+ All Categories
Home > Documents > Maximizing the Robustness of TDMA Networks with ... · Maximizing the Robustness of TDMA Networks...

Maximizing the Robustness of TDMA Networks with ... · Maximizing the Robustness of TDMA Networks...

Date post: 13-Aug-2020
Category:
Upload: others
View: 1 times
Download: 0 times
Share this document with a friend
35
Maximizing the Robustness of TDMA Networks with Applications to TTP/C Bruno Gaujal, Nicolas Navet To cite this version: Bruno Gaujal, Nicolas Navet. Maximizing the Robustness of TDMA Networks with Applica- tions to TTP/C. [Research Report] RR-4614, INRIA. 2002. <inria-00071971> HAL Id: inria-00071971 https://hal.inria.fr/inria-00071971 Submitted on 23 May 2006 HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destin´ ee au d´ epˆ ot et ` a la diffusion de documents scientifiques de niveau recherche, publi´ es ou non, ´ emanant des ´ etablissements d’enseignement et de recherche fran¸cais ou ´ etrangers, des laboratoires publics ou priv´ es.
Transcript
Page 1: Maximizing the Robustness of TDMA Networks with ... · Maximizing the Robustness of TDMA Networks with Applications to TTP/C Bruno Gaujal, Nicolas Navet To cite this version: Bruno

Maximizing the Robustness of TDMA Networks with

Applications to TTP/C

Bruno Gaujal, Nicolas Navet

To cite this version:

Bruno Gaujal, Nicolas Navet. Maximizing the Robustness of TDMA Networks with Applica-tions to TTP/C. [Research Report] RR-4614, INRIA. 2002. <inria-00071971>

HAL Id: inria-00071971

https://hal.inria.fr/inria-00071971

Submitted on 23 May 2006

HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, estdestinee au depot et a la diffusion de documentsscientifiques de niveau recherche, publies ou non,emanant des etablissements d’enseignement et derecherche francais ou etrangers, des laboratoirespublics ou prives.

Page 2: Maximizing the Robustness of TDMA Networks with ... · Maximizing the Robustness of TDMA Networks with Applications to TTP/C Bruno Gaujal, Nicolas Navet To cite this version: Bruno

ISS

N 0

249-

6399

ISR

N IN

RIA

/RR

--46

14--

FR

+E

NG

ap por t de r ech er ch e

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

Maximizing the Robustness of TDMA Networkswith Applications to TTP/C

Bruno Gaujal , Nicolas Navet

No 4614

November 7, 2002

THEME 1

Page 3: Maximizing the Robustness of TDMA Networks with ... · Maximizing the Robustness of TDMA Networks with Applications to TTP/C Bruno Gaujal, Nicolas Navet To cite this version: Bruno
Page 4: Maximizing the Robustness of TDMA Networks with ... · Maximizing the Robustness of TDMA Networks with Applications to TTP/C Bruno Gaujal, Nicolas Navet To cite this version: Bruno

�������������� �������������������� ���� �!�#"%$'&(��) *+���-,.��/102�!,3�4���)�5�5�6���78���1���# ��3���9$:$:;=<#>

?A@CBEDEFHGJI-BLKMIONQPSRUT�VXWYFZN[IO\2T�I^]O_a`�bc�d�egfShJikjmlknpo�hrqYsZtuhgvwo�xOoyv�egfShpo

z�{�|Y}4hgvkc�lw~y�lkqY���E|�{yv#��h2{�hp��d�hg{���d�h������L�Zig��j��k|���hgf��Ehg{��O���^�^�^��j��Zi�� qY��hpo

�����p�r�a�� ��^¡ ~¢��vyd�£¤o¥oyvys��Zx�¦§h#oyd�|�¦'d�|r¦.|���hw�rqY�¨s�o�h�© qYs�ª«v4¬­c®|�ª¤hg{�qY�-v�¯w��£«v�o#°±©Qc�¯2²8£«�JqY�|���vy£«fJqYªE¦�qpxuv�|�fJqY³�h2q�c�´�µ�¶3��hgv­¦§|�{y³u{�|���s�oyv�v�|���s�{�oyv­x�{�qY����|�f·�1hg{yvys�{y� qYvy£¤|���or¸¥¹ºh�p|���o�£¤��hg{kv¢¦§|u�1|^o�oy£«��ª¤h�|��Z}yhp�gvy£«��hpor¸A~¼»½|���h�¦¾qY�-v�o�v�|Jf¨£«��£«f¨£«¿rh�vyd�h���{�|�� qY��£«ª«£«v­x�|Y»¥ª¤|^o�£«���qYª«ªk{�hg��ª«£¤�rq�oÀ|Y»�q%�^£«��hg�.fShpo�o�qY��h^��vyd�hg�.vyd�hH|���vy£«fJqYªA�1|�ª«£¤�gx'£¤o�v�|Áoy��{�hrq��Âvyd�hÃ{�hg��ª«£¤�rq�o|���hg{Jvy£«fSh^¸!c�d�£¤oJ£¤oS��{�|r��hp�Âs�oy£«���Ä�p|��L��hMtZ£«v­x:��{�|��Ehg{yvy£¤hpo�|Y»2vyd�hŪ¤|^o�oS��{�|�� qY��£«ª«£«v­x�¸��k�vyd�h��p|��Lvy{�qY{yx�£Æ»§|���h�¦¾qY�Lv�o�v�|�f¨£«��£«f¨£«¿rh�vyd�h���{�|�� qY��£«ª«£«v­x�|Y»§ª¤|^oy£«���ÀqYvAª¤hrq�oyv�|���h�{�hg��ª«£¤�rqZ�vyd�hg�Çvyd�h�|���vy£«fJqYª¥o�|�ª«s�vy£¤|��Ç£¤owv�|u�^{�|�s��ºqYª«ªQ{�hg��ª«£¤�rq�oAv�|���hgvyd�hg{r¸�c�d�£¤oA£¤ok��{�|���hp�Ç�LxÃs�o�£«���fJqa}y|�{y£«¿�qYvy£¤|��Äv�hp��d���£¤È-s�hpor¸Ã©�£«� qYª«ª«x%¦§h�oyd�|r¦+d�|r¦�vyd�hpo�hS£¤��hrq�o��rqY�Á�Eh�q���qY��v�hp�Á»±|�{�vyd�hc�c�z¾É^Ê=��{�|�v�|-�p|�ª¼¸ËÃÌ�ÍEÎ�Ï�Ð �aÑ��^¡ lwhrqYªÆ¬­c�£«fShºÒ-xOo�v�hgfSor�#© qYs�ª«vÀc®|�ª¤hg{�qY���ph^�Ac�´�µ�¶��½lkhg��ª«£¤�rqZ�w~¢�Z¬¼Ó¥hgd�£¤�gª¤h�Ahgv­¦¾|�{y³1� c�c�z§É^ÊA¸

ÔÆÕ®Ö¢×¼ØrÙ#Ö§Ú®Û«×¼ÜÞÝpß

à�á âQãkä�åpæMçwè1äLé±êZë�ìgí�î®ïðïòñ4ó¥ô�õ éXö­÷CïòøòóMëYíMùCúMúgû�ä�åpæMçLëYü�ý­÷CçYþÿó���á � ÷�øðï�����ý��açYæ� Q÷�� �Þ÷�ï���ó4ç��±è ï åpæMç�� � ý� ä����®é±î¥ëOá�ç��Þó � ë��A÷��gó4ç��aó�ôaó�ïð÷2üYæ�ý��ÿö§ôaó�ïð÷ �Q÷4åpóCë�!�ì"!�#4í%$�÷CçYôaæpó&��pý¼óMë�üaý­÷�çYþ4ó �Aá � ÷Cøòï��§â�øòþ4æCèï ÷��&� âQ÷��góÿö'��ïòæCý¼øð÷�� � ý

Unite de recherche INRIA LorraineTechnopole de Nancy-Brabois, Campus scientifique,

615 rue de Jardin Botanique, BP 101, 54600 VILLERS L ES NANCY (France)Telephone : 03 83 59 30 30 - International : +33 3 3 83 59 30 30

Telecopie : 03 83 27 83 19 - International : +33 3 83 27 83 19Antenne de Metz, technopole de Metz 2000, 4 rue Marconi, 55070 METZ

Telephone : 03 87 20 35 00 - International: +33 3 87 20 35 00Telecopie : 03 87 76 39 77 - International : +33 3 87 76 39 77

Page 5: Maximizing the Robustness of TDMA Networks with ... · Maximizing the Robustness of TDMA Networks with Applications to TTP/C Bruno Gaujal, Nicolas Navet To cite this version: Bruno

������������ �������# ����+6��/1�#������������� ������/���� ���#��� $'&(��) ���#��7��5�5�6���78���1���# ��9$:$';=<#>

� � ��� ¡ ´�qY��o¾�phgvyv�hwngvys���h^�-��|�s�o§fS|��Lvy{�|���o��p|�f¨fShg�Lv�s�vy£«ª«£¤o�hg{¾��h�»¼q��p|���|���vy£«fJqYª¤hA��hpos���£«v�npo�v�|�ª¤ng{�qY�-v�hpo�qYsZt�»ÞqYs�v�hpo¨°±© qYs�ª«v4¬­c8|�ª¤hg{�qY�Lv�¯w��£«v�oA¬#©Qc�¯�²w�E|�s�{2o�h���{�ngf�s���£«{��p|��Lvy{�h��hpo��1hg{yvys�{y� qYvy£¤|���o�hg�%{�qa»ÞqYª¤hpo�oys�{2s��º{�npo�hrqYsºc�´�µH¶�¸®´AhgsZt%|��Z}yhp�gvy£Æ»±o�o�|��Lv�ngvys��Z£¤npor¸ÀÒ-£ª�� |�����hgs�v§f¨£«��£«f¨£¤o�hg{§ª q���{�|�� qY��£«ª«£«v�nk��hw�1hg{��Z{�hkv�|�s�v�hpo½ª¤hpo§{�ng��ª«£¤È-s�hpo§��� s��uf��gfSh#fShpo�oCqY��h^�ª q��E|�ª«£«vy£¤È-s�h2|���vy£«fJqYª¤h�hpoyv#��h�{�ng� qY{yvy£«{#ª q�vy{�qY��oyf¨£¤o�oy£¤|�� ��hpo�{�ng��ª«£¤È-s�hpow��qY��o�ª¤h�v�hgf¨��or¸��8q��{�hgs���hAs�vy£«ª«£¤o�hA��hpo¾��{�|���{y£¤ngv�npo���h��p|��L��hMtZ£«v�nA��hkª q���{�|�� qY��£«ª«£«v�n���hk�Ehg{yv�h^¸�¶ksu�p|��-vy{�qY£«{�h^��oy£ª�� |��Á��hgs�v2f¨£«��£«f¨£¤o�hg{�ª qu��{�|�� qY��£«ª«£«v�n���h¨�1hg{��Z{�h�s���hS{�ng��ª«£¤ÈLs�hS|�s%��ª«s�o��8qYª¤|�{�o�ª qÀo�|�ª«s�vy£¤|��|���vy£«fJqYª¤h�hpoyv���h¨vy{�qY��o�fShgvyvy{�hJª¤hpo2{�ng��ª«£¤È-s�hpo�{�hg�^{�|�s���nphpo���qY��o�ª¤h¨v�hgf¨��or¸��®qÅ��{�hgs���h�hpoyv� q�o�nph�oys�{#s���h�v�hp��d���£¤È-s�h���h2fJqa}y|�{�qYvy£¤|��U¸�©�£«� qYª¤hgfShg�Lvr����|�s�okq���qY��v�|���oA�phpo#{�npoys�ª«v�qYv�oAqYs�rq�o#� qY{yvy£¤�gs�ª«£¤hg{k�ZsÅ��{�|�v�|O�p|�ª¤h�c�c�z§É^ÊA¸��Ð �r� Î  �� ¡ Ò-xZoyv�egfShpoAc®hgf¨��o�lwnphgª¼�1c®|�ª¤ng{�qY���ph¨qYsZt © qYs�v�hpor�Ec�´�µH¶��EÒ-v�qYvy£¤|��Hlkhp��|��Z¬��qY�-v�h^��lwnpo�hrqYs���f�� qY{�È-s�n^� c�c�z¾É^ÊA¸

Page 6: Maximizing the Robustness of TDMA Networks with ... · Maximizing the Robustness of TDMA Networks with Applications to TTP/C Bruno Gaujal, Nicolas Navet To cite this version: Bruno

�������������� ��������������������� ���� �!�#"%$'&(�*),+-����./�10 2�� �

3 4 k� /E� ����7®�����# 5 Ð�6 � Ì87 � Ð:9 �<; Ì �r� �®Ñ Í µ s�ª«vy£Æ¬¢q��p�phpo�o���{�|�v�|-�p|�ª¤o�� q�o�hp�Ä|��'c�´�µ�¶ °±c�£«fShÀ´A£«�O£¤oy£¤|��µ�s�ª«vy£«��ª¤h�¶��p�phpo�oC²kqY{�h�¦�£¤��hgª«x�s�o�hp� £«�Ç�p|�f¨f�s���£¤�rqYvy£¤|���o�oyxOo�v�hgfSor¸wc�´�µH¶ � q�o�hp����{�|�v�|Y¬�p|�ª¤o�qY{�hS� qY{yvy£¤�gs�ª qY{yª«xº¦¾hgª«ª¾o�s�£«v�hp�%v�|Ã{�hrqYªÆ¬¼vy£«fShÀqY����ª«£¤�rqYvy£¤|���o�oy£«���phSvyd�hgxÇ��{�|r�O£¤��hu��hgv�hg{4¬f¨£«��£¤o�vy£¤�uq��p�phpo�o�v�|Ãvyd�h�fShp�Z£«s�f qY���Ávyd-s�o��E|�s�����hp� {�hpoy�1|���o�hSvy£«fShpor¸�µ�|�{�hp|���hg{r�Qvyd�hg£«{{�hg�^s�ª qY{�fShpo�o�qY��hJvy{�qY��oyf¨£¤o�oy£¤|���o��rqY�Á�EhSs�o�hp� q�o>=­d�hrqY{yvy�1hrqYv�o#?¨»Þ|�{���hgv�hp�gvy£«���Ç��|O��h¨»¼qY£«ªÆ¬s�{�hpo�¸wc�d�hg{�h�hMtO£¤oyv�oAo�hg��hg{�qYª���qY{y£ qY�-v�o�|Y»¥vyd�h�c�´�µH¶ o���d�hgfSh^�1£«�Hvyd�£¤ow� qY�Ehg{k¦¾h��p|���o�£¤��hg{vyd�h�oyxO����d�{�|���|�s�o�c�´AµH¶�o���d�hgfSh¨q�o�q���|���v�hp�Ç�Lx�vyd�h¨c�c�z§É^Ê ��{�|�v�|-�p|�ªA@ò�^�CB­¸�c�d�h¨oyv�qa¬vy£¤|���o�d qp��hJq��p�phpo�o�v�|Àvyd�h���s�o�£«�Áq�o�vy{y£¤�gv���hgv�hg{yf¨£«��£¤oyvy£¤�Jo�hpÈ-s�hg�Lvy£ qYª§|�{���hg{r�Uhrq���dÁo�v�qYvy£¤|���E|^o�o�hpo�o�hpowvyd�h���s�oA»Þ|�{�qu�p|���oyv�qY�-v2�1hg{y£¤|O�Ç|Y»§vy£«fSh��rqYª«ª¤hp�ºqD�FEG�1�Q�Zs�{y£«���Ŧ�d�£¤��d%£«v�d q�o�v�|vy{�qY��o�f¨£«v#|���hA»[{�qYfSh^¸¥c�d�h2o�hpÈLs�hg���ph2|Y»Qoyª¤|�v�o�oys���dÅvyd qYvAqYª«ªEo�v�qYvy£¤|���o�d qp��h�q��p�phpo�o�|����ph2v�|vyd�h���s�ow£¤o#�rqYª«ª¤hp� qH0I���J LK^¸

c�d�h�s�o�h�|Y»�c�´�µ�¶�� q�o�hp� ��{�|�v�|-�p|�ª¤ow£¤ow�p|���oy£¤��hg{�hp� £«� d�£«�^dZ¬­��hg�Ehg����qY��£«ª«£«v­xÃ{�hrqYªÆ¬¼vy£«fShqY����ª«£¤�rqYvy£¤|���o½¦�d�hg{�h¾»ÞqYs�ª«vQv�|�ª¤hg{�qY���phwqY�����^s qY{�qY�-v�hphp�S{�hpoy�E|���o�h¾vy£«fShpoQd q�o�v�|A�Eh§��{�|��-£¤��hp� ¸��t�qYf¨��ª¤hpo#|Y»Qoys���d qY����ª«£¤�rqYvy£¤|���oAqY{�hM=­��{�qY³�hM¬¼�Lx-¬¼¦�£«{�h�?�qY���%=¢oyv�hphg{4¬¼�Lx-¬¼¦�£«{�h�?A£«�Z¬¼��hgd�£¤�gª¤h�qY�Z¬��ª«£¤�rqYvy£¤|���o�°±o�hphM@ONPB[²E|�{�qr�-£¤|���£¤��qY����ª«£¤�rqYvy£¤|���or¸Q~¢��oys���d�o�|#�rqYª«ª¤hp�*Q�Rk¬¼�Lx-¬¼¦�£«{�h8Q§qY����ª«£¤�rqYvy£¤|���or�fShp��d qY��£¤�rqYª¾qY���Çd-xO�Z{�qYs�ª«£¤�S�p|�f¨�E|���hg�Lv�o�qY{�h�{�hg��ª q��php�%�LxÇ�p|�f¨��s�v�hg{��p|��-vy{�|�ª½¦�d�£¤��dÁd q�ov�|��1h�»ÞqYs�ª«v4¬¼v�|�ª¤hg{�qY�-vr¸½¶ © qYs�ª«v4¬­c®|�ª¤hg{�qY�-vw¯k��£«vA°±©Qc�¯�²�£¤o§q�o�hgv§|Y»Uv¢¦§|�|�{½fS|�{�h#��|O��hpo§vyd qYv�Ehg{4»Þ|�{yfSokvyd�h¨oCqYfSh�»±s����gvy£¤|��ºqY���ÇvydLs�o�fJqrx v�|�ª¤hg{�qYv�h¨vyd�h�»¼qY£«ª«s�{�h�|Y»�|���h�|�{AfS|�{�h�|Y»¾£«v�o�p|���o�vy£«vys�hg�Lv�oyv�qYvy£¤|���or¸2¶��gvys qYª«ª«x��Uvyd�h�{�|�ª¤h�|Y»§©Qc�¯AoA£¤okv¢¦§|Y¬Þ»Þ|�ª¤�º�p|���oy£¤��hg{y£«���Åvyd�h�v­xO�1h�|Y»»¼qY£«ª«s�{�h�|Y»½vyd�h�o�v�qYvy£¤|���or¸kc�d�hgx fJqY³�h�vyd�h�oyxZoyv�hgf {�hpoy£«ª«£¤hg�Lv�£«�Hvyd�h���{�hpo�hg���ph�|Y»M��0I�� S����T�FU�����1 V��0F0I��0F��°±o�|�fSh�»[{�qYfShpoAfJqpx �1h�oyvy£«ª«ª8�Eh��p|�{y{�hp�gv�¦�d�£«ª¤h�|�vyd�hg{�o2qY{�h��p|�{y{ys���v�hp� ²M¸�c�d�hgxqYª¤o�|���{�|��-£¤��h�qu¦¾qpxÇv�|HW��^d-v�qY�LqY£«��oyvX���I�P����0Y������ :���� LK[ZY���]\L���^�1�����1 _��0�0I��0 �2|O�p�gs�{y{y£«����EhM»Þ|�{�h¨vyd�hSvy{�qY��oyf¨£¤o�oy£¤|��=°±o�|�fShJ��|-��hJfJqpxºo�hg���ºvyd�hJ�p|�{y{�hp�gv���qYª«s�hpo�¦�d�£«ª¤hJ|�vyd�hg{�o�fJqpxfJqY³�h�hg{y{�|�{�oC²M¸�~­�Çvyd�h�»±|�ª«ª¤|�¦�£«���Ŧ§h�¦�£«ª«ª�o�hph�vyd qYv�q��p�p|�{��Z£«���Åv�|u¦�d�£¤��d%{�|�ª¤h�£¤oAvyd�h�fS|^oyv£«f¨�E|�{yv�qY�Lvr� vyd�h2|���vy£«f¨£«¿�qYvy£¤|���¦�£«ª«ª®ª¤hrq��Àv�|¨��hg{yxu�Z£a` hg{�hg�LvAo�|�ª«s�vy£¤|���or¸

��f��Ehp����hp��o�xOoyv�hgfSo#fJqpxÅoys�` hg{�»[{�|�f o�vy{�|���� �¥µH~A°±hgª¤hp�gvy{�|Y¬¼fJqY�^��hgvy£¤��£«�-v�hg{4»±hg{�hg���phpoC²¦�d�£¤��d.fJqrxÄ{�hg��{�hpo�hg�Lvuqºo�hg{y£¤|�s�oSvyd�{�hrqYv�v�|%vyd�hÃ�p|�{y{�hp�gvJ�Ehgd qp�O£¤|�{J|Y»2vyd�hÃoyxZoyv�hgfø:© |�{£«��o�v�qY���ph^�^£«�SqYs�v�|�fS|�vy£«��hwqY����ª«£¤�rqYvy£¤|���or��vyd�h �½µ�~b@«i�N����Zi�B��rqY�Shg£«vyd�hg{��Eh¾{�q��Z£ qYv�hp���Lx�o�|�fSh£«�Z¬¼��hgd�£¤�gª¤hÀhgª¤hp�gvy{y£¤�rqYª#��hg�O£¤�phpoÅ°±oy¦�£«v���d�hpor�¥{�hgª qpxZor¸«¸ð² |�{��p|�fSh�»±{�|�f q�o�|�s�{��phÀ|�s�v�oy£¤��h�vyd�h��hgd�£¤�gª¤h2°[{�q��Z£¤|��L{�q���qY{�or�Pc�q�oyd�hpo�|Y» ª«£«�^d-vy��£«����¸«¸ð²M¸��¥µH~U�p|�s�ª¤�SqC`Ehp�gvQvyd�h��p|�{y{�hp�gv�»[s����gvy£¤|���£«���|Y»EqYª«ª-vyd�h¾hgª¤hp�gvy{�|���£¤����hg�-£¤�phpo���s�vQvyd�h§vy{�qY��oyf¨£¤o�oy£¤|��¨oys����E|�{yv8£¤o�qw� qY{yvy£¤�gs�ª qY{yª«x*QM¦¾hrqY³�ª«£«��³'QqY���Svyd�h�s�o�h�|Y»UqY��qYª«ªÆ¬­|���vy£¤�rqYª ��hgv¢¦§|�{y³E�-¦�d�£¤��dJ|P` hg{�o¥��hg{yx�d�£«�^dS£«f¨f�s���£«v¢xSv�| �¥µH~C��£¤o¥��|�v��hg��hg{�qYª«ª«x�»Þhrq�oy£«��ª¤h��Ehp�rqYs�o�h¥|Y»Zvyd�h�ª¤|�¦�¬­�p|^oyv®{�hpÈ-s�£«{�hgfShg�Lv�£«f¨�1|^o�hp�2�Lx�vyd�h�£«���Zs�oyvy{yxJ°±o�hph�@ �1B»Þ|�{½fS|�{�h���hgv�qY£«ª¤o�|��uvyd�hkhgª¤hp�gvy{�|Y¬¼fJqY�^��hgvy£¤�2o�hg��o�£«vy£«�-£«v¢xJ|Y»®�Z£a`Ehg{�hg�-v�v­xO�1hpo�|Y»Uvy{�qY��oyf¨£¤o�o�£¤|��o�s����1|�{yvC²M¸ ����hg� ¦�£«vyd3qÁ{�hp�Zs�����qY�LvÅvy{�qY��oyf¨£¤o�o�£¤|�� o�s����1|�{yvr�koys���d3q�ou£«�!c�c�z§É^ÊA�¾vyd�h��hgv¢¦§|�{y³À£¤o���|�v#£«f¨f�s���h2v�|�vy{�qY��oyf¨£¤o�o�£¤|�� hg{y{�|�{�o#oy£«���ph�q��Ehg{yvys�{y� qYvy£¤|��ã¤o�ª«£«³�hgª«xuv�|SqC` hp�gv

� � çedMìgí #ÿì

Page 7: Maximizing the Robustness of TDMA Networks with ... · Maximizing the Robustness of TDMA Networks with Applications to TTP/C Bruno Gaujal, Nicolas Navet To cite this version: Bruno

� �����A�������1E�� +�'+>��P� �

�E|�vyd���d qY����hgª¤o�£«�HÈ-s�£«v�h�qSoy£«f¨£«ª qY{AfJqY����hg{2oy£«���ph�vyd�hgx�qY{�h�£¤��hg�Lvy£¤�rqYª¥qY������hg{yx �gª¤|^o�h�|���hv�|¨hrq���d |�vyd�hg{r¸¥¯k��ª«£«³�h�ʾ¶k�·°¼Ê¾|��Lvy{�|�ª«ª¤hg{A¶w{�hrq��Ahgv­¦¾|�{y³u¬ @ ��B[²M��c�´AµH¶ ��|���|�vw��{�|r�O£¤��hqYs�v�|�fJqYvy£¤��{�hgvy{�qY��oyf¨£¤o�o�£¤|��Á»Þ|�{��p|�{y{ys���v�hp�Á»[{�qYfShpo�qY���ºvyd�hg£«{���qYv�q�qY{�huq��gvys qYª«ª«x%ª¤|^oyv�»Þ|�{vyd�h�qY����ª«£¤�rqYvy£¤|��U¸

� Ð � � Ð:9 �<; Ì�� � �¥Ì � c�d�hH��{�|���ª¤hgf ¦§h%q����Z{�hpo�ou£«�=vyd�£¤oÅoyvys��Zx:£¤o�v�|VW����Âvyd�h��EhpoyvqYª«ª¤|O�rqYvy£¤|��º|Y»½vyd�h�oyª¤|�v2|Y»§hrq���dºoyv�qYvy£¤|��Ç£«�Hvyd�h�{�|�s����Ç£«�Çoys���dÁq�o2q�¦¾qrxHq�o�v�|ufJqatZ£«f¨£«¿rhvyd�h�{�|���s�o�vy��hpo�ow|Y»Qvyd�h�oyxZoyv�hgf qY�LqY£«��oyvwhg{y{�|�{�or¸¥¹ºh��p|���oy£¤��hg{#v¢¦§|J�Z£¤o�vy£«���gvw|��Z}4hp�gvy£«��hpo��

i^¸�� ��� Ì  ^����� Ì�� ¡ f¨£«��£«f¨£«¿rh^��»Þ|�{�hrq���d�©Qc�¯��Zvyd�h���{�|�� qY��£«ª«£«v­xÅvyd qYvwqYª«ª »±{�qYfShpo#|Y»�vyd�h©Qc�¯Â�rqY{y{yxO£«����vyd�h#o�qYfSh#£«�Z»±|�{yfJqYvy£¤|��S¦�£«ª«ª �1h��p|�{y{ys���v�hp� ¸�~¢�¨vyd�h�{�hpoyv½|Y»Evyd�h�� qY�Ehg{r�vyd�£¤o���{�|�� qY��£«ª«£«v­xŦ�£«ª«ª®�1h�v�hg{yfShp� vyd�h]=­ª¤|^o�o���{�|�� qY��£«ª«£«v­x�?�qY���Ã��hg��|�v�hp�Å�-x����! � ¢¸

�Z¸�� ��� Ì  ^����� Ì#" ¡ f¨£«��£«f¨£«¿rh^��»Þ|�{�hrq���dÄ©Qc�¯��®vyd�h���{�|�� qY��£«ª«£«v­x vyd qYv�|���hÇ°±|�{�fS|�{�h�²»[{�qYfShA|Y»®vyd�hA©Qc�¯3�rqY{y{yx-£«����vyd�hAo�qYfShk£«�Z»±|�{yfJqYvy£¤|��Ŧ�£«ª«ª1�1hkª¤|^oyvr¸Qc�d�h��p|�{y{�hpoy�E|����O¬£«���S��{�|�� qY��£«ª«£«v­xÅ£¤ow��hg��|�v�hp�Å�Lx���$&%('a¸

c�d�hJo�|�ª«s�vy£¤|��ºv�|Àvyd�£¤o�o�ª¤|�v�qYª«ª¤|-�rqYvy£¤|��º��{�|���ª¤hgf d q�o2v�|Åv�qY³�h¨£«�-v�| q��p�p|�s��Lv�vyd�h�»¼q��gv2vyd qYvqu��qYv�qÀ¦�£«ª«ª��Eh�o�hg�Lv��LxHfS|�{�h�vyd qY�º|���h���|-��h�£«�%vyd�h¨o�qYfSh�{�|�s����:°[�-xºqYª«ª���|O��hpo�|Y»¾vyd�h©Qc�¯2²½qY���uvyd qYv�£«v�f¨£«�^dLv��EhAo�hg�Lv�o�hg��hg{�qYªEvy£«fShpo¾�Lxuq�o�qYfShk��|O��h�°[£«�Åoys��p�phpo�oy£«��hw{�|�s�����oM²¦�d�hg� vyd�h���{�|-�Zs��gvy£¤|����1hg{y£¤|O�Å|Y»�vyd�h���qYv�q�£¤o��^{�hrqYv�hg{#vyd qY�Ãvyd�h�ª¤hg���^vyd |Y»�q�{�|�s���� ¸

¶koA£«v�¦�£«ª«ª��1h�»±s�{yvyd�hg{2�Z£¤o��gs�o�o�hp�H£«� Ò-s���o�hp�gvy£¤|��Á�Z¸ò�Z�Evyd�h�v¢¦§|Å|��Z}yhp�gvy£«��hpo��p|�{y{�hpoy�E|����v�|æ§hgª«ªÆ¬­��h�W���hp�Äoy£«vys qYvy£¤|���o�£«�%vyd�h W hgª¤� |Y»§»¼qYs�ª«v4¬¼v�|�ª¤hg{�qY���phSvyd qYv�qY{�hS�Z£¤oyvy£«���^s�£¤oyd�hp�Á¦�£«vyd{�hg�LqY{��ºv�|Åvyd�hS�p|����phg��v�|Y» =¼»¼qY£«ªÆ¬­oy£«ª¤hg���ph�?�¸J~¢v2¦�£«ª«ª¾qYª¤o�|À�1hSo�d�|r¦��ºvyd qYv�vyd�h�»[s�ªaW�ª«ª«fShg�-v�|Y»vyd�hpo�h�v­¦§|�|��Z}4hp�gvy£«��hpoAqYv�vyd�h�o�qYfSh2vy£«fSh�£¤o�£«���p|�f¨� qYvy£«��ª¤h^¸

���r� � � � ��� ÐL6 � Ð�6 �<; Ì¨Ì �a� Ð � � Ð Ñ Ì � ~­��vyd�£¤o�oyvys��Zx��^¦¾h�¦�£«ª«ª��p|���oy£¤��hg{§qY�¨hg{y{�|�{½qY{y{y£«�^qYª��{�|O�phpo�o�¦�d�hg{�h=­��s�{�o�v�o#?�|Y»�vy{�qY��oyf¨£¤o�oy£¤|��ºhg{y{�|�{�o�fJqpx%|-�p�gs�{r¸�c�d�£¤o2£¤o���hg{yxHª«£«³�hgª«xH£«�%vyd�h�p|��-v�hMtOvk|Y»�£«�Z¬¼��hgd�£¤�gª¤h�f�s�ª«vy£«��ª¤hMtZ£«���uqY����ª«£¤�rqYvy£¤|���or¸

~¼»�oys��p�phpo�oy£«��hÅvy{�qY��o�f¨£¤o�oy£¤|��.hg{y{�|�{�oJqY{�hÃ��|�vJ�p|�{y{�hgª qYv�hp� °[£¼¸ £¼¸ � ¸ð²M�¾£«v¨£¤oS�gª¤hrqY{Svyd qYvJvyd�hª¤|O�rqYvy£¤|��º|Y»�hrq���d%£«���Z£«�O£¤�Zs qYª½oyª¤|�v�|Y»�qY�º©Qc�¯�d q�o2��|À£«��c�s�hg���phS|��%vyd�h�ª¤|^o�o���{�|�� qY��£«ª«£«v¢xo�£«���ph2hrq���d�o�ª¤|�v�d q�o�vyd�h�o�qYfSh2��{�|�� qY��£«ª«£«v¢xÃ|Y»��Ehg£«���S�p|�{y{ys���v�hp�ã«����hg�Ehg����hg�Lvyª«x�¸*)A|r¦§hg��hg{£«����{�q��gvy£¤�ph�vy{�qY��oyf¨£¤o�oy£¤|��Shg{y{�|�{�o¥qY{�h�d�£«�^d�ª«x��p|�{y{�hgª qYv�hp�JqY����|���h�|���o�hg{y��hpoQ��s�{�o�v�o�|�{Qhg{y{�|�{�oª¤hrq��Z£«����v�|Ao�s��p�phpo�oy£«��h�vy{�qY��oyf¨£¤o�o�£¤|��Shg{y{�|�{�or¸Qc�d�hwq�o�oys�f¨��vy£¤|���o�fJq���h¾»Þ|�{�vyd�h�hg{y{�|�{½qY{y{y£«�^qYª��{�|O�phpo�o�¦�£«ª«ª vydLs�o¥£«��c�s�hg���phwvyd�h#o�|�ª«s�vy£¤|���v�|2vyd�h���{�|���ª¤hgf�|Y»Eª¤|O�rqYvy£«����vyd�h#©Qc�¯ oyª¤|�v�or¸8¹ºh¦�£«ª«ª��p|���oy£¤��hg{SqY� hg{y{�|�{�fS|-��hgª�vyd qYv¨�rqY� v�qY³�h�£«�-v�|%q��p�p|�s��Lv¨�E|�vyd hg{y{�|�{�»±{�hpÈLs�hg���gxÄqY���hg{y{�|�{A�^{�qp�O£«v­x ¦�d�£¤��dH��hg��hg{�qYª«£«¿rhpo�qJfS|O��hgª®��{�|��E|^o�hp� £«� @«ig��B­¸+)Ahg{�h�qY{�h�vyd�h�q�o�oys�f¨��vy£¤|���o|��Åvyd�h��1hg{yvys�{y� qYvy£¤|��Hhg{y{�|�{�o#fJq���h2£«�Åvyd�h2{�hpo�v#|Y»�vyd�h2� qY�Ehg{r¸

é[â��®é±î

Page 8: Maximizing the Robustness of TDMA Networks with ... · Maximizing the Robustness of TDMA Networks with Applications to TTP/C Bruno Gaujal, Nicolas Navet To cite this version: Bruno

�������������� ��������������������� ���� �!�#"%$'&(�*),+-����./�10 2�� N

������� �½q���d vy£«fShÁqY� �¥µH~�|O�p�gs�{�or�A£«vŦ�£«ª«ª��Ehg{yvys�{y�3vyd�hº�p|�f¨f�s���£¤�rqYvy£¤|���oH|��!vyd�hº��s�o�Zs�{y£«���Jq��phg{yv�qY£«�Ã�Zs�{�qYvy£¤|��U��hrq���d ��£«v¾vy{�qY��o�f¨£«vyv�hp�Ã�Zs�{y£«���¨vyd�£¤o��Ehg{yvys�{y� qYvy£¤|��Åd q�owq��{�|�� qY��£«ª«£«v­x��Hv�|��Eh��p|�{y{ys���v�hp� �Z£«����hg�1hg����hg�-vyª«xu|Y»Qhrq���dÅ|�vyd�hg{�°[vyd�hAhg{y{�|�{�fS|O��hgªE£«�@«ig�1B8q�o�oys�fShpo�� ��²M¸

������� c�d�h�o�v�qY{yvy£«���uvy£«fShpo�|Y»¥vyd�h �¥µH~���s�{�oyv�o2qY{�h�£«����hg�Ehg����hg�Lv�{�qY����|�f�^qY{y£ qY��ª¤hpor�1s���£Æ¬»±|�{yf¨ª«xÀ�Z£¤oyvy{y£«��s�v�hp��|r��hg{#vy£«fSh^¸

������� c�d�h�oy£«¿rh�|Y»�hrq���d �¥µH~¾��s�{�oyv#£¤okhMtO�E|���hg�Lvy£ qYª«ª«xÃ�Z£¤oyvy{y£«��s�v�hp�ÇqY���Å£¤ow£«����hg�1hg����hg�-vA|Y»hg��hg{yx-vyd�£«���uhgª¤o�h^¸

¹Â£«vyd�|�s�v�»[s�{yvyd�hg{�³-��|�¦�ª¤hp�Z��h�|��ºvyd�h��p|���oy£¤��hg{�hp� qY����ª«£¤�rqYvy£¤|��'qY���º£«v�o�hg�L�O£«{�|���fShg�Lvr�q�o�oys�f¨��vy£¤|���o�� ���� qY����� ����� qY{�h�{�qYvyd�hg{�{�hrq�o�|�� qY��ª¤h^¸¥¶Ao�oys�f¨��vy£¤|�� ������� £¤o�fS|�{�hv�hp��d���£¤�rqYª¥qY���æ�£«ª«ª��1h2s�o�hp�ã«�Hvyd�h���{�|O|Y»±ok|Y»¾ÒOhp�gvy£¤|��Ç�Z¸«iu°±|��Z}4hp�gvy£«��hui�²M¸�¹ºh�d qr��h�o�|�fShd�|��Eh�vyd qYv#vyd�h2{�hpo�s�ª«v#|Y»¥ÒOhp�gvy£¤|��Ç�Z¸«i�{�hgfJqY£«��o#�^qYª«£¤�À»±|�{#|�vyd�hg{k�Z£¤oyvy{y£«��s�vy£¤|���or¸

)k|�¦§hg��hg{r��vyd�h�{�hpoys�ª«v�oJ|Y»�ÒOhp�gvy£¤|��=�°±|��Z}yhp�gvy£«��hH��²S��|º��|�v�s�o�h�q�o�o�s�f¨��vy£¤|�� ��� � �qY���:qY{�hu�^qYª«£¤�Ä»±|�{SqYª«ª#�Z£¤oyvy{y£«��s�vy£¤|���o�|Y»kvyd�hÅoy£«¿rhÀ|Y»Avyd�hÀ��s�{�oyv�oð[��{�|r�O£¤��hp�'vyd�hgxÄ{�hgfJqY£«�£«����hg�Ehg����hg�LvA|Y»�vyd�h�oyv�qY{yvy£«���S�E|�£«�Lvw|Y»Qvyd�h �¥µH~4²M¸

uÌ �[�Z� Ì Ñ Ï�Ð ��� c�d�hÃc�£«fShM¬­c®{y£«�^��hg{�hp�¶k{���d�£«v�hp�gvys�{�h °±c�c®¶ ¬�o�hph @«ir����i^i�B[²�d q�oS�Ehphg���hpo�£«�^��hp�H»±|�{2d�£«�^dZ¬­��hg�Ehg����qY��£«ª«£«v­x%{�hrqYªÆ¬¼vy£«fShSo�xOoyv�hgfSo�oys���d q�o�qYs�v�|�fS|�vy£«��hJqY����ª«£¤�rqYvy£¤|���or¸c�d�h�c�c�z§É^Ê ��{�|�v�|O�p|�ªA@ò�^�PB­�E¦�d�£¤��d%£¤o�qu�phg�-vy{�qYª¥� qY{yv2|Y»§vyd�h�c�c®¶��E�1|^o�o�hpo�o�hpok�Ls�fShg{�|�s�o»ÞhrqYvys�{�hpoÅqY���Âo�hg{y�-£¤�phpo�{�hgª qYv�hp�.v�|Ä��hg�Ehg����qY��£«ª«£«v­xÂo�s���d3q�oJvyd�h���s�o��^s qY{��Z£ qY�,@«i ��B­��vyd�h�^{�|�s��=fShgf��Ehg{�oyd�£«�!qYª«��|�{y£«vyd�f @«i��PB�qY���.o�s����1|�{yvJ»±|�{�fS|-��hH��d qY����hpo�@«ir�CB­¸3c�d�h�c�c®¶qY���Ãvyd�h2c�c�z§É^Ê ��{�|�v�|O�p|�ªUd qp��h��1hphg� ��hpo�£«�^��hp� qY���ÃhMtZv�hg��oy£«��hgª«xuo�vys��Z£¤hp� qYv�vyd�h2ÓA£¤hg��� q¯k��£«��hg{�oy£«v¢xÂ|Y»�c8hp��d���|�ª¤|��^x�¸ ʧª¤|^o�hgª«x={�hgª qYv�hp� v�| |�s�{u��{�|��E|^o�qYª2£¤ouvyd�hǦ§|�{y³=��hpo��g{y£«�Ehp�£«� @ð�<B�¦�d�hg{�hÀvyd�hu{�hgª«£ qY��£«ª«£«v­x |Y»kvyd�huvy{�qY��oyf¨£¤o�o�£¤|�� |��:qHc�c�z ��hgv¢¦§|�{y³Ä£¤o¨oyvys��Z£¤hp�Ħ�£«vydvyd�h�v�qY³O£«���ã«�-v�| q��p�p|�s��Lv�|Y»¾vy{�qY��oyf¨£¤o�oy£¤|��Áhg{y{�|�{�o�|��ºvyd�h���s�o�q�o�¦§hgª«ª�q�o�»ÞqY£«ª«s�{�hpo�£«�%vyd�hc�c�z·��|O��hpor¸(¯k����hg{Àvyd�hºq�o�oys�f¨��vy£¤|�� vyd qYv qYª«ªA»ÞqY£«ª«s�{�hpoÃqY���=vy{�qY��oyf¨£¤o�oy£¤|��!hg{y{�|�{�oÃqY{�ho�v�qYvy£¤oyvy£¤�rqYª«ª«xÀ£«����hg�1hg����hg�-vr�1q�fShrq�o�s�{�h�|Y»�vyd�h2{�hgª«£ qY��£«ª«£«v­xÃ|Y»�vyd�h2vy{�qY��oyf¨£¤o�oy£¤|�� £¤o��^£«��hg��£«�v�hg{yfSo¾|Y»®µHhrqY��c�£«fShAc®|�© qY£«ª«s�{�h�°±µHc�c�©¥²�¦�d�hg{�h�q��p|�f¨f�s���£¤�rqYvy£¤|��À»ÞqY£«ª«s�{�hw»Þ|�{¾qY��©Qc�¯£¤o���h�W���hp�ºq�o�vyd�h�ª¤|^o�o2|Y»¾qYª«ªQfShpo�oCqY��hpo2|Y»�qY�%©Qc�¯(o�hg�Lv2£«�Çvyd�h�o�qYfSh�{�|�s���� ¸�©�{�|�f9vyd�hµHc�c�©=|Y»§hrq���d%£«���Z£«�O£¤�Zs qYª¥©Qc�¯�� qJ�^ª¤|�� qYª�fShrq�oys�{�h�|Y»§vyd�h�{�hgª«£ qY��£«ª«£«v­xÇ|Y»§vyd�h�oyxZoyv�hgf £¤o��hg{y£«��hp� ¸

c�d�hg{�h�hMtZ£¤oyvkv­¦§|ÀfJqY£«�Ç�Z£a` hg{�hg���phpo�¦�£«vydH|�s�{�¦§|�{y³E¸��k��h��p|����phg{y��o�vyd�h�q�o�oys�f¨��vy£¤|���ofJq���hA|��Àvyd�hw�Ehg{yvys�{y� qYvy£¤|���o#qY����vyd�hAo�hp�p|����Jvyd�hA��qYv�q���{�|-�Zs��gvy£¤|��U¸¥~¢��@ð�<B1vyd�hAhg{y{�|�{�o�qY{�hq�o�oys�fShp�uv�|��Ehk£«����hg�1hg����hg�-vr��vyd�hAª¤|-�rqYvy£¤|��Ã|Y»®vyd�h�©Qc�¯ o�ª¤|�v�o§d q�o�vyd-s�o���|�£«��c�s�hg���ph�qY���£¤oA��|�v2�p|���oy£¤��hg{�hp� ¸ )Ahg{�h�|��Hvyd�h��p|��Lvy{�qY{yx��U¦§h�v�qY³�h�£«�Lv�|Åq��p�p|�s��Lv2vyd�h���s�{�oyvy£«��hpo�oA|Y»§vyd�h

� � çedMìgí #ÿì

Page 9: Maximizing the Robustness of TDMA Networks with ... · Maximizing the Robustness of TDMA Networks with Applications to TTP/C Bruno Gaujal, Nicolas Navet To cite this version: Bruno

� �����A�������1E�� +�'+>��P� �

�Ehg{yvys�{y� qYvy£¤|��À��{�|-�phpo�or¸ )khg���phkvyd�h#vy£«fSh�qYª«ª¤|-�rqYvy£¤|���o�|Y»Uvyd�hA©Qc�¯ {�hg��ª«£¤�rq�o�¦�£«ª«ª d qp��h2q���£«�h�` hp�gv#|��Ãvyd�h2vy{�qY��o�f¨£¤o�oy£¤|��Ãhg{y{�|�{#��{�|�� qY��£«ª«£«vy£¤hpor¸

¶kok»Þ|�{Avyd�h¨��qYv�q���{�|O�Zs��gvy£¤|��º£¤o�oys�h^�E£«� @ð�<BQ»ÞqY£«ª«s�{�h�£¤o���hp�g£¤��hp�º|��ºq��Ehg{A{�|�s����Ç� q�o�£¤o¦�d�£«ª¤h§£«��vyd�£¤o8� qY�Ehg{�vyd�£¤o�hg��hg�-vQ¦�£«ª«ª-�Eh¾q�o�o�hpo�o�hp���p|���oy£¤��hg{y£«����vyd�h¥»±{�qYfShpoQo�hg�Lv�£«��qw��{�|O�Zs��M¬vy£¤|��H�gxO�gª¤h�|Y»§q¨��qYv�qZ¸#~¢����hphp� � vyd�h�o�qYfSh���qYv�qSf¨£«�^dLvA�1h�vy{�qY��oyf¨£«vyv�hp�H�Zs�{y£«���uoys��p�phpo�oy£«��h{�|�s�����o�qY���Hvyd�h�»¼q��gvAvyd qYv2��|�»±{�qYfSh�|Y»¾qY�%©�c�¯�d q�oA�1hphg�%oys��p�phpo�oy»[s�ª«ª«xÃvy{�qY��o�f¨£«vyv�hp�%£«�|���h�{�|�s����J��|-hpoQ��|�v���hp�phpo�oCqY{y£«ª«x�£«f¨��ª«x¨q��p|�f¨f�s���£¤�rqYvy£¤|��J»ÞqY£«ª«s�{�h#�1hp�rqYs�o�h�vyd�h�oCqYfSh���qYv�q£¤okqYª¤o�|¨o�hg�Lvw£«�À»±|�ª«ª¤|�¦�£«���J{�|�s�����o�°±o�hph�� qY{�qY�^{�qY��dH�Z¸ò��²M¸

c�d�h�o�hp�p|����H�Z£a`Ehg{�hg���ph�¦�£«vyd @ð�CB8£¤owvyd qYvk¦¾h���|���|�vwfShg{�hgª«x �p|�f¨��s�v�h�vyd�h�{�hgª«£ qY��£«ª«£«v¢x|Y»wqÃ�^£«��hg� oyxZoyv�hgf ��s�v�qYª¤o�|Ã��{�|��-£¤��hÀqÀ¦¾qrxºv�| |���vy£«f¨£«¿rh�£«v��O£ qÀvy£«fShuqYª«ª¤|-�rqYvy£¤|�� |Y»�vyd�h{�hg��ª«£¤�rq�o�¸�c�d�£¤o§��|Ohpo½��|�v§{�hpÈLs�£«{�h�qY�Lx¨fS|O�Z£aW �rqYvy£¤|��Å|Y»Uvyd�h#��{�|�v�|O�p|�ª1|�{¾|Y»Uvyd�h#� qY{�qYfShgv�hg{�o|Y»1vyd�h�oyxOo�v�hgfø��^s�o�v���ª qpxO£«���2¦�£«vyd¨vyd�h§vy£«fSh#qYª«ª¤|O�rqYvy£¤|��S|Y»�{�hg��ª«£¤�rq�o¥��{�|r�O£¤��hpo½qAoys���oyv�qY�Lvy£ qYª�LqY£«� £«�Å{�hpoy£«ª«£¤hg���phJ°ÞqY{�|�s������^���·£«�ÅfJqY�LxÅ�rq�o�hpoC²�q�owo�hphg�Å£«��ÒOhp�gvy£¤|��H�Z¸

©8£«� qYª«ª«x3qY��|�vyd�hg{Ã��|���hgª«v­x!¦�£«vyd3{�hpo��1hp�gvÅv�|'��{�hg�O£¤|�s�oŦ¾|�{y³ �p|�fShpoû[{�|�f vyd�hº��{�|-|Y»v�hp��d���£¤ÈLs�hpo�¸�c�d�hgx�qY{�h§� q�o�hp��|��%� �JE �����X��K��JEa��0������ qY����������0�� ���� ����� �� LZ ���®»Þ|�{QÒOhp�gvy£¤|��S�qY����|��*�X� � ��0F����8������ 2qY������Z�����0MZY�1 P�I�������¾»±|�{¥ÒOhp�gvy£¤|��¨��¸�c®|�vyd�h¾�1hpo�v�|Y»E|�s�{�³O��|r¦�ª¤hp�Z��h^�vyd�hpo�hk��|�vy£¤|���o#d qp��h2��hg��hg{��Ehphg� qY����ª«£¤hp�À£«�Åvyd�£¤o¾»[{�qYfShg¦¾|�{y³ÃqY���Àvyd�hgxJfJqpxu�Eh�s�o�hM»[s�ª1»Þ|�{o�hg��hg{�qYª®|�vyd�hg{#{�hgª qYv�hp�Å��{�|���ª¤hgfSor¸

� /E�w��Q,.��/10 �#"Ã�1���·���1� ���~¢�Åvyd�£¤o#o�hp�gvy£¤|��U��¦§h W�{�oyvw��hpo��g{y£«�1h�vyd�h��p|�f¨f�s���£¤�rqYvy£¤|�����{�|�v�|O�p|�ª¼��vyd�hg�Åvyd�h2fS|O��hgª®|Y»�vyd�hqY����ª«£¤�rqYvy£¤|��HqY���Àvyd�h���|�v�qYvy£¤|���o�s�o�hp� ¸Q¹ºh�vyd�hg��}4s�oyvy£Æ»[xJvyd�h�v­¦§|J�Z£¤oyvy£«���gv#|��Z}4hp�gvy£«��hpo#vyd qYv¦¾hg{�h2£¤��hg�Lvy£aW hp� ¦�£«vydÃ{�hg�LqY{��Ãv�|�vyd�h��p|����phg��vw|Y» =¼»¼qY£«ªÆ¬­oy£«ª¤hg���ph�?�¸

����� ������������� �"!$#&%'���)(+*$��(+� ,

c�d�{�|�s��^d�|�s�v¾vyd�£¤o¥� qY�Ehg{r�-¦§h#¦�£«ª«ª1�p|���oy£¤��hg{½vyd�hwoyx-����d�{�|���|�s�o§c�´�µH¶=��{�|�v�|-�p|�ª¼¸�c�d�hwvy£«fSh��hphp��hp�Hv�|�vy{�qY��oyf¨£«v2|���h���£«vA|���hg{Avyd�h���s�oA£¤oAv�qY³�hg�ºq�owvyd�h�vy£«fSh�s���£«vr¸�~­�Hvyd�h�»Þ|�ª«ª¤|r¦�£«���qYª«ª vy£«fSh�ÈLs qY�-vy£«vy£¤hpoAqY{�h��^£«��hg�Ãs�o�£«���Svyd�£¤o�vy£«fShM¬¼��£«vAq�o#s���£«vr¸

c�d�h2�Ls�f��1hg{�|Y»e���^�1��� �� S�M�&-���£¤o#oyv�qYvy£¤��qY���Ãvyd�h�oyv�qYvy£¤|���owd qp��h�q��p�phpo�owv�|¨vyd�h2��s�ow£«��qo�vy{y£¤�gv���hgv�hg{yf¨£«��£¤o�vy£¤�So�hpÈ-s�hg�Lvy£ qYª¥|�{���hg{r¸ �½q���dÁo�v�qYvy£¤|��%�1|^o�o�hpo�o�hpokvyd�h���s�oA»±|�{�qu�p|���oyv�qY�Lv�Ehg{y£¤|O��|Y»¾vy£«fSh��rqYª«ª¤hp�ºq ��Ea�1���Zs�{y£«���Ŧ�d�£¤��d%£«v�d q�oAv�|uvy{�qY��oyf¨£«v�|���h "F0I�����g¸�c�d�h�oy£«¿rh�|Y»vyd�h2o�ª¤|�v�o�£¤o���|�v#��hp�phpo�o�qY{y£«ª«xu£¤��hg�Lvy£¤�rqYª®»±|�{wqYª«ªUo�v�qYvy£¤|���o���s�vwoys��p�phpo�o�£«��h2oyª¤|�v�o��1hgª¤|����^£«����v�|vyd�h¨oCqYfSh¨oyv�qYvy£¤|�� qY{�h¨|Y»§vyd�hJo�qYfSh¨oy£«¿rh^¸�c�d�h¨o�hpÈ-s�hg���ph¨|Y»�oyª¤|�v�o2o�s���dºvyd qYv�qYª«ª¥oyv�qYvy£¤|���od qr��h�q��p�phpo�o#|����ph�v�|¨vyd�h���s�o#£¤o#�rqYª«ª¤hp��qH0#���J LK^��q�o#oyd�|r¦�� £«� ©�£«�^s�{�hSi^¸

é[â��®é±î

Page 10: Maximizing the Robustness of TDMA Networks with ... · Maximizing the Robustness of TDMA Networks with Applications to TTP/C Bruno Gaujal, Nicolas Navet To cite this version: Bruno

�������������� ��������������������� ���� �!�#"%$'&(�*),+-����./�10 2�� �

��FZBED���� ��FZBED���������� ����� ���

©�£«�^s�{�hSi ��¶ {�|�s����Å£¤o�fJq���h�|Y»$- oyª¤|�v�o�°[d�hg{�h�- ���²A� |���h2oyª¤|�v#�Ehg{#oyv�qYvy£¤|��U¸��� � � * * � (���� ��(+� ,�� ��!$#&�c8|�q���d�£¤hg��hk»ÞqYs�ª«v4¬¼v�|�ª¤hg{�qY���ph^�Zvyd qYv¾£¤o½vyd�hA�rqY� q��g£«v­x�|Y»8q�oyxZoyv�hgf�v�|���hgª«£«��hg{�£«v�o§o�hg{y�-£¤�phkhg��hg�£«�Âvyd�hÃ��{�hpo�hg���phH|Y»�»ÞqYs�ª«v�o���o�|�fShÃ��|O��hpouqY{�h 0I� \LE � ZY�1� �YKºqY���ÂqY{�h��gª«s�oyv�hg{�hp�.£«�-v�|�� ���JE ��U$L��E���0I�� :��� :������°±©Qc�¯koC²M¸k¶w�Ç©Qc�¯ £¤oAq�o�hgvA|Y»§o�hg��hg{�qYªQoyv�qYvy£¤|���oAvyd qYvk�Ehg{4»±|�{yf vyd�h�o�qYfSh»±s����gvy£¤|�� qY���%hrq���dÁ��|O��h¨|Y»�qY�º©Qc�¯��1|^o�o�hpo�o�hpo�£«v�o�|�¦��Áoyª¤|�v2£«�%vyd�h¨{�|�s����ºo�|Àvyd qYv2vyd�h»¼qY£«ª«s�{�hS|Y»�|���hS|�{�fS|�{�hSo�v�qYvy£¤|���o�£«�ºvyd�h¨o�qYfShJ©�c�¯+f¨£«�^d-v��1hSv�|�ª¤hg{�qYv�hp� ¸Jc�d�hSoyv�qYvy£¤|���o»Þ|�{yf¨£«���ºqY�Ä©Qc�¯ qY{�hu�rqYª«ª¤hp� 0I� \LE � ZY�P��£«� vyd�h�»Þ|�ª«ª¤|r¦�£«����¸%©�|�{�vyd�huo�qY³�hu|Y»koy£«f¨��ª«£¤�g£«v­x��¾q��|��Z¬¼{�hg��ª«£¤�rqYv�hp��oyv�qYvy£¤|��æ�£«ª«ª8qYª¤o�|��1h�v�hg{yfShp�HqY�é�c�¯·°±|Y»¥�rqY{��Z£«� qYª«£«v­xÃ|���h�²M¸

�k��h���hg��|�v�hpok�Lx�� vyd�h�o�hgvk|Y»�©Qc�¯Ao���� � �"!$#�!&%(')')'+* qY��� %-, £¤o#vyd�h��rqY{��Z£«� qYª«£«v¢x|Y»U©Qc�¯ � �:� ��� �Qvyd�h#�Ls�f��Ehg{�|Y» o�v�qYvy£¤|���o�»Þ|�{yf¨£«����©Qc�¯ � ¸Qc�d�h#oy£«¿rh�°[£«�J��£«v�oC²Q|Y» vyd�h#o�ª¤|�v�o|Y»¥qYª«ª vyd�h�oyv�qYvy£¤|���ow£«� � £¤o#vyd�h�o�qYfSh�qY���Å£¤o#��hg��|�v�hp� �Lx/. , ¸0§xÀ��h�W���£«vy£¤|��U��vyd�h�v�|�v�qYª®�Ls�f��Ehg{k|Y»Q��£«v�o#£«� q�{�|�s���� ����hg��|�v�hp�21¨�Z£¤o#hpÈ-s qYªUv�| �

1 43,6587 % , . , 'c�d�h���{�|���ª¤hgf �p|���oy£¤oyv�o�£«�Ã��d�|O|^oy£«���¨vyd�h��E|^oy£«vy£¤|��Å|Y»8vyd�h2oyª¤|�v�o�|Y»�qYª«ª oyv�qYvy£¤|���o�»Þ|�{yf¨£«���

qY�Ä©Qc�¯�£«� qÅ{�|�s���� ¸Åc�d�£¤o�£¤o���|���h�s�����hg{�vyd�hS»±|�{yf |Y»#qÃ��£«� qY{yx%��hp�gv�|�{"9 , |Y»#oy£«¿rh:1°±�rqYª«ª¤hp��qY��qYª«ª¤|-�rqYvy£¤|��û±|�{ � ²���h�W���hp�Å�-x

; �=<?>@<A1 ! 9 ,B DC � £Æ»�o�|�fSh2oyv�qYvy£¤|��ã«� � vy{�qY��oyf¨£«v�oAqYv�vy£«fShM¬¼��£«vE>F |�vyd�hg{y¦�£¤o�h '�A|�v�h�vyd qYv�vyd�hS�p|���o�vy{ys��gvy£¤|�� |Y»G9 , f�s�oyv2»Þ|�ª«ª¤|r¦�o�hg��hg{�qYª½�p|���oyvy{�qY£«�Lv�o�¸S©�£«{�oyv�vyd�h���£«� qY{yx��hp�gv�|�{"9 , f�s�oyv��1h¨fJq���hJ|Y» % , QM��ª¤|O��³Zo<Q¨|Y»�|���hpor�8hrq���d |Y»#oy£«¿rhH. , v�| �p|�{y{�hpoy�1|����Áv�|qY�%qYª«ª¤|O�rqYvy£¤|��H|Y»¾qYª«ª�vyd�h�oyª¤|�v�ok|Y» � ¸2ÒOhp�p|���� �Evyd�h�qYª«ª¤|O�rqYvy£¤|���o�|Y»§qYª«ª8vyd�h�©Qc�¯koAf�s�o�vk�EhZ ���]\��1���^��EG�p�8fShrqY��£«����vyd qYv�vyd�hSoCqYfSh¨��£«v��rqY����|�v��1hSqYª«ª¤|O�rqYv�hp�ºv�|Åv¢¦§|H�Z£a`Ehg{�hg�-v�©Qc�¯Aor¸©�£«� qYª«ª«x�qYª«ª���£«v�oA£«�ºqS{�|�s����Hf�s�o�vk�Eh�qYª«ª¤|O�rqYv�hp�Hv�|�o�|�fSh�©Qc�¯�¸�~¢��fJqYvyd�hgfJqYvy£¤�rqYª�v�hg{yfSovyd�hpo�h2�p|�f¨� qYvy£«��£«ª«£«v­x �p|���oyvy{�qY£«�-v�ow�rqY�Ã�Eh�¦�{y£«vyv�hg�

3,6587 9 , JI � !LKLKLKM! �8N '

� � çedMìgí #ÿì

Page 11: Maximizing the Robustness of TDMA Networks with ... · Maximizing the Robustness of TDMA Networks with Applications to TTP/C Bruno Gaujal, Nicolas Navet To cite this version: Bruno

� �����A�������1E�� +�'+>��P� �

©8£«� qYª«ª«x��Zvyd�hA»±{�qYfSh�o�hg�Lv��-xuq���|O��h��p|��Lv�qY£«��o#o�|�fSh2��qYv�q�¦�d�|^o�hk�^qYª«s�h�£¤o¾�1hg{y£¤|O�Z£¤�rqYª«ª«xs��E��qYv�hp��q�o#£«vw£¤o���hg��hg{�qYª«ª«xÅvyd�h��rq�o�h2£«� �Z£¤oyvy{y£«��s�v�hp���p|��Lvy{�|�ª�qY����ª«£¤�rqYvy£¤|���or¸�©�|�{w£«��o�v�qY���ph^�£«�Hq�v­xO��£¤�rqYª8�rqY{whg�-�-£«{�|���fShg�-vr� q�»[{�qYfSh�o�hg�Lvk�LxÀvyd�h�hg���^£«��h��p|��-vy{�|�ª«ª¤hg{kfJqrxÃ�p|��Lv�qY£«��vyd�hlkz¥µ �^qYª«s�h���ª«s�o2vyd�h¨hg���^£«��h�v�hgf¨�Ehg{�qYvys�{�h�qY���ºq���hg¦ »±{�qYfSh�£¤o���s�£«ª«v�hg��hg{yxÄir�YfSo�°[vyd�hfJqatZ£«f�s�f {�hM»[{�hpo�dÅ{�qYv�h�|Y»�vyd�h�v¢¦§| =¢oy£«�^� qYª¤o#?2�p|�f¨�1|^oy£«���Jvyd�h�»[{�qYfSh�²M¸

Ò-£«���ph�vyd�hgxÄqY{�hJ{�hg��ª«£¤�rq�or�§qYª«ª§��|O��hpo�|Y»AqY�Ä©Qc�¯ s��E��qYv�h�vyd�hg£«{���qYv�q�¦�£«vydÄvyd�hÀo�qYfSh�Ehg{y£¤|O�=��hg��|�v�hp� �-x�� , qY���!�rqYª«ª¤hp�3q \L0I��K��SZ ��� �� ,Z �1Z EG�g¸·c�d�hÇ��qYv�q o�hg�-vÃ�Zs�{y£«���:|���h��{�|O�Zs��gvy£¤|�� �gxZ�gª¤hS£¤o�qYª¤o�|Ã�rqYª«ª¤hp�Äq����F� � ������£«�ºvyd�h¨»Þ|�ª«ª¤|r¦�£«����¸À~¢v�£¤o�qYª¤o�|�q�o�oys�fShp�ºvyd qYvqYª«ª8��|-��hpoA|Y»§qS©Qc�¯ qY{�h�oyx-����d�{�|���£«¿rhp� s�o�£«����vyd�h��^ª¤|�� qYª8vy£«fSh�o�hg{y�O£¤�ph�{�hpÈ-s�hpoyv�hp�Ã�-xÅvyd�h�p|�f¨f�s���£¤�rqYvy£¤|�� ��{�|�v�|O�p|�ª½o�|Àvyd qYv�qYv�hrq���dÁ�E|�£«�Lv�£«�%vy£«fShShrq���d ��|-��h¨|Y»#qY�Á©Qc�¯�o�hg����ovyd�h2��qYv�qS�p|�{y{�hpoy�1|����Z£«����v�|�vyd�h�o�qYfSh2��{�|O�Zs��gvy£¤|�� �gxZ�gª¤h^¸

c�d�h�ª¤hg���^vyd%|Y»½vyd�h�c�´�µH¶ {�|�s���� 1�£¤o2qJ»[s����gvy£¤|��%|Y»½vyd�h��-s�f��Ehg{�|Y»½��|-��hpo��1|Y»§vyd�hfJqatZ£«fJqYª�oy£«¿rhÀ|Y»wvyd�hufShpo�o�qY��hÀo�hg�-v�£«�'hrq���d:oyª¤|�vr�½qY��� |��'o�|�fSh���d qY{�q��gv�hg{y£¤oyvy£¤�poJ|Y»kvyd�h��hgv¢¦§|�{y³ÃqY���Ã|Y»Qvyd�h2�p|�f¨f�s���£¤�rqYvy£¤|��Ç�p|��Lvy{�|�ª«ª¤hg{�or¸§c�d�h2�^qYª«s�h2|Y» 1�£¤o�vyd-s�o���|�v#��hg��hg{�qYª«ª«x�p|�{y{�hgª qYv�hp��¦�£«vyd�vyd�h���{�|O�Zs��gvy£¤|����Ehg{y£¤|O��|Y»�vyd�h¥��qYv�q��r¸®~­»�� ��� � or¸ vr¸�� , �A1'vyd�hg��o�|�fSh��qYv�q�fJqrx ��|�v��1h�vy{�qY��oyf¨£«vyv�hp�Ǧ�d�£¤��d%£¤ok��hg��hg{�qYª«ª«x�s�� q��p�phg��v�qY��ª¤h^¸�~­» ; �� � ! � , � 1vyd�hg�%vyd�h¨oCqYfSh¨��qYv�qÀ£¤o�vy{�qY��oyf¨£«vyv�hp�%£«�%fS|�{�h¨vyd qY�º|���hS{�|�s���� ¸�¶kª¤o�|�� £Æ»¾vyd�h¨�Ehg�^£«����£«���|Y»8vyd�hw��{�|O�Zs��gvy£¤|��Ã�gxO�gª¤hA��|Ohpo§��|�v��p|�{y{�hpoy�E|����uv�|�vyd�hw�Ehg�^£«����£«���S|Y»�q�{�|�s����ÀqY���u£Æ»8©Qc�¯� d q�o�fS|�{�hkvyd qY�Ã|���hA{�hg��ª«£¤�rqZ�Zvyd�hg�Å��qYv�q��p|�{y{�hpoy�E|����Z£«���¨v�|��Z£a` hg{�hg�Lv���{�|O�Zs��gvy£¤|��Ã�gxZ�gª¤hpofJqrxÅ�1h�vy{�qY��oyf¨£«vyv�hp��£«� vyd�h�oCqYfSh�{�|�s����Hq�ow£«vk£¤o#vyd�h��rq�o�h�£«��vyd�h(W�{�oyv�qY���Ãvyd�£«{���{�|�s����|Y»Qvyd�h2hMt�qYf¨��ª¤h��Z{�qp¦��H|�� ©�£«�^s�{�h��Z¸

���� �� $(��� ����� #&� ��(���#���(+�� �)#&% * #&� � ����� ��(���� %'(���#&,$��#��

c�d�h½�Ls�f��1hg{Q|Y»�{�hg��ª«£¤�rq�o8�Ehg{®©Qc�¯ ¦�d�£¤��d�£¤oU{�hpÈ-s�£«{�hp��v�|wv�|�ª¤hg{�qYv�h��2»ÞqYs�ª«v�o®d�hrqp�O£«ª«x���hg�1hg����o|�� vyd�hS�Ehgd qp�O£¤|�{�|Y»�vyd�hJ£«���Z£«�-£¤�Zs qYª��p|�f¨�1|���hg�-v�o*@ONPB­¸Å©�|�{�£«��o�v�qY���ph^�8£Æ»#vyd�hS»ÞqY£«ª«s�{�h�|Y»����|O��hpoQf�s�o�v��Eh¾v�|�ª¤hg{�qYv�hp� ��vyd�h�ª¤hrq�oyvQ��hp�phpo�oCqY{yx��Ls�f��1hg{¥|Y»1{�hg��ª«£¤�rqYv�hp�¨��|O��hpoQ£¤o � ! �¾¦�d�hg�qYª«ª ��|O��hpowqY{�h�" ����E U �F��EG�� :�Þ¸�¶ ��|O��h�£¤o#o�qY£¤�Å»ÞqY£«ªUoy£«ª¤hg�-v#£Æ»

i^¸wq�²�£«v§o�hg����o�»±{�qYfShpo§qYv½vyd�hw�p|�{y{�hp�gv§�E|�£«�Lv¾|Y»1vy£«fSh�°±�p|�{y{�hp�gvy��hpo�o¾£«�Svyd�h#vy£«fSh#��|�fJqY£«��²qY���Å��²§vyd�h��p|�{y{�hp�gv#�^qYª«s�h2£¤o�vy{�qY��o�f¨£«vyv�hp� °±�p|�{y{�hp�gvy��hpo�ow£«�Åvyd�h2��qYª«s�h2��|�fJqY£«��²M�

�Z¸#|�{w£«vwo�hg����ow��hgv�hp�gv�qY��ª«xã«���p|�{y{�hp�gvk»[{�qYfShpo�°±hg��¸¾¦�{�|����Àʾl#Ê�²�£«��£«v�ow|r¦��Hoyª¤|�vk|�{w��|»[{�qYfSh�qYvwqYª«ª¼¸

"$#&% ó§ïð÷�ö¼ó&�±ö �Móÿý��ÞøòæMç�æ �1ö % ó #'# ê�(*) ��+^ó4þÿø-,Yþy÷�ö¼øòæCç/. �Cú10�óÿç�÷12aïòó&��ö % ó§ôaó �Þø43CçYóÿý�ö¼æwøòç��ÞóÿýÞö�÷Cç�øðôaïòó¾ö¼ø � ó÷�� ö¼ó¢ý�ö % ó§öÞý­÷�ç�� � ø � �ÞøòæMç�æ � ÷ � ý­÷ � ó �Þækö % ÷�öQö % ó�ô �aý­÷�ö¼øòæCç�æ�� ÷wý¼æ �YçYô�þ4÷Cç�ö­÷65gó�÷�ç�÷6+7+aïðøòþy÷�ö¼øòæMç�ý¼óÿï ÷�ö¼óyô�M÷�ï �aó��ZéXç�+Y÷�ýÞö¼øòþ&�Yïð÷�ý�ø ö þ4æ �Yïðô�2^óEó98 �Y÷Cïrö¼æ�ö % óEïòó4ç:3Cö % æ �^ö % ó;+aý¼ærô�aþÿö¼øòæMç§þÿå�þ4ïòó æ ��÷Qôa÷�ö­÷�2�aö�ö % ó&+aý¼æ12Yïòó �ý¼ó � ÷Cøòç��&<Uø ö % ôa÷�ö­÷ % ÷���øòç:3�ôaø>=�óÿý¼óÿçrö?+�ý¼ærô�aþÿö¼øòæMç�þÿå�þ4ïòó&�&�

é[â��®é±î

Page 12: Maximizing the Robustness of TDMA Networks with ... · Maximizing the Robustness of TDMA Networks with Applications to TTP/C Bruno Gaujal, Nicolas Navet To cite this version: Bruno

�������������� ��������������������� ���� �!�#"%$'&(�*),+-����./�10 2�� �

� ,��� # ��� # ��� # ��� # ��� ���� # ��� # ��� ��

©�£«�^s�{�h�� �½c�d�{�hph�o�s��p�phpo�oy£«��h2{�|�s�����or¸��k��ª«xÅvyd�h�oyª¤|�v�oAqYª«ª¤|-�rqYv�hp� v�|¨vyd�h�©Qc�¯ � |Y»¥�rqY{��Z£Æ¬� qYª«£«v¢x��JqY{�h�oyd�|�¦��U¸kc�d�h�fShpo�o�qY��h��p|�{y{�hpoy�1|����Z£«���Àv�|�vyd�h:I >�! �8N����J��{�|O�Zs��gvy£¤|��H�gxZ�gª¤h�£¤oo�hg�Lvw|���hg{k��{�|�s�����or¸

c�c�z¾É^Ê3��{�|��-£¤��hpoA��hg{yxÃ��|O|O��oys����E|�{yvk»±|�{kvyd�h�{�hpÈ-s�£«{�hgfShg�Lv�oSi^¸òq�²kqY���Ç��²�°[¦�d�|^o�h�»±s�ªaW�ª«ªÆ¬fShg�-vk��{�|��-£¤��h�vyd�h�o�|Y¬­�rqYª«ª¤hp�D=¼»¼qY£«ªÆ¬­oy£«ª¤hg���ph�£«��vyd�h�v�hgf¨�E|�{�qYª���|�fJqY£«��?ÿ²whpoy�Ehp�g£ qYª«ª«xÅvyd�{�|�s��^dvyd�h���s�ow�^s qY{��Z£ qY� �p|����phg��vw¦�d�£«ª¤h�vyd�h2��qYª«s�h2��|�fJqY£«��£¤o�fJqY£«��ª«xÀvyd�h�{�hpoy�E|���oy£«��£«ª«£«v­xÅ|Y»Qvyd�hqY����ª«£¤�rqYvy£¤|��U¸ c�d�h {�hrq���hg{J£¤o¨{�hM»Þhg{y{�hp�:v�| @ON��#i �Z�#i �CB#»Þ|�{S��|O|-�:o�v�qY{yvy£«���Á�E|�£«�Lv�oJ|��.vyd�h��{�|���ª¤hgf·|Y»®hg��oys�{y£«����»ÞqY£«ªÆ¬­o�£«ª¤hg���ph^¸�©�|�{¾©Qc�¯Ao½�p|�f¨�E|^o�hp�u|Y»8q�o�hgv¾|Y» »ÞqY£«ªÆ¬­oy£«ª¤hg�-v§��|O��hpor�Lvyd�ho�s��p�phpo�o4»±s�ª®vy{�qY��oyf¨£¤o�o�£¤|��Ç|Y»¾|���h�oy£«���^ª¤h�»±{�qYfSh�»Þ|�{Avyd�h�¦�d�|�ª¤h�o�hgv�|Y»½{�hg��ª«£¤�rq�o�£¤oAoysJ�g£¤hg�Lvo�£«���phJvyd�h���qYª«s�hÀ�rqY{y{y£¤hp�Ä�LxÁvyd�hS»±{�qYfShu£¤o���hp�phpo�oCqY{y£«ª«xÁ�p|�{y{�hp�gvr¸%~­�Ävyd�£¤o��rq�o�h^�Qvyd�h�fJqY£«�|��Z}yhp�gvy£«��h�v�|Sq���d�£¤hg��h�¦�£«vydÅ{�hg�LqY{��Åv�|�vyd�h2{�|���s�oyvy��hpo�o#qY�LqY£«��o�v#vy{�qY��oyf¨£¤o�oy£¤|��Ãhg{y{�|�{�o#£¤o�vyd�hf¨£«��£«f¨£«¿p£«���S|Y» ���! ­�Ovyd qYv�£¤o�vyd�hk��{�|�� qY��£«ª«£«v¢xÀvyd qYv�qYª«ª1»±{�qYfShpo�|Y»®vyd�h2©Qc�¯�°±�rqY{y{yxO£«���S��qYv�q�p|�{y{�hpo��1|����Z£«���Sv�|Svyd�h�o�qYfSh���{�|O�Zs��gvy£¤|����gxZ�gª¤h�²§¦�£«ª«ªU�1h2�p|�{y{ys���v�hp� ¸

~­����{�q��gvy£¤�ph½{�hg��ª«£¤�rqYv�hp��o�hg��o�|�{�o8fJqpx2{�hgvys�{y��oyª«£«�^dLvyª«x��Z£a`Ehg{�hg�-v�|���o�hg{y�^qYvy£¤|���o�qY��� �Y¦�£«vydZ¬|�s�v�hMtZvy{�q��p|�f¨f�s���£¤�rqYvy£¤|��'»Þ|�{�qY�'qY�^{�hphgfShg�Lvr�½{�hg��ª«£¤�rqYv�hp� ��|-��hpo�|Y»Aq�o�qYfShu©�c�¯ fJqpxvy{�qY��o�f¨£«v��Z£a` hg{�hg�Lv¨��qYv�qZ¸ ~­»kqÃ��hp�g£¤o�£¤|��U�¥oys���d'q�o�qÅfJqa}y|�{y£«v­xÁ��|�v�h^�Q£¤o�v�qY³�hg�Ä�Lx q �p|��Z¬o�s�fShg{¨��|-��hŦ�£«vyd.{�hg�LqY{��'v�|Çvyd�hÅ�^qYª«s�hÅ|Y»Avyd�hÅvy{�qY��o�f¨£«vyv�hp�:��qYv�qZ�¾vyd�hÅ|��Z}4hp�gvy£«��h £¤o¨v�|f¨£«��£«f¨£«¿rh ��$ %�'kvyd�h2��{�|�� qY��£«ª«£«v­xÅvyd qYvw|���hS°±|�{wfS|�{�h�²¾{�hg��ª«£¤�rq¨|Y»¥q�fShpo�o�qY��h�¦�£«ª«ªU�Eh�ª¤|^oyvr¸

©�{�|�f qY�ã«f¨��ª¤hgfShg�-v�qYvy£¤|��Ç�1|�£«�-vw|Y»��-£¤hg¦���£«vw£¤o#��hg��hg{�qYª«ª«xÅ��{�hM»Þhg{�qY��ª¤h�v�|¨��{�hpo�hg�-vk|���ª«x|���h��p|��LxH|Y»¾q���qYv�qJv�|�vyd�h�qY����ª«£¤�rqYvy£¤|��%£«�Ç|�{���hg{Av�|Àoy£«f¨��ª«£Æ»±xÃvyd�h�qY����ª«£¤�rqYvy£¤|��º�p|O��h�qY���v�|�³�hphg�H£«vk£«����hg�Ehg����hg�Lv�|Y»½vyd�h�ª¤hg��hgª�|Y»¥{�hp�Zs�����qY���gx °[£¼¸ h^¸k�Ls�f��1hg{�|Y»¥��|O��hpo��p|�f¨�1|^o�£«���vyd�h�©�c�¯2²M¸�~¢���2Ò ����ÉaÓA´AR� 2v�hg{yf¨£«��|�ª¤|��^x���vyd�h�qYª«��|�{y£«vyd�f {�hpoy�1|���o�£«��ª¤hA»±|�{�vyd�h���d�|�£¤�ph�|Y»vyd�hA�^qYª«s�hAvyd qYv¾¦�£«ª«ªE�EhAvy{�qY��oyf¨£«vyv�hp�Àv�|�vyd�h�qY����ª«£¤�rqYvy£¤|��Å£¤o¾v�hg{yfShp� QMvyd�h2qY�^{�hphgfShg�LvAqYª«��|Y¬{y£«vyd�f Q�¸k~­vk£¤ows�o�s qYª«ª«xÃ��hp�phpo�o�qY{yxÅvyd qYvAvyd�h�qY�^{�hphgfShg�-v�qYª«��|�{y£«vyd�f ¦¾|�{y³OoA¦�£«vydÇ�p|���oy£¤oyv�hg�-v��qYv�q�vyd qYv2£¤o���qYv�qÀ�p|�{y{�hpoy�E|����Z£«���Åv�|uvyd�h�oCqYfSh¨�gxO�gª¤h�|Y»½��{�|O�Zs��gvy£¤|��º|�{�o�qYf¨��ª¤hp�%qYv�vyd�h

� ��ãaá�� (�$����Hø ��÷ +aý¼æ �±ó4þÿöQæ � ö % ó½÷ ��ö¼æ � æCö¼ø �gó¥øòç�ô ���±öÞýÞå�÷Cø � øòç73#÷�ö�2��aøðïðô�øðç:3 �±ö­÷Cç�ôa÷�ý­ô�÷�ý¼þ % ø ö¼ó4þ¢ö��aý¼ó�ÆæCý®øòçaè �gó % øòþ4ïòó�þ4æMçpöÞý¼æCï �açYø ö��&���Qóÿö­÷�øðïòóyôkøðç �Ææ�ý � ÷�ö¼øòæMç�þ4÷Cç�2^óQæ12aö­÷�øðçaóyô�÷�ö % öÞö +�� (1( <'<'< � æ��Þó 5rè ��ô�� � æCý 3 �

� � çedMìgí #ÿì

Page 13: Maximizing the Robustness of TDMA Networks with ... · Maximizing the Robustness of TDMA Networks with Applications to TTP/C Bruno Gaujal, Nicolas Navet To cite this version: Bruno

ir� �����A�������1E�� +�'+>��P� �

oCqYfShS�E|�£«�Lv�£«�Ävy£«fSh^¸ÅµÇqY�Lx qY�^{�hphgfShg�-v�o�vy{�qYv�hg�^£¤hpo�qY{�hJ�1|^o�o�£«��ª¤h ����£¤��³-¬¢qY�Lx=°X»¼qY£«ªÆ¬­oy£«ª¤hg�-v��|O��h�²M�#qp��hg{�qY��hM¬¼�^qYª«s�h^�#��£¤��³L¬¢qa¬¼� qY{yvy£¤�gs�ª qY{4¬­|���h:°[vyd�h�o�hgª¤hp�gv�hp�:�^qYª«s�hÃd q�o��Ehphg�.��{�|O�Zs��php��-xºvyd�hu�1hpo�v�o�hg��o�|�{C²M�QfJqa}4|�{y£«v¢xÁ��|�v�hÀhgv���¸p¸p¸y¶ ��{�|��1|^o�qYª�|Y»Aq�o�|Y»±v­¦�qY{�h�ª qrx�hg{¨{�hpoy�E|���oy£Æ¬��ª¤hû±|�{J£«f¨��ª¤hgfShg�Lvy£«��� vyd�h�qY�^{�hphgfShg�-vuoyvy{�qYv�hg�^x'd q�oJ�1hphg�ÂfJq���hÃ�-x'vyd�h��2Ò ����ÉaÓ�´�R�p|���o�|�{yvy£«s�f @«ir�PB­¸

� ���� ��������� ��9�����·6��w� �!5�/1�#��������6��y� �~­» |���h§¦¾qY�Lv�o¥v�|w��s�£«ª¤��vyd�h¾oyxOo�v�hgf £«��oys���dSqw¦¾qrx�q�o�v�|wf¨£«��£«f¨£«¿rh¾vyd�h¾ª¤|^o�o8��{�|�� qY��£«ª«£«v­x��-|���hf�s�oyv���|�v�v�q���³Oª¤h�vyd�h¾��{�|���ª¤hgf�|��JqA�Ehg{Q{�|�s����¨� q�oy£¤oQ��s�v�£«����hphp�S�p|���oy£¤��hg{�vyd�h���{�|�� qY��£«ª«£«v¢xvyd qYv2qYª«ª8»[{�qYfShpok»±{�|�f vyd�h�o�qYfSh�©�c�¯ £«�Ç|���h���{�|-�Zs��gvy£¤|��%�gxZ�gª¤h�qY{�h��p|�{y{ys���v�hp� �E�Ehp�rqYs�o�hvyd�£¤o#fShrqY��o#vyd qYvw|���h2��qYv�q�£¤ow�p|�f¨��ª¤hgv�hgª«xŪ¤|^oyvr¸

~­��vyd�£¤o¥o�hp�gvy£¤|��U��¦§h�£«�-��hpoyvy£«�LqYv�h�vyd�h���{�|���ª¤hgf�|Y»Ef¨£«��£«f¨£«¿p£«����vyd�h�ª¤|^o�oQ��{�|�� qY��£«ª«£«v¢x ���! � ¢¸~¢�ÄÒ-s���o�hp�gvy£¤|��:�Z¸«i^��¦§hJ»Þ|-�gs�o�|��Ávyd�hJ|���vy£«fJqYª¾�E|�ª«£¤�gxÇ»±|�{�|���h�©Qc�¯�¸®~¢�ÄÒ-s���o�hp�gvy£¤|��'�Z¸ò�¦¾h2�p|���oy£¤��hg{wqYª«ª ©Qc�¯ �p|�f���£«��hp��qY���ÃqY�ud�hgs�{y£¤o�vy£¤��£¤o���{�|��1|^o�hp��»Þ|�{�vyd�h2�rq�o�hA¦�d�hg{�h�fS|�{�hvyd qY�uv­¦¾|¨�Z£a`Ehg{�hg�-v��rqY{��Z£«� qYª«£«vy£¤hpo��p|-hMtZ£¤oyvr¸�~¢v�o½�Ehg{4»±|�{yfJqY���phpo�qY{�h2q�o�o�hpo�o�hp�S�-xJoy£«f�s�ª qYvy£¤|��U¸

���� � *$��( � � �E�����+����� ��(+� , �?�"� %'(�,����+#�����)Ahg{�h�¦§h¨»±|O�gs�o2|�� qu�^£«��hg�Á©Qc�¯�� o�qpx � fJq���hJ|Y»� � % , {�hg��ª«£¤�rq�o2�Ehg{2{�|�s���� ��qYª«ª¥|Y»o�£«¿rh .�� . , ¸�c�d�h2��{�|���ª¤hgf d�hg{�h2£¤o�v�| W����ÃqY��qYª«ª¤|O�rqYvy£¤|��29�|Y»�vyd�h� {�hg��ª«£¤�rq�ow|r��hg{k|���h{�|�s����Svyd qYv�f¨£«��£«f¨£«¿rhpo¥vyd�h���{�|�� qY��£«ª«£«v¢x ���! � �vyd qYv§qYª«ª�{�hg��ª«£¤�rq�o½�rqY{y{yxO£«����vyd�h�o�qYfSh�fShpo�oCqY��hqY{�h�ª¤|^o�vr��{�hg�LqY{��Zª¤hpo�ow|Y»Qvyd�h2|�vyd�hg{w©Qc�¯Aor¸

c�d�h ��{�|O|Y»�v�hp��d���£¤È-s�hÇs�o�hpouv­¦¾|Ä��|�vy£¤|���oÀ£«�Lvy{�|O�Zs��php� £«� @ò�CB­��f�s�ª«vy£«fS|-�Zs�ª qY{y£«v¢x3qY���Ò-vys�{yf¨£ qY�Ho�hpÈ-s�hg���phpor¸

��� � � � � � ��� �������Z��� ÐL6 �8��� 6�� � � � ��� � Ð Ñ � �[���(�X� Í � 6 Ñ��E� �®�����[� 6 � Ì�� � Ì�6   Ì ��®hgv 9 �1huqÅ��£«� qY{yx%��hp�gv�|�{�|Y»#oy£«¿rhH1¨¸Å~¢v�o%K��� S�F�����¨£¤o�I ��� 1 N�� �B"! � 9 B ¸Å¶���£«� qY{yxÁ��hp�gv�|�{£¤o¨q_��EG��Z�2<U P�IZ �^�10J¦�£«vyd ��ª¤|-��³Oo¨|Y»�oy£«¿rh/. £Æ» 9 B �À|���ª«x £«�'£«�Lv�hg{y�^qYª¤o¨|Y» .'�p|���o�hp�gs�vy£«��h�^qYª«s�hpor¸�¶ ��Ea�<Z�2*� ��� "��£¤o2q���hp�gv�|�{$# B oys���d%vyd qYv�# B I�%MN� F »±|�{�qYª«ª&%ÄhMt��phg��v�# B I > N ! �qY���'# B I > ! .�N )(��^¸ 0�q�oy£¤�rqYª«ª«xS£Æ» 9Å£¤o�q���ª¤|-��³L¬¼��hp�gv�|�{r�Zvyd�hg� 9�!*# B £¤o�qYª¤o�|�q���ª¤|-��³L¬¼��hp�gv�|�{o�£«f¨£«ª qY{wv�| 9H¦�£«vyd�|���h�|Y»Q£«v�ow��ª¤|-��³Ookoyd�£Æ»±v�hp�Ãv�|¨vyd�h�ª¤hM»[v#�-xÃ|���h2s���£«vAq�ow£«� vyd�h�»Þ|�ª«ª¤|r¦�£«���hMt�qYf¨��ª¤h2¦�£«vydÃ��ª¤|O��³Zo#|Y»�oy£«¿rh��Z¸

9 I F ! F ! F ! F ! � ! � ! � ! F ! � ! � ! � ! F ! F ! � ! � ! � ! F N9 !+#-, I F ! F ! F ! � ! � ! � ! F ! F ! � ! � ! � ! F ! F ! � ! � ! � ! F N '

é[â��®é±î

Page 14: Maximizing the Robustness of TDMA Networks with ... · Maximizing the Robustness of TDMA Networks with Applications to TTP/C Bruno Gaujal, Nicolas Navet To cite this version: Bruno

�������������� ��������������������� ���� �!�#"%$'&(�*),+-����./�10 2�� i^i

¶ ��EG���I��E �F��� "��/�I"]���GC���H�����A£¤o#qY�Å|��Ehg{�qYvy£¤|��Ã|��À��hp�gv�|�{�o�vyd qYv�oyd�£Æ»[v�o#qYª«ªE�^qYª«s�hpo�v�|�vyd�hª¤hM»±v��Lx���°[fS|-�Zs�ª¤|¨vyd�h�oy£«¿rh2|Y»�vyd�h���hp�gv�|�{C²�q�o�£«�Ãvyd�h�»Þ|�ª«ª¤|r¦�£«����hMt�qYf¨��ª¤h^¸

9 I F ! F ! F ! F ! � ! � ! � ! F ! � ! � ! � ! F ! F ! � ! � ! � ! F N� � I 9�N I F ! F ! � ! � ! � ! F ! � ! � ! � ! F ! F ! � ! � ! � ! F ! F ! F N '

¶ {�hrqYªU»±s����gvy£¤|����HI 9 N�£¤o ��EG��Z�2<U � ��Ea���T�X��K1��EG��0A¦�£«vyd���ª¤|O��³Zok|Y»½o�£«¿rh".H£Æ»¥vyd�h2»Þ|�ª«ª¤|r¦�£«���£«��hpÈ-s qYª«£«v­xÀd�|�ª¤��o#»±|�{wqYª«ª ��ª¤|O��³-¬¼��hp�gv�|�{�oE9�¸

; >�� �HI 9 ! # B N'!��HI 9 ! # � N ���HI 9 N?!��HI 9 !+# B !+# � N !q�owo�|-|�� q�o 9 !+# B ! 9 ! #�� ! 9 ! # B ! #���qY{�h�qYª«ª ��ª¤|O��³À��hp�gv�|�{�or¸

¶ �L���J0��� �� � ��� �� LZF���J¦�£«vyd ��hg��oy£«v¢x�� ���A£¤owq���£«� qY{yxÀ��hp�gv�|�{w|Y»�oy£«¿rh��Aoys���dÃvyd qYv��% �� %�������� (�� I�% ( �8N�������� ' °4i�²

©�|�{®hMt�qYf¨��ª¤h^��vyd�h§Ò-vys�{yf¨£ qY��o�hpÈLs�hg���ph¥¦�£«vyd���hg��oy£«v­x�� ����£¤o�� �! #" I F ! F ! � ! F ! F ! � ! F ! �8NM¸¶ ��EG��Z�2 �L���J0F� � �� P�IZ��^��0@9Ŧ�£«vydÅ��hg��oy£«v¢x .$� ��I%� !�I . ( �8N�� N§¦�£«vydÀ��ª¤|-��³Oo�|Y»�oy£«¿rh(.Å£¤o

�p|���o�vy{ys��gv�hp�À»[{�|�f&�J£«�Ãvyd�h�»Þ|�ª«ª¤|r¦�£«���J¦¾qpx�¸' Ò-v�qY{yv#¦�£«vyd29Hhgf¨��v­x�¸' ~¼»(� B�) i^��vyd�hg�29 � 9 ' � KLKLK �^�®°[¦�£«vyd .Ã|���hpow�p|����rqYv�hg� qYv�hp�HqYv#vyd�h2hg���Ã|Y» 9E²M¸' ~¼»(� B F ��vyd�hg�29 � 9 ' F ¸

ʾ|��-vy£«�Ls�£«����vyd�h�hMt�qYf¨��ª¤h^�^vyd�h���ª¤|O��³SÒ-vys�{yf¨£ qY�S��hp�gv�|�{�¦�£«vyd¨��ª¤|O��³Zo¥|Y»1o�£«¿rh��A¦�£«vydS��hg��o�£«v­x* � �,+�£¤o#��hg{y£«��hp�À»±{�|�f�� �! #" s�oy£«���¨vyd�h2��{�|O�php�Zs�{�h�qY�E|r��h �

9� I F ! F ! � ! � ! � ! F ! F ! � ! � ! � ! F ! � ! � ! �8N '�A|�v�h�vyd qYv 9�£¤o®��|�v8hpÈ-s qYªLv�|��- � , �avyd�h¾Ò-vys�{yf¨£ qY��o�hpÈ-s�hg���ph¥¦�£«vyd���hg��oy£«v­x * � �,+��^o�£«���ph��.- � ,��|Ohpo���|�vw�p|��-v�qY£«� ��ª¤|O��³Zo#|Y»Qo�£«¿rh�� �/��- � , JI F ! � ! F ! � ! � ! F ! � ! � ! F ! � ! � ! F ! � ! � NM¸#©�|�{wfS|�{�h��hgv�qY£«ª¤ok|��Åf�s�ª«vy£«fS|O�Zs�ª qY{y£«v­x�qY���HÒ-vys�{yf¨£ qY��o�hpÈ-s�hg���ph^�Zvyd�h2{�hrq���hg{#f¨£«�^d-vw{�hM»Þhg{�v�|D@ �Z���PB­¸

�k��h§�rqY��qY����ª«x�q#��hg��hg{�qYªZ|���vy£«f¨£«¿�qYvy£¤|��¨c�d�hp|�{�hgf �^£«��hg��£«�%@ò�PBOv�|#vyd�h½��ª¤|-��³��rq�o�h^¸�c�d�£¤ovyd�hp|�{�hgf {�hgª qYv�hpo#vyd�h2f¨£«��£«f¨£«¿p£«���u|Y»�f�s�ª«vy£«fS|-�Zs�ª qY{w»[s����gvy£¤|���ow¦�£«vydHÒ-vys�{yf¨£ qY� ��hp�gv�|�{�or¸0 ; Ì-Ð � Ì � � �2143�576 "�� �T�X����EG��Z�2<U � ��Ea���T�X��K1��EG��0 "F�J LZ ��� �� �]���S�� Z �� S�F� K���0%�����X� ��� "��A�T UP��0F� �� :�+P��0F�F� �� �I"8� �A L������E �D������"��J LZ ��� �� :9 I 9 N�� ��� 1 � �<; �B !>= �HI%� B I 9�N N ' $ �S�� ?9 � �� �� �����<�YK(��P��0A��E�ES��Ea��Z�2���IZ �^��0F��. �����K��� S�F�����A@*� �(�����]��EG��Z�2 �L���J0F� � �� P�IZ �^�10M�#" K��� S�F�����@ �

� � çedMìgí #ÿì

Page 15: Maximizing the Robustness of TDMA Networks with ... · Maximizing the Robustness of TDMA Networks with Applications to TTP/C Bruno Gaujal, Nicolas Navet To cite this version: Bruno

ir� �����A�������1E�� +�'+>��P� �

��� � � " � � �X��� � Ð Ñ ���[����� � Í'Ð:9 �<; Ì � Ð ��� � � Ð ����� � � �X� Í)Ahg{�h^�^¦¾h���{�|r��h#vyd qYv�vyd�h�ª¤|^o�oQ��{�|�� qY��£«ª«£«v­x ���! �£¤o���ª¤|O��³-¬¼f�s�ª«vy£«fS|O�Zs�ª qY{r¸��Uhgv¥s�o W�{�o�v�fJqY³�hvyd�hA»Þ|�ª«ª¤|r¦�£«���Àq�o�oys�f¨��vy£¤|����

��� , � ~­�'vyd�hÀ£«��£«vy£ qYª«£«¿�qYvy£¤|��Â��d q�o�h^�§¦§h q�o�o�s�fShÀvyd qYvSvyd�hÅoyv�qY{yvS|Y»�vyd�hHW�{�oyv¨��{�|-�Zs��gvy£¤|���gxO�gª¤h�£¤o�s���£Æ»±|�{yf¨ª«xÅ�Z£¤oyvy{y£«��s�v�hp��|r��hg{#vyd�h�W�{�o�v�{�|�s���� ¸

�T�A¹ºhJ�p|���oy£¤��hg{¨qÅoy£«���^ª¤h�hg{y{�|�{���s�{�o�v�vyd qYv��Ehg�^£«��o�qYv�qÅvy£«fShJs���£Æ»±|�{yf¨ª«xº�Z£¤oyvy{y£«��s�v�hp�|r��hg{#vyd�h�� q�oyvwqY���Ŧ�d�|^o�h�o�£«¿rh�£¤o#hMtO�E|���hg�Lvy£ qYª«ª«xÃ�Z£¤o�vy{y£«��s�v�hp�Ŧ�£«vydÃ� qY{�qYfShgv�hg{��®¸

�½Ì � �º� � ��� �� S����K���0��X�F�� ���E��-��0F0I��0!���J0 ���/���J�8�e�Y� ���� S�!�1�e�X�����X�(�� :� " ��0��E ��K��T����0�� ���J� �IK� P��0*�����>\��P�����1 LK . �J�P�����F��<��� � �I�F\��� �� :��� ��ETE � K�� ����0F�^����� �IK�. �����H\���0I������� ��0�� � � LK���0�1 LK �P� �����]\���� �� S��� � ��� �� � �� �� LK�� � , � ���I��P� ������!\L0I���I�����TE ����� ��� � �#"�EG�P�F�� �� �1ETE0Y� \LE � ZY�P��#" ��$ � � �T�-��Ea��Z�2X��JEa���T�X��K��JEG��0!�e. ����� ��EG��Z�2���I"��F��<� . , �� 0#���I" �w¹ºh2�p|���oy£¤��hg{ 9ÇqY� qY{y��£«vy{�qY{yxÅqYª«ª¤|O�rqYvy£¤|��À»Þ|�{ � ¸Q¹ºh�ª¤|O|�³ÀqYv�vyd�h2��{�|�� qY��£«ª«£«v¢xÀvyd qYvqY�Åhg{y{�|�{��p|�{y{ys���v�o#qYª«ªE{�hg��ª«£¤�rq�o��rqY{y{yx-£«���Sq��^£«��hg�ÀfShpo�o�qY��h� »±|�{�qYª«ª¤|-�rqYvy£¤|��/9�¸�c�d�h�o�qYfShfShpo�o�qY��hS°� �²§£¤o�hgf¨£«vyv�hp�Ã�-xÀq��-s�f��Ehg{w|Y»�{�hg��ª«£¤�rq�o#¦�d�£¤��d��rqY�Å�Eh�¦�{y£«vyv�hg��q�o�� ¦�d�hg{�h� £¤o�qY�Ä£«�Lv�hg��hg{r�½qY��� � % , £¤o�vyd�h��Ls�f��1hg{¨|Y»#{�hg��ª«£¤�rq�o��Ehg{�{�|�s���� ¸H©�|�{���|�v�qYvy£¤|��o�£«f¨��ª«£¤�g£«v­x���¦¾h�qYª¤o�|¨o�hgvE. � �. , ¸

~­� vyd�h¨»Þ|�ª«ª¤|r¦�£«���H¦§hÀqYª¤o�| ��hg��|�v�h��Lx % vyd�hS{�|�s���� ¦�d�hg{�h�fShpo�o�qY��h� �1hg�^£«��o�¸À¹ºh��hg��|�v�h��Lx�����vyd�h��1|^o�£«vy£¤|�� |Y»8vyd�h2ª q�o�v���£«v#|Y»�vyd�h �-¬¼vydÃ{�hg��ª«£¤�rq�»±|�{�vyd�h2©Qc�¯ � £«�/9ÇqY����-x�@ B � vyd�h!Qg�Z£¤o�v�qY���ph8Q#�Ehgv­¦¾hphg�u{�hg��ª«£¤�rq �JqY���J{�hg��ª«£¤�rq � ! >!�/@ B � ������ B (����-¸�¹ºhk��hg��|�v�h�-x ���! I 9 N¥vyd�hAª¤|^o�o¾��{�|�� qY��£«ª«£«v¢xu|Y»� s�����hg{#qYª«ª¤|O�rqYvy£¤|���9��O�Lx � = I 9 N½vyd�hkª¤|^o�o§��{�|�� qY��£«ª«£«v¢xs�����hg{JqYª«ª¤|O�rqYvy£¤|��?9:�^£«��hg� vyd qYv¨vyd�hÀ�Ehg{yvys�{y� qYvy£¤|��:oyv�qY{yv�o�£«�:q {�|�s���� ��{�hp�php�Z£«���Á{�|�s����% qY�����Lx � � I 9 N�vyd�h�ª¤|^o�o#��{�|�� qY��£«ª«£«v¢x�s�����hg{2qYª«ª¤|O�rqYvy£¤|�� 9Ç�^£«��hg�Hvyd qYvkvyd�h��1hg{yvys�{y� qYvy£¤|��o�v�qY{yv�o�£«�uvyd�h�o�qYfShA{�|�s����º° % ²M¸8¹=d�hg�Åvyd�h �¥µH~���s�{�oyv#�p|r��hg{�o�vyd�h�¦�d�|�ª¤hAfShpo�oCqY��h^�Zvyd�hg{�h£¤oSq�{�hgª qYvy£¤|��:�Ehgv­¦¾hphg�:vyd�hÀ{�qY����|�f �^qY{y£ qY��ª¤hpoS�p|�{y{�hpoy�E|����Z£«���ºv�|Hvyd�hÀ�1hg�^£«����£«���º|Y»Avyd�hfShpo�o�qY��h� °±�rqYª«ª¤hp� # ²�qY���Çvyd�h��1hg�^£«����£«��� |Y»�vyd�h¨hg{y{�|�{���s�{�oyvJ°±�rqYª«ª¤hp� -¥²k{�hpoy�Ehp�gvy£«��hgª«x�¸0�q�oy£¤�rqYª«ª«x��-vyd�hwhg{y{�|�{§f�s�oyv¾oyv�qY{yv½�EhM»Þ|�{�h�vyd�hwhg����|Y»Uvyd�h�W�{�oyv¥{�hg��ª«£¤�rq��rqY{y{yxO£«����fShpo�o�qY��h� º¸

ÒOhph�©�£«�^s�{�hpow�SqY���À�¨»Þ|�{wqY�Å£«ª«ª«s�oyvy{�qYvy£¤|���|Y»��rq�o�hpo�� = qY��� � � {�hpoy�1hp�gvy£«��hgª«x�¸

é[â��®é±î

Page 16: Maximizing the Robustness of TDMA Networks with ... · Maximizing the Robustness of TDMA Networks with Applications to TTP/C Bruno Gaujal, Nicolas Navet To cite this version: Bruno

�������������� ��������������������� ���� �!�#"%$'&(�*),+-����./�10 2�� ir�

������������ ������������ ������

������ ������������ ��

�� ������������

ãrö­÷�ýÞö®æ��Zö % ó +^ó¢ý 2��aý 2Y÷�ö¼øòæMç�� ��� ä-óÿç73Cö % æ �Zö % ó +^óÿýÞö��aý 2Y÷�ö¼øòæMç����ý¼æ��aç�ô�� ý¼æ��YçYô������

� ó&� �¼÷13Có�� ø � �Þóÿçrö®í½ö¼ø � ó �U÷Cç�ôkøòö�2^ó 3Cøòç��÷�ö8÷ �Yçaø �ÆæCý � ý­÷�ç�ô�æ � ö¼ø � óQøòçký¼æ �Yç�ô�� � ���çYó�ý¼ó +aïòøðþ4÷�æ �Zö % ó¥ü #�� ø �®öÞý­÷�ç�� � ø öÞö¼øòç73 � ó&� �¼÷13Có��

©�£«�^s�{�hÃ� �À¶ �1hg{yvys�{y� qYvy£¤|��:��s�{�o�v¨¦�d�£¤��d:�Ehg�^£«��o�£«�.qH{�|�s���� ��{�hp�php�Z£«���ºvyd�hÃoyv�qY{yv¨|Y»�qfShpo�o�qY��h2�p|r��hg{�owvyd�h2¦�d�|�ª¤h2fShpo�o�qY��h�°±�rq�o�h�� = ²M¸

������ � � !�!!�!"�""�" #�##�#

$�$$�$ %�%%�%&�&&�& '�''�'

(�((�()�))�)*�**�*ý¼æ �YçYô�� ý¼æ �YçYô������

� ó&� �¼÷63Mó��

� �

©�£«�^s�{�hA� ��¶!�Ehg{yvys�{y� qYvy£¤|��Å��s�{�oyvr�Z�Ehg�^£«����£«���¨£«��vyd�h2o�qYfShk{�|�s����Ãq�o¾vyd�hAfShpo�o�qY��h^���p|r��hg{�ovyd�h�¦�d�|�ª¤h�fShpo�o�qY��hS°±�rq�o�h � � ²M¸

� � çedMìgí #ÿì

Page 17: Maximizing the Robustness of TDMA Networks with ... · Maximizing the Robustness of TDMA Networks with Applications to TTP/C Bruno Gaujal, Nicolas Navet To cite this version: Bruno

ig� �����A�������1E�� +�'+>��P� �

0§xH�p|����Z£«vy£¤|���£«����|���hg{�vyd�h���qYª«s�hpo2|Y» # qY������°[¦�d�£¤��d qY{�h�£«����hg�Ehg����hg�Lv���qY{y£ qY��ª¤hpoM²M�¦¾h�|���v�qY£«���

� � I 9�N ��� � �3� ! ��� � �

� I � (�I � (��6N�� N� � �� I ��� ; � ( .�� # � ��� ( .�N �� I % 1J< -�<���� N�� I�� ! -���1 � ( @ �� ; � ( . ! ��� N�

I � (�I � ( �6N � N � �1 � ��� � �3

� ! ��� � � @�� ; �������� ������� I �MI ( 1 � !�@ �� ; � !�. ( ��� ! -�N N�@�-

I � (�I � ( �6N � N � �

� 1 � �3� ! �

@ �� ; � ����� I �MI ( 1 � !�@ �� ; � !?. N N I � ( ����� I ( � ��� N N !qY���

� = I 9 N ��� � �3� ! ��� � � I � ( I � (��6N � N � � �� I ��� ; � (E.�� # � ��� (E.�N �� I���! -���1 � !�����(@

�� ; � (E. NI � (AI � ( �6N � N � �1 ��� � �3

� ! ��� � � @�� ; � �� I�� ! -��A1 � ! ��� ( @ �� ; � N

I � ( I � ( �6N � N � �1 ��� � �3

� ! ��� � � @�� ; � � � �

= ����� I �6I ( 1 � ( ��� ! - !�@ �� ; � N N�@ -�� % 1� � � I � (�I � ( �6N � N � �

� % 1 ��3� ! �

@ �� ; � ����� I ( � 1 � ( � ��� ! � @ �� ; � N I � ( ����� I ( � % 1 N N '©8£«� qYª«ª«x��

��� � I 9�N ����I % ! �8N � � I 9 N?! % ��I % ! �8N � = I 9�N � �3

� ! �@ �� ; � I I � ( ����� I ( � ��� N N?! ����� I ( � ��� N I � ( ����� I ( � % 1 N N N ����� I �$@ �� ; � N

� �3� ! �

@ �� ; � I � ( ����� I ( � ��� ( � % 1 N N ����� I � @ �� ; � N !

é[â��®é±î

Page 18: Maximizing the Robustness of TDMA Networks with ... · Maximizing the Robustness of TDMA Networks with Applications to TTP/C Bruno Gaujal, Nicolas Navet To cite this version: Bruno

�������������� ��������������������� ���� �!�#"%$'&(�*),+-����./�10 2�� i�N

¦�d�hg{�h � ����� I (�� 1 � ! � . N I � ( I � ( �6N � N � �I % ! �8N � 1 � '¹ºh��p|���oy£¤��hg{woyd�£Æ»±v�o�v�|¨vyd�h�ª¤hM»±vw|Y»Qvyd�h���|�{yfJqYª«£«¿�qYvy£¤|�� �HI 9 N�� ��� � I 9 N ����¸

�:I 9 !+# B N �3

� ! ��� � �5�� � � � � ��� @�� ; � I � ( ����� I (�� ��� ( � % 1 N N ����� I �$@ �� ; � N

! I @ �� ; � ( �8N I � ( ����� I ( � � � ! � ( � % 1 N N ����� I �$@ �� ; � ( ��N! I @ �� ! �8N I � ( ����� I ( � � � � � ( � % 1 N N ����� I � @ �� ! ��N '¹ºh2��hphp�Ãv�|S�Z£¤o�vy£«���^s�£¤oyd vyd�h��rq�o�hpo�¦�d�hg{�h�� � ( �� �^¸§¹ºh2»Þ|-�gs�ow|��Ãvyd�h��rq�o�h�¦�d�hg{�h

� � ! �J°[vyd�h��rq�o�h2� � ! �2£¤o#oyxOf¨fShgvy{y£¤�rqYªU�LxÅhMt���d qY���^£«����vyd�h�{�|�ª¤hpow|Y» �uqY��� �p²M¸~¼» � � ��! �^�

�:I 9 !+# B !+#�� N �3

� ! ��� � �5�� � � � � ��� �� � ��� @�� ; � I � ( ����� I ( � ��� ( � % 1 N N ����� I �$@ �� ; � N

! I @ �� ; � ( �8N I � ( ����� I (�� � � ! � ( � % 1 N N ����� I �$@ �� ; � ( � N! I @ �� ! �8N I � ( ����� I (�� � � � � ( � % 1 N N ����� I �$@ �� ! ��N! I @ � ; � ( �8N I � ( ����� I ( � � !�� ( � % 1 N N ����� I �$@ � ; � ( ��N! I @ � ! �8N I � ( ����� I ( � � � � ( � % 1 N N ����� I �$@ � !�� N '~¼» �� ��! �^��|��Ãvyd�h�|�vyd�hg{#d qY��� ��¦¾h2��hgv

�HI 9 !+# B !+#�� N �3

� ! ��� � �5�� � � � � ��� � � ��� @�� ; � I � ( ����� I ( � ��� ( � % 1 N N N ����� I �$@ �� ; � N

! I @ �� ; � ( �8N I � ( ����� I ( � � � ! � ( � % 1 N N ����� I � @ �� ; � ( � N! I @ �� N I � ( ����� I ( � � � � � ! � ( � % 1 N N ����� I �$@ �� N! I @ �� � � ! �8N I � ( ����� I ( � � � � � ( � % 1 N N ����� I �$@ �� ; � ! � N '

� � çedMìgí #ÿì

Page 19: Maximizing the Robustness of TDMA Networks with ... · Maximizing the Robustness of TDMA Networks with Applications to TTP/C Bruno Gaujal, Nicolas Navet To cite this version: Bruno

ir� �����A�������1E�� +�'+>��P� �

~¼» ¦§hA�p|�f¨��s�v�h��)� ��:I 9 ! # B N ! �:I 9 ! #�� N ( �HI 9 ! # B ! #�� N ( �HI 9 NM�O¦§hk��hgv F ¦�d�hg�� � � ! ��qY���æ�d�hg� � ��! �^��¦§h2��hgv

� I @ �� ! �8N I � ( ����� I ( � � � � � ( � % 1 N N ����� I �$@ �� ! ��N( I @ �� N I � ( ����� I ( � � � � � ! � ( � % 1 N N ����� I �$@ �� N! I @ �� ( �8N I � ( ����� I ( � � � � � !�� ( � % 1 N N ����� I �$@ �� ( ��N( @ �� I � ( ����� I ( � � � � � ( � % 1 N N ����� I �$@ �� N '¶#»±v�hg{#o�|�fSh2oy£«f¨��ª«£aW �rqYvy£¤|���or� ¦§h2��hgv� ����� I �6I @ �� ! �8N N�@ �� !�@ �� ����� I �MI @ �� ( �8N N&(���@ �� ����� I �$@ �� N! ����� I �6I @ �� ! �8N N&( ����� I �6I @ �� ( �8N N! ����� I �6I @ �� ( � � � � ( 1 % N N ( ����� I �MI @ �� ( � � � � ( 1 % ! �8N N� F '

c�d�h W�{�oyv�ª«£«��h¨£¤o2��|��Z¬¼��hg�LqYvy£«��hJ�Lx%�p|��L��hMtZ£«v­xº|Y»�vyd�h�»±s����gvy£¤|������� ����� I �� NM¸�c�d�hJoys�f|Y»¥vyd�h�o�hp�p|����ÇqY���Hvyd�£«{�� ª«£«��hpoA£¤o2qYª¤o�|���|��Z¬¼��hg�LqYvy£«��h��Lx �p|��-��hMtO£«v¢x�|Y»½vyd�h�»±s����gvy£¤|������

����� I �� NM¸c�d�h2��{�|-|Y»��rqYª«ª¤o�»Þ|�{#o�hg��hg{�qYª®�p|�f¨fShg�-v�or¸' ¹ºh#q�o�oys�fShp��vyd qYv�vyd�h�o�qYfSh¾fShpo�o�qY��h�£¤o�o�hg�Lv¥�Zs�{y£«����qY��£«�Lv�hg��hg{¥�-s�f��Ehg{½|Y»�{�|�s�����o�¸~­�H»¼q��gvr�1vyd�h���{�|-|Y»§�rqY�%�Eh�q���qY��v�hp�H»±|�{�vyd�h��rq�o�h�¦�d�hg{�hSq�fShpo�o�qY��h�|���ª«x��p|���hg{�o�q»[{�q��gvy£¤|�� qYª �-s�f��Ehg{#|Y»®{�|�s�����o�° � �T� �½|Y»®vyd�hk»Þ|�{yf � ! �M�O¦�£«vyd��½oyfJqYª«ª¤hg{�vyd qY��º²M¸

' ¹ºh¾�1hgª«£¤hg��h�vyd qYv�vyd�h¾f�s�ª«vy£«fS|O�Zs�ª qY{y£«v­x¨d�|�ª¤��o�»Þ|�{QfS|�{�h���hg��hg{�qYª��Z£¤oyvy{y£«��s�vy£¤|���o½|Y»�vyd�hhg{y{�|�{Ao�£«¿rh^¸*)A|r¦¾hg��hg{�¦§h���d�hp��³�hp�Hvyd qYvA£«vk��|Ohpo#��|�vwd�|�ª¤�û±|�{Az�qY{�hgv�|À�Z£¤oyvy{y£«��s�vy£¤|���o�¸c�d�h¨hMtO�E|���hg�Lvy£ qYª§�Z£¤oyvy{y£«��s�vy£¤|��Äq�o�o�s�f¨��vy£¤|��%£¤o�qYª¤o�|Å�g{ys��g£ qYª¥£«�%vyd�h���{�|O|Y»§vyd�h¨��hMtZvvyd�hp|�{�hgfø

0 ; Ì-Ð � Ì � " � � LK���0A���S�M�P�F�F�J�]\������1 S� � ��� � � � ��� � � � � � � �1 LK� � , � � ���S� \�0#���Y������E �������� � >�#" EG�P���T �����E�Eb0Y� \LE ��Z �P�H�#" ��$ � � " ��0���� ��[���������1�X�����F�F�������%�T� � �T ��� �GC�IK � "X�����0Y� \LE � ZY�P���0Y���E�Ea��Z �1� �IK ��P��0 �I�8Z��D0#���J LK*�8ZYZ ��0#K1�T ��*�^�*�*��EG��Z�2 �L���J0F� � �� [� ��� �� LZF� �

� 0#���I" � 0¾x��p|���oy£¤��hg{y£«����|���ª«x���s�{�oyv�oQ�Ehgv­¦¾hphg�Svy£«fSh F qY����{�|�s���� % ��¦§h#�rqY�Jq�o�o�s�fSh§s�o�£«���� � � � vyd qYv¥qYª«ª-vyd�h§��s�{�oyv�oQoyv�qY{yv¥qYv8£«����hg�Ehg����hg�Lv�{�qY����|�f vy£«fShpor��s���£Æ»Þ|�{yf¨ª«x��Z£¤oyvy{y£«��s�v�hp� ¸c�d�h�»Þq��gv�vyd qYv½hrq���dS£«���Z£«�O£¤�Zs qYª���s�{�oyv¥£¤o¥|Y» hMtZ�1|���hg�-vy£ qYª oy£«¿rh�° � ��� � ²M�^fJqY³�hpo½£¤o��1|^o�oy£«��ª¤hv�|¨�Z£¤o��rqY{��Ã|r��hg{yª qY��or¸

c�d�hSo�hp�p|����ºoyv�hg� |Y»�vyd�hS��{�|-|Y»��p|���oy£¤oyv�o�£«�º��|�vy£¤�g£«��� vyd qYv ��� � ¾��|-hpo���|�v���hg�1hg���Á|��o�d�£Æ»[v�o8|Y»Zvyd�h¾qYª«ª¤|-�rqYvy£¤|���o�hpÈ-s�hg���ph�9�¸8c�d�£¤o8fShrqY��o®£«v8£¤o®hpÈ-s qYªLv�|w£«v�o8oyd�£Æ»±vU£«�-��qY{y£ qY�-v���hg{�o�£¤|��U¸

é[â��®é±î

Page 20: Maximizing the Robustness of TDMA Networks with ... · Maximizing the Robustness of TDMA Networks with Applications to TTP/C Bruno Gaujal, Nicolas Navet To cite this version: Bruno

�������������� ��������������������� ���� �!�#"%$'&(�*),+-����./�10 2�� i��

©8£«� qYª«ª«x��§c�d�hp|�{�hgf iJv�|���hgvyd�hg{¨¦�£«vyd �®hgf¨fJq iÇ°[¦�d�£¤��d'£¤o���|r¦ vy{ys�h�»±|�{¨hrq���dÄ��s�{�o�v£«����hg�Ehg����hg�Lvyª«x�²M��oyd�|�¦!vyd qYv�vyd�hA»[s����gvy£¤|�� ��� � £¤o���ª¤|O��³-¬¼f�s�ª«vy£«fS|O�Zs�ª qY{AqY���À£¤o�f¨£«��£«f¨£«¿rhp�£Æ»Qvyd�h�qYª«ª¤|O�rqYvy£¤|���|Y»Qvyd�h�{�hg��ª«£¤�rq�o�»Þ|�{yf q���ª¤|-��³ Ò-vys�{yf¨£ qY� o�hpÈ-s�hg���ph^¸

~­��|�{���hg{�v�|w£«ª«ª«s�oyvy{�qYv�h¾vyd�£¤o�{�hpoys�ª«vr�Y¦§h�oyd qYª«ª-��hg��hgª¤|��SqkoyfJqYª«ªZhMt�qYf¨��ª¤h^¸ �UhgvQs�o8�p|���o�£¤��hg{qÃ�rq�o�hS¦�£«vyd qY� ©Qc�¯ � fJq���hJ|Y»#�Ã{�hg��ª«£¤�rq�o�|Y»woy£«¿rh �Ã��£«v�o�|���hg{�qÅ{�|�s����Á¦�£«vydÄqÃv�|�v�qYª|Y»#i�N���£«v�o�¸½vyd�hg�H|���h�|���vy£«fJqYª�qYª«ª¤|O�rqYvy£¤|��H|Y» � £¤o�v�|uqYª«ª¤|-�rqYv�h�vyd�h2{�hg��ª«£¤�rq�ows�o�£«����q���ª¤|O��³Ò-vys�{yf¨£ qY�So�hpÈLs�hg���ph�¦�£«vyd���ª¤|O��³Zo¥|Y»�oy£«¿rh��Z�^£«���E|^oy£«vy£¤|���o � � � ! � � � ! � � * ! ��,� � �Z�q�owoyd�|r¦��ã«�é�£«�^s�{�h-NZ¸

©�£«�^s�{�h-N ��0¾ª¤|O��³ÃÒ-vys�{yf¨£ qY�HqYª«ª¤|O�rqYvy£¤|�� |Y»Qo�£«¿rh��¨|Y»�qYª«ª {�hg��ª«£¤�rq�ok|Y» � ¸c�d�£¤o¾qYª«ª¤|O�rqYvy£¤|��u|Y» vyd�hk{�hg��ª«£¤�rq�o½£¤o§vyd�h#�Ehpoyv§v�|(W��^dLv�qY�LqY£«��oyv �¥µ�~�oCqYvy£¤o4»[xO£«����q�o�o�s�f¨�Z¬

vy£¤|���o ����� � � ������� qY��� ������� ¸

�� � ���+���2��� �+����� ��(�� , �?� ����� ���� %~¢�Ávyd�£¤o�o�hp�gvy£¤|��U��¦§hu�p|���oy£¤��hg{¨qYª«ª¾©Qc�¯Ao�v�|���hgvyd�hg{SqY���º¦¾hJvy{yx%v�| W���� qY�'qYª«ª¤|O�rqYvy£¤|��Ä|Y»qYª«ª¥|Y»¾vyd�hgf oy£«f�s�ª«v�qY��hp|�s�o�ª«x�¸¨¶w�%|���vy£«fJqYª§qYª«ª¤|O�rqYvy£¤|��Ç»±|�{�hrq���dº©Qc�¯��p|���oyvy{ys��gv�hp�Çs�o�£«���vyd�hp|�{�hgf i�£¤o���|�v�qYª«¦¾qpxZo2»Þhrq�oy£«��ª¤h¨oy£«���ph�vyd�hSqYª«ª¤|O�rqYvy£¤|���o2fJqrx��Eh¨�p|���c�£¤�gvy£«���æ�£«vydÁhrq���d|�vyd�hg{�°[£Æ»�v¢¦§|�qYª«ª¤|O�rqYvy£¤|���o#d qr��h�qYv#ª¤hrq�oyv#|���h���£«v�£«� �p|�f¨fS|���²M¸

~­�Åvyd�hA»Þ|�ª«ª¤|r¦�£«����¦§h��Z£¤oyvy£«���^s�£¤o�dÅvyd�h��rq�o�h�¦�d�hg{�h2£«v�£¤o��E|^o�oy£«��ª¤h�v�|SqYª«ª¤|O�rqYv�h�qYª«ªU©Qc�¯Ao|���vy£«fJqYª«ª«xÀqY���Jvyd�h��rq�o�hk¦�d�hg{�hAvyd�£¤o¾£¤o§��|�v¾�E|^o�oy£«��ª¤h�qY���J¦�d�hg{�h��p|�f¨��{�|�f¨£¤o�hpo�d qr��h�v�|��Eh»Þ|�s���� ¸

��� " � � � Ð � ÌÃÐ � ��� �Á� �§ L��� Ì �¶Ao�oys�fSh�vyd qYv¨vyd�hu{�hg��ª«£¤�rq�oS|Y»�qYª«ª�©Qc�¯Ao�d qp��hÀvyd�hÀo�qYfShÀoy£«¿rh%° .�²M¸Á¹Âd�hg�'vyd�hu�Ls�f��1hg{|Y»#{�hg��ª«£¤�rq�o��Ehg{�©Qc�¯·v�qY³�hpo�s��Äv�| v¢¦§|Ç�Z£a`Ehg{�hg�-v���qYª«s�hpor��£«v�£¤o��E|^o�oy£«��ª¤hJv�|HqYª«ª¤|O�rqYv�h�vyd�ho�ª¤|�v�o�»±|�{wqYª«ªU©Qc�¯Aowq��p�p|�{��Z£«���Jv�|Sq¨��ª¤|-��³ÅÒ-vys�{yf¨£ qY�Ho�hpÈLs�hg���phA»Þ|�{#hrq���dH|Y»�vyd�hgf ¦�£«vydÅ��|�p|���c�£¤�gv�o

)khg{�h�£¤owvyd�h�¦¾qrx v�|���s�£«ª¤�HqY�%qYª«ª¤|O�rqYvy£¤|��H¦�d�£¤��dÇ£¤o�qJ��ª¤|O��³HÒ-vys�{yf¨£ qY�Ç��hp�gv�|�{���»±|�{�qYª«ª©Qc�¯Aor¸

� æCý8÷ � % ø � ö®æ � ÷�2aïòæpþ 5kãrö���ý � øð÷Cç �gó4þÿö¼æ�ý�< % øòþ % +^óÿý �Ææ�ý � �Uö % ó �¼÷ � ó �

� � çedMìgí #ÿì

Page 21: Maximizing the Robustness of TDMA Networks with ... · Maximizing the Robustness of TDMA Networks with Applications to TTP/C Bruno Gaujal, Nicolas Navet To cite this version: Bruno

i � �����A�������1E�� +�'+>��P� �

�Uhgv %!�1huvyd�hÀv�|�v�qYª#�-s�f��Ehg{J|Y»2©�c�¯Aoð��"� %8²M¸Ä¶Ao�oys�fShuvyd qYvSvyd�hÅ�rqY{��Z£«� qYª«£«v¢x|Y»kqYª«ª¾©Qc�¯ko�£¤o�hg£«vyd�hg{��J|�{ @ ¸Ãc�d�hu©Qc�¯ko�¦�£«vydÄ�rqY{��Z£«� qYª«£«v¢x�� °[{�hpoy�U¸ @º²�qY{�hu��hg��|�v�hp����L!LKLKLK6! ��� °[{�hpoy�U¸ #�� !LKLKLK6!$#�� ²M�Z¦�£«vyd !� %¥¸

i^¸ 0¾s�£«ª¤� qSÒ-vys�{yf¨£ qY��o�hpÈ-s�hg���ph2¦�£«vyd ��hg��o�£«v­x�� ��I�� !�@ 8NM¸�Z¸#lwhg��ª q��ph%qYª«ª�vyd�h_Q��8Q�� oÁ°[{�hpoy�U¸ Q F Q�� oC²J�-x ��� !LKLKLK ��� °[{�hpoy�U¸ # �L!LKLKLK6!$#�� ²J£«�3q

{�|�s����O¬¼{�|���£«�û¼q�oyd�£¤|��U¸¹=d qYvS£¤oXW�� qYª«ª«x:|���v�qY£«��hp�.£¤ouq%o�hpÈ-s�hg���ph |r��hg{ÀqYª«ªk©Qc�¯Ao ��� !LKLKLK ���"!$# � !LKLKLK6!$#�� ¦�£«vyd��ª¤|O��³ÃÒ-vys�{yf¨£ qY�HqYª«ª¤|O�rqYvy£¤|��Å»Þ|�{kqYª«ªU|Y»Qvyd�hgfø

c®|Å£«ª«ª«s�oyvy{�qYv�h�vyd�£¤o�qYª«��|�{y£«vyd�fÃ�®ª¤hgv2s�o2�p|���oyvy{ys��gv�qY�ºhMt�qYf¨��ª¤h^¸�¹ºh¨�p|���oy£¤��hg{�qu{�|�s����fJq���hk|Y» N�©Qc�¯Ao � �"!$#�!&%(!�� !�� �O¦�£«vyd�{�hpo��1hp�gvy£«��hk�rqY{��Z£«� qYª«£«vy£¤hpoANZ�ò�Z�ONZ�ò�Z�ò�Z¸Q¹ºhw¦�£«ª«ª���s�£«ª¤�qY�%qYª«ª¤|O�rqYvy£¤|��H¦�d�£¤��dÇ£¤o�Ò-vys�{yf¨£ qY�H»Þ|�{Ahrq���dÇ|���h�|Y»½vyd�hgf9qY��� d q�oA��|u�p|���c�£¤�gvr¸�©8£«{�o�vk¦¾h�p|���o�vy{ys��gv�qÀÒ-vys�{yf¨£ qY�%o�hpÈLs�hg���ph�¦�£«vydÇ��hg��oy£«v­xÄir��ÉOir�Z��vyd�£¤ok£¤o F � F � � F � � F � F � � F � ��¸��k|�¦¦¾h�{�hg��ª q��ph�hrq���d�� ���J�-x � |�{ % °[£«�Çq�{�|�s����O¬¼{�|���£«�ÇfJqY����hg{C²kqY��� hrq���d Q F Q��Lx � |�{ �|�{ # °ÞqY�LqY£«�ã«�Hq�{�|�s����O¬¼{�|���£«� fJqY����hg{C²M¸¥¹ºh���hgv�vyd�h�»±|�ª«ª¤|�¦�£«����|���vy£«fJqYª8qYª«ª¤|O�rqYvy£¤|�� �

´ ¶ � Ê ¶ 0 Ê ¶ ´ Ê � ¶ Ê 0 ¶ Ê~­� q����Z£«vy£¤|��U��vyd�hg{�h�hMtO£¤oyv�o�|�fShS�rq�o�hpo�¦�d�hg{�hSvyd�h��rqY{��Z£«� qYª«£«vy£¤hpo�v�qY³�hJfS|�{�hSvyd qY� v­¦§|

�^qYª«s�hpo�qY��� q�Ò-vys�{yf¨£ qY�ÄqYª«ª¤|O�rqYvy£¤|��Á£¤o�oyvy£«ª«ª¥�E|^o�oy£«��ª¤h�»Þ|�{�qYª«ª¾©�c�¯Aor¸�~¼»�vyd�hS�rqY{��Z£«� qYª«£«vy£¤hpoqY{�h2|Y»Qvyd�hA»Þ|�{yf � ! � ! + ! � ! � � !LKLKLK�! � � °ÞqYª«ªU�1|�¦§hg{�ok|Y»Q��²¾vyd�hg�U�Z£«v#£¤o��E|^o�oy£«��ª¤h�v�|W����HÒ-vys�{4¬f¨£ qY�HqYª«ª¤|-�rqYvy£¤|���ow»±|�{AqYª«ª®©Qc�¯kor¸§c�d�h�©�{�q�hg��³�hgªQ�p|���}4hp�gvys�{�hu°±o�hphX@«i�B[²�oCqpxZo#vyd qYv#vyd�hpo�h�qY{�hhpo�o�hg�Lvy£ qYª«ª«xÅvyd�h�|���ª«xÃ�rq�o�hpo#¦�d�hg{�h�vyd�h�oys��1hg{y�E|^oy£«vy£¤|��H|Y»�o�hg��hg{�qYª�Ò-vys�{yf¨£ qY�ÇqYª«ª¤|O�rqYvy£¤|���ok£¤o�E|^o�oy£«��ª¤h�¦�£«vyd�|�s�vk�p|���c�£¤�gv�or¸

¶�vkvyd�£¤oA�E|�£«�Lvr�1¦§h�oyd�|�s�ª¤���1|�£«�-vA|�s�v�vyd qYvAvyd�h��rq�o�h�¦�£«vydHs���v�|�v­¦¾|Å�Z£a`Ehg{�hg�-v2�rqY{��Z£Æ¬� qYª«£«vy£¤hpo�oyd�|�s�ª¤�Ç»±s�ªaW�ª«ª�fS|^oyv�|Y»�vyd�h���hphp��or¸S~¢�ÁqÅoyxZoyv�hgf9¦�d�hg{�hS|���ª«x%qÅoys���o�hgv�|Y»���|O��hpoqY{�hÀ�g{y£«vy£¤�rqYª¾»±{�|�f q vyd�hu�1|�£«�-v¨|Y»w�-£¤hg¦ |Y»wvyd�hÀ��hg�Ehg����qY��£«ª«£«v­x��¾©Qc�¯ko�¦�£«ª«ª���hg��hg{�qYª«ª«xÁ�Eh|Y»A�rqY{��Z£«� qYª«£«v­x |���hÇ°[��|��Z¬­�g{y£«vy£¤�rqYª���|O��hpoC²�qY���Äv­¦¾|:°±�g{y£«vy£¤�rqYª¾��|O��hpoC²M¸%~¢� vyd�h��p|��-v�hMtOvS|Y»Rk¬¼�Lx-¬¼¹=£«{�h�qY����ª«£¤�rqYvy£¤|���o�¦�d�hg{�h���hg�1hg����qY��£«ª«£«v¢x��p|���oyvy{�qY£«�-v�o�qY{�h�o�vy{y£«����hg�Lvr�1v­¦¾|u�Z£a` hg{�hg�Lv�rqY{��Z£«� qYª«£«vy£¤hpowoyd�|�s�ª¤�ÃqYª¤o�|���hg��hg{�qYª«ª«x��EhAoysJ�g£¤hg�Lvr¸§©�|�{�£«��oyv�qY���phAvyd�h���{�|�v�|�v­xO�1h2��hpo�£«�^��hp�£«�Ävyd�h�0§{y£«v�h���s�{�qYf ~�~�~A��{�|Y}yhp�gv>=4ÒZqa»Þhgv­x {�hgª qYv�hp� © qYs�ª«v¨c®|�ª¤hg{�qY�Lv�Ò-xZoyv�hgfSo�~­�ÄÓ¥hgd�£¤�gª¤h�?°±o�hph%@ONCB[²#£¤o��p|�f¨�1|^o�hp�H|Y»½��|O��hpo�|Y»½�rqY{��Z£«� qYª«£«vy£¤hpo2v­¦¾|Ç°[vyd�h�oyv�hphg{y£«���u¦�d�hphgª¥q��gvys qYv�|�{�qY���vyd�h2o�v�hphg{y£«���J�p|��Lvy{�|�ª8s���£«vC²�qY���Åvyd�{�hphJ°±oyv�hphg{y£«���uq��gvys qYv�|�{�oC²M¸

��� " � " � ÌJ6®Ì �a� �¾ ���� ̶AoAfShg�Lvy£¤|���hp�Á�1hM»Þ|�{�h^� vyd�h��rq�o�h�¦�£«vyd%v­¦¾|Ã�Z£a` hg{�hg�Lv��rqY{��Z£«� qYª«£«vy£¤hpo2£¤o�{�qYvyd�hg{��p|�f¨fS|��º£«���{�q��gvy£¤�ph^¸½�khg��hg{yvyd�hgª¤hpo�o���£«v��p|�s�ª¤�Àd qY���1hg�Ãvyd qYvwq�fS|�{�h��Z£ J�gs�ª«v#�p|���W��^s�{�qYvy£¤|��ÇqY{y£¤o�hpor¸�~¢�

é[â��®é±î

Page 22: Maximizing the Robustness of TDMA Networks with ... · Maximizing the Robustness of TDMA Networks with Applications to TTP/C Bruno Gaujal, Nicolas Navet To cite this version: Bruno

�������������� ��������������������� ���� �!�#"%$'&(�*),+-����./�10 2�� i �

��hg��hg{�qYª¼�¥£«v�£¤o���|�v��1|^o�o�£«��ª¤hJv�|%qYª«ª¤|O�rqYv�hJvyd�hÀoyª¤|�v�o�|Y»kqYª«ª�©Qc�¯ko�q��p�p|�{��Z£«���%v�|ÇÒ-vys�{yf¨£ qY�o�hpÈLs�hg���phpo�¦�£«vyd�|�s�vk��hgvyvy£«���J�p|���c�£¤�gv�or¸¥c®¦§|J�1|^o�o�£«��ª¤h2oyvy{�qYv�hg�^£¤hpow�rqY�Ã�Eh2�p|���oy£¤��hg{�hp� �

i^¸k�k��h��rqY�H��hgª«£«�1hg{�qYv�hgª«xûÞqr��|�{�q�oys���o�hgv -:|Y»¥� qY{yvy£¤�gs�ª qY{yª«x �g{y£«vy£¤�rqYª�©Qc�¯Ao#d qp�O£«���ÅqYª«ªvyd�hJo�qYfSh��rqY{��Z£«� qYª«£«v­x qY���Ávyd�hJo�qYfShJoy£«¿rh:.®¸u~¢�ºvyd�£¤o��rq�o�h^�8vyd�hJoyª¤|�v�o�|Y»�vyd�|^o�h©Qc�¯ko�qY{�h�qYª«ª¤|-�rqYv�hp� |���vy£«fJqYª«ª«x'°[{�hg�LqY{��Z£«����vyd�h¨ª¤|^o�o2��{�|�� qY��£«ª«£«v¢x�²�¦�d�£«ª¤h¨vyd�hSo�ª¤|�v�o|Y»®vyd�hA|�vyd�hg{�o�©�c�¯Ao�qY{�hMW�v¾£«�uvyd�hk{�hgfJqY£«��£«����»±{�hphw��ª q��phpo�¸¥c�d�hAqYª«ª¤|O�rqYvy£¤|��À£¤o¾�^£«��hg��Lx%qY�LxH��ª¤|-��³%Ò-vys�{yf¨£ qY� o�hpÈLs�hg���ph�°±o�hph�hpÈLs qYvy£¤|��.i�²A|Y»���hg��oy£«v­x�� �/I+-�N . � 1q�o#��|���h2£«�Åvyd�h2��{�hg�O£¤|�s�owo�hp�gvy£¤|��U¸

�Z¸#�k|J©�c�¯AokqY{�h�|Y»¥oy�1hp�g£ qYª®£«f¨�1|�{yv�qY���ph�qY����q¨o�|�ª«s�vy£¤|�� f¨£«��£«f¨£«¿p£«���uvyd�h�ª¤|^o�ow��{�|�� qa¬��£«ª«£«v­xu»Þ|�{�vyd�h�o�hgv#|Y»¥qYª«ªUfShpo�o�qY��hpo#|Y»Qvyd�h2oyxZoyv�hgf d q�o#v�|��EhA»Þ|�s���� ¸

��� " �"� � Ì �8���X�r���X  � Ð � �®��� Ð�6 � Ð �<; Ì � � � Ð  ��Z��� Ð�6#� � Ð � � Ì �~¢�Ãvyd�£¤ok� qY{�qY�^{�qY��dU�1¦§h���{�|��E|^o�h�q¨ª¤|r¦ �p|�f¨��ª¤hMtZ£«v­xÃd�hgs�{y£¤oyvy£¤��o�|�ª«s�vy£¤|���»±|�{kvyd�h��p|�f���£«��hp�qYª«ª¤|O�rqYvy£¤|��J��{�|���ª¤hgf qY����q�o�o�hpo�o�£«v�o½�1hg{4»Þ|�{yfJqY���phAqY�LqY£«��o�v½{�qY����|�f+qYª«ª¤|O�rqYvy£¤|���o§s�����hg{§�^qY{4¬£¤|�s�o��Ehg{yvys�{y� qYvy£¤|��'�p|����Z£«vy£¤|���or¸%c�d�hu�1hg{4»Þ|�{yfJqY���phufShgvy{y£¤��£¤o�vyd�huª¤|^o�o���{�|�� qY��£«ª«£«v­x ���! � ¢¸¶Ao�»Þ|�{kqJÒ-vys�{yf¨£ qY�ÇqYª«ª¤|-�rqYvy£¤|��U��vyd�h2£¤��hrqS|Y»�vyd�£¤owd�hgs�{y£¤oyvy£¤�2£¤o#v�|Joy��{�hrq��Ãvyd�h2{�hg��ª«£¤�rq�oA|Y»½qoCqYfSh2©�c�¯ q�o#hg��hg��ª«xÃq�o��E|^o�oy£«��ª¤h2|���hg{wvy£«fSh^¸

©�|�{½hrq���du©Qc�¯ � ¦�£«vydJ�rqY{��Z£«� qYª«£«v¢x % , qY����»[{�qYfShpo§|Y»Eo�£«¿rhE. , �^¦§hk��h�W���h#vyd�h#��hg��o�£«v­x|Y» »±{�qYfShpo§�1hg{¾��£«v ��� , � % , . , � 1¨¸�~¢�Lvys�£«vy£«��hgª«x���� , £¤o§vyd�hw�-s�f��Ehg{�|Y» »[{�qYfShpo¾�Ehgª¤|����^£«���v�|J©�c�¯ � vyd qYvwoyd�|�s�ª¤� �1h�vy{�qY��oyf¨£«vyv�hp� �1hg{w��£«vr¸½c�d�h�oys�f |Y»�vyd�h���hg��oy£«vy£¤hpo#s��Ãv�|S��£«v��»Þ|�{#©Qc�¯ � £¤o�� ,� � ��� , ¸¥¹ºh���hg��|�v�h��Lx'% ,B vyd�h��Ls�f��Ehg{k|Y»Q��£«v�ow©Qc�¯ � d q�okqYª«{�hrq��Zx�Ehphg�JqYª«ª¤|O�rqYv�hp�¨s��¨v�|2oyv�hg� >§°[£«���gª«s��Z£«����oyv�hg� >ÿ²M¸Q¶�v¥hrq���dJo�v�hg�U�-qY�S©Qc�¯.¦�£«ª«ª��Eh#qYª«ª¤|O�rqYv�hp�vyd�h��-s�f��Ehg{�|Y»½��£«v�o���hp�phpo�o�qY{yx�v�|Ào�hg���H£«v�oA»[{�qYfSh^¸�~¢�Çvyd�h�»±|�ª«ª¤|�¦�£«����� � I > Nk£«���Z£¤�rqYv�hpo�vyd�h©Qc�¯ ��d�|^o�hg� qYv#oyv�hg�À£ ¦�d�£«ª¤h�� I > N§£¤o�vyd�h�v�|�v�qYª �-s�f��Ehg{w|Y»8��£«v�owqYª«{�hrq��ZxÅqYª«ª¤|O�rqYv�hp�ÃqYv#oyv�hg�£¼¸

i^¸#~­��£«vy£ qYª«£«¿�qYvy£¤|��Hoyv�hg� � % ,= � F � � I F N � F qY��� > � �^¸�Z¸#¶�vAo�v�hg� >�vyd�h�©Qc�¯ »Þ|�{k¦�d�£¤��dHvyd�h��Z£a` hg{�hg���ph��1hgv¢¦§hphg�%vyd�h��Ls�f��1hg{2|Y» =¢�Zs�h�?���£«v�o

qY���Åvyd�h2��{�hg�O£¤|�s�owqYª«ª¤|O�rqYvy£¤|��ã¤o�fJqatZ£«f�s�f £¤owo�hgª¤hp�gv�hp� �� I > N�� 3qY{y�^fJqat , 5� I� ,�� B ; � � � ( % ,B ; � N '

�Z¸#c�d�h2��hMtZvE.�� � B ��£«v�oAqY{�h�qYª«ª¤|-�rqYv�hp�û±|�{#©Qc�¯ � I > NM¸��¸#z8hg{4»Þ|�{yf�vyd�h#s��1��qYv�hpo � I > N � � I >�( �8N !/. � � B � % � � B B � % � � B B ; � !/. � � B qY��� % ,B � % ,B ; �£Æ» � � I > NM¸

� � çedMìgí #ÿì

Page 23: Maximizing the Robustness of TDMA Networks with ... · Maximizing the Robustness of TDMA Networks with Applications to TTP/C Bruno Gaujal, Nicolas Navet To cite this version: Bruno

�^� �����A�������1E�� +�'+>��P� �

NZ¸�£Æ» � I > N 1 oyv�|��Ãhgª¤o�h2£ � ) £���i^�Z��|¨v�|�£«v�hgf �Z¸c�d�hÃqYª«��|�{y£«vyd�f¨£¤�Ã�p|�f¨��ª¤hMtO£«v¢x'|Y»kvyd�hÀd�hgs�{y£¤oyvy£¤�ÃqYª«ª¤|O�rqYvy£¤|��:£¤o�ª«£«��hrqY{¨£«�'vyd�hÀ�Ls�f��1hg{

|Y»���£«v�oÅ|Y»�q {�|�s���� ¸��A|�v�h�vyd qYvÃqÄoy£«f¨£«ª qY{Å�p|���oyvy{ys��gvy£¤|��!� q�o�hp�!|��!��hg��oy£«v¢x.d q�oÀ�Ehphg�o�s��p�phpo�o4»±s�ª«ª«xJs�o�hp�À»Þ|�{#��h�W���£«���uq��E|�ª«£¤�gxuvyd qYvwoyd qY�Ehpo�{�hrqYªÆ¬¼vy£«fSh2vy{�q J��£«� @ò�CB­¸

��� " ��� � Ì � 9¢Ð ���º� 6   Ì   Ð � � ���(�[� Ð�6'Ï �X�<;:�Y� 6 Ñ Ð � � � � Ð  L�Z��� Ð�6 �c8|Åq�o�o�hpo�oAvyd�h�{�|���s�oyvy��hpo�o2|Y»§vyd�hSqYª«ª¤|-�rqYvy£¤|���o��^£«��hg�%�-x vyd�h�d�hgs�{y£¤oyvy£¤�^� oy£«f�s�ª qYvy£¤|���o�¦¾hg{�h�Ehg{4»Þ|�{yfShp�ÄqY�LqY£«��oyv�{�qY����|�f qYª«ª¤|-�rqYvy£¤|���o�¸H¶ �p|���W��^s�{�qYvy£¤|��'£¤o���h�W���hp�Ä�Lx q �Ls�f��1hg{¨|Y»©Qc�¯ qY���Ävyd�hÀ�rqY{��Z£«� qYª«£«v­x |Y»khrq���d ©Qc�¯�¸�¹ºhu�Z£¤o�vy£«���^s�£¤oyd v­¦§|%�gª q�o�o�hpo�|Y»w��{�|���ª¤hgf q��M¬�p|�{��Z£«����v�|Svyd�h��Ls�f��Ehg{�|Y»�©Qc�¯Aow|���vyd�h���hgv­¦¾|�{y³ �§»Þ|�{Aq!=­fShp�Z£«s�foy£«¿rh���{�|���ª¤hgf ?�vyd�hg{�hqY{�hSqYv2ª¤hrq�o�v��À©Qc�¯ko�qY���ÁqYv2fS|^o�v��À©�c�¯Ao2¦�d�£«ª¤h�£«� q�=­ª qY{y��hJoy£«¿rh���{�|���ª¤hgf ?¨vyd�hg{�hSqY{�hs���v�|�ir�J©Qc�¯kor¸wcU¦§|ud-s����Z{�hp��o2�p|���W��^s�{�qYvy£¤|���o�¦§hg{�h�{�qY����|�f¨ª«xÃ��hg��hg{�qYv�hp�H¦�£«vydH©Qc�¯Aod qr�-£«���Àq¨�rqY{��Z£«� qYª«£«v­x �1hgv¢¦§hphg�H�JqY���Ã��¸¾©�|�{whrq���dÇ�p|���W��^s�{�qYvy£¤|��U��¦§h�{�qY����|�f¨ª«xÅ��£¤��³Às��ir�^��d-s����Z{�hp�Ãoyª¤|�v�o2°[£«�Àvyd�h¨ir�^�^�>W�{�o�v¾{�|�s�����oM²§¦�d�hg{�h�q���qYv�q�£¤o¾vy{�qY��oyf¨£«vyv�hp�À»Þ|�{¾vyd�h W�{�oyvvy£«fSh^¸ c�d�hÅ�Zs�{�qYvy£¤|��:|Y»kvyd�hÀ��{�|O�Zs��gvy£¤|��:�gxO�gª¤hÀ|Y»Avyd�hu��qYv�qH£¤o¨hpÈ-s qYª�v�|%�H{�|�s�����oSqY���£¤o2��hg��|�v�hp�º�Lx ��¸�c�d�hg�Ç»Þ|�{2hrq���dÁo�hgª¤hp�gv�hp�%o�v�qY{yv�|Y»§vy{�qY��oyf¨£¤o�o�£¤|��U� N^�^�u��s�{�oyv�o2|Y»¾hg{y{�|�{�oqY{�h���hg��hg{�qYv�hp�Ħ�£«vyd � �uqY��� qÃo�£«¿rh�hMtZ�1|���hg�-vy£ qYª«ª«x �Z£¤oyvy{y£«��s�v�hp� |Y»#fShrqY� � K �·¦�£«vyd� � � F ' � ! � ! � ' � ! � * ¸�~­»Uvyd�h#��s�{�o�v¾|Y»®hg{y{�|�{�o�oyv�qY{yv�o§�1hM»Þ|�{�h#vyd�hAhg���u|Y»Evy{�qY��o�f¨£¤o�oy£¤|��À|Y»Uvyd�hW�{�o�vk{�hg��ª«£¤�rqÅqY���[W���£¤oyd�hpo�qa»±v�hg{kvyd�h�oyv�qY{yv2|Y»½vy{�qY��o�f¨£¤o�oy£¤|��%|Y»½vyd�h�ª q�oyv�{�hg��ª«£¤�rqZ�Evyd�h���qYv�q£¤o#ª¤|^oyvr¸�c�d�h2{�hpoys�ª«v�o#|Y»Qvyd�hpo�h�hMtZ�Ehg{y£«fShg�Lv�okqY{�h�oyd�|r¦�� |��é�£«�^s�{�h��Z¸

c�d�hJs�o�h�|Y»#vyd�hJ��{�|��E|^o�hp� d�hgs�{y£¤oyvy£¤�po��^{�hrqYvyª«xÁ�Z£«f¨£«��£¤oyd�hpo�vyd�h�v�|�v�qYª§�-s�f��Ehg{S|Y»#ª¤|^oyv��qYv�qÁ°[s��ºv�|������S²k³O��|r¦�£«����vyd qYv�vyd�hg{�h�qY{�hS�rq�o�hpo2¦�d�hg{�h¨vyd�hSoy£«¿rhJ|Y»¾vyd�h¨��s�{�o�v�£¤o�oys���dvyd qYv½vyd�hw��qYv�q2�rqY����|�v§�1h�vy{�qY��o�f¨£«vyv�hp�S¦�d qYv�hg��hg{¾vyd�hwqYª«ª¤|-�rqYvy£¤|��U¸�c�d�£¤o¥»Þq��gv§hMtO��ª qY£«��o½¦�dLxvyd�h�h J�g£¤hg���gx�|Y»�vyd�h�d�hgs�{y£¤oyvy£¤��v�hg����oQv�|��Eh¾ª¤|r¦¾hg{¥¦�d�hg��vyd�h�oy£«¿rh�|Y»�vyd�h���s�{�oyvQ£¤oQ�Ehp�p|�f¨£«���ª qY{y��hg{r¸À~¢v�£¤o�qYª¤o�|Å��|�v�hg¦§|�{yvyd-xÁvyd qYv�vyd�hSd�hgs�{y£¤oyvy£¤�S£¤o�{�|���s�oyv�v�|HqY�º£«���g{�hrq�o�hJ|Y»�vyd�h�oy£«¿rh|Y»#vyd�hu��s�{�oyv�oy£«���ph�vyd�hu�1hg{4»Þ|�{yfJqY���phÀ|�� vyd�hX=­ª qY{y��hÀoy£«¿rhJ��{�|���ª¤hgf ?u£¤o��gª¤|^o�hJv�|�vyd�hu|���h|���v�qY£«��hp� |��Ãvyd�h =­fShp�Z£«s�f oy£«¿rh���{�|���ª¤hgf ?�¸

��� " ��� � Ì � 9¢Ð ���º� 6   Ì   Ð � � ��� Ì Ñ Ï �X�<; Ð � ��� �Á� �½� � � Ð  ��Z��� ÐL6 �¹ºh���|r¦ hg�^qYª«s qYv�h2vyd�h��1hgd qr�-£¤|�{w|Y»�vyd�h2d�hgs�{y£¤o�vy£¤��¦�£«vydÃ{�hg�LqY{�� v�|¨vyd�h�|���vy£«fJqYªQÒ-vys�{yf¨£ qY�qYª«ª¤|O�rqYvy£¤|��U¸Ã¹ºh��p|���oy£¤��hg{¨qÃ�rq�o�hJ¦�£«vydÄ|���ª«x%v¢¦§|�{�hg��ª«£¤�rq��rqY{��Z£«� qYª«£«vy£¤hpo�¸ ¯Aoy£«��� vyd�hJ��{�hM¬�O£¤|�s�o2o�hp�gvy£¤|��U� ¦¾h�³-��|�¦ vyd qYv2¦§h¨�rqY�º�p|���oyvy{ys��gv�|��º|���vy£«fJqYª½qYª«ª¤|O�rqYvy£¤|��U¸�c�d�h�d�hgs�{y£¤o�vy£¤�qYª«ª¤|O�rqYvy£¤|��%¦�£«ª«ª���|�v���hp�phpo�o�qY{y£«ª«x W����Çvyd�£¤o2|���vy£«fJqYª½qYª«ª¤|-�rqYvy£¤|��ÁqY���Ǧ¾h�¦¾qY�Lv�v�|�fShrq�o�s�{�hd�|�¦3¦§hgª«ª®£«v#�1hg{4»Þ|�{yfSo#�p|�f¨� qY{�hp� v�|�vyd�h�|���vy£«fJqYª¼¸

¹ºh��p|���oy£¤��hg{S�^�^� {�qY����|�f �p|���W��^s�{�qYvy£¤|���oS|Y»#vyd�hufShp�Z£«s�f o�£«¿rhJ��{�|���ª¤hgf »Þ|�{�¦�d�£¤��dvyd�h�|���vy£«fJqYª¾qYª«ª¤|O�rqYvy£¤|��Á£¤o�³-��|�¦�� °[£¤h^¸u�Ls�f��1hg{�|Y»#©Qc�¯ko��rqY{��Z£«� qYª«£«vy£¤hpo�£¤o�ª¤hpo�o2vyd qY� ��²M¸

é[â��®é±î

Page 24: Maximizing the Robustness of TDMA Networks with ... · Maximizing the Robustness of TDMA Networks with Applications to TTP/C Bruno Gaujal, Nicolas Navet To cite this version: Bruno

�������������� ��������������������� ���� �!�#"%$'&(�*),+-����./�10 2�� �Zi

0

20

40

60

80

100

0.5 1 1.5 2

Red

uctio

n of

lost

dat

a (%

)

Mean size of a burst (unit: duration of a production cycle)

medium size problem: up to 6 FTUslarge size problem: up to 12 FTUs

©�£«�^s�{�h2� �Qlwhp�Zs��gvy£¤|��Å|Y»®vyd�hA�Ls�f��1hg{#|Y»Uª¤|^o�v���qYv�q�¦�d�hg�Àvyd�hAd�hgs�{y£¤oyvy£¤�A£¤o§s�o�hp�J£«��oyv�hrq��À|Y»qJ{�qY����|�f qYª«ª¤|-�rqYvy£¤|��U¸�c�d�h�fShrqY�Ç��s�{�oyvAo�£«¿rh�{�qY����hpok»±{�|�f�Z¸ONJv�|u�Jvy£«fShpoAvyd�h�ª¤hg���^vyd%|Y»q���{�|O�Zs��gvy£¤|����gxZ�gª¤h�¦�d�£¤��d�£¤o#��d�|^o�hg� hpÈLs qYª®v�|S��c�´AµH¶ {�|�s�����or¸

c�d�hÀ�p|����Z£«vy£¤|���o�|Y»wvyd�hÅhMtO�Ehg{y£«fShg�LvJqY{�huvyd�hÅo�qYfShÅq�o�£«� � qY{�qY�^{�qY��dÂ�Z¸ò�Z¸ �ºhMt��phg��v�vyd qYvvyd�h��Ls�f��Ehg{�|Y» W�{�o�vwvy{�qY��oyf¨£¤o�o�£¤|���oyª¤|�v�owvyd qYvAqY{�h�o�hgª¤hp�gv�hp� £¤owhpÈ-s qYª®v�|Ãir�^�^��°[£«� vyd�h>W�{�oyv�^�^�^��{�|�s�����oC²�qY���Ävyd qYvXN^�^�^�Ã��s�{�oyv�o�|Y»khg{y{�|�{�oSqY{�h�{�qY����|�f¨ª«xÁ��hg��hg{�qYv�hp� ¸Ác�d�hJª¤|^o�o�|Y»�Ehg{4»Þ|�{yfJqY���ph�qY�LqY£«��oyv#vyd�h�|���vy£«fJqYª8o�|�ª«s�vy£¤|��ã¤o#oyd�|�¦�� |�� ©�£«�^s�{�h��O¸

0

5

10

15

20

25

0.5 1 1.5 2

Incr

ease

of l

ost d

ata

(%)

Mean size of a burst (unit: duration of a production cycle)

medium size problem: up to 6 FTUs

©�£«�^s�{�h�� ��~¢���g{�hrq�o�h2|Y»Qvyd�h2ª¤|^oyv#��qYv�q�¦�d�hg� vyd�h�d�hgs�{y£¤oyvy£¤�2£¤o�s�o�hp�À£«��o�v�hrq��Ã|Y»�vyd�h�|���vy£«fJqYªqYª«ª¤|O�rqYvy£¤|��U¸

� � çedMìgí #ÿì

Page 25: Maximizing the Robustness of TDMA Networks with ... · Maximizing the Robustness of TDMA Networks with Applications to TTP/C Bruno Gaujal, Nicolas Navet To cite this version: Bruno

�^� �����A�������1E�� +�'+>��P� �

c�d�h�ª¤|^o�o�|Y»§�Ehg{4»±|�{yfJqY���ph�¦�£«vyd%{�hg�LqY{��Çv�|Àvyd�h¨|���vy£«fJqYª¥o�|�ª«s�vy£¤|��%£¤o2o�fJqYª«ª�°[ª¤hpo�o�vyd qY�i^i �9|��ºvyd�£¤o�hMtZ�1hg{y£«fShg�-vC²�qY���%£«v�ª¤|��^£¤�rqYª«ª«xÇ��hp�g{�hrq�o�hpo�¦�d�hg�ºvyd�hSo�£«¿rhS|Y»¾vyd�hS��s�{�oyv�o��EhM¬�p|�fShpowª qY{y��hg{r¸§c�d�h���|-|O�À�Ehgd qp�O£¤|�{k|Y»�vyd�h2d�hgs�{y£¤oyvy£¤��|�� �p|���W��^s�{�qYvy£¤|���oA¦�£«vyd ª¤hpo�o�vyd qY�Ç��Z£a` hg{�hg�Lv��rqY{��Z£«� qYª«£«vy£¤hpo�£¤o�qu�E|^oy£«vy£«��h¨hgª¤hgfShg�Lv�¦�£«vyd%{�hg�LqY{��%v�|À£«v�o2�Ehg{4»±|�{yfJqY���phJ|��ÁqY{y��£Æ¬vy{�qY{yxÀ�p|���W��^s�{�qYvy£¤|���or¸

� ���� ��������� �� ������5�/1�#��������6��y� � ������� ��� 6����#��� �# ���/1�Q5�6���7�� ���78�#/�/E��5������

¯k��ª«£«³�h�vyd�hJ��{�hg�O£¤|�s�o¨�rq�o�h^�Qvyd�h�v�hp��d���£¤È-s�hÀs�o�hp�Áv�| W���� vyd�hu|���vy£«fJqYª#qYª«ª¤|O�rqYvy£¤|��'|Y»#vyd�h{�hg��ª«£¤�rq�ok|Y»Q|���h2©Qc�¯�£¤o�� q�o�hp�Å|�� fJqa}4|�{y£«¿�qYvy£¤|��ÇqY��� ÒO��dLs�{A�p|��L��hMtZ£«v­x�¸

�$��� ��� �� � ��� , � #�� (+��� ��,$!J� � � � �)( � ��(�� ,�®hgv � DI � � !LKLKLKM! � % NwqY��� � DI � � !LKLKLKM! ��%�N#�1h�v­¦§|�{�hrqYª���hp�gv�|�{�oA|Y»¥oy£«¿rh %¥¸�¹ºh���hg��|�v�h�-x I ��� �� KLKLK ! ��� % N�qY��� I ��� �� !LKLKLK ! ��� % N�vyd�h��1hg{yf�s�v�qYvy£¤|���o¨|Y» �.qY��� �ºoys���d vyd qYv ��� �� <KLKLK < � � % qY���?� � �� < KLKLK < � � % ¸Jc�d�h¨��hp�gv�|�{ � �X� � ��0��GC�F���'° ��� ��²k£Æ»¾vyd�h�»Þ|�ª«ª¤|r¦�£«����p|����Z£«vy£¤|���owd�|�ª¤� �

%3 B"! � � B %3 B"! � � B ! °Þ��²

�3 B"! � ��� B <�3 B"! � ��� B ' °Þ��²

¶:»±s����gvy£¤|�����»±{�|�f�� % v�|��.£¤o � Z����J0 ZY�� P�I��°[{�hpoy�U¸ ��Z����J0 Z �� LZY�����²�£Æ» ������£«f¨��ª«£¤hpo� I � N ��� I ��N�°[{�hpoy�U¸�� I ��NE<�� I ��Ny²M¸�©�|�{AfS|�{�h���hgv�qY£«ª¤o�|�� vyd�hpo�h���|�vy£¤|���or� vyd�h�{�hrq���hg{A�rqY�{�hM»Þhg{�v�|D@«ir�CB­¸

�$� � ��� �� � ��� , ��� � ( ��� ����� $&%('~¢�Åvyd�£¤o#o�hp�gvy£¤|�� ¦§h�¦�£«ª«ª8oyd�|r¦3vyd qYvwvyd�h2��{�|�� qY��£«ª«£«v­xÅvyd qYv#vyd�h�qY�Ãhg{y{�|�{w��s�{�oyv#�p|�{y{ys���v�o�qYvª¤hrq�o�v�|���h�{�hg��ª«£¤�rqu¦�£«vyd�£«�Áq���{�|O�Zs��gvy£¤|��Á�gxO�gª¤hÅ°���$&%�'M²w£¤o�qÅÒO��dLs�{��p|����rqp��h�»±s����gvy£¤|��%¦�£«vyd{�hpo��1hp�gv§v�|�vyd�h�qYª«ª¤|-�rqYvy£¤|��À|Y»Uvyd�hk{�hg��ª«£¤�rq�or¸�¯ko�£«����vyd�hk��h�W���£«vy£¤|��À|Y»�ÒO��dLs�{#�p|����rqr�-£«v¢x��Zvyd�£¤o¦�£«ª«ª®��{�|r�O£¤��h��Z£«{�hp�gvyª«xÀvyd�h2�1hpo�vwqYª«ª¤|-�rqYvy£¤|��Ãf¨£«��£«f¨£«¿p£«��� ��$&%('a¸

¹ºhk��hg��|�v�hA�Lx � � vyd�hk�Ls�f��1hg{�|Y» »±{�qYfShpo�°±|Y»®o�£«¿rh(.1²¥�p|�f¨�E|^oy£«���¨q�fShpo�oCqY��h#»Þ|�{©Qc�¯ � ¸ �Uhgv 9��1h�qY�JqYª«ª¤|O�rqYvy£¤|��J|Y»�vyd�h {�hg��ª«£¤�rq�o�»Þ|�{yf¨£«����©Qc�¯ � ¸�c�d�h�È-s qY�Lvy£«v¢x�� B I 9 N��hg��|�v�hpowvyd�h2£«�Lv�hg{y�^qYª®�Ehgv­¦¾hphg� vyd�h�hg��� |Y»�{�hg��ª«£¤�rq B ; � qY���Åvyd�h2�Ehg�^£«����£«����|Y»Q{�hg��ª«£¤�rq B ¸

é[â��®é±î

Page 26: Maximizing the Robustness of TDMA Networks with ... · Maximizing the Robustness of TDMA Networks with Applications to TTP/C Bruno Gaujal, Nicolas Navet To cite this version: Bruno

�������������� ��������������������� ���� �!�#"%$'&(�*),+-����./�10 2�� �^�

¹ºhJ��hg��|�v�hS�Lx ��I 9�NAvyd�hSo�hpÈLs�hg���phS|Y»�£«�-v�hg{y��qYª¤o�I � � !LKLKLK � � N2qY���%�-x � ��I 9�N � vyd�h¨��hp�gv�|�{�|Y»vyd�hwª¤hg���^vydu|Y»Uvyd�hw£«�Lv�hg{y�^qYª¤or� � � I 9 N � JI � � � � !LKLKLKM! � � � � NM¸��k|�v�hwvyd qYv � � � I 9�N � ! KLKLK ! � � � I 9�N � ��I 1 (* . N���|-hpo���|�vw��hg�Ehg���Ã|��Ãvyd�h�qYª«ª¤|O�rqYvy£¤|�� 9�¸�½Ì � �º� " ��� ��� ���(ZY�1 S�F� K���0(��F�� ���E��>��0�0I��0-����0F��� ���^��0����T J�X�1�b� 0I�� LK8�1� ������� �J � " ��0�� E �K1� ����0F�^����� �IK �����0 �� �� 0I���J LK �1 LK �P�F�F�J�X� � � � � ��� 9 �� LK 9�� �Y� ��./� ��E�EG��ZY�1��� �� S��I" � � 6 " � ��I 9�N ��� � ��I 9�� N � ������ �����\L0#���I������E �����^�F� �#"%EG�P���T �� �1�(E��I�P���!�� ���"�0I�����H���8��� ��" ���$&%�' I 9 NA� ��$&%(' I 9 � N �� 0#���I" �k©�£«{�oyv���|�v�hÀvyd qYv���s�{�o�v�o¨oyv�qY{yvy£«���%�Zs�{y£«���%vyd�h�vy{�qY��o�f¨£¤o�oy£¤|��'|Y»k|���hÀ{�hg��ª«£¤�rqH|Y» ��p|�{y{ys���v�o�vyd�£¤o�{�hg��ª«£¤�rq�¦�£«vyd ��{�|�� qY��£«ª«£«v¢xÂi^���1hp�rqYs�o�h � �^¸�c�d�£¤o�fShrqY��o�vyd qYv�vyd�h��Z£Æ»[¬»Þhg{�hg���phA�1hgv¢¦§hphg� ��$&%�' I 9 N�qY��� ��$&%�' I 9��)N§�rqY�Ã|���ª«xu�p|�fShA»[{�|�f hg{y{�|�{�o�oyv�qY{yvy£«���¨£«�À�Ehgv­¦§hphg�vyd�h�{�hg��ª«£¤�rq�o�¸

¶wª¤o�|u��|�v�h�vyd qYv2£Æ» � ��vyd�hg� � � I 9 N � � � � I 9�N � ��I 1 (� .�N � � � I 9 � N � � � I 9 � N �qY����qYª«ª®qYª«ª¤|-�rqYvy£¤|���oAqY{�h�hpÈLs�£«�^qYª¤hg�Lvkoy£«���ph2vyd�h2hg{y{�|�{#fS|O��hgªU£¤o�vy£«fSh2d�|�fS|���hg��hp|�s�o�¸

~¼» ��� �Z��¦§h�{�hg�Ls�f��1hg{2vyd�h�£«�Lv�hg{y�^qYª¤o2|Y»�9%qY��� 9 � o�s���dHvyd qYv � � � �� � < KLKLK < � � � � � qY���� ��� �� ��< KLKLK < � ��� � � ¸½¯Aoy£«���¨vyd�h2fJqa}y|�{y£«¿�qYvy£¤|��Ç�p|����Z£«vy£¤|��U� |���h2��hgv�o�»Þ|�{kqYª«ª � � � � B"! � � � � B ���� � B ! � � � �� B � '¹ºh§��|�¦'��{�|r��h��-x�£«���Zs��gvy£¤|��¨vyd qYvQ»Þ|�{QqYª«ª �=< �:< ��|���h¾�rqY�¨�p|���oyvy{ys��gv½qA�p|�s���ª«£«���2�EhM¬v¢¦§hphg� � � �� !LKLKLK6! � � � qY��� ��� �� !LKLKLK ! ��� � oys���d¨vyd qYvQvyd�h§��{�|�� qY��£«ª«£«v¢x ��� vyd qYv¥qY��hg{y{�|�{¥o�v�qY{yvy£«���£«� ���� �� !LKLKLKM! ���� � qY���Ã�p|�{y{ys���vy£«���ÅqYv#ª¤hrq�oyv#|���h2{�hg��ª«£¤�rqS£¤o#oyfJqYª«ª¤hg{#vyd qY�Ãvyd�h��p|�{y{�hpoy�1|����Z£«�����{�|�� qY��£«ª«£«v¢x � �A£«� � � �� !LKLKLK ! � � � ¸�© |�{ � �^�Ovyd�h��p|�s���ª«£«���¨£¤o���|���h�q��p�p|�{��Z£«���¨v�|�©�£«�^s�{�h �Z¸

��� ��

� � �

� � ��

©8£«�^s�{�h�� �½Ê¾|�s���ª«£«���S»Þ|�{�vyd�h�oyfJqYª«ª¤hpoyv�£«�-v�hg{y��qYª¼¸¶#»±v�hg{�vyd�h��p|�s���ª«£«�����Lvyd�h#£«�Lv�hg{y�^qYª � � �� £¤o¥oy��ª«£«v�£«�Lv�|�v­¦§|�£«�Lv�hg{y�^qYª¤or�� � qY����� � oys���dJvyd qYv

� � �� � ��� � � qY��� � � �� �� � ��� � � ¸�¶(��s�{�oyv�o�v�qY{yvy£«���Å£«��� � d q�o�vyd�hSo�qYfSh���{�|�� qY��£«ª«£«v­x%|Y»�p|�{y{ys���vy£¤|���vyd qYvkq���s�{�oyvkoyv�qY{yvy£«���J£«� ��� �� ���Ehp�rqYs�o�h2£«v#£¤owhg��|�s��^d vyd qYv#vyd�h2��s�{�o�vw|r��hg{yª qY��ovyd�hÀ£«�-v�hg{y��qYª�v�|%�p|�{y{ys���vJvyd�hÀ{�hg��ª«£¤�rqHvyd qYv�»±|�ª«ª¤|�¦#or¸ c�d�hÀ{�hgfJqY£«��£«���º¿r|���hº°� � ²�£¤o¨oys���dvyd qYvJqY�:hg{y{�|�{�oyv�qY{yvy£«���%£«�� � �p|�{y{ys���v�oS|���hÅ{�hg��ª«£¤�rqH¦�£«vyd=qH��|��Z¬¼��hg�LqYvy£«��hÅ��{�|�� qY��£«ª«£«v­x�¸c�d�hg{�hM»Þ|�{�h^� � � � �� � ¸c�d�hÅ��{�|-|Y»��p|��-vy£«�Ls�hpou�Lx £«���Zs��gvy£¤|��Â|�����¸Âc�d�hã«���Zs��gvy£¤|��.��{�|��Ehg{yv­x'£¤oSvyd qYvS»Þ|�{�q�^£«��hg� ��|���h¨�rqY�º�p|���oyvy{ys��gv�quoy��ª«£«vyvy£«����|Y» � � �� !LKLKLK6! � � � £«�-v�| I�� � ! � N !LKLKLKM! I�� � ! � NAoys���d

� � çedMìgí #ÿì

Page 27: Maximizing the Robustness of TDMA Networks with ... · Maximizing the Robustness of TDMA Networks with Applications to TTP/C Bruno Gaujal, Nicolas Navet To cite this version: Bruno

�Y� �����A�������1E�� +�'+>��P� �

vyd qYvAvyd�h���{�|�� qY��£«ª«£«v­xHvyd qYv�qJ��s�{�oyv2oyv�qY{yvy£«���u£«� � � � KLKLK�� � ��£¤owª qY{y��hg{2|�{2hpÈLs qYª�vyd qY�Ç£«�� �� �� ��KLKLK�� � �� � qY��� vyd�hJ¿r|���h � ��KLKLK�� ����Qd q�o�q ��|��Z¬¼��hg�LqYvy£«��huv�|�v�qYª¾��{�|�� qY��£«ª«£«v¢x |Y»�p|�{y{ys���vy£«���Àq�{�hg��ª«£¤�rqZ¸

¹ºh2��|r¦ q���� � � � � �� qY��� � �� � � �� ¸¥c®¦§|S�rq�o�hpow�rqY� |-�p�gs�{r¸i�²w~­» � � � � �� � ���� � � �� vyd�hg�º|���h�o���ª«£«v�o�� � � � �� q�o�£«v�d q�o��1hphg�º��|���h�»Þ|�{�� � �� qY��� ��� �� £«�©�£«�^s�{�h��Z¸�¹ºhJ��hgv���hg¦�£«�Lv�hg{y�^qYª¤o � � � qY��� � � � � qY���Ávyd�hS£«���Zs��gvy£¤|��Ä{�hgfJqY£«��o�vy{ys�hJ�Lx

s�o�£«���¨vyd�h�qY{y�^s�fShg�Lvk�^£«��hg�Å»±|�{ � �^¸��²¥~­» � � � � �� < � �� � � �� �-¦§hA�p|�s���ª¤h2q��p�p|�{��Z£«����v�|�vyd�h#»Þ|�ª«ª¤|r¦�£«������{�|O�php�Zs�{�h^¸¥c�d�hk£«�Lv�hg{y�^qYª

��� � � �� £¤o#oy��ª«£«vk£«�Lv�|Sv¢¦§|J£«�Lv�hg{y�^qYª¤o �(qY������oys���d vyd qYv ��� � � � � � � �� � ��¦�d�£¤��dHqY{�h��p|�s���ª¤hp�v�|���hgvyd�hg{r¸

�� � ��

� � �

� � � � ��

���� �

���� � � ��

���� ��

� �

� �

1�� � �

���� � � � � � � �

©�£«�^s�{�h�� �¾Ê¾|�s���ª«£«���S¦�d�hg� � � � � �� < ��� � � �� ¸�k|�v�h�vyd qYv¥�-x�vyd�h�fJqa}4|�{y£«¿�qYvy£¤|�����{�|��1hg{yv¢x��� � � � ���� � � �� � ( � � � � � �� ��< � � ��! KLKLK ! � � � ¸

�®hgv ��� ���� �:�'� � � �6! KLKLK ! � � � � � � � * ¸�¹ºh�oy��ª«£«v�vyd�hk£«�-v�hg{y��qYª ��£«�-v�|�v­¦§|�£«�-v�hg{y��qYª¤o1�� ! � �Joys���dÇvyd qYv ����� � � � � ( I � � � ! KLKLK ! � � ; � � NM¸2c�d�h��p|�s���ª«£«���Å£¤oA£«ª«ª«s�oyvy{�qYv�hp�Ç£«�©�£«�^s�{�h��Z¸

' ¶w�Ähg{y{�|�{�oyv�qY{yvy£«���H£«��� d q�o�vyd�hJoCqYfShS��{�|�� qY��£«ª«£«v­x v�| �p|�{y{ys���v¨qÀ»±{�qYfShSvyd qY� qY�hg{y{�|�{woyv�qY{yvy£«���S£«� � � � � �� ¸

é[â��®é±î

Page 28: Maximizing the Robustness of TDMA Networks with ... · Maximizing the Robustness of TDMA Networks with Applications to TTP/C Bruno Gaujal, Nicolas Navet To cite this version: Bruno

�������������� ��������������������� ���� �!�#"%$'&(�*),+-����./�10 2�� �1N

' ¶w�Ähg{y{�|�{�oyv�qY{yvy£«����£«� � d q�o�q oyfJqYª«ª¤hg{���{�|�� qY��£«ª«£«v¢x |Y»#�p|�{y{ys���vy£¤|�� vyd qY�ÄqY�Ähg{y{�|�{oyv�qY{yvy£«���S£«� � � KLKLK � ; � � � ���Ehp�rqYs�o�h ��� � � ��� B �L»±|�{kqYª«ª >-<��E¸

' ¶w�ºhg{y{�|�{�o�v�qY{yvy£«���Å£«� ���� �� � KLKLK� ��� � d q�o�qu��{�|�� qY��£«ª«£«v¢xº|Y»¾�p|�{y{ys���vy£¤|��Áo�fJqYª«ª¤hg{�|�{hpÈLs qYª®vyd qY��qY�Ãhg{y{�|�{woyv�qY{yvy£«���S£«� � � � KLKLK � � ���Lxuvyd�h2£«���Zs��gvy£¤|�� d-x-�E|�vyd�hpoy£¤or¸

' ¶w�'hg{y{�|�{Jo�v�qY{yvy£«���%£«�A1�� � ��� � ��KLKLK � ��Åd q�oJq���|��Z¬¼��hg�LqYvy£«��hÃ��{�|�� qY��£«ª«£«v­x'|Y»�p|�{y{ys���vy£¤|��U¸

~¢�Åv�|�v�qYª¼� � � � � � ��� � � ¸©8£«� qYª«ª«x��wvyd�hH£«���Zs��gvy£¤|��!q�o�oys�f¨��vy£¤|�� £¤ou�rqY{y{y£¤hp� |���h�fS|�{�hÇoyv�hg�Â�-x:s�oy£«��� vyd�h���hg¦o���ª«£«vyvy£«���J|Y»�� � �� !LKLKLK6! � � � � �� £«�Lv�|I I�� � !�� N !LKLKLK I�� � ; � !�� N ! I�� � ! 1�� N ! I�� ��� � ! ��� � N !LKLKLKM! I�� � ! �� N ! I � � � � �� � � KLKLK � � ; � � � � !�� N N '

c�d�h2��{�|-|Y»�£¤o#�p|����gª«s���hp���LxÀ��|�vy£¤�g£«����vyd qYv ��$&%(' I 9�N � � ! �§°ÞqY��hg{y{�|�{koyv�qY{yv�ok�Zs�{y£«���q�»±{�qYfSh�²�qY��� ��$ %�' I 9 � N � �

�! �§°ÞqY��hg{y{�|�{koyv�qY{yv�o#�Zs�{y£«���Àq�»[{�qYfSh�²#qY���Ãs�oy£«���¨vyd�h2{�hpo�s�ª«v

|Y» vyd�hw£«���Zs��gvy£¤|��uoyd�|�¦�£«����vyd qYv � � � ���qY����vyd�h�»¼q��gv¥vyd qYv§vyd�h#��{�|�� qY��£«ª«£«v­x�vyd qYv¾qY�uhg{y{�|�{

o�v�qY{yv�o#�Zs�{y£«���Svyd�h2vy{�qY��oyf¨£¤o�oy£¤|�� |Y»¥q�»[{�qYfSh2£¤owhpÈLs qYª £«�Ã�E|�vydÃ�rq�o�hpor¸0 ; Ì-Ð � Ì � � � � LK���0 ��� �F�J�]\���� �� S� � ����� �� LK � � � � �1 LKH� "�� �(�A������ X" ��0 �I�8Z����$ � � �M�������F\������X��Eb��ETEG��ZY�8������ 9 $&%�'�� �� �������� �����$&%(' ��0I���<\:�X�^�Y�������S��0%��E�E 0I� \�E � ZY�P� �� ���S�(� �����(0I���J LK �

� 0#���I" �k¯k����hg{ � � � � hrq���dÁ��s�{�oyv��rqY�Á�1h¨�p|���oy£¤��hg{�hp�Á£«����hg�1hg����hg�-vyª«x%�1hp�rqYs�o�h � �^¸c�d�hg{�hM»Þ|�{�h^� ��$&%�'u£¤oJvyd�hHoys�f |Y»�vyd�h ª¤|^o�o���{�|�� qY��£«ª«£«vy£¤hpou»Þ|�{uhrq���dÂ��s�{�oyvr¸ �Uhgv�9 �1hHqY�qY{y��£«vy{�qY{yx¨qYª«ª¤|O�rqYvy£¤|��U¸�c�d�h�{�hpo�vy{y£¤�gvy£¤|���o�|r��hg{¥|���h�{�|�s���� 1=|Y» 9JqY����9 $&%�'¥qY{�h���hg��|�v�hp��9 � �qY��� 9 $ %�' � � {�hpoy�1hp�gvy£«��hgª«x�¸2c�d�hgxH|��L�O£¤|�s�oyª«x o�qYvy£¤oy»[x ��I 9 � � N � ��I 9 $&%(' � � NM¸=0¾xÃ�Ehg{y£¤|O�Z£¤�g£«v­x��I 9�N I � I 9 � � N ! � I 9 � � N !LKLKLK6! ��I 9 � � N N½°[{�hg�EhrqYv�hp���·vy£«fShpoC²M¸8c�d�£¤o8£«f¨��ª«£¤hpo�� I 9 N�� � I 9 $&%(' NM¸©�£«� qYª«ª«x�� qY����ª«xO£«��� �Uhgf¨fJq����p|����gª«s���hpo#vyd�h2��{�|O|Y»y¸

c�d�£¤o�{�hpoys�ª«vw�rqYª«ª¤o�»Þ|�{wo�hg��hg{�qYªU�p|�f¨fShg�-v�or¸

' c�d�h�q�o�o�s�f¨��vy£¤|��Ãvyd qYvAq���{�|O�Zs��gvy£¤|����gxZ�gª¤h�£¤okq�f�s�ª«vy£«��ª¤h�|Y»�vyd�h�oy£«¿rh�|Y»½q�{�|�s���� £¤ohpo�o�hg�-vy£ qYªUd�hg{�h^��s���ª«£«³�h2£«�Ãvyd�h���{�|-|Y»�|Y» �®hgf¨fJqÅi^¸

' ~¼»§vyd�h�q�o�o�s�f¨��vy£¤|��Hvyd qYv�� ��£¤ok{�hgfS|r��hp�ºvyd�hg� �®hgf¨fJqu��£¤oA��|�vAvy{ys�h�qY�-x-fS|�{�h^¸)k|�¦§hg��hg{r��c�d�hp|�{�hgf ��fJqpxJoyvy£«ª«ª��1hk�^qYª«£¤�S»Þ|�{§o�|�fShA�Z£¤oyvy{y£«��s�vy£¤|���o�|Y»®vyd�hk��s�{�oyv�oy£«¿rh^¸

� � çedMìgí #ÿì

Page 29: Maximizing the Robustness of TDMA Networks with ... · Maximizing the Robustness of TDMA Networks with Applications to TTP/C Bruno Gaujal, Nicolas Navet To cite this version: Bruno

�^� �����A�������1E�� +�'+>��P� �

' c�d�h�|���vy£«fJqYª�qYª«ª¤|O�rqYvy£¤|��¨»±|�{ ��� � ®°[vyd qYv½¦§h��rqYª«ª 9 �! � [²8£¤o½Ò-vys�{yf¨£ qY��q�o���{�|r��hp�S£«�Sc�d�hM¬|�{�hgf �Z¸Q¶Ao½oyd�|�¦��S£«� @ò�PB­�ZÒ-vys�{yf¨£ qY��o�hpÈ-s�hg���phpo§qY{�h�f¨£«��£«fJqYª »±|�{½vyd�h�fJqa}4|�{y£«¿�qYvy£¤|��|�{���hg{y£«��� qY��� vyd�hg{�hM»Þ|�{�h/9 � � �d q�o¨vyd�hÀ¦¾|�{�oyvS�E|^o�oy£«��ª¤hu�Ehg{4»±|�{yfJqY���ph q��p�p|�{��Z£«��� v�|��$&%('a¸ c�d�£¤o�oyd�|r¦#o�vyd qYv�vyd�h�v­¦¾|Ç|��Z}4hp�gvy£«��hpoÀ°[f¨£«��£«f¨£«¿p£«��� ��$&%('¨qY�������! � ±²�qY{�hJ£«�Z¬�p|�f¨� qYvy£«��ª¤h�£«��q�o�vy{�|����Jo�hg��o�hS°[vyd�h2�Ehpoyv#qYª«ª¤|O�rqYvy£¤|�� ��$&%('w£¤o�vyd�h2¦¾|�{�o�h�»Þ|�{����! � ±²M¸

' ¯w��ª«£«³�h¥»±|�{����! ­�avyd�h½�p|�f���£«��hp��f¨£«��£«f¨£«¿�qYvy£¤|��¨|Y» � $&%(' »Þ|�{�qYª«ª-©Qc�¯AoU£¤o8��|�v�q���{�|���ª¤hgfoy£«���ph2vyd�h2|���vy£«fJqYª�o�|�ª«s�vy£¤|��Ã�^{�|�s���okqYª«ª {�hg��ª«£¤�rq�ok|Y»Qhrq���d ©Qc�¯�v�|���hgvyd�hg{r¸

� )�5�5�6���7��������w �1�$.$';=<#>~¢�Svyd�£¤o§o�hp�gvy£¤|��J¦¾h#£«�Lv�hg�^{�qYv�hkvyd�hwhg{y{�|�{§d qY���Zª«£«����fShp��d qY��£¤oyfSo¾|Y» c�c�z¾É^Ê'£«�Svyd�hAqY� qYª«xOo�£¤o��hg��hgª¤|��Ehp�Ão�|�»¼qY{wqY���Å£«�-��hpoyvy£«�LqYv�h�£«�æ�d�£¤��d hMtZv�hg�Lv#vyd�h2{�hpo�s�ª«v�o�{�hgfJqY£«� �^qYª«£¤� ¸

����� � ������� # ����� � � , !$� (�,�� � #&� ��,$(�% � %c�d�hJc�c�z§É^Ê���{�|�v�|O�p|�ª¥£«���gª«s���hpo��E|r¦¾hg{4»[s�ª§��s�v��p|�f¨��ª¤hMtºqYª«��|�{y£«vyd�fSo�oys���d q�o2vyd�h¨�gª«£¤È-s�hqr��|�£¤��qY���phJqY���%fShgf��1hg{�oyd�£«�ÄqYª«��|�{y£«vyd�fSor¸¨~­�Çvyd�£¤o2� qY{�qY�^{�qY��dU�®¦§h¨�^£«��hSqÀo�£«f¨��ª«£aW hp�%��hM¬o��g{y£«��vy£¤|��=|Y»�vyd�h »[s����gvy£¤|���£«��� o���d�hgfShpoÅ|Y»�c�c�z§É^Ê ��hg{�o�£¤|���i^¸ò�Ávyd qYvÀqY{�h�{�hgª qYv�hp�¦�£«vydvy{�qY��o�f¨£¤o�oy£¤|�� hg{y{�|�{kd qY���Zª«£«���uqY���Åvyd qYvkf¨£«�^dLvAq¨��{y£¤|�{y£®£«�Lv�hg{4»Þhg{�h�¦�£«vydH|�s�{kqY� qYª«xZoy£¤or¸½© |�{£«��o�v�qY���ph^��c�c�z§É^Ê ��h�W���hpoAvyd�h��p|����phg��v�|Y»]Qgoyd q���|r¦-Q���|O��h^¸�¶ oyd q���|r¦���|O��h�{�hg��ª q��phpo2q��hM»Þhp�gvy£«��hS��|O��h¨��s�v���|-hpo���|�v��1|^o�o�hpo�o�hpo2£«v�o�|r¦��Äoyª¤|�v�£«�Ávyd�hS{�|�s���� ¸uc�d�£¤o�{�hp�Zs�����qY���gxo���d�hgfSh���|Ohpo#��|�vk��{�|�v�hp�gv�qY�LqY£«��o�vwvy{�qY��oyf¨£¤o�o�£¤|���hg{y{�|�{�okqY��� ¦§h�¦§|�� � v��p|���oy£¤��hg{wvyd�hgf £«�vyd�h�{�hpo�vw|Y»�vyd�h�� qY{�qY�^{�qY��dU¸

i^¸#µHqatZ£«f�s�f·µHhgf��Ehg{�oyd�£«�u©�qY£«ª«s�{�h�ʾ|�s��-v2°±µHµH©8Ê�²���d�hp��³ �8£Æ»8q2��|O��h#ª¤|^o�hpo½£«v§fShgf�¬�1hg{�o�d�£«�%£«�Áµ�µH©8Ê�oys��p�phpo�oy£«��hSo�hg���Z£«���Ãoyª¤|�v�o��Uvyd�hg�ºvyd�h¨�p|��-vy{�|�ª«ª¤hg{�v�hg{yf¨£«� qYv�hpo�£«v�o|��1hg{�qYvy£¤|�� �LxÀhg�Lv�hg{y£«���Jvyd�h]=¼»±{�hphg¿rh�oyv�qYv�h�?�¸�~¢v�£¤o#qY� |���vy£¤|�� qYª »ÞhrqYvys�{�h2oy£«���ph2µHµ�©�Ê�rqY�Ã�Eh2o�hgv�v�|�¿rhg{�|¨¦�d�£¤��d�fShrqY��o#��|���hg{y£aW �rqYvy£¤|��U¸

�Z¸ �U|^oyv�|Y»¾fShgf��Ehg{�oyd�£«�Á�Zs�h�v�|Ãqu£«���p|�{y{�hp�gv2vy{�qY��oyf¨£¤o�o�£¤|��#�w£Æ»#qJ»[{�qYfSh�£¤o2�p|�{y{ys���v�hp��Zs�{y£«���S£«v�o�vy{�qY��oyf¨£¤o�o�£¤|��Åvyd�h�o�hg����hg{�ª¤|^o�hpo�£«v�o�fShgf��1hg{�o�d�£«��qY���Åhg�Lv�hg{�owv�|�vyd�h�� q�o4¬oy£«��hÃoyv�qYv�h^¸!~¢vS¦�qY£«v�oJ£«�:vyd�h � q�o�oy£«��h�oyv�qYv�hÅs��Lvy£«ªA£«vJ�rqY�Â{�hM¬¢q��pÈLs�£«{�h�£«v�oJoyª¤|�vr¸ c8|{�hM¬¢q��pÈLs�£«{�huqÅoyª¤|�v�vyd�hJ�p|��Lvy{�|�ª«ª¤hg{�f�s�oyv�d qr��hS{�hp�phg£«��hp�Ávyd�h =­f¨£«��£«f�s�f £«�Lv�hg�^{�qYvy£¤|���p|�s��Lv#?J�p|�{y{�hp�gv�»[{�qYfShpou°±oyd�|�s�ª¤�%�EhSo�hgv�qYv�ª¤hrq�o�v�v�| qÀ�^qYª«s�hS|Y»�v­¦§|L²M��v�hg{yfShp� =­vyd�hfJqa}4|�{y£«v¢xÀ{ys�ª¤h�?�¸

�Z¸#lwhM¬¼£«�-v�hg�^{�qYvy£¤|��u|Y»®q2��|O��h�°[vy{�qY��oy£«v½»[{�|�f(»[{�hphg¿rhkoyv�qYv�h�v�|�� q�o�oy£«��h#oyv�qYv�h�²+��q]=¼»±{�|�¿rhg��?��|-��h�f�s�oyvA¦¾qY£«vAs��Lvy£«ª�vyd�h�qY����ª«£¤�rqYvy£¤|��Ho�hgv�okvyd�h�ʾ|��-vy{�|�ª«ª¤hg{��k� °¼Ê��2² W hgª¤� v�|Jvyd�h

é[â��®é±î

Page 30: Maximizing the Robustness of TDMA Networks with ... · Maximizing the Robustness of TDMA Networks with Applications to TTP/C Bruno Gaujal, Nicolas Navet To cite this version: Bruno

�������������� ��������������������� ���� �!�#"%$'&(�*),+-����./�10 2�� ���

��qYª«s�h%=¢|���?�¸ c�d�hg�'£«v¨f�s�oyv¨ª«£¤oyv�hg�'v�|ºq���qYª«£¤�Ä»±{�qYfShÅ�p|��Lv�qY£«��£«���ÄhMtO��ª«£¤�g£«v�ʽ¬­o�v�qYv�h�1hM»Þ|�{�h�hg�Lv�hg{y£«���Ãvyd�h�� q�o�oy£«��h�oyv�qYv�h^¸�c�d�hg�%vyd�h���|O��h�d q�o2v�|u{�hM¬¢q��pÈLs�£«{�hS£«v�o2oyª¤|�v�q�o��hpo��g{y£«�Ehp�Å£«�Å�1|�£«�-vk�Z¸

��¸kʧª«£¤ÈLs�h�qp��|�£¤��qY���phÇqYª«��|�{y£«vyd�f �Å�EhM»±|�{�h oyv�qY{yvy£«���ºv�|ºo�hg���ÂqH»[{�qYfSh^��qÇ��|O��hÅf�s�o�v��hg{y£Æ»[x%¦�d�hgvyd�hg{�vyd�hJ�Ls�f��Ehg{¨|Y»�»[{�qYfShpo�vyd qYv�d qp��hu�1hphg�Äoys��p�phpo�o4»±s�ª«ª«xºo�hg�-v�£«� vyd�hª q�oyv"-Äoyª¤|�v�o2°[¦�d�hg{�h -Á£¤o�vyd�hk�-s�f��Ehg{w|Y»8oyª¤|�v�o�£«�Àvyd�hA{�|�s����Ào�|�vyd qYv�£«v¾£«���gª«s���hpo#£«v�o|r¦��Hª q�oyvwvy{�qY��o�f¨£¤o�oy£¤|���²#£¤ow�^{�hrqYv�hg{Avyd qY��vyd�h��Ls�f��1hg{2|Y»¥£«���p|�{y{�hp�gvA»[{�qYfShpo�¸�~­��vyd�hª qYvyv�hg{k�rq�o�h^� vyd�h���|O��h�hg�Lv�hg{�okvyd�h%QC»±{�hphg¿rh�o�v�qYv�h8Q�|�vyd�hg{y¦�£¤o�h�£«v#vy{�qY��oyf¨£«v�ok£«v�o�»±{�qYfShqY���Å{�hpo�hgv�£«v�o#�p|�s��-v�hg{�or¸½c�d�£¤ow{ys�ª¤h�¦�£«ª«ªU�Eh�v�hg{yfShp�Ãvyd�h�fJqa}y|�{y£«v­xÀ{ys�ª¤h^¸

c�d�h¾{ys�ª¤hpo�i^�ò��qY������q��gvys qYª«ª«x�qC`Ehp�gv�vyd�h¾��qYª«s�h¾|Y» ���! Z��s�vQ��|�v�¦�d�£¤��dSqYª«ª¤|-�rqYvy£¤|��So���d�hgfSh¾£¤o|���vy£«fJqYª¼¸ )A|r¦¾hg��hg{r��vyd�h�fJqa}y|�{y£«v­x�{ys�ª¤h�|Y» c�c�z§É^Ê!°[£«v�hgf ��qY�E|r��h�²Q��d qY����hpo¥vyd�h��p|�f���£«��hp�f¨£«��£«f¨£«¿�qYvy£¤|��º|Y» ���! � 8»Þ|�{�qYª«ªQ©Qc�¯koA¦�£«vydÇ{�hpoy�Ehp�gvkv�|�vyd�h���hg��hg{�qYªQc�´�µH¶(�rq�o�h^¸�~¢� »¼q��gvr�£«vwfJqY³�hpow£«vwhrq�o�£¤hg{wv�|¨{�hrq���d�|���vy£«fJqYª�qYª«ª¤|-�rqYvy£¤|�� »±|�{kqYª«ª�©Qc�¯ko�v�|���hgvyd�hg{2q�owo�d�|r¦�� £«� vyd�h��hMtZv#o�hp�gvy£¤|��U¸

¶ko½»±|�{¾vyd�hkf¨£«��£«f¨£«¿�qYvy£¤|��Ã|Y» ��$&%('���|�� qY{yvy£¤�gs�ª qY{�{ys�ª¤hk|Y»�c�c�z§É^Ê=qYª«v�hg{�o�vyd�hw{�hpoys�ª«v�o�|Y»c�d�hp|�{�hgf �Z¸

��� � � (�,$( � ( (�, � �� # �+� %'% * ���?� �&� (���(+���~¢�Åvyd�h�o�hp�gvy£¤|��U��¦§h�q�o�o�s�fSh��� � �½q���£«vk�p|r��hg{�hp�Ã�-xÃq��1hg{yvys�{y� qYvy£¤|���¦�£«ª«ªU�1h2�p|�{y{ys���v�hp�¦�£«vydÅ��{�|�� qY��£«ª«£«v¢x%i�qY���Àvyd�hk¦�d�|�ª¤h2»[{�qYfShA£¤owqYª¤o�|��p|�{y{ys���v�hp� ¸§¯w����hg{�vyd�£¤o#q�o�o�s�f¨��vy£¤|��U��£«v£¤o§�E|^o�oy£«��ª¤hkv�|>W����ÅqY�À|���vy£«fJqYª qYª«ª¤|O�rqYvy£¤|��Ã|Y»8qYª«ª ©Qc�¯ko§£«��vyd�hA��hg��hg{�qYª �rq�o�h�°[¦�£«vyd�|�s�v�vyd�h�p|����Z£«vy£¤|��H|��Åvyd�h��rqY{��Z£«� qYª«£«vy£¤hpok�^£«��hg�ã«��ÒOhp�gvy£¤|��H�Z¸ò��²M¸

)khg{�h�£¤owvyd�h�¦¾qrxÃv�|u�p|���oyvy{ys��gvy£¤|��ºqY�H|���vy£«fJqYª��^ª¤|�� qYª¥qYª«ª¤|-�rqYvy£¤|�� 9 � � ¼¸w¹ºh��p|���o�vy{ys��gvv¢¦§|So�v�q���³Zo - � qY��� - � |Y»�oyª¤|�v�or¸�»Þ|�{#hrq���d ©Qc�¯�>½¦�£«vyd % B {�hg��ª«£¤�rq�or����s�oyd � % B � � ��o�ª¤|�v�o�£«�vyd�h�ª qY{y��hpo�vwoyv�q���³ÃqY����� % B � ����o�ª¤|�v�o�£«�Åvyd�h�oyfJqYª«ª¤hpoyv#o�v�q���³E¸

c�d�h½qYª«ª¤|-�rqYvy£¤|�� 9 � � ��� �§£¤oU�p|���oyvy{ys��gv�hp���Lx2�p|����rqYv�hg� qYvy£«��� - � qY��� - � ¸®c�d�h¥�p|���oyvy{ys��gvy£¤|��£¤o#£«ª«ª«s�oyvy{�qYv�hp�Å�LxÅ©�£«�^s�{�hSir�Z¸0 ; Ì-Ð � Ì � ��� � �F�� �� $�$ ��� � �� �K �� �K���0������" ��0I�^�8���� �� �P�F�F�J�]\������1 S�!������� ��ETEG��ZY�8������ 9 � � �� �*� �T :�T� ��<�F�*���! �

� 0#���I" �wvyd�hJqYª«ª¤|O�rqYvy£¤|�� 9 � � ��� �Jd q�o�vyd�h�»±|�ª«ª¤|�¦�£«��� ��{�|��Ehg{yv­x ��hrq���dÄ©Qc�¯�¦�£«vydºfS|�{�hSvyd qY�v¢¦§|Á{�hg��ª«£¤�rq�oSd q�o¨v¢¦§|º{�hg��ª«£¤�rq�o�o�hg� qY{�qYv�hp�'�-x'qYvJª¤hrq�oyv � -�� � ��o�ª¤|�v�or¸'�k|�¦���q�oSo�|O|��.q�ov¢¦§|'{�hg��ª«£¤�rq�o�|Y»�vyd�hºoCqYfShÇfShpo�o�qY��hºqY{�hºqYª«ª¤|O�rqYv�hp�!fS|�{�h%vyd qYv � -�� � �Áo�ª¤|�v�o qY� qY{yvr�A��|�Ehg{yvys�{y� qYvy£¤|��Â�rqY�Â��hpoyvy{�|rx �E|�vyd.|Y»Avyd�hgf ¦�£«vyd�|�s�vS»±{�hphg¿p£«���ºvyd�hÃoyxZoyv�hgfø:c�d�£¤o¨fShrqY��ovyd qYvE9 � � ��� ��£¤o#|���vy£«fJqYª¼¸

� � çedMìgí #ÿì

Page 31: Maximizing the Robustness of TDMA Networks with ... · Maximizing the Robustness of TDMA Networks with Applications to TTP/C Bruno Gaujal, Nicolas Navet To cite this version: Bruno

��� �����A�������1E�� +�'+>��P� �

�����

�����

�����

�����

����������������������������������������

��������

���������������

���������������

- � - �

¶wª«ª¤|O�rqYvy£¤|�� |Y»¥q�{�|�s����©8£«�^s�{�hJir� �§Ê¾|���oyvy{ys��gvy£¤|���|Y»Qvyd�h�|���vy£«fJqYª8qYª«ª¤|O�rqYvy£¤|��29 � � �� �-¸

�A���X��0F2��Q¹=£«vydJ|�s�{§hg{y{�|�{½fS|-��hgª¼�ZqY���¨�Ehp�rqYs�o�h#|Y» vyd�h�fJqa}y|�{y£«v­x¨{ys�ª¤h^��£«v½£¤o¥s�o�hgª¤hpo�o¥vyd�hd qr��h�fS|�{�h2vyd qY� v­¦§|J{�hg��ª«£¤�rq�ow�Ehg{#©Qc�¯�£Æ»�vyd�h�|��Z}yhp�gvy£«��h�£¤o#v�|¨f¨£«��£«f¨£«¿rh�vyd�h��p|�{y{ys���vy£¤|��|Y»¥qYª«ª vyd�h2{�hg��ª«£¤�rq�or¸

>.�# �786�� ���� �� /1�¥���/102�c�d�£¤o¾oyvys��Zx¨oyd�|�¦#o§vyd qYv½»±|�{¾c�´�µ�¶w¬¼� q�o�hp�Jo�xOoyv�hgfSo½¦�£«vydJ��s�{�oyv¢xS�Ehg{yvys�{y� qYvy£¤|���o¾��d�|-|^o�£«���vyd�hÀ�E|^oy£«vy£¤|��:|Y»�vyd�hÀ{�hg��ª«£¤�rq�o¨£«��oy£¤��hÅvyd�hÀ{�|�s���� d q�o�qH��hg{yx £«f¨�E|�{yv�qY�LvS£«f¨� q��gv�|��'vyd�hh J�g£¤hg���gxÅ|Y»Qvyd�h2{�hg��ª«£¤�rqYvy£¤|��U¸

c�d�hHW�{�oyv�{�hpoys�ª«vS|Y»kvyd�£¤o¨oyvys��ZxÄ£¤o�v�|H�^£«��hÃqY�:|���vy£«fJqYª�¦¾qrxÄv�|Çoy��{�hrq�� vyd�hÀ{�hg��ª«£¤�rq�o£«�Ä|�{���hg{�v�|Ãf¨£«��£«f¨£«¿rhuvyd�hJª¤|^o�o���{�|�� qY��£«ª«£«v¢x |Y»kqYª«ª¾{�hg��ª«£¤�rq�or¸�c�d�£¤o�{�hpoys�ª«v�£¤o��^qYª«£¤� ¦�d�hg�vyd�hÀ�rqY{��Z£«� qYª«£«vy£¤hpoS|Y»Avyd�hÀ©�c�¯Ao�v�qY³�hÀs�� v�|�v¢¦§|Ç�^qYª«s�hpo¨|�{�¦�d�hg�'vyd�hu�rqY{��Z£«� qYª«£«vy£¤hpo�qY{�hqYª«ª8�1|�¦§hg{�oA|Y»�v­¦§|�¸w©�|�{wvyd�h�|�vyd�hg{A�rq�o�hpo���¦§h���{�|r�O£¤��h�qSª¤|r¦�¬­�p|�f¨��ª¤hMtZ£«v­x d�hgs�{y£¤o�vy£¤��¦�d�£¤��d��{�|���hpo#v�|¨�Eh���hg{yxÀh J�g£¤hg�Lvk|�� vyd�h2oy£«f�s�ª qYvy£¤|���okvyd qYv#¦§hg{�h��1hg{4»Þ|�{yfShp� ¸

~­��q#o�hp�p|������ qY{yvr�Y£«v8d q�o8�1hphg����{�|r��hg��vyd qYvQ�gª«s�oyv�hg{y£«���kv�|���hgvyd�hg{�qYª«ªO{�hg��ª«£¤�rq�o8f¨£«��£«f¨£«¿rhpovyd�h���{�|�� qY��£«ª«£«v¢x�v�|kª¤|^o�h�|���h¾|�{�fS|�{�h�{�hg��ª«£¤�rq�o¥s�����hg{½qAfS|�{�h���hg��hg{�qYª���s�{�oyv­x��1hg{yvys�{y� qYvy£¤|��fS|O��hgª�°[vyd�hwª¤hg���^vydu|Y» vyd�hw��s�{�oyv�o¾qY{�hw��|�v½��hp�phpo�oCqY{y£«ª«xShMtO�E|���hg�Lvy£ qYª«ª«xJ�Z£¤o�vy{y£«��s�v�hp� ²M¸�~­v§d q�o

é[â��®é±î

Page 32: Maximizing the Robustness of TDMA Networks with ... · Maximizing the Robustness of TDMA Networks with Applications to TTP/C Bruno Gaujal, Nicolas Navet To cite this version: Bruno

�������������� ��������������������� ���� �!�#"%$'&(�*),+-����./�10 2�� � �

�Ehphg��o�d�|r¦��Àvyd qYv§vyd�hkv­¦¾|�|��Z}4hp�gvy£«��hpo¾vyd qYv¾¦§hg{�hk£¤��hg�Lvy£aW hp�ÃqY{�hk£«���p|�f¨� qYvy£«��ª¤hk£«�Åq�o�vy{�|����o�hg��o�h^¸

~­�Àq2»±s�vys�{�hw¦¾|�{y³1�Z|���hAfJqpx��p|���oy£¤��hg{§vyd�h��rq�o�hw¦�d�hg{�h2q�oys���o�hgv¾|Y»®©Qc�¯Ao½{�hpÈ-s�£«{�hpo¾vyd�hf¨£«��£«f¨£«¿�qYvy£¤|��Å|Y» vyd�h#ª¤|^o�o¥��{�|�� qY��£«ª«£«v­xJ¦�d�£«ª¤hwvyd�hw{�hpoyv§|Y» vyd�hw©Qc�¯Ao¥��hphp�Jv�|�f¨£«��£«f¨£«¿rhkvyd�h��{�|�� qY��£«ª«£«v¢x�vyd qYv�qYv®ª¤hrq�oyv8|���h½{�hg��ª«£¤�rq�£¤oUª¤|^o�vr¸8c�d�£¤o®fJqpx2�Eh½q#oy£«vys qYvy£¤|���qY{y£¤oy£«���k|���oyxZoyv�hgfSofJq���h�|Y»Q»ÞqY£«ªÆ¬­oy£«ª¤hg�-vAqY��� ��|��û¼qY£«ªÆ¬­oy£«ª¤hg�Lvk��|-��hpor¸¾¶k��|�vyd�hg{#»±s�vys�{�h�¦§|�{y³Å£¤owv�|S�p|���oy£¤��hg{kvyd�hs�o�hS|Y»�© |�{y¦¾qY{�� ��{y{�|�{�ʾ|�{y{�hp�gvy£¤|��Äv�hp��d���£¤È-s�hpou°±o�s���d q�o�lwhphp�O¬ÿÒZqYª¤|�fS|�� �p|O��hpoC²A£«��o�v�hrq��|Y»�{�hg��ª«£¤�rq�oS£«�Â|�{���hg{Sv�|%fJqY³�hÃvyd�h oyxOo�v�hgf hg��hg�:fS|�{�hÅ{�|���s�oyvSv�|ºvy{�qY��oyf¨£¤o�oy£¤|��.hg{y{�|�{�o�¸©�£«� qYª«ª«x��1¦§h�£«�-v�hg���Çv�|�oyvys��Zx vyd�h�{�|���s�o�vy��hpo�o2qY�LqY£«��oyv�vy{�qY��oyf¨£¤o�oy£¤|��Çhg{y{�|�{�o2|Y»§qY�ÇdLxO��{y£¤�hg��hg�-v4¬¼vy{y£«�^��hg{�hp� Éavy£«fShM¬¼vy{y£«�^��hg{�hp�Á��hgv¢¦§|�{y³ o�s���d%q�ok©�ª¤hMtZ{�qpxŦ�d�£¤��dÇ£¤oAqYª¤o�|��p|���o�£¤��hg{�hp�ûÞ|�{s�o�h�£«� Rw¬¼�-xL¬¼¹=£«{�h�qYs�v�|�fS|�vy£«��h�qY����ª«£¤�rqYvy£¤|���o�¸

����"p�Q/E�� �7����@«i�B �#¸U¶kª«vyfJqY�U�M0�¸��2qYs�}�qYª¼�QqY���º¶�¸ )A|�{��Z£ }4³1¸�¶��Zf¨£¤o�oy£¤|�� �p|��-vy{�|�ª½£«�Áo�v�|-��d q�oyvy£¤�Shg��hg�Lv

�^{�qY��d�or¸ 6������ $ 0#�� ���8Z ��� �� �� D)]�J�^���X�1��� Z � �� :��0I��EÆ�Z��N�° N�²�� �1NY��� �^���Z�1�^�^�^�Z¸@ò�CB �#¸®¶kª«vyfJqY�U� 0�¸��2qYs�}�qYª¼��qY���Ķ�¸ )k|�{��Z£ }4³E¸Åµ s�ª«vy£«fS|O�Zs�ª qY{y£«v­x��¥�p|��-��hMtO£«v¢xÄqY��� |���vy£Æ¬

f¨£«¿�qYvy£¤|��Å��{�|��1hg{yvy£¤hpor¸/� �1���S���X�1����Z��(�#"� \S��0I�1��� �� S�M�A�F���I��0IZ��-���1N�°Þ��²��ò�^�Y���O�Y�-�O���^�^�^�Z¸@ò�CB ��£«v�qY��¶kª«vyfJqY�U� 0¾{ys���|�2qYs�}�qYª¼� qY���ök{y£¤h )A|�{��Z£ }ÿ³E¸¥lkhg�^s�ª qY{w|�{���hg{y£«���uqY��� qY����ª«£¤�rqa¬

vy£¤|���o8£«���p|��Lvy{�|�ªZ�E|�ª«£¤�g£¤hpor¸������J0F ���E �#" &>�T��Z�0Y��� � � P�� :��& �� L��� � Z �&�P��� ����g�Oir��°Þ��²��«i ��� ��Zir�Z� �^�^�^�Z¸

@ ��B ��¸ 0�qY{y{�hg��o���d�hphg� qY�����¨¸ �kvyv�h^¸�¶k� qYª«xOoy£¤o2|Y»¾vyd�h���dLxZoy£¤�rqYª¾Ê¾¶A� ��s�o2ª qpx�hg{r¸�~¢��� � ��T :� ��0F L�8������ ���E � )M+ � �� �"���0Y�� LZ � � 6 ����� ��� �O� qY��hpow�^�Z¸ò�^���O�^�Z¸ò���Z�E�A�gv�|���{�hJi � ���O¸

@ONCB �#¸Z´k£«ª«��hg{r��c2¸Z©���d�{�hg{r��0�¸Oµ���ª«ª¤hg{r��qY���ÃÒ1¸Zz8|�ª¤hp�Z� qZ¸Qc�d�hkt-¬¼�-xL¬¼¦�£«{�h2�p|����phg��v �½c�£«fShM¬vy{y£«�^��hg{�hp��£«�Z»Þ|�{yfJqYvy£¤|��ÃhMt���d qY����h2qY���S»¼qY£«ªEoy£«ª¤hg���ph�oys����1|�{yv¾�LxJ��hg¦=oyxZoyv�hgf·o�hg{y�O£¤�phpor¸c®hp��d���£¤�rqYªQlwhg�E|�{yvk�^ÉOi � ���Z� c8hp��d���£¤o���d�h�¯k��£«��hg{�oy£«v��Yv#¹=£¤hg�U��~¢��oyvy£«vys�v�»���{#c®hp��d���£¤o���d�h~­�Z»Þ|�{yfJqYvy£«³1� i � ���Z¸§qYª¤o�|Jqp�^qY£«ª qY��ª¤h�q�okÒO¶ �.c8hp��d���£¤�rqYªQz�qY�Ehg{ ���^�1N1NZ¸

@ò�CB 0A¸���qYs�}yqYª#qY���'��¸��Aqr��hgvr¸Äc®{�q J�Ãoyd qY��£«���%£«�'{�hrqYªÆ¬¼vy£«fShÃ�Z£¤o�vy{y£«��s�v�hp�:oyxZoyv�hgfSo �Åqª¤|r¦�¬­�p|�f¨��ª¤hMtZ£«v­x qY����{�|�q���dU¸ � �1�M\��J� ��0 � �1�� �J � ZY�1��� �� S�M���^��°4i��^²��«i�N^�^����i�N����Z�8i � � �Z¸

@ð�<B���¸��A{�����o�v�hg£¤�Zª¼�+)�¸ ��qY�Lvy¿��kqY��� )�¸ �k|��1hgvy¿�¸ ʾ|�f¨f�s���£¤�rqYvy£¤|��!{�hgª«£ qY��£«ª«£«v¢x.£«� �Z£¤o4¬vy{y£«��s�v�hp��{�hrqYªÆ¬¼vy£«fSh2o�xOoyv�hgfSo�¸U~­� �"!����$#��0F2��F�J�F\ �� H&>� ����0F�^����� �IK � ���]\L��� ��0 � �� :��0I��E�&�P��� ����M� i � �Zi^¸

� � çedMìgí #ÿì

Page 33: Maximizing the Robustness of TDMA Networks with ... · Maximizing the Robustness of TDMA Networks with Applications to TTP/C Bruno Gaujal, Nicolas Navet To cite this version: Bruno

�^� �����A�������1E�� +�'+>��P� �

@ �CB 0A¸ )�qa}yhg³1¸ �QtZvy{�hgfJqYªwoy��ª«£«vyvy£«����o�|Y»k�E|�£«�LvS��{�|O�phpo�o�hpor¸ � �8�������X�1��� Z�� �#"�� \ ��0#�8������ �A�F� �Y��0#Z��-�8ir��°[�-²��ONY�L���JN1N^�Z�8i ���1NZ¸

@ �CB�~­�-v�hg{y� qYvy£¤|�� qYª1Ò-v�qY����qY{��J�k{y�LqY��£«¿�qYvy£¤|��J~yÒ���¸:�����8K�� �F��� Z�E��F��U � ��. �8\S�Y�IK!����0F� ��E:K8�8�^�ZY��� � �� � ZY�1��� �� U ����0�� 3�� � ��. �1\S�Y�IK � �� :��0I��E�EG��0()]0I�Y��+-����./�10 2a¸k~4Ò����8i � �Y��¸w~yÒ��i^i�NZi �a¬¢�Z¸

@«ir�CB )�¸ �k|��1hgvy¿�¸ �A�I��E U�$ ����� � ��� ���!� � &!���F�G�� � 0F�� LZ�� \LE��F�("���0%&>�T����0�� ����� �IK � ���Y�IK�K��IK)e\8\LE � ZY�1��� �� S�M¸ ��ª«s-¦§hg{�¶��rq���hgf¨£¤��z�s���ª«£¤oyd�hg{�o�� 0�|^oyv�|��U� i � ���O¸

@«i^i�B )�¸ �k|��1hgvy¿�� ��¸ 0�qYs�hg{r� qY����Ò1¸�z�|�ª¤hp�Z� qZ¸§c8|�ª¤hg{�qYvy£«���uqY{y��£«vy{�qY{yxÅ��|O��hA»ÞqY£«ª«s�{�hpow£«�Åvyd�hvy£«fShM¬¼vy{y£«�^��hg{�hp��qY{���d�£«v�hp�gvys�{�h^¸1~¢���S) ��3 !"!"� #�10FEGK � �1 ���0Y�F� � � � ��0IZ�� 3 !"!"�(� &!� ��0#����� �� 6 �=� ��)���µHqY{r¸ �^�^�Zi^¸

@«ir�CB )�¸��k|��1hgvy¿���l�¸��A|^o�o�qYª¼� l�¸ )AhMtZhgª¼� ¶�¸���{�����hg{r��´�¸�µ £«ª«ª«£«����hg{r�1l�¸�zQqYª«ª«£¤hg{�hg{r�UÊA¸�c®hgf�¬��ª¤h^�8qY���ºµÄ¸ ��{ys���¸¨µ�|O��h�d qY���Zª«£«��� £«�%vyd�hSvy£«fShM¬¼vy{y£«�^��hg{�hp� qY{���d�£«v�hp�gvys�{�h^¸ � �� :��0I��E� ����� ��Y��0��� �� � 0#��Z ����ZF�g�1��°4i � ����²��ò�Zi �O�^�Z��µHqY{r¸®i � ���Z¸

@«ir�CB�¶�¸¾¹ ¸�µHqY{�oyd qYª«ª�qY���!~M¸#�kª«³-£«�U¸ 6 �� ����1E ����� ��� � $ ���Y��0 � �I"[��� ���10F����1��� �� �� �K �����)e\8\LE � ZY�1��� �� S�M����|�ª«s�fSh2ig�L��|Y» � �1���S���X�1����Z�� �� �� Z�� �� LZ � �� LK � ����� ��Y��0F�� ��^¸�¶��rq���hgf¨£¤�z�{�hpo�or� i �����Z¸

@«ig��B���¸r��qp��hgvr���¨¸ ¬���¸YÒO|������^qY����©�¸aÒ-£«fS|���|�vr¸L¹º|�{�o�v4¬­�rq�o�h§��hrq��Zª«£«��h½»¼qY£«ª«s�{�h¥��{�|�� qY��£«ª«£«v­x�£«�{�hrqYªÆ¬¼vy£«fSh�qY����ª«£¤�rqYvy£¤|���o��Z£¤o�vy{y£«��s�v�hp��|���hg{½Ê¾¶k� °¼Ê¾|��-vy{�|�ª«ª¤hg{¥¶w{�hrqk�Ahgv­¦§|�{y³�²M¸ �����J0� L��E�#" �&�P��� ����])]0IZ������ �IZ ���J0I�g���L��°¼�^²��ò�^��� �O�Zi �Z�E�^�^�^�Z¸

@«i�NCB�~C¸ �#¸1�A|���ª¤h^¸ �¥µÇÊ qY���Hvyd�h¨qYs�v�|�fS|�vy£«��h�£«���Zs�oyvy{yx�¸ � E��IZ ��0I�� � Z��� � �1�� �J � ZY�1��� �� � ����� ��Y��0��� �������J0F ���E ��� qY��hpok�^�^���O���Oi^���A�gv�|���{�h�i � �^�Z¸

@«ir�CB¨�2Ò ��� �A{�|�s��U¸S�2Ò ����ÉaÓ�´�R+»ÞqYs�ª«v4¬¼v�|�ª¤hg{�qY�-v��p|�f¨f�s���£¤�rqYvy£¤|��U� �^s�ª«x%�^�^�Zi^¸�Ó¥hg{�o�£¤|��i^¸ò�Z� qp�^qY£«ª qY��ª¤h�qYv#d-vyvy� �ðÉ^Éa¦�¦�¦�¸ |^o�hg³L¬¼�Z�Ot ¸ |�{y�-ÉO¸

@«i��<B )�¸�zQ»Þhg£Æ»±hg{r¸+©�|�{yfJqYª���hg{y£aW �rqYvy£¤|�� |Y»�vyd�hÇvyvy�!�^{�|�s��!fShgf��Ehg{�oyd�£«� qYª«��|�{y£«vyd�fø·~¢�� �/��$ � � � � $� 3 !"! !�� �^�^�^�Z¸@«i �CB¨Ò-v�hM»ÞqY�:z�|�ª¤hp�Z� qZ�§z�hgv�hg{H0�qY{y{�hgvyvr�¾¶wª qY�?0¾s�{y��or�§qY���'¶k���ZxÁ¹ºhgª«ª«£«����o�¸Älkhg��ª«£¤�rqH��hM¬

v�hg{yf¨£«��£¤oyf qY���*c hMtZ£«��ª¤h�o���d�hp�Zs�ª«£«���u£«�Ãd qY{��Å{�hrqYªÆ¬¼vy£«fSh���hg�1hg����qY��ª¤h�oyxOo�v�hgfSor¸ 6������$S0I�� S� �8Z ��� �� S�-�� � ���]\L�J� ��0 �g�Z� ��°Þ��²��«ir�^����i^i^i^��©�hg�U¸��^�^�^�Z¸

@«i �CB¨ÊA¸ c®hgf¨��ª¤h^¸�¶8��|�£¤�Z£«���Jvyd�h�� qY����ª«£«����¬¼£¤�Z£¤|�vA»ÞqY£«ª«s�{�h�£«��q¨vy£«fShM¬¼vy{y£«�^��hg{�hp�H�p|�f¨f�s���£¤�rqa¬vy£¤|���o�xOoyv�hgføZ~¢� 6 L� ��0F ��1�����1 L��E �&���]\��P�F���J� �1 �� �1��Ea��U�$L��E���0#�1 :� � �1�M\��J���� ���� � $ � �� Y�� qY��hpow�Zi ���O�^���O�®i � ���Z¸

é[â��®é±î

Page 34: Maximizing the Robustness of TDMA Networks with ... · Maximizing the Robustness of TDMA Networks with Applications to TTP/C Bruno Gaujal, Nicolas Navet To cite this version: Bruno

�������������� ��������������������� ���� �!�#"%$'&(�*),+-����./�10 2�� �Zi

@ò�^�CB�c�c�c®hp��d�ʾ|�f¨��s�v�hg{yv�hp��d���£«³���f�� )�¸ �1\ �IZ�� �eZY�1��� �� �#" ���S�-$�$ ��� � � 0I�1�^��Z ��ESU�P��0 ������ � � !^� �^s�ª«xÃ�^�^�^�Z¸

@ò�Zi�B �#¸��EqY��|���£AqY���.z8¸§z�qr��qY�U¸=~­f¨��{�|��-£«��� vyd�hÅ{�hgª«£ qY��£«ª«£«v­x=qY���:o�qa»Þhgv­x |Y»�qYs�v�|�fS|�vy£«��hhgª¤hp�gvy{�|���£¤�por¸ 6������ � � Z�0I�^�Uir��°4i�²��ò�^���L�)�Z�8i � �^�Z¸

� � çedMìgí #ÿì

Page 35: Maximizing the Robustness of TDMA Networks with ... · Maximizing the Robustness of TDMA Networks with Applications to TTP/C Bruno Gaujal, Nicolas Navet To cite this version: Bruno

Unit e de recherche INRIA Lorraine, Technop ole de Nancy-Brabois, Campus scientifique,615 rue du Jardin Botanique, BP 101, 54600 VILLERS LES NANCY

Unit e de recherche INRIA Rennes, Irisa, Campus universitaire de Beaulieu, 35042 RENNES CedexUnit e de recherche INRIA Rh one-Alpes, 655, avenue de l’Europe, 38330 MONTBONNOT ST MARTIN

Unit e de recherche INRIA Rocquencourt, Domaine de Voluceau, Rocquencourt, BP 105, 78153 LE CHESNAY CedexUnit e de recherche INRIA Sophia-Antipolis, 2004 route des Lucioles, BP 93, 06902 SOPHIA-ANTIPOLIS Cedex

EditeurINRIA, Domaine de Voluceau, Rocquencourt, BP 105, 78153 LE CHESNAY Cedex (France)��������� ���� ���������� ��� ���

ISSN 0249-6399


Recommended