Appel/dices 42/
A.J
ERLANG B TABLE (FOR BLOCKED·CALLS·CLEARED)
A.3.J
O\'erview of Basic Concepts
The Erlang 8 table is based o n Erlang's Loss Formula. also c alled Erlang 8 Fommla (8
for blocked). given by equation A. I . The fommla provides nn estimate of a c all blocking
probability. PH' used as a measure of grade of service (GoS) in te le phone systems whe n a
mean traffic load of A Erlangs is applied to a mmked system of C channels.
(A.I)
The IOlal applied traffic A in Erlnngs is AT where A is the arriva l rate in Cl/1Is/fIO/tr and T
is the m'('ragl' call duration o r call holding lime. Fo r instance. an emergency centre
receivingl50 call!;/hour (A). wilh an average call holding time T of 30 sec is handling a
load traffic 150 x 130 s/3600 sihour) = 1.25 Erlangs. While calculating the GoS is
straightforward from equation A.I. often. the exercise is to detcmline the other
paramctcr(s). i.e. offered traffic A and/or trunk size C. for a given value of GaS. A typical
problem could therefore be stated as: What trunk size would suppon a 1% GoS? The
example used III Section 1.21 illustrmes Ihis point. In this situation calculating for A or C
becomes cumbersome and the table of values as sho\\Tl o n the following pages will be
useful.
The reader should however be aware of the underlying assu mpti ons for this fommla . as
cnution should be exercised in its application for c apacity estimatio n:
•
The call arri\'als A are assumed to be random and independent. For instance. the
Erlang B form ula could overestimate the capacity for systems with correlaled calls.
e.g. TETRA e mergenq' calls Ihut ensue an incident. as opposed to random calls.
•
The number of traffic sources nre assumed sufficientl} l:trge compared 10 the number
of channels C. This conditio n is the basis for assuming 3 constanl arrival rate it
regardless of number of call requests. A finite number of call requests therefore lends
to provide a consen'ative estimate of the blocking probability.
The table on the following pages gives the offered traffic load A corresponding to call
blocking probability P B and number of c hannels C in the trunk. The actual carried load is
A.( I· PRJ which should always be less than C even if the offered load A > C. The truoking
efficiency. "Ill' is therefore given by the ratio of the actual traffic carried to the mink size.
••
".r I-P" J
C
(A2)
The mtio p = NC represents the offered traffic per channel and is known as the chal/lIl'f
oCl·"{J(lI/(T·
411
Digilal Mobi/~ Commulljcatio" s and the TETRA System
A.3.2
Erlang B Table
The table gives the offered traffic load A in Erlangs corresponding to the number of traffic
channels C in a trunk (column I ) and call blocking probability PB in percentage (top row).
Blocking ProbabililY, PB
Channels
c
OSI.
I<;f,
,
0.01
0.1 I
OJS
0 .01
O. IS
om
om
0.46
0.B7
1.36
0""
0."
0."'
•S
0.70
I,n
l S I>
0.19
1.S2
.
,.,.".. ,."
0.22
0.03
0.Q3
0.25
0.28
,
1.111
•."
' .H
"
4.81
S.96
6.6 1
7.I1S
6.66
7JS
7.82
1.33
8. 11
B.61
11.10
8.118
9.04 1
10.2
6.6 1
7."
• .20
9 .01
9 .83
!J. I
.4.0
'U
03 S
14.0
14.9
IH
14.9
15.11
16.7
17.6
16.2
IS.4
16.2
17. 1
, 1$
IB.O
17.0
17.8
18.04
".9
14.3
IS .2
18.6
,OS
19.3
20.2
22.8
21.5
22 .9
23.9
22.)
23.11
JS
2J.2
24.6
24.11
2S.6
36
J7
J8
24.0
25.S
26.4
.."
.."""
so
HO
31.0
28.2
29.1
29.9
29.9
30.8
3 1.7
)2.5
33.4
l l .O
l l .9
32.8
M .3
3S.2
36.1
35.6
33.4
34.2
n .l
36.0
37.0
l1.9
"S
,.A
29.2
.1<7.1
28.1
29.0
n.s
2).7
24.6
27.4
28.3
26.S
31.7
21.9
27.3
28.3
29.2
26.5
21.4
)(1.8
"S
12.7
'"
10.7
22.1
""
"
JO.S
7m
1M
16.6
HJ
' .M
8.&.1
9 .73
10.6
•..,
12.3
1}.2
16.0
16.9
24.8
2S.7
,."
11 .8
12.7
12.8
n .7
14.5
1s.J
16.1
19.9
20.7
H4
1).4
11 .9
12.6
n .4
IB.2
19.0
7.,"
11.4
12.0
11.11
18.6
19.5
20J
6.11
12.2
11.1
IS.O
3.74
11.9
104
11 .2
IH
16.6
17.04
4.75
..s.n"
I I.~
9.65
21.2
22.0
"'"
'JS
8.8l
21.0
"
"
13
"
7.70
832
3.2S
10.2
11 .0
"'.,
2 1.2
JO
6.10
9.S8
10.)
14.2
..so
S.S3
S,.
"03
""
""
.,""
""
13
""
"
"'"
"
S."'
S.l.!
S.16
6.29
s.oo
S."
4.6 1
HB
S. II
4.14
'M
l ."
'"
.'" S.'" .,.
S.28
4.01
3.76
4.67
'-"
1.74
..
3AS
'.J<l
l .82
2.50
3.U
) .78
4.46
33.7
34.6
lOS
37.4
380
,,~
J<U
31.9
32.8
m
" .7
35.6
"S
37.S
" .l
.,.
B.61
8.49
9 .47
,OS
11.4
"S
12.5
12.4
13.4
14.3
15.3
16.3
,OS
,.,
,
..
18S
19.6
22.8
20.0
19.4
20.9
2 1.9
22.9
:!.3.8
26.7
2S.3
2H
27.7
16.11
18.7
27.1
27.7
28.0
29.0
28.6
29.6
JO.'
"'S
,<S
32.7
)3.4
29.7
"'.7
3S.S
36.'
37.4
J8J
3s.J
36.2
22.1
2.1.9
26.4
23.2
24 ..!
25.2
26.2
24.9
16.0
27. 1
~7 .6
28.9
XI.2
28.7
3 1.4
27.2
28.2
".,
29.2
JO. ~
,OJ
"'0
3U
32.3
34.5
J5.6
31.3
)3.6
3H
3S.6
37.4
37.1
37.6
38.6
39.6
41.1
38. 1
41 .5
41.5
4 1.9
~S
2S.l
21.2
.n .J
'M
24. '
24.2
.\4.4
32,4
3JA
""
". "" ""'..•
, ....
".• ".,
"S
"S
13.'
no
11.8
22.8
31.6
J2.6
31.4
IS.6
16.B
17.6
".S
2H
14.4
18.0
19.2
20.4
2 1.6
19.0
24.0
10.9
12.0
D .l
IS.S
20.7
14.9
7.37
8.52
is ..!
16.3
17.4
03S
192
20.2
16.2
31.7
10.8
11 .9
13.0
14.1
18.'
24.8
6.23
•.'" ."
~. 8
"3
' 00
1.93
2.95
8.62
21.9
"S
".,
"'.0
"A
"0
,,,
4 1.0
40.2
"0
36.S
7.'"
6.78
1$S
IB.7
2 1.2
21.'
2..1 . 1
29.9
6.SS
7S 1
S."
19.7
20.7
11.6
24.4
IS.!
S.S8
L",
,.so
18.1
2Q.l
=
23A
11.5
U.
17.1
19.8
0.2S
,.so
J.63
4.34
•
••
0."
1.27
2.05
! .88
2.96
2.28
0.18
0."'
us
' .S8
2.11
",.
0 .11
0.47
2.42
3. 10
1.9 1
l A.
1.26
m.
0 .38
0 .90
1.S2
2.22
1.62
2. 16
2.73
3.33
) .96
7
0.12
0."
..
37.7
..
4!.0
·1.1. 1
4U
....
42.6
43 .6
45 .7
" .7
'
45.1
.".,.•
<6.,
47,4
2S_'
29.9
11.0
32. '
n .3
"A
3M
26.S
27.7
32.6
3J.8
35.1
36'
J7S
JU
36.7
"'.0
37.9
)9.0
40.2
4 1.2
41.3
42.5
43.6
+U
4S.9
47.1
48.2
42.4
43.7
44.9
".,
.....
47.4
49.9
S I.I
S2.3
"so..•• ".•
53.6
S I.?
S2.9
,.0
Sl.O
,"S
S7J
Appclldius ..J.23
Erlang B Table (continued fro m page 422)
Channels
c
""
"
""
""
""
'"
""
"
""
'"
'"
,.""
"
"
...
n.7
",
....,
38.6
41.~
42.1
H.O
43.9
44.8
....
47.4
66
"n
"19
.,
""
""
."
85
45.6
~5
,,-'
.."
.,"
OXl
"'.,
.....
4 1.5
4!.4
41.1
44.2
45.1
.....
.....
".,
.... 0
479
50.'
40,1
4 1.0
4 ~ .0
4U
4 2.1
43.1
42. 1
43.0
44.0
4~.9
" .0
~.
.• .....
.....
,,,
"., "'.•
4.1.&
~
~ ,
45.9
45.1
41i.4
SO.3
50.'
SO,
5 1.~
r.~
~2.4
".0
53.4
55.0
R9
\4 ..1
~S9
5~.8
55,2
5.l.?
56'
'"
~7.8
';6.4
5Ll
"2
"'.,
60.0
"'.•
6 1.8
61.7
.......,
6.1.6
M.'
672
M.'
"'0
W.•
.... "'.•
'"
~8.7
".0
511.7
60.'
6 1.6
62.5
..
"
6.1.4
., ""
61 .6
6.1.5
65 ,3
".
66.1
67.1
66 ..1
67.2
68.2
"""'.0
"'.•
7 1.9
7 1.0
7 1.9
12.11
73.7
74.7
73.&
14.&
75.1
76.7
156
no
n.8
n,
'"
",
75.4
"A
19.,
"19.'.•
"'.,
8 1.5
77.1
82,4
7M
76.3
78.1
19,
".0
W .•
SO,3
52.5
,..,
51.4
& I.~
&.lA
82.2
8·0
OJ.'
".,
&5.3
86.1
50.'
'"
4Q.6
50'
5 1.6
~1 , 5
~5 ~
59~1
60.'
'"
5.1.5
:\4.5
58.1
57.1
~,
47.7
~ 1 .6
'"
5Ll
41.9
43.9
44.8
45.8
51.6
U.5
SH
...,
57.4
~8, I
SS.4
5'"
574
""
~9.1
"-.
66.'
61.7
.."'".•
7 1.6
12.5
7.1.5
",
15.4
'"
nJ
",,
"'0
"'.9
7111
7!.1l
7.l.8
74.8
75.8
76.8
77.7
19.'
...
&1.6
62.6
6.l.1
8~.1
83. 1
".
57.6
59.6
"'. ...""
6.'-1
61.6
626
6J.7
.'-'
66.4
675
"'68'.•
"'.
71.7
12.7
"'.,
""
7U!
12.&
748
7.l.&
71.1
74 .8
12.1
73.0
74.0
7S.0
..
75.8
,
no
".0
78.9
19.'
W .•
&1.9
&1.9
&.1.9
85.5
86.8
87.8
"'.•
76.9
."
87.0
", "'.•
89.'
1.'.8
"'.•
M.'
."
7U
n."
7J.5
74.6
75.6
76.7
n.,l
711-1
.,.
19.'
8 1.9
831
84~
"'., .....
8JJ
90'
.6.6
9.16
86.,
"0
"',
~.2
'"
"'.0
88-8
94.1
75,Q
76.1
84..1
85 ..1
11 1. 1
112.2
14.6
70.1
"'.9
88. '
89.'
12. 1
68.0
76.11
78.0
OJ.O
81.4
"-,
90'
9 1.6
91.7
917
".
95.8
96.9
97,9
"'.0
"'.,
"'66.'.• ""
73.4
"'"'-,. ,
.66
81.1
67.2
68.,
"'.•
87.7
...".,
64.5
,
M.9
12.6
73.8
&.1.2
85.0
'"
6n
6~ , 2
6J.S
68.0
66.'
19.'
&1.0
61.0
6 1.0
n'
8~.!
".
..
".,
"
".
78.9
80.0
8 1.1
82.2
7S.1l
59'
1 1.4
."
n.'
113.2
86.0
".0
.0>
"'.1
81.6
8.1,6
8H
59. .1
~S.6
.....
"'.,
".
"., ..,
.., ".
8 1.1
57.1
58.2
56.
6.1.2
w.o
""
"'.,
66.7
67.7
67.2
68.0
~7.5
~S.O
~1.6
6>.1
670
680
52.8
" .•
6~ .3
M'
S5.2
.56..1
53.9
~ I ..'i
52.6
6.1.2
64.1
65 .2
".,
".
,,-' ",.,
"'2
.50'
m
61.~
".
51.7
50.'
45.S
...,
60.1
66.'
M.'
....,
60.3
61.3
..., ...,
....
...
6 1.0
60.0
"""'. , "'.•
68.1
41.8
49.7
49.6
~I.O
~7.0
~.
45,11
47.8
,g.,
50. '
71.8
.,"
.111.8
~l.I
'"
m
or.
1.5<:<-
m
49.1
"''" "'.
..'" n.7
"•••
Blocking ProbabililY. p.
0 .5'1
89.9
IlL O
91. 1
85.1
81.7
'"
91.~
91,4
'"
".
117. 1
.,..
98.2
IlU
100.6
~.2
101.7
101.9
IOU
IQ.D
,,-'
"'..
~75
78 ..1
8O •
....".•
8~.1
S.Ll
".0
lIS.]
89.'
90.'
91.0
9.1.3
.
~ .'\
".,
".0
911.1
W.,
100.7
1 0~.0
10.1.1
lOB
IOS.7
106.9
108.2
109 ..1
110.7
111 .9
I 1 3.~
08.'
106 .1
114 '*
W.,
107 ,6
108.8
109,9
I IS.7
116.9
I t8.~
J ill ...
110.6
100.8
101 .9
10.1,0
104.1
111.1
111. .1