Fascicle II.3 _ Rec. E.524 5
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Recommendation E.524
OVERFLOW APPROXIMATIONS FOR NON_RANDOM INPUTS
1 Introduction
This Recommendation introduces approximate methods for the
calculation of blocking probabilities blocking probabilities for
individual traffic streams in a circuit group arrangement. It is
based on contributions submitted in the Study Period 1984_1988 and
is expected to be amended and expanded in the future (by adding the
latest developments of methods).
The considered methods are necessary complements to those
included in the existing Recommendation E.521 when it is required
to take into account concepts such as cluster engineering with
service equalization, service protection and end_to_end grade of
service. Recommendation E.521 is then insufficient as it is
concerned with the grade of service for only one non_random traffic
stream in a circuit group.
Design methods concerning the above_mentioned areas are
subject to further study and this Recommendation will serve as a
reference when, in the future, Recommendation E.521 is complemented
or replaced.
In this Recommendation the proposed methods are evaluated in
terms of accuracy, processing time, memory requirements and
programming effort. Other criteria may be relevant and added in the
future.
The proposed methods are described briefly in 2. Section 3
defines a set of examples of circuit group arrangements with
exactly calculated (exact resolution of equations of state)
individual blocking probabilities, to which the result of the
methods can be compared. This leads to a table in 4, where for
each method the important criteria are listed. The publications
cited in the reference section at the end contain detailed
information about the mathematical background of each of the
methods.
2 Proposed methods
The following methods are considered:
a)Interrupted Poisson Process (IPP) method,
b)Equivalent Capacity (EC) method,
c)Approximative Wilkinson Wallstr”m (AWW) method.
2.1 IPP method
IPP (Interrupted Poisson Process) (Interrupted Poisson
Process) is a Poisson process interrupted by a random switch. The
on_/off_duration of the random switch has a negative exponential
distribution. Overflow traffic from a circuit group can be
accurately approximated by an IPP, since IPP can represent bulk
characteristics of overflow traffic. IPP has three parameters,
namely, on_period intensity and mean on_/off_period durations. To
approximate overflow traffic by an IPP, those three parameters are
determined so that some moments of overflow traffic will coincide
with those of IPP.
The following two kinds of moment match methods are considered
in this Recommendation:
_ three_moment match method [1] _ where IPP parameters are
determined so that the first three moments of IPP will
coincide with those of overflow traffic;
_ four_moment ratio match method [2] where IPP parameters are
determined so that the first moment and the ratios of the
2nd/3rd and 7th/8th binomial moments of IPP will coincide
with those of overflow traffic.
To analyze a circuit group where multiple Poisson and overflow
traffic streams are simultaneously offered, each overflow stream is
approximated by an IPP. The IPP method is well suited to computer
calculation. State transition equations of the circuit group with
IPP inputs can be solved directly and no introduction of equivalent
models is necessary. Characteristics of overflow traffic can be
obtained from the solution of state transition equations. The main
feature of the IPP method is that the individual means and
variances of the overflow traffic can be solved.
2.2 EC method
The EC (Equivalent Capacity) method (Equivalent Capacity)
method [3] does not use the traffic_moments but the transitional
behaviour of the primary traffic, by introducing a certain function
r(n) versus the equivalent capacity (n) of the partial overflow
traffic, as defined by the recurrent process:
(2_1)
if n is a positive integer and approximated by linear
interpolation, if not.
A practical approximation, considering the predominant
overflow congestion states only, leads to the equations:
(2_2)
with:
Di(n) = 1 + ai (2_3)
defining the equivalent capacity (ni) of the partial overflow
traffic labelled i, and influenced by the mutual dependency between
the partial overflow traffic streams.
The mean value of the partial second overflow is:
Oi = ai p ri(ni) (2_4)
where p is the time congestion of the overflow group.
The partial GOS (grade of service) equalization is fulfilled
if:
ri (ni) = C (2_5)
C being a constant to be chosen.
2.3 AWW method
The AWW (Approximative Wilkinson Wallstr”m) method
(Approximative Wilkinson Wallstr”m) method uses an approximate ERT
(Equivalent Random Traffic) model based on an improvement of Rapp's
approximation. The total overflow in traffic is split up in the
individual parts by a simple expression, see Equations (2_7) and
(2_9). To calculate the total overflow traffic, any method can be
used. An approximate Erlang formula calculation for which the speed
is independent of the size of the calculated circuit group is given
in [4].
The following notations are used:
M mean of total offered traffic;
V variance of total offered traffic;
Z V/M;
B mean blocking of the studied group;
mi, vi, zi, bi corresponding quantities for an individual
traffic stream;
~ is used for overflow quantities.
2.3.1Blocking of overflow traffic
For overflow calculations, an approximate ERT_model is used.
By numerical investigations, a considerable improvement has been
found to Rapp's classical approximation for the fictitious traffic.
The error added by the approximation is small compared to the error
of the ERT_model. It is known that ERT underestimates low blockings
when mixing traffic of diverse peakedness [2]. The formula, which
was given in [4] (although with one printing error), is for Z > 1:
A* V + Z(Z _ 1) (2 + gá)
where
g = (2.36 Z _ 2.17) log {1 + (z _ 1)/[M(Z + 1.5)]}
and
á = Z/(1.5M + 2Z _1.3) (2_6)
2.3.2Wallstr”m formula for individual blocking
There has been much interest in finding a simple and accurate
formula for the individual blocked traffic mœi. Already in 1967,
Katz [5] proposed a formula of the type
(2_7)
with w being a suitable expression. Wallstr”m proposed a very
simple one but with reasonable results [6], [2]:
w = 1 _ B (2_8)
One practical problem is, however, that a small peaked
substream could have a blocking bi > 1 with this formula. To avoid
such unreasonable results a modification is used in this case. Let
zmax be the largest individual zi.
Then the value used is
w = (2_9)
2.3.3Handling of overflow variances
For the calculation of a large network it would be very
cumbersome to keep track of all covariances. The normal case is
that the overflow traffic from one trunk group is either lost or is
offered to a secondary group without splitting up. Therefore it is
practical to include covariances in the individual overflow
parameters so that they sum up to the total variance. The
quantities vi are obtained from the total overflow variance by a
simple splitting formula:
(2_10)
One can prove that Wallstr”m's splitting formula (2_8) and
formula (2_10) together with the ERT_model satisfies a certain
consistency requirement. One will obtain the same values for the
individual blocked traffic when calculating a circuit group of N1 +
N2 circuits as when calculating first the N1 circuits and then
offering the overflow to the N2 circuits.
Since the individual variances are treated in this manner,
they are not comparable with the results reported in Table 2/E.524.
3 Examples and criteria for comparison
The defined methods are tested by calculating the examples
given in Table 1/E.524.
The calculation model is given in Figure 1/E.524.
For comparison, the following criteria are established:
_ accuracy of the overflow traffic mean and variance (mean
and standard deviation),
_ computational criteria (processor time, memory
requirements, programming effort).
Figure 1/E.524 - T0200630-87
TABLE 1a/E.524
Exactly calculated mean and variance of individual overflow traffic
_ Three first choice circuit groups
Case A1 A2 A3 a1 a2 a3 A0 N O0 O1 O2 O3
N1 N2 N3 Z1 Z2 Z3 V0 V1 V2 V3
7.03 26.6 64.1 3.00 3.00 3.00 _ 0.43 0.74 1.09
1 6 88 69 3 1 0 37 90 1
5 28 70 1.57 3.02 4.52 _ 11 _ 0.76 2.11 4.44
3 2 7 56 0 1
7.03 26.6 64.1 3.00 3.00 3.00 _ 0.11 0.27 0.49
2 6 88 69 3 1 0 49 58 44
5 28 70 1.57 3.02 4.52 _ 16 _ 0.24 0.73 1.91
3 2 7 36 28 1
7.03 26.6 64.1 3.00 3.00 3.00 _ 0.01 0.02 0.06
3 6 88 69 3 1 0 369 846 627
5 28 70 1.57 3.02 4.52 _ 25 _ 0.02 0.06 0.22
3 2 7 041 461 05
7.03 10.1 13.2 3.00 5.00 7.00 _ 0.74 1.26 1.78
4 6 76 50 3 3 2 59 2 5
5 6 7 1.57 1.56 1.55 _ 14 _ 1.19 2.29 3.62
3 7 9 3 2 4
7.03 10.1 13.2 3.00 5.00 7.00 _ 0.28 0.48 0.68
5 6 76 50 3 3 2 84 57 32
5 6 7 1.57 1.56 1.55 _ 19 _ 0.46 0.90 1.46
3 7 9 36 89 0
7.03 10.1 13.2 3.00 5.00 7.00 _ 0.03 0.05 0.08
6 6 76 50 3 3 2 570 915 237
5 6 7 1.57 1.56 1.55 _ 26 _ 0.05 0.10 0.16
3 7 9 358 26 21
7.03 32.3 77.6 3.00 5.00 7.00 _ 0.45 1.17 2.34
7 6 95 17 3 2 1 16 6 4
5 31 77 1.57 3.02 4.51 _ 16 _ 0.74 3.46 10.3
3 9 1 34 6 9
7.03 32.3 77.6 3.00 5.00 7.00 _ 0.15 0.42 0.97
8 6 95 17 3 2 1 38 94 39
5 31 77 1.57 3.02 4.51 _ 23 _ 0.24 1.20 4.21
3 9 1 27 0 9
7.03 32.3 77.6 3.00 5.00 7.00 _ 0.01 0.03 0.10
9 6 95 17 3 2 1 303 984 06
5 31 77 1.57 3.02 4.51 _ 35 _ 0.18 0.09 0.36
3 9 1 41 378 90
64.1 32.3 13.2 3.00 5.00 7.00 _ 1.15 1.45 1.32
10 69 95 50 0 2 2 7 6 0
70 31 7 4.52 3.02 1.55 _ 15 _ 4.44 4.25 2.85
7 9 9 2 6 0
64.1 32.3 13.2 3.00 5.00 7.00 _ 0.55 0.58 0.47
11 69 95 50 0 2 2 64 49 49
70 31 7 4.52 3.02 1.55 _ 21 _ 2.02 1.67 1.02
7 9 9 6 5 3
64.1 32.3 13.2 3.00 5.00 7.00 _ 0.06 0.05 0.03
12 69 95 50 0 2 2 907 265 848
70 31 7 4.52 3.02 1.55 _ 32 _ 0.21 0.12 0.07
7 9 9 67 95 165
7.03 26.6 64.1 3.00 3.00 3.00 0.40 0.50 0.82 1.16
13 6 88 69 3 1 0 64 38 74 0
5 28 70 1.57 3.02 4.52 3.00 13 0.55 0.85 2.24 4.57
3 2 7 0 78 66 3 4
7.03 26.6 64.1 3.00 3.00 3.00 0.14 0.18 0.33 0.57
14 6 88 69 3 1 0 60 40 84 29
5 28 70 1.57 3.02 4.52 3.00 18 0.19 0.30 0.87 2.16
3 2 7 0 92 43 79 3
TABLE 1a/E.524 (cont.)
Case A1 A2 A3 a1 a2 a3 A0 N O0 O1 O2 O3
N1 N2 N3 Z1 Z2 Z3 V0 V1 V2 V3
7.03 26.6 64.1 3.00 3.00 3.00 0.01 0.01 0.03 0.07
15 6 88 69 3 1 0 170 506 086 035
5 28 70 1.57 3.02 4.52 3.00 28 0.01 0.02 0.06 0.22
3 2 7 0 472 218 861 87
7.03 32.3 77.6 3.00 5.00 7.00 0.12 0.44 1.15 2.30
16 6 95 17 3 2 1 53 51 6 4
5 31 77 1.57 3.02 4.51 1.00 17 0.13 0.72 3.36 10.1
3 9 1 0 92 66 6 0
7.03 32.3 77.6 3.00 5.00 7.00 0.04 0.15 0.42 0.96
17 6 95 17 3 2 1 250 36 75 74
5 31 77 1.57 3.02 4.51 1.00 24 0.04 0.24 1.18 4.14
3 9 1 0 696 09 3 8
7.03 32.3 77.6 3.00 5.00 7.00 0.00 0.01 0.05 0.12
18 6 95 17 3 2 1 4542 687 106 82
5 31 77 1.57 3.02 4.51 1.00 35 0.00 0.02 0.12 0.47
3 9 1 0 4891 398 14 51
64.1 32.3 13.2 3.00 5.00 7.00 1.76 1.25 1.65 1.63
19 69 95 50 0 2 2 1 1 4 0
70 31 7 4.52 3.02 1.55 9.00 21 3.05 4.51 4.40 3.10
7 9 9 0 2 7 6 3
64.1 32.3 13.2 3.00 5.00 7.00 0.67 0.65 0.73 0.64
20 69 95 50 0 2 2 61 01 89 27
70 31 7 4.52 3.02 1.55 9.00 28 1.25 2.22 1.95 1.27
7 9 9 0 3 5 6 9
64.1 32.3 13.2 3.00 5.00 7.00 0.06 0.09 0.07 0.06
21 69 95 50 0 2 2 219 577 978 069
70 31 7 4.52 3.02 1.55 9.00 40 0.10 0.28 0.18 0.10
7 9 9 0 54 84 87 99
TABLE 1b/E.524
Exactly calculated mean and variance of individual overflow traffic
_ Two first choice circuit groups
A1 N1 A2 N2 N O1 V1 O2 V2
8.2 5 30.0 30 10 0.6155 1.1791 1.1393 3.4723
5 1.8068 3.2634 2.4656 7.4312
21 0.0188 0.0304 0.0485 0.1240
14 0.2108 0.3898 0.4624 1.3701
14.3 7 22 0.0470 0.0771 0.0929 0.1983
16 0.3743 0.6602 0.7546 1.7626
12 0.9282 1.6137 1.8320 4.2120
7 2.0023 3.2718 4.0953 7.8064
42.0 37 27 0.0230 0.0354 0.0978 0.2984
19 0.2136 0.3683 0.8356 2.9450
8 1.4984 2.6161 4.4363 14.601
8
13 0.6940 1.2375 2.4148 8.4923
30.0 30 14.3 7 25 0.0653 0.1613 0.0541 0.1112
18 0.4664 1.2990 0.4662 1.0879
12 1.3746 3.9321 1.7390 4.0015
7 2.4255 6.9941 3.8063 7.6277
8.2 5 67.9 65 30 0.0160 0.0242 0.0979 0.3548
20 0.1839 0.3141 0.9739 4.1953
14 0.5385 0.9676 2.4438 10.720
8
8 1.3598 1.4401 4.7035 19.710
9
51.5 54 14.3 7 27 0.0735 0.2239 0.0399 0.0802
19 0.6404 1.2499 0.4699 1.1030
13 1.4033 5.0795 1.3609 3.2229
7 2.5873 9.6136 3.6744 7.5139
TABLE 1c/E.524
Exactly calculated mean and variance of individual overflow traffic
_ One first choice circuit group
A1 N1 A0 N O1 V1 O0 V0
8.2 5 4.0 16 0.0499 0.0872 0.0331 0.0479
11 0.4859 0.9154 0.3494 0.5382
9 1.1692 2.1202 0.9011 1.3274
5 2.1422 3.5883 1.8018 2.3694
30.0 30 20 0.0601 0.1565 0.0167 0.023
13 0.5804 1.7427 0.1990 0.3062
9 1.3997 4.2546 0.5988 0.9338
5 2.5579 5.6196 1.5661 2.1991
51.5 54 22 0.9751 0.2497 0.0144 0.0197
15 0.5141 1.8924 0.1209 0.1819
10 1.8820 5.3004 0.4297 0.6790
5 2.4294 3.2974 1.1450 1.7255
4 Summary of results
The available methods and the performance measures with
respect to the criteria are listed in Table 2/E.524.
TABLE 2/E.524
Comparison of different approximation methods
Functi Input Outpu Comparison
ons t
Highe Overflow traffic error Computational
st effort
Requi momen
red ts of Mean Variance
highe
r
momen overf Memor Progr
ts low Mean Stan Mean Stan Proce y am-
Method traff dard dard ssor requi ming
ic devi devi time re- effor
atio atio ments t
n n
IPP
method
a) 3 3 3 _ 0.05 _ 0.09
moment 0.00 85 0.02 22
match 45 10
b) 4 8 0.00 0.02 _ 0.03
moment 08 55 0.00 73
ratio 53
EC 1 1 _ 0.15
method 0.06 27
61
AWW 2 2 _ 0.16
method 0.04 47
48
References
[1] MATSUMOTO (J.) and WATANABE (Y.): Analysis of individual
traffic characteristics for queuing systems with multiple
Poisson and overflow inputs. Proc. 10th ITC, paper 5.3.1,
Montreal, 1983.
[2] RENEBY (L.): On individual and overall losses in overflow
systems. Proc. 10th ITC, paper 5.3.5, Montreal, 1983.
[3] LE GALL (P.): Overflow traffic combination and cluster
engineering. Proc. 11th ITC, paper 2.2B_1, Kyoto, 1985.
[4] LINDBERG (P.), NIVERT, (K.), SAGERHOLM, (B.): Economy and
service aspects of different designs of alternate routing
networks. Proc. 11th ITC, Kyoto, 1985.
[5] KATZ (S.): Statistical performance analysis of a switched
communications network. Proc. 5th ITC, New York, 1967.
[6] LINDBERGER (K.): Simple approximations of overflow system
quantities for additional demands in the optimization. Proc.
10th ITC, Montreal, 1983.