Performance Analysis of Markov Modulated 1-Persistent CSMA/CA Protocols with Exponential Backoff Scheduling

Transcription

1 Performance Analyss of Markov Modulated -Persstent CSMA/CA Protocols wth ponental Backoff Schedulng Pu ng Wong, Dongje Yn, and Tony T. Lee, Abstract. Ths paper proposes a Markovan model of -persstent CSMA/CA protocols wth -ponental Backoff schedulng algorthms. The nput buffer of each access node s modeled as a Geo/G/ queue, and the servce tme dstrbuton of each ndvdual head-of-lne packet s derved from the Markov chan of the underlyng schedulng algorthm. From the queung model, we derve the characterstc equaton of network throughput and obtan the stable throughput and bounded delay regons wth respect to the retransmsson factor. Our results show that the stable throughput regon of the exponental backoff scheme exsts even for an nfnte populaton. Moreover, we fnd that the bounded delay regon of exponental backoff s only a sub-set of ts stable throughput regon due to the large varance of the servce tme of nput packets caused by the capture effect. All analytcal results presented n ths paper are verfed by smulatons. eywords: queung analyss, performance analyss, -persstent CSMA, ponental Backoff. Introducton Carrer Sense Multple Access (CSMA) s a standardzed Meda Access Control (MAC) protocol for the transmsson of network statons over a commonly shared medum, such as 80. wreless LANs or Ethernet networks. It s a set of rules for resolvng collsons when two or more nodes attempt to use a transmsson channel smultaneously. The network statons governed by CSMA wth collson avodance (CSMA/CA) check the busy/dle status of the transmsson channel before attemptng to transmt. If the channel s detected busy, the packets nvolved n the collson attempt to retransmt after a random tme nterval. Wth exponental backoff schedulng, ths random watng tme nterval for retransmsson geometrcally ncreases wth the number of collsons encountered by the packet. The p-persstent CSMA protocols try to maxmze the channel utlzaton by contnuously montorng the transmsson medum. An actve node wth a packet ready to transmt constantly checks the channel status. If the channel s sensed dle, the node transmts the packet wth a probablty p. Otherwse, the node wll persstently montor the channel untl t becomes dle, and then transmts agan wth the same probablty p. Ths process repeats untl the packet s dspatched. Collsons may stll occur when two or more packets are smultaneously transmtted. The -Persstent Carrer-Sense Multple Access (P-CSMA) protocol s a specal case of the p-persstent CSMA protocol wth p =.. Dept. of Informaton Engneerng, The Chnese Unversty of Hong ong, Hong ong (emal: {wpk007, ydj008, ttlee}@e.cuhk.edu.hk). Dept. of Electroncs Engneerng, Shangha Jao Tong Unversty, Shangha, Chna.

2 The complete analyss of a buffered random-access network has long been consdered to be a challengng problem even for the smplest case. The throughput analyss of slotted -persstent CSMA for a network wth an nfnte populaton of statons was frst nvestgated n [] by lenrock and Tobag. Many follow-up researches were proposed to analyze the performance of -persstent CSMA protocol. For example, a fnte populaton system wth the slotted -persstent CSMA, whch can also be extended to nfnte populaton cases, was analyzed n []. The characterstc equaton for network throughput was re-derved n [3] by usng a state-transton dagram of the transmsson channel. However, these results are lmted to the maxmum throughput analyss of a saturated network n whch the schedulng of collded packets was not consdered. The -ponental Backoff algorthm s a wdely used collson resoluton scheme n most MAC protocols such as the 80. [4]-[8]. A packet encountered collsons tmes s scheduled for retransmsson wth probablty q, for =,,. The parameter q s called the retransmsson factor wth 0 < q < and s the cut-off phase. Many publshed works [9]-[3] studed the performance of MAC protocols n conjuncton wth the exponental backoff algorthm under a saturaton traffc condton. A two-dmensonal Markov chan that descrbed the exponental backoff algorthm was proposed by Banch n [9] to obtan the network throughput of the 80. Dstrbuted Coordnaton Functon (DCF) protocol. Authors n [0]-[3] appled the two-dmensonal Markov chan to analyze the performance of network throughput of the 80. DCF under dfferent scenaros. Although these works, together wth the prevous results n []-[3], have revealed the connecton between the maxmum throughput and the underlyng backoff algorthm, ssues on throughput stablty and bounded delay reman open under the saturaton assumpton. Some non-saturated analyses of 80. protocols under a smplfed buffer-less assumpton were reported n [4]-[6], n whch the queung behavors of nput buffers were completely gnored. The am of ths paper s to provde a complete performance analyss of the P-CSMA protocol under a non-unsaturated traffc envronment. Our approach s an extenson of the Markov model for non-persstent CSMA networks proposed n [7], whch ncludes both the stable throughput regon and the correspondng delay analyss of NP-CSMA. The throughput of a buffered random-access network s determned by the actvtes of Head-Of-Lne (HOL) packets n the entre network. In ths paper, we assume that the nput to each node s a Bernoull process wth rate λ packet/tmeslot and each nput buffer s modeled as a Geo/G/ queue. The servce tme of HOL packets s descrbed by a Markov chan that represents the backoff schedulng algorthm from whch the steady-state network throughput can be expressed as a functon of the aggregate attempt rate G of HOL packets. Usng an extenson of the methodology developed n [7], the complete queung analyss enables us to specfy the stable regons of both throughput and bounded mean delay for P-CSMA. Specfcally, we characterze the stable throughput regon of retransmsson factor q from the frst moment of

3 packet servce tme wth a gven aggregate nput rate ˆ = nλ packet/tmeslot, where n s the number of nodes n the system. Furthermore, the Pollaczek-hnchn (P-) formula of Geo/G/ queue reveals that a bounded mean delay requres a bounded second moment of servce tme, whch also depends on the retransmsson schedulng algorthm employed n P-CSMA protocols. The exponental backoff schedulng algorthm wth nfnte cutoff phase ( = ) s of specal nterest n practcal operatons of P-CSMA protocols. Our result on the throughput of exponental backoff scheme agrees wth that reported n [4] that the network can have non-zero throughput even f the number of nodes n. Moreover, the bounded delay regon determned by our queung model s only a subset of the stable throughput regon. For any retransmsson factor q nsde the stable throughput regon but not wthn the bounded delay regon, we show that a stable throughput of the entre network can be acheved wth a large delay varance due to the capture effect descrbed n [8] and [9]. Thus, the maxmum throughput of the network wthn the bounded delay regon s slghtly smaller than that achevable n the stable throughput regon. The rest of ths paper s organzed as follows. The channel model and nput buffer model that characterze the network throughput of P-CSMA protocols are presented n secton and secton 3, respectvely. Secton 4 descrbes the general requrements of stable condtons. Based on these condtons, the stable throughput and bounded delay regons of the P-CSMA protocol wth exponental backoff scheme are specfed n secton 5. Secton 6 provdes a concluson. Markov chan of Slotted P-CSMA In a slotted P-CSMA network, the tme axs s slotted and the network s synchronzed. Packets can be sent only at the begnnng of a tmeslot, and each packet s dentcal and takes one slot tme for transmsson. Suppose the rato of propagaton delay to packet transmsson tme s a, then the tmeslot can be further dvded nto mn-slots of slot sze a. Accordng to the channel status dsplayed n Fg., the busy perod s a seres of transmsson perods each wth + a tmeslots, and pure dle perod s the tme n whch the channel s dle and no packet presents awatng transmsson. Then the tme axs can be consdered as a sequence of alternatng busy perods and pure dle perods. There are two types of transmssons wth dfferent probabltes of success. The leadng transmsson perod n each busy perod s called the Type I transmsson perod, n whch the packet s sent successfully f, and only f, no one s scheduled to access the channel n the prevous mn-slot. All subsequent transmsson perods are defned as Type II transmssons. Due to the persstence property of P-CSMA, packets scheduled to access the channel n any transmsson 3

4 perod accumulate at the end of the prevous transmsson perod; hence, the second type of transmsson s successful f, and only f, no one attempts to access the channel durng the prevous transmsson perod. In each transmsson perod, a packet s ether successfully transmtted or collded. Therefore, the channel has three fundamental states: pure dle (Idle), successful transmsson (Suc), and collson (Col). For the P-CSMA, the successful transmsson state should be dvded nto two sub-states Suc and Suc correspondng to the Type I and Type II transmsson perods, respectvely. The state transtons of the channel can be descrbed by the Markov chan shown n Fg.. The lmtng probabltes Idle, Suc and Col of the Markov chan satsfy the followng set of equatons: P P P P P Idle, Idle Idle Suc, Idle Suc Suc, Idle Suc Col, Idle Col Suc Idle, Suc Idle P P P Suc, Suc Suc Suc, Suc Suc Col, Suc Col PCol, Col Col PIdle, Col Idle PSuc, Col Suc PSuc, Col Suc. () Snce packets attempt to access the channel only when the channel s detected as dle, the attempt rate n any busy perod s zero. Durng a pure dle perod, the aggregate attempts generated by all fresh and re-scheduled HOL packets form a Posson stream wth rate G; therefore, the probablty that no attempt s generated n a mn-slot s e -ag. From a system pont of vew, the Type I transmsson s successful only f there s exactly one attemptng packet n a mn-slot wth the probablty age -ag. All packets scheduled to access the channel n any transmsson perod are accumulated at the end of the prevous perod, and then attempt to access the channel at the same tme. Thus, the transton probablty from transmsson perod to dle perod s equvalent to that when there s no scheduled packets wthn (+a) slot tmes,.e., e -(+a)g. Smlarly, the probablty of havng a successful Type II transmsson s (+a)ge -(+a)g. Hence, the transton probabltes of the Markov chan are gven as follows: P Idle, Idle ag PIdle, Suc age ag P e age Idle, Col e ag ag P P P e Suc, Idle Suc, Idle Col, Idle Suc, Suc Suc, Suc Col, Suc Suc, Col Suc, Col Col, Col ag P P P a Ge ag ag ag P P P e a Ge. () We obtan the followng lmtng probabltes from () and () n a straghtforward manner: 4

5 Idle ag e ag ag e e ag ag age e e e ag ag age e e e ag e age Ge e ag ag e e Suc ag ag Suc ag ag Col ag ag ag. (3) The tme-average probabltes of each state can be easly obtaned from sojourn tmes of the Idle, Suc, and Col states tidle a, and t t t a, respectvely. Of partcular mportance, the probablty that the channel s ether n Suc Suc Col Suc or Suc states s gven by asucsuc ag ag a Ge a e ag Suc Suc ag a Suc Suc Col a Idle a e ae. (4) Snce the successful transmsson of a packet only takes / (+a) of the tme n a transmsson perod, the followng network throughput s defned by the fracton of the tme that the channel s productve: ˆ Suc Suc ag ag Ge a e ag a a e ae ag, (5) whch s consstent wth lenrock and Tobag s result n []. Although the network throughput can be obtaned from the above model, but t s by no means a comprehensve performance analyss of the system because the re-schedulng of collded HOL packets s not consdered n the channel model. The next secton descrbes a more detaled queung model of the nput buffer n whch the servce tme dstrbuton s derved n the context of the underlyng backoff schedulng algorthm. 3 Queung Model of Input Buffer The nput buffer of each node s modeled as a Geo/G/ wth a Bernoull arrval process of rate λ packets/tmeslot. As we mentoned before, we consder the P-CSMA protocol wth the -ponental Backoff algorthm for contenton resolutons. The behavor of each HOL packet wth ths backoff algorthm s descrbed by a smple flow chart shown n Fg. 3. A fresh HOL packet s ntally n phase 0 and sent only when an dle channel s detected. If the channel s busy, t wats untl the channel becomes dle and then transmts the packet mmedately. If the transmsson s successful, a new fresh HOL packet starts from phase 0. Otherwse, the packet s n backoff mode n whch the random watng tme s determned by the current backoff phase. A packet s n phase, f t has encountered collson tmes. The -ponental Backoff algorthm allows an HOL packet n phase to retransmt wth probablty q, for =,,, where 0 < q < s 5

6 the retransmsson factor and s the cut-off phase. That s, the retransmsson probablty decreases exponentally wth the number of collsons, up to tmes, experenced by the backlogged HOL packets. Based on the flow chart of the P-CSMA protocol shown n Fg. 3, the correspondng state transton dagram wth the -ponental Backoff algorthm s shown n Fg. 4. An HOL packet s n one of the three fundamental states: sensng (S ), transmsson, and watng (W ), 0,...,, at any tme. Note the transmsson state s splt nto two sub-states: F and F', correspondng to the two types of transmssons, respectvely. If channel dle s detected n the sensng state S, an HOL packet s sent and moved to state F wth probablty q. Otherwse, the correspondng node contnually montors the channel actvty untl an dle channel s sensed. Ths persstent sensng acton s represented by the watng state W. When the channel turns dle, the packet moves to the Type II transmsson state F' wth probablty q. If the packet s successfully sent, a fresh HOL packet starts wth the ntal sensng state S 0 ; otherwse, the collded packet moves to phase + and repeats the above process startng from the sensng state S +. Moreover, for the sake of smplcty, we assume that a new packet at an empty buffer arrves at the begnnng of the transmsson perod f the channel s busy, so that the watng tme s slot tme for any phase. The transton probabltes of the Markov chan are derved below. Let α be the probablty that the channel s n pure dle state. Then t can be calculated from the lmtng probabltes gven by (3) as follows: Idle Suc Suc Col ag a Idle ae a a a e ae ag ag. (6) It s llustrated n Fg. 4 that the Type I transmsson s successful only f all other nodes do not attempt to send n the frst mn-slot of the busy perod. Thus, the probablty of a successful Type I transmsson s p e ag. (7) Whle the success of the Type II transmsson requres that no other nodes attempt to send n the prevous transmsson perod; hence, the probablty of a successful Type II transmsson s ( ) p e ag. (8) Type I transmssons lead busy perods; therefore, the probablty that a transmsson perod s Type I s equvalent to the probablty α gven by (6) that the system s n a pure dle perod. It follows that the probablty of successful transmsson of P-CSMA s gven by 6

7 p Pr{Type transmsson} p Pr{Type transmsson} p p ( ) p a p p. a p ap (9) Substtute (7) and (8) nto (9), and the probablty of successful transmsson can be expressed by the followng functon of the attempt rate G: p G ag ag e a e ag a e ae ag. (0) It should be noted that the probablty of success gven by (0) s consstent wth (5) because p ˆ G by defnton. Let s, f, f ', and w be the respectve lmtng probabltes of states S, F, F', and W, of the Markov chan shown n Fg. 4, from whch we obtan the followng set of state equatons: p f0 f f + p f0 ' f ' f ',for 0 s q s q w p f p f ',for,..., q s q w p f f p f ' f ',for. w s, for 0,..., f q s, for 0..., f ' q w, for 0,..., It can be proven from () that f p + q >, then all states of the Markov chan are postve recurrent and aperodc. () Thus, the tme-average probabltes of those states can be determned from () wth the sojourn tme ts a, t, F t, and t of state S, F, F', and W, respectvely, for = 0,, as follows: ' F W a p, 0,,..., D q s a p, pd q p, 0,,..., D q w p, pd q, () p, 0,,..., f D p, pd p, 0,,..., f ' D p, pd 7

8 where q a q p D. p q p q p The offered load ρ of each nput queue s the probablty that the queue s non-empty. It s the basc measure for analyzng the performance of each nput buffer. The nput rate λ of the Bernoull arrval process can be nterpreted as the probablty of fndng a packet arrved at nput n any tme slot. Each nput packet wll eventually become a fresh HOL packet, and vst the transmsson states n phase 0 for one slot tme. Therefore, the nput rate λ should be equal to f, the probablty of fndng a phase 0 HOL packet n ether Type I or Type II transmsson states n any tme f ' 0 0 slot. Wth the tme-average probablty of state F0 and F'gven 0 n (), the expresson of the offered load can be obtaned as follows: a q p q. (3) f 0 f 0' p q p q p We show n the next theorem that the network throughput derved from the servce tme of HOL packets s the same as (5), whch was prevously obtaned from the channel model shown n Fg.. Although the servce tme s obvously dependent on the retransmsson factor q of the backoff schedulng algorthm, the fact that the throughput s nvarant wth respect to q mples that the stablty of the system cannot be determned by the characterstc equaton of throughput alone; t s manly related to the queung behavor of each nput buffer. Theorem. For buffered -persstent CSMA wth -ponental Backoff, the throughput n equlbrum s gven by ˆ out ˆ ln p p ag ag Ge a e ag a a e ae ag. (4) Proof: A partcular HOL packet s ready to be sent only f an dle channel s detected. The probablty of successful Type I transmsson p for a desred node s the condtonal probablty that none of the other nodes accesses the channel gven that all nodes sense the channel dle, whch s gven by p Pr{none of other n nodes access the channel channel s sensed dle} Pr{none of other n nodes access the channel}. Pr{channel s sensed dle} (5) If no one accesses the channel, t means that all the other n nodes are ether empty, or n sensng states but not accessng the channel. Thus, we have Pr{none of other n nodes access the channel} Pr{node s empty} Pr{node s n sensng but not scheduled to be sent} for large n n s 0 q n s 0 q exp. n (6) 8

9 The probablty that the node senses an dle channel s gven by Pr{channel s sensed dle}= Pr{node s empty} Pr{node s n sensng state} for large n n s 0 n s 0 exp. n (7) Substtutng (6), (9), (), and (3) nto (6) and (7), the probablty of successful transmsson p defned by (5) can be expressed as exp exp n s 0 q p expn s 0 n s q 0 an exp p p p 0 D a ˆ exp. p (8) In equlbrum, the network throughput ˆout should be equal to the aggregate nput rate ˆ. It s easy to show that the network throughput (4) n equlbrum can be obtaned from (8). Note that the throughput gven by (4) also agrees wth that obtaned by lenrock and Tobag n [] and our channel model n secton. The consstency between these approaches ndcates that t s approprate to adopt the ndependence assumpton among nput buffers. Ths s because the correlaton among n nput queues becomes weak when n s large [0]. Furthermore, let random varables S, F, F, and W be the servce completon tme of an HOL packet, startng from the states S, F, F', and W respectvely, untl t s successfully transmtted. We assume, wthout loss of generalty, that M = /a s an nteger. It s straghtforward to show from the Markov chan of Fg. 4 that the generatng functons S( z), F( z), F'( z ), and W ( z) of these servce completon tmes can be found by solvng the followng set of equatons: S S ( z) E z q zs ( z) q zf ( z) zw ( z), for 0,,..., W M M W ( z) E z q z F '( z) q z S ( z), for 0,,..., F M M E z p z p z S ( z), for 0,,..., F ( z) F ( z), for F E z pz M p z M S ( z), for 0,,..., F '( z) F '( z), for. (9) Snce the servce of each HOL packet always starts from the state S 0, the frst and second moments of servce tme can be derved from the set of generatng functons n (9) and are gven n Appendx I. Note that the mean servce tme 9

10 f f 0 ' 0 derved n Appendx I s consstent wth the expresson gven n (3). Based on the queung model of the nput buffer, we wll nvestgate varous stablty ssues of P-CSMA concernng throughput and delay n the followng sectons. 4 Stable Regons For a -persstent CSMA network wth n nodes, suppose that there are total n b n backlogged HOL packets n the mn-slot before transmsson, n whch n packets are n the sensng state of phase, for =,,. The followng HOL packets may attempt to transmt durng the mn-slot of a sensng state: An empty node may send a newly arrved packet wth probablty aλ; An HOL packet n phase 0 wll be transmtted mmedately; A backlogged HOL packet n phase wll be transmtted wth probablty q, for =,,. Hence, the attempt rate n ths mn-slot s gven by The mean number of empty nodes n the system s whle the mean number of backlogged nodes n phase s ag a E n n q E n. (0) b 0 b E n n n ; () E n s n. () s j0 j It follows from () that the attempt rate n a mn-slot defned (0) can be gven as follows: n ag a ˆ a ˆ n 0 j0 s s q j pq. (3) p pq p q pq q By solvng equaton (3), the retransmsson factor q can be expressed as a functon of the attempt rate G, denoted as q h( G). For any fnte cut-off phase, ths functon hg ( ) s on the order /n,.e., q = O(/n), whch mples that the network s ntrnscally unstable when the number of nodes n s large. Ths pont can be explaned by consderng the saturated case of Geometrc Retransmsson ( = ) when all nodes have backlogged HOL packets. In ths worst-case scenaro, the probabltes of successful transmsson for the Type I and Type II are p e anq and a nq p e. 0

11 Consequently, ether the retransmsson factor q or the probabltes of successful transmsson p and p approach zero when the number of nodes n. Ths nherently nstable property of Geometrc Retransmsson has also been prevously reported n [4] and [5]. On the other hand, ponental Backoff mtgates the contenton problem by pushng packets to deeper phases; therefore, the retransmsson factor q wth ponental Backoff ( = ) expressed n terms of hg ( ) s on the order of O(), whch suggests that the network can be stablzed even for an nfnte populaton. A detaled study of ponental Backoff s presented n secton 5. The characterstc equaton (4) of the throughput versus attempt rate G s a curve that frst ncreases and then decreases wth G as plotted n Fg. 5, n whch max ˆ denotes the maxmum throughput of equaton (4). Ths ndcates that to have optmal throughput, the attempt rate G cannot be too small or too large. For a throughput smaller than the maxmum throughput, ˆ ˆ max, the throughput equaton (4) has two roots; the smaller and larger roots are denoted as G ( ˆ S ) and G ( ˆ ) L, respectvely. Consderng the tradeoff between G and p, we know G should be bounded n the range between G ( ˆ S ) and G ( ˆ ) L to ensure a stable network throughput where ˆ out ˆ. amples n Fg. 5 llustrate ths pont. For a gven nput rate ˆ 0.3 and a = 0., the network has a stable throughput ˆ ˆ 0.3 when G s wthn the range [ 0.347,.98]. Thus, a necessary condton of stable throughput of the entre system can be stated as follows: Stable Throughput Condton (STC): For any nput rate ˆ ˆ max out, the attempt rate G should satsfy G ( ˆ ) G G ( ˆ ). (4) S In general, the attempt rate G s an mplct functon of the retransmsson factor q assocated wth the underlyng schedulng algorthm. The retransmsson factor q = h(g) can be obtaned by solvng equaton (3).Ths functonal relatonshp, together wth the nequalty (4), determnes a stable regon of q. It s easy to show that functon h(g) s monotonc ncreasng wth respect to G. Thus, the stable throughput regon R T of q correspondng to the stable throughput condton (4) can be defned as follows, L qr h( G ), h( G ). (5) T S L Furthermore, the network throughput s defned by ˆ out n f0 f0 mn{ ', ˆ } and the stable throughput condton STC ensures that ˆ out ˆ. It follows that the STC mples n f0 f0' ˆ n, whch means the offered load ρ. On the other hand, t can be shown from (3) that the offered load ρ s a monotonc ncreasng functon of the retransmsson

12 factor q f the attempt rate s bounded n the range G S G G L, or equvalently q R h( G ), h( G ). In partcular, T S L the attempt rate G wll reach G L when the offered load ρ =. Therefore, f the retransmsson factor q s chosen from S L q h( G ), h( G ), the offered load ρ of Geo/G/ queue of each nput buffer s strctly less than, whch s smply the stable condton of any queung system, that the arrval rate should be strctly less than the servce rate. As we mentoned before, the stable throughput condton (4) s not suffcent to guarantee a bounded mean delay of packets queued n each nput buffer. Let X be the servce tme of HOL packet, the mean delay s determned by the frst and second moments, E[X] and E[X ], of the servce tme. To complete the analyss of stable regons, we deduce the followng addtonal constrant from our queung model of the nput buffer. Bounded Delay Condton (BDC): The Pollaczek-hnchn formula for mean queueng delay E[T] of Geo/G/ queue [] E X E[ T ] E[ X ] ( EX [ ]) [ ] E[ X ] (6) requres bounded second moment of servce tme 0 < E[X ] <. The condton BDC s more restrctve than the condton STC. A regon of the factor q, denoted R D, that guarantees bounded mean delay can be determned by the second moment of servce tme E[X ]. In general, the bounded delay regon s a subset of the stable throughput regon, RD RT. Detaled dscussons on these stable regons wth the exponental backoff schedulng algorthm are provded below. 5 Analyss of ponental Backoff The exponental backoff scheme has been studed n many prevous papers [4]-[8]. In ths secton, we dscuss the stablty and delay performance of ths schedulng algorthm based on the condtons specfed n secton 4. The analytc results are verfed by the smulatons wrtten n MATLAB. 5. Stable Throughput Condton of ponental Backoff The attempt rate G of exponental backoff can also be derved from (3) and expressed as follows: ˆ a p q ag a q n a ˆ p q p pq. (7) or, equvalently, the retransmsson factor q can be formulated as a functon of attempt rate G as

13 where ˆ 4 ˆ A p ˆ a ˆ a p n 3 A p A p a a p n q h G p, (8) A p ˆ a p a G ˆ p ap ˆ n. For retransmsson factor q n the range 0 < q <, ths functon q = h (G) s monotoncally ncreasng wth respect to the attempt rate G. It follows that the followng stable throughput regon RT of exponental backoff can be obtaned from (5) and (8): R [ h ( G ), h ( G )]. (9) T S L The area under the lower and upper bounds shown n Fg. 6.a s the stable throughput regon RT of exponental backoff wth n = 0 and a = 0., whle that n Fg. 6.b s the stable regon wth n = 50 and a = 0.. The maxmum throughput wth exponental backoff can be acheved such that ˆ ˆ max as the stable regon (9) shrnks to a sngle max pont when G ˆ ˆ S( max ) GL( max ). If we gnore hgher order terms of ˆ and p n (8), the retransmsson factor q s approxmately equal to p: q h G p, (30) whch mples that the stable regon of exponental backoff s almost ndependent of the populaton sze when n s large enough. For example, the stable throughput regons for n = 0 and 50 shown n Fg. 6 are very close to each other. Therefore, the stable throughput regon R T can be approxmately gven as follow, R [ p( G ), p( G )]. (3) T S L The equalty n (3) holds when the retransmsson factor q n (8) s equal to p as n, whch concdes wth Song s results proved n [4], [5], and [8] that network throughput can be non-zero even when the number of nodes goes to nfnty. Our stablty analyss s confrmed by the smulaton results shown n Fg. 7 wth n = 0 and a = 0., whch closely follows the protocol detals for -persstent CSMA. The 95% confdence ntervals are shown for all the smulatons ponts. For fxed aggregate nput rate ˆ 0.3, Fg. 7 dsplays that stable throughput can be acheved f the retransmsson factor q s properly chosen from the stable throughput regon R T = [0.35, 0.849]. The smulaton results match exactly wth the analytcal one when the retransmsson factor q s wthn the stable regon. Whle outsde the 3

14 stable regon, the throughput mmedately drops and the sze of the confdence ntervals ncreases due to the unstable queung behavor of nput buffers. 5. Bounded Delay Condton of ponental Backoff The second moment of servce tme of the exponental backoff scheme s derved from (37) of Appendx I as follows: j p [ ],, lm,, j q E X B p p q C p p q, (3) where B(p, p, q) and C(p, p, q) are two polynomals gven n (38) and (39) of Appendx I. It follows that the bounded p delay condton requres the convergence of the nfnte geometrc sum lm, whch mples j q j q p. (33) For bnary exponental backoff,.e., q = 0.5, the above condton (33) s consstent wth Yang s result gven n []. To acheve stable throughput, the attempt rate G should be larger than or equal to the small root G S as descrbed n (4). Combne the condton (33) wth the stable throughput regon R T (9), and the bounded delay regon s gven as follows: RD p( GS ), h ( GL ). (34) The bounded delay regon R of exponental backoff s a subset of the stable throughput regon R. As shown n Fg. D T 8, the shaded regon s a bounded delay regon; outsde ths regon,.e., R \ R T D, the system has stable throughput but cannot guarantee bounded delay. In ths undesred regon, predomnatng backlogged packets are pushed to deep phases wth very low retransmsson probabltes. If a node tres to send ts HOL packet, the successful probablty wll be very hgh. Once the backlogged HOL packet s cleared, then the channel may be captured by subsequent packets n the nput buffer, whch are all n phase 0, untl the queue s cleared. Durng ths perod, t appears that the network throughput s stll stable, but the varance of the servce tme of each ndvdual packet can be nfntely large due to ths unfarness of servces caused by the capture effect descrbed n [8] and [9]. Fg. 9 and Fg. 0 llustrate the packet-queung delay, n terms of number of packets n the system, versus the retransmsson factor q and the nput rate ˆ, respectvely. Our analyss s confrmed by smulaton as shown n both fgures. In Fg. 9, f the retransmsson factor q s chosen wthn the regon of bounded delay, then there are nearly zero backlogged packets n the system. However, f the retransmsson factor q les outsde the bounded delay regon, the number of backlogged packets n the system would become larger even though t s stll wthn the stable throughput 4

15 regon. For any retransmsson factor q outsde the stable throughput regon, the number of backlogged packets n the system mmedately becomes much larger, whch mples an unacceptable mean queung delay. Other examples of the bounded delay regon are shown n Fg. 0, whch exhbts the number of packets n the system versus nput rate ˆ for fxed retransmsson factor q = 0. and q = 0.5. It can be clearly seen that the number of packets n the system s nearly zero wthn the bounded delay regon. Ths result concdes wth [5] n whch the authors clamed that the exponental backoff scheme can be stable f the nput rate s suffcently small. Snce the bounded delay regon R D specfed by (30) s a subset of the stable throughput regon R T, the maxmum throughput max ˆ may not be achevable f q R D. As shown n Fg.8, the network throughput ncreases wth the lower bound of the bounded delay regon pg, and decreases wth the upper bound pg S L n equaton (3). Therefore, the maxmum throughput wthn the bounded delay regon RD can be acheved when the lower bound and upper bound of the retransmsson factor q gven by (30) are equal. That s, pg p G. (35) S In (35), we plot the respectve maxmum throughput wthn the throughput stable regon and bounded delay regon. For 0 < a <, the maxmum throughput max ˆ gven n Fg. s always larger than that obtaned from the bounded delay L regon. In other words, the absolute maxmum throughput max ˆ n the stable throughput regon cannot be acheved wth bounded mean delay guarantee. 6 Concluson We have analyzed the stablty condtons n terms of stable throughput and bounded delay for slotted -persstent CSMA/CA networks. The queung model of nput buffer wth the -ponental Backoff collson resoluton algorthm s proposed to conduct the throughput analyss of the entre system and the performance of each ndvdual nput buffer. Based on ths model, the exponental backoff schedulng algorthm has been used to establsh the stable throughput regon and bounded delay regon wth respect to aggregate nput rate ˆ and retransmsson factor q. It s shown that the network throughput of the exponental backoff scheme can always be stablzed wth the proper selecton of q; whle the regon shrnks remarkably subject to the bounded delay requrement. Consequently, the maxmum achevable throughput of the network wthn the bounded delay regon s slghtly smaller than the absolute maxmum throughput. The proposed methodology can also be appled to other CSMA-based networks, such as IEEE 80., n the future. 5

16 Appendx. Servce Tme Dstrbuton for P-CSMA Protocol For exponental backoff scheme as, the frst and second moments of scaled servce tme, E[X] and E[X ], can be derved from (9) and gven as follows. The frst moment: am aq M M EX [ ]. (36) p q p The second moment: where B p, p, q a M M and j p [ ],, lm,, j q E X B p p q C p p q, (37) p q p q, (38) pq p q qp p q 3 M M M q M M M M qm q M M C p, p, q a { p p q p q p q p M M qm q M M M Mp pm } E[ X ]. p p q p pq (39) References [] L. lenrock and F. A. Tobag, Packet Swtchng n Rado Channels: Part -Carrer Sense Multple-Access Modes and Ther Throughput-Delay Characterstcs, IEEE Trans. Commun., vol. COM-3, pp , 975. [] H. Takag and L. lenrock, Throughput analyss for persstent CSMA systems, IEEE Trans. Commun., vol. 33, no. 7, pp , Jul [3] R. Rom and M. Sd, Multple Access Protocols Performance and analyss, Sprnger-Verlag New York, June 990. [4] B-J wok, N-O Song, and L. E. Mller, Analyss of the stablty and performance of exponental backoff, IEEE WCNC, vol.3, pp , 003. [5] N.O. Song, B. J. wak, and L. E. Mller, On the Stablty of ponental Backoff, Journal of Research of the Natonal Insttute of Standards and Technology, vol. 08, pp , 003. [6] D. J. Aldous, Ultmate nstablty of exponental backoff- protocol for acknowledgment-based transmsson control of random access communcaton channels, IEEE Trans. Inf. Theory 33 (), pp. 9-3, 987. [7] J. Goodman, A. G. Greenberg, N. Madras, and P. March, Stablty of bnary exponental backoff, J. ACM 35 (3), pp ,

17 [8] B-J wok, N-O Song, and L. E. Mller, Performance analyss of exponental backoff, IEEE/ACM Trans. Networkng, pp , Aprl 005. [9] G. Banch, Performance Analyss of the IEEE 80. Dstrbuted Coordnaton Funton, IEEE Journal of selected areas n COMMUN., Vol. 8, No. 3, pp , 000. [0] P. Chatzmsos, V. Vtsas, and A. C. Boucouvalas, Throughput and delay analyss of IEEE 80. protocol, n Proc. 5th IEEE Workshop Networked Applances, Oct. 003, pp [] Z. Hadz-Velkov and B. Spasenovsk, Saturaton throughput: delay analyss of IEEE 80. DCF n fadng channel, n Proceedngs of IEEE ICC003, May, 003. [] H. Wu, Y. Peng,. Long, S. Cheng, and J. Ma, Performance of relable transport protocol over IEEE 80. wreless LAN: analyss and enhancement, n Proceedngs of IEEE INFOCOM00, June 00. [3] H. Wu, S. Cheng, Y. Peng,. Long, and J. Ma, IEEE 80. dstrbuted coordnaton functon (DCF): analyss and enhancement, n Proceedngs of IEEE ICC00, pp , 00. [4] M. Ergen and P. Varaya, Throughput analyss and admsson control n IEEE 80.a, Moble Networks and Applcatons, vol. 0, no. 5, pp , Oct [5]. Duffy, D. Malone, and D. J. Leth, Modelng the 80. dstrbuted coordnaton functon n non-saturated condtons, IEEE Commun. Lett., vol 9, pp. 75, 005. [6] D. Malone,. Duffy, and D. J. Leth, Modelng the 80. dstrbuted coordnaton functon n nonsaturated heterogeneous condtons, IEEE/ACM Transactons on Networkng (TON), v.5 n., p.59-7, February 007. [7] Dongje Yn, Pu ng Wong, and Tony T. Lee, Analyss of Non-Persstent CSMA Protocols wth ponental Backoff Schedulng, Networkng and Internet Archtecture, May 00. [8].. Ramakrshnan and H. Yang, The Ethernet capture effect: analyss and soluton, Proc. 9th Local Computer Networks Conf., Oct [9]. Medepall and F. A. Tobag, On optmzaton of CSMA/CA based wreless LANs: Part I mpact of ponental Backoff, Proc. IEEE Globecom, 006. [0] J. Y. Hu, Swtchng and Traffc Theory for Integrated Broadband Networks, luwer Academc Publshers, 990. [] Hdeak Takag, Queueng Analyss, A Foundaton of Performance Evaluaton, Volume 3: Dscrete-Tme Systems, NORTH-HOLLAND, 993. [] Y. Yang and T. S. P. Yum, Delay Dstrbutons of Slotted ALOHA and CSMA, IEEE Trans. Commun, vol. 5, No., pp ,

18 Lst of Fgure Fg.. Fg.. Fg. 3. Fg. 4. Fg. 5. The renewal process of P-CSMA wth two types of transmsson perods. Markov chan of P-CSMA. Flow chart of access behavors for HOL packets. Markov chan of HOL packet. Throughput versus attempt rate for P-CSMA. Fg. 6. Stable throughput regons of exponental backoff : a) n = 0 and b) n =50. Fg. 7. Fg. 8. Fg. 9. Smulaton results of throughput for exponental backoff wth fxed nput rate. Stable throughput regon and bounded delay regon of exponental backoff. Number of packets n the system versus retransmsson factor q for exponental backoff. Fg. 0. Number of packets n the system versus nput rate for exponental backoff. Fg.. Maxmum throughputs n stable throughput and bounded delay regons. 8

19 Transmsson perod wth duraton +a Busy perod Pure dle perod a Transmsson Transmsson Transmsson Transmsson a Transmsson Type I: no node s scheduled to access the channel for a mn-slot Type II: If no node attempts to access the channel durng ths +a slots, the next transmsson perod s successful Fg.. The renewal process of P-CSMA wth two types of transmsson perods. Idle p Idle, Idle p Suc, Idle p Suc, Idle p Idle, Suc p Col, Idle p Suc, Suc Suc Suc p Suc, Col p Idle Col p Suc p, Suc Suc, Col p Col, Suc, Col p Col, Col Fg.. Markov chan of the P-CSMA channel. Backoff: Random watng tme determned by current backoff phase Sensng Busy Watng: Untl the channel turns dle Idle Transmsson Backoff phase s ncreased by At least one other node sends packet No other node sends packet Return to phase 0 Collson Success Fg. 3. Flow chart of access behavors for HOL packets. 9

20 p p p F0 q S0 W0 F' 0 p p p p p p p F F p F q q q q S... S... S q q p p W W W q F' p p p q F' p p p q q q F ' p Fg. 4. Markov chan of HOL packet. ˆ 0.47 max a 0. Network Throughput ˆ out a 0.3 Stable range when a 0.3 a 0. Stable range when a 0. Stable range when a 0. GS GL.98 Attempt Rate G Fg. 5. Throughput versus attempt rate for P-CSMA. 0

21 a) n0 and a0. b) n50 and a 0. Lower bound h ( G S ) Upper bound h ( G L ) Lower bound h ( G S ) Upper bound h ( G L ) Network Throughput ˆ out ˆ max Stable throughput regon when n 0 R T Network Throughput ˆ out ˆ max Stable throughput regon when n 50 R T Retransmsson Factor q Retransmsson Factor q Fg. 6. Stable throughput regons of exponental backoff: a) n = 0 and b) n =50. ˆ 0.3, n0 and a 0. Analytcal Results Smulaton Results (95% confdence nterval) Network Throughput ˆ out Stable throughput regon when ˆ Retransmsson Factor q Fg. 7. Smulaton results of throughput for exponental backoff wth fxed nput rate.

22 ˆ max 0.47 Lower bound h ( G S ) RT \ RD :Stable throughut regon wthout guarantee bounded delay n0 and a0. Upper bound h ( G L ) Network Throughput ˆ out Lower bound of Bounded delay regon R D Bounded delay regon when ˆ 0.3 R D Stable throughput regon when ˆ Retransmsson Factor q Fg. 8. Stable throughput regon and bounded delay regon of exponental backoff. ˆ 0.3, n0 and a 0. Analytcal results Smulaton results Number of packets n the system Lower bound h G S Upper bound h ( G L ) Stable throughput regon Bounded delay regon R T R D Retransmsson Factor q Fg. 9. Number of packets n the system versus retransmsson factor q for exponental backoff.

23 q 0. n0 and a0. Analytcal results for q 0. Smulaton results for q 0. Analytcal results for q 0.5 Smulaton results for q 0.5 Number of packets n the system Wthout bounded delay guarantee when q 0.5 Wthout bounded delay guarantee when q 0. q 0.5 Stable throughput regon for q 0. Stable throughput regon for q 0.5 ˆ Input Rate Fg. 0. Number of packets n the system versus nput rate for exponental backoff. max. throughput n stable throughput regon max. throughput n bounded delay regon Maxmum Throughput max. throughput ˆ wthout bounded delay guarantee max. throughput wth bounded delay guarantee max Mn-slot Sze a Fg.. Maxmum throughputs n stable throughput and bounded delay regons. 3