Lecture 5. ALOHAnet. ALOHA protocols. Client. Client. Hub. Client

Transcription

1 Lecture 5 ALOHA protocols ALOHAnet Aloha was a poneerng computer networkng system developed at the Unversty of Hawa n 97 s. The dea was to use rado to create a computer network lnkng the far-flung campuses of the Unversty. The orgnal verson of ALOHA used two dstnct frequences n a hub/star confguraton, wth the hub machne broadcastng packets to everyone on the downlnk channel, and the varous clent machnes sendng data to the hub on the uplnk channel. Clent Hub Clent Clent Data receved was mmedately re-sent, allowng clents to determne whether or not ther data had been receved properly. Any machne notcng corrupted data would wat a short tme and then re-send the packet

2 Infnte user populaton Users wll generate packets accordng to Posson process The aggregate packet arrval rate generated by the users s λ Pure Aloha If staton has a packet t wll mmedately try to transmt t Staton A Packet length s T Staton B t t + T t + T t + 3T 2 The packet generated by staton B colldes f any staton generates a packet on the tme nterval t <t<t +2T

3 Pure Aloha Packets arrve to the system wth ntensty λ Packets collde wth probablty p In case of collson, the packet wll be retransmtted after random back-off tme Aggregate of new packets and faled packets returnng to the transmtter buffer s stll a Posson process wth ntensty g = ( p) λ = λ λ 2 ( ) λ = λ+ λ g = p λ λ 2 μ λ = pλ Pure Aloha Packets arrval process s Posson wth ntensty g packets per second. Transmsson tme of the packet s T Probablty that k packets arrve n tme wndow Δt k gδt( gδt) Pr { k arrvals n Δ t} = e k! Probablty that there s no collson s equal to the probablty that no other packets arrve durng the tme nterval t : < t t + 2T 2 Pr{ no collson} = Pr{ arrvals n 2T} = e g T Normalzed load, average number of arrvals n tme T G=gT Throughput per tme T S = Ge 2G

4 Slotted Aloha If staton has a packet t wll mmedately n the begnnng of the next slot Departures Tme slot T Successful transmsson Collson Collson Successful transmsson Arrvals ~ ~ Busy perod B Idle perod I Cycle Slotted Aloha ~ The pdf for the length of the dle perod I s gven by Pr I = = Pr Some packets arrve durng the frst dle slot { } { } { } = Pr No arrval durng the frst slot = Pr Expected length of the dle perod G e { I = k} = Pr{ No arrvals durng k- frs slots and some arrvals at slot k} k ( ) e e = k {} { } ( ) I = E I = kpr I = k = k e e = e k= k=

5 Slotted Aloha The pdf for the length of the busy perod s gven by Pr B = = Pr No arrval durng the fst busy slot = e gt Pr { } { } { B = k} = Pr{ No arrval durng the fst k- busy slots and some arrvals at slot k} k ( e ) ( e ) = Expected length of the busy perod k B= E{ B } = kpr{ B = k} = k( e ) ( e ) = e k= k= G Slotted Aloha ~ Let U denote the number of useful slots durng the busy perod. That s, the number of slots durng whch there were no collsons. The probablty that a slot n busy perod s successful s Pr{ Busy slot s useful} = Pr{Number of arrvals = Number of arrvals>} Pr{ Number of arrvals=, Number of arrvals >} = Pr{ Number of arrvals>} Pr{ Number of arrvals =} Ge = = Pr Number of arrvals> e { } Expected number of useful slots s G U = Pr{ Busy slot s useful} B= G e

6 Throughput G U S = = e = Ge B+ I + e e Slotted Aloha Slotted Aloha.4.35 Pure ALOHA Slotted ALOHA.3.25 S G

7 Rado channel Consder a rado transmsson system n whch large number of transmtters are sendng packets to a base staton. P 625P α = 4 5 km km P Transmt power tx P,,2, 2 5 = Ptx = α < Receved power r α Power capture phenomena: The receved sgnal-tonterference rato (SIR) from one of the colldng packets s large enough so that t can be decoded successfully Rado channel Packet s receved correctly f the Sgnal-to-Interference Rato (SIR) s larger β P α tx r SIR = > β P α tx r j j Assume that capture occurs only f up to two packets collde and the SIR of the other packet s greater than β: { β } { β } { β } Pcapture = Pr P2 P + Pr P P2 = 2Pr P2 P 2 α α Pcapture 2Pr r β r2 = β

8 Rado channel Probablty of successful transmsson { } = { } + P { } Pr success Pr arrvals n T Pr arrval n T = + e PcaptureGe = e + β 2 α Ge capture Throughput per tme nterval T: 2 α S= GPr{ Success } = G e + β Ge 2 α = Ge + β G Rado channel α=3.5 β= db β=5 db β= db Slotted ALOHA.5 S G

9 Fnte user populaton M users share the channel An user can be n two states Thnkng The user has no packets n the transmsson buffer. The user generates a packet n a gven slot wth probablty σ Once a packet s generated, ts transmsson s attempted mmedately Backlogged If the packet transmsson was unsuccessful, the user enters the backlogged state In a gven slot a backlogged user tres to retransmt wth probablty ν Ths could be nterpreted as defnng geometrcally dstrbuted backoff tme n case of collson Slotted Aloha Fnte number of users Slots are numbered sequentally k=,, Let N(k) denote the number of backlogged users at the begnnng of slot k. N(k) defnes the state of the system The acton of an user depends on the current state of the system, but not on the past states. Hence, the system can be modeled as a Markov chan

10 Slotted Aloha Fnte number of users In steady state, the state probabltes converge to π j = lmk Pr { Nk ( ) = j} and the state transton probabltes converge to p = lm Pr Nk ( ) = jnk ( ) = j M k π = = { } The probabltes fulfll π= πp π = ( π π π ) P = p j M R. Rom and M. Sd, Multple Access Protocols, Sprnger-Verlag, Slotted Aloha Fnte number of users Let us defne j B j N j M j T j N j (, ) = Pr { backlogged users attempt transmsson = } = ν ( ν) (, ) = Pr { thnkng users attempt transmsson = } = σ ( σ) Transton from state to j<- s not possble, snce only one packet can be transmtted n a slot. Hence p j = for j=,,,-2 Transton from stat to tself, p can happen n two ways No new packet s generated No new packet n generated by a thnkng user and the transmsson of the backlogged users result n collson p = B(, ) T(, ) + B( k, T ) (, ) k= j M j

11 Slotted Aloha Fnte number of users Transton from state to j=+, can happen f one new packet s generated and there s collson p+ = B( k, T ) (, ) k= Transton from state to j>+ happens f j- thnkng users generated packet causng collson. The behavor of the backlogged users does not matter n ths case. p = T( j, ), j > + j The transton from state to state j=- occurs f no thnkng users generate packet and there s no collson p ( T ) B = (, ) (, ) Slotted Aloha Fnte number of users Fnally, we get j< M υ( υ) ( σ) j = M M υ( υ) ( σ) + ( M ) ρ( σ) ( υ) j = pj = M ( M ) ρ( σ) ( ( υ) ) j = + M j M j ρ ( σ) j > + j Now, the state probabltes can be solved from π= πp M π = =

12 Slotted Aloha Fnte number of users Probablty of successful transmsson when the number of backlogged users s Psuc () = T(, ) Psuc () = T(,) B(,) + T(,) B(,) M ( ) ( ) M = M σ σ ( υ) + ( σ) υ( υ) Total throughput M S E{ P () } P () π = = In a specal case where ν=σ, we get clearly suc suc = G lmm G = Ge M M M G S = E{ Psuc () } = Mσ ( σ) = G, G = Mσ M M Slotted Aloha Fnte number of users The throughput defnes the departure rate In state there are M- thnkng users generatng packets wth the rate (M-)σ. In stable system the nput and output rates must be equal. Thus we must requre that S = E{ ( M ) σ} = ( M N) σ where N s the average number of backlogged users N N = π =

13 Slotted Aloha Fnte number of users Let b denote the rate packets jon the backlog Snce packets depart from the system at rate S then a fracton of (S-b)/S of the packets are newer backlogged and suffer a delay of tme slot. The fracton of b/s of the packets that are backlogged wll frst have to wat n the backlog for w=n/b (b=nw) slots and thus see an overall delay of N/b+ slots. Hence the average delay n the system s ˆ S b b N D = + + S S b Recall that. S = ( M N) σ S Hence N = M σ and S M + b ˆ S b b M D σ = + = + S S b S σ Slotted Aloha Fnte number of users In a specal case where ν=σ, we get M ˆ ( σ ) D = + M σ ( σ) When M, also D so n the nfnte populaton case, the Aloha protocol becomes unstable When σ, we have D M. Ths mples that although most packets get through wth delay, some packets stll collde and see very large delays. On average the packet delay becomes equal to M

14 Slotted Aloha Fnte number of users The curve s parameterzed wth respect to σ Delay ncreases monotoncally as a functon of σ Throughput ncreases up to a pont after whch t starts to decrease agan Slotted Aloha Capture phenomenon If Mν<< and <<σ<, then capture phenomena s manfested Throughput Mσ S M + ( M ) σ ncreases as a functon of σ! Delay M D M + σ decrease as a functon of σ! R. Rom and M. Sd, Multple Access Protocols, Sprnger-Verlag, 989 The system eventually becomes backlogged, but the probablty that a backlogged user transmts s small. Hence, t s very lkely that a new packet gets transmtted rght away. One user captures the channel for a whle untl there s collson agan

15 Slotted Aloha Capture phenomenon Example packet trace M= σ=.7 ν=. Backlogged state, no transmsson Collson Successful transmsson from backlogged state Successful transmsson from thnkng state Due to hgh transmsson probablty, all users eventually become backlogged Capture Collson Backlogged user has low transmt probablty, but when t fnally transmts and becomes thnkng, she can capture the channel for long tme due to hgh σ (In)Stablty Stablty condton, for the nfnte number of users case: S=Ge -G G => G e In practce the assumpton that the aggregate rate of new packets and backlogged packets s Posson does not hold. Consder the case, n whch the traffc source s generatng packets wth arrval rate λ The arrval rate s only an average rate. The actual arrvals wll fluctuate around ths mean. If the tme average of the mean rate exceeds e - the throughput starts to degrease and the number of backlogged users growng wthout bound. Ths wll happen wth probablty! That s, the ALOHA protocol s nstable

16 (In)Stablty.45 Tme average over 5 slots Arrval rate exceed e - and thus the queues start to grow wthout bound Tme average G Throughput (S) λ Tme wndow Offered Traffc (G) G = λt Arrval process Backlogged users Stablzng Aloha { } a = Pr new packets arrve at slot = a λ = a = { } b( n) = Pr backlogged users transmt at slot n backlogged users Number of backlogged users at slot k when Nk ( ) = b( ) a [ ] + () [ ] wth probablty wth probablty b( ) a b a Nk ( + ) = + wth probablty b( ) a + j, j 2 wth probablty aj 6

17 Stablzng Aloha Expected change n system state durng one tme slot { } ( ) () () ( ) ( ) ( ) ( ) () ( ) j j= 2 E N k+ N k N k = = b a + b a + b a + + j a () () = λ b a b a System s stable f E N ( k+ ) N ( k) N ( k) = = λ b a b a { } ( ) ( ) () () λ < b a + b a Stablzng Aloha Assume that we somehow know how many packets are backlogged and choose the retransmsson probablty based on the state n n b ( n) = υ( n) υ( n) The maxmum arrval rate under whch the system s stable s gven by { ( ) ( ) } λ = lmsup n b n a + b n a n The stablty lmt yelds λ < e log a a + a

18 Defne Stablzng Aloha ( ) ( ) ( ) ( ) n ( ) ( ) Sn υ = b n a + b n a = υ n a + nυ n υ n a Maxmzng the above wth respect to ν yelds * a a ν ( n) = na a The maxmum throughput for gven n s thus n * * n Sn ( υ ) = a a n a The maxmum arrval rate must correspond to the system throughput a ln a + * * a λ < lmn Sn υ = e 35 ( ) n Stablzng Aloha For Posson arrval of packets, we have a ( ) λ λ = e! a = λ a Hence the stablty condton becomes a ln a + a λ < e = e And the probablty for backlogged user to transmt becomes a λ ν ( ) = = for small λ a n n λ n a * a n

19 Stablzng Aloha The mert of ALOHA s beng very smple protocol Its major drawback s that s nstable Stablzaton of the protocol requres knowledge on the system state n whch requres Coordnaton between the transmtters Or Some form of feedback mechansm to be appled ν k + = f ( vk, Feedback of slot k) Stablzng Aloha In practce, the number of retransmsson attempts s lmted. If the maxmum number of transmsson attempts s reach the packet s rejected. In ths manner the system can be stablzed, but the we have to cope wth occasonal packet drop. For smplcty, let us consder the case n whch σ=ν. The probablty of successful transmsson s M M G Psuc = E{ Psuc () } = Mσ ( σ) = G = S M Maxmum number of transmsson attempts s r. Probablty that packet s rejected s R. If retransmsson r attempts are ndependent R= ( S ) + Offered traffc G = RMσ

20 Reverse engneerng ALOHA Strategc games One shot random access game Stategc games Game theory s a form of nteractve decson theory that provdes a rgorous mathematcal formulaton for how decson-makers behave when they nteract. Players n a strategc game choose ther actons smultaneously and ndependently wthout knowng the selectons done by others nor past outcomes of the game. Defnton: A strategc game s the tuple G = N, { A, N},{ u, N} where N denotes the set of players For every N, A s the set of actons avalable to player. For every N, u : x A j R j N s the utlty functon of player showng hs preference over the set of acton profles A

21 Stategc games If the set of actons for every player s fnte, then the game s fnte An acton profle a=(a ) N s an outcome. That s, t defnes the actons taken by the players. An acton profle a A s preferred over acton profle b A only f u (a) u (b) Let Δ(A ) denote the set of all probablty dstrbutons σ on A (σ s the probablty that acton a s selected) There are two types of strateges avalable for the users Pure strategy: Player chooses an acton a A wth probablty. Mxed strategy: Player chooses an acton a A wth probablty σ (a ) Stategc games Defnton: The mxed strategy profle σ* n a strategc game wth s a mxed strategy Nash equlbrum f, for each player and every mxed strategy σ of player, the expected payoff to player of (σ *, σ* - ) s at least as large as the expected payoff to player of (σ, σ* - ). That s, σ = arg max σ, σ ( ) u A ( ) * * σ Δ Furthermore, the mxed strategy Nash equlbrum s sad to be fully mxed f σ non-degenerate. That s for all σ (a )

22 One shot random access game Each player can choose from two actons: a =: Transmt a =: Stay dle The utltes are gven by a = u( a) = a = N c a > Cost nduced by unsuccessful transmsson N One shot random access game Accordng to Inealtekn and Wcker (25), the one shot random access game wtn n players has 2 n - Nash equlbrum ponts: n pure strateges n whch only one node transmts There s one fully mxed nash equlbrum (FMNE) wth c n a = c + Stay dle σ ( a) = c n a = c + Transmt The rest are combnatons of pure and mxed strateges where some of the players choose to stay dle all the tme and the remanng n users use mxed strategy

23 One shot random access game The throughput of the FMNE s gven by c c n Sn ( ) = n c + c + Throughput s maxmzed f n * n c = n n The optmal throughput s gven by * lm n S ( n) e = whch corresponds to the maxmum channel utlzaton of the slotted ALOHA Reverse engneerng ALOHA Furthermore, when n the number of players choosng to transmt converge to Posson dstrbuton For gven c, the throughput S converges to c c S = lm n S( n) = ln + c + c The throughput of slotted aloha n case of nfnte user populaton s S = Ge c The throughput becomes the same f = e. That s, + c e c = e

24 Reverse engneerng ALOHA Convergence of the channel throughput Inaltekn, H.; Wcker, S., "A one-shot random access game for wreless networks, Wreless Networks, Communcatons and Moble Computng, 25 Internatonal Conference on, vol.2, no., pp vol.2, 3-6 June Applcatons of ALOHA ALOHA s utlzed n applcatons where carrer sensng s dffcult or not possble. E.g. In Frequency Dvson Duplexng (FDD), systems transmsson and recevng takes place smultaneously on dfferent frequency bands In under water acoustc communcatons, the propagaton delays are very long (long tme would be needed for carrer sensng n order to detect ongong transmssons)