Statistical Model for Link Lengths in Optical Transport Networks

Sudhir K. Routray; Rui M. Morais; José R. Ferreira da Rocha; Armando N. Pinto

doi:10.1364/JOCN.5.000762

Journal of Optical Communications and Networking
Vol. 5,
Issue 7,
pp. 762-773
(2013)
•https://doi.org/10.1364/JOCN.5.000762

Statistical Model for Link Lengths in Optical Transport Networks

Sudhir K. Routray, Rui M. Morais, José R. Ferreira da Rocha, and Armando N. Pinto

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Sudhir K. Routray, Rui M. Morais, José R. Ferreira da Rocha, and Armando N. Pinto, "Statistical Model for Link Lengths in Optical Transport Networks," J. Opt. Commun. Netw. 5, 762-773 (2013)

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Abstract

From the analysis of statistical properties of real networks, it is found that general extreme value (GEV) distribution provides an accurate model for link length statistics of optical transport networks (OTNs). The parameters of GEV distribution can be estimated from the average link length of the OTNs. Expressions for average link lengths, based only on the knowledge of network coverage area and number of nodes, are improved for better accuracy. It is shown that the optimized GEV distribution estimates the link statistics of OTNs with good accuracy (Kolmogorov–Smirnov statistic less than 0.18).

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No.	Distribution	Avg. KSS	L. KSS	H. KSS
1	GEV	0.1061	0.0360	0.1785
2	Log-logistic (3P)	0.1096	0.0476	0.2154
3	Pearson 5 (3P)	0.1098	0.0411	0.1877
4	Log-normal (3P)	0.1123	0.0411	0.1997
5	Log Pearson 3	0.1133	0.0469	0.2101
6	Frechet (3P)	0.1145	0.0413	0.2034
7	Inverse Gaussian (3P)	0.1159	0.0403	0.2276
8	General Pareto	0.1164	0.0610	0.2107
9	Burr	0.1181	0.0337	0.3539
10	Pearson 6	0.1226	0.0419	0.2246
11	Weibull (3P)	0.1251	0.0531	0.2581
12	Weibull	0.1318	0.0429	0.2567
13	Gamma	0.1324	0.0389	0.4169
14	Log-gamma	0.1359	0.0731	0.2461
15	Log-logistic	0.1392	0.0709	0.2815
(3P) indicates the three-parameter version of the distribution

No.	Network	$N$	$L$	$〈 δ 〉$	$〈 l 〉$	$σ$	${KSS}_{G}$
1	VIA network [3]	9	12	2.67	571	338	0.1744
2	BREN [3]	10	11	2.20	94	33	0.1785
3	RNP [3]	10	12	2.40	748	517	0.1709
4	Abilene core [14]	10	13	2.60	1067	516	0.1266
5	LEARN [12]	10	12	2.40	189	111	0.1476
6	CompuServe [13]	11	14	2.55	1161	859	0.1543
7	vBNS [3]	12	17	2.83	965	563	0.1005
8	CESNET [3]	12	19	3.17	91	31	0.1425
9	NSFNET [3]	14	21	3.00	1086	707	0.1014
10	ITALY [3]	14	29	4.14	280	146	0.0783
11	ACONET [3]	15	22	2.93	119	111	0.1397
12	MZIMA [3]	15	19	2.53	852	430	0.0896
13	GARR-B [3]	16	27	3.37	224	140	0.1354
14	ARNES [3]	17	20	3.25	38	19	0.1388
15	GERMANY [3]	17	26	3.06	143	78	0.1057
16	REDIRIS [3]	17	28	3.29	319	131	0.0820
17	LambdaRail [3]	19	23	2.42	671	378	0.0901
18	MEMOREX [3]	19	24	2.53	137	68	0.0865
19	CANARIE [3]	19	26	2.74	668	632	0.0685
20	EON [3]	19	37	3.89	754	356	0.1043
21	ARPANET[3]	20	32	3.20	839	491	0.0739
22	OPTOSunet [16]	20	24	2.40	100	34	0.1219
23	Hibernia USA [17]	20	27	2.70	279	207	0.0823
24	PIONIER [3]	21	25	2.38	131	55	0.1418
25	COX [3]	24	40	3.33	662	428	0.0673
26	SANET [3]	25	28	2.24	36	14	0.1553
27	NEWNET [3]	26	31	2.38	528	268	0.1183
28	PORTUGAL [3]	26	36	2.77	203	325	0.1593
29	RENATER [3]	27	35	2.59	155	66	0.0790
30	IBN31 [18]	31	51	3.29	131	59	0.0914
31	BULGARIA [19]	32	33	2.06	51	21	0.1039
32	GEANT2 [3]	32	52	3.25	661	455	0.0716
33	LONI [3]	33	37	2.24	62	15	0.0886
34	METRONA [3]	33	41	2.48	73	40	0.0798
35	COST37 [20]	37	57	3.08	439	248	0.0840
36	CERNET [21]	37	53	2.86	636	503	0.0698
37	OMNICOM [3]	38	54	2.84	298	153	0.0796
38	INTERNET2 [3]	56	61	2.18	334	186	0.0694
39	CORONET [22]	75	99	2.64	326	265	0.0546
40	USA100 [1]	100	171	3.42	310	174	0.0360
Average:					410.78		0.1061
Standard deviation:					332.83		–

No.	Network	$A_{c}$	$A_{e}$	$A_{g}$
1	VIA network [3]	1,267,876	824,464	1,915,456
2	BREN [3]	42,196	33,780	110,883
3	RNP [3]	1,814,676	1,077,222	8,514,965
4	Abilene core [14]	5,698,507	3,728,429	8,080,464
5	LEARN [12]	183,296	148,137	696,241
6	CompuServe [13]	5,442,148	4,661,688	8,080,464
7	vBNS [3]	6,279,993	4,182,861	8,080,464
8	CESNET [3]	45,957	32,357	78,909
9	NSFNET [3]	6,007,605	5,005,030	8,080,464
10	ITALY [3]	436,577	291,715	301,230
11	ACONET [3]	52,826	39,297	83,858
12	MZIMA [3]	6,805,153	4,278,891	8,080,464
13	GARR-B [3]	476,100	242,064	301,230
14	ARNES [3]	12,645	7112	20,273
15	GERMANY [3]	196,675	157,319	357,021
16	REDIRIS [3]	463,539	367,094	504,882
17	LambdaRail [3]	6,899,337	4,393,019	8,080,464
18	MEMOREX [3]	253,850	141,524	981,834
19	CANARIE [3]	3,462,688	2,095,793	9,985,140
20	EON [3]	5,800,056	2,524,156	6,550,943
21	ARPANET [3]	5,975,012	5,640,345	8,080,464
22	OPTOSunet [16]	122,980	81,063	449,964
23	Hibernia USA [17]	1,257,762	531,658	2,697,216
24	PIONIER [3]	194,178	163,888	312,685
25	COX [3]	4,870,065	3,531,799	8,080,464
26	SANET [3]	24,336	18,496	48,845
27	NEWNET [3]	7,033,452	4,200,490	8,080,464
28	PORTUGAL [3]	888,824	612,952	92,391
29	RENATER [3]	462,096	320,900	642,346
30	IBN31 [18]	426,658	200,382	300,448
31	BULGARIA [19]	68,008	52,069	111,910
32	GEANT2 [3]	7,703,318	3,383,604	770,326
33	LONI [3]	68,008	52,069	129,578
34	METRONA [3]	158,837	75,727	244,820
35	COST37 [20]	5,371,917	3,329,573	6,228,201
36	CERNET [21]	6,719,547	2,628,942	9,644,552
37	OMNICOM [3]	2,596,279	2,091,020	2,646,447
38	INTERNET2 [3]	7,169,590	5,974,161	8,080,464
39	CORONET [22]	7,903,482	6,046,394	8,080,464
40	USA100 [1]	8,191,044	5,992,704	8,080,464

No.	Network	$〈 l_{c} 〉$	$〈 l_{e} 〉$	$〈 l_{g} 〉$	$〈 l 〉$	$E_{c} (%)$	$E_{e} (%)$	$E_{g} (%)$	Diff.1	Diff.2	Best Area
1	VIA network [3]	563	454	692	571	1.40	20.49	$- 21.19$	19.09	19.79	$A_{c}$
2	BREN [3]	95	85	154	94	$- 1.06$	9.57	$- 63.83$	7.86	62.77	$A_{c}$
3	RNP [3]	623	480	1350	748	16.71	35.83	$- 80.48$	19.12	63.77	$A_{c}$
4	Abilene core [14]	1104	893	1315	1067	$- 3.47$	16.31	$- 23.24$	12.84	19.77	$A_{c}$
5	LEARN [12]	198	178	386	189	$- 4.76$	5.82	$- 104.23$	1.06	99.47	$A_{c}$
6	CompuServe [13]	1007	932	1227	1161	13.26	19.72	$- 5.68$	6.46	$- 7.58$	$A_{g}$
7	vBNS [3]	1017	830	1154	965	$- 5.39$	13.99	$- 19.59$	8.60	14.20	$A_{c}$
8	CESNET [3]	87	73	114	91	4.40	19.78	$- 25.27$	15.38	20.87	$A_{c}$
9	NSFNET [3]	894	816	1037	1086	17.68	24.86	4.51	7.18	$- 13.17$	$A_{g}$
10	ITALY [3]	241	197	200	280	13.93	29.64	28.57	15.71	14.64	$A_{c}$
11	ACONET [3]	80	69	101	119	32.77	42.02	15.13	9.25	$- 17.64$	$A_{g}$
12	MZIMA [3]	908	720	989	852	$- 6.57$	15.49	$- 16.08$	8.92	9.51	$A_{c}$
13	GARR-B [3]	230	164	183	224	$- 2.68$	26.79	18.30	24.11	15.62	$A_{c}$
14	ARNES [3]	36	27	46	38	5.26	28.95	$- 21.05$	23.69	15.79	$A_{c}$
15	GERMANY [3]	142	127	191	143	0.70	11.19	$- 33.57$	10.49	32.87	$A_{c}$
16	REDIRIS [3]	218	194	228	319	31.66	39.18	28.53	7.52	$- 3.13$	$A_{g}$
17	LambdaRail [3]	782	624	846	671	$- 16.54$	7.00	$- 26.08$	$- 9.54$	9.54	$A_{e}$
18	MEMOREX [3]	150	112	295	137	$- 9.49$	18.25	$- 115.33$	8.76	105.84	$A_{c}$
19	CANARIE [3]	554	431	941	668	17.07	35.48	$- 40.87$	18.41	23.80	$A_{c}$
20	EON [3]	717	473	762	754	4.91	37.27	$- 1.06$	32.36	$- 3.85$	$A_{g}$
21	ARPANET [3]	704	684	819	839	16.09	18.47	2.38	2.38	$- 13.71$	$A_{g}$
22	OPTOSunet [16]	101	82	193	100	$- 1.00$	18.00	$- 93.00$	17.00	92.00	$A_{c}$
23	Hibernia USA [17]	323	210	473	279	$- 15.77$	24.73	$- 69.53$	8.96	53.76	$A_{c}$
24	PIONIER [3]	123	113	156	131	6.11	13.74	$- 19.08$	7.63	12.97	$A_{c}$
25	COX [3]	566	482	729	662	14.50	27.19	$- 10.12$	12.69	$- 4.38$	$A_{g}$
26	SANET [3]	39	34	55	36	$- 8.33$	5.56	$- 52.78$	$- 2.77$	44.45	$A_{e}$
27	NEWNET [3]	647	500	693	528	$- 22.54$	5.30	$- 31.25$	$- 17.24$	8.71	$A_{e}$
28	PORTUGAL [3]	230	191	74	203	$- 13.30$	5.91	63.55	$- 7.39$	50.25	$A_{e}$
29	RENATER [3]	162	135	191	155	$- 4.52$	12.90	$- 23.23$	8.38	18.71	$A_{c}$
30	IBN31 [18]	143	98	120	131	$- 9.16$	25.19	8.40	16.03	$- 0.76$	$A_{g}$
31	BULGARIA [19]	65	57	71	51	$- 27.45$	$- 11.76$	$- 39.22$	$- 15.69$	11.77	$A_{e}$
32	GEANT2 [3]	596	395	596	661	9.83	40.24	9.83	30.41	0.00	$A_{g} = A_{c}$
33	LONI [3]	56	49	77	62	9.68	20.97	$- 24.19$	11.29	14.51	$A_{c}$
34	METRONA [3]	84	58	104	73	$- 15.07$	20.55	$- 42.47$	5.48	27.40	$A_{c}$
35	COST37 [20]	456	359	491	439	$- 3.87$	18.22	$- 11.85$	14.35	7.98	$A_{c}$
36	CERNET [21]	510	319	611	636	19.81	49.84	3.93	30.03	$- 15.83$	$A_{g}$
37	OMNICOM [3]	312	280	315	298	$- 4.70$	6.04	$- 5.70$	1.34	1.00	$A_{c}$
38	INTERNET2 [3]	413	377	438	334	$- 23.65$	$- 12.87$	$- 31.14$	$- 10.78$	7.49	$A_{e}$
39	CORONET [22]	367	321	371	326	$- 12.58$	1.53	$- 13.80$	$- 11.05$	1.22	$A_{e}$
40	USA100 [1]	318	272	316	310	$- 2.58$	12.26	$- 1.94$	9.68	$- 0.64$	$A_{g}$
Average:		396.53	322.38	477.60	410.78	0.53	18.99	$- 22.09$	8.97	19.99	$A_{c}$
Absolute average:		396.53	322.38	477.60	410.78	11.26	20.22	31.25	12.69	24.03	$A_{c}$

No.	Network	$E_{c}^{'} (%)$ ( $k_{c} = 0.97$ )	$E_{e}^{'} (%)$ ( $k_{e} = 1.22$ )	Diff.	Best Area	$〈 l_{c}^{'} 〉$	$α_{c}^{'}$	$β_{c}^{'}$	$ξ_{c}^{'}$	${KSS}_{GE}$	Acceptable? ( $CI = 0.95$ )
1	VIA network [3]	4.38	2.98	$- 1.40$	$A_{e}$	546	368	228	0.167	0.2692	Yes
2	BREN [3]	2.13	$- 10.64$	8.51	$A_{c}$	92	69	28	0.167	0.3084	Yes
3	RNP [3]	19.25	$- 21.66$	2.41	$A_{c}$	604	406	254	0.167	0.2231	Yes
4	Abilene core [14]	$- 0.37$	$- 2.06$	1.69	$A_{c}$	1071	713	460	0.167	0.1351	Yes
5	LEARN [12]	$- 1.59$	$- 14.81$	13.22	$A_{c}$	192	135	72	0.167	0.1243	Yes
6	CompuServe [13]	15.85	2.07	$- 13.78$	$A_{e}$	977	651	418	0.167	0.1872	Yes
7	vBNS [3]	$- 2.18$	$- 4.97$	2.79	$A_{c}$	986	657	422	0.167	0.1438	Yes
8	CESNET [3]	7.69	2.20	$- 5.49$	$A_{e}$	84	64	25	0.167	0.3254	No
9	NSFNET [3]	20.17	8.29	$- 11.88$	$A_{e}$	867	579	370	0.167	0.1849	Yes
10	ITALY [3]	16.43	$- 14.29$	$- 2.14$	$A_{e}$	234	163	91	0.167	0.1951	Yes
11	ACONET [3]	34.45	$- 29.41$	$- 5.04$	$A_{e}$	78	60	22	0.167	0.4421	No
12	MZIMA [3]	$- 3.40$	$- 3.05$	$- 0.35$	$A_{e}$	881	588	376	0.167	0.1468	Yes
13	GARR-B [3]	0.45	10.71	10.26	$A_{c}$	223	155	86	0.167	0.1551	Yes
14	ARNES [3]	7.89	$- 13.16$	5.27	$A_{c}$	35	32	3	0.167	0.5283	No
15	GERMANY [3]	3.50	$- 8.39$	4.89	$A_{c}$	138	99	48	0.167	0.1731	Yes
16	REDIRIS [3]	33.86	25.71	$- 8.15$	$A_{e}$	211	147	81	0.167	0.3782	No
17	LambdaRail [3]	$- 13.11$	$- 13.41$	0.30	$A_{c}$	759	508	322	0.167	0.1235	Yes
18	MEMOREX [3]	$- 6.57$	0.00	$- 6.57$	$A_{e}$	146	105	52	0.167	0.1203	Yes
19	CANARIE [3]	19.61	21.26	1.65	$A_{c}$	537	362	224	0.167	0.1170	Yes
20	EON [3]	7.82	23.47	15.65	$A_{c}$	695	466	294	0.167	0.1223	Yes
21	ARPANET [3]	18.59	0.60	$- 17.99$	$A_{e}$	683	458	289	0.167	0.2062	Yes
22	OPTOSunet [16]	2.00	0.00	$- 2.00$	$A_{e}$	98	73	31	0.167	0.2619	Yes
23	Hibernia USA [17]	$- 12.19$	8.24	$- 3.95$	$A_{e}$	313	215	126	0.167	0.1756	Yes
24	PIONIER [3]	9.16	$- 5.34$	$- 3.82$	$A_{e}$	119	87	40	0.167	0.2047	Yes
25	COX [3]	17.07	11.18	$- 5.89$	$A_{e}$	549	370	230	0.167	0.0956	Yes
26	SANET [3]	$- 5.56$	$- 13.89$	8.33	$A_{c}$	38	34	4	0.167	0.3885	No
27	NEWNET [3]	$- 18.94$	$- 15.53$	$- 3.41$	$A_{e}$	628	422	265	0.167	0.1178	Yes
28	PORTUGAL [3]	$- 9.85$	$- 14.78$	4.93	$A_{c}$	223	155	86	0.167	0.6653	No
29	RENATER [3]	$- 1.29$	$- 6.45$	5.16	$A_{c}$	157	112	57	0.167	0.1488	Yes
30	IBN31 [18]	$- 6.11$	8.40	2.29	$A_{c}$	139	100	49	0.167	0.1067	Yes
31	BULGARIA [19]	$- 23.53$	$- 37.25$	13.72	$A_{c}$	62	49	15	0.167	0.2929	No
32	GEANT2 [3]	12.56	27.08	14.52	$A_{c}$	578	389	243	0.167	0.1053	Yes
33	LONI [3]	12.90	3.23	$- 9.67$	$A_{e}$	54	44	11	0.167	0.0902	Yes
34	METRONA [3]	$- 10.96$	2.74	$- 8.22$	$A_{e}$	81	62	23	0.167	0.2278	Yes
35	COST37 [20]	$- 0.68$	0.23	$- 0.45$	$A_{e}$	442	299	183	0.167	0.1447	Yes
36	CERNET [21]	22.17	38.84	16.67	$A_{c}$	495	334	206	0.167	0.1726	Yes
37	OMNICOM [3]	$- 1.68$	$- 14.77$	13.09	$A_{c}$	303	208	122	0.167	0.1411	Yes
38	INTERNET2 [3]	$- 20.06$	$- 37.72$	17.66	$A_{c}$	401	272	164	0.167	0.1615	Yes
39	CORONET [22]	$- 9.20$	$- 20.25$	11.05	$A_{c}$	356	243	145	0.167	0.1221	Yes
40	USA100 [1]	0.65	$- 7.10$	6.45	$A_{c}$	308	212	123	0.167	0.0707	Yes
Average:		$- 3.53$	1.13	1.76	$A_{c}$	384.70	–	–	–	0.2076	Yes
Absolute average:		10.90	12.65	7.27	$A_{c}$	384.70	–	–	–	–	Yes
Standard deviation:		13.72	16.70	8.86	–	303.25	–	–	–	–	–

Abstract

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Figures (12)

Tables (5)

Equations (23)

Journal of Optical Communications and Networking