Maciej Stasiak, Mariusz Głąbowski Arkadiusz Wiśniewski, Piotr Zwierzykowski Basic Definitions and...

Maciej Stasiak, Mariusz GłąbowskiArkadiusz Wiśniewski, Piotr Zwierzykowski

Basic Definitions and Terminology

Modeling and Dimensioning of Mobile Networks: from GSM to

Modeling and Dimensioning of Mobile Networks: from GSM to LTE

Course – assigned reading

• M. Stasiak, M. Głąbowski, P. Zwierzykowski: Modeling and Dimensioning of Mobile Networks: from GSM to LTE, John Wiley and sons Ltd., January 2011.

• Iversen V.B., ed., Teletraffic Engineering, Handbook, ITU, Study Group 2,Question 16/2 Geneva, January 2005, published on-line.

Arrival stream

Stochastic point process

• Possible realization of the stochastic point process

Process parameters

• Λo intensity of arriving calls

• Pk(t)

o probability of k calls arrival within time interval of length t

• f(t)o inter-arrival time distribution

Arrival Poisson processes properties

• Stationarityo stream intensity is not time-dependable

o λ(t)= λ =const

• Memorylessness (independence of all time instants)

o number of arrivals occurring within the time interval t1 is independent of the number of arrivals occurring within the time interval t2

• Singularityo in a given time point only one arrival can occur

Lack of memory property

• Distribution function of time interval between consecutive calls (inter-arrival time) is exponential function:

• We assume that inter-arrival time interval is equal to t. Let us determine the conditional probability so that this interval lasts for at least time τ.

• So, we can

)( tTTP

)()()( tTTPtTPtTP

tetTPtF 1)()(

0 t+t T

Lack of memory property

• Taking into account the distribution function we have

• The conditional probability have to receive the following value:

)()( TPetTTP

)( tTTP

)()( tTTPee tt >>= -+- tltl

)()(1 ttPi

)()(1 tttP

)(1)(0 tttP

Singularity property

• Let us consider time interval Δt → 0. It results from the singularity property that probability of appearance of more then one arrivals within the time interval Δt is going towards 0:o where q(Dt) is infinitely small value if compared with Δt

• Elementary probabilities:

Poisson stream parameter

• The flow parameter L(t) at time point t is defined as the limit of quotient:

Probability of appearing at least one arrival within time interval Dt+ t

time interval length Dt ® 0:

)(lim)(

Poisson stream - characteristics

• Probability of appearance of k arrivals at time t:

• for k=0 i k=1 we receive:

)()( t

tetP )(0ttetP )(1

Poisson stream – characteristics

• Inter-arrival time distribution:

• Mean value and variance of inter-arrival time:

• Peakedness coefficient:

,)( tetf

.1/2 TT mZ

dttftmT

22 /1)(

TT mdttft

Streams operations

• Superposition of Poisson streams

• Random decomposition of Poisson stream

Strumieñ 1

Strumieñ 2

Strumieñ 3

Stream 1

Stream 2

Stream 3

Strumieñ 1p p p p

1-p1-p 1-p1-p

Stream 1

Streams operations

• Erlang k-decomposition

o inter-arrival time distribution

o mean value of inter-arrival time

o variance of inter-arrival time

o disorder coefficient

Strumieñ 1

Stream 1

Stream 2

,/ kmT

,/ 22 kT

./1/2 kmZ TT

Markov process definition

• A stochastic process is called the Markov process when the future trajectory of the process depends only on the present state S(t0) at the time point t0 , but is independent of how this state has been obtained.

przeszłość przyszłośćt0

t < t t > t0 0

past future

Markov process in the M/M/2 system

• A service process in the M/M/2 system (trunk group with two channels) is the Markov process when:o Arrival process is the Poisson process, o Service time has exponential distribution.

States

Trajectory of the service process in M/M/2 system

Service stream

Trajectory of the Markov process

Exponential service time

• Distribution function:

• Density function:

• Mean value and variance:

tetTPtF 1)()(

dttfth

22 /1)(

hdttfth

Service stream

• At time point t there are k servers busy. The probability of service termination in i servers within Δt time-interval can be determined on the basis of Bernoulli distribution for i successful events, when total number of events is equal to k:

• Probability of service termination in one server within Δt time-interval:

ikii PP

)1(),(

tetTPtFP 1)()(

Service stream

• For i=0 we obtain the probability of the event that within, time interval Δt, there are no terminations among k busy servers:

• Termination probability by at least one server:

• P1 (Δt ) decomposition (into series):

• Service stream parameter:

tketkP ),(0

tketkPtP 1),(1)( 01

)()()( ttkj

1tk11e1t

ùêë

)(lim)(

Markov proces

Birth and death process in M/M/2 system

• state 0 all links are free• state 1 one busy link• state 2 two links are busy

blocking state

Birth and death process in M/M/2 system

• Infinite number of traffic sources• Finite number of busy servers

Kolmogorov equations

• Determination of the probability P0(t + Δt)

• Events within time Δt:o Was in state "0" and transferred into state "1": λ Δt o Was in state "0" and remained in state "0": 1- λ Δt o Was in state "1" and transferred into state "0": μ Δt

0 1 1- t

Was in state "0" and remained in state "0"

).()()(

),()()()(

,)(1)()(

tPtPdt

ttPttPttP

0 1 1- t

Was in state "1" and transferred into state "0"

• Solution:

Solution of Koplmogorov equations in M/M/2/0 system for λ=μ=1, P0(0)=1

P (t)0

P (t)1

P (t)2t0 1 2

Steady-state distribution

)(lim dt

tdPtPp i

222120

Probability calculations: Solution

In the steady-state regime of the process, the state probabilities arenot time-dependable

Steady states

• Interpretation of the probability [Pi]V:

• The state probability is interpreted as the proportion of the time in which the system remains in state i:

n timeobservatio

statein spent timelim

timenobservatioi

Stream value in state i

Stream value in the state i for : t

n timeobservatio

statein spenttimeintensity stream

ii Pll =

State equations in the system M/M/2

• For the M/M/2/0 system state equations take the following form:

222120

In state i:

Sum of incoming streams = sum of outgoing streams

Generalized birth and death process

• State transition diagram:

2211100

ViiViiiVii

ViiVii

Local balance equation

• Streams between neighboring states are in equilibrium

• Process solution

ViiVii pp 11

kVk pCppp

kV Cpwhere:

Local balance equation in M/M/2

222120

Concept of Traffic

Telecommunication traffic

• Traffic as a process of capacity units occupancy

where n(t) – number of occupied units at time T

• Units: o 1 SM (speech-minutes)

o 1 Eh (Erlang-hour)

o 1 Eh = 60 SM

ttnTA0

vol d)()(

Telecommunication traffic intensity

• Traffic intensity:

o where n(t) – number of occupied units at time T

• Units: 1 Erlang ( 1 Erl.) o 1 Erlang = 1 call serviced during time t when observation

time is equal to t

Telecommunication traffic and traffic intensity

vol d1d tttttttnTATT

Traffic volume:

Traffic intensity:

TAA 321vol

service time service time

t1 t3t2

service time

Traffic intensity

service time service time

t1 t3t2

service time

t1 t3t2

busy time idle time

T=100% =time unit

% of idle time% of busy time

unittime

occupancyunittimeoffraction321

tttA busy

Traffic intensity

• Parameters:o V=4 - number of channels,o N=5 - number of time periods,o tobs=5T - period under

consideration,o ti,j - occupancy of the j-th

channel during the i-th time period

o - call intensityo h - mean service time.

)12214(/)1.(

1 1., Erl

TttDefA

jobsji

1T 2T 3T 4T 5T0

Traffic intensity Def. 1

• Traffic intensity is equal to the average number of simultaneously occupied channels during a given period of time under considerations.

)3223(/)2.(

1 1., Erl

TttDefA

jobsji

1T 2T 3T 4T 5T0

• Traffic intensity is the ratio of the sum of channel occupancy time during a given period of time under considerations with respect to this period.

1T 2T 3T 4T 5T0

8)3.( Erl

TchDefA

• The product of the average number of o calls (offered traffic)

o connections (carried traffic)

• per time unit and the average time of connection.

1T 2T 3T 4T 5T0

The mean number of calls (connections) per mean service time

offered traffic carried traffic

8)4.( Erl

TsDefA

SystemtelekomunikacyjnyRuch oferowany Ruch obsłużony

Ruch tracony

SystemOffered traffic Carried traffic

Rejected traffic

ATTENTION!Conventionally, under the notion of traffic we understand traffic intensity

Kinds of traffic

• Carried traffic o the traffic carried by the group of servers during the time

interval T

• Offered traffic o the traffic which would be carried if no calls were rejected

due to lack of the capacity, i.e. unlimited number of servers. The offered traffic is a theoretical value and it cannot be measured

• Lost (rejected) traffic o the difference between offered traffic and carried traffic

Quality of service in telecommunication

systems

Call and packet level in networks

),(),(

2121 ttN

ttNttN

ttNttB

offerd

carriedoffered

offered

Concept of blocking

• Call congestion (Call loss probability) B(t1, t2 ) in time interval (t1, t2) is the fraction of all calls which are rejected due to lack of capacity Nlost(t1, t2 ) with respect to all calls which are offered in the system Noffered(t1, t2 )

),(),(

2121 ttT

ttTttE blocking

Concept of blocking

• Time congestion (Blocking probability) E(t1, t2 ) in time interval (t1, t2) is the fraction of the time Tblocking(t1 , t2) when all servers are busy with respect to the total time of observation T(t1, t2)

Basic notions and parameters

• Traffic load capacityo the value of the offered traffic (traffic intensity) which can be

serviced with the adopted value of blocking probability (loss probability)

• Loado the value of the carried traffic (traffic intensity) in the system

• Blocking o the state of system in which a call arriving at the input of the

system cannot be serviced due to occupancy of all servers in the system

• Throughput o the probability of event that the given call will be serviced in the

system

Quality of service in communication systems (QoS)

• Packet delay (cell delay)o A delay considered between the moment of sending and

receiving the packet (in appropriate nodes)

• Delay parameters (for example of ATM network)

o CDTmean - Mean Cell Transfer Delay - statistical average delay of packet

o CTDmax - Maximum Cell Transfer Delay – maximum delay of packet, guaranteed by network with probability 1-α

o CDVpeak-peak - Peak to Peak Cell Delay Variation – maximum delay decreased by constant system delay (i.e. propagation time, processing time in node)

constant max delay variation

max delay

α= loss ratio

Quality of service in communication systems

• Interpretation of delay parameters

Reasons for networks delay

• Constant and independent of network loado propagation time in physical layer

o processing time in network node

o minimum time the node wait for packet acknowledgement

o bit rate of outgoing link bigger than incoming link

Reasons for networks delay

• Dependent on network loado queuing in the buffers

o queuing discipline,

o priorities for given packet classes

o mechanisms for packet streams shaping

o resources reservation for given packet classes

server

outgoing stream

buffer

incoming stream

Kinds of traffic at the packet level

• In majority of packet networks kinds of traffic are associated with parameters of offered services

• We can always distinguish the following traffic streamso Constant bit rate traffic

o Variable bit rate traffic• stream traffic, constant parameters of transmission• adaptive traffic• elastic traffic

Maciej Stasiak, Mariusz Głąbowski Arkadiusz Wiśniewski, Piotr Zwierzykowski Basic Definitions and...

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