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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 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) 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 [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 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.
2.3.1	Blocking 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)
2.3.2	Wallström 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)
2.3.3	Handling 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	
1	7.036	26.688	64.169	3.003	3.001	3.000			–	0.4337	0.7490	1.091	
	5	28	70	1.573	3.022	4.527	–	11	–	0.7656	2.110	4.441	
2	7.036	26.688	64.169	3.003	3.001	3.000			–	0.1149	0.2758	0.4944	
	5	28	70	1.573	3.022	4.527	–	16	–	0.2436	0.7328	1.911	
3	7.036	26.688	64.169	3.003	3.001	3.000			–	0.01369	0.02846	0.06627	
	5	28	70	1.573	3.022	4.527	–	25	–	0.02041	0.06461	0.2205	
4	7.036	10.176	13.250	3.003	5.003	7.002			–	0.7459	1.262	1.785	
	5	6	7	1.573	1.567	1.559	–	14	–	1.193	2.292	3.624	
5	7.036	10.176	13.250	3.003	5.003	7.002			–	0.2884	0.4857	0.6832	
	5	6	7	1.573	1.567	1.559	–	19	–	0.4636	0.9089	1.460	
6	7.036	10.176	13.250	3.003	5.003	7.002			–	0.03570	0.05915	0.08237	
	5	6	7	1.573	1.567	1.559	–	26	–	0.05358	0.1026	0.1621	
7	7.036	32.395	77.617	3.003	5.002	7.001			–	0.4516	1.176	2.344	
	5	31	77	1.573	3.029	4.511	–	16	–	0.7434	3.466	10.39	
8	7.036	32.395	77.617	3.003	5.002	7.001			–	0.1538	0.4294	0.9739	
	5	31	77	1.573	3.029	4.511	–	23	–	0.2427	1.200	4.219	
9	7.036	32.395	77.617	3.003	5.002	7.001			–	0.01303	0.03984	0.1006	
	5	31	77	1.573	3.029	4.511	–	35	–	0.1841	0.09378	0.3690	
10	64.169	32.395	13.250	3.000	5.002	7.002			–	1.157	1.456	1.320	
	70	31	7	4.527	3.029	1.559	–	15	–	4.442	4.256	2.850	
11	64.169	32.395	13.250	3.000	5.002	7.002			–	0.5564	0.5849	0.4749	
	70	31	7	4.527	3.029	1.559	–	21	–	2.026	1.675	1.023	
12	64.169	32.395	13.250	3.000	5.002	7.002			–	0.06907	0.05265	0.03848	
	70	31	7	4.527	3.029	1.559	–	32	–	0.2167	0.1295	0.07165	
13	7.036	26.688	64.169	3.003	3.001	3.000			0.4064	0.5038	0.8274	1.160	
	5	28	70	1.573	3.022	4.527	3.000	13	0.5578	0.8566	2.243	4.574	
14	7.036	26.688	64.169	3.003	3.001	3.000			0.1460	0.1840	0.3384	0.5729	
	5	28	70	1.573	3.022	4.527	3.000	18	0.1992	0.3043	0.8779	2.163	

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	
15	7.036	26.688	64.169	3.003	3.001	3.000			0.01170	0.01506	0.03086	0.07035	
	5	28	70	1.573	3.022	4.527	3.000	28	0.01472	0.02218	0.06861	0.2287	
16	7.036	32.395	77.617	3.003	5.002	7.001			0.1253	0.4451	1.156	2.304	
	5	31	77	1.573	3.029	4.511	1.000	17	0.1392	0.7266	3.366	10.10	
17	7.036	32.395	77.617	3.003	5.002	7.001			0.04250	0.1536	0.4275	0.9674	
	5	31	77	1.573	3.029	4.511	1.000	24	0.04696	0.2409	1.183	4.148	
18	7.036	32.395	77.617	3.003	5.002	7.001			0.004542	0.01687	0.05106	0.1282	
	5	31	77	1.573	3.029	4.511	1.000	35	0.004891	0.02398	0.1214	0.4751	
19	64.169	32.395	13.250	3.000	5.002	7.002			1.761	1.251	1.654	1.630	
	70	31	7	4.527	3.029	1.559	9.000	21	3.052	4.517	4.406	3.103	
20	64.169	32.395	13.250	3.000	5.002	7.002			0.6761	0.6501	0.7389	0.6427	
	70	31	7	4.527	3.029	1.559	9.000	28	1.253	2.225	1.956	1.279	
21	64.169	32.395	13.250	3.000	5.002	7.002			0.06219	0.09577	0.07978	0.06069	
	70	31	7	4.527	3.029	1.559	9.000	40	0.1054	0.2884	0.1887	0.1099	

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	14.6018	
				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	10.7208	
				8	1.3598	1.4401	4.7035	19.7109	
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
Functions	Input	Output	Comparison	
		Highest	Overflow traffic error	Computational effort	
	Required higher	moments of 	Mean	Variance		
Method	moments	overflow traffic	Mean	Standard deviation	Mean	Standard deviation	Processor time	Memory require-ments	Program-ming effort	
IPP method										
a) 3 moment match	3	3	– 0.0045	0.0585	– 0.0210	0.0922				
b) 4 moment ratio	8		0.0008	0.0255	– 0.0053	0.0373				
EC method	1	1	– 0.0661	0.1527						
AWW method	2	2	– 0.0448	0.1647						
	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.