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1
Non-profit
Joint stock
company
Department of
Telecommunication
Networks and
Systems
TELETRAFFIC THEORY
Lecture notes for students majoring in specialty
5В071900 – Radio engineering, electronics and telecommunications
Almaty 2019
ALMATY
UNIVERSITY OF POWER
ENGINEERING &
TELECOMMUNICATIONS
2
ISSUED BY: Tumanbaeva K. Kh. Teletraffic theory. Lecture notes for
students majoring in specialty 5В071900 specialty - Radio engineering, electronics
and telecommunications. - Almaty: AUPET, 2019. – 61 p.
The lecture notes contain a summary of the topics of the discipline
"Teletraffic theory". They present the basics of the theory of message service in
switching systems and communication networks, methods for predicting the load,
calculating the quality of service in telecommunications systems. The approximate
engineering methods of calculation are presented.
Fig.31, table 1, bibliogr. – 11 references.
Reviewer: Kim Yelena Sergeevna, senior Lecturer, master of Technical
Sciences
Published according to the publishing plan of a Non-profit joint-stock
company “Almaty university of power engineering and telecommunications”.
© NJSC «Almaty university of power engineering and telecommunications»,
2019.
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Introduction
The purpose of studying the discipline "Teletraffic theory" is to prepare
students who are familiar with the basic principles of constructing mathematical
models of servicing the message flows necessary to solve problems of optimal
design of switching systems and communication networks and their qualified
service.
Having studied the basic methods of the teletraffic theory, a student will be
able to calculate the characteristics of the quality of service in telecommunication
systems, manage the basic parameters of the quality of service of real networks and
systems, and measure them, as well as offer optimal technical solutions in terms of
the quality of service when designing new networks and systems.
The objectives of studying the discipline are mastering students of methods
for predicting the load, calculating the quality of service in switching systems with
losses and waiting, methods for analyzing various models of call flows.
4
Lecture 1. Goals and main tasks of the theory of teletraffic
The purpose of the lecture is to familiarize students with the purpose and
tasks of the discipline, with the basic messaging service model studied in teletraffic
theory, with the basic terms of the theory, properties and characteristics of call
flows.
Content:
a) the goals and objectives of the discipline;
b) the basic concepts of teletraffic theory;
c) properties and characteristics of call flows.
The foundations of the theory were set in the works of the Danish
mathematician, an employee of the Copenhagen telephone company A.K. Erlang.
Works were published in 1908-1918. The new scientific discipline was called the
theory of teletraffic, the subject of its study were the processes of servicing by
systems information distribution of incoming message flows and their numerical
characteristics. Information distribution systems include: communication networks,
a switching station, a switching center and their separate parts serving incoming
communications.
The results obtained by A.K. Erlang, served as the basis of the theory of
queueing system , known to you from the course "Modeling of telecommunications
systems". Now the theory of queueing system, in addition to telecommunication, is
effectively used to solve problems of trade, transport and other spheres of economic
activity.
The basic mathematical model of the service process, studied in teletraffic
theory, contains the following components: the flow of incoming messages, the
service system, discipline and quality of service characteristics.
The discipline of serving incoming messages is understood as the way of
servicing (without loss of messages, with obvious losses, with expectation,
repetition, combination), order of service (in order of priority, in random order, with
priority), way of seeking free outputs (ordered or random).
Quality of service characteristics of the incoming message flow include: the
probability of message loss, the average message service time, the average wait
time for service start, the capacity of the service system, and others.
The main task of teletraffic theory is the problem of analysis, namely the
purpose of determining the quality of service characteristics depending on the
properties and parameters of the incoming call flow, the system and the service
discipline. Along with this task, the reverse is also solved - finding the parameters
of the service system depending on the parameters and properties of the call flow
and the quality of the service.
The main directions of the theory include the development of methods for
constructing optimal structures for switching circuits, methods for calculating
throughput, tasks for optimizing the structure of the communication network, and
calculating the quality of service indicators.
5
Rationing and optimal distribution of network segments and stages of
connection of quality of service indicators is another class of problems in teletraffic
theory.
The mathematical apparatus of the theory of teletraffic is based on probability
theory, combinatorics, and mathematical statistics.
Let's consider the basic terms of the theory.
A message is a form of information representation that has signs of beginning
and end and intended for transmission through a communication network or a
switching system.
The message can be a telephone conversation, a facsimile transmission, a
tele-radio transmission, a computer file. The message is characterized by the
volume, occupied channel, the transmission time, the addresses of the source and
the receiver, the form of the information (analog, digital).
A call is a requirement of a source to establish a connection that has entered a
communication network or a switching system for the purpose of transmitting or
maintaining a message [10].
The call is characterized only by the moment of its receipt. The source of the
call can be telephones, fax machines or a computer, they can also be receivers. Calls
are divided into several types:
- serviced which received the connection and at the same time by the fully
serviced called received the connection with the required receiver, and the call that
received the connection on some part of the network is called partially served;
- successful, ended by sending a message to the receiver;
- lost, denied service;
- delayed, waiting to start the connection, waiting in line;
- primary, the first call for this message;
- repeated, entered the communication network through a random or
deterministic period of time, after the previous one was lost.
Occupation is any use of the device, line, device to establish a connection,
whether it ends with a message or not. Characterized by the moment of occupation
and duration.
Release - return the device, line, device to the original, inoperative state.
Characterized by the moment of onset.
A lot of consecutive moments of incoming calls form a Call flow.
A call flow is called deterministic (non-random) if this sequence of moments
is predetermined in advance. If this sequence is random, then the call flow is called
random.
Deterministic flows are rarely found in practice. An example of deterministic
flows can be the streams of the moments of the beginning and the end of the
programs of radio and television broadcasts, the flow of communication sessions
with artificial space satellites. In communication networks, in general, there are
random call flows.
6
To specify random flows, are used the probabilistic laws of distribution of the
following random variables: the moments of arrival of calls, the intervals between
neighboring calls (zk) and the number of incoming calls in the interval [0,t].
Random call flows are classified according to the presence or absence of the
following three properties: stationarity, aftereffect, and ordinary[1,2,4].
Stationarity means that with time, the probabilistic characteristics of the flow
do not change, in other words, for a stationary flow the probability of entering i-
calls for a time interval t depends only on the length of this interval and does not
depend on its location on the time axis.
This means that for a stationary flow the probability of the arrival of a certain
number of calls for a certain time interval depends on the length of this gap and
does not depend on its origin. Otherwise, the flow is non-stationary. The intensity of
call flows on telephone networks varies greatly depending on the time of day; the
number of calls per unit of time in daytime and evening hours reaches a maximum
value, and decreases at night. This means that the call flow that arrives during the
day is non-stationary. But within a limited period of a day, for example, an hour, the
nonstationarity of the telephone call flow is not very noticeable, which makes it
practical to consider it stationary.
Ordinarity (simplicity) means the impossibility of group coming of calls, that
is, the probability of entering of two or more calls for any infinitesimal interval is
infinitesimal. In communication networks, call flows are ordinary.
An example of a single flow is a call flow that arrives at the PBX from a
subscriber group of any capacity. Flows of telephone calls to the subscribers of the
dispatching or conference connection, the flows of telegrams are not ordinary.
An aftereffect (independent) means the dependence on the probabilistic
characteristics of calls from previous events.
An example of a flow without consequences is the flow of telephone calls
coming from a large group of sources. Indeed, only a small part of the subscriber
group simultaneously participates in telephone connections. Therefore, the
probability of receiving any number of calls from a large group of sources at any
time interval is practically independent on the process of receiving calls before the
beginning of this interval.
A call flow is a thread with a consequence if the probability of receiving a
certain number of calls for a certain period of time depends on the process of
receiving calls before the beginning of this interval.
The main characteristics of the flow of calls include the leading function of
the flow, its parameter and intensity.
The leading function ),0( tх of a random stream is the mathematical
expectation of the number of calls in the interval [0, t). The function ),0( tх is
nonnegative, non - decreasing.
Under the flow parameter λ (t) at time t is meant the limit of the probability of
the arrival of at least one call in the interval [t,t+t] to the length of this interval t
at t → 0:
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λ(t) = .
According to the definition of a stationary flow, the probability of a certain
number of calls reaching a certain time interval is the same, does not depend on the
location on the time axis of this interval. Consequently, the probability density of
the calls to the stationary flow, that is, its parameter λ(t) is a constant value,
independent of the moment t, that is, λ(t) = λ .
The flow parameter λ(t) characterizes not the flow of calls, but the flow of the
triggering moments, and this characteristic does not apply to the entire interval [0,
t], but only to the fixed moment t.
The intensity of the stationary flux μ is the mathematical expectation of the
number of calls arriving per unit time. This parameters are called as arrival rate λ
and service rate μ.
For ordinary flows, μ = λ = const.
Lecture 2. Call flows
The purpose of the lecture is to familiarize students with the main types of
call flows used in Teletraffic theory.
Content:
a) Poisson call flow;
b) properties and characteristics of the Poisson call flow;
c) a nonstationary Poisson flow;
d) Bernoulli flow;
e) the flow of release.
The Poisson call flow..
A stationary, ordinary flow without aftereffect (independent) is called the
Poisson call flow.
The Poisson call flow is given by the family of probabilities
(t) of the
arrival of i calls in the interval t.
The probability Pi(t) is calculated by the formula:
Pi(t) = t
i
ei
t
!, (2.1)
where λ is the arrival rate, a constant value, since the flow is stationary;
λ=μ, because the flow is ordinary.
The formula (2.1) is called the Poisson formula or the Poisson distribution.
Consider the formula (2.1). Let us calculate the ratio of two neighboring
probabilities.
8
t
et
et
t
t
t
t
1
1)(
)!1()(
)(
)( .
For I ≤ λt→1
(t)<
(t), and for i>λt→1
(t)>
(t). Consequently, as i
increases, the probability Pi (t) increases as i ≤ λt and begins to decrease for i> λt.
The function
(t) for the whole value λt reaches the maximum at 2 points,
for i = λt, i = λt-1, and for a fractional value at one point i = [λt].
When solving λt practical problems, the probability of receipt of at least i
calls for the interval t is used.
Р≥i (t)=
(t).
The probability values Р≥i (t) for some values of x = λt and i are tabulated in
the annexes of the textbooks [1, 2]. Using the data in the table, you can then
calculate the probability of receiving no more than i calls for the interval t and the
probability of receiving i calls for a given period.
P≤i(t) = 1 - P≥i+1 (t) and Pi(t) = P≥i (t) - P≥i+1(t). (2.2)
Figure 2.1
Example. Calculate the probabilities of entry 5, at least 5, and no more than 5
calls for a time interval t = 180 seconds, if λ = 160 calls per hour. 5
(t), 5
(t), 5
(t)-?
We calculate
λt= 81803600
1160 .
Using the table presented in (4), we find 5
(t)=0,9004, 6
(t)=0,8088.
Then by the formula (2.2):
5
(t)=1-0,8088=0,1912;
5
(t)=0,9004-0,8088=0,0916.
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The Poisson flow can be defined by the distribution function of the interval
between neighboring calls z:
F (t) = P (z <t) = 1-P (z> t).
The probability P (z> t) is equivalent to the probability that there is not a
single call within the interval of length t.
F(t)=P(z>t)=1-0
(t)=1- е . (2.3)
The distribution law (2.3) is called exponential, and λ is its parameter.
Let us consider the properties and characteristics of the simplest flow. The
mathematical expectation of the gap between neighboring calls z is Mz = 1 / λ. The
dispersion of this quantity is 1/ 2 , therefore,
The standard deviation σz = 1 / λ, that is, equality
Mz = σz = 1 / λ.
The coincidence of these quantities is used to verify that the random variable
z is distributed according to exponential law.
The mathematical expectation of the number of calls i for the time interval t
is equal to λt, the variance of the number of calls for the interval t is also equal to λt,
that is,
Mi = Di = λt.
The coincidence of these quantities is used in practice when checking the
correspondence of the real flow to the simplest one.
When several Poisson flows are combined, Poisson call flow is formed with a
parameter equal to the sum of the parameters of the input flows:
λ = λ1 + λ2 + ... + λn.
When the incoming Poisson flow is disconnected from the parameter λ to n
directions, n Poisson flows are formed.
These properties and characteristics of the Poisson flow are widely used in
the calculation of station equipment and communication networks.
The Poisson flow model has the largest distribution in the Teletraffic theory
in relation to all other models.
Briefly consider other types of call flow.
A non-stationary Poisson flow is an ordinary flow without aftereffect, for
which at any time t there exists a finite parameter λ (t).
There are flows with variable and random parameters. In the first case, λ (t) is
known, in the second case, λ (t) is a random function.
10
Consider λ (t) with a variable parameter, in this case λ (t) can be continuous
or stepwise. In the latter case, the function changes its value by jumps to the already
known or random times t1, t2, ... tn.
The model of a Poisson flow with a variable parameter is as follows
Pi
t0
t
0
i
ie
t 0
,
where Pi (t0, τ) - is the probability of entering i calls in the interval (t0, t0 + τ);
λ(t0,τ) - is the mathematical expectation of the number of calls received in
the interval (t0, t0 + τ);
,0
t - the average intensity of the call flow in the interval τ.
This model allows to describe quite well the real unsteady flow of calls, for
example, calls to the telephone exchange within 24 hours.
A primitive flow is an ordinary flow that parameter λi is directly proportional
to the number of free sources of Ni in the state of the serviced system i:
λi=αNi=α(N-i),
where α - is the call flow parameter from one free source;
N - is the total number of sources;
i - number of occupied sources.
The mathematical model of a primitive flow is the Bernoulli distribution:
Pk = kN
kk
NaaС
)1( ,
where is k the number of incoming calls;
N - total number of sources;
a - the load coming from one source (
1a , the intensity of the flow
from one source);
kP - the probability of receiving calls.
In the calculation of k
P it is convenient to use the recurrence formula of
Bernoulli:
. a
a
k
kNPP
kk
111
.
The flow model takes into account that calls can only come from free
sources. The flow parameter assumes the greatest value exist when all sources are
free, and the smallest when i reaches the maximum.
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This model of the call flow is applied when the number of sources is not very
large, when N> 300-500 the flow turns into the Poisson flow.
The flow of release
The sequence of call termination moments forms a release flow. The
properties of the release flow generally depend on the properties of the incoming
call flow, the quality of the switching system, and the law of service time
distribution [1,10].
The service time can be deterministic or random. The deterministic service
time is characteristic for markers of coordinate PBXs and PBX processors with
program control. In other cases, the service time is specified by the distribution law.
In communication networks and switching systems, the call processing time is in
most cases distributed according to the exponential law.
Let there be occupied k lines in the switching system, then the probability
that i lines will be released during the time t is determined by the formula:
ikii
kppCtkiP
)1(),,( , (2.4)
where
)!(!
!
iki
kС
i
k
, k>i ;
p - is the probability that one line will be released for the interval t. The
formula (2.4) is called the Bernoulli formula.
Under the exponential distribution law of the service time, the probability p is
defined as h
t
ep
1 ,
1h , therefore,
h
ikt
ih
t
i
k
ikh
t
ih
t
i
keeCeeCtkiP
)(
)1()11()1(),,(
.
The probability that no line will be released in time t:
h
tk
h
kt
h
t
keeeCtkP
)0(
00)1(),,0( .
The probability that at least one line will be released:
h
tk
etkiP
1),,1( .
The parameter of the release flow for busy k lines is:
h
k .
The flow of releases is ordinary and its parameter is proportional to the
number of occupied lines. If the switching system works in such a way that the
released line immediately deals with the incoming call, then the release flow has a
constant parameter υ / h and is the simplest in its properties.
12
Lecture 3. Load
The aim of the lecture is to familiarize students with the important concept
that characterizes the functioning of the switching system, the load, the units for
measuring the intensity of the load, the main types of load, and the method for
calculating the intensity of the subscriber load.
Content:
a) determination of load, load intensity, unit of measurement;
b) types of load;
c) factors affecting the change in load;
d) calculation of the incoming load intensity.
The load or more precisely the load serviced at time t, is the number of
simultaneously serviced calls or the number of occupied inputs, outputs, lines and
devices at this point in time.
Since the load is a random quantity, the mathematical expectation is used in
the calculations:
)()(
1
tPitY
V
i
i
,
where Pi (t) - is the probability of occupying i lines from v possible at time t.
For a constant probability Pi within a certain time interval, the value of Y (t)
will also be constant.
The mathematical expectation of a load is called the intensity of the load. The
load and its intensity are measured in Erlang.
The intensity of the load 2.5 Erl. Means that the number of occupied lines
during the period under consideration is either on the average 2.5, or 2.5 lines on the
average for the considered interval.
There are three main types of load: offered traffic (incoming), carried traffic
(served) and lost.
Figure 3.1
In figure 3.1, A is the offered traffic, Y is the carried traffic and - the
lost traffic.
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The offered traffic A can only be predicted, the carried traffic Y can be
measured, and the lost one can be defined as the difference between the incoming
and the served
.
When measuring the intensity of the load, it is defined as the average number
of occupied lines in the beam over a period of time (21
, tt ).
The following factors affect the intensity of the load: the structural
composition of subscribers (the share of the apartment, national economic and
administrative sectors), the rhythm of local life (the beginning and end of the
working day), the time of day, the day of the week, the day of the month, the month
of the year.
During the examination of the daily load distribution when the load reaches
its maximum value a gap of 1 hour is selected. This hour is called the Busy Hour of
the greatest load (BH).
Figure 3.2
Figure 3.2 shows the approximate daily distribution of the load intensity for
two sectors of subscribers.
In addition when the load reaches its maximum value a gap of 3 hours is
allocated. It is called the period of the greatest load or Busy Period (BP).
The degree of load concentration in BH and PH is estimated using the
appropriate concentration coefficients.
Concentration coefficient in BH:
,
where .
Contraception coefficient in BP:
.
14
The lower the concentration ratio is the more switching equipment is loaded
and the less volume it takes to perform the same work.
Calculation of the intensity of the offered traffic.
Calculation of the intensity of the offered traffic is carried out according to
the following formula:
A = iii
tCN , (3.1)
where A - is the magnitude of the incoming load, (erl.);
- number of subscribers of the i-th category;
n - number of category;
С - the number of calls in BH coming from one i-category subscriber;
ti - the average duration of one seizure (occupation) for the i-category.
When designing telephone exchange the values
and
С are determined
specifically for each telephone area using statistical observations.
The following main categories of subscribers are considered in the
calculation:
- subscribers of the housing sector;
- subscribers of the business sector;
- subscribers of payphones;
- subscribers from trunks established by PBX.
Now let’s consider how the value of ti is determined. For this purpose we
define types of seizures.
Sessions are successful and unsuccessful. Successful sessions include ended
conversation.
The reasons for the unsuccessful seizure are:
a) busyness of the called subscriber;
b) absence of a response from the called subscriber;
c) errors when dialing the subscriber's number;
d) technical reasons.
When calculating the load we take the following parts of different
occupations:
- a share of occupations that ended with a conversation (0,4 ÷ 0,5);
- the percentage of lessons that did not end with the call due to the
busyness of the called party (0.15 ÷ 0.3);
- the share of unsuccessful occupation due to improper recruitment (0.1 ÷
0.03);
- the share of unsuccessful occupation due to non-response (0.1 ÷ 0.2);
- the share of unsuccessful occupation for technical reasons (0,01 ÷ 0,02).
Thus the sum of all accepted values should be equal to one 1 .
Let us denote average duration of the corresponding occupations by these
values and determine them.
15
Duration of successful occupation:
,
where - duration of station response (≈3s.);
- duration of connection establishment (1.5m + 2.5c), where m - number
of digits in the phone number;
- call signal with ≈7s;
T - duration of the conversation (for each category of its own);
– habg-up (k, Oc).
The length of the occupation when the subscriber is busy:
,
where - the duration of the signal is busy (≈5 s).
Average duration of the occupation when there is no answer:
,
where is the average duration when there is no response (≈30 s.).
The average duration of an unsuccessful of the occupation in dialing the
number due to an error is approximately 7s.
The average duration of the unsuccessful of the occupation for technical
reasons is approximately 7c.
The total average duration of one of the occupation is calculated by the
formula:
.
This value is calculated for each category separately.
The obtained values of ti are substituted into formula (3.1). We get the value
of the offered traffic in Erlang.
In practice a simplified method of calculating the load is used in which the
average duration of the occupation is determined by the formula:
α ,
where α is the coefficient of non-productive occupation of the switching
system. It is determined from the diagram for known T and [10].
16
Lecture 4. Methods for calculating single server full accessible switching
loss system
The aim of the lecture: get students acquaint with Markov processes, with the
Kolmogorov-Chapman equation, with the derivation of Erlang's first formula for
calculating a one-link, full accessible CS with obvious losses.
Content:
a) the Markov process, the Kolmogorov-Chapman equation;
b) quality of service characteristics of a full accessible system with obvious
losses;
c) derivation of Erlang's first formula (B – formula).
Consider the switching system (SS).
Figure 4.1
The number of occupied lines x from v at time t is a random variable.
Let {x} be the state of the system when x lines from v are occupied. When a
call (line occupation) or end of its service (line release) arrives, the system moves
from one state to another. And if the system was in the state {x} then it can go to
one of the following states: {x - 1}, {x}, {x + 1}. But the state in the future depends
only on the present and does not depend on the past. Such random processes are
called Markov processes, in honor of the outstanding Russian mathematician A.A.
Markov.
Suppose that for a time interval τ the system could go from the state k to the
state i with the probability pki (τ). The probability that the system was in the state k
at time t - τ is denoted by Pk (t – τ).
Then the system of equations for the probability of states has the form
Pi (t) = ∑ Pk (t – τ) • pki (τ), i = 1,2, …, v , (4.1)
where Pi (t) - is the probability of finding the state system i at time t. The
summation in (4.1) is carried out over all possible states of the system.
Equation (4.1) is called the Kolmogorov-Chapman equation and is the main
one in the analysis of queuing systems.
The Kolmogorov-Chapman equation belongs to the class of recurrence
relations that allow to calculate the probability of states of the Markov random
process at any step (stage) in the presence of information at the previous stage.
17
There are many ways to solve this system of equations. Next we will consider the
so-called steady-state regimes when the probability is independent on time.
Consider a one-piece full accessible switching system (FA SS) with obvious
losses.
A full accessible switching system is a system in which each input is
accessible for any free output (figure 4.2).
Figure 4.2
Quality characteristics of service, QoS (Quality of Service) for a system with
obvious losses are the following indicators:
1) The probability of losing a call that has arrived in a certain period equal to
the ratio of the average intensities of the lost and incoming calls flows in this
interval.
2) The probability of time losses in a certain interval equal to the probability
of employment in this interval of all available connecting paths in the required
direction.
3) The probability of loss by load in a certain time interval equal to the ratio
of the intensities of the lost and incoming loads in this time interval.
Formation of the problem. Let the simplest call flow with parameter λ arrive
at the FA SS with v outputs. The service time of one call is a random variable
distributed according to the exponential law with the average value taken per unit of
time (h = 1 c.t.u.). Discipline of service is with obvious losses.
In Kendall's notation the described system can be represented as M / M / v /
L. The problem was firstly posed and solved by Erlang.
It is necessary to determine the probabilities of call loss, time loss and load
loss: . Let {x} be the state of the system when x lines from v are occupied and P (x)
is the probability of the system being in the state x.
Figure 4.1 shows the Markov chain. In the case under consideration λ1 = λ2 =
. . . = λn = λ because the flow is the simplest and for it λ – const.
Figure 4.3
18
That is we consider the steady-state regime when the flow parameter does not
depend on time. In this mode the probabilities describing the state of the system also
do not change with time. The parameter μ is the release parameter, μ = x.
Then equation (4.1) assumes in the conditions of statistical equilibrium:
λP(х-1)=хP(х) . (4.2)
The intensity of the transition to a higher state multiplied by the probability
of the system staying in this state is equal to the intensity of the transition to a lower
state multiplied by the probability of the system being in this state.
1 xPx
xP
;
01
1 PP
;
0!2
012
12
2
2
PPPP
;
0!3
0!23
23
3
3
PPPP
;
---------------------------------------------------
0!
Px
xP
x
;
---------------------------------------------------
0!
Pv
vP
v
;
__________________________________
v
x
v
x
x
xPxP
1 1 !0
. Let’s add Р(0) to both parts.
v
x
xv
x
v
x
x
xP
xPxP
00 1 !0
!10
v
x
x
x
P
0 !
10
v
x
x
x
x
xxP
0 !
!
– First Erlang distribution.
In solving practical problems
h
A ( 1h ), where A – offered traffic
AE
x
A
vAvPP
vv
x
x
v
0
0
!
! – first Erlang formula. In some sources it is called B-
Erlang formula.
Using this formula for known values of the incoming load A and the number
of lines v it is possible to calculate the probability of incoming call losses.
This formula is tabulated. To calculate Ev(A) for large values of v use the
recurrence formula:
19
AEAi
AEAAE
AE
i
i
i
1
1
01
.
When fully accessible system receives a simple flow of calls the quality of
service characteristics can be calculated using the first Erlang formula. The main
one is the probability of losing the incoming call among the characteristics listed
below:
a)probability of loss by time:
Pt = Ev (A);
b) probability of call loss:
Pc = Ev (A);
c) probability of load loss:
Pl = Ev (A);
d) the throughput is defined as follows (the value of Y is tabulated):
e) the capacity of individual outputs in case of accidental occupation:
v
y ,
in ordered occupation: )]()([
1AEAEA
iii
.
Lecture 5. Maintenance of primitive flow by a full-access system with
explicit losses
The purpose of the lecture: familiarization with methods of calculating the
quality of service characteristics of a full-access system with explicit losses, when
the incoming flow is primitive.
Content:
a) maintenance of primitive flow;
b) Engset formula, quality of service characteristics;
c) Erlang's second formula, quality of service characteristics;
d) the models of Crommelin and Burke.
Formulation of the problem. A full-access system with explicit losses
receives a primitive call flow. Call service time is distributed according to the
exponential law. It is necessary to determine the characteristics of the quality of call
service, namely: the probability of call loss Рв, the probability of loss in time Pt and
the probability of loss in load Рн (in Kendall's symbols Mi/M/v/L).
The parameter of the primitive flow is determined by the formula:
λк=α•(n-k), (5.1)
20
where λк - the parameter of the primitive flow;
α - intensity of the flow from one source;
n - the total number of sources;
к - the number of occupied sources.
Having considered the Markov chain, similar to the solution of the problem in
the previous lecture, we determine the probability of occupying the lines:
Pi =
V
j
j
k
k
i
k
k
j
i
0
1
0
1
0
!
!
. (5.2)
Substituting into the formula (5.2) the value of λк from (5.1), we obtain:
Pi =
V
j j
jnnnn
i
innnn
0 !
))1(...()2()1(
!
))1(...()2()1(
.
Considering that:
!
))1(...()1(
)1...(21!
...321
)!(!
!
i
innn
ni
n
ini
nС
j
n
,
we transform the formula to the form:
Pi =
V
j
jj
n
ii
n
aC
aС
0
;
(5.3)
Pt =PV=
V
j
jj
n
VV
n
aC
aС
0
.
(5.4)
The formula (5.3) is the distribution of Engset. The formula (5.4) is called the
Engset formula. It should be noted that when calculating α is defined as the ratio of
α = λ/µ, where μ is the service intensity. The value of μ calculate as µ=1/t, where t
is the average service time.
QoS (Quality of Service) characteristics of a primitive flow by a full-access
system with explicit losses are calculated using the following formulas:
a) the probability of call loss in time:
21
Pt =PV =
V
j
jj
n
VV
n
aC
aС
0
;
b) probability of incoming call loss:
PV =
jj
n
VV
n
aC
aC
1
1 .
The value v
P can be determined from the table: vnfPv
,, . vt
PP always
takes place ,,,,1 vnPvnPvt
;;
c) probability of loss in load:
Pl = tP
n
Vn
;
d) system throughput:
Y = al
P 1( ) ,
where 1
a ;
e) intensity of incoming load:
ynA ;
f) intensity of potential load:
Apot =n∙a.
Full-access system with waiting.
Formulation of the problem. The Poisson call flow with parameters λ arrives
at the full-access system with waiting. The length of service is distributed by
exponential law t
etF
1)( ,
where 3
t =
1- is the average service time.
If all lines are busy at the time of the call, the call is queued and waiting for
service. Calls from the queue are selected in the order they are received. In
Kendall's symbols we have a system M / M / v / r=∞ / FF.
It is required to determine the probability of waiting for an incoming call in
the queue, the mathematical expectation of the waiting time and the length of the
queue.
The problem in this form was first delivered and solved by Erlang. In the
problem delivered by Erlang, the number of waiting places r = ∞.
22
The solution of this problem allows us to evaluate the quality of service
characteristics of a full-access system with waiting:
a) the probability of waiting in the queue for an incoming call:
P = Pt = P(γ>0) =
AD
AEV
A
AE
V
V
V
11
)(, (5.5)
where A is the incoming load, and (5.5) is called the second Erlang formula;
b) the mathematical expectation of the length of staying in queue:
PA
Lt
; t =
Av
1;
c) mathematical expectation of queue length:
L = PAV
A
.
Maintenance of the Poisson calls flow with a constant duration of the busy
session.
In a full-access system with waiting, the amount of incoming load is equal to
the serviced, as it is considered that in this case there is no lost load.
Figure 5.1
Let {i} be the state of the system at time t, and v is the number of outputs in
the system.
If i <= v, then all calls are in service.
If i> v, then v calls are in service, r = i-v are waiting in the queue.
There are three ways to select a call from a queue:
1) FIFO - ordered choice;
2) RANDOM - selection from the queue is random;
3) LIFO - the last received call is serviced first.
Crommelin model.
The full-access system with waiting receives the simplest calls flow, the
service time of one call is constant and equal to h. Number of places for waiting r =
∞. The selection method in queue is FIFO. In Kendall's symbols, the condition of
the problem can be written in the form M / D / v / r=∞ / FF.
Determine the probability of waiting for a call in the queue, more than the
permissible time. The solution was found by Crommelin.
In engineering practice, the Crommelin diagrams are used to solve similar
problems, which reflect the relationship between the conditional time t* and the
probability of waiting beyond the permissible time P(t*>γ*). When solving such
problems, it is necessary to determine η=А/v - the intensity of the load arriving on
one line. Then we determine t*=tD/h, where TD is the admissible waiting time.
23
Figure 5.2
The Burke model.
The full-access system with waiting receives the simplest calls flow, the
number of lines at the output v = 1. The service time of one call is constant and
equal to h. Number of places for waiting r = ∞. The selection method in queue is
RANDOM.
Determine the probability of waiting for a call in the queue more than the
admissible time. The problem was delivered and solved by Burke.
The results obtained by Burke are also presented in the form of a diagram [4].
Lecture 6. Limited-access systems
The purpose of the lecture: familiarization of students with limited-access
systems, with an algorithm of constructing an optimal circuit of limited-access
switching and with methods for its calculating.
Content:
a) structure of limited-access systems;
b) types of limited-access circuits;
c) a cylinder and a connectivity matrix;
d) the algorithm of constructing of the optimal circuit;
e) methods for calculating limited-access circuits.
Limited-access switching of lines is a simple economical way of combining
small full-access beams into one large limited-access one, which allows to increase
the use of lines.
The method is most widely used in electromechanical automatic telephone
stations. Currently, the principles of limited-access switching have found
application in mobile radio communication systems with dynamic channel
distribution between cells. Therefore, the method is relevant today, perhaps, will be
used in future telecommunications systems.
The limited-access switching system consists of g-load groups, each is a full-
access system of n-inputs and D-outputs, and circuits of limited-access switching of
v lines.
24
Figure 6.1
In such a switching system, each of the n inputs have only to D lines accessed
from V. The parameter D is called the accessibility of the circuit.
Constructing a circuit of limited-access switching means, in a certain way, to
connect v lines to D*g-outputs. With given parameters g, D, V, we can construct a
set of circuits of limited-access switching
A necessary condition for constructing the circuit of limited-access switching
is the inequality: D<V<D*g.
An important characteristic of the limited-access switching circuit is the
densification factor: γ=g*D/V, which shows how many different outputs of different
groups are connected to one line.
There are three types of limited-access circuits:
a) stepped;
b) uniform;
c) ideally symmetrical.
In practice, the first two are often used.
Stepped circuit - the number of outputs of load groups served by one line
varies and increases monotonically with the output number.
Figure 6.2
Uniform circuit - the number of outputs of load groups served by one line is
the same. The throughput of a uniform circuit is higher than that of a stepped one.
25
Figure 6.3
In limited-access circuits, three types of line switching are used: direct,
intercepted and shifted.
When direct connected, the same outputs of neighboring load groups are
connected.
Figure 6.4
When intercepted switched, the outputs of each load group are connected, if
possible, with the same outputs of the remaining load groups.
Figure 6.5
When switched, the outputs of one group are connected to the opposite
outputs of the other groups. Shift can be with interception and without it. An
interception shift is called a cyclic circuit.
Cylinder.
A cyclic circuit is called a cylinder if the condition V=g is satisfied.
Figure 6.6
The coefficient of densification is γ=g*D/V= D.
26
The cylinder is an elementary, uniform circuit for which V=g (the number of
outputs is the same as the number of load groups). The number D for the cylinder is
called the step, the step indicates how many groups are connected to one line.
In addition to the size, called the step, each cylinder is characterized by an
inclination.
Slope = 1
a)
Slope = 2
b)
Figure 6.7
Optimal limited-access circuits.
As noted earlier, for any g, V, and D, several circuits of limited-access
switching can be constructed. But from them it is necessary to choose the optimal
one. That is, one that will ensure maximum throughput. To verify the optimality of
the circuits, a connectivity matrix is used.
The connectivity matrix is a square matrix of g order (the number of load
groups). The elements of the main diagonal of the matrix are the numbers D -
accessibility. The element j
а
of the matrix is equal to the number of the connection
between and j load group. It is obvious that j
а
=ji
а . The matrix is symmetric with
respect to the main diagonal.
Example. Five load groups are given, D=4, v=8.
Figure 6.8
The connection matrix is optimal if for any two of its elements, except for
the elements of the main diagonal, the following condition meets:
27
|aij - aim| < 1. (6.1)
And also for any bi = ∑ aij , i # j, meet the condition:
|bk - br| < 1.
(6.2)
We construct the connectivity matrix for circuit on figure 6.8:
M =
In the example considered, condition (6.2) does not meet, therefore the
connectivity matrix is not optimal.
Lecture 7. Algorithm for constructing an optimal limited-access scheme
and calculation methods
The purpose of the lecture: familiarizing students with the algorithm for
constructing the scheme of incomplete access (IA) and methods for calculating it.
Content:
a) an algorithm for constructing an optimal limited-access scheme;
b) calculation of the scheme (IA) using the third Erlang formula;
c) approximate methods of calculating the circuit (IA);
d) two-link systems.
Algorithm for constructing an optimal limited-access scheme and calculation
methods
Initial data: V, g and D.
1. Determine the pitch of the cylinder used in the circuit:
V
Дgr , where -
integer part. In the scheme, we use r and (r + 1) - step cylinders.
2. Determine the total number of cylinders in the circuit: g
V .
3. Determine the number of r-step cylinders in the circuit: g
ДgrV
r
)1(.
4. Determine the number of (r + 1) - stepping cylinders: rr
KKK 1
28
5. Determine the slope of the cylinder, it can be determined using the tables of
the textbook application [1]. It is necessary to choose a sequence of cylinders, so that
the connectivity matrix is optimal.
Methods for calculating incomplete schemes.
Formulation of the problem. Let the flow with a parameter λ enter the
incomplete access system. It is required to determine the probability of losing a call.
Assume that the number of inputs of the load group is greater than the
number of outputs, i.e. N> D.
We introduce the following notation:
Let {x} be the state of the system when x lines from V are busy.
P (x) is the probability that the system will remain in the state x.
When x = D or x> D, there are losses.
Denote the blocking ratio as Тз =Д
хС . If x = D, then one load group is
blocked:
Тз= Д
ДС = 1.
Denote the probability that the call will go to the blocked group as Т(х)=Тз/g
Denote the probability that the call will go to an unlocked group as
μ (x)=1-T (x).
Having constructed the Markov chain, we will compose the equation of
statistical equilibrium. The solution of the equation is obtained as:
P(x) =
V
j
j
i
j
x
i
x
ij
A
ix
A
0
1
0
1
0
)(!
)(!
.
The resulting equation is called the third distribution of Erlang.
The probability of loss is easily determined for Perfectly symmetrical scheme
(PSS).
Perfectly symmetrical scheme is a uniform, incomplete access in which each
load group has its own, different from other combination of D lines, and the total
number of load groups g =D
VС .
The total probability of losing a call in an incomplete access system is
V
x
xTxPP
0
)()( .
If x <D, then T (x) = 0 (none of the groups will be blocked) P =
V
D
xTxP
)()(
.
29
P x( )
Ax
x
0
x
i
i( )
0
v
j
Aj
j
0
j 1
j
i( )
D
V
D
x
C
CxT )( .
Perfectly symmetrical scheme have not received practical application, since a
very large number of load groups are required for their construction. However, the
results of their calculation are used to estimate losses in real incomplete inclusion
systems. Perfectly symmetrical scheme is used in theoretical calculations.
Approximate methods for calculating limited-access schemes.
1. The simplified Erlang formula:
DP
YV ,
where V is the number of lines;
Y - intensity of the serviced load;
D - availability;
P -the probability of loss.
2. O'Dell's formula:
D
D
P
YYDV
,
where V is the number of lines;
Y - intensity of carried traffic;
D - availability;
P is the probability of loss;
YD is the load serviced by a full – access circuit group of D lines (defined in
tabular form).
3. The Palm-Jacobaeus formula:
)(
)(
AE
AEP
DV
V
,
The third Erlang formula
30
where EV (A) - is the first Erlang formula;
A – offered traffic;
P is the probability of loss.
4. The modified Palm - Jacobaeus formula.
Instead of the incoming load A, the value of the fictitious incoming load Аf is
taken, and the value of the fictitious load (Аf <A), satisfying the equality:
Аf=У/(1-Еv(Аф)),
where У – services load.
Obtained Аf is substituted in the Palm - Jacobaeus formula:
P=EV (A)/ EV-D (A).
5. Engineering method:
V=α*A+β ,
where D
P
1 ,
D
D
P
YD .
Lecture 8. Two-link and multi-link systems
The purpose of the lecture is to familiarize students with the structures of
two-link and multi-link systems.
Content:
a) two-link switching systems;
b) multi-link circuits;
Two-link switching systems.
So far, we have considered fully accessible and incomplete single-link
systems. Now, we turn to the consideration of multi-link systems. One of the
elements of the multi-link system is the switch. It is an elementary fully accessible
system.
Figure 8.1
The service of the incoming call consists in connecting this input to an open
output at one switching point.
Consider the structure of a two-link switching system.
31
Figure 8.2
Let us introduce the following notation:
ni - the number of inputs in the switch of the i-th link;
mi - the number of outputs in the switch of the i-th link;
Ki - the number of switches the i-th link;
h- number of directions;
Vh - number of exits in h-th directions;
qr - the average number of outputs in the switch of the last link connected to
the r-th direction.
The following relations hold:
11KnN - number of inputs;
22KmM - number of outputs;
22112,1nKmKV - number of intermediate lines;
1
2
2
1
2,1
K
n
K
mf - connectivity between links.
In a two-link switching system, two switching points and one intermediate
line are required to establish an input-output connection.
Multi-link circuits.
Consider three-link schemes.
Figure 8.3 - Three-link scheme
32
If there is no more than one connecting path in the circuit between any of the
inputs and any of the switches of the last link, this structure is called a fan path
(figure 8.3).
If the number of connecting paths is more than 1, then such a structure is
called connected (figure 8.4).
Figure 8.4 – Connected scheme
Three-link circuits can be used in individual and group-search modes.
Individual search mode - a specific input must be connected to the specified
output.
Group search mode - a specific input must be connected to one of the free
outputs, to the specified group - to the direction.
Let us consider 4-link switching schemes.
Figure 8.5 – The scheme with an indivisible structure
In switch circuits, the ratio of the number of outputs to the number of inputs
on each link is called the coefficient of expansion if it is> 1; concentration
coefficient, if it is <1.
The probability of losses in switching circuits depends on many factors, such
as the nature of the call flow, the magnitude of the incoming load, the structure of
the circuit diagram, the number of links in the circuit.
Lecture 9. Methods for calculating two-link and multi-link systems
The purpose of the lecture: acquaint students with the basic methods of
calculating two-link and multi-link systems.
Content:
33
a) combinatorial method for calculating two-link systems;
b) method of effective accessibility;
c) methods for calculating multi-link circuits.
Combinatorial method for calculating two-link systems.
The method was developed by the Swedish scientist Jacobus. He assumed
that call loss in a two-link scheme occurs in three cases:
a) if all the intermediate lines that can be used for the incoming call are
occupied;
b) if all exits in the required direction are occupied;
c) when there are unsuccessful combinations of free intermediate lines and
free outputs.
Let the call enter the input of the first switch, curci in intermediate lines from
m are busy. If the outputs of the desired direction corresponding to m-i lines are
occupied, then losses will appear.
Denote by - the probability of occupying i intermediate lines belonging to one
commutator of the first link. The probability of taking m-i outputs is denoted by
imH
. Then
m
mH
0
. (9.1)
This formula is valid when the following conditions are met:
a) events described by probabilities
and im
H
are independent;
b) the employment of intermediate lines and free outputs is random and
equiprobable.
When the number of call sources is unlimited, then the incoming stream is
considered to be the simplest, and the probabilities are defined as:
m
j
j
A
А
0!
!
;
AE
AEH
m
m
1
.
Method of effective accessibility.
Consider a two-linked scheme (figure 9.1). In the first link, k switches in each
of them have n inputs and m outputs. In the switch of the second link, q outputs.
This system is fully accessible, since any input is accessible to any free output, but
in this scheme, intermediate lines that can be occupied also participate in the
connection of the switching points.
Therefore, in comparison with a one-part, fully accessible system, the loss
probability calculation becomes more complicated, since the system arrives at a
large number of states. At present, there are many approximate methods for
calculating two-link schemes. Consider one of them.
34
The effective accessibility method is based on the concept of an accessibility
variable. In the scheme under consideration, if all intermediate lines are free, then
any output is accessible to each input. In this case:
Dmax=m•q.
Figure 9.1
In the presence of i-occupied intermediate lines, accessibility decreases by
i*q, then
Di=m•q-i•q=(m-i)•q.
Now define the minimum availability, when n ≤ m:
Dmin=m•q-(n-1)•q=(m-n+1)•q. (9.2)
Thus,
Dmin ≤ Di ≤ Dmax .
After this, consider a two-linked system as a one-part incomplete. The
availability of such system will be called effective accessibility De. It is
mathematically proved that De <Ď, where Ď is the mathematical expectation of the
availability value, then we have
Dmin < Dэ < Ď
or
De = Dmin + Θ•(Ď-Dmin) , (9.3)
where Θ is the coefficient determined depending on the losses due to internal
locks.
Ď=q•(m-Ym) , (9.4)
where Ym is the intensity of the load served by m intermediate lines.
Calculating the values of Dmin according to the formula (9.2) and Ď by the
formula (9.4) and substituting the values obtained in the formula (9.3), we obtain
effective accessibility of De. After that, the calculation of the probability of losses
35
can be carried out according to approximate formulas for incomplete available
single-unit schemes, such as the O'Dell method, the Palma-Jacobus method, the
engineering one, and others. In these formulas, instead of D, one must take the
value of De.
Multi-link scheme.
Multi-link switching systems are formed by connecting a number of "building
blocks".
Calculating service quality indicators for multi-link systems is a very difficult
task. There is (in addition to simulation) a number of analytical methods for
analyzing the relevant models. The choice of method depends on the properties of
the link systems used.
Multi-link systems can be considered as queuing systems (QS). Analysis of
QS is one of the most difficult problems in the theory of teletraffic. In automatic
telephone exchange of electromechanical type the algorithm of service of calls with
obvious losses is used. Therefore, it is necessary to consider QS without waiting.
QS is often considered with expectation. They serve as good models for modern
switching systems.
One way of analyzing multi-link schemes is to construct probability graphs.
The following figure shows the possible probability graphs for a three-link scheme.
Figure 9.2
The structure of each graph shows possible ways of establishing connections.
If for each edge of the graph you determine the probability of a successful path,
then between points "A" and "B" you can determine the possible losses. Let us
establish the following conditions for the graph shown in the left part of the sixth
figure:
1p - probability of successful creation of a path between points "A" and "2-1";
2p - probability of successful creation of a path between points "A" and "2-2";
3p - probability of successful creation of a path between points "2-1" and "3";
4p - probability of successful creation of a path between points "2-2" and "3";
5p - probability of successful creation of a path between points "3" and "B".
The probability of successful establishment of the path between the points
"A" and "3" - 3A
P can be determined by the following formula:
)1)(1(142313
ppppPA
.
36
Then the required probability of losing a call between points "A" and "B" -
BAP
is calculated as follows:
)]1)(1(1[142315
pppppPBA
.
This probability is equal to one under these conditions: there are no paths
between the points "3" and "B" (this is equivalent), there are no paths between the
points "A" and "2-1", and between the points "2-2" and "3 "(This is equivalent to
the coincidence of events). Obviously, for the model under consideration the most
important is the availability of the path between points "3" and "B".
There are other methods for calculating multi-link circuits [1,2].
The tasks of the theory of teletraffic are also the methods of load distribution,
the normalization of losses at various stages of the search, the measurement of load
parameters and losses. With them, students need to get acquainted independently
[1,2,3,4].
In conclusion, it should be noted that new theories and methods appear in the
theory of teletraffic. Characteristic features of the problems of the theory of
teletraffic, which appeared in connection with the use of new technologies, is the
need for multilevel consideration of problems.
The development of teletraffic theory can be viewed from various angles. The
evolution of teletraffic theory is associated with a change in the principles of
building an infocommunication system. The main driving forces that stimulated
changes in these principles can be considered:
a) intensive development of new types of communications, among which
should be allocated mobile networks and the Internet;
b) changing the methods of transmission and switching caused by the
transition to IP technology.
Lecture 10. Quality of service of calls
The purpose of the lecture: to acquaint students with the basic indicators of
the quality of service calls.
Content:
a) call quality indicators;
b) calculation of the probability of call service;
c) admissible losses;
d) calculation of the availability factor.
To assess the quality of call service in PSTN (Public Switched Telephone
Network), two measures are most often used: the probability of the event being
investigated and the time of the corresponding process. The length of execution of
most processes related to call handling is a random variable. For this reason, the
evaluation of the time under study is carried out using the characteristics adopted to
describe the random variables.
37
The figure shows a hypothetical connection between two telephones installed
in PSTN. The caller's telephone is shown to the left. A user who initiates a
connection to a PSTN is usually referred to as an "A" subscriber. Subscriber "B",
respectively, refers to the called user. His telephone is shown on the right side of the
picture. The subscript of the local station indicates the type of subscriber. It is
assumed that the connection is established through transit stations, and the inclusion
of both telephones is carried out on individual two-wire subscriber lines.
Figure 10.1 - Established connection between telephones
Based on theoretical studies and measurement results in PSTN, norms were
established that determine the quality of call service performance for the network as
a whole. Further, the corresponding norms are indicated with the lower index "0".
The figure shows only two such indicators: 0
P - the probability of losing a call and (1 )
0T - the average time (mathematical expectation) of establishing a connection.
For the local station, in which subscriber "A" is switched on, two possible
outcomes of the connection establishment process are shown. The call to the MCA is
lost with probability j
P . This means that the connection continues to be established
with probability 1j j
Q P . If the probability of call maintenance in all switching
stations is mutually independent random variables, then the value 0
P is determined
by such a formula:
0
{ }
1 (1 )j
J
P P . (10.1)
In each switching station, the time is spent for establishing connection
between the subscriber terminals "A" and "B". The value (1 )
0T is defined as the
mathematical expectation of the sum of random variables:
(1) (1)
0
{ }
k
K
T T . (10.2)
The permissible loss value is chosen taking into account two main
considerations. On the one hand, large losses render service unacceptable from the
point of view of subscribers. On the other hand, when constructing a PSTN with
very low losses, the costs of the Operator are significantly increased. As a result, he
38
is forced to set high tariffs, which is also unacceptable for subscribers. This means
that it is necessary to find a compromise solution.
The regulatory documents currently in force determine the value 0
P for
typical connections between the subscriber terminals "A" and "B". In particular, for
the connections within the local telephone communication, the following admissible
losses were established:
- when two subscriber terminals of one CTE (City Telephone Exchange) are
connected, 2.0%;
- when the subscriber terminal of the CTE communicates with the special
services unit (SSU) - 0.1%;
- when communication SSU with the workplace of the operator of emergency
services - 0.1%;
- at connection of SSU with the workplace of the operator of information and
reference services - 3.0%;
- at communication of two subscribers of one RTN (Rural Telephone
Network) - 7,0%.
Therefore, if the PSTN subscribers are connected, losses, measured in
percentage terms, are allowed. When contacting an emergency services operator, it
is assumed that the losses that make up a percentage share are normalized.
The choice of the mean values of the connection establishment time and the
individual stages of the call maintenance is carried out taking into account the same
considerations that guide the Operator to establish the allowable losses. In the
normalization of quantities k
T , along with the mean value - (1 )
kT , the quantile of the
corresponding distribution function is sometimes set. This means that the
probability is determined from which the random variable under consideration
should not exceed a certain threshold - x
t . As a rule, this probability - the value of
the distribution function of a random variable - is chosen at the level of 0.95 or
more. Then such an inequality holds (1 )
k xT t .
The call begins by raising the handset. At random time - O C
t the subscriber
will hear an acoustic signal: "Station answer". ITU recommends that the following
standards be set for the reference load "A":
- the average value of the time interval O C
t should not exceed 400 ms;
- with a probability of 95%, the duration of the time interval O C
t should not
exceed 600 ms.
The acoustic signal "Station answer" is sent to the subscriber of its MS,
therefore, there is only one term (1 )
0T in the calculation formula. When analyzing the
norm for the time of establishing the connection, the same formula includes the
maximum number of terms. The considered time interval starts after dialing the last
digit of the number of the called subscriber. The time for establishing a connection
is completed by receiving an acoustic signal ("Call Control" or "Busy"). This signal
determines the status of the terminal of the called subscriber. In a number of foreign
PSTNs for this period of time - tec when intercity connection is chosen such norms:
39
- the average value of the time interval tec should not exceed 2.5 s;
- with a probability of 95%, the duration of the time interval У C
t should not
exceed 4.0 s.
The numerical values of these norms are defined for a country with a small
territory (when the propagation time of the signal can be neglected) and under the
condition of transmission of control signals and interaction over the ACS network.
On the other hand, the values given are chosen taking into account the subscriber's
response to the duration of the connection establishment time. For this reason, they
can be considered close to those that are universal for PSTN of any country.
An important feature of the quality of service indicators in PSTN is their
gradual change. This process is due to two main trends. The first trend is connected
with the fact that the majority of subscribers make increasingly stringent
requirements to the quality of traffic servicing. The second trend is formed due to
the stratification of the client base. Some groups of subscribers that bring the
highest income to the PSTN Operator have special requirements for traffic service
indicators. Operators of PSTN are certainly interested in ensuring that such
subscribers do not go to competitors. As a measure of retention of subscribers with
high incomes (increasing their loyalty), the practice of concluding service level
agreements, better known in the English-speaking abbreviation SLA (Service Level
Agreement) is used.
In particular, for the availability factor - A at the conclusion of the SLA
agreement the level of 0.99999 is set. In the technical literature appeared the
expression "Rule of Five Nines". The coefficient of readiness for a period of time -
XT - is determined by the ratio of the time of finding the object in the operable state
AT to the value
XT . It is assumed that during the time
XT the object under study is
either in an operable state, or decommissioned. The length of the period when the
object under consideration is not operated is equal to F
T . Then the expression for
calculating the availability factor can be represented as follows
A
A F
TA
T T
. (10.3)
By substituting a value 0 , 99999A , you can define a valid value F
T for the
selected period of operation. For the year the required value is about 5.3 minutes.
To implement such a norm, redundancy of many elements of the network is often
required. This means that the forecasting of those changes that are related to the
quality of service indicators becomes one of the important tasks facing PSTN
Operators.
Lecture 11. Measurement of parameters of load and losses
The purpose of the lecture: familiarization of the students with the task of
measuring the load and losses parameters.
Contents:
40
a) the purpose of measurement of parameters of load and losses;
b) the organization of the process of measurement;
C) the objects and types of measurements;
g) principles of measurement of the load parameters.
Measurement of parameters of load and losses are carried out with the aim of
solving a number of practical and theoretical tasks:
- design of telecommunication networks;
- managing telecommunication networks;
- load forecasting;
- signing of SLA;
- hypothesis testing on the quantitative and qualitative properties of the load;
- other tasks.
Efficiency of capital investments in network development and the quality of
their functioning depend essentially on the regularity of the measurements of the
correct processing and analysis of statistical data load parameters.
For the measurement of traffic (including specific measurement objectives),
you need to choose:
- object (or set of objects) of measurement;
- the duration of the measurement period;
- the microstructure of the measurement period;
- the type and amount of data collected;
- the value of the permissible error;
- other attributes.
Objects of dimensions that are selected to solve the problem, can be:
- total number of incoming calls;
- the number of calls from a specific traffic source;
- share held conversations;
- duration of service calls;
- latency, and loss challenges;
- other objects.
All kinds of measurements of parameters of the traffic can be classified as
follows:
- according to the method of obtaining data (automatic and manual);
- according to the method of data logging (direct and indirect);
- by type of organization dimensions (continuous, periodic);
- on coverage of studied objects (inline and selective).
An example of a continuous load measurement can be considered a method,
based on the control current strength.
In practice, the continuous measurement – before switching stations
controlled – not produced due to problems of financial and organizational nature. In
mathematical statistics the entire target population of homogeneous elements is
called a population. The part of the population selected for measurement, is called
41
the sampled population. Usually, we study the behavior of a sampling. There are
three ways of measurement:
- continuous monitoring;
- scan of the investigated process;
- analysis of random events.
The use of diagrams ammeter – an example of continuous monitoring of the
studied quantity.
Figure 11.1 – Diagram of the continuous measurement of the served load
In the process measurements, one can obtain the expectation for the General
population ( ) and sample ( ). Let’s J
N and – number -th group elements in
the General population and stratified sampling. The volume elements is determined
according to the following scheme:
K
J
JNN
1
and .
The degree of divergence between the individual values of the studied
process is characterized by the variance for the General and sample population:
K
J
JJ
X
N
NXX
1
2
2 )( and
M
J
JJ
x
n
nxx
1
2
2
)1(
)( .
An example of recommendations for the measurement of traffic-blocking
beam serving devices. We assume that the time classes equal to one, i.e. If
the measurement time is more than 20 times more than the average time of classes,
for the distribution of statistical estimates, you can use the normal law.
Some important postulates:
Jn j
M
J
Jnn
1
.1
42
1) The measurement Precision is proportional Т
1.
2) Absolute standard error of measurements for the serviced load (Y) at low
loss probability (0.01) is determined by the following formula:
Y
T
21
.
3) Relative root mean square error of measurements for the serviced load (Y)
is determined by the following formula:
YT
21.
4) Absolute standard error of measurement at low loss probability (0.01) is
determined by the following formula:
)1(21
N
.
5) Relative standard error of measurement if a low probability loss is defined
by the following formula:
)1(21
N
.
Guidelines for designing networks are contained in ITU recommendations
(ITU) and national standards of the Administration.
ITU recommends that, in international calling for 30 Busy Hour the
maximum loss does not exceed 0,01. At the same time for 5 of these Busy Hour
allowed to set the rate of loss of 0.07.
Exemplary rules for loss "call from the subscriber to the subscriber" (end-to-
end) for PSTN is given in table 11.1.
T a b l e 11.1
Type of established connections Acceptable losses
Within the limits city telephone network 0,03 – 0,05
Within the limits rural telephone network 0,12
Intrazonal communication 0,07
Long-distance communication (through city
telephone network)
0,07
Long-distance communication 0,13
Exemplary rules for an efficiency rate of calls to STOP is given in table 11.2.
T a b l e 11.2
View established connections Efficiency challenges
Local service (CTN or RTN) 0,5 – 0,6 (1,6 – 2,0 on conversation)
Intrazonal communication 0,4 – 0,5 (2,0 – 2,5 on conversation)
Long distance (through city
telephon network)
0,4 (2,5 on conversation)
43
One should distinguish between probabilistic and time characteristics that are
important to the field of switching and control devices.
In this example we are talking about the control unit. For bandwidth
estimation and switching field can use the first formula of erlang.
The same magnitude of load intensity can be obtained by different
combinations of the factors in the formula:
Y = NCT.
When a large value of the number of processed calls (C) and small time
service (a typical example is the reference service) should be carefully calculate the
VIN associated with the control device. When the value of service time and a small
number of calls (e.g., modem pool) the greatest interest is the calculation of the VIN
for the field switching and transport resources.
Lecture 12. The concept of self-similar processes. The definition of
fractals
The aim of the lecture: familiarization of the students with self-similar
processes and their properties.
Contents:
a) definition of a fractal;
b) the concept of self-similar processes;
c) properties of self-similar processes;
d) job self-similar processes;
e) the Hurst parameter.
In the design, start-up and operation of telecommunication networks one of
the main problems is the task of ensuring the quality of service. Until recently the
solution of this problem in the design of the information distribution system
provided the sections of the teletraffic theory which was the result of the work of A.
K. Erlang, T. Engset, G. O'dell, C. Palm, A. Y. Khinchin, etc. [1,2,10,11].
This part of the theory describes well the processes occurring in such systems
the distribution of information, like the telephone network, built on the principle of
switching channels. The most common model of a call flow, as you know, is the
simplest flow. The present period of rapid development of high technology has led
to the emergence and widespread distribution networks with packet data
transmission, which gradually began to displace the system switched, but still, they
were designed on the basis of the General provisions of the teletraffic theory.
However, in 1993 a group of American researchers W. Leland, M. Taqqu, W.
Willinger and D. Wilson published the results of his new work, which radically
changed the existing ideas about the processes occurring in telecommunication
networks with packet switching. These researchers studied the traffic in information
networks Corporation and Bellcore found that the flows cannot be approximated by
44
a simplest and as a result, they already have a completely different structure than it
is in the classical theory of teletraffic. In particular, it was found that traffic such a
network has a so-called property of self-similarity., ie looks qualitatively the same
at almost all scales the time axis, has a memory (delay), and is characterized by high
pasechnogo. As a result of theoretical calculation of parameters of system of
information distribution, intended for processing of such traffic, according to the
classical formula gives an incorrect and unduly optimistic results [3].
Moreover, the usual algorithms for traffic handling, designed to work with the
simplest flows are insufficient for flows with self-similarity. Thus was formed the
problem of self-similarity of teletraffic, which over the last 16 years dedicated to
more than a thousand works, and which still has not lost its relevance.
The concept of fractal was introduced by Benoit Mandelbrot in 1975. The
word is derived from the Latin word fractus – consisting of fragments. From a
mathematical point of view, a fractal object, first of all, has a fractional
(nonintegral) dimension.
It is known that a point has dimension zero. Line segment and a circle,
characterized by a length (length), have dimension equal to one. The circle and the
sphere, characterized by the square, have a dimension of two. For descriptions of a
variety with dimension 1.5 requires something between length and area.
Another important property possessed by almost all fractals the property of
self-similarity (scale invariance). A fractal can be split into arbitrarily small pieces
so that each piece will be just a smaller part of the whole. In other words, if you
look at a fractal through a microscope, we will see the same picture that without a
microscope (figure 12.1). Nature created fractals for millions of years. In fact, most
objects in nature are not circles, squares or lines.
Figure 12.1 - Examples of fractal objects
In self-similar traffic present a number of rather strong emission at the
relatively low average level, significantly increasing delays and jitter in the self-
45
similar traffic passing through the network, even in cases when the average traffic
rate is far below the potentially achievable transmission rate in this channel.
Self-similar processes are processes with long memory that allows you to
predict their future, knowing the relatively recent past. Note that the prediction of
teletraffic is extremely important in the development of algorithms networks,
providing improved quality of service (QoS). For service providers forecasting
download networking allows you to plan for timely development.
To date, it is shown that the structure has a self-similar traffic in wired
networks using common Ethernet protocols, OKC 7, VoIP, TCP, etc. Similar to
effects found in cellular telephone networks with packet switching. In a published
study, the results of which confirm the existence of self-similar properties and the
traffic in modern telecommunication networks that use wireless access technology
IEEE 802.116.
In the technical literature instead of the term "fractality" sometimes use the
word "self-similarity" – English "self-similarity". Similarly, instead of the term
"durable relationship" (long-term dependence) used the word "persistence" (tracing
the English "persistence").
The Hurst parameter.
For practical identification of the properties of fractality proposed the Hurst
exponent (Hurst). It is named by the name of the author of this idea. The Hurst
exponent (H) determines the degree of self-similarity.
Exploring the Chronicles 800 years of Nile floods, Hurst found that there has
been a tendency, when the year is good for Vodnany was followed by another fertile
year, and, on the contrary, after year of low water was followed by another
"hungry". In other words, it seemed that the appearance of the hungry and fertile
years for a reason. To confirm this fact HART introduced a factor 0 < N < 1, which
in his honor is now called the parameter (exponent) Harsta. In the case of
independence on each other levels of the annual floods, it would be logical to
present the process spills the ordinary Brownian motion with independent
increments, with Hurst parameter H = 0.5 in. However, as found Herst, for Neal N =
0,7.
Check for self-similarity and estimation of the Hurst exponent H is a complex
task. In real conditions always operate with finite data sets, so it is impossible to
verify whether or not the traffic is self-similar by definition. Therefore, it is
necessary to investigate various properties of self-in real measured traffic data. This
raises the following problem.
1. Even if confirmed the properties of self-that you cannot conclude that
the analysed data have a self-similar structure. We should talk about self-similar
structure in a predetermined scale range for a given set of data.
2. Estimation of the Hurst exponent depends on many factors, such as the
estimation technique, sample size, time scale, etc.
One way to calculate H is to analyze the so-called R/S statistic (normalized
range) [3].
46
For a random sample set ( 1, 2 , .. . , )j
X j N determine the sample mean, sample
variance and integrated variance:
2 2
1 1 1
1 1, ( ) ,
jN N
k N j j k
k k k
M X S X M D X jMN N
(12.1)
The variability of a random process on the interval is defined as a non-
decreasing function of the length of the next interval (for 1 j N )
m a x m inN j j
R D D . (12.2)
Hurst showed that for many natural processes, the relation is valid:
then
,
where H is the Hurst parameter, 0.5<H<1.
If the condition H<0.5 then the process does not have the property of self-
similarity, and if H>0.5, the process is self-similar.
There is another quantity that characterizes the degree of self-process is β, the
value associated with the Hurst exponent H by the relation:
.
In practice, the Hurst exponent is determined graphically. For this plot the
values of log(RN / SN) depending on log(N). For each value of N is the appropriate
point built video. The tangent of the angle of tilt built directly will be the parameter
H. It should be noted that the found point is not immediately possible to build
direct, it is necessary to approximate the points.
Lecture 13. The application of the QS models for the analysis of
functioning of the IP network
The purpose of the lecture: familiarization of the students with the task of
calculating data networks.
Contents:
a) calculate the delays in packet switching node;
b) calculation of the probability of losses in packet switching node.
47
The main parameters of quality of service (QoS) in networks of PD on the
basis of packet switching are computed delay and loss (and in the network nodes,
and end-to-end).
Calculate the delays in packet switching node.
We consider the problem of calculating the average duration of the delays in
packet switching node. The term "packet switching node" here means the
concentrator (statistical multiplexer), and a virtual node packet switching network
(X. 25, Frame Relay, ATM network), and router (IP network). The packet switching
node may be represented in the form element with multiple input channels and one
output channel (the hub) or element with multiple input and output channels
(switch/ Router). Using Kendall notation such network elements can be represented
by queueing systems of G/G/1 or G/G/n (arbitrary probability distribution
describing and the incoming flow of requests (in our case packets or Protocol
blocks), and the time of their service (note that in the analysis of the switching
nodes of the packet are often used models with one servicing device, that is, system
G/G/1) [8].
The average queue length in the system M/G/1 (Poisson stream of packets at
the input, an arbitrary distribution of service time) under the infinite buffer size is
calculated by the classic Pollaczek – Khinchin formula:
q =
,12
12
2
sC
ρ 1, (13.1)
where
- the load of the Queuing system (the ratio of the intensity of
the incoming flow of requests to the intensity of their service);
2
2
s
s
s
t
tDC - the quadratic coefficient of variation of the distribution of
service time;
D(ts) - the variance of the distribution of service time;
st - the average time of service Protocol unit (datagram, packet, frame, cell)
in the system.
To determine the average delay in the system M/G/1 we use the Little's
formula:
q = qt .
Then the average delay will be defined as:
48
qt =
12
11
2
ss
Ct
q. (13.2)
To calculate the average queue length and average delay it is necessary to
know the variance and the mathematical expectation (or coefficient of variation) of
the time distribution Protocol service unit (the service time is proportional to the
length of Protocol block). Table 8.1 shows the expression to calculate the quadratic
coefficients of variation of some distributions used in the estimation of the average
delay in networks of the Internet.
T a b l e 13.1 - Quadratic coefficients of variation for some distribution
Parameters of systems of the form G/G/1 with infinite memory cannot be
precisely calculated when the distributions of the parameters of the incoming flows
other than Poisson. However, there is a set of approximate formulas allowing to
calculate queues and delays. The following is the formula to calculate the average
queue length in the system G/G/1 where easily obtained in the average duration of
delay [8]:
121
22
1
saCC
q ; (13.3)
122
122
2
saCC
q ; (13.4)
12
22
3
saCC
q , (13.5)
Distribution Coefficient С
Exponentional (М) С2 = 1
Erlang’s
kC
12 (k – Eralng’s order distribution)
Hyperexponential
(H) ,
12
2212
SS
SS
0 S ≤
2
1
(S - option hyperexponential distribution for
the case of a sum of two exponentials)
) Geometrical C2 = ρi, 0 < ρi <1
(ρi – parameter of geom. distribution)
A constant service time of the
application
(D)
С2 = 0
49
where Са and Cs - the quadratic coefficients of the distribution of the
incoming stream of Protocol blocks and the time of their service, respectively.
From formulas to estimate the average lengths of the queues (delays) it is
seen that the denominator of each formula there is a multiplier (1 - ρ), which is the
pole of the equation.
The approximation (8.3) reduces to the formula Khinchin-Polacca that is
accurate for the system M/G/1. The use of a particular approximate formula for
calculating the queue is determined by how the distribution of the incoming stream
differs from the Poisson, as well as the load-serving device of ρ [24].
Calculation of the probability of losses in packet switching node.
Another important parameter QoS in data networks is the probability of
packet loss.
There are a number of factors that packages are not delivered at the
destination. Among the main reasons, we note the distortion of the packets during
transmission through the network, exceeding the "time of life" packets and discard
packets at nodes in the absence of free space in the buffer memory node.
The last phenomenon occurs in the case if the drive has a finite storage
capacity. The probability of loss is defined as the probability of overflow of the
buffer of the drive.
In this section we consider the problem of calculating the probability of
memory overflow in the node, which in General form is described by the queueing
system of the G/G/1/N. let's start with the simplest model of the system with
Poisson input flow and the exponential distribution of service time, and then
consider a more General model of Queuing system [8].
The system M/M/1/N. the Probability of memory overflow is determined on
the basis of the processes of death and reproduction, and is equal to:
N
NlossP
11
1 (13.6)
is obvious that the values of ρ<< 1 for the system M/M/1/N can be used the
following approximation:
Ploss ≈ PN.
. (13.7)
From equation (13.7) can also obtain the required buffer size in the node,
based on probabilities of losses. The solution of the equation relative to the capacity
of the buffer N is expressed by the following formula:
ln
lnloss
PN . (13.8)
System G/G/1/N. Obtaining exact solutions in closed form for systems of this
type with known distributions of the input flow and service time, especially at the
50
end of storage capacity, is associated with significant difficulties. More effective is
the use of approximate but simple to use estimates based on quadratic coefficients
of variation of input flow and service time. The approximate formula for estimating
the probability of losses in the system, if these distribution parameters of input flow
and service time are known, was proposed in the mid 70-ies of the last century
Vladimir Lipaev and S. F. Laskovym and has the following form:
22
22
2
12
1
1sa
sa
CC
N
CC
lossP
. (13.9)
Knowing the distribution of input flow and service time and, thus, having the
values of the quadratic coefficients of variation, one can calculate the probability of
losses in a rather complex system. Of course, note that these estimates are
approximate, but you can always estimate the error calculations after the simulation
of the selected system, for example, with the use of GPSS World.
Lecture 14. The mobile communication system as a Queuing system
The purpose of the lecture: to acquaint students with the methods of
calculation in the mobile communication system.
Contents:
a) organization of cells in a mobile communication network;
b) the calculation of the load on the network of mobile communication of
GSM standard;
c) assessment of the capacity of the transport network in the GPRS network.
The organization of cells in mobile communication network.
In the mobile communication system as a transmission medium using digital
or analog wireless radio channels.
As you know, the range of frequencies of electromagnetic waves are divided
into frequency bands reserved for specific purposes. Some of these bands are
reserved for mobile communication.
Each band corresponds to a limited number of telephone radio, and this fact
restricts the resources of the mobile communication systems. Optimal use of this
resource is the main task of provider.
If a particular geographical area should be provided with mobile
communication in the territory must be equipped with a corresponding number of
base stations.
The base station is an antenna receiving and transmission equipment or radio
channel associated with the mobile telephone switch (MTS), which is part of the
traditional telephone network. Mobile telephone switch common to all base stations
in a given field of communication.
51
Radio waves propagating in the atmosphere is damped, so the base station
becomes the only way to cover a predetermined geographic area called a cell.
You cannot use the same frequency to two adjoining base stations, but at two
base stations that do not have a common border, using the same frequencies are
possible, which allows to reuse channels. Figure 14.1 cell with the same frequency
are colored the same and they are not adjacent.
Figure 14.1 – The Organization of cells in mobile communication network
Thus, it becomes possible to determine the number of channels per cell, based
on a specified amount of transferred data (traffic). The size of the honeycomb will
depend on the volume of traffic.
Load estimation in mobile communication network GSM.
The analysis of the load (traffic) mobile network is a fundamental task, since
information about the magnitude of the load necessary for the efficient operation of
the network for the configuration of its bandwidth.
When calculating the load of mobile communication network using the data
display in the Recommendations of the International telecommunication Union
(ITU).
The Recommendations of rationing of the load is set depending on the size of
loss probability of call in the busy hour for 30 more years. The probability of failure
of a radio link in the service is 5-10%. In practice, this value may be lower (in the
range from 3 to 5%).
Subscriber load received by one base station, consists of calls from
subscribers located within the coverage area of the station and calls from
subscribers in the coverage areas of the neighboring BS, in other words, the
handover calls.
The results of the research works devoted to the analysis of laws of
distribution of incoming calls to demonstrate that the time intervals between the
moments of receipt of calls are distributed by the exponential law, in other words
the incoming call flow is the simplest or Poisson.
The main parameter of the incoming call flow is its intensity λ. The intensity
of the elementary stream is equal to the average number of calls per unit of time.
52
Denote by λ1 - challenges-of-area coverage, λ2 - handover calls. Out of the
teletraffic theory it is known that the sum of the elementary flow is the simplest
flow, therefore total flow of calls to a base station that is the simplest with intensity:
λ = λ1 + λ2 .
The service time of the base station of the received call has an exponential
distribution. Denote by µ1 is the mean service time for a call, your coverage area,
and using µ2 is the mean service time of a handover call. Then the duration of the
radio channel is a random variable, distributed on exponential law with intensity:
µ = µ1 + µ2 .
Let the number of radio channels in a cell is equal to v.
You must enter the following designations:
ρ - is the total load on the GSM system, created its own challenges and
handovers of the calls, that is, ρ = λ/µ.
You need to consider how to calculate the load or capacity of the cell in GSM
technology.
Mobile communication system, as with any telephone system is an example
of a Queuing system with random flow of requests (calls) in a random length of
service (sessions) and a finite number of service channels (physical channels).
In the GSM network it is transmitted in the mode switching channels, so in
this case the mobile network is considered as a QS type M/M/N/L, that is, the
incoming call flow is the simplest, service time call a random variable, distributed
exponentially, the number of service channels – v. The system is full accessibility,
discipline and service, with obvious losses.
Under the capacity of mobile networks is the number of subscribers And that
network is able to service at given [8]:
- loss probability P0 (the probability of failure in connection);
- the number of physical channels per cell v;
- number of per cell M on-site coverage.
Since the GSM network is considered as an QS type M/M/N/L, the loss
probability P0 is determined by the Erlang B - formula (first formula of Erlang):
AE
x
A
vAvPP
vv
x
x
v
0
0
!
!.
The objective of the calculation capacity of the mobile communication system
is solved in the following order.
1) Knowing the number of channels per cell, can be determined from the
tables for In a formula of admissible value of erlang traffic Mustache for Erlang
53
serviced in the cell for a given probability P0. In GSM networks this value is
typically set in the range from 0.01 to 0.05.
2) The load intensity per subscriber Yi is estimated in the period of maximum
load at a known average duration of one lesson and the number of calls. This option
is commonly known in the calculations of load and its value in the initial stages of
development of mobile communication networks is equal to 0, 015 Earl.
3) The number of subscribers that can be serviced in one cell, estimated the
ratio of the following form:
i
c
Y
Ya .
4) The number of subscribers that can be served whole set M SOT (assuming
uniform loading) is:
A = aM.
The estimated bandwidth of the transport network in GPRS
Calculation of the capacity of the transport network (PS domain) defined the
requirements for service quality indicators, in particular, to the delay.
GPRS system, like any network communication, is modeled by Queuing
system (QS), and when calculating the bandwidth used formula corresponding to
the selected model. Because GPRS uses packet switching of packets to model such
a system used a system with queues.
Used to simulate the switch GPRS system M/G/1 (Poisson input, General
distribution, service times, one service device, infinite buffer size). The average
delay of the Protocol unit in this system is calculated by the formula Pollaczek -
Khinchin:
12
11
2
ss
q
Ct
qt , (14.1)
where q - the average queue length in the system (in number of Protocol blocs);
- the intensity of the load of the system M/G/1, of course, ρ<1 λ, µ -
intensity of service receipt and PB in the system, respectively;
st - the average service time of PB in the system;
2
2
s
s
s
t
tDC - the quadratic coefficient of variation of time of service, equal
to the ratio of the variance of service time to the square of its expectation. The
intensity of the service node in the GPRS is defined as:
54
μ=B
l,
where l is the average packet length;
In the data transfer rate. If we consider GPRS as the switch QS of the form
m/M/1, given the values of the quadratic coefficient of variation formula (14.1)
takes the form:
11s
qt
qt , (14.2)
where ρ<1.
By transformation we can prove that then:
B>λl.
As the node serving GPRS packets, it is possible to simulate the QS as M/D/1
(the time of servicing a constant value). Equation (14.1) then takes the form:
)1(21
s
qt
qt , (14.3)
where ρ<1.
From the expression (14.3) it is possible to obtain an approximate estimate of
the transmission rate at the output node GPRS:
B> (λl)/2.
Lecture 15. Characteristics of digital networks
The purpose of the lecture: familiarization of students with methods of
determining the characteristics of digital networks
Content:
a) definition of the characteristics of the digital network;
b) analysis of digital networks with a simple input flow and unlimited buffer
capacity.
When analyzing digital communication systems, the most important
characteristics are the average volume of buffer usage and the average time of the
packet stay in the system. We concretize these concepts. To do this, consider a
digital system consisting of a buffer capable of storing a queue of infinite length,
and the server processing information arriving at its input (figure 15.1).
55
Figure 15.1 - The scheme of a single-server digital system
The input load, i.e. the number of packets arriving at the input of such a
system in time t, we denote by a(t), and the number of served packets as b(t). Then
the total amount of data that is in the system at the time is equal to:
N(t) = a(t) – b(t).
Obviously, the total time spent by all applications in the system over time ,
can be calculated as the area under the function graph N(t):
t
dttNty
0
)()( .
An analysis of this expression shows that if the function y(t) divided to the
integration time t, we get the average number of packets in the system:
Ñt = y(t)/t.
If the function y(t) divided to the total number of packets received in time,
then you can get the average time of finding the package in the system:
Tt = y(t)/a(t).
From the last two expressions follows:
Ñt = Tt a(t)/t =Ttλt ,
where λt = λ(t) - intensity of the input stream. If there is a stationary mode of
operation in the communication system, the resulting expression is independent on
time and can be written in the form of Little's formula:
Ñ = λT.
This formula describes the relationship between the two most important
characteristics of digital systems, which means that the average number of packets
in the system is proportional to the average intensity of the input stream and the
average time of the packet's stay in the system. This expression plays an important
56
role in the analysis of digital communication systems. In this case, it remains valid
for any distribution of the input stream and the service time.
It is also interesting that if in the digital system only the data buffer is
considered, then the form of Little's formula is preserved, only the meaning of the
variables changes: Ñq - average queue length; W - average time of the package stay
in the queue:
Ñq = λW.
Conversely, if we consider only the server, the Little formula takes the
following form:
ÑS – λx ,
where ÑS - the average number of packets on the server (or servers); x -
average processing time on the server of one package. And the total average time of
finding the packet in the system is:
T =W + x.
Analysis of digital networks with a Poisson input flow and unlimited buffer
capacity.
Since the main difference between digital and analogue communication
networks is that under the applications entering the digital system we mean a data
packet of a certain length, rather than a call from the subscriber, then it is possible
to apply the previously considered Erlang formulas to the analysis of digital
communication systems.
Consider the single-server system shown in Figure 5.1. We will assume that
its buffer is able to store a queue of infinite length, and the server processes
incoming packets with an average intensity µ. If we assume that the time intervals
between packets are distributed according to an exponential law with intensity λ:
t
etf
)( .
And the processing time of one packet with the same PRV, but with a
parameter µ: t
etf
)( .
Then for the analysis of communication systems, one can use the Erlang
formula for systems with expectation for an infinite queue length:
1
0 11
!
n
j
nj
nkn
k
Zn
n
n
Z
j
Z
n
Z
n
Z
p ,
57
where Z=λ/µ; n - number of servers. In the case of a single server, we have:
k
k
kZZ
Z
Z
Zp )1(
11
, at k =0,1, …
The resulting expression can be considered as the PRV of the number of
packets in the system. Consequently, their average number in the system can be
found as the mathematical expectation:
Ñ = .1
)1(
00 Z
ZkZZkp
k
k
k
k
.
.
To find the mean time of the packet stay in the system, we use the Little
formula, from which we obtain:
T = Ñ/λ = (
11
1
Z
Z.
.
Figure 15.2 shows the average number of packets Ñ and average time of
stay of packets T in the system, depending on the input load Z.
Figure 15.2 - Dependence of the average number of packets and the average
time from the input load
It should be noted that the average number of packets Ñ in the system
consists of the average number of packets in the buffer and the average number of
packages on the server ÑS, i.e. Ñ = Ñq + ÑS . From this expression it follows that
the average number of packets in the buffer can be found by the formula:
Ñq = Ñ - ÑS = Z
ZZ
Z
Z
111
2
.
58
And the average waiting time in the queue is:
W = Ñq /λ = )1( Z
Z
.
Consider now the case when there are two servers in the system, both of
which are equally accessible to incoming packets from the communication channel
(figure 5.3). Systems with several servers of this type are called fully available.
Obviously, the performance of this system will be higher, compared to the single-
server system.
Figure 15.3 - The scheme of a two-server digital system
Using formula (15.1), we calculate the probability of finding k packages in
the system:
Pk = 1
1
2
1
22
2
2
2
2
2
2
21
2
k
kk
k
k
k
Z
Z
Z
Z
Z
Z
Z
ZZ
Z
.
Similarly, we find the average number of packets in the system and the
average waiting time:
Ñ2 =
2
0 )2/(1
2
Z
Zkp
k
k Ñ =
Z
Z
1;
T2 = Ñ2/λ = Z
T
Z
1
/1
2/1
2
2
.
Thus, in a system with two servers, the average number of packets and the
waiting time are shorter than the one-server scheme. It is interesting in compare the
two-server system with the performance of each µ and one-server with a capacity
twice as large 2µ. That is, to answer the question: what is better than parallelization
of data streams or faster processing? Using the formula (5.1), we obtain the average
waiting time of packets in a system with the processing intensity of data 2µ:
59
T= Ñ/λ =
2/1
2/11
2/1
2/1
. .
Figure 15.4 shows the average packet wait time in a system for a two-server
system with a parameter µ (graph 1) and single-server with parameter 2µ (graph 2).
Figure 15.4 - Average packet wait time for different digital systems
Analysis of figure 15.4 shows that doubling the speed of the server is a more
efficient solution than using another parallel server.
60
Glossary
Stationarity - стационарность
Ordinarity (simplicity) - ординарность
An aftereffect (independent) – последействие
Poisson call flow – простейший поток вызовов
Offered traffic (incoming) – поступающий трафик (входящий)
Carried traffic (served) – обслуженный трафик
Lost – потерянный трафик
Busy Hour – час наибольшей нагрузки (ЧНН)
Busy Period – период наибольшей нагрузки
Subscriber – абонент
Full access system (FAS) – полнодоступная система
Switching system (SS) – коммутационная система
Primitive flow – примитивный поток
Markov chain – марковская цепь
Quality of Service (QoS) - качество обслуживания
Limited-access systems – неполнодоступная система
Stepped circuit – ступенчатая схема
Uniform circuit – равномерная схема
Ideally symmetrical – идеально-симметричная схема
Load groups – нагрузочные группы
Connectivity matrix – матрица связности
Multi-link system – многозвенная система
Self-similar process – самоподобный процесс
Telephone exchange – телефонная станция
List of abbreviations
QS - Queueing system – Система массового обслуживания
QoS – Quality of Service – Качество обслуживания
PSTN – Public Switched Telephone Network – Телефонная сеть общего
пользования
CTE – City Telephone Exchange – Городская телефонная станция
CTN - City Telephone Network - Городская телефонная сеть
RTN – Rural Telephone Network – Сельская телефонная сеть
SLA – Service Level Agreement – Соглашение об уровне обслуживания
TCP –Transmission Control Protocol – Протокол управления передачей
VoIP – Voice over IP – Голос над IP
61
References
Basic
1 Stepanov S.N. Teletraffic theory: concept, models, applications. –
Moscow: Hot line – Telekom, 2015.
2 Villy B. Iversen. Teletraffic engineering and Network planning: manual /
V.B. Iversen. Transl. engl. with edit. A.N.Berlin. – Moscow: National Open
University "INTUIT": BINOM. Laboratoryof knowledge, 2011.
3 Shelukhin O. I. Modeling of information systems: Manual. – M.: Hot line –
Telecom, 2011.
4 Teletraffic theory. Lecture notes [In Russian]/Tumanbaeva K.Kh. - Almaty,
AUPET, 2014.
5 Teletraffic theory. Methodological instructions to course paper
implementation for students majoring in speciality 5B071900 – Radio engineering
and telecommunications/Tumanbaeva K.Kh. - Almaty, AUPET, 2018.
6 Bitner V.I. Rationing of service quality in telecommunication. Moscow:
Hot line – Telekom, 2009.
7 Berlin A.N. Switching in communication systems and networks - Moscow:
Eco-Trends, 2016.
8 Goldstein B.S. Communication networks. - Sr.Petrsburg: BHV-Petersburg,
2011.
Additional
9 Stepanov S.N. Basics of the multi-service networks teletraffic - Moscow:
Eco-Trends-2010.
10 Kornyshev Yu.N., Pshenichnikov AP, Kharkevich A.D. Teletraffic theory
– Moscow, Radio and Communication, 1996.
11 Krylov B.V., Samokhvalova S.S. Teletraffic theory and its applications. -
Sr.Petrsburg: BHV-Petersburg, 2005.
62
Summary Plan 2019, pos.259
Kumisay Khasenovna Tumanbayeva
TELETRAFFIC THEORY
Lecture notes for students majoring in specialty
5В071900 – Radio engineering, electronics and telecommunications
Editor M.D. Kumanbekova
Specialist in standartization G.T. Mukhametsarieva
Signed in print 18.12.19 Format 60x84 1/16
Circulation 30 copies Typographical paper №1
Volume 3,4 Order 937 Price 1850 тг.
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63