TRANSMISSION OF A MESSAGE DURING LIMITED TIME

WITH THE HELP OF TRANSPORT CODING

Evgenii Krouk

St.-Petersburg State University of Aerospace Instrumentation, Bolshaia Morskaia str., 67, St.-Petersburg, 190000, Russia

Sergei Semenov

Nokia Technology Platforms, P.O. Box 86, FIN-24101 Salo, Finland

Keywords: packet switching network, coding, transport protocols, telecommunication traffic, redundancy, message

delay, transport coding

Abstract: A fundamental characteristic of the majority of communications networks is the mean message delay. In a

packet-switching network, the mean packet and message delay may differ considerably from each other, and

their distribution will often take different forms. The mean message delay depends both on the mean packet

delay and on the dispersion of the packet delay. Obviously, by reducing the mean packet delay one can also

reduce the message delay. However, it is not always possible to decrease the mean packet delay in the

network. A proposed method of transmitting data in a network is based on the use of error-correcting

coding, which reduces the dispersion of the packet delay with some increase in the mean packet delay. The

conditions were obtained for which an increase in the mean packet delay with simultaneous reduction in the

dispersion leads to a reduction in the mean message delay. In many real time networks there are exist some

restrictions on the message delay. Use of the transport coding makes it possible to deliver the messages over

the network during some limited time with high probability.

1 INTRODUCTION

In this paper we will consider the applications of

error correcting codes to the data network. It is well

known, that using of error controlling codes adapted

to typical errors in a defended system is the

universal method of error protection. However in

modern data networks error correcting (or

controlling) codes are used only as means of

increasing the reliability of information during the

data transmission over the different channels; and no

correlation between coding and other network

procedures is considered. The application of coding

not only to the physical layer but also to the

procedures at higher layers (e.g. transport layer)

gives us some unexpected results indicating that

coding in network helps not only to increase the

reliability of the transmitted information, but also

can be used to improve such important characteristic

of a network, as the mean message delay. In this

chapter we will consider mostly the packet switching

network with datagram routing (or in datagram

mode). Packet switching is switching in which

messages are broken into packets and one packet at a

time is transmitted on a communication link. Thus,

when a packet arrives at a switching node on its path

to the destination site, it waits in a queue for its turn

to be transmitted on the next link in its path. The

datagram routing is packet switching in which each

packet finds its own path through the network

according to the current information available at the

nodes visited (Bertsekas, Gallager, 1992). To be

more precise, there is only one restriction on the

considered network model, which is essential for the

exposition of this chapter. This is the possibility to

get the packets out of order at the destination node.

It is shown in (Bertsekas, Gallager, 1992) that not

only the datagram networks but also the virtual

circuit networks have this feature as well. However,

for simplicity we assume that we are dealing with a

datagram network and that packets can get out of

order arbitrarily on the network consider a packet

switching network.

88

Krouk E. and Semenov S. (2005).

TRANSMISSION OF A MESSAGE DURING LIMITED TIME WITH THE HELP OF TRANSPORT CODING.

In Proceedings of the Second International Conference on e-Business and Telecommunication Networks, pages 88-93

DOI: 10.5220/0001419300880093

Copyright

c

SciTePress

The outline of the paper is as follows. Section 2

describes the concept of transport coding. Section 3

analyses the possibility of using transport coding for

improving the probability of message delivery

during limited time. Section 4 discusses an

interpretation of the results.

2 DECREASING THE MESSAGE

DELAY WITH THE HELP OF

TRANSPORT CODING

The one of the most important measure of the

effectiveness of a data network is the information

delay. The mean packet delay has been subject to

many studies, for example (Kleinrock, 1975),

(Kleinrock, 1964), (Kleinrock, Naylor, 1974).

However, in a packet-switching network, the

parameter of interest is not the delay of a separate

packet but the delay of a message as a whole. And

the mean message delay can differ from the mean

packet delay, as the assembly of a message at a

destination node can be delayed due to the absence

of a small number of packets (for example one).

This section deals with an analysis of the method of

decreasing the mean message delay with the help of

error-correcting code at the transport level of

network. This method was suggested in

(Kabatianskii, Krouk, 1993) and generalized in

(Krouk, Semenov, 2002). The possibility of using

error-correcting code in a bipolar network was

described in (Maxemchuk, 1975).

Let us consider a model of a network having M

channels, in which the capacity of the ith channel is

C

i

. The time taken to transmit a packet over a

channel has an exponential distribution with the

expectation

µ

1 . When the servicing device is busy,

the packet may be placed in a queue. Each message,

arriving in the network, is divided into

K

similar

packets. The length of each packet is

s

bits. The

traffic arriving in the network from external sources

forms a Poisson process with the intensity

γ

(packets per second). We will denote the mean

number of packets passing through the ith channel

per second by

i

λ

. The total network traffic is then

.

1

∑

=

=

M

i

i

λλ

,

(1)

If the packets arrive to a node via different

routes, we can assume that the dependence between

packet delays is negligible. Hence, the model of the

network turns out to be close to the Kleinrock

model, for which the Kleinrock «assumption of

independence» holds (Kleinrock, 1975), (Kleinrock,

1964). According to this assumption, the packet

delays can be regarded as independent random

variables. This statement was proved in

(Vvedenskaya, 1998) for some network types. Then

the ith channel can be represented in the form of a

queuing system with a Poisson flow of intensity

i

λ

at the input and an exponential servicing time with

mean

i

C⋅

µ

1

. In this case we can assume that the

packet delays in the network have an exponential

distribution with the expectation

),(

µ

λ

t , where

.

1

),(

1

ii

M

i

i

C

t

λµγ

λ

µλ

−

⋅=

∑

=

,

(2)

If we consider a case where all

M

channels

have the same carrying capacity while the external

traffic is uniformly distributed between the channels

(so that the intensity of the packet flow for all

channels is the same), expression (2) can be written

as follows:

,

1

1

),(

ρµ

µλ

−

⋅=

C

l

t

(3)

where

γ

λ

γ

λ

i

M

l

⋅

==

is the mean path length

traversed by a packet along the network,

C⋅

=

µ

λ

ρ

is the network load, and

∑

=

=

M

i

i

CC

1

is the overall

capacity of the network channels. The value of the

network load in this case is identical with the

i

i

i

C⋅

=

µ

λ

ρ

, the load of a single channel. In fact, as

it will be shown later, all needed assumptions are as

follows: the packet delays are independent random

variables with the exponential distribution and with

expectation of form

ρ

−1

a

, where

ρ

is the network

load and a is the constant for the given network.

The delay

T

of an uncoded message in the

network is determined by the maximum delay

among the

K

packets of the given message

TRANSMISSION OF A MESSAGE DURING LIMITED TIME WITH THE HELP OF TRANSPORT CODING

89

},,...,max{

1 K

ttT =

where

i

t is the delay of the ith packet of the

message; i.e., the message delay is equal to the delay

of the packet which arrives last. If we redenote the

packet delays of the message in increasing order

KKKK

ttt

::2:1

...≤≤≤ , we have

.

:KK

tT =

We can apply now the coding at the transport

level of the network and to encode the message,

which consists of K packets with the help of an

MDS (N, K) code (for example Reed-Solomon

code). In case of using the Reed-Solomon code each

of K packets is considered an element of a field

GF(2

s

) (s is the packet length), and after encoding

the original message consisting of K packets is

replaced by a message consisting of N packets.

When the encoded messages are transmitting over

the network, the traffic increases by a factor of 1/R

(R=K/N is the rate of the code used). This naturally

leads to an increase in the mean packet delay in the

network. However, at the node-addressee, to

reconstruct the message (in view of properties of

MDS codes), only K packets need to be received, as

against all N packets. We will show that with some

restrictions on the operation of the network this

method leads to a decrease of the mean message

delay. We will denote this method further on as

transport coding.

In case of using transport coding, the delay of the

encoded message is

.

:NKcod

tT

=

Using the apparatus of order statistics (David,

1981), the mathematical expectation of the delay of

the ith packet (for an overall number of packets

N )

can be written as follows:

[] [ ] [ ]

)()(1)(

1

:

tdPtPtPtBtE

iNi

iNi

−

∞

∞−

−

−⋅⋅⋅=

∫

(4)

or

[] [ ]

,1)(

1

1

0

1

:

duuuuPBtE

iN

i

iNi

−

−−

−⋅⋅⋅=

∫

(5)

where

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

−

−

⋅=

1

1

i

N

NB

i

, )(tP is the distribution

function of packet delay and

)(

1

uP

−

is the inverse

function of

)(tP

. In the case of an exponential

distribution of the packet delay in the network

equations (4), (5) can be rewritten as follows:

∑

+−=

−

⋅=

N

iNj

Ni

jttE

1

1

:

][

(6)

where

t is the mean packet delay in the network

(depends on

λ

and

µ

). The mean delay of the

uncoded message in the network is then

TEt t j

KK

j

K

1

1

1

==⋅

−

=

∑

[](,)

:

λµ

(7)

where

),(

µ

λ

t is defined by (3). The sum on the

right-hand side of (7) can be represented as follows:

,

)1(...)1(2

1

ln

21

1

∑∑

∞

==

−

−+⋅⋅−⋅

−++=

i

i

K

j

iKKK

A

K

Kj

ε

∫

−−⋅⋅−⋅⋅=

1

0

,)1(...)1(

1

dxxixx

i

A

i

where

...577.0

=

ε

is Euler’s constant. Hence we

obtain the following estimate for the mean delay of

the uncoded message

1

T :

)ln(

1

1

)ln(),(

1

K

C

l

KtT +⋅

−

⋅=+⋅≥

ε

ρµ

εµλ

(8)

We can write the mean delay of the coded

message

2

T

for given

N , in accordance with (6), as

follows:

,),/(][

1

1

:2

∑

+−=

−

⋅==

N

KNj

NK

jRttET

µλ

(9)

where

),/(

µ

λ

Rt is the mean packet delay for traffic

that has been increased as a result of using coded

messages by a factor of

R1 ; R = K/N is the rate of

the code used. The sum on the right-hand side of (9)

can be represented as follows:

+

−

−+

⎟

⎠

⎞

⎜

⎝

⎛

−

=−=

∑∑∑

=

−

=

−−

+−=

−

N

j

KN

j

N

KNj

KNNKN

N

jjj

11

11

1

1

)(2

1

2

1

ln

≤

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

−+

−

−+−−

+

∑

∞

=2

)1...()1)...((

i

ii

iNN

A

iKNKN

A

⎟

⎠

⎞

⎜

⎝

⎛

−

≤

KN

N

ln

(10)

ICETE 2005 - GLOBAL COMMUNICATION INFORMATION SYSTEMS AND SERVICES

90

0.2 0.4 0.6 0.8 1.0

0

2

4

6

8

10

12

14

K = 10, exact calculation

K = 10, estimation

K = 100, exact calculation

K = 100, estimation

T

1

/T

2

ρ

Figure 1: A gain of transport coding vs. network load.

It is possible to choose the code rate R in such a

way as to minimize the mean message delay

2

T . For

the best-chosen code, we obtain

.),/(min

/

1/

1

2

⎭

⎬

⎫

⎩

⎨

⎧

⋅=

∑

+−=

−

RK

KRKj

R

jRtT

µλ

(11)

Then, with the help of (10) we can estimate (9)

as

.

1

1

ln),/(min

2

⎭

⎬

⎫

⎩

⎨

⎧

⎟

⎠

⎞

⎜

⎝

⎛

−

⋅≤

R

RtT

R

µλ

(12)

The mean packet delay for traffic which has been

increased by a factor of

R1 can be written in

accordance with (3), as

,),/(

ρµ

µλ

−

⋅=

R

R

C

l

Rt

(13)

where

γ

λ

γ

λ

==

R

R

l

is the mean path length

traversed by a packet along the network and

C⋅

=

µ

λ

ρ

is the load of the network when using the

uncoded messages. Minimizing (12) with respect to

R

, obtain

.

)1(

4

2

2

ρ

ρ

µ

−

⋅≤

C

l

T

(14)

The gain of using coding at the transport level of

the network can be obtained when the following

condition is satisfied:

0 200 400 600 800 1000

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

exact calculation

estimation

T

1

/T

2

K

Figure 2: A gain of transport coding vs. number of

information packets in message

.

.0

21

>−TT

(15)

Substituting (8) and (14) into (15), we obtain the

following condition for gain of using coding at

transport level

.

1

4

ln

ρ

ρ

ε

−

>+

K

(16)

The plots of gain of transport coding (in the

sense of decrease the mean message delay) are

shown in Fig. 1, 2.

As one can see from Fig. 1. exact calculation

shows that gain of transport coding can be obtained

with wider range of network load than it follows

from condition (16). However, the estimation

reflects the proper tend of changing gain versus

network load. The plot in Fig. 2 shows that increase

of number of information packets in message leads

to the gain increase that has logarithmic behaviour.

3 MESSAGE DELIVERY DURING

LIMITED TIME

For many data networks the probability )(

0

TP of

message delivery during the time no more than

0

T

has the same importance as the mean message delay.

Let us show that in this case transport coding also

can be used to increase

)(

0

TP . Let

{}

0

Pr Ttp

≤

=

denote the probability of message delivery during

the time no more than

0

T and let

R

p denote the

same probability for the encoded message having

regard to the increased network load. Then for the

uncoded messages we have

TRANSMISSION OF A MESSAGE DURING LIMITED TIME WITH THE HELP OF TRANSPORT CODING

91

0

0.2

0.4

0.6

0.8

1

10 20 30 40 50 60 70 80 90 100

K

P(To), PR(To)

P(To)

PR(To), N=K+1

PR(To), N=K+2

PR(To), N=K+10

Figure 3: A probability of message delivery during time T

0

vs. number of information packets in message, a = 4, ρ =

0.2

.

K

pTP =)(

0

(17)

and in case of using the encoded messages (code

length is N) the probability

)(

0

)(

TP

R

that the

encoded message is delivered during the time no

more than

o

T can be written as follows

()

PT

N

i

pp

R

R

i

i

NK

R

Ni()

()

0

0

1=

⎛

⎝

⎜

⎞

⎠

⎟

⋅− ⋅

=

−

−

∑

(18)

Now if we will use the same assumptions as in

Section 1 about exponential distribution of packet

delay and about dependence of the mean packet

delay on the network load (3) we obtain the

following expression for p:

pe

Tt

=−

−

1

0

/( ,)

λµ

(19)

where

t(, )

λ

µ

is the mean packet delay defined by

(3). Let denote as a the ratio

Tt

0

/(,)

λ

µ

and as

ξ

the ratio of the mean packet delay in the ordinary

network without transport coding to the mean packet

delay in the network with transport coding,

ξλµλµ ρ ρ

==−−

−

ttR R(, ) (/ , ) ( )( )11

1

. Then

formulas (17) and (18) can be rewritten as follows:

()

K

a

eTP

−

−= 1)(

0

(20)

()

PT

N

i

ee

Ra

Ni

i

NK

ia()

()

0

0

1=

⎛

⎝

⎜

⎞

⎠

⎟

⋅− ⋅

−

−

=

−

−

∑

ξξ

(21)

0

0.2

0.4

0.6

0.8

1

10 20 30 40 50 60 70 80 90 100

K

P(To), PR(To)

P(To)

PR(To), N=K+1

PR(To), N=K+2

PR(To), N=K+10

Figure 4: A probability of message delivery during time T

0

vs. number of information packets in message, a = 4, ρ =

0.6

.

It is easy to verify that

0)(

0

→TP with

increasing the number of packets in the message K.

From the other hand, in case

ξ

a

eR

−

>− )1( ,

cTP

R

→)(

0

)(

,

)10(

≤

<

c

with increasing K. The

condition

ξ

a

eR

−

>− )1( with the restriction

ρ

>R

can be written as the following inequality

ρ

ρ

ρ

<<−

−

−

−

Re

a

R

1

1

1

.

(22)

0.9

0.91

0.92

0.93

0.94

0.95

0.96

0.97

0.98

0.99

1

10 20 30 40 50 60 70 80 90 100

K

P(To), PR(To)

P(To)

PR(To), N=K+1

PR(To), N=K+2

PR(To), N=K+10

Figure 5: A probability of message delivery during time T

0

vs. number of information packets in message, a = 7, ρ =

0.6

.

Thus, for any R satisfying (22) the addition of N-

K redundant packets to the message leads to the fact

that the probability

)(

0

)(

TP

R

tends to the constant

greater than zero, whilst the probability

)(

0

TP tends

to zero with increasing K. The plots of

)(

0

TP and

)(

0

)(

TP

R

against K are represented in Fig. 3 - 5.

ICETE 2005 - GLOBAL COMMUNICATION INFORMATION SYSTEMS AND SERVICES

92

4 CONCLUDING REMARKS

The given estimates of mean message delays in the

network are rather rough. However, even these

estimations show that transport coding leads to a

significant decrease of mean message delay under

conditions of moderate network load. Moreover, it is

possible to use transport coding not only to decrease

the mean message delay, but also to increase the

reliability of message delivery during limited time,

which also is a question of great interest. All given

estimations and exact formulas given are based on

the assumption that packet delay in a network has an

exponential distribution. Although this assumption

has some grounding in (Kleinrock, 1975),

(Vvedenskaya, 1998) for many kinds of networks, it

is possible that in some networks the distribution of

the packet delay differs from the exponential one.

However, this assumption was used only for the

simplification of calculations. It is necessary to note

that an exponential distribution of packet delay is

not the best case for use of transport coding because

the probability of a high packet delay is very small.

We can therefore expect that the gain of transport

coding could be more significant for another

distribution of packet delay.

REFERENCES

Bertsekas, D., Gallager, R., 1992, Data Networks. Prentice

Hall.

Kleinrock, L., 1975, Queuing Systems. Theory, vol.1, John

Wiley & Sons.

Kleinrock, L., 1964, Communication nets; stochastic

message flow and delay. N.Y., Dover.

Kleinrock, L., Naylor, W., 1974, On Measured Behavior

of the ARPA Network, AFIPS Conf. Proc., National

Computer Conf., vol. 43, pp. 767-780.

Kabatianskii, G.A., Krouk, E.A., 1993, Coding Decreases

Delay of Messages in Networks. IEEE International

Symposium on Information Theory. Proceedings.

Maxemchuk, N.F., 1975, Dispersity routing, IEEE Conf.

Commun., San Francisco, 1975, N.Y., vol. 3.

David, H.A., 1981, Order statistics, .1, John Wiley &

Sons.

Vvedenskaya, N.D., 1998, Large Queuing System where

Messages are Transmitted via Several Routes,

Problems Of Information Transmission, Vol. 34, Nmb.

2, April–June, pp. 180–189

Krouk, E., Semenov, S., 2002, Application of Coding at

the Network Transport Level to Decrease the Message

Delay. Proceedings of Third International Symposium

on Communication Systems Networks and Digital

Signal Processing. 15-17 July 2002 Staffordshire

University, UK, pp. 109-112.

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