slotted aloha as a stochastic game with partial information

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inria-00466286, version 1 - 23 Mar 2010 Author manuscript, published in "N/P"

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put

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� dq�zo�ltkº¶j÷���g]n�d � i|�In�c]l�yzo��zc�g�o]n�{Ry|lCn�c�k�n*kri �v6i|e*kCi|e�i[{¿o]}]v=n�dqyz}sy|{�n�c�k�ijl�lxdq�Li|�Ng�l�y|u�iju�df�4dqn�dqkre #",{Ry|lc� ��¸ � � �*´

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Arriaval probability qa

Ret

rans

mis

sion

pro

babi

lity

qr

m=2m=3m=4

� dq�zo�ltk�¸&÷0b.c]k�y|g�n�d � i|��l�k6n*lti|}]e � dfe�e�dqyz}¨g]l�y|u�iL~u�d4�fdqn�dqkre.d4}mn�c�k;n*kri � v6i|e*k�i|e�i¨{¿o]}]v=n�dfyz}my|{0n�c�kijl�lxdq�Li|�Ng�l�y|u�iju�df�4dqn�dqkre #"�{Gy|l�� ��¸ � � �*´��

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Arrival probability qa

Number of mobiles=2Number of mobiles=2Number of mobiles=3Number of mobiles=3

Retransmmission qr

Total throughput

� dq�zo�ltk �&÷�b.c�k�kr¢&o]df�fdqu�ltdfo � ltk6n*lti|}]e � d4e�e�dqyz}Cg]l�y|u�iju�df�·~dqn�dfkre�df}�n�c�k��zi � k¨v6i|e*ksi|e,iº{Go�}]v=n�dqyz}�y|{�n�c�k�ijltltdq�ji|�g]lty|u�iju�df�fdfn�dqkre� "|�

� W�¯�¯±² ­ @ `I®�Z�\Z �r®�^ ^ ��\^@_ ­ Z@� ²tZ �Z²t® ­ Z

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§¿dî�èk|��g�i|vt�|k6n�e�n�c�ijn.kr}Àn*k6l�n�c�k,e��&e*n*k � i|}]pºijltk,}�y|n�ltk=~ÿ*krv=n*krp�ª;dfe � 6�7!8 §N zªx�w§¿b�c]dfe[dfe;uNkrv6i|o]e*ksn�c�ksk=û&gNkrv=n*krp}&o � uk6l�y|{�ijl�ltdf�Li|�¼g�i|vt�|k6n�e�kr¢&o]i|�fe[n*yHn�c�k(k=û&gNkrv=n*krp}&o � uk6lCy|{�p�k6g�ijltn�df}��ºg�i|v��|k6n�e�ijn[e*n*kri|p���~»e*n�ijn*k|��i|}]pkri|vxcHn�d � k±i(g�i|v��|k6n[ijl�ltdf�|kre�ijn�n�c�k�e��&e*n*k � dqn�dfe�d � ~� krp�dfijn*kr�q�hn*lti|}]e � dfn*n*krp@�ïª

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� = °�� �0^�²t`I_ �>² ­?> �0Z�\N²A@�_�\@²�® ­½¾}Hn�c�dfe,e*krv=n�dfyz}H��khetc]i|�f�0y|u�n�i|df}Hn�c]k�l�k6n*lti|}�e � dfe�etdqyz}g]lty|u�iju�df�fdfn�dqkre±��c�dfvtc¡e*yz�q�|k�n�c]k�n*kri � ijl�k�n�c�k��zi � k

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½¾} � dq�zo�ltkre�´�i|}]p£òº��khp�k6g�dfv=n�n�c�khn�c�l�yzo��zc�~g�o]n,y|u]n�i|df}]krpHijn[n�c�k�y|g�n�d � i|��e*yz�fo�n�dqyz}Pi|}]pPn�c�k�y|g�~n�d � i|��l�k6n*lti|}]e � dfe�e�dqyz}>g]l�y|u�iju�df�fdqn�dfkre6��ltkre*gNkrv=n�dq�|kr�q�|��i|eiº{Go]}�v=n�dqyz}�y|{�n�c�k�ijl�lxdq�Li|�0g]l�y|u�iju�d4�fdqn»�|��{Ry|lCn�c�k�n*kri �g]lty|u��qk � ��dqn�c2� � ¸&��{Gy|l,�jijltdqyzo]e��Li|�fo]kre�y|{ ���±¤�ke*k6k�n�c]ijn�uNy|n�c,n�c]k�n�c�ltyzo��zc�g�o�nai|ea��kr�f�&i|ean�c�k¼l�k6n*lti|}]e*~� d4e�e�dqyz}wg]l�y|u�iju�df�fdqn�dqkre¨ijl�k � yz}�y|n*yz}]k9p�krv=l�kri|e�df}]�¦d4}n�c�k(v=yze*n6�£b.c]dfe[v6i|}¦uNkhk=û&gNkrv=n*krp e�df}�v=khltk6n*lti|}]e � d4e¾~e�dfyz}]e±uNkrv=y � k � y|l�k>v=yze�n��q� ��dqn�c¹d4}]v=l�kri|e�df}]� ������}df}�n*k6l�kre*n�d4}��¨{Gkrijn�o�l�k�dfe�n�c]ijn�{Gy|l�i|}��mü�û&krp � �q���Nn�c�kl�k6n*lxi|}]e � dfete�dqyz};g]l�y|u�iju�df�fdqn�dqkre�ü�lte�n0df}]v=l�kri|e*k�df}[n�c�k�ijl*~ltdf�Li|��g]l�y|u�iju�df�fdqn©�¹i|}]p�n�c�kr}¡p�krv=l�kri|e*k|� � y|l�� � ���df}hv=yz}�n*lti|e*n6�]n�c�k�y|g]n�d � i|��ltk6n*lti|}]e � d4e�e�dqyz}�g]l�y|u�iju�d4�fdqn»�p�krv=ltkri|e*kreCd4}9n�c�khijl�ltdq�ji|�Ig]l�y|u�iju�df�fdqn©�£§G��c�dfvtcPdfe,}]ijn¾~o�lxi|��e�df}]v=kCv=yz}��|kre*n�dqyz}¥df}(n�c�k;e*��e*n*k � d4}]v=l�kri|e*kre�i|e #"df}�v=l�kri|e*krexªx�

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Arrival probability qa

Thr

ougt

put

θ = 0θ=0.2θ=0.4θ=0.6θ=0.8

� dq�zo�ltkö´�÷ b.c�ltyzo��zc�g�o�n�ijn�y|g]n�d � i|� ( {Ry|ln�c�k n*kri � v6i|e*k i|eYi�{¿o]}]v=n�dqyz} y|{¡n�c�k�ijl*~ltdf�Li|�Cg]l�y|u�iju�df�fdqn�dqkre " {Ry|l � � ¸�i|}]p �]������ � ¸ ��� � ´ ��� � � ��� � ���

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

Arrival probability qa

Retr

ansm

issi

on p

robabili

ty q

r

θ=0θ=0.2θ=0.4θ=0.6θ=0.8

� dq�zo�ltk¨ò&÷�b.c�k±y|g]n�d � i|�0l�k6n*lxi|}]e � dfete�dqyz}�g]l�y|u]~iju�d4�fdqn�dqkremd4}�n�c�k£n*kri � v6i|e�k£i|e¥iÁ{¿o]}]v=n�dfyz}�y|{n�c�k9ijltltdq�ji|��g�l�y|u�iju�df�4dqn�dqkre "�{Ry|l^� � ¸¦i|}]p�v�#����� � ¸ ��� � ´ ��� � � ��� � �&�

ø�k=û&n>��kHv=yz}�e�dfp�k6lmn�c]k9�zi � kHg�l�y|u��qk � ��dfn�c¸ � y|u�d4�qkre6� � dq�zo�l�k �¡e�c�y���e¥n�c�k d � g�i|v=n�y|{ �wyz}n�c�kHkr¢�o]d4�fdqu]ltd4o � l�k6n*lti|}�e � dfe�etdqyz}¡g]l�y|u�iju�df�fdqn©� ( ��i|ei£{¿o]}]v=n�dqyz}Áy|{,n�c�k�ijl�ltdq�ji|�C #"|��¤�k�n�c]ijn(df}]v=l�kri|e�d4}��n�c�k[v=yze*n �¨l�kre�o]�qn�e�df}>p�krv=l�kri|e�df}]�¨n�c�k;l�k6n*lti|}�e � dfe�etdqyz}g]lty|u�iju�df�fdfn�dqkre6�

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Arrival probability qa

Retr

ansm

mis

sion p

robabili

ty q r

θ=0θ=0.2θ=0.4θ=0.6θ=0.9θ=1

� dq�zo�ltk �&÷ b.c�kTkr¢&o]df�fdfu]ltdfo � l�k6n*lti|}�e � dfe�etdqyz}g]lty|u�iju�df�fdfn�dqkre�df}>n�c�kC�zi � k[v6i|e�k[i|e�{¿o]}]v=n�dqyz}ºy|{n�c�k±ijltltdq�ji|��g�l�y|u�iju�df�4dqn�dqkre "�{Ry|l`� �Y¸�§¿}�o � ~uNk6l�y|{ � y|u�df�fkrexª�i|}]p��v�#����� � ¸ ��� � ´ ��� � � ��� � ���r¶z�

� dq�zo�l�k =�g]lty-��dfp�kre¨n�c�kºn*y|n�i|��n�c�l�yzo]�zc�g�o�n�ijnkr¢&o]df�fdfu]ltdfo � i|eai.{¿o]}]v=n�dqyz}�y|{�n�c]k�ijl�ltdf�Li|�|g�l�y|u�iju�df�4dqn�dqkre{Gy|l�n�c�k�v6i|e*k�y|{Nn»��y � y|u�df�qkrer��¤�k�e*k6k.n�c]ijn�df}]p�k6krp�n�c]k

n�c�ltyzo��zc�g�o�nCdfe,d � g]l�y-�|krpPv=yz}]e�dfp]k6ltiju��q�¥u��9i|p�p]df}��ºiv=yze*nCyz}�l�k6n*lti|}�e � dfe�etdqyz}@�@kre*gNkrv6dfi|�f�f�º{Gy|l,�fijl��|k�ijltltdq�ji|�g]lty|u�iju�df�fdfn�dqkre6�

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Arrival probability qa

Thro

ughput

θ=0θ=0.2θ=0.4θ=0.6θ=0.9θ=1

� dq�zo�ltk =�÷�b�y|n�i|�¨n�c�l�yzo]�zc�g�o�n9{Gy|lHn�c]kÁ�zi � kv6i|e*kPi|eºi {¿o]}]v=n�dqyz}¡y|{[n�c�k9ijltltdq�ji|��g]l�y|u�iju�d4�fd·~n�dqkre� "({Ry|l � ��¸ §¿}�o � uNk6l�y|{ � y|u�df�qkre'ª�i|}]p�v�#����� � ¸ ��� � ´ ��� � � ��� � � �r¶|�

¤�k;n�c�kr}ºv=y � g�o�n*k,n�c�kCg]ltdfv6d4}�� �±n�c]ijn�dfe�}�krv'~kre�etijl��±{Gy|l�n�c�k�kr¢&o]df�4dqu]ltdfo � l�k6n*lti|}]e � dfe�e�dqyz}�g]l�y|u�iju�df�·~dqn�dfkre�n*yºv=yzdf}]v6dfp]k;��dqn�c�n�c�yze*k±y|u�n�i|df}�krp�{Gy|l�n�c�k±n*kri �g]lty|u��qk � ��b�c]dfe�d4e�n�c�k;�ji|�fo�k,y|{ ��n�c]ijn���df�4�@�&dfkr�fpmn�c�ky|g]n�d � i|�¼e*�&e�n*k � n�c]l�yzo��zc�g�o]n6�9b.c�khl�kre�o]�qn�e[ijl�ksg]ltk=~e*kr}�n*krpºd4} � df��� �&�

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Arrival probability qa

extra θretransmmission q

rthroughput

� dq�zo�ltk�&÷�b.c]kCl�k6n*lxi|}]e � dfete�dqyz}mv=yze*n �he�o]vxc>n�c�ijn�n�c�ky|g]n�d � i|�@l�k6n*lti|}�e � dfe�etdqyz}(df}sn�c�kC�zi � kCv=yzdf}�v6dfp�kre���dfn�cn�c]ijn[y|{�n�c�ksy|ltdq�zdf}]i|��n*kri � g]lty|u��qk � �Ii|e[{¿o]}]v=n�dqyz}Py|{n�c�k�ijltltdq�ji|�Ng]l�y|u�iju�d4�fdqn�dqkre #"�{Gy|l�� ��¸&�

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