ee 489 traffic theory university of alberta dept. of electrical and computer engineering
DESCRIPTION
EE 489 Traffic Theory University of Alberta Dept. of Electrical and Computer Engineering Wayne Grover TR Labs and University of Alberta. Traffic Engineering. One billion+ terminals in voice network alone Plus data, video, fax, finance, etc. - PowerPoint PPT PresentationTRANSCRIPT
EE 489EE 489Traffic TheoryTraffic Theory
University of AlbertaUniversity of Alberta
Dept. of Electrical and Computer EngineeringDept. of Electrical and Computer Engineering
Wayne GroverWayne GroverTRLabs and University of Alberta
Material prepared by W. Grover (1998-2002)
2
Traffic Theory, W. Grover–University of Alberta, Dept. of Electrical and Computer 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
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!
• How is this possible?
Traffic theory !!
Material prepared by W. Grover (1998-2002)
3
Traffic Theory, W. Grover–University of Alberta, Dept. of Electrical and Computer Engineering
Traffic Engineering – Trade-offsTraffic Engineering – Trade-offs
• 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? – E.g.
• High-usage trunk groups• Overflow trunk groups• Where should traffic flows be combined or kept separate?
• Design network topology– Number and sizing of switching nodes and locations– Number and sizing of transmission systems and locations– Survivability
Material prepared by W. Grover (1998-2002)
4
Traffic Theory, W. Grover–University of Alberta, Dept. of Electrical and Computer Engineering
Characterization of Telephone Characterization of Telephone TrafficTraffic• Calling RateCalling Rate () – also called arrival rate, or attempts rate,
etc.– Average number of calls initiated per unit time (e.g. attempts per
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:
If receive calls from mm terminals in time T:
Tαγ g
Group calling rateTm
αγ
Per terminalcalling rate
Material prepared by W. Grover (1998-2002)
5
Traffic Theory, W. Grover–University of Alberta, Dept. of Electrical and Computer Engineering
Characterization of Telephone Traffic Characterization of Telephone Traffic (2)(2)• Calling rate assumption:
– Number of calls in time T is Poisson distributed:
– In our case
...2 ,1 ,0!
)(
xx
exp
x
Time between calls is “-ve exponentially” distributed:
tetf t 0)( 1
mean
T
• Class Question: What do these observations about telephone traffic imply about the nature of the traffic sources?
Material prepared by W. Grover (1998-2002)
6
Traffic Theory, W. Grover–University of Alberta, Dept. of Electrical and Computer Engineering
-ve Exponential Holding Times-ve Exponential Holding Times
• Implies the “Memory-less” propertyImplies the “Memory-less” 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 T1
tTPTTtTTP 11
1
1111
TTP
TTtTTPTTtTTP
• Proof:
1
1
TTP
tTTP
hT
hthT
e
ee/
//
1
1
hte /
hT
htT
e
e/
/)(
1
1
tTP
htetTP /)(
Recall:
Material prepared by W. Grover (1998-2002)
7
Traffic Theory, W. Grover–University of Alberta, Dept. of Electrical and Computer Engineering
Characterization of Telephone Traffic Characterization of Telephone Traffic (3)(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:
– 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 ():
0)( / tetTP ht
00)( ttTP
h
1
Material prepared by W. Grover (1998-2002)
8
Traffic Theory, W. Grover–University of Alberta, Dept. of Electrical and Computer Engineering
Some Real Holding Time DataSome Real Holding Time Data
Material prepared by W. Grover (1998-2002)
9
Traffic Theory, W. Grover–University of Alberta, Dept. of Electrical and Computer Engineering
Traffic Traffic Volume (V)Volume (V)
hV = # calls in time period T
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 circuit busy for 100 seconds, or– two circuits busy for 50 seconds, or– 100 circuits busy for one second, etc.
Material prepared by W. Grover (1998-2002)
10
Traffic Theory, W. Grover–University of Alberta, Dept. of Electrical and Computer Engineering
TrafficTraffic Intensity (A) Intensity (A)
• Also called “traffic flowtraffic flow” or simply “traffictraffic”.
= # calls in time period T
h = mean holding time
T = time period of observations
hA
T
• Units:– “ccs/hourccs/hour”, or– dimensionless (if h and T are in the same units of time)
“ErlangErlang” unit
h
= # calls in time period T
h = mean holding time
T = time period of observations
= calling rate
= # calls in time period T
h = mean holding time
T = time period of observations
= calling rate
= departure rate
T
Recall:
h
1
Recall:
V
T
V h
Recall:
= # calls in time period T
h = mean holding time
T = time period of observations
= calling rate
= departure rate
V = call volume
Material prepared by W. Grover (1998-2002)
11
Traffic Theory, W. Grover–University of Alberta, Dept. of Electrical and Computer Engineering
The The ErlangErlang
• Dimensionless unit of traffic intensity
• Named after Danish mathematician A. K. Erlang (1878-1929)
• Usually denoted by symbol EE.
• 1 Erlang is equivalent to traffic intensity that keeps:– one circuit busy 100% of the time, or– two circuits busy 50% of the time, or– four circuits busy 25% of the time, etc.
• 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– 104 circuits busy 25% of the time, etc.
Material prepared by W. Grover (1998-2002)
13
Traffic Theory, W. Grover–University of Alberta, Dept. of Electrical and Computer Engineering
Erlang (2)Erlang (2)
• How does the ErlangErlang unit correspond to ccsccs?
• 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 E– interoffice trunk -> 0.70 E
0.027E
1E
100 call seconds1 ccs hour
1 hour × 60 min hr × 60 sec min
3600 call seconds36 ccs hour
1 hour × 60 min hr × 60 sec min
× 60 min hr × 60 sec min
× 60 min hr × 60 sec min
Material prepared by W. Grover (1998-2002)
14
Traffic Theory, W. Grover–University of Alberta, Dept. of Electrical and Computer Engineering
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, and uses their expected holding times
• Also Carried Traffic Carried Traffic (TTC C ) and Lost TrafficLost Traffic (TTL L )
• Consider a group of 150 terminals, each with 10% utilization (or in other words, 0.1 E per source) and dedicated servicededicated service:
1
150
1
150
each terminal has anoutgoing trunk
(i.e. terminal:trunk ratio = 1:1)
TO = A = 150 x 0.10 E = 15.0 E
TC = 150 x 0.10 E = 15.0 E
TL = 0 E
Material prepared by W. Grover (1998-2002)
15
Traffic Theory, W. Grover–University of Alberta, Dept. of Electrical and Computer Engineering
Traffic Offered, Carried, and Lost Traffic Offered, Carried, and Lost (2)(2)• A = TO = TC + TL
TrafficIntensity Offered
Traffic
CarriedTraffic
LostTraffic
• 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
Material prepared by W. Grover (1998-2002)
16
Traffic Theory, W. Grover–University of Alberta, Dept. of Electrical and Computer Engineering
Grade of Service (gos)Grade of Service (gos)
• In general, the term used for some traffic design objective
• Indicative of customer satisfaction
• In systems where blocked calls are cleared, usually use:
L L
O L C
T T( )
T T + TP Bgos
• 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 higher
• In systems with queuing, gos often defined as the probability of delay exceeding a specific length of time
Material prepared by W. Grover (1998-2002)
17
Traffic Theory, W. Grover–University of Alberta, Dept. of Electrical and Computer Engineering
Grade of Service Related TermsGrade of Service Related Terms
• Busy HourBusy Hour– One hour period during which traffic volume or call
attempts is the highest 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 day
• 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
Material prepared by W. Grover (1998-2002)
18
Traffic Theory, W. Grover–University of Alberta, Dept. of Electrical and Computer Engineering
Grade of Service Related Terms Grade of Service Related Terms (2)(2)
HighestHighestABSBHABSBH
• Average Busy Season Busy Hour (ABSBH)Average Busy Season Busy Hour (ABSBH)– One hour period with highest average daily busy hour
during the busy season
• Average Busy Season Busy Hour (ABSBH)Average Busy Season Busy Hour (ABSBH)– One hour period with highest average daily busy hour
during the busy 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 Mean
00:00 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 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 4.215:00 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 2.022:00 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
Note: Red indicatesdaily busy hour
Material prepared by W. Grover (1998-2002)
19
Traffic Theory, W. Grover–University of Alberta, Dept. of Electrical and Computer Engineering
Hourly Traffic VariationsHourly Traffic Variations
Material prepared by W. Grover (1998-2002)
20
Traffic Theory, W. Grover–University of Alberta, Dept. of Electrical and Computer Engineering
Daily Traffic VariationsDaily Traffic Variations
Material prepared by W. Grover (1998-2002)
21
Traffic Theory, W. Grover–University of Alberta, Dept. of Electrical and Computer Engineering
Seasonal Traffic VariationsSeasonal Traffic Variations
Material prepared by W. Grover (1998-2002)
22
Traffic Theory, W. Grover–University of Alberta, Dept. of Electrical and Computer Engineering
Seasonal Traffic Variations (2)Seasonal Traffic Variations (2)
Material prepared by W. Grover (1998-2002)
23
Traffic Theory, W. Grover–University of Alberta, Dept. of Electrical and Computer Engineering
Typical Call Attempts Typical Call Attempts BreakdownBreakdown• Calls Completed - 70.7%
• 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%
Material prepared by W. Grover (1998-2002)
24
Traffic Theory, W. Grover–University of Alberta, Dept. of Electrical and Computer Engineering
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
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
approximation only)– Tries to approximate call reattempt efforts
• 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
Material prepared by W. Grover (1998-2002)
25
Traffic Theory, W. Grover–University of Alberta, Dept. of Electrical and Computer Engineering
Source #1
Offered Traffic
Source #2
Offered Traffic
1
2
3
4
10 minutes
Total Traffic Offered:TO = 0.4 E + 0.3 E
TO = 0.7 E
2 sources
Blocked Calls Cleared (BCC)Blocked Calls Cleared (BCC)
Only one server
Traffic
Carried
1st call arrives and is served
1
2nd call arrives but server already busy
22nd call is cleared
1
3rd call arrives and is served
3
4th call arrives and is served
4
Total Traffic Carried:TC = 0.5 E
Material prepared by W. Grover (1998-2002)
26
Traffic Theory, W. Grover–University of Alberta, Dept. of Electrical and Computer Engineering
Source #1
Offered Traffic
Source #2
Offered Traffic
1
2
3
4
10 minutes
Total Traffic Offered:TO = 0.4 E + 0.3 E
TO = 0.7 E
2 sources
Blocked Calls Held (BCH)Blocked Calls Held (BCH)
Traffic
Carried 1 21 2 3 4
Only one server1st call arrives and is served
2nd call arrives but server busy
2nd call is served
3rd call arrives and is served
4th call arrives and is served
Total Traffic Carried:TC = 0.6 E
2nd call is held until server free
Material prepared by W. Grover (1998-2002)
27
Traffic Theory, W. Grover–University of Alberta, Dept. of Electrical and Computer Engineering
Source #1
Offered Traffic
Source #2
Offered Traffic
1
2
3
4
10 minutes
Total Traffic Offered:TO = 0.4 E + 0.3 E
TO = 0.7 E
2 sources
Blocked Calls Wait (BCW)Blocked Calls Wait (BCW)
Only one server
Traffic
Carried
1st call arrives and is served
1
2nd call arrives but server busy
2
2nd call waits until server free
2nd call served1 2
3rd call arrives, waits, and is served
3
4th call arrives, waits, andis served
4
Total Traffic Carried:TC = 0.7 E
Material prepared by W. Grover (1998-2002)
28
Traffic Theory, W. Grover–University of Alberta, Dept. of Electrical and Computer Engineering
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 calls arrive faster that they depart?• If calls depart faster than they arrive?
Material prepared by W. Grover (1998-2002)
29
Traffic Theory, W. Grover–University of Alberta, Dept. of Electrical and Computer Engineering
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
m
A
m
A
k
mkP
1)(
kmk
m
A
m
A
kmk
m
1)!(!
!
Material prepared by W. Grover (1998-2002)
30
Traffic Theory, W. Grover–University of Alberta, Dept. of Electrical and Computer Engineering
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
kmk
m
A
m
A
k
mkP
1)(
m
Nk
kmk
m
A
m
A
k
m1
1
0
11N
k
kmk
m
A
m
A
k
m
Remember:
Material prepared by W. Grover (1998-2002)
31
Traffic Theory, W. Grover–University of Alberta, Dept. of Electrical and Computer Engineering
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?– Blocked Calls Held (BCH)
Material prepared by W. Grover (1998-2002)
32
Traffic Theory, W. Grover–University of Alberta, Dept. of Electrical and Computer Engineering
Time Congestions Time Congestions vs.vs. Call Call CongestionCongestion• 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 blocked– Probability of blocking from point of view of calls
• Why/How are they different?
Time Congestion:
)()( NkPBP
Probability that allservers are busy.
Call Congestion:
)()( NkPBP
Probability that there aremore sources wanting servicethan there are servers.
Material prepared by W. Grover (1998-2002)
33
Traffic Theory, W. Grover–University of Alberta, Dept. of Electrical and Computer Engineering
Poisson Traffic ModelPoisson Traffic Model
• Poisson approximates Binomial with large mlarge m and small A/msmall A/m
!)(
k
ekP
k = Mean # of
Busy Sources
Note: )(lim BinomialPoissonm
• What is ?– Mean number of busy sources = A
!)(
k
AekP
kA
Material prepared by W. Grover (1998-2002)
34
Traffic Theory, W. Grover–University of Alberta, Dept. of Electrical and Computer Engineering
Poisson Traffic Model (2)Poisson Traffic Model (2)
• Now we can calculate probability of blocking:
)()( NkPBP )(...)1()( PNPNP
Remember:
!)(
k
AekP
kA
Nk
kA
k
Ae
!
Nk
Ak
ek
A
!
AN
k
k
ek
A
1
0 !1
),()( ANPBP “P” = Poisson
“N” = # Servers
“A” = Offered Traffic
Example:
)10,7(P
PoissonPoisson P(B) with 10 E10 Eoffered to 7 servers7 servers
Material prepared by W. Grover (1998-2002)
35
Traffic Theory, W. Grover–University of Alberta, Dept. of Electrical and Computer Engineering
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
ek
Ae
k
A
9
010 !1
!01.0
• How do we calculate A?– Very carefully, or– Use traffic tables
Material prepared by W. Grover (1998-2002)
36
Traffic Theory, W. Grover–University of Alberta, Dept. of Electrical and Computer Engineering
Traffic Tables (2)Traffic Tables (2)P(B)=P(N,A)P(B)=P(N,A)
NN
AA
Material prepared by W. Grover (1998-2002)
37
Traffic Theory, W. Grover–University of Alberta, Dept. of Electrical and Computer Engineering
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 trunks
has P(B) = 0.01:has P(B) = 0.01:
System can handleSystem can handle
Offered traffic (A) = 4.14 EOffered traffic (A) = 4.14 E
Material prepared by W. Grover (1998-2002)
38
Traffic Theory, W. Grover–University of Alberta, Dept. of Electrical and Computer Engineering
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 trunks
has P(B) = 0.01:has P(B) = 0.01:
System can handleSystem can handle
Offered traffic (A) = 4.14 EOffered traffic (A) = 4.14 E
Material prepared by W. Grover (1998-2002)
39
Traffic Theory, W. Grover–University of Alberta, Dept. of Electrical and Computer Engineering
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?– Called efficiency of large groupsefficiency of large groups:
For N = 10, A = 4.14 E efficiency %4.4110
14.4
N
A
For N = 100, A = 78.2 E efficiency %2.78100
2.78
N
A
The larger the trunk group, the greater the efficiency
Material prepared by W. Grover (1998-2002)
40
Traffic Theory, W. Grover–University of Alberta, Dept. of Electrical and Computer Engineering
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
Material prepared by W. Grover (1998-2002)
41
Traffic Theory, W. Grover–University of Alberta, Dept. of Electrical and Computer Engineering
TrafCalc Software (2)TrafCalc Software (2)
• How do we calculate P(32,20)?
Material prepared by W. Grover (1998-2002)
42
Traffic Theory, W. Grover–University of Alberta, Dept. of Electrical and Computer Engineering
TrafCalc Software (3)TrafCalc Software (3)
• How do we calculate A for which P(32,A) = 0.01?
Material prepared by W. Grover (1998-2002)
43
Traffic Theory, W. Grover–University of Alberta, Dept. of Electrical and Computer Engineering
Erlang B ModelErlang B Model
• More sophisticated model than Binomial or Poisson
• Blocked Calls Cleared (BCC)
• Good for calls that can reroute to alternate route if blocked
• No approximation for reattempts if alternate route blocked too
• Derived using birth-death processbirth-death process– See selected pages from Leonard Kleinrock, Queueing
Systems Volume 1: Theory, John Wiley & Sons, 1975
Material prepared by W. Grover (1998-2002)
44
Traffic Theory, W. Grover–University of Alberta, Dept. of Electrical and Computer Engineering
Erlang B Birth-Death ProcessErlang B Birth-Death 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 call– Note: if kk calls in system, departure rate is kk
• Steady State Diagram:
0 1 2 N-1 N……
2
N(N-1)3
Immediate Service
Blockage
Material prepared by W. Grover (1998-2002)
45
Traffic Theory, W. Grover–University of Alberta, Dept. of Electrical and Computer Engineering
Erlang B Birth-Death Process (2)Erlang B Birth-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
0 1 2 N-1 N……
2
N(N-1)3
P0 P1 P2 PN-1 PN
Probability of movingfrom state 1 to state 2?PP11
Probability of movingfrom state 2 to state 1?22PP22
Material prepared by W. Grover (1998-2002)
46
Traffic Theory, W. Grover–University of Alberta, Dept. of Electrical and Computer Engineering
Erlang B Birth-Death Process (3)Erlang B Birth-Death Process (3)
• Set up balance equations:
0 1 2 N-1 N……
2
N(N-1)3
P0 P1 P2 PN-1 PN
0 1P P
1 1 2 02P P P P
2 2 3 12 3P P P P
3 3 4 23 4P P P P
1 1 2( 1) N N N NN P P N P P
1N NN P P
0 1P P
1 22P P
2 33P P
1k kP k P
1N NP N P
1 0P P
2 12P P
2
0
2
P
3 23P P
3
0
6
P
0
!
k
k
PP
k
Material prepared by W. Grover (1998-2002)
47
Traffic Theory, W. Grover–University of Alberta, Dept. of Electrical and Computer Engineering
Erlang B Birth-Death Process (4)Erlang B Birth-Death Process (4)
Rule of Total Probability:
0
1N
ii
P
0
0 !
iN
i
P
i
0
0
1
1!
iN
i
P
i
0
!
k
k
PP
k
Recall:
0
1!
1!
k
k iN
i
kP
i
A h
Recall:
0
!
!
k
iNk
i
AkP
Ai
For blocking, must be in state k = N:
( ) ( , ) NP B B N A P
“B” = Erlang B
“N” = # Servers
“A” = Offered Traffic
0
!
!
N
iN
i
AN
Ai
Material prepared by W. Grover (1998-2002)
48
Traffic Theory, W. Grover–University of Alberta, Dept. of Electrical and Computer Engineering
Erlang B Traffic TableErlang B Traffic Table
Example: In a BCC system with m= sources, we can accept a 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 0.5% chance of blocking.
How many outgoing trunks do we need?
B(N,A)=0.001B(N,A)=0.001
A=40 EA=40 EN=59N=59
Nominal design: 59 trunks
B(N,A)=0.005B(N,A)=0.005
AA48 E48 E
N=64N=64
Overload design: 64 trunks
Requirement: 64 trunks
Material prepared by W. Grover (1998-2002)
49
Traffic Theory, W. Grover–University of Alberta, Dept. of Electrical and Computer Engineering
Example (2)Example (2)P(N,A)=0.01P(N,A)=0.01
N=32N=32
A=20.3 EA=20.3 E
Material prepared by W. Grover (1998-2002)
50
Traffic Theory, W. Grover–University of Alberta, Dept. of Electrical and Computer Engineering
P(N,A) & B(N,A) - High BlockingP(N,A) & B(N,A) - High 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
(1 ( ))CT A P B 10 (1 0.64666)
3.533CT E
3.5330.88
4
PoissonPoisson
(4,10) 0.98966P
(1 ( ))CT A P B 10 (1 0.98966)
0.103CT E
0.1030.026
4
How can 4 trunks handle 10E offeredHow can 4 trunks handle 10E offered
traffic and be busy only 2.6% of the time?traffic and be busy only 2.6% of the time?
Material prepared by W. Grover (1998-2002)
51
Traffic Theory, W. Grover–University of Alberta, Dept. of Electrical and Computer Engineering
P(N,A) & B(N,A) - High Blocking P(N,A) & B(N,A) - High Blocking (2)(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– 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
Material prepared by W. Grover (1998-2002)
52
Traffic Theory, W. Grover–University of Alberta, Dept. of Electrical and Computer Engineering
Engset Distribution ModelEngset Distribution Model
• BCC model with small number of sources (m > N)
= mean departure rate per call
= mean arrival rate of a single source
k = arrival rate if in the system is state k
kk = = (m-k)(m-k)
0 1 2 N-1 N……P0 P1 P2 PN-1 PN
m (m-1)
2
(m-2) [M-(N-2)] [m-(N-1)]
N(N-1)3
Immediate Service
Blockage
Material prepared by W. Grover (1998-2002)
53
Traffic Theory, W. Grover–University of Alberta, Dept. of Electrical and Computer Engineering
Engset Traffic Model (2)Engset Traffic Model (2)
• Balance equations give:
0
!
!( )!
k
k
mP P
k m k
and 0
0
1iN
i
Pm
i
therefore:
0
k
k iN
i
m
kP
m
i
but can show that: A
m A
( ) ( )P B P k N
0
( , , )
N
iN
i
mAm A N
E m N AmA
m A i
“E” = Engset
Material prepared by W. Grover (1998-2002)
54
Traffic Theory, W. Grover–University of Alberta, Dept. of Electrical and Computer Engineering
Engset Traffic TableEngset Traffic TableM = 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)
Example: 30 terminals each provide 0.16 Erlangs to a concentrator with a goal of less than 1% blocking.
How many outgoing trunks do we need?A = 30 x 0.16 = 4.8 E
A=4.8 EA=4.8 E
P(B)<0.01P(B)<0.01
N=10N=10
Requirement: N = 10 TrunksN = 10 Trunks
Check m < 10 x N?M=30 < 10 x 10 = 100
Material prepared by W. Grover (1998-2002)
55
Traffic Theory, W. Grover–University of Alberta, Dept. of Electrical and Computer Engineering
Erlang C Distribution ModelErlang C Distribution Model
• BCW model with infinite sourcesinfinite sources (m) and infinite queue infinite queue lengthlength
= arrival rate of new calls
= mean departure rate per call
0 1 2 N Q1 Q2…… ……P0 P1 P2 PN PQ1 PQ2
2
NNNN3
Immediate Service
Blockage
Material prepared by W. Grover (1998-2002)
56
Traffic Theory, W. Grover–University of Alberta, Dept. of Electrical and Computer Engineering
Erlang C Distribution Model (2)Erlang C Distribution Model (2)
• Balance equations give:
0 ,!
k
k
A PP k N
k and
0 ,!
k
k k N
A PP k N
N N and0 1
0
1
! !
N iN
i
PA N AN N A i
• But P(B) = P(kN):
0( )!
k
k Nk N
A PP B
N N
0
!
k
Nk N
PA
N N N
00!
kN
k
A AP
N N
but can show that:
0
k
k
A N
N N A
0( )!
NA NP B P
N N A
1
0
!( , )
! !
N
N iN
i
A NN N AC N A
A N AN N A i
“C” = Erlang C
Material prepared by W. Grover (1998-2002)
57
Traffic Theory, W. Grover–University of Alberta, Dept. of Electrical and Computer Engineering
Erlang C Traffic TablesErlang C Traffic Tables
# trunks # trunks (N)(N)
Traffic offered (A)Traffic offered (A)
P(B)=C(N,A)P(B)=C(N,A)
Example:
What is the probability of blocking in an Erlang C system with 18 servers offered 7 Erlangs of traffic?
N=18N=18
A=7 EA=7 E C(18,7)=0.0004C(18,7)=0.0004
Material prepared by W. Grover (1998-2002)
58
Traffic Theory, W. Grover–University of Alberta, Dept. of Electrical and Computer Engineering
Delay in Erlang CDelay in Erlang C
• Expected number of calls in the queue?
( ) kk N
k N P
0( )!
k
k Nk N
Ak N P
N N
0!
kN
kk
A AP k
N N
0
!
NP A A N
N N A N A
( , )A C N A
N A
( , )
hC N A
N A
Mean #Calls DelayedMean Delay over All Calls =
Arrival Rate of Calls( , )
hC N A
N A
T
Recall:
T
Mean Delay of Delayed Calls = h
N A
Also:
( ) ( , )h
TN AP delay T C N A e
Material prepared by W. Grover (1998-2002)
59
Traffic Theory, W. Grover–University of Alberta, Dept. of Electrical and Computer Engineering
Comparison of Traffic ModelsComparison of Traffic Models
Offered Traffic (A)
P(B)
Binomial (BCH, m sources)Binomial (BCH, m sources)
Poisson (BCH, Poisson (BCH, sources) sources)
Erlang B (BCC, Erlang B (BCC, sources) sources)
Engset (BCC, m sources)Engset (BCC, m sources)
Erlang C (BCW, Erlang C (BCW, sources) sources)
Material prepared by W. Grover (1998-2002)
60
Traffic Theory, W. Grover–University of Alberta, Dept. of Electrical and Computer Engineering
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
example 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
Material prepared by W. Grover (1998-2002)
61
Traffic Theory, W. Grover–University of Alberta, Dept. of Electrical and Computer Engineering
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
N2=?5 E
How many trunks total?
From traffic tables, find B(13,5) 0.002
N1=13
N2=13 Ntotal = 13 + 13 = 26 trunks
Trunk efficiency?
CT
N 10(1 0.002)
0.38426
38.4% utilization 38.4% utilization
Material prepared by W. Grover (1998-2002)
62
Traffic Theory, W. Grover–University of Alberta, Dept. of Electrical and Computer Engineering
Efficiency of Large Groups (3)Efficiency of Large Groups (3)
• One trunk group offered 10 Erlangs, and B(N,A)=0.002
N=?10 E
How many trunks?
From traffic tables, find B(20,10) 0.002 N=20
N = 20 trunks
Trunk efficiency?
CT
N 10(1 0.002)
0.49920
49.9% utilization 49.9% utilization
For same gos, we can save 6 trunks!
Material prepared by W. Grover (1998-2002)
63
Traffic Theory, W. Grover–University of Alberta, Dept. of Electrical and Computer Engineering
Efficiency of Large Groups (4)Efficiency of Large Groups (4)
N
B=0.1B=0.1
B=0.01B=0.01
B=0.001B=0.001
N
A B=0.1B=0.1
B=0.01B=0.01
B=0.001B=0.001
Material prepared by W. Grover (1998-2002)
64
Traffic Theory, W. Grover–University of Alberta, Dept. of Electrical and Computer Engineering
Sensitivity to OverloadSensitivity to Overload
• Consider 2 cases:
Case 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
Material prepared by W. Grover (1998-2002)
65
Traffic Theory, W. Grover–University of Alberta, Dept. of Electrical and Computer Engineering
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
• What does this mean?
– Random HuntingRandom Hunting: Increase in trunk group’s total carried traffic after adding an Nth trunk
– Sequential HuntingSequential Hunting: Actual traffic carried by the Nth trunk in the group
( ) ( , )NA N A B N A for very low blocking
Material prepared by W. Grover (1998-2002)
66
Traffic Theory, W. Grover–University of Alberta, Dept. of Electrical and Computer Engineering
Incremental Traffic Carried by NIncremental Traffic Carried by Nthth Trunk (3) Trunk (3)
N
AN
Fixed B(N,A)
Material prepared by W. Grover (1998-2002)
67
Traffic Theory, W. Grover–University of Alberta, Dept. of Electrical and Computer Engineering
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 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 economical trunk is not economical