424 Digital Mobilr Commu.

llicatiollS alld thr TETRA SYSleItI

A.4 ERLANG C TABLE (FOR BLOCKED-CALLS- QUEUED)

A.4.1 Overview of Basic Concepts

Call request queuing is supported wilhin TETRA and Erlang C table is therefore useful
for trunking capacity estimation under call queuing strategy. With the Erlang C system. a
call arriving at a busy system is queued rather than blocked immediately as in the Erlang
B system. A queued call wi ll only be lost if it cannot be connected within a certain
queuing time To which is usually imposed by the switching system. The queued call
requests do nol. unlike Erlang B. immediately contribute to a ncw arrival process and
therefore a modified equation applies to Erlang C formu la as shown below .
•c

( A.3)

NOIe thm the Erlang C equation gives the probability of queue. PQo that an arriving caU
finds a busy system and has 10 wail. (I · Po is the probabi lity that call requests are being
serviced). The equatio n does not say how long the wait time is and for this another
equation is employed. The probability that a qllelled call waits for more than I is given by

(A.4)

where A. C and T are as defined for Erlang B fo rmula. On the o ther hand, the probability
that all cal/s, including those serviced immediately. are delayed for more than I is given
by PQ- PU" The unde rlying assumptions of traffic source behaviour for Erlang B also apply
to Erlang C. with some additional poi nts 3 5 noted be low.
• The queue size is assumed to be sufficiently large (ideally infinite) and queued calls
are served o n first-come first-served basis.
• Erlang C function makes sense only if p < I (or A < C) otherwise the queue would
overflow as the system cannot cope with the call requests.
£r:ample: An 8-channe l trunked TETRA system operates under a call queuing strategy
with up to 5% of call connections delayed. Determine the additional number of channe ls
required for lowering the de layed calls to I % o r less. What percent of the delayed calls
waits beyond the mean call holding time T. before and after increasing the channel?

Soll/tio,,: Given C = 8 and Po = 0 .05 (5%). from the Erlang C table on the next page, the
offered traffic can be found to be A = 3.87 . If Po has to be reduced to 0.01 ( 1%). for
approximately the same offered traffic the number of channels can be found to be 10. This
corresponds to the nearest safe value of A = 4.08 from the table. Hence two additionaJ
channels would be required. The probability of queued calls waiting for more than Ti s
given by equation A.4. Since the waiting time is equal to the mean call holding time T.
equation A.4 reduces to ~K""'j giving. Pw = 0.016 for C = 8, and Pw = 0.0022 for C = 10.
Hence. with 8 channels 1.6% of the queued calls wait for more than the mean call holding
ti me. As expected. this figu re is reduced to 0.22% after the channel is increased to 10.
 Ap~tld jces 425

A.4.2 Erlang C Table

The table gives the o ffered traffic load A in Edongs corresponding to lhe number of Iraffi c
channe ls C in a lrunk (column I) and C3.11 queuin g probabililY PQ in percentage (lOp row).

Channels Delay Probability. PQ

c O.~':f, 1'1> ",.
0.00 om 0.02 om 0.02 0 .03 0'" 0.., 0 .10 O. I ~ 000
0.10 O. I ~ 0.18 0.21 0.24 0.16 0.4 1 0." 0.74
OJ) 0.4) 050 05S 0.60 0.65
0.>0
0.79 0." '"
,." ,...
L.. 1.23 L )9

•, 0.66
1.07
0.81
I .~
0.91
1...19
0.99
150
1.06
1.59
1.12
1.66
1J2
1.91
1.47
2.31 2.6 1
2.10
2.8S

,• ""
.... ....
I.~~ 1.76 1.92 2.~ 2. IS 2.24 2.53 2.7S 3.01 J.>o 3.62
2,0 1 2.30 2.49 2.6.\ 2.7S 2.86 3. 19 ).4) 3.73 4.10 4.41

9
• 2.S4
3. 10
2.87
3.46
3.08
).70
3.25
3.88
J.)8
4.Q.l
3.S(!
4. 17
3.87
4S7
4. 14
4.87 S.22 5.67
S.2 1
6.OJ

" ).68 4.08 4.34 4.54 4,11 4, 85 ~.29 S.61 '.99 6.47 6.8S
4.28 4.71 S.OO S.21 S.39 SSS
.,.
6.02 ." '" ."
.-"
"""
U"
4.90
S.53
6.17
B6
6.03
6.71
S.67
6')S
S.90
6.60
7.) 1
6.09
6.8 1
7.53
6.26
6.98
7.71 8.27
...
'" .... '3'
, 7.1.1
."
7.~S

9. 16
8. 10
8.93
9.76 10.2

""
1.~
6.&.1 7.39 1.76 8.Q.l S.26 1.46 9." 9.41 9.97 10.6 I Ll

"" s.n ,.,

...
1.50 8.(19 ,""S 9.00 9.21 9.82 10.8 1l.4 12.0
8.18 8.80 9.20 9.5 1 9.7S 9.91 10.6 11. 1 11.6 ,,-' 12.8
8.87 952 9.94 10hl 10.5 10.7 11.-1 11.9 12.4 13.1 13.7

"" 957
10.)
10.2
11.0
10.7
IIA
11.0
II.S
11.3
12.0
11.5
12.)
122
13.0
12.7
I3.S
13.3
14.1 14.9
14.6
OS,
"'"22
.
11 .0 11.7 ]2.2 125 12.8 13.1 13.8 143 15.0 IS.7 "3
11.7 125 12.9 In 13.6 13.9 14.6 IS .2 IS.8 16.6 11.1
2J 12.4 1).2 13.7 14.1 14.4 14.6
,,
IS.4 16.0 16.7
'" 18.1

"" 13.2 14.0 14.S 14.9 15.2 1M 16.8 I1.S "3 19.0

,. 13.9
14.6
14.7
IS5
IS.2
16.0
IS.6
16.4
16.0
16.S
16.2
17.1
17.1
17.9
11.7
II.S
18.4
19.2
19.2
::0.1
19.9
2'0.8
" IS.4 16J 16.8 11.2 17.6 17.9 18.7 19A
::!o.2
"'. , ~ I .o
21.9
21.7

"'.•
16.1 11.0 17.6 IS.O 18.4 18.7 19.6 ::0.9 22.6

""30 16.9
11.1
11.8
18.6
18.4
19.2
18.8
19.6
192
::0.0
19.5
::0,) 21.2
21.1
21.9
2 1.8
22.7
22.8
2.\.6 '"
24.4
18.4 19.-1 W.O ::!O.4 ::!O.I 2 1.1 22.1 22.8 2.3.6 "3
"" 19.~ W.2 20.8 21.3 21.6 2!.0 22.9 !..1.6 :!4.4 '"
2.S.4 26.2

"""
W.O
W .1
21.0
21.7
21.6
22.4
22.1
22.9
22.S
2.1.3
22.8
2.1.6
".
24.6 '"
25.4
2.!'i.)
26.2
'63
~7.2
21. 1
28.0

,. 215
22.)
2~5

!.1.3
23.2
24.0
:!.l.7
24.S
24.1
~.9
24.5
2S.3
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26.3
26.2
27.1
27.1
27.9
28.1
29.0
28.9
29.8

"
)8
23. 1
23.9
24.2
25.0
24.8
25.7
25.4
~. 2
2S.8
26.6
26. 1
27.0
27.2
28.0 "29.7..
" .0 28.8
29.1 ".•
29.9 30.7
~ 1 .7

.,'" ~.6 25.8 ::!6..'l 27.0 21.4 27.8 28.9 30.6 JJ.7 32.6

.. 25.4
~.~
26.6
21.4
21,)
21. 1
21.8
28.7
28.3
29.1
28.7
29.5
29.S
30.'
30.'
)1.4
)I.S
)2.4
32.6
HS
""3.. ,.,
'"..
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...,
42 21.0 28.2 29.0 295 JO.O 32.) )J.)
.lO')
'"
.."
'J 27.1 29.0 29.8 JO.) JO.S 3 1.2 32.4 )).2 34. 1
28.6
29.4
29.8
30.7
30.6
31.4
31.2
32.0
3 1.1
32.S
32.1
)2.9
3).2
.W.l
".,
".0
~.O
35.9
36.2
)1.1
37..2
38.1

,., ,., " .• "'.0 .,.•

,,.
...
30.2 315 3D 32.9 3).4 )).S )S.O )S.8
J6.' "'.0 38.1
3 1.1 32,) )). 1 B.7 34.2 .W.6 )7.7 39.9

,... .,"'.,.• .,.

3 1.9 33.1 34.0 34.6 3S.1 )5.5 )7.6 39.9

"so n.1 34.0 34.8 3M 3S.9 J6_' 31.6".,

33.5 34.8 3S.6 36.3 36.8 37.2
'" 4 1.7 4 2.7
426 Digital Mobile C Ql1IlnulficlUiOlIS alld till' TETRA Systt m

Erlang C Table (continued from page 425)

Channels Delay Probabilily. Po

c 2.Yl- 15'.l

"" .'I.i ..\ 15.() 36..'> '7. 1 .17 .6 .18.1 40 ..1 ·H ..1 4.1.6

....
.15.1 .\7..\ \8 ,0 ,OS .18.'1 4 1.! 4~.1 ~,

"" "" ..••

........
J7-' 1II.3 .W.8 42.0 JJ. I
"'.•
J5.Q .".~ ~8.' ."
\8 .1 30' "'.7 "'.7 4:'9
",
40~ ~

"S6 \7 .6 79.' ." ." 43 .8 ~ 47~1

.• ...,
......, .,7
\8 .4 " .7 41 .4 41.9 42.4 4.\.11 ~ .7 4~.1I 47.1 4S.1

" .W.~ 4 1.6 J2 ,! 42.8 4.\ ..1 ~ 45.6 " .7 "., 49.:

"" 4 1.5 42.4

..••
J.\. I 4.1.7 ~. , ." 017.6 49.0
SO.,
SO,

...
4~." 4.1..\ ~ ~, 45.0 46A 47A 48.5 51.0
00 41.7 43.!
.• ~. , ~ 45 .4 45.9 4 7~1 48 . .1 49.5 SO. 51.0

""
...
4~.6 ~ 4.'>.0 4S.7 " .J ".7 48.~ 4'1.2 SO.. 5 1.8 52.9

..
63

65
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4S .1
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47.6

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49.8
SO.,
47.6
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SO..
5 1.1
49.1
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52.6
SO. ,
5 1.0
5 1.9
5:"
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5.1. 1

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51.'
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55.7

57.6

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59.4
50.11 51.9 SH 55..\ 56 ... 57.7 59.2
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,... ",SH ".7

SI.7 S~.8 56.2 57 ..\ 58.6 60 ' 6U

" "'".•, 60.'

SO., S!.O 59.5 6 1.0
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5.\.6 55.0 55.s SII.! 6!.~
5 1.8 5.U 55 ..1 55.9 S6.'
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S6.' 57.1 H.II 5'1.1 6 1.9 6.1.2
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55.! 50.9 511.0 58.8 59'" 6~ . 8 ." M.'
TI S6.0 57.8 58.8 "'.7 60 ..\ 60.' 6!3 6.\.7 6.~.0
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"., 67.9
56.9
57.7
58.6
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59.7
60.
60.5 6 1 .~ 6 1.7
6! .6
6.1A

".' 655
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67.6
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6 1A 62.1 68.'
'""" 60.' £ ..1 6S.2
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511.0 63' .U 67.8 70.7

"'. ... .,.

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60.'
6 1.2
6~. 1
...,
62.,\
6.1 . ~
.. ,,
63.2 6.1.9
6"-1 M .'
"'-, 67~1

...68.:,
68.7 ro.'
11.3
7 1.6
72.6

,.,
........
61 .1 6.\.0 M.' 66. ~ 67.9 10.' 72.2 7.\.s
£ .0 6.1.S
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MO 65.'
66.7 ."
67.4
67.1 68.'
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n.4
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74. 1

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6~ .9 M.' 7 1.0 75.4
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66A
66.7
67.6
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68.S
68 ..\
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7.\..1
74.2
75.0
76,0
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".,"" "70....\
773
6.~~~ 6L\ ro , 7:A 7.\.7 75.1 76.9 78 . ~

"'-.' 68.2 7 1.0 7 1.6 73A 74.6 76.1 71.&

...,
'N.:
67.!
'". ro.. 7 1. 1 7 1.9 ", 74~\ 7" no 78.'

"" ..•• "10. 71.1 no 71.11 73.4

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n.' "'.7 81.1

""'J r-.O n9 73.6 n.

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81.0
., "'ro.'.•
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71.7 n.Q 7.\.' 74.5 7.'>.! 7,. 78 ~\ 81.6 8.1.0

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79.2
",
74.7 75.4 76.1 8:.5 8.1.9

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75.6
7S.6
76. ~
76 ..\
77.2
77.0
77.'1
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82.~ ....
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.... ...OS....
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77.4 82.() 8.1.5 85 .)
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7.\.~ 75 . ~ 78.1 78.8 116.7
.," 74.1 76. 1 773 78..1 "'.7 8!.9 so.) 87.7

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75.0 77.0 78.~ 79. ~ .!.S &.\.8 87.! 88.'
'00 75.8 no "'., 8O.' 83.4 "'7 86..1 88. 1
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