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An Analytical Model of a new Packet

Marking Algorithm for TCP flows

Giovanni Neglia, Vincenzo Falletta

Dipartimento di Ingegneria Elettrica, DIE

Universit` a degli Studi di Palermo

Palermo, Italy

Giuseppe Bianchi

Dipartimento di Ingegneria Elettronica, DIE

Universit` a degli Studi di Roma - Tor Vergata

Roma, Italy

Abstract

In Differentiated Services networks, packets may receive a different treatment ac-

cording to their Differentiated Services Code Point (DSCP) label. As a consequence,

packet marking schemes can be devised to differentiate packets belonging to a same

TCP flow, with the goal of improving the experienced performance. This paper

presents an analytical model for an adaptive packet marking scheme proposed in

our previous work. The model combines three specific sub-models aimed at de-

scribing i) the TCP sources aggregate ii) the marker, and iii) the network status.

Preliminary simulative results show quite accurate predictions for throughput and

average queue occupancy. Besides, the research suggests new interesting guidelines

to model queues fed by TCP traffic.

Key words: TCP Marking, Differentiated Services, Models

1 Introduction

Differentiated Services (DiffServ) networks provide the ability to enforce a dif-

ferent forwarding behavior to packets, based on their Differentiated Services

Email addresses: giovanni.neglia@tti.unipa.it (Giovanni Neglia),

vincenzo.falletta@tti.unipa.it (Vincenzo Falletta),

bianchi@elet.polimi.it (Giuseppe Bianchi).

Preprint submitted to Elsevier Science12 September 2005

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Code Point (DSCP) value. A possible way to exploit the DiffServ architec-

ture is to provide differentiated support for flows belonging to different traffic

classes, distinguished on the basis of the DSCP employed. However, since it

is not required that all packets belonging to a flow are marked with the same

DSCP label, another possible way to exploit DiffServ is to identify marking

strategies for packets belonging to the same flow.

Several packet marking algorithms have been proposed for TCP flows. The

marking strategy is enforced at the ingress node of a DiffServ domain (edge

router). Within the DiffServ domain, marked packets are handled in an aggre-

gated manner, and receive a different treatment based on their marked DSCP.

Generally, a two-level marking scheme is adopted, where packets labelled as

IN receive better treatment (lower dropping rate) than packets marked as

OUT. Within the network, dropping priority mechanisms are implemented in

active queue management schemes such as RIO - Random Early Discard with

IN/OUT packets [1].

The basic idea of the proposed algorithms is that a suitable marking profile

(e.g. a token bucket which marks IN/OUT profile packets) may provide some

form of protection in the case of congestion. A large number of papers [1–16]

have thoroughly studied marking mechanisms for service differentiation, and

have evaluated how the service marking parameters influence the achieved

rate.

More recently, TCP marking has been proposed as a way to achieve better

than best effort performance [17–19]. The idea is that packet marking can be

adopted also in a scenario of homogeneous flows (i.e. all marked according

to the same profile), with the goal of increasing the performance of all flows.

Our algorithm was first proposed in [20] and share this aim. An introductory

comparison with the other marking algorithms is presented in section 2.

In this paper we slightly modify the mechanism proposed in [20], and we de-

scribe an analytical model to evaluate the network performance. This model

can be employed to study possible variants of the algorithm. By the way,

the network sub-model exhibits some novelty in comparison to previous ap-

proaches and could be useful in different network scenarios where TCP traffic

is considered.

The rest of this paper is organized as follows. After an overview of proposed

marking schemes in section 2, section 3 describes our adaptive packet mark-

ing algorithm, focusing on some changes to the previous version. Section 4

presents the analytical model which relies on the Fixed Point Approximation,

whose rationale and whose employment in computer networks field are shortly

introduced in subsection 4.1. The three submodels are detailed respectively in

subsections 4.2, 4.3, 4.4, while in subsection 4.5 existence and uniqueness of

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a solution are proven. Section 5 deals with validation of the proposed model.

A simple application of the model to evaluate the performance of a variant of

the algorithm is presented in section 6. Finally, conclusive remarks and further

research issues are given in section 7.

2 Related works

The idea to employ marking mechanisms for service differentiation was first

introduced in [1], where the authors propose a time-sliding window marker. In

[2] token bucket appears to achieve better performance in comparison to time-

sliding window. At the same time the authors claim that marking cannot offer

a quantifiable service to TCP traffic due to the interaction of TCP dynamics

with priority dropping: when IN packets and OUT packets are mixed in a single

TCP connection, drops of OUT packets negatively impact the connection’s

performance. Afterwards token bucket and time-sliding window markers have

been extended to three colors [3–5].

Following studies confirm the difficulty of marker configuration. A detailed

experimental study of the main factors that impact the throughput of TCP

flows in a RIO based DiffServ network is provided in [6]. The article shows that

in an over-provisioned network all target rates are achieved, but unfair shares

of excess bandwidth are obtained. However, as the network approaches an

under-provisioned state, not all target rates can be achieved. In [7] it is shown

that it is possible to improve the throughput significantly even when a small

portion of traffic is sent as in-profile packets. At the same time the authors

observe that, in order to fully utilize the benefit of out-profile packets, the

amount of out-profile packets sent in addition to the in-profile packets has to

be carefully determined. In [8] a set of experimental measures is presented. The

main result is that the differentiation among the transmission rates of TCP

flows can be achieved, but it is difficult to provide the required rates with a

good approximation. In [9] the limits of token bucket are deeply investigated.

It appears that (i) the achieved rate is not proportional to the assured rate,

(ii) it is not always possible to achieve the assured rate and, (iii) there exist

ranges of values of the achieved rate for which token bucket parameters have

no influence.

These results suggested the need to introduce some adaptivity in order to

cope with TCP dynamics. In [10] the Packet Marking Engine monitors and

sustains the requested level of service by setting the DS-field in the packet

headers appropriately. If the observed throughput falls below the minimum

target rate the Engine starts prioritizing packets until the desired target rate is

reached. Once the target is reached, it strives to reduce the number of priority

packets without falling below the minimum requested rate. The Active Rate

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Management is proposed in [11] in order to provide minimum throughputs to

traffic aggregates. It is a classical, linear, time-invariant controller, which sets

the token bucket parameters (specifically the token bucket rate) adapting to

changes in the network. The same issue is tackled by [12]. The adaptive dual

token bucket in [13] regulates the amount of OUT packets in order to prevent

TCP packet losses caused by excess low-priority traffic in the network. This

adaptive technique requires a congestion signaling procedure from internal

routers to border routers.

The Equation-Based Marking [14] is someway similar to ours because it senses

the current network conditions, in particular it estimates the loss probability

and the Round Trip Time (RTT) experienced by a TCP flow (without any

signaling with core routers), and adapts the packet marking probabilities ac-

cordingly. In particular it uses the TCP model in [21] and these estimates in

order to identify the target loss probabilities, corresponding to target through-

put rates. Then, it uses the current loss probability estimate as well as these

target loss probabilities to calculate the packet-marking probabilities. Main

targets are fairness among heterogeneous TCP flows and protection against

non-assured traffic. Fairness is also the main focus of [15] and [16]: the first

concentrates on the effect of different RTTs, the second propose the Direct

Congestion Control Scheme to achieve fairness between responsive and unre-

sponsive aggregates.

The proposals described above share the purpose to assure a minimum through-

put to TCP individual flows or aggregates. As we said in the previous section,

TCP marking has also been proposed as a way to achieve better than best

effort performance [17–19]. In particular [17] focuses on WWW traffic and

proposes two packet marking schemes. The first one is tightly integrated with

the TCP protocol: the source is allowed to send up to NsIN packets when it

starts, and then up to Na= sstresh at the beginning of a Slow Start phase,

and up to Na= cwnd at the beginning of a Fast Recovery phase. The second

scheme does not require the knowledge of internal TCP variables, but it uses a

constant value Na= Ns= 5, hence this scheme can be implemented at ingress

router. The rationale behind the schemes in [17] is that packets marked as IN

will be protected against network congestion, hence marking can be useful

employed to protect flows with small window or retransmitted packets, when

packet losses cannot be recovered via the fast retransmission algorithm but

trigger timeouts, which reduce TCP source throughput.

The TCP-friendly marker in [18,19] considers long lived flows and adopt good-

put and loss as performance metrics. The main guidelines are: 1) to protect

small-window flows and retransmitted packets from losses by marking them

IN; 2) to avoid, if possible, to mark OUT consecutive packets in order to re-

duce the possibility of burst loss of packets. Our approach share the purpose

to space as much as possible packet losses, at the same time many differences

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hold. In the TCP-friendly marker a fixed number of IN tokens is available for

each time interval and it has to be distributed among the flows, on the contrary

our scheme adaptively set the length of IN packets burst (i.e. the number of

flow consecutive packets that are marked IN) according to the network status.

Besides, all the marking schemes share the idea that packets marked IN will be

protected against network congestion, while our algorithm operates according

to the someway opposite philosophy to employ OUT packets as probes (see

section 3). Finally, our approach is much simpler.

We present some further remarks about the previous algorithms in order to

stress the novelty of our approach. In a DiffServ Assured Forwarding (AF)

scenario, the differentiation between traffic classes is relative. For example

usual RIO configuration [1] assures that IN packets dropping probability is

lower than OUT packets one, but no bound is guaranteed. For this reason

the protection of IN packets in [17] relies on the assumption that most of the

packets in the network are of type OUT, hence IN packets will receive a “good-

enough” service. In fact in [17] the authors show that a throughput reduction

may be encountered as long as the percentage of IN traffic becomes greater

than a given threshold. The authors claim that the problem is interleaving IN

and OUT packets, when the loss rate of the OUT traffic is much larger than

that of the IN traffic. We want to stress that the IN packet protection vanishes

as IN traffic increases. Indeed, we too have observed performance impairments

for both a token-bucket marker and for a marking scheme very similar to the

one proposed in [18,19] (protection of small window and retransmitted packets,

an OUT packet inserted every n IN packets).

Hence our approach shows two main differences [20]: 1) the majority of packets

are IN, 2) the performance takes advantage of a very high OUT packet loss

rate. The apparent conflict with results in [17] and with similar results for

the marker proposed in [18,19] relies on the adaptivity. These schemes are

not designed to be adaptive to the network congestion status, while ours uses

some heuristics to provide adaptivity.

3 The Packet Marking Algorithm (PMA)

In [20,22] we proposed a new marking algorithm, able to achieve better per-

formance in terms of average queueing delay and flow completion time versus

link utilization. According to this marking scheme “long” IN-packets bursts

are interleaved with a single OUT packet. The OUT packet is thence employed

as a probe to early reveal a possible seed of congestion in the network. The

algorithm dynamically updates the length of IN-packets bursts by a heuristic

estimation of the experienced packet loss ratio.

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The idea of marking the majority of packets as IN seems to be in contrast with

some results found with other marking scheme [18,19,17], but the intrinsic

adaptivity of our algorithm is something all these models lack.

If we think about Active Queue Management (AQM) techniques such as Ran-

dom Early Detection (RED) we observe the same idea of dropping some pack-

ets when signals of an incoming congestion are received. Our algorithm moves

further: it reallocates losses among the OUT packets, so it spaces them as

much as possible, avoiding consecutive losses for a flow and assuring a more

regular TCP adaptation behavior.

By simulative evaluation we found better performance when OUT-packets

dropping probability is near 100%, while IN packets are not dropped at all.

SN > SNh?

AIN:= (1 - α α α α)AIN+α α α α Lseq

Lseq:= 0

CIN := CIN + 1

MARK

IN

SNh:= SN

Lseq:= Lseq+ 1

CIN> AIN?

CIN:= 0

AIN:= AIN+1

MARK

OUT

YES

NO

NO

YES

CIN := CIN + 1

Arriving

Packet

Fig. 1. PMA Flow diagram.

The algorithm flowchart is shown in Fig. 1. Now we will explain how this

procedure works. Each time a new SYN packet arrives at the edge router a

new state vector is set, containing the following variables:

SNh: This counter stores the highest Sequence Number (SN) encountered in

the flow. It is initially set to the ISN (Initial Sequence Number) value. It is

updated whenever a non-empty packet (i.e. non ACK) arrives with a higher

SN1.

Lseq: It is initially set to zero. It is increased by one unit for each new arrived

packet (i.e. in-sequence packet), while is reset to zero every time an out-of-

sequence packet arrives.

CIN: It counts the number of IN-packets in the burst. It is reset to zero when

it exceeds AINand an OUT packet is sent.

1in a cyclical sense - recall that sequence numbers wrap when the value 232− 1 is

reached.

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AIN: It stores the number of packets which will be marked IN, it tracks the

average length of in-sequence packet bursts through autoregressive filtering.

The algorithm has been slightly changed in comparison to the version pre-

sented in [20,22]. In the previous algorithm a single variable (LIN) was taking

into account the number of in-sequence packets (as Lseq actually does) and

the number of IN packets of the actual IN-packets burst (as CIN actually

does). This coupling required an artificial increase of the variable AIN after

marking an OUT packet, we chose AIN := 2AIN+ 1 but its correct amount

was dependant from network condition as it is discussed in [20,22]. After the

introduction of the new variable CIN, a small increase of AIN has been left:

it assures better fairness among the flows, allowing flows with underestimated

AINvalues to faster reach the correct estimate.

4 The analytical model

The algorithm has shown good performance, but it essentially relies on a

heuristic. In order to achieve a deeper understanding and to establish RIO

setting criteria, we have developed an analytical model.

The model assumes n long-lived homogeneous flows sharing a common bottle-

neck, whose capacity is C. The model is based on a Fixed Point Approximation

(FPA), a modeling technique described in the following subsection. Accord-

ing to FPA the system is divided into its three main components as shown

in Fig. 2: the TCP sources, the network and the marker. Each element is

modeled separately, taking into account the effects of the others through the

parameters shown in figure. For example TCP sources depend on the network

by the RTT and the dropping probabilities pinand pout, and on the marker

by the length of IN packet bursts (AIN).

After an overview of FPA methods in section. 4.1, the submodels for the TCP

sources, for the marker and for the network are respectively presented in sec-

tions 4.2, 4.3 and 4.4. Each of them could be replaced by a more sophisticated

one.

Sources MarkerNetwork

T

outinpp RTT

,,

seq

L

IN

A

IN

A

Fig. 2. The three-block model.

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4.1 About Fixed Point Approximations

The expression Fixed Point Approximation (FPA) refers to a particular mod-

eling technique, which we are going to describe in this section. This name is

quite spread in scientific literature [23–25], but also other names appear: fixed

point models [26,27], fixed point approach [28], reciprocal model tuning [29].

Other papers [30,31] refer the expression “fixed point” to the specific method

employed to solve the model system of equations, rather than to the modeling

technique.

This section is organized as it follows. Firstly we introduce the idea of FPA

with reference to our specific problem, and we explain the origin of the ex-

pression fixed point. Secondly we briefly present telecommunications works

employing this kind of modeling technique. For a more detailed overview of

FPA in the field of computer networks refer to [32].

Let us consider a single bottleneck network, where a single TCP flow is marked

at the edge and feed the queue at the bottleneck. Suppose we are interested

into some average values, like TCP throughput or queue occupancy. If we know

all the parameters characterizing the network (i.e. link capacities, link delays,

buffer size) the TCP sender (e.g. the TCP version, the maximum congestion

window size, the timer granularity), the TCP receiver and the marker, we

are able to describe exactly the behavior of each element of the network and

to evaluate the throughput of the TCP sender or the queue occupancy at

each time. If we were able to describe the evolution of these quantities in a

closed form, we could evaluate their average value, by integrating the analytical

expressions, but in general it is not the case.

In order to achieve our purpose we have to sacrifice the exact description of

the system. A way to make the problem analytically tractable is to divide the

system into three parts (e.g. the TCP source, the queue at the bottleneck and

the marker), to assume some simplifying assumptions about their interaction,

and then to develop an analytical model for each part.

According to the FPA approach, the main assumptions are that we model each

part considering the other in a steady state, and that this state is independent

by the behavior of the part we are modeling. In our example we know that the

throughput of the TCP source is dependent from the path current RTT, from

packet discard at the queue and from the marking pattern (characterized by

AIN). At the same time the TCP traffic generates the queue in the network

and causes eventually packet discard when the buffer is full. Nevertheless, in

order to model the TCP behavior, we assume that the network and the marker

are in a steady state: specifically we consider that RTT and AINare constant

(AIN = A), and that the packet discard for both IN and OUT packets are

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bernoullian processes respectively with mean values pin and pout. Different

assumptions can be done. Anyway these allows us to derive an expression for

the long-term steady-state TCP throughput as a function of RTT, p, A, say:

T = f(RTT,pin,pout,A) (1)

and an expression for the average number of in-sequence packets as:

L = g(pin,pout,A) (2)

In the same manner, in order to model the network, we assume the TCP

source offers a constant traffic intensity to the network, independently from

the present network status (queue occupancy and packet discard probability),

with a ratio of IN packets to OUT ones equal to A. If we add some further

hypothesis about the statistical characterization of packet arrivals at the buffer

and the way IN and OUT packets are interspersed, we are able to derive the

mean number of packets in the router and hence the average RTT and the

mean dropping probabilities (pin,pout), i.e.:

pin=hin(T,A)

pout=hout(T,A)

RTT =l(T,A)

(3)

(4)

(5)

Finally given the average number of in-sequence packet (L), we can derive the

average length of IN packet bursts.

A = m(L) (6)

In order to determine T, RTT, pin, pout, L, A, we need to solve the system of

equations (1),(2),(3), (4),(5) and (6). If we define the function U : ?6→ ?6

as

U(T,L,pin,pout,RTT,A) =

= [f(RTT,pin,pout,A),g(pin,pout,A),hin(T),hout(T),l(T),m(L)]

(7)

then we can note that a solution of such system ([T∗,L∗,p∗

if any, satisfies the following relation:

in,p∗

out,RTT∗,A∗]),

[T∗,L∗,p∗

in,p∗

out,RTT∗,A∗] = U(T∗,L∗,p∗

in,p∗

out,RTT∗,A∗), (8)

i.e. the point [T∗,L∗,p∗

in,p∗

out,RTT∗,A∗] is a fixed point for the ?6→ ?6

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mapping, established by the function U2. This remark justifies the name of

FPA.

Under some proper conditions about the function U and its definition set,

fixed-point theorems can be used to conclude that at least a solution exists,

like the Bolzano’s theorem, the Brouwer’s one and the Kakutani’s one (see for

example [33] or [34]). The question of uniqueness is more difficult, eventual

monotonicity greatly constraints the possible dynamics.

Different methods can be employed in order to solve Eq. (8). In particular

repeated substitution takes into account the following relation:

[Ti+1,Li+1,pin,i+1,pout,i+1,RTTi+1,Ai+1] = U(Ti,Li,pin,i,pout,i,RTTi,Ai),(9)

assuming that

lim

i−>∞[Ti,Li,pin,i,pout,i,RTTi,Ai] = [T∗,L∗,p∗

in,p∗

out,RTT∗,A∗].

This kind of solution is particularly appealing, because Eq. (9) can be read as

a dynamical system, describing the network operation [24,35]: in our example

if the network is unloaded (pin= pout= 0) and the TCP source starts injecting

traffic in the network, the buffer provides new (different) value of pinand pout

by dropping packets. The source reacts to this packet loss probability adjusting

its sending rate and at the same time the marker changes its marking profile

until convergence is reached. Despite this striking interpretation, it is not

clear how close Eq. (9) actually describes the network operation. By the way,

convergence of Eq. (9) is not guaranteed.

Some other kind of approximations are often employed together with FPA and

they cannot often be easily distinguished from a FPA approach extended to all

the network elements, i.e. when we divide the network into as many parts as

the number of TCP sources plus the number of the network routers. One exam-

ple is the Mean Field Theory (also known under many names and guises, e.g.

Self Consistent Field Theory, Bragg-Williams Approximation, Bethe Approx-

imation, Landau Theory) whose simplifies a many-body interactions problem

by replacing all interactions to anyone body with an average or effective in-

teraction. The Mean Field Theory is explicitly referenced in [36,37] as a way

to model the interaction of many TCP flows. Another common assumption

concerns the networks of queues, and it is known as Kleinrock’s independence

approximation [38]. Also in a network of queues there is a form of interaction,

in the sense that a traffic stream departing from one queue enters one or more

2Note that in what follows we will introduce for convenience other variables, but

nothing changes as regards the idea of FPA described here.

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other queues, perhaps after merging with portions of other traffic streams de-

parting from yet other queues. Analytically, this has the unfortunate effect

of complicating the character of the arrival process at downstream queues.

Kleinrock suggested that merging several packet streams on a transmission

line has an effect akin to restoring the independence of interarrival times and

packet lengths. It was concluded that it is often appropriate to adopt M/M/1

queueing model for each communication link regardless of the interaction of

traffic on this link with traffic on other links.

The employment of FPA techniques to model networks is not a novelty. For

example there is a considerable body of literature on the application of fixed

point methods to estimating blocking probabilities in circuit switched networks

(see [39] for some applications). More recently FPA has been widely used to

model the interaction of TCP sources with the network (see [26,31,35,40,23]

[27,41,29,28,24]). [26] has probably the merit to be the first paper where the

FPA approach is clearly stated and presented as a method “which allows the

adaptive nature of TCP sources to be accounted for”. Regarding network mod-

els [31,35,40,23,27,41] do not need stochastic queue model, but they essentially

rely on the assumption that long-lived TCP flows are able to achieve full band-

width utilization. Aside from [27,41] they consider AQM mechanisms relating

the dropping probability and the queue occupancy. [27] considers zero buffer

queue and [41] considers large delay-bandwidth networks in order to neglect

queueing delay. Multi-bottleneck networks are considered in [40,23,27,41], and

the existence of a solution is proved in [41] under the above simplification. The

hypothesis of full bandwidth utilization is removed in [26,29,28,24], which

consider Poisson arrival at the queue. In [28] each buffer is modeled as a

M/M/1/K queue or as a M[X]/M/1/K queue with batch arrivals. The paper

discusses the admissibility of the Poisson hypothesis and prove the existence

and the uniqueness of the solution when the nominal load is less than one

for short and long lived TCP flows. A more detailed investigation of the ex-

istence, the uniqueness and the stability of equilibrium points appears in [24]

for a single-bottleneck scenario and short-lived flows.

As a final remark we note that there has been related work focusing on the

development and solution of a set of differential equations describing the tran-

sient behavior of TCP flows and queue dynamics [42]. FPA complements this

approach. The fixed point approach is much more efficient computationally as

the number of unknowns equals the number of links in the network, whereas

the differential equations approach requires the solution of a number of equa-

tions equal to the number of links plus the number of TCP flows. On the

other hand, the differential equations approach can be used to study transient

behavior.

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packets

sent

IN

IN

OUT

Correctly received

packet

Retransmitted packet

OUT packet

1

2

3

4

6

…

…

…

…

?i???9

?i???8?

?i???7

?i???6

?i???3

?i? 2

?i? 1?

…

Yi - 1

Yi

?i + 1

?i? 5

…

…

?i? 4

…

12

…

Xi??1

Xi

?i

5

…

…

…

…

…

…

…

…

…

…

……

Wi-1/2

Wi

…

…

...

Wi-1

no. of rounds

?i

Lost packet

b

i-th period

Fig. 3. Timeline and transmitted packets.

4.2The Sources Model

According to the previous description, we aim to obtain an expression of the

average TCP throughput (T, the input to the Network block) and of the aver-

age length of the in-sequence packet burst (L, the input to the Marker block),

given the marking profile (A) and the network status (RTT, pin, pout). We

have conjectured a regenerative process for TCP congestion window (cwnd),

thus extending the arguments in [21] to include two different service classes,

with different priority levels.

In our analysis we neglect slow start operation and time-out events, we only

consider loss indications due to triple duplicated acks, which turn on (always

successful) TCP fast retransmit mechanism. As regards time-out neglecting,

this approximation appears to be not critical because PMA spaces OUT pack-

ets and hence loss events. For this reason errors are usually recovered by fast

retransmission, not by time-out. Such intuition is confirmed by our simulation

results, where the number of time-outs appear to be significantly reduced in

comparison to a no-marker scenario.

A period of our regenerative process starts when the sender congestion window

is halved due to a loss indication. Fig. 3 shows cwnd trend as rounds succeed.

Wi−1is the cwnd value at the end of the (i − 1)-th period, hence in the i-th

period cwnd starts from Wi−1/2 and it is incremented by one every b rounds

(b is equal to 2 or 1, respectively if the receiver supports or not the delayed

ack algorithm). Notice that, due to our assumptions on TCP operation, each

period starts with an IN retransmitted packet, hence the number of packets

sent in the period (Yi) is equal to Lseq+1, according to the marker description

in section 4.3.

In the i-th period we define also the following random variables: Iiis the length

of the period; βiis the number of packets transmitted in the last round; αiis

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the number of the first lost packet since the beginning of the period, while γi

is the number of packets transmitted between the two losses occurred in the

(i−1)-th and in the i-th period. We get Yi= αi+Wi−1 and αi= γi−(Wi−1−1).

Due to the renewal-reward theorem we can obtain the expression for the av-

erage throughput of n sources sharing the same path:

T(A,RTT,pin,pout) = nE[Yi]

E[Ii]

We first compute E[Yi]. The relation between αiand γiallows us to explicit

E[Yi] as a function of the marking profile (A) and the network status (in

particular pin, pout). In general Yi ?= γi, however if we consider their mean

values, it holds:

E[Yi] = E[αi] + E[Wi] − 1 = E[γi] − (E[Wi−1] − 1) + E[Wi] − 1 = E[γi]

Let us denote by N the expected value E[γi]. We compute N as:

N =

∞

?

n=0

np(n) =

∞

?

n=0

(1 − P(n)) =

∞

?

n=0

Q(n)

where p(n) is the probability of losing the n-th packet after (n−1)-th successful

transmission, P(n) =

?n

Q(n) = 1 − P(n) represents the probability of not losing any packet among

these n. If we put n as n = k(A + 1) + h, with 0 ≤ h < (A + 1) we can write

Q(n) as

l=0p(l) is cumulative distribution function, and so

Q(n) = skA+h

in

sk

out

where sin= 1 − pin, sout= 1 − pout. The expression of N can be rewritten as

N =

∞

?

k=0

A

?

h=0

skA+h

in

sk

out

and can be solved in a close form:

N =sA+1

in

sin− 1

− 11

1 − sA

insout

(10)

Now we compute E[Ii]. Denoting with Xithe round in the i-th period when a

packet is lost, we obtain the period length as Ii=?Xi+1

j=1 ri,j, where ri,jis the

13

Page 14

j-th round trip time length. Supposing rijindependent of the round number

j (i.e. independent of cwnd size), taking expectation we find

E[Ii] = (E[X] + 1)E[r]

where E[r] = RTT is average round trip time.

In the i-th period cwnd size grows from Wi−1/2 to Wiwith linear slope 1/b,

so3

Wi=Wi−1

2

+Xi

b

− 1

and taking expectation we get

E[W] =2

b(E[X] − b)

To simplify our computations we assume Wi−1/2 and Xi/b to be integers. Now

let us count up all the packets:

Yi=

Xi/b−1

?

=Xi

2

k=0

?Wi−1

Wi−1+Xi

2

+ k

?

b + βi=XiWi−1

2

+Xi

2

?Xi

b

− 1

?

+ βi

?

b

− 1

?

+ βi=Xi

2

?

Wi+Wi−1

2

?

+ βi

and taking again expectation it follows

N =E[X]

2

?

E[W] +E[W]

2

?

+ E[β]

Assuming β identically distributed between 1 and Wi−1 we can write E[β] =

E[W]/2; therefore, solving for E[X]:

?

then it follows

E[X] =b

2

−2+3b

3b+

?

8N

3b+

?2+3b

3b

?2+ 2

?

=3b−2

6

+

?

2bN

3+

?2+3b

6

?2

E[Ii] = RTT

3b − 2

6

+

?

?

?

?2bN

3

+

?2 + 3b

6

?2

+ 1

3The formula is a linear approximation of the exact relation Wi = Wi−1/2 +

?Xi/b? − 1. In [21] a different approximation has been considered.

14

Page 15

Now we can write down the throughput formula:

T(N,RTT) = n

N

RTT(E[X] + 1)=

nN

RTT

1

3b−2

6

+

?

2bN

3+

?2+3b

6

?2+ 1

(11)

Throughput dependance from A, pinand poutis included in N through Eq. (10).

Note that if AIN= A = 0 (i.e. there is only one class of packets) and pout=

p → 0 we get the well-known formula [21]:

T(p,RTT) ?

n

RTT

?

3

2bp

Finally, as regards the average length of the in-sequence packet burst (L),

from previous remarks it simply follows:

L = E[Yi] − 1 = N − 1 (12)

4.3The Marker Model

We have discussed before about PMA in this paper, and we have seen how the

procedure acts marking one packet OUT every AININ packets, where AINis

obtained filtering Lseqwith an autoregressive unitary-gain filter. Hence, given

A and L respectively the average values of AINand Lseq, they are tied by the

relation4:

A = L (13)

The relation between AINand Lseqhas been chosen according to the rationale

discussed in section 3. Anyway the relation between A and L can be considered

a project choice:

A = m(L) (14)

A change of the m() law leads to a different marking algorithm, for example

pursuing a different target. We are going to show an example in section 6.

As regards the fixed-point approach approximation, we observe that the pre-

vious relation looks more suitable as long as the system reaches the state

4A closer look to the algorithm reveals that this is an approximation due to the

update A := A + 1 after each OUT-packet transmission.

15

Page 16

where pin ? 0 and pout ? 1. In fact, in the case of pin = 0,pout = 1 we

would have AIN = Lseq, not simply A = L. In [20] and [22] we have shown

that the algorithm exhibits optimal performance under hard differentiation

setting, which leads to pin? 0 and pout? 1. Hence fixed-point approximation

appears justified for PMA.

4.4The Network Model

SourcesNetwork

T

p RTT,

Fig. 4. Interaction between the Network Model and the Sources Model.

In [43] we have proposed a network submodel, extending the approach pro-

posed in [35] for a best-effort scenario to a DiffServ one, where routers deploy

RIO (we indicate the configuration parameters as (minout,maxout,Pmaxout)

and (minout,maxout,Pmaxout) respectively for OUT and IN packets [1]). A

limit of that approach is that TCP sources are intrinsically assumed to achieve

full bottleneck utilization (assumption also in [31,35,40,23,27,41]), hence the

model is able to predict average queue occupation, not link utilization. Besides

the model in [43] predicts a range of solutions when maxout< minin. These

problems could be overcome introducing in the model queue variability.

Anyway in this paper the approach is radically different, we consider that the

queue can be modelled as a M/M/1/K queueing system. This allows us to

evaluate the stationary distribution of the queue for a given offered load T,

and then the average values we are interested in, i.e. RTT, pinand pout.

As regards the assumption of Markovian arrivals, it seems to be justified

when the TCP connection rate increases [44]. Anyway M/M/1/K models

have been widely employed in literature and have shown good performance

[45,26,30,46,47]. In particular our framework is similar to those of [30] and [47],

which model respectively Token Bucket and Single Rate Three Color Marker,

but it differentiates because it assumes state dependent arrivals, rather than

uniform ones.

These models take into account the presence of different class of traffic and

the effect of AQM mechanism like RIO, but they assume that dropping prob-

ability depends only on the instantaneous queue size, disregarding the effect

of filtering.

According to [30], the stationary distribution of the queue can be evaluated

16

Page 17

as:

π(i) = π(0)

?T

C

?i i−1

j=0

?

(1 − p(j)),i = 1,2,...maxin

where C is the bottleneck capacity, π(0) is given by the normalization equation

π(0) =

1 +

maxin

?

i=1

?T

C

?i i−1

j=0

?

(1 − p(j))

−1

and

p(i) =Tinpin(i) + Toutpout(i)

Tin+ Tout

=Apin(i) + pout(i)

A + 1

Note that we assumed maxout < maxin, and that it is useless considering

queue values greater than maxinbecause RIO drops all the incoming packets

when the instantaneous queue is equal to maxin.

Once π(i) has been obtained RTT, pinand poutcan be evaluated as

RTT = R0+ q/C = R0+1

C

maxin

?

i=0

iπ(i) (15)

pin=

maxin

?

i=0

pin(i)π(i) (16)

pout=

maxout

?

i=0

pout(i)π(i) (17)

where R0is the sum of propagation and transmission delays.

We have followed such approach, but results are unsatisfactory. The physical

explanation appears from figures 5(b) and 5(a), which show the empirical

distribution coming from simulations and the queue distribution predicted by

the model given the same average load, for three different configurations. The

RIO settings in the legend are given in the form (minout,maxout,Pmaxout)−

(minin,maxin,Pmaxin). According to the model the queue should exhibit a

spread distribution, with high probability for low queue values (in particular

the probability density is strictly decreasing if T < C), while the empirical

distribution looks like a gaussian one: the dynamic adaptive throughput of the

TCP sources, which increase their throughput when RTT decreases and vice

versa, appear to be able to create a sort of “constant bias”.

17

Page 18

020 40 60 80 100120 140

0

0.02

0.04

0.06

0.08

0.1

0.12

Queue size

(a)

Probability

RIO settings=(2,6,0.2)−(8,24,0.05)

RIO settings=(8,24,0.2)−(32,96,0.05)

RIO settings=(24,72,0.2)−(96,288,0.05)

0 2040 6080100 120 140

0

0.02

0.04

0.06

0.08

0.1

0.12

Queue size

(b)

Probability

RIO settings=(2,6,0.2)−(8,24,0.05)

RIO settings=(8,24,0.2)−(32,96,0.05)

RIO settings=(24,72,0.2)−(96,288,0.05)

0 20 40 60 80100 120140

0

0.02

0.04

0.06

0.08

0.1

0.12

Queue size

(c)

Probability

RIO settings=(2,6,0.2)−(8,24,0.05)

RIO settings=(8,24,0.2)−(32,96,0.05)

RIO settings=(24,72,0.2)−(96,288,0.05)

Fig. 5. (a) Queue distribution predicted by the model with uniform arrivals. (b)

Queue distribution obtained by simulations. (c) Queue distribution predicted by

the model with state dependant arrivals.

In order to capture this behavior, we have modified the model in [30], by

introducing arrival dependence from the network status. The input to the

sub-model is

18

Page 19

F(N)=T ∗ RTT =

N

3b−2

6

+

?

2bN

3+

?2+3b

6

?2+ 1

(18)

and the arrival rate when the there are j packets in the queue is:

T(j) =

F

R0+

j

C

Now the stationary distribution can be evaluated as:

π(i) = π(0)

i−1

?

j=0

T(j)

C

(1 − p(j)),i = 1,2,...maxin

Fig. 5(c) shows the queue distribution evaluated by the new model. The sim-

ilarity with figure 5(c) is impressive, the only difference is for the first config-

uration ((2,6,0.2) − (8,24,0.05)), as regards low queue occupancy. The peak

for q = 0 is probably due to timeouts, which are more common with low RIO

settings, and make TCP throughput less uniform and hence the markovian

arrival assumption less accurate.

4.5About the solutions of the system

Summarizing, our model has 8 variables (N,A,T,L,RTT,pin,pout,F) and 8

equations (10), (11), (12), (13), (15) (16), (17) and (18). In this section we

afford existence and uniqueness of solutions for this system. We are going to

reduce the system to a simpler one with two variables (F and q).

First, let us note that F can be expressed as an increasing function of A by

equations (18), (12) and (13). Besides it can be proven that π(i + 1)/π(i)

increases with F and A alike, being pin(i) < pout(i). Hence q, pinand poutare

continuous increasing function of F and A and by the relation between F and

A, we can express them as increasing function of F (e.g. from q = q(A,F) and

A = A(F), q = q(A(F),F) = q(F)). Being these function invertible, we can

express pinand poutas (increasing) functions of q.

Besides, the following results hold:

q(F = 0) = 0

lim

F→+∞q(F) = maxin

19

Page 20

As regards A, from equations (10) and (13) (A = L), we obtain that A is the

solution (if any) of the following equation

A + 1 =sA+1

in

sin− 1

− 1

1

1 − sA

insout

(19)

The right member of equation (19) (i.e. N) is an increasing function of A, sin

and soutas it appears immediately from the same definition of N, taking into

account that pin< pout. Being 1/pout≤ N ≤ 1/pinthe curve represented by

the right member always intersects the line represented by the left member

only in one point (because N increases with A). Hence equation (19) admits

one and only one solution A and this solution increases with sin and sout

(because N increases with sinand sout). From the relation between pin, pout

and q it follows that A is a decreasing function of q. Besides when q converges

to zero poutconverges to zero and A > 1/poutdiverges, i.e.

lim

q→0A(q) = +∞

Let us focus on the expression of F (18). From the relation between F and A

and the relations established above, it appears that F is a decreasing function

of q and:

lim

q→0F(q) = +∞

lim

q→+∞F(q) = 0

From the previous considerations and hypotheses it follows that the simplified

system in F and q admits one and only one solution, as it is qualitatively shown

in figure 6. Being all the function monotone, the original system admits only

one solution.

It is possible to set up an iterative procedure to find numerically this solution,

and this is just what we did using MATLAB.

5 Model validation

To validate our model we considered the network topology showed in Fig. 7,

consisting of a single bottleneck link with capacity equal to 6Mbps. Considering

both the transmission and the propagation delay of packets and acks in the

network, the average Round Trip Time is R0∼= 138ms. The IP packet size is

chosen to be 1500 Bytes, for a bottleneck link capacity of c = 500packets/s.

20

Page 21

maxin

F

q

F(q) q(F)

Fig. 6. Existence and uniqueness of the solution.

Table 1

Model vs Simulation with 10 flows

RIOT (pkt/s)G (pkt/s)q (pkt)A (pkt)

modsim modsimmod sim modsim

(2,6)(8,24)478.02474.87467.57 467.82 7.868.5451.5157.35

(3,9)(12,36)494.22492.59 485.02486.29 12.4912.13 60,42 67.70

(4,12)(16,48)499.25 499.76490.94494.27 16.72 14.9067.17 79.50

(6,18)(24,72) 501.29504.15 494.25499.2724.52 22.1778.83 90.76

(8,24)(32,96)501.10 504.39 495.02499.8832.04 29.2490.33 97.96

(12,36)(48,144)500.39 503.83 495.68499.9846.63 44.01 114.59113.96

(16,48)(64,192)499.93503.23 496.15 499.9960.89 58.57 141.06132.07

(24,72)(96,288)499.44502.37496.84 500.0089.0386.67 201.26169.65

mean error (%) -0.28 -0.65 5.00-3.92

max error (%) -0.68-1.00 12.22 18.63

RIO Pdrop (%) Pdropin(%)Pdropout (%)

mod sim modsimmodsim

(2,6)(8,24)2.185 1.4551.009 0.194 62.79175.099

(3,9)(12,36)1.860 1.2880.731 0.13170.108 80.599

(4,12)(16,48)1.665 1.1350.580 0.090 74.59084.961

(6,18)(24,72)1.4051.017 0.4080.05179.978 89.252

(8,24)(32,96) 1.2140.939 0.3030.03783.494 89.453

(12,36)(48,144)0.942 0.7920.1840.017 87.80589.628

(16,48)(64,192)0.7570.670 0.1210.012 90.45288.069

(24,72)(96,288) 0.5220.5060.060 0.01193.531 85.179

mean error (%)30.52 648.84-6.02

mean error (%) 50.24991.51 -16.39

As regards RIO configurations we considered non overlapping the ones in

which maxout< minin, more precisely we choose maxout= 3minout, maxin=

3mininand minin= 4maxout. In previous performance evaluation this kind

of settings showed better results in comparison with a overlapping RIO con-

figuration in which, maxout >= minin. We tested seven different configura-

tions, varying minoutfrom 2 up to 24, and for each configuration we gathered

statistics from 10 trials of 1000 seconds each. We chose Pmaxout= 0.2 and

Pmaxin= 0.05. We ran our simulations using ns v2.1b9a, with the Reno ver-

21

Page 22

C1

C2

E1

E2

E3

E4

6Mbps

19ms

30Mbps

5ms (avg)

30Mbps

5ms (avg)

60Mbps

19ms

19ms

60Mbps

S5

S6

S7

8

S10

S1

S2

3

S

D1

D2

3

D

D4

D5

D10

D9

10

D

D7

D6

S

S4

9S

Fig. 7. Network topology.

sion of TCP.

Table 1 compares model predictions with simulation results when the number

of flows is equal to n = 10, as regards throughput (T), goodput5(G), queue

occupancy (q), the dropping probability for the generic packet, for IN packets

and for OUT packets (respectively Pdrop,Pdropin, Pdropout), and the average

length of IN packets bursts. The average mean error over the different settings

and the maximum error are shown in the last two rows. The model appears to

be able to predict with significant accuracy throughput, goodput and queue

occupancy, which are the most relevant performance indexes when we consider

TCP long lived performance flows. On the contrary dropping probability es-

timates are very inaccurate, in particular as regards Pdropin. We think the

reason is that the model neglects the effect of filtering on dropping probability

calculation from RIO routers. In fact some preliminary results which take into

account filtering seem to suggest that filtering: i) can be neglected in order

to evaluate the dynamic of the instantaneous queue, ii) it is significant for

the evaluation of the dropping probabilities. In particular probabilities esti-

mates look better. At the moment we have introduced the effect of filtering

by considering a two dimensional Markov chain where the status is the pair of

instantaneous queue and filtered queue (whose values have been quantized).

This approach is particularly heavy from the computational point of view, for

this reason, at the moment, we have not adopted it.

The goodput/delay tradeoff is presented in Fig. 8, where each point corre-

sponds to a different threshold setting.

We evaluated also the model with the same network topology with a different

number of flows (n = 6,n = 20). The differences between model predictions

and simulation results are similar to those observed for n = 10 flows. The

relative errors for these two scenarios are shown in Table 2.

5The goodput is estimated as G = T(1 − Pdrop).

22

Page 23

Table 2

Model vs Simulation with 6 and 20 flows

n = 6

T (pkt/s)

G (pkt/s)

q (pkt)

Pdrop (%)

Pdropin(%)

Pdropout (%)

A (pkt)

mean error (%) -0.65-0.74 -1.0112.12649.54 -1.60 25.98

max error (%)-0.86-0.95 -22.4623.71 1141.3028.5858.46

n = 20

T (pkt/s)

G (pkt/s)

q (pkt)

Pdrop (%)

Pdropin(%)

Pdropout (%)

A (pkt)

mean error (%) 1.25-0.27 2.1783.70 598.05-3.23 -36.48

max error (%)4.431.58 -18.42120.97729.45 -13.51-46.13

0

0.05

0.1

0.15

0.2

0.25

80.00% 82.00% 84.00%86.00% 88.00% 90.00%92.00%94.00% 96.00%98.00% 100.00%

Goodput %

Queue Delay (sec)

PMA - model

PMA - ns

best effort

Fig. 8. Queue Delay vs Goodput for the PMA model and the corresponding ns2 sim-

ulations, together with simulation results of a standard best effort service (without

marking)

6 A model application

Here we want to show a possible application of our model. In particular we

want to evaluate a new variant of the algorithm where a higher number of

packet is marked OUT. Intuitively this new version should be able to react

more quickly to traffic changes, by allowing more probes. With reference to the

algorithm flowchart in Fig. 1, in the new marking scheme a packet is marked

OUT every time CIN exceeds AIN/2. From the modeling point of view we

have only to change the marking law m() (see section 4.3) as it follows:

A =1

2L

The model predictions and the simulation results are shown in Fig. 9 as perfor-

mance frontiers. The same RIO configurations have been considered for both

the original algorithm and the variant, with minoutranging from 2 to 24 while

the other parameters have been chosen according to section 5. It appears that

the model is able to capture the main change: the curve of the new variant

23

Page 24

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

85.00%87.00% 89.00%91.00% 93.00% 95.00%97.00% 99.00%

Goodput %

Queue Delay (sec)

A=L matlab

A=L/2 matlab

A=L ns

A=L/2 ns

Fig. 9. A study of a variant of the PMA

is shifted towards lower utilization because of higher sensibility to congestion,

but its shape is almost unchanged. In order to stress this point, two pairs of

points are circled in the figure: they correspond to the minout= 12 configura-

tion for the original algorithm and minout= 16 for the new variant. It appears

that the two algorithms are able to achieve almost the same performance with

different configurations.

At the same time simulation results show another effect that the model is not

able to catch: the new variant exhibits also a higher queuing delay for average

utilization. We think the reason is a higher traffic variability (the dropping

probability of OUT packets decreases from about 80% to about 50%) which

produces larger queues. This effect is not addressed by the model because

the network sub-model takes into account mainly the average throughput as-

suming the same markovian arrival process independently from the specific

marking strategy.

7 Conclusions and further research issues

In this paper we have presented an analytical model for our adaptive packet

marking scheme proposed in previous works. From preliminary simulative re-

sults, model predictions about throughput and average queue occupancy ap-

pear to be quite accurate. We have also shown that the model can be employed

to evaluate variants of the original marking algorithm. We are going to extend

simulative evaluation and to employ such model to study possible variants of

24

Page 25

the marking algorithm and to establish optimal RIO settings.

Besides our network sub-model exhibits some novelty and seems to be more

suited than traditional M/M/1/K proposals to capture the behavior of long

lived TCP flows. We are going to study it deeply and to evaluate it in a simpler

best effort scenario. We want to evaluate the effect of filtering, which is usually

neglected in M/M/1/K models, but it appears to have a deep impact on the

performance.

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