0) = l, W (n + l) = f (W (n), ACK (n)),

E 3 .. "/„ = (int) W (n) = \ W {n).

E 5 d: {n: ΛCJf (n) = 1}, 0 </ (W (n), 1) - W (») <L

ς- 2: < ^n: ^ ϋf (n) = 0}, 1 </ flW, 0) <W (n).

TCP, Tahoe:

Efi! /(W(n),l.W.(n)) = W (n) + 1; W (n) <W. (n),

= W (n) + - _t ; ' W "(n) ^/ ≥ ^: VT. (N),

Ê6: ^~, * "'^ _Wl

EC: f (W (n), Q, W.n)) = i;

F65: W. {n + 1) _: = ol („) j.

TCP, Reno £ ^6: / Wn), l) = W (n) + -,

EC; / (W (n), 0) = L W (n) J.

Ê ^: V "> 0, ^σ ° ( ⁿ ) = °; V <(") = ** (") + θi {) J _Λ "(") = y ("- w _n -ι) ® dκ, ₀ (n - w„ _ι), "<(") = i -ι (") ® * -ι. * ( ⁿ ) © V <( ⁿ ~ *)) ® ^σ * ( ⁿ )>^" = 1> ^{• • •} ι- * C

1 "= (yι (n), ï _f e (n), .., wc (n)) e" * & (n) = (K (n), r (n - l), .., Y ( n- w * + 1)) 'GR ^ ïf' ¹ . 3: Z (0) = (0, ... 0) ^e , Z (n) = ..., (n) ® Z (n - 1), Vn> 1,) © D, - ••, A ".(not)

HJ | W'- ( _l '(n)) = ∑ (d * -ι, fc (n) + "fc (n)) + dκ.o (n- [i = ^ î ^" ∞ [i <

E ^ 4 {τϋj, ιι> _a , ...} = {1,2,2,3,3,3, ..., UJ * - 1, .., m ^* - l, wι ^* , l, 2.2, ~}.

• ^• -i $ - Vi € {l, .., w ^* -l}, = i _} * .. = 1; T = "'^ ^• - ^ï ) + 1

Z (nx T + 1) = (A _V .AZ.'Z ₁ ^1. -Aι) ⁿ ® Λ _x ® Z (0). Vi> 1,

V "> l Vne {l, .., i-l}

_. ! " ^• ("^" -1) + 2 E ^ • ' ^{7 ~} 2 (uι * -l) S + σ ^"'

EI αr ^ → + oo ^ ^{7 "1} ~! ^ Τ-

vτ> ^• τOO ^ ³ J

4σ *

P) _Λ . 7 = ∑ _β6Δ ατr (α), = ^• ^ ^U 'Δ = {S- (Al) σ ^* , A = l, .., _w * -l} u {σ *}.

A =. {»* ( ^N +!) ^_ F ( ^Λ )) €

£ <Ll • _• W (n)) = W („) + -L. ^ ≈o.

EZ3'- ACK {n) = l, Uw _n <b + -, 0, if w _n > b + -. at*

S {n) ≈ T. ^σ * ⁿ⁾

S = sup {5 (n)}; σ * = sup {σ * (n)}. r (n) = ^{S <} w * = b + sup {r (n)}

ACK (n) vin lι vin <b + r (), 0, uι _n > b + _r (n). εi ^• • ¹ = ' ^{+ Σ} * ^≈ ' ∑ o Pu (i + 1 + ∑ ^k ∑l (b + j)) p ^{u =} (s ^{p (r (n)>} H ^{(r (n)} > * ww ≤ *).

Eif:

P (T,> n, Z _n = z) = P (nj _{= 1} {»* (*) - _yκ ( _k _ _u , _t _ ₁ ) < _{TQ Zn β z)}

= 13 ( ⁿ * -1 ("* (*) - J / X (k -" * _!) <TO, Z _n - ₁ ≈ z Z _n = z)

** 6Δ

~ jC ( ^Z "= *> Vκ (n) - _yκ (", „_,) _< T _O I _Zn _, _{= Z} ' _{) P {Ά> n} _ _{lni = zl)} tVî ^' . ^Um ”→ c ₀ P ((y _κ n + _l) - _{Vκ (n))} - _{1 <} . _. _τ

APPENDIX IT

[1] Altman, E., Bolot, J., Nain, P., Elouadghiri, D., Erramdani, M., Brown, P. and Collange, D. (1997) Performance Modeling of TCP / EP in a Wide- Area Network. INRIA Report, March, N "3142, INRIA.

[2] Baccelli, F., Cohen, G. Olsder, G.J. and Quadrat, J.P. (1992) Synchronization and Linearity, Wiley.

[3] Baccelli, F. and Hong, D. (1998) Analytic Expansions of (max, +) yapunov Exponents. INRIA Report, May, N ° 3427, INRIA Sophia-Antipoliβ, To appear in Ann ls of AppL Prob.

[4] Baccelli, F., Gaubert, S., Hong, D. (1999) Representation and Expansion of (MaxJPlus) - Lyapunov Exponents. Proc. of 37th Annual Allerton Conf. on Communication, Control and Computing, September, Allerton Park.

[5] Baccelli, F., Bonald, T. (1999) Window flow control in FIFO networks with cross tra e. Queueing Systems, 32, 195-231.

[6] Chang, C.S. (1999) Performance guarantees in Communication Networks. Springer Verlag.

[7] Chaintreau, A, Baccelli, F., Diot, C. (2000) Impact of Network Delay Variation on Multicast Sessions Performance With TCP-like Congestion Control, submission SIGCOMM 2000.

[S] Hong, D. (1999) Equality of the maximum and minimum throughputs of irreducible stochastic max-plus linear Systems. In preparation.

[9] Keshav, S. (1997) An engineering approach to computer networking. Addison-Wesley Praf s- sional Computing Series.

[10] Lakshman, T.V., Madhow, U. (1997) The performance of TCP / IP for networks with high bandwidth-delay products and random loss. IEEE / ACM Trans. Networking, June.

[11] Mathis, M. Semske, J. Mahdavi, J and Ott, T. (1997) The Macroscopic Behavior of the TCP Congestion Avoidance Algorithm. Computer Communication Reυiew, 27 (3), July.

[12] Padhye, J., Firiou, V-, Towsley, D. and Kurose, J. (1998) Modeling TCP throughput: a simple model and its empirical validation. Proc. of ACM SIGCOMM.

[13] Padhye, J-, Firiou, V., Towsley, D. (1999) A Stochastic Model of TCP Reno Congestion Avoidance and Control. Technical Report, 99-02, CMPSCI, Univ. of Massachusetts, Amherst.

Claims

claims

1. Device for assisting in monitoring and / or simulating a complex system, in particular a communication network, characterized in that it comprises:

a memory (ROM, RAM) for storing first data representative of network parameters, as well as for receiving at least second data representative of events in the network, a portion of said memory being reserved for storing data in matrix form ,

- a calculation module (CAL), capable of performing on at least two matrices of dynamically variable structure, an operation forming a product according to the so-called MAX-PLUS algebra,

a modeling module (MOD) for constructing at least a first matrix (Z (n)) and a current matrix (A (n)) respectively as a function of the first data and of the second data, according to a chosen model, and

- a pilot module (PIL) for repeatedly applying the first matrix and the current matrix to the calculation module, the product matrix obtained (Z (n + 1)) becoming a new first matrix.

2. Device according to claim 1, characterized in that the first data comprise data relating to the topology of the network.

3. Device according to claim 2, characterized in that the first data comprise the number (K) of network routers crossed by a connection to be monitored or simulated and properties of said routers.

4. Device according to claim 3, characterized in that the first data comprise memory sizes of the routers (b _κ ).

5. Device according to one of claims 2 to 4, characterized in that the first data comprise statistical properties of the traffic offered in the network. „.

34

6. Device according to one of the preceding claims, characterized in that the modeling module comprises:

- a static modeling sub-module (ST), to build the first matrix according to the first data, and

- a dynamic modeling sub-module (DYN), to build at least one current matrix as a function of the second data.

7. Device according to one of the preceding claims, characterized in that the second data comprise information relating to losses (p_, p ₊ ) in the network.

8. Device according to one of the preceding claims, characterized in that the second data comprise information relating to transverse flows in the network, with respect to a controlled connection to be monitored or simulated.

9. Device according to one of claims 7 and 8, characterized in that the second data comprise information relating to congestion in the network.

10. Device according to one of claims 7 to 9, characterized in that the second data comprise information relating to timeouts (TO) in the network.

11. Device according to one of the preceding claims, characterized in that said product matrix is representative of a throughput in the network associated with the connection to be monitored or simulated.

12. Device according to claim 11, characterized in that said produced matrix is representative of an average speed in the network.

13. Device according to claim 11, characterized in that said produced matrix is representative of fluctuations in an instantaneous flow rate.

14. Device according to one of the preceding claims, characterized in that the modeling module is arranged to build a plurality of matrices, in number corresponding substantially to the number of packets in the network.

15. Device according to one of the preceding claims, characterized in that said chosen model includes consideration of the variable size of a window ( _n ) used to control the number of packets in the network.

16. Device according to claim 15, characterized in that said chosen model includes the consideration of a network comprising routers of discipline of the "first come first served" type.

17. Device according to one of claims 15 and 16, characterized in that said chosen model includes the consideration of a network comprising routers with discipline of type FQ.

18. Device according to one of the preceding claims, characterized in that the chosen model includes consideration of a network controlled by a TCP type protocol.

19. Device according to one of claims 15 to 18, characterized in that said chosen model includes consideration of a deterministic service (s).

20. Device according to one of claims 15 to 18, characterized in that said chosen model includes consideration of a random service (s). 3b

21. Device according to one of the preceding claims, characterized in that the current matrix at least comprises dynamically variable coefficients (A _ik ).

22. Device according to one of the preceding claims, characterized in that the product matrix obtained is a vector represented by a single column matrix (Z (n)).

23. Method for assisting in monitoring a complex system, in particular a communication network, characterized in that it comprises the following steps: a) obtaining first data representative of network parameters, b) constructing a first matrix (Z (n)), according to a chosen model, as a function of said first data, c) receive, at a chosen time, at least second data representative of events in the network, d) construct at least a second matrix (A (n)) of dynamically variable structure, depending on the model chosen, as a function of the second data, and e) performing on said matrices a product forming operation according to the so-called MAX-PLUS algebra, the product matrix obtained (Z (n +1)) being representative of the state of the network at said selected instant.

24. The method of claim 23, characterized in that it further comprises the following step: f) repeating, at selected times, steps c), d) and e), while the product matrix obtained becomes the first matrix after step e), which makes it possible to follow a temporal evolution of the state of the network at said selected times.

25. Method for simulating a complex system, in particular mechanisms and controls for flow and congestion in a communication network, characterized in that it comprises the following steps: a) obtaining first data representative of parameters specific to the network, b) construct a first matrix (Z (n)), according to a chosen model, as a function of said first data, c) simulate events in the network and provide at least second data representative of said events, d) construct at least one second matrix (A (n)) according to the chosen model, as a function of said second data, and e) perform on said matrices a product forming operation according to the algebra called MAX-PLUS, the product matrix obtained (Z (n + 1)) being representative of a state of the network undergoing said events.

26. The method of claim 25, characterized in that it further comprises the following step: f) repeating, for successive events, steps c), d) and e), while the product matrix obtained becomes the first matrix after step e), which makes it possible to predict a change in the state of the network as a function of said events.