Instantaneous traffic intensity x
t
1
2
3
Basic Concepts of
C
xt
Teletraffic Theory
t
Measure of telephone traffic
called instantaneous traffic intensity
in moment  t is
number of simultaneous calls  xt .
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Traffic volume  an example Traffic volume Q
Q = 1call " 5min
1
5min 2
3
Q = 5calls " 1min
L
C
1min
Q = " xi" "
" "ti
" "
" "
i = 1
Q = 5 PM [PM] = callminute xt
4min
"t1 "t2 & "ti & "tL
t
xt 1min 3min
Q = 2calls " 1min + 1call " 3min
2
1
0
t
L
Q = x1" " + x2" " + ... + xL" " = " xi" "
"t1 "t2 "tL " "ti
" " " " "
" " " " "
i = 1
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Stanisław Stoch
Traffic volume Q Traffic volume Q
1 1
2 2
3 3
L L
C C
Q = " xi" " Q = " xi" "
" "ti " "ti
" " " "
" " " "
i = 1 i = 1
xt xt
M M
Q
Q = " xj " "t Q = " xj " "t
" " " "
" " " "
" " " "
"t "t "t "t "t "t "t "t "t dt dt dt dt dt dt dt dt dt
j = 1 j = 1
t t
T
T
Q = +" xt dt
+"
+"
+"
0
Representation of traffic volume is the area on the
graph of instantaneous traffic intensity  xt .
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1
" " "
" " "
" " "
" " "
calls
calls
calls
calls
Average traffic intensity A  the definition Units of traffic intensity
The unit of traffic intensity  A is 1 erlang.
1
2
3
Q = A " T
Interpretation of big values:
A = 10 erl means  in every moment there are
Q
C (averagely) 10 simultaneous calls (10 circuits busy)
A =
T
xt Interpretation of small (fractional) values:
A = 0,1 erl means  probability of existence of one
A
T
1 Q
call (of occupation of some circuit in a group)
A = +" xt dt dt dt dt dt dt dt dt dt dt
t
T 0 in every moment equals 0,1
T
L
x1" "t1 + x2" "t2 + ... + xL" "tL 1
1erl = 1call  for simultaneous calls only
A = = " xi" "ti
T T i = 1
In USA: CCS (hundred call seconds per hour)
x1 + x2 + x3 + ... + xL
A = for equal "t only, where: T = L " "t
1 erl = 36 CCS
L
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Stanisław Stoch
Units of traffic volume Q Daily variations of traffic
Units of traffic volume Q: Example typical for Poland.
number of
PM = callminute
occupied
Q [PM] = A [erl] " T [min]
devices
callhour
1 erl " 1 h = 1 erl " 60 min = 60 PM
In USA:
pcm (paid call minutes) = 1PM (conversation)
0 2 4 6 8 10 12 14 16 18 20 22 0
hours of day
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Busy Hour Weakly variations of traffic
Time period 10:00  11:00
Busy hour is the uninterrupted period of
60 min during which the traffic is a maximum.
The busy hour is defined as
that four consecutive quarter hours
whose traffic intensity is the greatest.
Mon Tu We Th Fr Sat Su
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2
" " "
calls
Seasonal variations of traffic Long-term variations of traffic
The traffic shows generally a consistent
tendency to increase. This increase is not
uniform. Decline is also possible.
Therefore, high and low values may have
different relative increase. It is generally
01 02 03 04 05 06 07 08 09 10 11 12 Month
rather difficult to distinguish between
growth and seasonal variations.
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Stanisław Stoch
Average holding time h (call duration) Calls overstepping the T-period boundary
(eg.billing)
1 1
q1=1" h1 1
2 2 2
3 3
q2=1" h2 3
C C
qC=1" hC C
Q xt
xt
h
A
t
T T T
Q = C " h
dt dt dt dt dt dt dt dt dt
t
Don't split calls into parts.
T
Q = A " T
Count each call only once  even if it appears in two
A " T = C " h (or more) subsequent observation periods.
Units are important:
Assign it to that observation period, where the call
10 erl " 1 h = 10 erl " 60 min = 600 PM = 200 calls " 3 min
begins (count  call attempts ).
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Mean calling rate (mean call intensity) Traffic intensity in subscriber line
C C
 
A " T = C " h A = " h =  " h A = " h =  " h
 
 
T T
C
 H" 2 calls/h h H" 2,4 min for local calls
 = [ calls / h ]



T
 H" 0,1 call/h h H" 3,8 min for long distance calls
"  is basic measure for control devices /



 H" 0,1 call/h h H" 3,2 min for calls to special
processors (setting time of a call is
services (9xxx)
independent on call duration)
" BHCA (Busy Hour Call Attempts)  maximal
A = A1 + A2 + A3 = 2 " 2,4 + 0,1 " 3,8 + 0,1 " 3,2 H"
allowed vallue of 



H" 6 (calls/h) " min = 6 (calls / 60min) " min = 0,1 erl
" A is measure of traffic intensity
Stanisław Stoch Switching Systems 19 Switching Systems 20
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3
" " "
" " "
" " "
" " "
calls
calls
poł
Ä…
czenia
Exercise 1. Exercise 2.
In the observation time of T = 2h In the trunk group, average number of busy
a subscriber group made C = 400 calls trunks equals A = 20.
with average call duration of h = 3 min. In the observation time of T = 1h the
Calculate average traffic intensity A for that number of C = 600 calls has been counted.
subscriber group. Calculate mean holding time h.
A " T = C " h
CÅ"h 400Å"3
A " T = C " h
A = = = 600
WRONG
T 2
CÅ"h 400callsÅ"3min 400callsÅ"3min
A Å"T 20erlÅ"1h 20callsÅ"60min
A = = = = 10erl
h = = = = 2min
T 2h 120min
C 600calls 600calls
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Total occupation time of i:th circuit Äi Average occupation time Ä
Ä Ä
Ä Ä
Ä Ä
1 1
q1=1" Ä1 1
Ä
Ä
Ä
2 2 2
q2=1" Ä2
Ä
Ä
Ä
N N N
qN=1" ÄN
Ä
Ä
Ä
Q
Äi
Ä
Ä
Ä
xt xt
Ä
Ä
Ä
Ä
A A
Q = N " Ä
Ä
Ä
Ä
dt dt dt dt dt dt dt dt dt dt dt dt dt dt dt dt dt dt
t t
T T
Q = A " T Q = A " T
A " T = N " Ä
Ä
Ä
Ä
Units are important:
10 erl " 1 h = 10 erl " 60 min = 600 PM = 20 trunks " 30 min
Stanisław Stoch Switching Systems 24 Switching Systems 25
Stanisław Stoch
Mean occupancy (load per device)
Ä
A
a = = ( 0 d" a d" 1 )
N T
Interpretations of  a :
Ä [min]
Ä
Ä
Ä
a [%] = - (average) fraction of observation
T [min]
time when one circuit is occupied
blank
A [erl]
a [%] = - (average) fraction of
N [=Amax]
simultaneously occupied circuits
in a group, same as probability of
occupation of one circuit
A [erl]
a [erl] = - (average) traffic intensity per
N [dim less]
circuit (also called: load per device)
Stanisław Stoch Switching Systems 26
4
" " "
" " "
" " "
" " "
circuits
circuits
circuits
Queuing system
Aof Ac circuit Ac
group
Basic Concepts of
Ar
Teletraffic Theory
( part II ) Aof - offered traffic
Ac - carried traffic
Ar - rejected traffic
Aof = Ar + Ac
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B - call congestion E - time congestion
Calls are rejected, when all circuits are occupied.
The measure of quality of service is
Another measure of quality of service is
call congestion B.
time congestion.
Ar number of rejected calls
xt
B = =
Aof number of offered calls
number of circuits N
(0 d" B d" 1)
d" d"
d" d"
d" d"
t
congestion congestion
observation time
Because measuring of Ar is in many cases
difficult, another measure has been
congestion time
d" d"
E = (0 d" E d" 1)
d" d"
d" d"
introduced (follows on next slide).
observation time
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Comparison of B and E Comparison of B and E (cont.)
xt xt
number of circuits N number of circuits N
t t
CALLS CALLS



congestion congestion congestion congestion
observation time observation time
If calls are not independent (distribution different to
If calls are independent (Poisson distribution) their time
Poisson), in the congestion interval may appear more
distribution is independent on congestion intervals.
(as on the picture above) or less calls.
Fraction of rejected calls is proportional to congestion
Ratio of rejected calls to all calls is than not equal to
time, so:
ratio of congestion time to observation time, so:
B = E
B `" E (on the picture B > E)
`"
`"
`"
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Assumptions of Erlang Theory Assumptions of Erlang Theory (cont.)
" TRAFFIC OFFERED: The individual traffic " TRAFFIC REJECTED: Calls originating when
source acts independently of the other all devices are busy are lost (i.e. rejected calls
sources and is independent of the number of do not change the call intensity = no repeated
occupations. We assume Poisson arrival attempts! ).
process.
" EQUILIBRIUM:
" TRAFFIC CARRIED: Calls are carried by
Traffic process is in statistical equilibrium
full availability group of  N devices.
(calling rate is equal to termination rate).
The holding times are distributed with mean  h
" BH: Observation time assumed is Busy Hour.
(for telephony - exponentially distributed,
for data-transmission - constant).
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Stanisław Stoch
State diagram (equilibrium) State diagram (equilibrium)
Coin sides marked as: 0, 1. Coin sides marked as: 0, 1.
State for two coins = sum of points State for two coins = sum of points
We invert random chosen coin. We flip random chosen coin.
1 0,5 0,5 0,25
0 1 2 0 1 2
0 0 0 1 1 1 0 0 0 1 1 1
or or
vice versa vice versa
0,5 1 0,25 0,5
p(0)" 1 = p(1)" 0,5 p(1)" 0,5 = p(2)" 1 p(0)" 0,5 = p(1)" 0,25 p(1)" 0,25 = p(2)" 0,5
2 2
Solution: p(1) = 1/2 Solution: p(1) = 1/2
p(x) = 1 p(x) = 1
" "
p(0) = p(2) = 1/4 p(0) = p(2) = 1/4
x=0 x=0
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Erlang model Erlang model
Probability of termination of one specified call
0 1 2 " " " x x+1 " " " N
"t
in time interval  "t equals:
h
If during observation time  T there were  C
"t
h
calls, than probability of arriving of new call in
If in some moment there exist  x+1 calls,
time interval  "t
than probability of termination of one of them
C
"t "t
equals: (x +1)
equals: C
"t
h
T
T
Probability of state transition from  x+1 to  x
Probability of state transition from  x to  x+1
"t
"t
equals:
p(x +1)Å"(x +1)
equals:
p(x)Å"C
h
T
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6
Erlang model Solution checking
Equilibrium equation:
A Å" p( x ) = p( x +1)Å"( x +1)
"t "t
p(x) Å"C = p(x +1) Å"(x +1)
Ax Ax+1
T h
( x +1)!
x!
h
A Å" = Å"( x +1)
Substituting we get: N
C = A
Ai N Ai
T
" "
i! i!
A Å" p(x) = p(x +1)Å"(x +1) i =0 i =0
N
Ax
Ax Ax Å" A
"p(x) = 1
A Å" = Å"( x +1)
x!
x=0
Solution: p(x) =
x! x!Å"( x +1)
N
Ai
"
left side = right side
i!
i =0
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Stanisław Stoch
Erlang formula Traditional Erlang table (table of E)
E(A,N) N
N
During congestion all circuits are occupied, so A 1 2 3 4 5 10 15 30 60 120 240 480 960
0,01 ,0099
x = N , time congestion E is equal to probability 0,02 ,0196 ,0002
0,04 ,0385 ,0008
of congestion p(N) in arbitrary moment: 0,06 ,0566 ,0017
0,08 ,0741 ,0030 ,0001
0,1 ,0909 ,0045 ,0002
0,2 ,1667 ,0164 ,0011 ,0001
0,4 ,2857 ,0541 ,0072 ,0007 ,0001
0,6 ,3750 ,1011 ,0198 ,0030 ,0004
0,8 ,4444 ,1509 ,0387 ,0077 ,0012
AN
1 ,5000 ,2000 ,0625 ,0154 ,0031
2 ,6667 ,4000 ,2105 ,0952 ,0367
A 3 ,7500 ,5294 ,3462 ,2061 ,1101 ,0008
EN( A)
4 ,8000 ,6154 ,4507 ,3107 ,1991 ,0053
N!
5 ,8333 ,6757 ,5297 ,3983 ,2849 ,0184 ,0002
EN( A) = E1,N( A) =
10 ,9091 ,8197 ,7321 ,6467 ,5640 ,2146 ,0365
N
15 ,9375 ,8755 ,8140 ,7532 ,6932 ,4103 ,1803 ,0002
Ai
30 ,9677 ,9356 ,9034 ,8714 ,8394 ,6813 ,5272 ,1325
60 ,9836 ,9672 ,9508 ,9345 ,9181 ,8365 ,7553 ,5149 ,0963
120 ,9917 ,9835 ,9752 ,9669 ,9587 ,9174 ,8762 ,7527 ,5078 ,0694
"
first Erlang formula
240 ,9959 ,9917 ,9876 ,9834 ,9793 ,9585 ,9378 ,8756 ,7514 ,5040 ,0498
i! 480 ,9979 ,9958 ,9938 ,9917 ,9896 ,9792 ,9688 ,9376 ,8753 ,7507 ,5020 ,0355
i =0
B-Erlang formula
960 ,9990 ,9979 ,9969 ,9958 ,9948 ,9896 ,9844 ,9688 ,9376 ,8751 ,7503 ,5010 ,0253
Stanisław Stoch Switching Systems 42 Switching Systems 43
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Inverted Erlang table (table of A) Summary of basics
E 0,5 % 0,6 % 0,7 % 0,8 % 0,9 % 1 % 2 % 3 % 4 % 5 %
E
N 0,005 0,006 0,007 0,008 0,009 0,01 0,02 0,03 0,04 0,05 Ac circuit Ac
A
1 ,00503 ,00604 ,00705 ,00806 ,00908 ,01010 ,02041 ,03093 ,04167 ,05263
2 ,10540 ,11608 ,12600 ,13532 ,14416 ,15259 ,22347 ,28155 ,33333 ,38132
group
3 ,34900 ,37395 ,39664 ,41757 ,43711 ,45549 ,60221 ,71513 ,81202 ,89940
4 ,70120 ,74124 ,77729 ,81029 ,84085 ,86942 1,0923 1,2589 1,3994 1,5246
5 1,1320 1,1870 1,2362 1,2810 1,3223 1,3608 1,6571 1,8752 2,0573 2,2185
Ar
6 1,6218 1,6912 1,7531 1,8093 1,8610 1,9090 2,2759 2,5431 2,7649 2,9603
7 2,1575 2,2408 2,3149 2,3820 2,4437 2,5009 2,9354 3,2497 3,5095 3,7378
8 2,7299 2,8266 2,9125 2,9902 3,0615 3,1276 3,6271 3,9865 4,2830 4,5430
9 3,3326 3,4422 3,5395 3,6274 3,7080 3,7825 4,3447 4,7479 5,0796 5,3702
A = Ar + Ac
10 3,9607 4,0829 4,1911 4,2889 4,3784 4,4612 5,0840 5,5294 5,8954 6,2157
N
11 4,6104 4,7447 4,8637 4,9709 5,0691 5,1599 5,8415 6,3280 6,7272 7,0764
12 5,2789 5,4250 5,5543 5,6708 5,7774 5,8760 6,6147 7,1410 7,5727 7,9501
13 5,9638 6,1214 6,2607 6,3863 6,5011 6,6072 7,4015 7,9667 8,4300 8,8349
14 6,6632 6,8320 6,9811 7,1154 7,2382 7,3517 8,2003 8,8035 9,2977 9,7295
15 7,3755 7,5552 7,7139 7,8568 7,9874 8,1080 9,0096 9,6500 10,174 10,633
Ar Ac
16 8,0995 8,2898 8,4579 8,6092 8,7474 8,8750 9,8284 10,505 11,059 11,544
EN (A) = B = a =
17 8,8340 9,0347 9,2119 9,3714 9,5171 9,6516 10,656 11,368 11,952 12,461
18 9,5780 9,7889 9,9751 10,143 10,296 10,437 11,491 12,238 12,850 13,385
A N
19 10,331 10,552 10,747 10,922 11,082 11,230 12,333 13,115 13,755 14,315
A
20 11,092 11,322 11,526 11,709 11,876 12,031 13,182 13,997 14,665 15,249
Stanisław Stoch Switching Systems 44 Switching Systems 45
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7