Traffic Engineering

Traffic EngineeringTraffic Engineering• One billion+ terminals in voice network alone

– Plus data, video, fax, finance, etc.

• Imagine all users want service simultaneously…its not even nearly possible (despite our common intuition)– In practice, the actual amount of equipment provisioned is vastlyIn practice, the actual amount of equipment provisioned is vastly

less than would support all users simultaneously

• And yet, by and large, we get the impression of phone and data networks that work very well!networks that work very well!

• How is this possible?

Traffic theory !!y

Traffic Engineering Traffic Engineering –– TradeTrade--offsoffs• Design number of transmission paths, or radio channels?

– How many required normally?– What if there is an overload?

• Design switching and routing mechanisms– How do we route efficiently?How do we route efficiently? – E.g.

• High-usage trunk groups -Trunk group that is the primary direct route between two switching systems. The group is provided with an alternate route for overflow traffic in order to provide an acceptable level of blocking.

• Overflow trunk groups• Where should traffic flows be combined or kept separate?Where should traffic flows be combined or kept separate?

• Design network topology– Number and sizing of switching nodes and locations

N b d i i f t i i t d l ti– Number and sizing of transmission systems and locations– Survivability

Characterization of Telephone TrafficCharacterization of Telephone Traffic

•• Calling RateCalling Rate () – also called arrival rate, or attempts rate, etc.– Average number of calls initiated per unit time (e.g. attempts per g p ( g p p

hour)– Each call arrival is independent of other calls (we assume)– Call attempt arrivals are random in time– Until otherwise, we assume a “large” calling group or source pool

Tαγ If receive calls from a terminal in time TT:T

If receive calls from mm terminals in time T:

Tαγ g

Group calling rateTm

αγ

Per terminalGroup calling rate Per terminalcalling rate

Characterization of Telephone Traffic (2)Characterization of Telephone Traffic (2)• Calling rate assumption:

– Number of calls in time T is Poisson distributed: e x

– In our case

...2 ,1 ,0!

)(

xx

exp

T

– x – No of Call Arrivals– λ – average call arrival rate

Ti b t ll i “ ti ll ” di t ib t d• Time between calls is “-ve exponentially” distributed:

tetf t 0)( 1

mean

• In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time and/or space if these events occur with a known average rate

• Class Question: What do these observations about telephone traffic imply about the nature of the traffic sources?

p gand independently of the time since the last event.

--ve Exponential Holding Timesve Exponential Holding Times•• Implies the “MemoryImplies the “Memory--less” propertyless” property

– Prob. a call last another minute is independent of how long the call has already lasted! Call “forgets” that it has already survived to time T1a eady asted Ca o gets t at t as a eady su ed to t e 1

tTPTTtTTP 11

• Proof:

1

1111

TTP

TTtTTPTTtTTP

1

1 TTP

tTTP

hT

htT

ee

/

/)(

1

1

htetTP /)(

Recall:

hT

hthT

eee/

//

1

1

hte / tTP

etTP )(

•In probability theory and statistics, the exponential distribution (a.k.a. negative exponential distribution) is a family of continuous probability distributions. It describes the time between events in a Poisson process.

Characterization of Telephone Traffic (3)Characterization of Telephone Traffic (3)

•• Holding TimeHolding Time (hh)– Mean length of time a call lasts– Probability of lasting time t or more is also –ve exponential in

nature:0)( / tetTP ht

– Real voice calls fits very closely to the negative exponential form

00)( ttTP

Real voice calls fits very closely to the negative exponential form above

– As non-voice “calls” begin to dominate, more and more calls have a constant holding time characteristic

•• Departure RateDeparture Rate ():

h1

Some Real Holding Time DataSome Real Holding Time Data

Traffic Traffic Volume (V)Volume (V)

hV = # calls in time period T

h h ldi tihV h = mean holding time

V = volume of calls in time period T

• In N. America this is historically usually expressed in terms of “ccsccs”:– Hundred call seconds

“cc” “cc” “ss”

– 1 ccs is volume of traffic equal to:one ci c it b s fo 100 seconds o

Centum Call Seconds

– one circuit busy for 100 seconds, or– two circuits busy for 50 seconds, or– 100 circuits busy for one second, etc.

TrafficTraffic Intensity (A)Intensity (A)• Also called “traffic flowtraffic flow” or simply “traffictraffic”.

= # calls in time period Th = # calls in time period T = # calls in time period TV = # calls in time period T # calls in time period T

h = mean holding time

T = time period of observations

hA

Th

# calls in time period T

h = mean holding time

T = time period of observations

# calls in time period T

h = mean holding time

T = time period of observationsR ll R ll

V

T

R ll

# calls in time period T

h = mean holding time

T = time period of observationspp

= calling rate

p

= calling rate

= departure rateT

Recall:

h1

Recall:

V h

Recall:p

= calling rate

= departure rate

• Units:– “ccs/hourccs/hour”, or

dimensionless (if h and T are in the same units of time)

V = call volume

– dimensionless (if h and T are in the same units of time)

“ErlangErlang” unit

The The ErlangErlang• Dimensionless unit of traffic intensity

• Named after Danish mathematician A. K. Erlang (1878-1929)Named after Danish mathematician A. K. Erlang (1878 1929)

• Usually denoted by symbol EE.

• 1 Erlang is equivalent to traffic intensity that keeps:g q y p– one circuit busy 100% of the time, or– two circuits busy 50% of the time, or– four circuits busy 25% of the time, etc.y ,

• 26 Erlangs is equivalent to traffic intensity that keeps :– 26 circuits busy 100% of the time, or

52 circuits busy 50% of the time or– 52 circuits busy 50% of the time, or– 104 circuits busy 25% of the time, etc.

Class Class • Could 4 E be produced as a traffic intensity by:

– 16 sources? (What is the utilization?)– 4 sources (same)– 1 source?

• What is special about the traffic intensity if it pertains to one source or terminal only?

Erlang (2)Erlang (2)• How does the ErlangErlang unit correspond to ccsccs?

100 call seconds 0.027E100 call seconds1 ccs hour1 hour × 60 min hr × 60 sec min

3600 call seconds

× 60 min hr × 60 sec min

• Percentage of time a terminal is busy is equivalent to the traffic

1E3600 call seconds36 ccs hour1 hour × 60 min hr × 60 sec min

× 60 min hr × 60 sec min

• Percentage of time a terminal is busy is equivalent to the traffic generated by that terminal in Erlangs, or

• Average number of circuits in a group busy at any time

• Typical usages:– residence phone -> 0.02 E– business phone -> 0 15 Ebusiness phone > 0.15 E– interoffice trunk -> 0.70 E

Traffic Offered, Carried, and LostTraffic Offered, Carried, and Lost•• Offered TrafficOffered Traffic (TTO O ) equivalent to Traffic Intensity (AA)

– Takes into account all attempted calls, whether blocked or not, d h d h ldand uses their expected holding times

• Also Carried TrafficCarried Traffic (TTC C ) and Lost TrafficLost Traffic (TTL L )

• Consider a group of 150 terminals each with 10% utilization (or• Consider a group of 150 terminals, each with 10% utilization (or in other words, 0.1 E per source) and dedicated servicededicated service:

11 each terminal has an1

150

1

150

outgoing trunk(i.e. terminal:trunk ratio = 1:1)

150150TO = A = 150 x 0.10 E = 15.0 E

TC = 150 x 0.10 E = 15.0 E

T 0 ETL = 0 E

Traffic Offered, Carried, and Lost (2)Traffic Offered, Carried, and Lost (2)• A = TO = TC + TL

TrafficLost

TrafficIntensity Offered

Traffic

CarriedTraffic

Traffic

• T = T x Prob Blocking (or congestion)• TL = TO x Prob. Blocking (or congestion)

= P(B) x TO = P(B) x A

•• Circuit UtilizationCircuit Utilization () - also called Circuit EfficiencyCircuit Efficiency– proportion of time a circuit is busy, or– average proportion of time each circuit in a group is busy

CT # of Trunks# of Trunks

Grade of Service (gos)Grade of Service (gos)• In general, the term used for some traffic design objective

• Indicative of customer satisfactionIndicative of customer satisfaction

• In systems where blocked calls are cleared, usually use:

L LT T ( )L L

O L C

T T ( )T T + T

P Bgos

• Typical gos objectives:• Typical gos objectives:– in busy hour, range from 0.2% to 5% for local calls, however– generally no more that 1%– long distance calls often slightly higherlong distance calls often slightly higher

• In systems with queuing, gos often defined as the probability of d l di ifi l th f tidelay exceeding a specific length of time

Grade of Service Related TermsGrade of Service Related Terms•• Busy HourBusy Hour

– One hour period during which traffic volume or call attempts is the h h ll d dhighest overall during any given time period

•• Peak (or Daily) Busy HourPeak (or Daily) Busy Hour– Busy hour for each day, usually varies from day to dayy y, y y y

•• Busy SeasonBusy Season– 3 months (not consecutive) with highest average daily busy hour

•• High Day Busy Hour (HDBH)High Day Busy Hour (HDBH)– One hour period during busy season with the highest load

Grade of Service Related Terms (2)Grade of Service Related Terms (2)•• Average Busy Season Busy Hour (ABSBH)Average Busy Season Busy Hour (ABSBH)

– One hour period with highest average daily busy hour during the b

•• Average Busy Season Busy Hour (ABSBH)Average Busy Season Busy Hour (ABSBH)– One hour period with highest average daily busy hour during the

bbusy seasonbusy season– For example, assume days shown below make up the busy season:

1-Apr 2-Apr 3-Apr 4-Apr 5-Apr 6-Apr 7-Apr 8-Apr 9-Apr 10-Apr 11-Apr 12-Apr 13-Apr 14-Apr 15-Apr 16-Apr 17-Apr 18-Apr 19-Apr 20-Apr 21-Apr Mean00:00 to 01:00 1.4 1.4 1.2 1.5 1.1 1.5 1.7 1.5 1.0 1.0 1.8 1.5 1.8 1.6 1.2 1.9 1.8 1.6 1.4 1.5 1.2 1.501 00 t 02 00 1 2 1 8 1 6 1 3 1 0 1 6 1 1 1 1 1 0 1 2 1 7 2 0 2 0 1 8 1 3 1 7 1 4 1 9 1 1 1 4 1 5 1 5

HighestHighestABSBHABSBH

01:00 to 02:00 1.2 1.8 1.6 1.3 1.0 1.6 1.1 1.1 1.0 1.2 1.7 2.0 2.0 1.8 1.3 1.7 1.4 1.9 1.1 1.4 1.5 1.502:00 to 03:00 1.4 1.8 1.5 1.9 1.2 1.0 1.2 1.1 1.1 1.7 1.5 1.5 1.9 1.9 1.3 1.5 1.8 1.1 1.1 1.2 1.5 1.403:00 to 04:00 1.2 1.8 1.7 1.4 1.7 1.1 1.5 1.6 1.1 1.9 1.0 1.0 1.4 1.5 1.6 1.1 1.4 1.9 1.4 1.2 1.1 1.404:00 to 05:00 1.8 1.8 2.3 2.2 2.0 1.7 2.3 1.6 2.2 1.5 2.1 1.6 2.3 2.1 1.7 2.5 1.6 2.0 1.7 1.5 2.3 1.905:00 to 06:00 2.2 2.3 1.9 2.4 2.5 2.0 2.0 1.7 1.8 1.6 2.0 2.0 2.2 2.2 2.1 1.8 1.6 1.7 2.0 2.3 2.1 2.006:00 to 07:00 1.7 2.2 1.7 2.5 2.2 2.1 2.2 2.0 2.3 1.6 2.4 2.2 1.5 2.1 2.2 1.8 1.8 1.7 2.1 2.0 2.1 2.007:00 to 08:00 2.0 2.8 2.2 2.4 2.3 2.4 2.9 2.0 2.4 2.4 2.1 2.9 2.3 2.1 2.9 2.7 2.8 2.3 2.1 2.1 2.7 2.408:00 to 09:00 3.4 3.1 2.8 2.9 2.5 2.7 2.9 3.0 3.4 3.4 3.1 2.9 2.9 2.9 3.3 3.2 3.5 3.1 3.1 3.1 2.5 3.009:00 to 10:00 3.4 3.4 4.0 3.2 3.5 3.4 3.1 3.7 3.3 3.3 3.5 3.9 3.4 4.0 3.7 3.7 3.1 3.4 3.9 3.9 3.4 3.510:00 to 11:00 5.0 4.4 4.8 4.9 4.1 3.0 4.0 4.9 4.2 4.9 4.7 4.2 3.8 3.0 4.6 4.9 4.4 5.0 4.7 3.6 3.8 4.311:00 to 12:00 4.8 5.0 4.7 4.3 4.5 3.8 3.4 4.2 5.0 4.6 5.0 4.7 3.2 3.4 5.0 4.8 4.1 4.3 4.4 3.6 3.7 4.312:00 to 13:00 4.5 4.2 4.1 4.8 4.6 3.8 3.3 4.0 4.2 4.6 4.7 4.0 3.3 3.1 5.0 4.9 4.6 4.1 4.2 3.2 3.6 4.113:00 to 14:00 4.3 4.2 4.7 4.5 4.8 3.2 3.1 4.1 4.5 4.6 4.9 4.7 3.6 3.6 4.8 4.2 4.8 4.9 4.4 3.3 3.0 4.214:00 to 15:00 4.8 4.7 4.5 4.1 4.4 3.6 3.7 4.5 4.3 4.3 4.9 4.5 3.5 3.5 4.3 4.3 4.3 4.5 4.3 3.3 3.2 4.214:00 to 15:00 4.8 4.7 4.5 4.1 4.4 3.6 3.7 4.5 4.3 4.3 4.9 4.5 3.5 3.5 4.3 4.3 4.3 4.5 4.3 3.3 3.2 4.215:00 to 16:00 4.4 4.9 4.4 4.8 4.5 3.8 3.2 4.1 4.8 4.4 4.5 4.2 3.3 3.9 4.3 4.9 4.4 4.3 4.5 3.7 3.3 4.216:00 to 17:00 3.2 3.2 3.8 3.5 3.7 3.1 3.5 3.5 3.2 3.2 3.8 3.4 3.2 4.0 3.3 4.0 3.9 3.0 3.3 3.5 3.3 3.517:00 to 18:00 2.7 2.6 2.7 2.9 3.3 3.1 3.4 2.9 3.2 2.8 2.7 3.0 3.3 3.2 2.5 2.9 2.8 3.4 3.5 2.9 3.2 3.018:00 to 19:00 3.0 2.9 3.0 2.7 2.9 3.4 3.3 3.4 2.7 3.3 3.5 3.5 2.7 3.1 3.1 3.3 3.4 3.1 3.0 3.3 3.3 3.119:00 to 20:00 3.3 3.3 2.6 3.4 3.2 2.7 2.7 3.4 3.4 3.0 3.0 3.4 3.1 2.8 3.2 3.4 3.0 3.4 3.4 3.1 2.9 3.120:00 to 21:00 2.9 2.3 2.1 2.9 2.9 3.0 3.0 2.4 2.3 2.9 3.0 2.1 2.2 2.9 3.0 2.6 2.4 2.5 2.7 2.7 2.6 2.621:00 to 22:00 2 1 1 6 2 3 1 6 2 2 2 1 2 4 1 9 1 6 2 1 2 4 1 7 1 8 2 4 1 8 1 9 2 2 1 9 2 2 2 2 1 6 2 0

Note: Red indicatesdaily busy hour

21:00 to 22:00 2.1 1.6 2.3 1.6 2.2 2.1 2.4 1.9 1.6 2.1 2.4 1.7 1.8 2.4 1.8 1.9 2.2 1.9 2.2 2.2 1.6 2.022:00 to 23:00 1.5 2.1 1.9 1.6 1.7 1.6 2.3 2.5 2.4 1.7 2.1 1.8 2.0 2.4 1.7 1.9 2.2 2.3 1.7 2.4 1.8 2.023:00 to 00:00 1.5 1.0 1.1 1.1 1.5 1.8 1.5 1.4 1.8 1.1 1.9 1.2 1.6 1.9 1.8 1.1 1.5 2.0 1.8 1.6 1.4 1.5

Hourly Traffic VariationsHourly Traffic Variations

Daily Traffic VariationsDaily Traffic Variations

Seasonal Traffic VariationsSeasonal Traffic Variations

Seasonal Traffic Variations (2)Seasonal Traffic Variations (2)

Typical Call Attempts BreakdownTypical Call Attempts Breakdown• Calls Completed - 70.7%

• Called Party No Answer - 12.7%Called Party No Answer 12.7%

• Called Party Busy - 10.1%

• Call Abandoned - 2.6%

• Dialing Error - 1.6%

• Number Changed or Disconnected - 0.4%

• Blockage or Failure - 1.9%

3 Types of Blocking Models3 Types of Blocking Models• Blocked Calls Cleared (BCCBCC)

– Blocked calls leave system and do not return– Good approximation for calls in 1st choice trunk group

• Blocked Calls Held (BCHBCH)– Blocked calls remain in the system for the amount of time it wouldBlocked calls remain in the system for the amount of time it would

have normally stayed for– If a server frees up, the call picks up in the middle and continues– Not a good model of real world behaviour (mathematical g (

approximation only)– Tries to approximate call reattempt efforts

• Blocked Calls Wait (BCWBCW)• Blocked Calls Wait (BCWBCW)– Blocked calls enter a queue until a server is available– When a server becomes available, the call’s holding time begins

Traffic formula selection Decision treeTraffic formula selection Decision tree

Blocked Calls Cleared (BCC)Blocked Calls Cleared (BCC)

10 minutes

2 sources

Source #1

Offered Traffic 1 3

Total Traffic Offered:

Source #2

Offered Traffic 2 4

TO = 0.4 E + 0.3 ETO = 0.7 E

Only one server

Traffic

1st call arrives and is served

2nd call arrives but server already busyTraffic

Carried 1

server already busy

22nd call is cleared

1

3rd call arrives and is served

3 4

Total Traffic Carried:4th call arrives and is served

Total Traffic Carried:TC = 0.5 E

Blocked Calls Held (BCH)Blocked Calls Held (BCH)

10 minutes

2 sources

Source #1

Offered Traffic 1 3

Total Traffic Offered:

Source #2

Offered Traffic 2 4

TO = 0.4 E + 0.3 ETO = 0.7 E

Traffic

Only one server1st call arrives and is served

2nd call arrives but server busy

Traffic

Carried 1 21 2 3 42nd call is served

3rd call arrives and is servedTotal Traffic Carried:

2nd call is held until server free

4th call arrives and is served

Total Traffic Carried:TC = 0.6 E

Blocked Calls Wait (BCW)Blocked Calls Wait (BCW)

10 minutes

2 sources

Source #1

Offered Traffic 1 3

Total Traffic Offered:

Source #2

Offered Traffic 2 4

TO = 0.4 E + 0.3 ETO = 0.7 E

1st ll i d i d

Only one server

Traffic

1st call arrives and is served

2nd call arrives but server busy

2nd call waits until server freeTraffic

Carried 1 2 2nd call served1 2

3rd call arrives, waits, and is served

3

th ll i i d

4

Total Traffic Carried:4th call arrives, waits, andis served

Total Traffic Carried:TC = 0.7 E

Blocking ProbabilitiesBlocking Probabilities• System must be in a Steady StateSteady State

– Also called state of statistical equilibrium–– Arrival RateArrival Rate of new calls equals Departure RateDeparture Rate of

disconnecting calls– Why?

If ll i f h h d ?• If calls arrive faster that they depart?• If calls depart faster than they arrive?

Binomial Distribution ModelBinomial Distribution Model• Assumptions:

–– mm sources–– AA Erlangs of offered traffic

• per source: TO = A/m• probability that a specific source is busy: P(B) = A/m

• Can use Binomial Distribution to give the probability that a certain number (kk) of those m sources is busy:

kmk

mA

mA

km

kP

1)(

kmk

mA

mA

kmkm

1)!(!

!mmk mmkmk )!(!

Binomial Distribution Model (2)Binomial Distribution Model (2)• What does it mean if we only have N serversN servers (N<m)?

– We can have at most N busy sources at a time– What about the probability of blocking?

• All N servers must be busy before we have blocking

)()( NkPBP )(...)1()( mkPNkPNkP

m kmk AAm

kmk

mA

mA

km

kP

1)(

m

Nk mA

mA

km

1

kk

Remember:

1

011

N

k

kmk

mA

mA

km

Binomial Distribution Model (3)Binomial Distribution Model (3)• What does it mean if k>N?

– Impossible to have more sources busy than servers to serve them– Doesn’t accurately represent reality

• In reality, P(k>N) = 0

– In this model, we still assign P(k>N) = A/m – Acts as good model of real behaviour

• Some people call back, some don’t

• Which type of blocking model is the Binomial Distribution?yp g– Blocked Calls Held (BCH)

Time Congestions Time Congestions vs.vs. Call CongestionCall Congestion• Time Congestion

– Proportion of time a system is congested (all servers busy)– Probability of blocking from point of view of servers

• Call Congestion– Probability that an arriving call is blockedProbability that an arriving call is blocked– Probability of blocking from point of view of calls

• Why/How are they different?

Time Congestion:

)()( NkPBP

Call Congestion:

)()( NkPBP )()(

Probability that allservers are busy.

)()(

Probability that there aremore sources wanting serviceth ththan there are servers.

Poisson Traffic ModelPoisson Traffic Model• Poisson approximates Binomial with large mlarge m and small A/msmall A/m

k

!)(

kekP

k = Mean # of

Busy Sources

Note: )(lim BinomialPoissonm

• What is ?• What is ?– Mean number of busy sources

– = A

!)(

kAekP

kA

Poisson Traffic Model (2)Poisson Traffic Model (2)• Now we can calculate probability of blocking:

)()( NkPBP )(...)1()( PNPNP

Remember:

AkA kA Ae

A

kA!

)(kAekP

kA

Nk k!

Nk

Aek!

N kA1A

N

k

k

ekA

1

0 !1

)()( ANPBPExample:

)107(P),()( ANPBP “P” = Poisson “A” = Offered Traffic

)10,7(P

PoissonPoisson P(B) with 10 E10 Eoffered to 7 servers7 servers

“N” = # Serversoffered to 7 servers7 servers

Traffic TablesTraffic Tables• Consider a 1% chance of blocking in a system with N=10 trunks

– How much offered traffic can the system handle?

A

k

k

k

Ak

ekAe

kA

9

010 !1

!01.0

• How do we calculate A?– Very carefully, or

ff bl– Use traffic tables

Traffic Tables (2)Traffic Tables (2)P(B)=P(N,A)P(B)=P(N,A)

NN

AA

Traffic Tables (3)Traffic Tables (3)P(N,A)=0.01P(N,A)=0.01

N=10N=10

A=4.14 EA=4.14 E

If system with N = 10 trunksIf system with N = 10 trunkshas P(B) = 0.01:has P(B) = 0.01:

System can handleSystem can handleSystem can handleSystem can handleOffered traffic (A) = 4.14 EOffered traffic (A) = 4.14 E

Poisson Traffic TablesPoisson Traffic TablesP(N,A)=0.01P(N,A)=0.01

N=10N=10

A=4.14 EA=4.14 E

If system with N = 10 trunksIf system with N = 10 trunkshas P(B) = 0.01:has P(B) = 0.01:

System can handleSystem can handleSystem can handleSystem can handleOffered traffic (A) = 4.14 EOffered traffic (A) = 4.14 E

Efficiency of Large GroupsEfficiency of Large Groups• What if there are N = 100 trunks?

– Will they serve A = 10 x 4.14 E = 41.4 E with same P(B) = 1%?– No!– Traffic tables will show that A = 78.2 E!

• Why will 10 times trunks serve almost 20 times traffic?Why will 10 times trunks serve almost 20 times traffic?– Called efficiency of large groupsefficiency of large groups:

For N = 10 A = 4 14 E efficiency%44114.4

AFor N = 10, A = 4.14 E efficiency %4.4110

N

For N = 100, A = 78.2 E efficiency %2.78100

2.78

NA

100N

The larger the trunk group, the greater the efficiency

TrafCalc SoftwareTrafCalc Software• What if we need to calculate P(N,A) and not in traffic table?

–– TrafCalcTrafCalc: Custom-designed software • Calculates P(B) or A, or• Creates custom traffic tables

TrafCalc Software (2)TrafCalc Software (2)• How do we calculate P(32,20)?

TrafCalc Software (3)TrafCalc Software (3)• How do we calculate A for which P(32,A) = 0.01?

Erlang B ModelErlang B Model• More sophisticated model than Binomial or Poisson

• Blocked Calls Cleared (BCC)Blocked Calls Cleared (BCC)

• Good for calls that can reroute to alternate route if blocked

• No approximation for reattempts if alternate route blocked toopp p

• Derived using birthbirth--death processdeath process– See selected pages from Leonard Kleinrock, Queueing Systems

Volume 1: Theory John Wiley & Sons 1975Volume 1: Theory, John Wiley & Sons, 1975

Erlang B BirthErlang B Birth--Death ProcessDeath Process• Consider infinitesimally small time tt during which only one

arrival or departure (or none) may occur

• Let be the arrival rate from an infinite pool or sources

• Let = 1/h= 1/h be the departure rate per callN t if kk ll i t d t t i kk– Note: if kk calls in system, departure rate is kk

• Steady State Diagram:

Blockage

0 1 2 N-1 N……

2 N(N-1)3

Immediate Service

Erlang B BirthErlang B Birth--Death Process (2)Death Process (2)• Steady State (statistical equilibrium)

– Rate of arrival is the same as rate of departure– Average rate a system enters a given state is equal to the average

rate at which the system leaves that state

Probability of moving

from state 1 to state 2? PP11

0 1 2 N-1 N……P0 P1 P2 PN-1 PN

2 N(N-1)3

Probability of movingProbability of movingfrom state 2 to state 1? 22PP22

Erlang B BirthErlang B Birth--Death Process (3)Death Process (3)

0 1 2 N-1 N……

P0 P1 P2 PN-1 PN

• Set up balance equations: 2 N(N-1)3

0 1P P 0 1P P 1 0P P

0 1P P

1 1 2 02P P P P

2 2 3 12 3P P P P

0 1P P

1 22P P

2 33P P

2 12P P

20

2P

3 3 4 23 4P P P P

3

1k kP k P

3 23P P

30

6P

1 1 2( 1) N N N NN P P N P P

N P P

1k k

P N P

0

!

k

kPPk

1N NN P P 1N NP N P !k

Erlang B BirthErlang B Birth--Death Process (4)Death Process (4)

Rule of Total Probability:

N iN P 1P

0

!

k

kPPk

Recall:

01

N

ii

P

0

0 !

N

i

Pi

0

0

1!

iN

i

P

i

1

k

A h

Recall:

kA

0

!

1!

k iN

i

kP

i

0

!

!

iNk

i

AkP A

i

0 !i i For blocking, must be in state k = N:

( ) ( ) !

NAN( ) ( , ) NP B B N A P

“B” = Erlang B

“N” = # Servers0

!

!

iN

i

NAi

N = # Servers“A” = Offered Traffic

Erlang B Traffic TableErlang B Traffic Table

Example: In a BCC system with m= sources, we can accept a

B(N,A)=0.001B(N,A)=0.001

0.1% chance of blocking in the nominal case of 40E offered traffic. However, in the extreme case of a 20% overload we can accept a

B(N,A)=0.005B(N,A)=0.005

20% overload, we can accept a 0.5% chance of blocking.

How many outgoing trunks do we need? A 40 EA 40 Eneed? A=40 EA=40 EN=59N=59Nominal design: 59 trunks

AA48 E48 EOverload design: 64 trunks

N=64N=64

Requirement: 64 trunks

Example (2)Example (2)P(N,A)=0.01P(N,A)=0.01

N=32N=32

A=20.3 EA=20.3 E

P(N,A) & B(N,A) P(N,A) & B(N,A) -- High BlockingHigh Blocking• We recognize that Poisson and Erlang B models are only

approximations but which is better?– Compare them using a 4-trunk group offered A=10E

Erlang BErlang B

(4 10) 0 64666B

PoissonPoisson

(4 10) 0 98966P(4,10) 0.64666B

(1 ( ))CT A P B 10 (1 0.64666)

3 533T E

(4,10) 0.98966P

(1 ( ))CT A P B 10 (1 0.98966)

0 103T E3.533CT E

3.533 0.884

0.103CT E

0.103 0.0264

4 4

How can 4 trunks handle 10E offeredHow can 4 trunks handle 10E offeredtraffic and be busy only 2.6% of the time?traffic and be busy only 2.6% of the time?

P(N,A) & B(N,A) P(N,A) & B(N,A) -- High Blocking (2)High Blocking (2)• Obviously, the Poisson result is so far off that it is almost

meaningless as an approximation of the example.– 4 servers offered enough traffic to keep 10 servers busy full time

(10E) should result in much higher utilization.

• Erlang B result is more believable.– All 4 trunks are busy most of the time.

• What if we extend the exercise by increasing A?Erlang B result goes to 4E carried traffic– Erlang B result goes to 4E carried traffic

– Poisson result goes to 0E carried

• Illustrates the failure of the Poisson model as valid for situations with high blocking– Poisson only good approximation when low blocking– Use Erlang B if high blocking

Engset Distribution ModelEngset Distribution Model• BCC model with small number of sources (m > N)

= mean departure rate per call mean departure rate per call

= mean arrival rate of a single source

k = arrival rate if in the system is state k Blockagek y

kk = = (m(m--k)k)

m (m-1) (m-2) [M-(N-2)] [m-(N-1)]

Blockage

0 1 2 N-1 N……P0 P1 P2 PN-1 PN

2 N(N-1)3

Immediate Service

Engset Traffic Model (2)Engset Traffic Model (2)• Balance equations give:

k 10

!!( )!

k

kmP P

k m k

and 0

0

1iN

i

Pmi

therefore:k

k iN

mk

Pm

but can show that: Am A

0i

mi

N mA

( ) ( )P B P k N ( , , ) iN

m A NE m N A

mAA i

0i m A i

“E” = Engset

Engset Traffic TableEngset Traffic Table M = 30 sourcesM = 30 sources

# trunks (N)# trunks (N)

Traffic offered (A)Traffic offered (A)

P(B)=E(m,N,A)P(B)=E(m,N,A) N=10N=10

Example: 30 terminals each provide

AA==4.8 E4.8 E

P(B) 0 01P(B) 0 01Example: 30 terminals each provide 0.16 Erlangs to a concentrator with a goal of less than 1% blocking.

How many outgoing trunks do we

P(B)<0.01P(B)<0.01

y g gneed?

A = 30 x 0.16 = 4.8 E

Requirement: N = 10 TrunksN = 10 TrunksCheck m < 10 x N?M=30 < 10 x 10 = 100Requirement: N = 10 TrunksN = 10 Trunks M=30 < 10 x 10 = 100

Erlang C Distribution ModelErlang C Distribution Model• BCW model with infinite sourcesinfinite sources (m) and infinite queue lengthinfinite queue length

= arrival rate of new calls arrival rate of new calls

= mean departure rate per callBlockage

0 1 2 N Q1 Q2…… ……P0 P1 P2 PN PQ1 PQ2

2 NNNN3

Immediate Service

Erlang C Distribution Model (2)Erlang C Distribution Model (2)• Balance equations give:

10 ,

!

k

kA PP k Nk

and0 ,

!

k

k k N

A PP k NN N and 0 1

0

1

! !

N iN

i

PA N AN N A i

• But P(B) = P(kN):

0( )k

k N

A PP B

0k

N

PA 0

kNA AP

but can show that:kA N

A

( )!k N

k N N N !N

k N N N N

00! k

PN N

0k N N A

NA NNA N

0( )!

A NP B PN N A

1

0

!( , )

! !

N iN

i

N N AC N AA N AN N A i

“ ” l 0! !iN N A i “C” = Erlang C

Erlang C Traffic TablesErlang C Traffic Tables N=18N=18

# trunks (N)# trunks (N) P(B)=C(N,A)P(B)=C(N,A)

Traffic offered (A)Traffic offered (A)A 7 EA 7 E

( )( )

Example:

h h b b l f bl k

A=7 EA=7 E C(18,7)=0.0004C(18,7)=0.0004

What is the probability of blocking in an Erlang C system with 18 servers offered 7 Erlangs of traffic?

Delay in Erlang CDelay in Erlang C• Expected number of calls in the queue?

( )k N P

( )kAk N P

kNA AP k

( ) k

k Nk N P

0( )!k N

k Nk N P

N N

0! k

kP k

N N

0NP A A N

( , )A C N A

( )h C N A

!N N A N A

N A

( , )C N A

N A

Mean #Calls DelayedMean Delay over All Calls ( )h C N AT Recall:

yMean Delay over All Calls = Arrival Rate of Calls

( , )C N AN A

T

Mean Delay of Delayed Calls = hMean Delay of Delayed Calls N A

Also:

( ) ( )hT

N AP delay T C N A e

( ) ( , ) N AP delay T C N A e

Comparison of Traffic ModelsComparison of Traffic Models

Erlang C (BCW,Erlang C (BCW, sources)sources)

i i l ( C )i i l ( C )

Poisson (BCH, Poisson (BCH, sources)sources)

Erlang B (BCC, Erlang B (BCC, sources)sources)

Erlang C (BCW, Erlang C (BCW, sources)sources)

P(B)

Binomial (BCH, m sources)Binomial (BCH, m sources)

Engset (BCC, m sources)Engset (BCC, m sources)

Offered Traffic (A)

Efficiency of Large GroupsEfficiency of Large Groups• Already seen that for same P(B), increasing servers results in

more than proportional increase in traffic carried

example 1: (10, 4.14) 0.01P and (100,78.2) 0.01P

l 2example 2: (32, 20.3) 0.01P (33, 20.1) 0.005P and

example 3: (8, 2.05) 0.001B (80,57.8) 0.001B and

• What does this mean?– If it’s possible to collect together several diverse sources, you can

• provide better gos at same cost, or• provide same gos at cheaper cost

Efficiency of Large Groups (2)Efficiency of Large Groups (2)• Two trunk groups offered 5 Erlangs each, and B(N,A)=0.002

N1=?5 E How many trunks total?

From traffic tables, find B(13,5) 0.002 N1=13

N2=?5 E N2=13 Ntotal = 13 + 13 = 26 trunks

Trunk efficiency?

CT 10(1 0.002) 0 384CTN

10(1 0.002) 0.384

26

38.4% utilization 38.4% utilization

Efficiency of Large Groups (3)Efficiency of Large Groups (3)• One trunk group offered 10 Erlangs, and B(N,A)=0.002

How many trunks?

N=?10 E

How many trunks?

From traffic tables, find B(20,10) 0.002 N=20

N 20 kN = 20 trunks

Trunk efficiency?

CTN

10(1 0.002) 0.499

20

49.9% utilization 49.9% utilization

For same gos, we can save 6 trunks!

Efficiency of Large Groups (4)Efficiency of Large Groups (4)

B=0.1B=0.1

B=0.01B=0.01

A B=0.1B=0.1

B=0.01B=0.01

B=0.001B=0.001 B=0.001B=0.001

N N

Sensitivity to OverloadSensitivity to Overload• Consider 2 cases:

Case 1: N = 10 and B(N,A) = 0.01Case 1: N 10 and B(N,A) 0.01

B(10,4.5) 0.01, so can carry 4.5 E

What if 20% overload (5.4 E)? B(10,5.4) 0.03( ) ( )

3 times P(B) with 20% overload

Case 1: N = 30 and B(N,A) = 0.01

B(30,20.3) 0.01, so can carry 20.3 E

What if 20% overload (24.5 E)? B(30,24.5) 0.08

8 times P(B) with 20% overload!

“Trunk Group Splintering”“Trunk Group Splintering”• if high possibility of overloads, small groups may be better

Incremental Traffic Carried by NIncremental Traffic Carried by Nthth TrunkTrunk

• If a trunk group is of size N-1, how much extra traffic can it carry if you add one extra trunk?

– Before, can carry: TC1 = A x [1-(B(N-1,A)]

– After, can carry: TC2 = A x [1-(B(N,A)]

2 1N C CA T T 1 ( , ) 1 ( 1, )A B N A B N A

( 1, ) ( , )A B N A B N A ( ) ( )A N A B N A

• What does this mean?

–– Random HuntingRandom Hunting: Increase in trunk group’s total carried traffic

( ) ( , )NA N A B N A for very low blocking

after adding an Nth trunk

–– Sequential HuntingSequential Hunting: Actual traffic carried by the Nth trunk in the group

Incremental Traffic Carried by NIncremental Traffic Carried by Nthth Trunk (3)Trunk (3)

Fixed B(N A)

AN

Fixed B(N,A)

N

ExampleExample• Individual trunks are only economic if they can carry 0.4 E or

more. A trunk group of size N=10 is offered 6 E. Will all 10 t k b i l?trunks be economical?

( 1, ) ( , )NA A B N A B N A

10 6 (9,6) (10,6)A B B

6 0.07514 0.04314

0.192 E 0.4 E

At least the 10At least the 10thth trunk is not trunk is not economicaleconomical