A Transport Coding Gain Estimation in the Conditions of Time Limitation for Maximum Acceptable Message Delay

Abstract

Recently, low delay becomes more and more important. It allows the implementation of new applications and services interacting in almost time that was impossible before. A requirement of low message delay now is a part of many communication standards as 5G and recommendation papers of future standards such as Tactile Internet proposed by ITU-T. In this research, we perform an efficiency estimation of the transport coding for packets with an exponentially distributed delay. We consider a traffic model that is critical to message delay, and an influence of the transport coding on such traffic types and on the message delay jitter. Our research shows that the transport coding can be used not only for decreasing the average message delay but for decreasing the message delay jitter.

Full text

A Transport Coding Gain Estimation in the
Conditions of Time Limitation for Maximum
Acceptable Message Delay
Evgenii Krouk1, Anton Sergeev2and Mikhail Afanasev3
1National Research University Higher School of Economics,
Moscow Institute of Electronics and Mathematics (MIEM HSE)
Tallinskaya Str. 34, 123458, Moscow
1)ekrouk@hse.ru
2,3Saint-Petersburg State University of Aerospace Instrumentation,
B.Morskaya 67, 190000, Saint-Petersburg, Russia
2)slaros@k36.org,3)af-mm@yandex.ru
Abstract. Recently, low message delay becomes more and more impor-
tant. It allows an implementing of new applications and services inter-
acting in almost real time that was impossible before. A requirement of
low message delay now is a part of many communication standards such
as 5G and recommendation papers of future standards such as Tactile
Internet proposed by ITU-T. In this research we perform an efficiency
estimation of the transport coding for packets with an exponentially dis-
tributed delay. We consider a traffic model, that is critical to message
delay, and an influence of the transport coding on such traffic types and
on the message delay jitter. Our research shows that the transport cod-
ing can be used not only for a decreasing of the average message delay
but and for a decreasing of the message delay jitter.
Keywords: transport coding, latency, delay, lattency critical applica-
tions, error correcting coding, network coding, Kleinrock’s model
1 Introduction
There are types of applications and services that are critical to message delay. For
instance, VoIP, remote control systems, spatially distributed security systems,
etc. These systems use audio, video and control commands. When something
happens these systems should response immediately in order to prevent possible
consequences or support suitable quality of service. We can underline several
parameters of such systems:
1. tmax is the maximum acceptable message delay (e.g. 250 ms for VoIP [1]);
2. Message delay jitter is a difference between average message delay and max-
imum and minimum message delay: t0
avg −t0
min, t0
max −t0
avg [1].
Let us consider application, tranport and network levels of OSI. Usual ap-
proaches to decrease message delay are:

1. Adjusting packet routing procedures in the network;
2. Adjusting a packet prioritization in the network (QoS);
3. Using buffers in order to fight with the message delay jitter;
4. etc.
An alternative approach is the transport coding first proposed by Krouk,
E.A., and Kabatiansky, G.A., in [2][3]. The main purpose of the transport coding
is a decreasing of the average message delay in the network. But it also allows
to organize a prioritization of urgent messages [4].
In initial research the simplified Kleinrock’s network model was considered.
In subsequent research, some simplified assumptions were dropped in order to
consider more realistic cases, like non-exponential packet delay [5], different chan-
nel capacities [6], non-heterogenious network structure [7], delivery of a message
during limited time [8], application of the transport coding [9].
The difference of this research from others about the transport coding is
that we add a new realistic assumption of the network model: maximum time
limit of message delivery and the volume of messages that we can afford to
loose. We perform an efficiency analysis of the transport coding for packets
with exponentially distributed delay in the network with the added realistic
assumption.
First we need to consider a network model.
2 The Transport Coding
2.1 The Network Model
In [2], Kleinrock’s network model [10] is considered with additional assumptions.
The network consists of Mchannels and Nnodes. All channels are absolutely
reliable and do not make errors. They have the same channel capacities C. Nodes
perform a processing of received packets, including routing procedures, storing
packets, controlling of packet queues. Nodes are absolutely reliable and do not
make errors. The time of processing at nodes is performed instantly and is not
taken into account in the consideration.
Packets arriving in the ith network node and moving towards to the jth
network node form a Poisson process with the equal average value γ00 (packets
per seconds) for every pair of iand j. The value
γ0=Nγ00
is an external traffic of the ith node, and the value
γ=Nγ0=N2γ00
is the total external traffic of the network.
The network has buffers for packets with infinite capacity. It means that
packets in the network can never be lost bacause of buffer overflow.

All messages arriving in the network consist of kpackets with length m. The
length of packets inside the network satisfies an assumption of independence
[10]: every time when a packet has been received at a node, a new length of this
packet b0is chosen independently with pdf
P(b) = µe−µb, b ≥0,
where b(the size of a packet in bits) is a parameter of the pdf function.
The packet flow going through the channels is a Poisson process with the
equal average value λ0(packets per a second) for every channel of the network.
The total internal traffic in the network is
λ=Mλ0.
Let us denote ρas the value of channel load in the network, then the average
packet delay is:
t(ρ) = λ
µCγ
1
1−ρ.(1)
where ρis
ρ=λ0
µC .
Let us make two additional assumptions that are not contained in the original
Kleinrock’s network model:
1. Packet delays in the network are random and independent values distributed
according to an exponential distribution law with the average value t(ρ)
Fρ(t)=1−e−t/t(ρ), t ≥0,(2)
where t(time units) is a parameter of the cdf function. An exponential dis-
tribution is used for describing of data flows in many researches, in particular
in [11].
2. The routing procedure in the network is chosen in such a way that an in-
creasing of an external traffic leads to the uniform increasing of an internal
traffic of each channel.
2.2 The Message Encoding Procedure
In [2], the following procedure named as the transport coding was proposed:
Initially, original packets arriving in the network have the length m. Let us
consider each character of a packet as an element of GF (2m). It allows us to
encode each message by a 2mMDS code (e.g. a Reed-Solomon code). Then each
original message consisting of kpackets maps to an encoded message consisting
of npackets. After encoding procedure, we will send the encoded message instead

of the original message. It leads to an increasing of the network load ρby a factor
n/k = 1/R and the average packet delivery time becomes t(ρ/R).
It is well known that the word of MDS(n, k) code can be decoded by any k
from ninformation characters. Hence, kfirst arrived encoded packets are enough
to obtain our original message.
The message delay of original messages equals to the kth order statistic from
k–Tk:k. While the message delay of encoded messages equals to the kth order
statistic from n–Tk:n.
In the case of when mis too long, we need to deal with big fields GF, that is
not desirable because of processing difficulty and it is unacceptable for low power
devices like Internet Of Things. In [2], the solution of this case is described.
3 The Problem Statement
Perform efficiency analysis of the transport coding for packets with an exponen-
tially distributed packet delay (2) in the network model described above and
traffic types that are critical to message delay.
Let us denote the direct problem statement and the inverse problem state-
ment:
1. The direct problem statement: minimize the delivery time of packets arrived
later than tmax;
2. The inverse problem statement: find optimal code parameters when tmax is
fixed value.
3.1 The Traffic Model That Is Critical to Message Delay
Network delay is one of the main factors which can degrade the Quality of Expe-
rience (QoE) of network services [12]. So to prevent the degradation of the per-
ceived quality of the services with delay constraints, a maximum limit is defined
in the most of communication systems and protocols. This applies both real-time
(VoIP, RDP, gaming, remote control, video [13] and audio [14] streaming) and
non real-time services (instant messaging, M2M metering etc.). A guidelines for
the maximum allowed latency and proposed multiplexing periods for different
services/scenarios can be found for example in [15] and [16]. Now there are many
research papers considering delay limit issues for video surveillance, real-time
traffic over TCP [17] and other applications. Servide latency limits are in the
focus of 3GPP 5G recommendations. In particular 3GPP Technical specification
”Mission Critical Video over LTE” [18] defines end-to-end maximum latency for
different scenarios: urgent real time video transmissions, Robots video remote
control etc.
We consider this fact in our research and network model and show that
the transport coding can be used both for significant decreasing of the average
message delay and jitter.
First we start with a traffic model which can be described by the following
parameters:

1. tmax is the maximum acceptable packet delivery time;
2. p=P r{Tk:n> tmax}is the volume of messages that have arrived later than
tmax;
3. kis the amount of packets of an uncoded message.
Taking into account the assumption (2) for considered network model, packet
delivery time is a random variable distributed according to an exponential dis-
tribution law. For this traffic model, it is impossible to talk about methods that
can guarantee that all messages will be delivered for the time less than tmax,
but we can talk about a minimizing of the value p.
3.2 The Average Packet Delivery Time
The average packet delivery time of an individual packet (1) depends on many
parameters, such as the total intensity of the internal traffic λ, the total intensity
of the external traffic γ,µ, the capacity of all channels C. We can see that when
ρ= 0 then λ
γµC is an initial packet delivery time in the empty network, and
1
1−ρis the increasing coefficient depending only on ρ. Thus, absolute values of
these parameters are not important. Only the initial packet delivery time in the
empty network is of interest. Let us denote it as tsand rewrite (1) as
t(ρ) = ts
1
1−ρ.(3)
For all next calculations we consider ts= 2. This value was taken thus that
for tmax = 9 the considered network model will not be able to deliver messages
for the average time less than tmax with the intensity of the channel traffic ρ. We
need to note that using time units are abstract values and they can be expressed
in needed units for a particular task.
3.3 The Direct Problem Statement
Let us fix the value tmax, what is the probability p(ρ) of that the message will
be delivered later than tmax when ρis known?
For the next calculations let us set k= 8.
t(ρ/R) = t(ρ)1−ρ
1−ρ
R
p= 1 −P r {Tk:n≤tmax}= 1 −
n
X
i=k
Ci
n(1 −ea)iea(n−i), n =k
R
where
a=−tmax
t(ρ/R)=−tmax(1 −ρ
R)
t(ρ)(1 −ρ)
In Fig. 1, we can see the probability of message lost for different code rates
for the case when tmax = 9 and the intensity of channel load ρ= 0.6:

Fig. 1. The direct problem statement when tmax = 9
1. In the case without the transport coding (R= 1) the probability of message
lost is p≈0.76;
2. In the case R= 0.89, p = 0.45;
3. In the case R= 0.80, p = 0.22;
4. In the case R= 0.67, p = 0.04.
When p= 0.2, without the transport coding the maximum channel load is
ρmax ≈0.2, but in the case with transport coding and the code rate R= 0.8,
the maximum channel load can be increased to ρmax ≈0.6.
3.4 The Influence of the Transport Coding on the Message Delay
Jitter
First we need to start with determining a dispersion and a standard deviation
for the average message delay. Taking into account the assumption (2) about an
exponential distribution law of packet delay, we can obtain:
M[T] = t(ρ)
D[T] = t(ρ)2
The dispersion of the kth order static [19]:
D[Tk:n] = D[T]
n
X
i=n−k+1
i−2=t(ρ\R)2
n
X
i=n−k+1
i−2

D[Tk:k] = D[T]
k
X
i=1
i−2=t(ρ)2
k
X
i=1
i−2
The gain in the standard deviation is
f(R) = σ[Tk:k]
σ[Tk:n]=pD[Tk:k]
pD[Tk:n]=t(ρ)qPk
i=1 i−2
t(ρ\R)qPn
i=n−k+1 i−2
=(1 −ρ\R)qPk
i=1 i−2
(1 −ρ)qPn
i=n−k+1 i−2
Fig. 2. The gain in the standard deviation
In Fig. 2, we can see a curve line of the gain in the standard deviation
depending on the intensity of channel load and the assessment of the gain in
the standard deviation obtained in [2]. The following cases were depicted in that
figure:
1. σ[Tk:n] when ρ= 0.2;
2. M[Tk:n] when ρ= 0.2;
3. σ[Tk:n] when ρ= 0.4;
4. M[Tk:n] when ρ= 0.4;
5. σ[Tk:n] when ρ= 0.6;
6. M[Tk:n] when ρ= 0.6.
We can conclude from obtained plots that the transport coding can decrease
the standard deviation and it means that the transport coding can also decrease

jitter of message delay. In particular, in the case when ρ= 0.2 and R= 0.36,
the gain of the standard deviation is ≈4.5 times, while the gain of the average
message delay [2] is ≈3.5 times.
3.5 The Inverse Problem Statement
Let us fix the value tmax and the probability p, that the message has arrived
later than tmax. What is a code rate we need to use in order to keep chosen p(ρ)?
p≥1−
n
X
i=k
Ci
n(1 −ea)iea(n−i), n =k . . . nmax
where nmax is a maximum amount of packets from which our message consists
of, and
a=−tmax
t(ρ/R)=−tmax(1 −ρ
R)
t(ρ)(1 −ρ)
For next calculations let us set nmax = 100.
Fig. 3. The inverse problem statement when tmax = 9

In Fig. 3, the following curves were depicted for the case tmax = 9:
1. tmax;
2. t(ρ);
3. M[Tk:n](ρ), σ[Tk:n](ρ), R(ρ) when p= 0.1;
4. M[Tk:n](ρ), σ[Tk:n](ρ), R(ρ) when p= 0.2;
5. M[Tk:n](ρ), σ[Tk:n](ρ), R(ρ) when p= 0.5;
6. M[Tk:n](ρ) when R= 1;
7. σ[Tk:n](ρ) when R= 1.
We can conclude that when the channel load ρincreases, the code rate de-
creases too in order to maintain chosen parameters of the traffic model, and
then after some ρmax, the transport coding will not be able to maintain chosen
parameters.
Conclusion
In this research we suggested new traffic model that is critical to the delay. The
traffic model consists of the maximum acceptable message delay tmax and the
probability that a message has been delivered later than tmax, p =P r {Tk:n>
tmax}. We performed efficiency analysis of the transport coding for considered
traffic model.
For considered network model parameters (ts= 2) and the traffic model
(tmax = 9, k = 8, nmax = 100), we concluded that the transport coding allows:
1. Decreasing of the message delay jitter. In the case of ρ= 0.2 and R= 0.36,
the gain of the standard deviation is ≈4.5 times, while the gain of the
average message delay [2] is ≈3.5 times;
2. Increasing of the maximum intensity of channel load ρmax in order to main-
tain chosen parameters of traffic and network models. In the case of p= 0.2
and without transport coding, we can maintain chosen parameters of chan-
nel and network models with the maximum intensity of channel load is
ρmax ≈0.2, while in the case with transport coding and the code rate
R= 0.8, this parameter can be increased to ρmax ≈0.6;
3. Decreasing of the probability pof that the message has arrived later than
tmax in the case of a fixed intensity of the channel load ρ. In the case without
the transport coding when ρ= 0.6 the probability of message lost is p= 0.76,
while the transport coding allows decreasing of this value to p= 0.04 with
a coderate R= 0.67.
Further possible research directions of the transport coding can include:
1. Efficency analysis of the transport coding in more realistic cases including
parameters such as: MTU size, packet’s length in bytes, packet delay in
milliseconds, etc.;
2. Considering of real protocols that are critical to the message delay.

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