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
Dipartimento di Ingegneria Elettronica, DIE
Universit` a degli Studi di Roma - Tor Vergata
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
Differentiated Services (DiffServ) networks provide the ability to enforce a dif-
ferent forwarding behavior to packets, based on their Differentiated Services
Email addresses: email@example.com (Giovanni Neglia),
firstname.lastname@example.org (Vincenzo Falletta),
email@example.com (Giuseppe Bianchi).
Preprint submitted to Elsevier Science12 September 2005
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 .
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
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  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 , 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
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
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 , where the authors propose a time-sliding window marker. In
 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 . 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  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  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  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
These results suggested the need to introduce some adaptivity in order to
cope with TCP dynamics. In  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
Management is proposed in  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 . The adaptive dual
token bucket in  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  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  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  and : the first
concentrates on the effect of different RTTs, the second propose the Direct
Congestion Control Scheme to achieve fairness between responsive and unre-
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  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  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
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  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  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  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 : 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  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.
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
CIN := CIN + 1
Lseq:= Lseq+ 1
CIN := CIN + 1
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
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
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
Fig. 2. The three-block model.
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 , reciprocal model tuning .
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
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 .
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
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.:
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
then we can note that a solution of such system ([T∗,L∗,p∗
if any, satisfies the following relation:
out,RTT∗,A∗] = U(T∗,L∗,p∗
i.e. the point [T∗,L∗,p∗
out,RTT∗,A∗] is a fixed point for the ?6→ ?6
mapping, established by the function U2. This remark justifies the name of
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  or ). 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)
i−>∞[Ti,Li,pin,i,pout,i,RTTi,Ai] = [T∗,L∗,p∗
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 . 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.
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  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]).  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.  considers zero buffer
queue and  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  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  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 
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 . 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
Yi - 1
?i + 1
no. of rounds
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  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
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]
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:
(1 − P(n)) =
where p(n) is the probability of losing the n-th packet after (n−1)-th successful
transmission, P(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
l=0p(l) is cumulative distribution function, and so
Q(n) = skA+h
where sin= 1 − pin, sout= 1 − pout. The expression of N can be rewritten as
and can be solved in a close form:
1 − sA
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
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,
and taking expectation we get
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:
b + βi=XiWi−1
and taking again expectation it follows
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[Ii] = RTT
3b − 2
?2 + 3b
3The formula is a linear approximation of the exact relation Wi = Wi−1/2 +
?Xi/b? − 1. In  a different approximation has been considered.
Now we can write down the throughput formula:
T(N,RTT) = n
RTT(E[X] + 1)=
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 :
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
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.
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  and  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
Fig. 4. Interaction between the Network Model and the Sources Model.
In  we have proposed a network submodel, extending the approach pro-
posed in  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 ). 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  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 . 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  and ,
which model respectively Token Bucket and Single Rate Three Color Marker,
but it differentiates because it assumes state dependent arrivals, rather than
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
According to , the stationary distribution of the queue can be evaluated
π(i) = π(0)
(1 − p(j)),i = 1,2,...maxin
where C is the bottleneck capacity, π(0) is given by the normalization equation
(1 − p(j))
p(i) =Tinpin(i) + Toutpout(i)
=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
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”.
020 40 60 80 100120 140
0 2040 6080100 120 140
0 20 40 60 80100 120140
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 , by
introducing arrival dependence from the network status. The input to the
F(N)=T ∗ RTT =
and the arrival rate when the there are j packets in the queue is:
Now the stationary distribution can be evaluated as:
π(i) = π(0)
(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
F→+∞q(F) = maxin
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
1 − sA
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.
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:
q→0F(q) = +∞
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
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.
Fig. 6. Existence and uniqueness of the solution.
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-
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).
Model vs Simulation with 6 and 20 flows
n = 6
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
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
80.00% 82.00% 84.00%86.00% 88.00% 90.00%92.00%94.00% 96.00%98.00% 100.00%
Queue Delay (sec)
PMA - model
PMA - ns
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
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:
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
85.00%87.00% 89.00%91.00% 93.00% 95.00%97.00% 99.00%
Queue Delay (sec)
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
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
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
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
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