bianchi模型

bianchi模型，用来分析802.11性能的理论模型
fered load lead to an eventually significant decrease in the sys 0.90 tem throughput. This results in the practical impossibility to op erate the random access scheme at its maximum throughput for 0.85 Ideal Offered Load Growth ○ Offered load a"long "period of time, and thus in the practical meaningless Throughput of the maximum throughput as performance figure for the ac 0.80 20 stations CWmin=32 Cwmax =256 cess scheme. The mathematical formulation and interpretation of this instability problem is the object of a wide and general 己0.75 discussion in [13 Indeed, the 802. 11 protocol is known to exhibits some form 0.70 of instability (see for example [5], [11]. To visualize the un stable behaviour of 802.ll, in figure 3 we have run simulations in which the offered load linearly increases with the simulation 0.65 time. The general simulation model and parameters employed are summarized in section V. The results reported in the figure 0.60 100.0 200.0 300.0 400.0 are obtained with 20 stations The straight line represents the time(seconds) ideal offered load, normalized with respect of the channel capac ity. The simulated offered load has been generated according to Fig. 3. Measured Throughput with slowly increasing offered load a Poisson arrival process of fixed size packets(payload equal to 8184 bits), where the arrival rate has been varied throughout the with the name RTS/CTS, is shown in figure 2. A station that simulation to match the ideal offered load. The figure reports wants to transmit a packet, waits until the channel is sensed idle both offered load and system throughput measured over 20 sec or a DiFS, follows the backoff rules explained above, and then, onds time intervals, and normalized with respect to the channel instead of the packet, preliminarily transmits a special short rate frame called Request To Send(rts). when the receiving station From the figure, we see that the measured throughput follows detects an RTS frame, it responds, after a SIFS, with a Clear To closely the measured offered load for the first 260 seconds of f Send(CTS)frame. The transmitting station is allowed to trans simulation, while it asymptotically drops to the value 0.68 in the mit its packet only if the cts frame is correctly received second part of the simulation run. This asymptotic throughput The frames rts and CTS carry the information of the length value is referred to, in this paper, as Saturation Throughput, and of the packet to be transmitted This information can be read represents the system throughput in overload conditions. Note by any listening station, which is then able to update a Network than, during the simulation run, the instantaneous throughput Allocation Vector(NAV) containing the information of the pe temporarily increases over the saturation value(up to 0.74 in the riod of time in which the channel will remain busy. Therefore. example considered ), but ultimately it decreases and stabilizes when a station is hidden from either the transmitting or the re to the saturation value. Queue buildup is observed in such a ceiving station, by detecting just one frame among the rTS and condition CTS frames, it can suitably delay further transmission, and thus IV. THROUGHPUT ANALYSIS avoid collision The RTS/CTS mechanism is very effective in terms of system The core contribution of this paper is the analytical evaluation performance, especially when large packets are considered, as it of the saturation throughput, in the assumption of ideal channel reduces the length of the frames involved in the contention pro conditions (i.e. no hidden terminals and capture [6]) . In the cess. In fact, in the assumption of perfect channel sensing by ev analysis, we assume a fixed number of stations, each always ery station, collision may occur only when two(or more) pack having a packet available for transmission. In other words, we ets are transmitted within the same slot time. If both transmitting operate in saturation conditions, i.e. the transmission queue of stations employ the rtS/CtS mechanism, collision occurs only each station is assumed to be always nonempty on the RTS frames, and it is early detected by the transmitting The analysis is divided into two distinct parts. First, we study stations by the lack of CTS responses. a quantitative analysis the behavior of a single station with a Markov model, and we will be carried out in section VII obtain the stationary probability T that the station transmits a packet in a generic (i.e. randomly chosen)slot time. This prob IIL MAXIMUM AND SATURATION THROUGHPUT ability does not depend on the access mechanism (i.e. Basic or PERFORMANCE RTS/CTS) employed. Then, by studying the events that can oc In this paper we concentrate on the "Saturation Throughput cur within a generic slot time we express the throughput of both This is a fundamental performance figure defined as the limit Basic and rTS/CtS access methods (as well as of a combination reached by the system throughput as the offered load increases, of the two)as function of the computed value T and represents the maximum load that the system can carry in stable conditions A. Packet Transmission Probability It is well known that several random access schemes exhibit Consider a fixed number T of contending stations. In satura an unstable behavior. In particular, as the offered load increases, tion conditions, each station has immediately a packet available the throughput grows up to a maximum value, referred to as for transmission, after the completion of each successful trans Maximum Throughput". However, further increases of the of mission. Moreover, being all packets"consecutive", each packet needs to wait for a random backoff time before transmitting (1p)wo Let b(t) be the stochastic process representing the backoff time counter for a given station. A discrete and integer time 0,1 0,W02 0.W01 scale is adopted: t and t l correspond to the beginning of two consecutive slot times, and the backoff time counter of each sta pWI tion decrements at the beginning of each slot time. Note that this discrete time scale does not directly relates to the system time In fact, as illustrated in figure l. the backoff time decrement pWi is stopped when the channel is sensed busy, and thus the time O,W;2 0,w;1 nterval between two consecutive slot time beginnings may b much longer than the slot time size o, as it may include a packet p/Wi transmission. In what follows, unless ambiguity occurs, with the term slot time we will refer to either the(constant)value o p:Wm and the(variable) time interval between two consecutive backoff m.0 m w time counter decrements Since the value of the backoff counter of each station depends p/W also on its transmission history (e.g. how many retransmis m Fig. 4. Markov Chain model for the backoff window size sion the headofline packet has suffered), the stochastic pro cess b(t)is non markovian. However, define for convenience CWmin. Let m, maximum backoff stage", be the Letb,k=limt→∞P{(t)=i,b(t)=k},∈(0,m value such that CWmax 2w, and let us adopt the nota k E(0, Wi1 be the stationary distribution of the chain.We tion Wi= 2 w, where i E(0, m) is called"backoff stage now show that it is easy to obtain a closedform solution for this Let s(t be the stochastic process representing the backoff stage Markov chain. First, note that (O,..., m) of the station at time t b21,0·p →b,.=p2b000<8<m The key approximation in our model is that, at each transmis sion attempt and regardless of the number of retransmissions suffered, cach packet collides with constant and independent bm10·P=(1p)bm,0→bn0==boo probability p. It is intuitive that this assumption results more (2) accurate as long as w and n get larger. p will be referred to as Owing to the chain regularities, for each k e(1, Wi1), it i conditional collision probability, meaning that this is the prob ability of a collision seen by a packet being transmitted on the (1p)∑=ob,0 b channel b2 1,0 0<<m(3) Once independence is assumed, and p is supposed to be a p·(bnm1,0+bn,o constant value, it is possible to model the bidimensional process B By means of relations (2), and making use of the fact that s(t),b(t)) with the discretetime Markov chain depicted in fig b, o/(1p), equation 3 rewrites as ure4. In this markov chain the only non null onestep transition probabilities are2 bm2b.0i∈(0,m),k∈(0,W21)(4) P{,i,k+1}=1 k∈(0,Wz2)i∈(0,m) P10,ki, 0=(1p)/wo kE(0, Wo1) iE(0, m) Thus, by relations(2)and(4), all the values bi, k are expressed as Pli, ki1, 0p/Wi kE(0, Wi1) iE(1, m) function of the value bo,o and of the conditional collision proba Pim, k/m, 0f=p/wm kE(0,Wm1 bility p. boo is finally determined by imposing the normalization (1)condition, that simplifies as follows The first equation in(1)account for the fact that at the begin ning of each slot time, the backoff time is decremented The second equation accounts for the fact that a new packet follow1 ing a successful packet transmission starts with backoff stage 0 m=∑4+1 W;1 ∑∑b,=∑b Wk 2 and thus the backoff is initially uniformly chosen in the range m1 (0, Wo1). The other cases model the system after an unsuc cessful transmission. In particular, as considered in the third ∑(2 元=0 equation of (1), when an unsuccessful transmission occurs at backoff'stage i1, the backoff stage increases, and the new ini from which tial backoff value is uniformly chosen in the range(0, Wi.Fi nally, the fourth case models the fact that once the backoff stage 2(12p)(1p) 0 (W+1)+pW(1(2p) (6) reaches the value m, it is not increased in subsequent packet transmissions We can now express the probability T that a station transmits we adopt the short notation in a randomly chosen slot time. As any transmission occurs Pi1, k1 io, kol= Ps(t+1)=i1,b(t+1)=k1 s(t)=i0, b(t)=ko]. when the backoff time counter is equal to 0, regardless of the backoff stage, it is PHY MAC PAYLOAD SIFS ACK DIFS ∑b0 0,0 2(12p) T success basic access P(12p)(W+1)+pW(1(2p) PHY MAC ()Lhd hdr PAYLOAD DIFS s a side note, it is interesting to highlight that, when m I collision basic access 0, i.e. no exponential backoff is considered, the probability T results to be independent of p and equation (7) becomes the much simpler one independently found in [9] for the constant LRTS sIFSL CTS SIFs I hdr hdr PAYLOAD SIFS ACK DIFS backoff window problem T success rts/cts W+1 s L DIFS However, in general, T depends on the conditional collision probability p, which is still unknown. To find the value of p Fig. 5. Is and Tc for Basic Access and RTS/CTS mechanisms it is sufficient to note that the probability p that a transmitted packet encounters a collision, is the probability that in a time transmits on the channel, conditioned on the fact that at least one slot, at least one of the nl remaining stations transmit. The station transmits 1e undamental independence assumption given above implies that each transmission sees" the system in the same state, i.e. in 1 nT steady state. At steady state, each remaining station transmits a (17) (11) packet with probability T. This yields We are now able to express s as the ratio p=1(17)1 (9) e payload information transmitted in a slot time Equations(7)and(9)represent a non linear system in the two E length of a slot time (12) unknowns T and p, which can be solved using numerical tech Being e[P the average packet payload size, the average amount niques. It is easy to prove that this system has a unique solution. of payload information successfully transmitted in a slot time In fact, inverting (9), we obtainT"(p)=1(1p)/eD). is PPsEjPl since a successful transmission occurs in a slot This is a continuous and monotone increasing function in the time with probability PtPs. The average length of a slot time is range p E(, 1), that starts from T*(O)=0 and grows up to readily obtained considering that, with probability 1Ptr,the T"(1)=l. Equation T(p) defined by(7)is also continuous in slot time is empty; with probability Ptr Ps it contains a success the range p∈ : continuity in correspondence of the criti ful transmission, and with probability Ptr(1ps)it contains a cal value p= 1/2 is simply proven by noting that (p) can b collision. Hence, (12) becomes alternatively written as P。Pt"E[P] TP ++pW∑=0 Pn)+PnPT+P(P。(3 Here, Ts is the average time the channel is sensed busy (i.e and therefore T(1/2)=2/(1+W+mw/2). Moreover,T(p)is the slot time lasts) because of a successful transmission, and Tc trivially shown to be a monotone decreasing function that starts is the average time the channel is sensed busy by each station from T(0)=2/W+1) and reduces up to T(1)=2/(1 during a collision. o is the duration of an empty slot time. Of 2mw). Uniqueness of the solution is now proven noting that course, the values EPl. T 8, Te, and o must be expressed with 7(0)>r*(0)andr(1)<r*(1) the same unit Note that the throughput expression(13)has been obtained B. Throughput without the need to specify the access mechanism employed o specifically compute the throughput for a given DCF access Let S be the normalized system throughput, defined as the mechanism it is now necessary only to specify the correspond fraction of time the channel is used to successfully transmit pay ing values Ts and Tc load bits. To compute S, let us analyze what can happen in Let us first consider a system completely managed via the randomly chosen slot time. Let Ptr be the probability that there Basic Access mechanism. Let H= PHYndr+MAChdr be is at least one transmission in the considered slot time. Since n the packet header, and S be the propagation delay. As shown in stations contend on the channel, and each transmits with proba figure 5, in the Basic Access case we obtain bility T, (10) T0a8= H+E[P]+ SIFS +8+ACK+DIFS+8 The probability Pg that a transmission occurring on the channel H+EIP+ DIFS+& is successful is given by the probability that exactly one station (14) where E[P* is thethe average length of the longest packet pa T如3+O5(1F(P) (19) load involved in a collision In the case all packets have the same fixed size, ELP* o compute TC=Tc(P)in the case of the Hybrid Access ElPI= P In the general case, the payload size of each collided scheme, we rely on the simplifying assumption that the proba bility of a collision of more than two packets in the same slot packet is an independent random variable Pi. It is thus necessary to assume a suitable probability distribution function F( for time is negligible. Hence, three possible collision cases may the packets payload size. Let Pmna be the maximum payload occur:(i) collision between two RTS frames, with probabil size. Taking the conditional expectation on the number k of ity(1F(P))2 (ii) collision between two packets transmitted colliding packets, ElP" writes as follows: via Basic Access, with probability F(P)2,and(iii)collision be tween a basic access packet and an rts frame Hence indicat E[P]=E[Emax(1,…,PA川k]= ing with Tcts/rts, Tbas/bas and rb ns/rts the respective average collision durations we obtain (3)+0a=7y*“=F T(P)=(1F(P) ts/rts (15) (1Tm +2F(P)(1F(P)rrts/as+F2(P)rbas/bas(20) When the probability of three or more packets simultaneously The average collision durations adopted in equation(20)de colliding is neglected, expression (15)simplifies to tail as follows. Let On =(TbasPTrts)=(HRTS be the extra length of the packet header with respect of the rts (1F(x)2)d (16) frame, and let a=I DIFS +d. The value TC rts/rts has been already computed in the case Tr of(17), and can be rewritten T is the period of time during which the channel is sensed with new notation as busy by the non colliding stations. We neglect the fact that the Trts/rts= rtS+ DIFS +6=aOn (21) two or more colliding stations, before sensing the channel again need to wait an ACK Timeout, and thus the le for these collid To compute the average length of a collision between an rTS ing stations is greater than that considered here( the same ap frame and a Basic Access packet let us note that, according to proximation holds in the following RTS/CTS case, with a cts the numerical values provided by the standard [3], the length of Timeout instead of the acK timeout) an RTS frame is always lower than the packet header size, or, Let us now consider a system in which each packet is trans in other words, the value On defined above is strictly positive mitted by means of the rTs/CTS Access mechanism. As, in Thus the average length of such a collision is given by the aver such a case, collision can occur only on rTs frames, it is(see age amount of time the channel is kept busy by the unsuccessful figure 5): transmission of the Basic Access Packet. Since F(a/F(P ts & C E(0, P)is the conditional probability distribution function RTS+ SIFS+6+ctS+ SiFS+d+H+ of the payload size of the packets transmitted according to the +ELP SIFS +d+ACK DIFS +d Basic Access mechanism, we readily obtain Trts= RTS+ DIFS +d Trts/bas=a+ a F(P (22) and the throughput expression depends on the packet size distri bution only through its mean Finally, noting that in the case of collision between two Ba Finally, formula(13)can be also adopted to express the sic Access packets, the probability distribution function of the throughput of an "Hybrid!"system in which, as suggested in the length of the longest packet payload involved in a collision is standard [3], packets are transmitted by means of the rts/ cts the square of the conditional probability distribution function of mechanism only if they exceed a given predetermined thresh the packet size distribution old P on the packet's payload size. More specifically, being, again, F( the probability distribution function of the packet dr size, F(P) is the probability that a packet is transmitted accord F2(P (23) ing to the Basic access mechanism (i.e. the packet size is lower than P), while 1F(P) is the probability that a packet is trans By substituting(21),(22)and (23 )in equation(20),we tinally mitted via the rtS/Cts mechanism. For convenience let us obtain indicate with T(P)=a(1F(P))Oh+ Orts=TTsT90s=RTS+SIFS++CTS+SIFS+8(18) +2F(P)(1F(P) F(a F( the rts/cts overhead for a successful packet transmission. It Is easy to recognize that, for the described hybrid access scheme, +F2(P) 7F2 (24) It P2(P) For simplicity, in the rest of this paper, we restrict our numer T=T(P)=1m8F(P+T8(1F(P)= ical investigation to the case of fixed packet size, and therefore packet payload 8184 bits MAC header 272 bits PHY header 128 bits 0.85 ACK 112 bits PhY header RTS 160 bits PhY header 0.80 CTS 112 bits phy header Channel Bit Rate 1 Mbit/ g0.75 Propagation delay1 us Slot time 50S SIFS 28S 0.65 △ basic,W=32,m=3 DIFS 128s ACK Timeout 300/uS 0.60 ○ basic.W=128.m=3 CTS_ Timeout 300s 米rtscts,W=32, 0.55 ◆rtscts,W=128 TABLE II 0.50 10 40 FHSS System parameters and additional parameters used to obtain numerical Number of stations results Fig. 6. Saturation Throughput: analysis versus simulation analysis simulation we will evaluate the performance of systems in which all sta n=2,BAS0.84730.846±0.001 tions operate either according to the Basic Access mechanism n=2,RTs0.81980.817±0.001 or according to the ris/cTs mechanism(1.e. never operating n=3,BAS0.83680835±0.001 in the hybrid mode) n=3,RS0.82790823±0001 V. MODELⅤ ALIDATION TABLE III To validate the model, we have compared its results with that Analysis versus simulation: comparison for a very low numberofstations obtained with the 802. 11 DCF Simulator used in [9]. Ours is an eventdriven custom simulation program, written in the c+t programming language, that closely follows all the 802. 11 pI tocol details for each independently transmitting station. In par cases. All simulation results in the plot are obtained with a 95% ticular, the simulation program attempts to emulate as closely as confidence interval lower than 0.002. Negligible differences possible the real operation of each station, including propaga well below 1%, are noted only for a small number of stations tion times turnaround times etc (results for the extreme case of as low as 2 and 3 stations are TE he values of the parameters used to obtain numerical results, tabulated in table IIl) for both the analytical model and the simulation runs, are sum marized in table ll The system values are those specified for the VI MAXIMUM SATURATION THROUGHPUT FHSS (Frequency Hopping Spread Spectrum) PHY layer [3] The analytical model given above is very convenient to de The channel bit rate has been assumed equal to 1 Mbit/s. The termine the maximum achievable saturation throughput. Let us frame sizes are those defined by the 802.11 MAC specifications, rearrange(13)to obtain and the phy header is that defined for the fhss phy. the val ues of the aCK Timeout and CTSTimeout reported in table II ELP TT t o(Ptr)/Pir+Tc (25) and used in the simulation runs only (our analysis neglects the effect of these timeouts) are not specified in the standard, and they have been set equal to 300 us. This numerical value has As Ts, TC,ElP], and o, are constants, the throughput S is max been chosen as it is sufficiently long to contain a SIFS, the ack imized when the following expression is maximized transmission and a round trip delay 尸s 77(17)y1 Unless otherwise specified, we have used in the simulation (26) runs a constant packet payload size of 8184 bits, which is about (1n)/ftr+/aT(1T)y(Te1) one fourth of the maximum MPDU size specified for the FHss where Tx=Tc o is the duration of a collision measured in slot PHY, while it is the maximum MPDU size for the DSSS PhY. time units o. Taking the derivative of (26)with respect to T, and Figure 6 shows that the analytical model is extremely accu imposing it equal to 0, we obtain, after some simplifications, the rate: analytical results (lines practically coincide with the sim following equation ulation results(symbols), in both Basic Access and RTS/CTS (17)”T{n7[1(17)2] (27) s A detailed performance analysis of the hybrid mode requires to assume one or more suilable probability distribution functions for the packel's payload size, Under the conditionT<< 1, and also to determine the sensitivity of the throughput on the assumed distri butions. Such a straightforward, but lengthy, study is out of the scopes of the present work (1T)≈1m7(m BASIC ACCESS Max Throughput Max Throughput Approx 0.9 50.832827(70.02869)0.832662(70.021426) approximate max throughput a maximum throughput 100.828279(70.010848)0.828272(70.010713) 08 200826111(=005294)0.826105(7=00535 n=5 500.824841(7=0.002089)0.824814(7=0.002143) 0.823957 当0.7 RTS/CTS ACCESS n=10 n Max Throughput Max Throughput Approx 50.838511(7=0.090399)0.838436(70.097940) ￡0.6 100.837281(7=0.043712)0.837129(7=0.048970) n=20 200.836686(7=0021520)0.836490(7=0.024485) 0.5 500.836335(7=0.008532)0.836110(7=0.00794) =50 0.835859 0.4 0.02 004006 0.10 TABLE IV Comparison between maximum throughput and throghput resulting from approximate solution (28)the case n= oo is obtained from equation(31) Fig. 7. Throughput versus the transmission probability T for the Basic Access 0.9 approximate max through holds, and yields the following approximate solution 口是口·一 o, maximum throughput +2(m1)(x1)/n 0.8 5 (1)(Tx1) n=10 0.7 quation(27)and its approximate solution(28 )are of funda mental theoretical importance. In fact, they allow to explicitly compute the optimal transmission probability that each sta ￡0.6 tion should adopt in order to achieve maximum throughput per n=20 formance within a considered network scenario(1.e. number of 0.5 stations n). In other words, they show that(within a PhY and n=50 an access mechanism, which determine the constant value Tx) maximum performance can be, in principle, achieved for every network scenario through a suitable sizing of the transmission 00 0.05 0.10 0.15 probability T in relation to the network size However, equations(7) and(9) show that T depends only on Fig 8. Throughput versus the transmission probability T for the RTS/CTS mech the network size and on the system parameters m and w. as /s n is not a directly controllable variable, the only way to achieve optimal performance is to employ adaptive techniques to tune igures have been obtained assuming the system parameters re the values m and w (and consequently r) on the basis of the ported in table Il. The figure reports also the different through estimated value of n put values obtained in the case of exact and approximate solu This problem has been specifically considered in 19] for the tion for T. As the maximum is very smooth, even a non neg case of fixed backoff window size (i. e. Tl= 0). In such a ligible difference in the estimate of the optimal value T leads case, T is given by( 8), and therefore the backoff window that to similar throughput values. The accuracy of the throughput maximizes the system throughput is readily found as obtained by the approximate solution is better testified by the Wpt≈m√2T numerical values reported in table IV. note that the agreement is greater in the Basic Access case, as Tc is greater. Refer to [9] for a large discussion related to the problem of esti A surprising result is that the maximum throughput achiev mating the value n. able by the basic access mechanism is very close to that achiev Unfortunately, in the 802.11 standard, the values W and m able by the rTS/CTS mechanism. Moreover, the maximum are hardwired in the PhY layer details(see table I for the stan throughput is practically independent of the number of sta dardized values), and thus they cannot be made dependent on tions in the wireless network. This is easily justified by not n. As a consequence of this lack of flexibility, the throughput ing that the throughput formula can be approximated as fol in some network scenarios can be significantly lower than the lows. Let K=VTs/ 2, and let us use the approximate solution maximum achievable T=1/(nK). For n sufficiently large Figures 7 and 8 show the maximum throughput theoretically achievable by the dcf protocol in both the cases of Basic Ac cess and RTS/CTS mechanisms. The values reported in these bits or ps slot time units(o=50 us Packet payload 8184 163.68 8982 179.64 naximum backoff stage m=6 8713 174.26 9568 191.36 0.80 t 417 8.34 TABLE V 0.70 Values lo and lc measured in hits and in 50 ps slot time units, for the considered system parameters, for both Basic and RTS/CTS access methods ●n=50 0.60 C 00=10 P=m27(1 匚—口n=5 N(nK1)(e1/K1)K(el/k1) 0.50 (30) 2 41282565121024 The maximum achievable throughput Smaa can thus be approx initial size of the backoff window(W) imated as Fig. 9. Saturation Throughput versus initial contention window size for the EIPI Basic access mechanism n a T8+aK+T(K(el11)1) 0.90 which results to be independent of n. Using the numerical val maximum backoff stage m=6 ues of table V we obtain K=9.334 for the Basic Access mech 8 anism and K= 2.042 for the rts/ctS mechanism. The re 0. sulting maximum throughput approximation values are reported in table iv under the lab An advantage of the rts/CTS scheme is that the throughput is less sensitive on the transmission probability T. In fact, we see from figures 7 and 8 (note the different xaxis scale) that a small ■■n=20 10 variation in the optimal value of T leads to a greater decrease in 口n=5 the throughput for the basic access case than for the rts/cts case. Hence, we expect(see quantitative results in the following section VIl)a much lower dependence of the rts/Cts through put on the system engineering parameters with respect of the 0.50 32 641282565121024 Basic Access throughput initial size of the backoff window(w) VIL. PERFORMANCE EVALUATION Fig. 10. Saturation Throughput versus initial contention window size for the RTS/CTS mechanism Unless otherwise specified, the following results have been obtained assuming the parameters reported in table Il and,in particular, assuming a constant payload size P=8184 bits Figure 9 shows that the throughput of the Basic Access mech Figure 6 shows that the throughput for the Basic Access anism highly depends on W, and the optimal value of w de scheme strongly depends on the number of stations in the net pends on the number of terminals in the network For example work. In particular, the figure shows that, in most cases, the an high value of w(e.g. 1024)gives excellent throughput per greater is the network size. the lower is the throughput. The only formance in the case of 50 contending stations, while it drasti partial exception is the case W=128. For such an initial con cally penalizes the throughput in the case of small number(e.g tention window size, the throughput is comparable in networks 5)of contending stations. This behavior is scen also in figure with 5 to 10 stations, although it smoothly decreases as the net 10, where the RTS/CTS mechanism is employed. Large val ork size increases. The same figure shows that performance ues of w may, in fact, limit the throughput of a single station, impairment does not occur for the RTS/CTS mechanism when which, when alone in the channel is bounded by n increases. In fact, the throughput is practically constant for EL W=32. and even increases with the number of mobile stations when W= 128 Ts+a(W1)/2 (32) To investigate the dependency of the throughput from the ini where e[P and Ts are the average packet payload and the av tial contention window size, w, we have reported in figures 9 erage channel holding time in case of successful transmission and 10 the saturation throughput versus the value w for, respec Equation(32)is directly obtained from equation(13)of section tively, the Basic Access and the rts/cts mechanisms. In both IvB by observing that, as there are no other stations which can figures, we have assumed a number of backoff stages equal to 6, collide with the considered one, the probability of success Ps i. e. CWmax 2W. The figures report four different network is equal to 1. In addition, the probability Ptr that a transmis sizes, i.e. number of stations n equal to 5, 10, 20 and 50 sion occurs on the channel is equal to the probability T that the station transmits. Being the conditional collision probability p equal to o, Ptr= T is given by formula(8 Of more practical interest is the case of small values of w and particularly in correspondence of the values W= 16, 32 2 and 64 (i.e. those standardized for the three Physee table D) Figures 9 and 10 show that the two access mechanisms achieve 00×英 W=256 a significantly different operation. In the case of the Basic Ac cess mechanism, reported in figure 9, the system throughput in 15 ng as w gets closer to 64. Moreover, the throughput a significantly decreases as the number of stations increases. On 10 the contrary, figure 10 shows that the throughput obtained with ① W=64 the rts/cts mechanism is almost independent of the value W <64, and, in this range, it is furthermore almost insensi 5 W=16 tive on the network size This surprising independence is quantitatively explained as 0 510152025 35404550 follows. Dividing numerator and denominator of (13)by Ptr Pa number of stations we obtain. Fig. 11. Average number of idle slot times per successfil packet transmission Pt S=EIPl/Ts+o +T P (33) The denominator of formula 33 expresses the average amount  Basic Access Mechanism 100 of time spent on the channel in order to observe the successful RTS/CTS mechanism transmission of a packet payload. This time is further decom posed into three components Ts is the time spent in order to successfully transmit a packet Table V reports the numerical values for Tbas and Tats, com puted according to equations(14)and(I7), in the assumption of system and channel parameters of table Il. The difference between Tots and Tbas(586 bits)is the additional overhead in W=256 troduced by the rtS,cTs mechanism The second term at the denominator of (33) does not depend W=16,64,256 on the access mechanism employed, and represents the amount a 0 of time the channel is idle, per successful packet transmission 5101520253035404550 In fact, 1/(Ptr Ps)is the average number of slot times spent on number of stations n the channel in order to have a successful transmission. Of those Fig. 12. Average number of slot time units wasted on the channel because of slot times, a fraction(1Ptr) is empty and each empty slot time packet collision, per successful packet transmission lasts o. The average number of idle slot times per packet trans mIssion, 1.e (1 Ptr)/(Ptr Ps), is plotted in figure 1l versus the network size. for three different values of the initial contention that, for the Basic Access mechanism, the amount of channel window W. We see that, for W= 16 and w=64, the amount time wasted in collisions is extremely large for a small value of idle slot times per packet transmission is very low, particu w and a large number of stations in the network. Conversely the additional amount of time wasted in collisions is negligible larly when compared with the values Ty given in table V. This for the rTS/CTS mechanism, regardless of the values T and value bccomes significant only when w gets greater(the case W 256 is reported in the figure)and the number of stations This explains the surprising constant RTS/CTS throughput in in the network is small any practical system and network operation conditions Finally, the third term at the denominator of (33)represents Figure 13 shows that the dependence of the throughput from the time wasted on the channel because of collisions, per suc the maximum number m of backoff stages is marginal. The cessful packet transmission. In fact, 1/Ps1 is the average figure reports the cases of both Basic and RTS/CTS access number of collided transmissions per each successful transmis schemes, with W =32(similar behaviour is observed for other sion, which is multiplied by Tc, i.e. the amount of time the chan values of the parameter w)and n =10,50. The points in the nel is held by a collision. Table v shows that the the rtS/cts box indicate the throughput achieved when m =5,i.e. in cor mechanism significantly reduces the time spent during a colli respondence of the standardized engineering parameters of the sion, with respect to the Basic Access mechanism. This reduc DSSS PHY (table D). We see that the choice of m does not prac tion is extremely effective when the system parameter W and tically affect the system throughput, as long as m is greater than the network size Tu lead to a large collision probability. This fact 4 or 5. The only case in which the throughput still grows, for is graphically shown in figure 12. This figure reports the av m relatively large, is the Basic Access mechanism with a large erage amount of time spent in collisions, per successful packet network size transmission, normalized with respect to the value o. It shows Our model allows to obtain other measures of interest. The
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Bianchi I模型作为周期性宇宙的原型
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论文研究  可变减速参数的Bianchi V型宇宙中相互作用的全息暗能量
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