Network Planning Network Planning VITMM215 VITMM215 Markosz - - PDF document

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Network Planning Network Planning

VITMM215 VITMM215

Markosz Maliosz Markosz Maliosz 1 11 1/ /0 04/2013 4/2013

Outline

Telephone network dimensioning

– Traffic modeling – Erlang formulas – Exercises

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– Exercises

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Telephone Network Telephone Network

• Circuit switching

Circuit switching

• Each voice channel

Each voice channel is identical is identical

• For

For each each call call one

• ne channel

channel is is allocated allocated

• A call is accepted if at least one channel is idle

A call is accepted if at least one channel is idle Goal Goal: : network dimensioning network dimensioning

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• Goal

Goal: : network dimensioning network dimensioning

• Question to answer

Question to answer: : How many circuits are required How many circuits are required to satisfy subscribers’ needs to satisfy subscribers’ needs? ?

• Input

Input: : traffic statistics traffic statistics

– – subscribers’ behavior: when, how often are calls arriving? subscribers’ behavior: when, how often are calls arriving? how long are the call durations? how long are the call durations?

Arrival Process Arrival Process

• In our case: telephone calls arriving to a

In our case: telephone calls arriving to a switching system switching system

• described as stochastic point process

described as stochastic point process

• we consider simple point processes, i.e. we

we consider simple point processes, i.e. we exclude multiple arrivals exclude multiple arrivals

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exclude multiple arrivals exclude multiple arrivals

• the i

the ith

th call arrives at time T

call arrives at time Ti

i

• N(t)

N(t): the cumulative number of calls in the half : the cumulative number of calls in the half-

• pen interval [0; t[
• pen interval [0; t[
• N(t)

N(t) is a random variable with continuous time is a random variable with continuous time parameter and discrete space parameter and discrete space

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t t N N(t) (t)

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Arrivals and Departures

N(t) D(t)

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N(t) to be the cumulative number of arrivals up to time t D(t) to be the cumulative number of departures up to

time t

L(t) = N(t) - D(t) is the number of calls at time t

Equations

Average arrival rate: λ

λ(t) (t) = = N(t)/t

F(t) = area of shaded region from 0 to t in

the figure

N(t) D(t) N(t) D(t)

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the figure

= total service time for all customers = carried traffic volume

Average holding time: W(t) = F(t)/N(t) Average number of calls: L(t) = F(t)/t

= W(t)N(t)/t = W(t)λ λ(t) (t)

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Traffic Volume

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Volume of the traffic: the amount of traffic

carried during a given period of time

Traffic volume in a period divided by the length

• f the period is the average traffic intensity in

that period = average number of calls

Traffic Variations

Traffic fluctuates over several time scales

– Trend (>year)

Overall traffic growth: number of users, changes in usage Predictions as a basis for planning

– Seasonal variations (months) – Weekly variations (day)

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– Weekly variations (day) – Daily profile (hours) – Random fluctuations (seconds – minutes)

In the number of independent active users: stochastic process

Except the last one, the variations follow a given

profile, around which the traffic randomly fluctuates

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Traffic Variations

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Source: A. Myskja, An introduction to teletraffic, 1995.

Busy Hour

• It is not practical to dimension a network for the largest traffic peak

It is not practical to dimension a network for the largest traffic peak

• describe the peak load, where singular peaks are averaged out

describe the peak load, where singular peaks are averaged out

• Busy Hour = The period of duration of one hour where the volume

Busy Hour = The period of duration of one hour where the volume

• f traffic is the greatest.
• f traffic is the greatest.
• Operator’s intention: spreading the traffic

Operator’s intention: spreading the traffic

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– – By service tariffs By service tariffs

• busy hour period is the most expensive

busy hour period is the most expensive

• less important calls are started outside of the busy hour, and typically last

less important calls are started outside of the busy hour, and typically last longer longer

• Recommendations define how to measure the busy hour traffic

Recommendations define how to measure the busy hour traffic

– – There are several definitions (ITU E.600, E.500) There are several definitions (ITU E.600, E.500) – – An operator may choose the most appropriate one An operator may choose the most appropriate one

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Busy Hour Measurements

ADPH (Average Daily Peak Hour)

– one determines the busiest hour separately for each day (different time for different days), and then averages over e.g. 10 days

TCBH (Time Consistent Busy Hour)

– a period of one hour, the same for each day, which gives the greatest average traffic over e.g. 10 days

FDMH (Fixed Daily Measurement Hour)

– a predetermined, fixed measurement hour (e.g. 9.30-10.30); the

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– a predetermined, fixed measurement hour (e.g. 9.30-10.30); the measured traffic is averaged over e.g. 10 days

Traffic Model

• Average arrival rate:

Average arrival rate: λ λ(t) (t) – – depends on time, however it depends on time, however it has a very strong deterministic component according to has a very strong deterministic component according to the profiles the profiles

• In the busy hour period the average arrival time is

In the busy hour period the average arrival time is considered stationary: considered stationary: λ λ, , and the arrival process is and the arrival process is considered as a Poisson process with intensity considered as a Poisson process with intensity λ λ

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considered as a Poisson process with intensity considered as a Poisson process with intensity λ λ

– – Time homogenity Time homogenity – – Independence Independence

• The future evolution of the process only depends upon the actual

The future evolution of the process only depends upon the actual state. state.

• Independent of the user

Independent of the user(!) (!) – – modeling all users in the same way modeling all users in the same way

• The average holding time (

The average holding time (W(t) W(t)) is also considered to be ) is also considered to be stationary, and exponentially distributed with intensity stationary, and exponentially distributed with intensity µ µ

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Traffic Model Traffic Model

• N(t)

N(t) – – Poisson Poisson process process: :

– – in time interval ( in time interval (t t, ,t t + τ] the + τ] the number of calls follows a Poisson number of calls follows a Poisson distribution with parameter distribution with parameter λτ λτ

Poisson Poisson

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– – Expected number of calls Expected number of calls = = λ λτ τ – – λ λ = = arrival intensity arrival intensity [1/ [1/hour hour] ]

• W(

W(t) = t) = W W – – exp exp. . distribution distribution

– – Expected value Expected value = = 1/ 1/µ µ = = h h – – h h – – average holding time average holding time [ [min min] (!) ] (!) f(x; f(x; µ µ) = ) = µ µe e-

• µ

µx x

Exp. Exp.

Traffic Intensity Traffic Intensity

• A

A – – traffic intensity traffic intensity

– – A A = = λ λ * * h h – – A [1], A [1], often written as

• ften written as Erl (Erlang)

Erl (Erlang)

• Example

Example: : individual subscriber individual subscriber

– – λ λ = 3 = 3 [1/ [1/hour hour] ]

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– – λ λ = 3 = 3 [1/ [1/hour hour] ] – – h h = 3 [ = 3 [min min] ] = = 0.05 [ 0.05 [hour hour] ] – – A = A = 3 3 [1/ [1/hour hour]* 0.05 [ ]* 0.05 [hour hour] ] = = 0.15 0.15 [Erl] [Erl]

• Example

Example: 10 000 : 10 000 line switch line switch

– – λ λ = 20 000 = 20 000 [1/ [1/hour hour] ] – – h h = 3 [ = 3 [min min] ] = = 0.05 [ 0.05 [hour hour] ] – – A = 20 000 [1/ A = 20 000 [1/hour hour]* 0.05 [ ]* 0.05 [hour hour] ] = = 1000 1000 [Erl] [Erl]

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Typical Traffic Intensities Typical Traffic Intensities

• Typical traffic intensities per a single

Typical traffic intensities per a single source are (fraction of time they are being source are (fraction of time they are being used) used)

– – private subscriber 0.01 private subscriber 0.01 -

• 0.04 Erlang

0.04 Erlang

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– – private subscriber 0.01 private subscriber 0.01 -

• 0.04 Erlang

0.04 Erlang – – business subscriber 0.03 business subscriber 0.03 -

• 0.06 Erlang

0.06 Erlang – – mobile phone 0.03 Erlang mobile phone 0.03 Erlang – – PBX (Private Branch Exchange) 0.1 PBX (Private Branch Exchange) 0.1 -

• 0.6 Erlang

0.6 Erlang – – coin operated phone 0.07 Erlang coin operated phone 0.07 Erlang

Traffic Modeling

Agner Krarup Erlang (1878 – 1929)

– Danish mathematician, statistician and engineer

Conditions:

– n identical channels – Blocked Calls are Cleared (BCC)

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– Blocked Calls are Cleared (BCC) – The arrival process is a Poisson process with intensity λ – The holding times are exponentially distributed with intensity µ (corresponding to a mean value 1/µ)

The traffic process then becomes a pure birth and

death process, a simple Markov process

– A= λ/µ

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Infinite number of channels

State diagram – nr. of busy channels:

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If the system is in statistical equilibrium, then the system

will be in state [i] the proportion of time p(i), where p(i) is the probability of observing the system in state [i] at a random point of time, i.e. a time average

When the process is in state [i] it will jump to state [i+1]

λ times per time unit and to state [i-1] iµ times per time unit

Infinite number of channels

In equilibrium state

– Node equations:

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– Cut equations:

Normalization restriction:

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Infinite number of channels

Derivation of cut equations:

A= λ/µ

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Infinite number of channels

Using the normalization constraint:

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State probabilities: Carried traffic = offered traffic = A No congestion, no traffic loss

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Limited number of channels

State diagram:

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Normalization condition becomes: State probabilities:

Erlang B formula

Time congestion:

– The probability that all n channels are busy at a random point of time is equal to the portion of time all channels are busy (time average)

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Call congestion:

– The probability that a random call attempt will be lost is equal to the proportion of call attempts blocked. If we consider one time unit, we find by summation over all possible states:

Carried traffic = Lost traffic =

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Erlang Erlang B formula B formula

• Conditions for applicability

Conditions for applicability: :

– – Gives good results if number of subscribers is much Gives good results if number of subscribers is much greater, than the number of channels greater, than the number of channels ( (around around 10x) 10x) – – Subscribers initiate calls independently from each other Subscribers initiate calls independently from each other ( (not applicable e.g. if a TV advertisement presents a not applicable e.g. if a TV advertisement presents a

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( (not applicable e.g. if a TV advertisement presents a not applicable e.g. if a TV advertisement presents a phone number and many people call it) phone number and many people call it) – – The only reason for blocking is if all channels are busy The only reason for blocking is if all channels are busy – – Blocked Calls are Cleared, no waiting queue Blocked Calls are Cleared, no waiting queue – – Subscribers do not repeat call attempt, if call was blocked Subscribers do not repeat call attempt, if call was blocked – – The channel is occupied only by the particular subscribers, The channel is occupied only by the particular subscribers, no resource sharing no resource sharing

Erlang B formula Erlang B formula

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Erlang Erlang B formula B formula

• Example

Example: 3 : 3 employees employees in an office in an office, , each of each of them calls 3 times in an hour with 3 minutes them calls 3 times in an hour with 3 minutes talking time talking time. .

• Question

Question: : How many channels are needed How many channels are needed for for max

• max. 5%

. 5% blocking blocking? (1? 2? 3??) ? (1? 2? 3??)

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for for max

• max. 5%

. 5% blocking blocking? (1? 2? 3??) ? (1? 2? 3??)

• A

Answer nswer: :

– – λ λ = 3* = 3*3 3 [1/ [1/hour hour] ] – – h h = 3 [ = 3 [min min] ] = = 0.05 [ 0.05 [hour hour] ] – – A = 3* A = 3*3 3 [1/ [1/hour hour]* 0.05 [ ]* 0.05 [hour hour] = 0 ] = 0. .45 [ 45 [Erl Erl] ]

• E

E(1) (1)=31% =31%

• E

E(2) (2)=6 =6. .5% ( 5% (not enough not enough!) !)

• (

( E E(3) (3)=1%, =1%, in reality in reality: : E E(3 (3)=0) )=0)

– – 3 channels are needed 3 channels are needed

Erlang B formula Erlang B formula

• E.g

E.g. 1000 . 1000 subscriber subscriber and and n n channels channels: : – – λ λ = 1000*3 [1/ = 1000*3 [1/hour hour] ] – – h h = 3 [ = 3 [min min] ] – – A = 1000*3 [1/ A = 1000*3 [1/hour hour]* 0.05 [ ]* 0.05 [hour hour] = 150 [ ] = 150 [Erl Erl] ] – – E(n) E(n): :

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– – E(n) E(n): :

• If the number of subscribers are large, the required

If the number of subscribers are large, the required number of channels (n) for a satisfactory blocking ratio number of channels (n) for a satisfactory blocking ratio converges to A converges to A

n n 100 100 150 150 155 155 160 160 200 200 E E(n) (n) 34% 34% 6,2% 6,2% 4,3% 4,3% 2,8% 2,8% 0,0015% 0,0015%

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Erlang B formula Erlang B formula

If A and achievable blocking is given, how to If A and achievable blocking is given, how to calculate n? calculate n?

• By probing

By probing

• Recursive method

Recursive method: : ∑

=

=

n i i n n

i A n A A E ! ! ) (

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• Recursive method

Recursive method: :

– – Def Def.: I .: In

n(

(A A) ) = 1 / E = 1 / En

n(A)

(A) – – I I0

0(A) = 1, that is with 0 channel the blocking = 1

(A) = 1, that is with 0 channel the blocking = 1 – – I In

n(A) = I

(A) = In

n-

• 1

1(

(A A) * n / A + 1 ) * n / A + 1 – – E.g. if the goal is E.g. if the goal is: : E En

n(A)

(A) = = 1 / I 1 / In

n(

(A A) ) < 0.05 < 0.05 – – I In

n(A) > 1/0.05 = 20

(A) > 1/0.05 = 20

Extended Erlang B

• Extended

Extended Erlang B: Erlang B: a certain percentage of a certain percentage of blocked calls blocked calls are reattempted are reattempted

– – Iterative calculation with extra parameter, the Recall Iterative calculation with extra parameter, the Recall Factor: R Factor: Rf

f

– – A A0

0:initial traffic intensity

:initial traffic intensity

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– – A A0

0:initial traffic intensity

:initial traffic intensity 1.

• 1. Calculate E

Calculate En

n(A

(A0

0) with Erlang B

) with Erlang B 2.

• 2. Nr. of blocked calls: B = A
• Nr. of blocked calls: B = A0

0 E

En

n(A

(A0

0)

) 3.

• 3. Nr. of recalls: R = R
• Nr. of recalls: R = Rf

f B

B 4.

• 4. New offered traffic: A

New offered traffic: A1

1 = A

= A0

0+ R

+ R 5.

• 5. Return to step 1 and iterate until value of A is

Return to step 1 and iterate until value of A is stabilized stabilized

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Erlang C formula

• Erlang C:

Erlang C: blocked calls remain in the system blocked calls remain in the system (waiting in a queue), until they get served (waiting in a queue), until they get served

– – E.g E.g. call center . call centers s

• Probability of waiting:

Probability of waiting:

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• Probability of waiting:

Probability of waiting:

∑

− =

− + − =

1

) ( ! ! ) ( ! ) , (

n i n i n w

A n n n A i A A n n n A A n P

Exercises Exercises Exercises Exercises

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Problems Problems

• What is the blocking probability if traffic intensity is 2 Erl and 5 lines are

available?

• Analyze the following diagram!

– Examine the network utilization if the number of available channels is low! – Examine the network utilization depending on the blocking ratio!

100%

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0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 A/n n 1% blocking 10% blocking

Problems Problems

How many lines are required for 100 subscribers, when

they individually generate 0.04 Erl traffic intensity?

– if the allowed blocking ratio is 20%? – if the allowed blocking ratio is 1%?

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20 employees work in an office with 2 lines. What is the

blocking ratio if employees call with 0.1 Erl intensity?

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Problems Problems

• 10 employees work in an office with 3 lines. What is the blocking ratio if

employees initiate once a 15 min long call in the busy hour?

– Average utilization of lines? – Blocking ratio? – Is it a well dimensioned system?

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• A subscriber generates 0.1 Erl traffic intensity. How many lines are

required, if the blocking requirement is 1% and the number of subscribers are

– 10? (5) – 100? (18) – 1 000? (117) – 4 000? (426) – 10 000? (1029)