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Fascicle II.3 _ Rec. E.524 5 All drawings appearing in this Recommendation have been done in Autocad. 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 Wallström (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 Wallström) method (Approximative Wilkinson Wallström) 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 + gß) where g = (2.36 Z _ 2.17) log {1 + (z _ 1)/[M(Z + 1.5)]} and ß = Z/(1.5M + 2Z _1.3) (2_6) 2.3.2Wallström formula for individual blocking There has been much interest in finding a simple and accurate formula for the individual blocked traffic m£i. Already in 1967, Katz [5] proposed a formula of the type (2_7) with w being a suitable expression. Wallström 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 Wallström'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.