A General Model of Wireless Interference

Transcription

1 A General Model of Wreless Interference Ll Qu Yn Zhang Feng Wang M Kyung Han Ratul Mahajan Unversty of Texas at Austn Mcrosoft Research Austn, TX 7872, USA Redmond, WA 9852, USA {ll,yzhang,wangf,hanm2}@cs.utexas.edu ratul@mcrosoft.com ABSTRACT We develop a general model to estmate the throughput and goodput between arbtrary pars of nodes n the presence of nterference from other nodes n a wreless network. Our model s based on measurements from the underlyng network tself and s thus more accurate than abstract models of RF propagaton such as those based on dstance. The seed measurements are easy to gather, requrng only O(N) measurements n an N-node networks. Compared to exstng measurement-based models, our model advances the state of the art n three mportant ways. Frst, t goes beyond parwse nterference and models nterference among an arbtrary number of senders. Second, t goes beyond broadcast transmssons and models the more common case of uncast transmssons. Thrd, t goes beyond homogeneous nodes and models the general case of heterogeneous nodes wth dfferent traffc demands and dfferent rado characterstcs. Usng smulatons and measurements from two dfferent wreless testbeds, we show that the predctons of our model are accurate n a wde range of scenaros. Categores and Subject Descrptors C.4 [Performance of Systems]: Modelng technques General Terms Measurement, Performance Keywords Model, Wreless Interference. INTRODUCTION Interference s fundamental to wreless networks. Due to the broadcast nature of the medum, transmssons from one sender nterfere wth the transmsson and recepton capabltes of other nodes. Understandng and managng nterference s essental to the performance of wreless networks. For nstance, t can drectly beneft channel assgnment [2, 25], transmt power control [2], routng [5, 6], transport protocols [8], and network dagnoss [4]. Unfortunately, the state of the art n estmatng the mpact of nterference s rather prmtve. Much of the exstng work s based on smple, abstract models of rado propagaton (e.g., the nterference range s twce the communcaton range). Whle such models may predct the asymptotc behavor, they can be hghly naccurate n any gven network [4, ]. Permsson to make dgtal or hard copes of all or part of ths work for personal or classroom use s granted wthout fee provded that copes are not made or dstrbuted for proft or commercal advantage and that copes bear ths notce and the full ctaton on the frst page. To copy otherwse, to republsh, to post on servers or to redstrbute to lsts, requres pror specfc permsson and/or a fee. MobCom 7, September 9 4, 27, Montréal, Québec, Canada. Copyrght 27 ACM /7/9...$5.. Ths has prompted researchers to devse models that are seeded usng measurements from the underlyng network [, 24]. These measurements are usually collected n a smple confguraton, such as each node sendng by tself. They are then used to predct the mpact of nterference n more complex confguratons such as multple transmttng nodes. Ths s a promsng drecton because t makes no assumptons about the nature of rado propagaton whch has proven dffcult to model n real envronments. However, the exstng measurement-based models are qute lmted. They do not apply to confguratons that have more than two senders or two flows, have uncast traffc, or have senders wth fnte demands. The only way today to predct network behavor under these general confguratons s to actually measure t. (Indeed, most expermental research today s forced to adopt ths methodology.) But measurement alone s nsuffcent because t lacks predctve power and scalablty. Whle t can accurately predct the performance of the measured confguraton, t cannot predct performance for other confguratons. To optmze network performance, one often needs to predct the performance of many alternatve confguratons. Snce measurng all possble confguratons s not feasble, t s necessary to develop a model to estmate network performance under arbtrary confguratons (e.g., to perform what-f analyss). In ths paper, we develop a general model of nterference n heterogeneous multhop wreless networks wth asymmetrc lnk qualty and non-bnary nterference relatonshps. Our model takes as nput traffc demand and receved sgnal strength (RSS) between pars of nodes, whch requres only O(N) measurements n an N- node network. It then estmates the rate at whch each sender wll transmt and the rate at whch each recever wll successfully receve packets. Compared to exstng measurement-based models [, 24], we advance the state of art n three mportant ways. Frst, we go beyond the case of two senders (or flows) and model nterference among an arbtrary number of senders. Ths s challengng due to complex nteractons among nodes. For nstance, the sendng rate of node m depends on those of all other nodes, whch n turn depend on the sendng rate of m tself. Second, we go beyond broadcast transmssons and model the more common case of uncast transmssons. Uncast transmssons ntroduce addtonal complextes due to retransmssons, exponental backoff, possbly asymmetrc lnk qualtes, and collsons wth not only data packets but also ACK packets. Thrd, we go beyond the case of nfnte traffc demands and model the more realstc case of fnte demands. Most real networks have heterogeneous nodes wth varyng traffc demands. Our model conssts of three major components:. An N-node Markov model for capturng nteractons among an arbtrary number of broadcast senders. The model provdes a smple yet accurate approxmaton to the 82. dstrbuted coordnaton functon (DCF). It s more general than prevous models (e.g., [2]) and can support multhop wreless networks, unsaturated demands, asymmetrc lnk qualty, and non-bnary nterference relatonshps.

2 2. A recever model of packet-level loss rates. In partcular, we fnd that slot-level and packet-level loss rates can be qute dfferent dependng on how losses are generated. Hdden termnals can sgnfcantly ncrease the packet-level loss rates well beyond the slot-level loss rates by spreadng the lossy tme slots across many packets. Based on ths observaton, our recever model captures both synchronzed and unsynchronzed packet-level collson losses. 3. Uncast sender and recever models. We further extend the above broadcast sender and recever models to capture nteractons among uncast transmssons. We develop two major extensons for ths purpose. The frst extenson models the retransmsson and exponental backoff at the sender sde, and the second extenson models data/data, data/ack, and ACK/ACK collson losses at the recever sde. We evaluate our model usng both extensve smulatons and real measurements over two dfferent wreless testbeds. Our results show that the model gves accurate predctons over a wde range of scenaros for both broadcast and uncast traffc, wth both saturated and unsaturated demands, and across dfferent number of senders. In smulatons, where accurate RF profle s avalable, our model s root mean square error (RMSE) s less than.5 for both throughput and goodput predctons. In the testbeds, where the RF profle s emprcally measured and subject to measurement nose and bas due to lost packets, our model s RMSE s less than.2. Whle our model s more general, we fnd that ts accuracy s hgher than the state-of-art model that consders the specal case of two broadcast senders wth nfnte demands [24]. Paper organzaton: The rest of the paper s organzed as follows. In Secton 2, we revew the background of IEEE 82.. In Secton 3, we gve an overvew of our nterference model. We present broadcast models n Secton 4, and uncast models n Secton 5. In Secton 6, we descrbe how to obtan model nputs. We evaluate our model usng smulatons n Secton 7 and usng testbed experments n Secton 8. We dscuss the related work n Secton 9 and conclude n Secton. 2. BACKGROUND ON 82. The IEEE 82. standard [22] specfes two types of coordnaton functons for statons to access the wreless medum: dstrbuted coordnaton functon (DCF) and pont coordnaton functon (PCF). In ths paper, we focus on DCF, whch s much more wdely used than PCF. DCF s based on CSMA/CA. Before transmsson, a staton frst checks to see f the medum s avalable by usng vrtual carrer-sensng and physcal carrer-sensng. The medum s consdered busy f ether carrer-sensng ndcates so. Vrtual carrer-sensng consders the medum to be dle f the Network Allocaton Vector (NAV) s zero, otherwse t consders the medum to be busy. Only when NAV s zero, physcal carrersensng s performed. A staton determnes the channel to be dle when the total energy receved at a node s less than the CCA (clearchannel assessment) threshold. In ths case, a staton may begn transmsson usng the followng rule. If the medum has been dle for longer than a dstrbuted nter-frame spacng tme (DIFS) perod, transmsson can begn mmedately. Otherwse, a staton that has data to send frst wats for DIFS and then wats for a random backoff nterval unformly chosen between [,CW mn ], where CW mn s the mnmum contenton wndow. If at any tme durng the perod above the medum s sensed busy, the staton freezes ts counter and the countdown resumes only after the medum becomes dle agan for DIFS. When the counter decrements to zero, the node transmts the packet. In the case of uncast, f the recever successfully receves the packet, t wats for a short nterframe spacng tme (SIFS) and then transmts an ACK frame. If the Model nputs: measured R mn RSS from node m to n B n Background nterference at n d mn Traffc demand from m to n Model nputs: rado-dependant constants β m CCA threshold of m γ n Rado senstvty of n δ n SINR threshold of n W n Thermal nose of n Model outputs t mn Normalzed throughput: rate of traffc sent by m to n g mn Normalzed goodput: rate of traffc receved by n from m L mn Packet loss rate from m to n Other varables S Subset of nodes that are transmttng n state π Probablty that the network s n state M Matrx of transton probabltes among states C(m S ) Probablty that channel s clear at m n state Q(m) Probablty that m has data to send when backoff counter = OH(m) Average overhead from DIFS, SIFS, and ACK at sender m CW(m) Average contenton wndow of m T µ (m) Average packet transmsson tme for m Table : A summary of key notatons. sender does not receve an ACK, t doubles ts contenton wndow to reduce ts access rate. When the contenton wndow reaches ts maxmum value, t stays at that value untl a transmsson succeeds, n whch case the contenton wndow s reset to CW mn. 3. OVERVIEW OF OUR MODEL Our model takes traffc demands and RF profle as nput and outputs the estmated sendng and recevng rates for each node. Such a model s a powerful tool for performng what-f analyss and facltatng network optmzaton and dagnoss. More specfcally, consder a network wth N nodes. The nputs to the model are: ) traffc demand from each sender m to each recever n, and ) RF profle, whch refers to the receved sgnal strength (RSS) between every par of nodes, denoted as R mn. The outputs are: ) normalzed throughput t mn,.e., the fracton of tme when m s sendng traffc to n (ncludng header overhead and retransmssons), ) normalzed goodput g mn,.e., the fracton of tme when n s recevng useful data from m (excludng header overhead and duplcate traffc), and ) the packet loss rate L mn. In ths paper, we focus on one-hop traffc demands, whch means that traffc s only sent over one hop and not routed further. If n cannot hear from m, ts recevng rate s zero. Modelng network performance under one-hop traffc demands s an mportant and necessary step towards estmatng end-to-end throughput over multhop paths, whch we plan to nvestgate n the future. Our model operates as follow. Frst, we measure the RF profle of the network by lettng each sender broadcast n turn and havng the other nodes measure receved RSSI values and loss rates. From these measurements, we recover parwse RSS (R mn ) and background nterference (B n ) due to external sources other than nodes n the modeled network (Secton 6). Whle we use custom traffc for our experments, t may be feasble to perform these measurements usng normal applcaton traffc. Then, we apply our sender model to estmate the amount of traffc sent by each sender under the gven demand and our recever model to estmate the amount of traffc successfully receved. Our key contrbutons le n the generalty and accuracy of the sender and recever models. They apply to both broadcast and uncast transmssons for an arbtrary number of senders, wth and wthout saturated traffc demands. For saturated broadcast demands, our model can drectly estmate throughput and goodput by computng the statonary probabltes of a Markov model. For uncast demands or unsaturated broadcast demands, the transton matrx of the Markov model tself nvolves addtonal unknown varables

3 to be estmated. As a result, the statonary probabltes cannot be drectly solved. We solve the problem by applyng an teratve framework, where we frst ntalze the unknown varables n the transton matrx and then compute statonary probabltes, whch are then used to update the transton matrx. Our results show that the teraton framework s effectve and converges quckly (wthn teratons n our evaluaton). We assume the followng rado behavor. A transmtter m determnes the channel s clear when the total energy t receves s below the CCA (clear-channel assessment) threshold, β m. A recever n correctly decodes a transmsson from a sender m when ) ts sgnal strength s at least rado senstvty, γ n ; and ) the sgnal to nterference-plus-nose rato (SINR) s at least the SINR threshold, δ n. We denote the thermal nose experenced by n as W n. The values of β m, γ n, δ n, and W n are constant but rado-dependent. The key notatons used n ths paper are summarzed n Table. We explan each term when t s frst encountered. 4. BROADCAST TRAFFIC In ths secton, we present our model for broadcast traffc. Extensons to handle uncast traffc are presented n the next secton. 4. Sender Model The goal of the broadcast sender model s to estmate how much each sender can transmt gven traffc demand. The classc Banch model [2] and ts extensons (e.g., [2]) model the behavor of 82. DCF by constructng a dscrete Markov chan. To make the model tractable, all packet transmssons are assumed to be synchronzed,.e., there are no partally overlappng transmssons. In a general multhop wreless network, however, partally overlappng transmssons can be common because not all nodes can carrer sense each other. Thus, these models do not drectly apply. We develop a general N-node broadcast sender model based on Markov chans. We present t ncrementally. Frst, we present the model for saturated traffc demands wth varable packet szes. Then, we extend t to handle fxed packet szes and unsaturated demands n Sectons 4.. and Fnally, we descrbe technques to enhance the scalablty of the model n Secton At a hgh level, we construct a Markov chan where each state represents a set of nodes (denoted by S ) that are transmttng smultaneously n a tme slot. Gven N senders, the Markov chan has 2 N possble states (whch we prune n Secton 4..3). We derve the transton matrx M for the Markov chan based on 82. DCF and use t to compute the statonary probablty π of each state. The throughput of node m s then smply t m = m S π. Dervng the transton matrx M: In ths secton, we assume that nodes send varable-length packets wth exponental dstrbuton, and that the state transtons of dfferent nodes are ndependent. We relax these assumptons n Secton 4... Under the ndependence assumpton, we can focus on computng the transton probabltes of an ndvdual node, say m. Ths nvolves computng four transton probabltes for every state : (a) stayng n dle mode, P (m S ); (b) enterng transmsson mode, P (m S ); (c) extng transmsson mode, P (m S ); and (d) stayng n transmsson mode, P (m S ). The probablty M(, j) for the network to transton from state to j s: M(, j) = m S S j P (m S ) m S S j P (m S ) m S S j P (m S ) m S S j P (m S ), () where S denotes the complement of a gven set S. We compute the four per-node probabltes based on 82. DCF. A node can begn transmsson when the followng three condtons are satsfed: ) ts random backoff counter reaches ; ) the medum s clear; and ) the node has data to transmt. Therefore: P (m S ) = Pr[medum s clear counter = m has data] = Pr[medum s clear] Pr[counter = medum s clear] Pr[m has data medum s clear counter = ] = C(m S ) Q(m), (2) CW(m)+OH(m) where C(m S ) s the probablty for m s medum to be clear whle n state, whch we wll compute below. OH(m) (for overhead) denotes the extra clear tme slots that node m needs to wat n addton to CW(m), the average contenton wndow. For broadcast transmssons, we have CW(m) = CW mn 2 and OH(m) = T DIFS T slot, where T DIFS s the DIFS duraton and T slot s the duraton of a tme slot. Q(m) s the probablty that m has data to send gven that the medum s clear and the backoff counter s zero. For saturated demands, Q(m) =. We derve Q(m) for unsaturated demands n Secton For the stayng dle probablty, we set P (m S )= P (m S ). To compute P (m S ) and P (m S ), assume that both transmsson and dle tmes are exponentally dstrbuted. (We relax ths assumpton n Secton 4...) Let T µ (m) be m s average packet transmsson tme, computed based on m s packet sze and transmsson rate, and T slot denote the duraton of a tme slot. We have: P (m S ) = T slot /T µ (m) (3) P (m S ) = P (m S ) = T slot /T µ (m) (4) Computng the clear probablty C(m S ): We have C(m S ) = Pr{I m S β m }, where I m S s the total nterference at m n state and β m s the CCA threshold. I m S s the sum of constant thermal nose W m, the background nterference B m, and nterference due to data transmssons by nodes n S (except for m tself). Thus, I m S = W m + B m + s S \{m} R sm. To estmate ths sum, we model each term as a lognormal random varable our testbed measurement results (omtted for lack of space) suggest that the lognormal dstrbuton fts the measured RSSI well. The standard approach for analyzng the sum of lognormal random varables s to approxmate the sum tself by a lognormal random varable [7, 26]. Followng Fenton [7], we fnd a lognormal random varable that matches the mean and the varance of I m S. Formally, assumng that R sm ( s S ) and B m are all ndependent, we have: E[I m S ]=W m + B m + s S \{m} R sm, and Var[I m S ]= B var m + s S \{m}r var sm. Let e Z be a lognormal random varable wth Z N(µ,σ 2 ). The frst two moments of e Z are E[e Z ] = e µ+σ2 /2 and E[e 2Z ] = e 2µ+2σ2. Equatng the frst two moments of e Z and I m S gves: e µ+σ2 /2 = E[I m S ], and e 2µ+2σ2 = E[Im S 2 ] = Var[I m S ] + (E[I m S ]) 2. Therefore, we have: µ= 2logE[I m S ] 2 loge[i2 m S ] (5) σ 2 = loge[i 2 m S ] 2logE[I m S ] (6) We can then approxmate C(m S ) as: [ ] C(m S ) Pr{e Z logβm µ β m } = Pr{Z logβ m } = Φ, σ where Φ[x] = 2π R x e u2 2 du s the standard normal CDF.

4 Computng statonary probabltes π : Havng derved the transton matrx M, we can compute the statonary probabltes π by solvng the followng system of lnear equatons: π M(, j) = π j ( j) (7) π = (8) where (7) comes from the property that the statonary probabltes of the current and next states are equal, and (8) normalzes the sum of the statonary probabltes to. When M s sparse, π can be effcently solved n Matlab usng lsqr [23]. Secton 4..3 descrbes how to make M sparse to enhance scalablty. 4.. Handlng Smlar Packet Szes The prevous secton assumes varable packet szes and ndependent transton probabltes for varous nodes. But when all nodes use smlar packet szes, the ndependence assumpton no longer holds. Specfcally, gven two nodes wthn carrer sense range from each other, unless ther random backoff counters both reach together, one node wll start transmttng earler and the other node wll sense the carrer and defer to the earler node [22]. As a result, overlappng transmssons from the two nodes wll have almost dentcal start tmes. Wth smlar packet szes, such transmssons wll also end at smlar tmes. Such synchronzaton clearly volates the ndependence assumpton. To handle such scenaros, we construct a synchronzaton graph G syn (S ) for each state as follows. Two nodes m,n S are connected ng syn (S ) f and only f C(m {n}) <. and C(n {m}) <., where C(s {t}) s the clear probablty at node s when node t alone s transmttng. We fnd all the connected components n G syn (S ) and defne each component as a synchronzaton group. We then make two adjustments to the transton probabltes M(, j) to account for the synchronzaton effect. Frst, f there exst two nodes m and n n the same synchronzaton group ofg syn (S ) such that m S j and n S j, then M(, j) =. Ths s because all nodes n a synchronzaton group must ext the transmsson mode together. Second, the probablty for all nodes n a synchronzaton group G to ext the transmsson mode together s (T slot /T µ ). In contrast, under the ndependent transmsson model, the probablty s m G (T slot /T µ (m)) = ( T slot /T µ ) G, whch s much smaller when G 2. Here G s the number of nodes n G Handlng Unsaturated Demands The man challenge n handlng unsaturated demands s estmatng Q(m), whch s the probablty that m has data to send when ts backoff counter s and the channel s clear at m. Wth saturated demands, Q(m) has a constant value of, but wth unsaturated demands t must be computed to ensure that the traffc demands d m are not exceeded. Computng Q(m) s dffcult due to strong nterdependency among nodes. Specfcally, Q(m) depends on how often the channel s clear at m, whch depends on the amount of traffc generated by other nodes and thus ther Q values, whch n turn depends on the traffc generated by m and Q(m). We develop an teratve algorthm to compute Q. The algorthm ntalzes Q to for all senders. In each teraton, the algorthm frst derves the transton matrx M based on the old Q values and computes the statonary probabltes π and the acheved throughput tm prev = :m S π. It then updates Q based on ther values n the prevous teraton. For ths, we use the followng relatonshps: Q prev (m) T µ (m) Q prev T µ (m)+t gap (m) Q(m) T µ (m) Q(m) T µ (m)+t gap (m) = t prev m (9) d m () Q(m) () where Q prev (m) s the value of Q(m) n the prevous teraton, T gap (m) s the average tme gap between two consecutve transmssons from m, and T µ (m) s the average packet transmsson tme of m. Equaton (9) captures the relatonshp between Q(m) and the m s sendng rate n the prevous teraton. Equaton () ensures that the total amount of traffc sent by m does not exceed ts demand d m. We then compute Q new (m) as the largest possble Q(m) value that satsfes all three constrants (9), () and (). Ths gves: { Q new (m) = mn, Q prev (m) d m tm prev d m tm prev }. (2) At ths pont, we could drectly use Q new (m) as our estmate of Q(m) for the next teraton. For quck convergence, we apply a relaxaton procedure that s commonly used n equlbrum computaton [5]. Specfcally, we set the new Q(m) to be a lnear combnaton of Q new (m) and Q prev (m): Q(m) = α Q new (m) + ( α) Q prev (m). Our evaluaton uses α =.9, though we fnd that the model converges quckly for a wde range of α Enhancng Scalablty The general sender model, as presented earler, requres 2 N states and 2 N 2 N state transtons for N senders. To enhance scalablty, we use two technques that prune the states and transtons. Frst, we prune all those states that nvolve too many synchronzed transmssons, whch should occur wth low probablty. Specfcally, gven state and the correspondng S, we elmnate f the number of edges n the correspondng synchronzaton graphg syn (S ) exceeds a gven threshold, whch s set to n our evaluaton. Second, we prune all those state transtons whose transton probabltes are too low. Specfcally, we reset the transton probablty M(, j) to f t falls below a threshold, whch s set to. n our evaluaton. For example, under common 82. settngs, we can elmnate transtons that nvolve 2 synchronzaton groups extng the transmsson mode together. By prunng unlkely transtons, we can reduce the number of non-zero entres n M, thus mprovng the effcency of sparse lnear solvers such as lsqr n computng the statonary probabltes. The combnaton of these two technques s hghly effectve. For example, consder senders n a 5 5 grd topology, where any two drect horzontal, vertcal, or dagonal neghbors can hear each other. Wthout prunng, the transton matrx has 24 states and more than a mllon transtons. After prunng, t has only 37 states and 736 transtons. 4.2 Recever Model We now present our recever model for broadcast traffc. Our goal s to estmate the goodput g mn (.e., the recevng rate). We have g mn = η t m ( L mn ), where L mn s the packet loss rate T from m to n, and η = payload T payload +T header +T preamble represents the fracton of packet transmsson tme for the payload (excludng header and preamble overhead). A key challenge n estmatng L mn s how to translate slot-level loss rates (derved from our slot-level Markov chan) to packet loss rates. Our experments show that slot-level loss rates (.e., the fracton of tme slots n whch loss occurs) can be qute dfferent from packet loss rates. For example, when loss comes from hdden termnals, where senders do not sense each other and cause collsons, a packet s usually corrupted partally. In ths case, the packet loss rate can be sgnfcantly hgher than slot-level loss rate. Consder transmsson of packets, whch contan altogether tme slots. Even f only around % slots ( slots) are lossy, they can cause a packet loss rate as hgh as % f these lossy slots are dstrbuted across all packets. Below we frst analyze the slot-level loss rates and then convert them nto packet loss rates.

5 4.2. Estmatng Slot-Level Loss Probabltes We frst estmate the slot-level loss rate from broadcast sender m to recever n n state (m S ). Let I mn S = W n +B n + t S \{m} R tn be the total nterference at recever n. Note that we allow t = n because n and m may be transmttng at the same tme. At the slot level, a loss occurs whenever the SINR falls below δ n and/or the RSS falls below γ n. Let l mn S = Pr{ R mn I mn S < δ n } be the slot-level loss rate caused by low SINR n state. Let l rss mn = Pr{R mn < γ n } be the slot-level loss rate caused by low RSS. Computng l mn S : Smlar to Secton 4., we approxmate I mn S = W n +B n + t S \{m} R tn wth a sngle moment-matchng lognormal random varable e Z, where Z N(µ,σ 2 ). Snce the rato of two ndependent lognormal random varables R mn and e Z s also a lognormal random varable, let e Z = R mn, where Z e N(µ,σ 2 ). We Z have µ = E[logR mn ] µ and σ 2 = Var[logR mn ]+σ 2. Thus: { } [ R mn logδn µ ] l mn S = Pr < δ n Pr{e Z < δ n } = Φ I mn S σ (3) Computng l rss mn S : There are two ways to compute l rss mn S. When the dstrbuton of R mn s known, we can drectly compute l rss mn S = Pr{R mn < γ n }. In practce, R mn has to be estmated and s subject to estmaton error. To mnmze error, we observe that when there s only a sngle sender and no external nterference, all losses are due to low RSS. Specfcally, ( l rss mn S ) T µ(m)/t slot gves the packet loss rate (assumng ndependent slot-level loss wthn a packet). Thus we can drectly use the measured packet loss rate under transmssons from a sngle sender m to estmate l rss mn S Computng Packet Loss Probabltes L mn Packet losses can be broadly dvded nto three categores. Frst, packet losses can stem from low RSS. Second, packet losses can stem from collson wth packets from the same synchronzaton group. In ths case, the fracton of lost packets s close to the fracton of lost slots. Thrd, packet losses can stem from collsons wth asynchronous transmssons (e.g., from hdden termnals). In ths case, the packet loss rate can be much hgher than the slot-level loss rate. Let Lmn, rss Lmn syn, and Lmn asyn denote the probabltes for these three types of packet losses, respectvely. Assumng ndependence among them, the combned packet loss rate s: L mn = ( L rss mn) ( L syn mn ) ( L asyn mn ) (4) Computng Lmn: rss We estmate Lmn rss as ( l rss mn) T µ(m)/t slot. Note that when measured packet loss rate from a sngle sender m s avalable, we can drectly use t as Lmn rss wthout convertng t to l rss mn. Computng Lmn syn and Lmn asyn : Let SS(m) be the set of all those states whose synchronzaton graph has at least one edge nvolvng m: SS(m) = { m S G (S ) has an edge nvolvng m} (5) Let the synchronous and asynchronous slot-level loss rates be: l syn mn = SS(m) π l mn S, and l asyn mn :m S π = SS(m) π l mn S :m S π (6) To estmate Lmn syn from l syn mn, we smply set Lmn syn = l syn mn. To estmate Lmn asyn from l asyn mn, we assume that packet losses are generated by collson of foreground traffc (from m to n) and background traffc that arrve ndependently. We further model background traffc as an ON/OFF process wth exponentally dstrbuted ON and OFF perods. Let T on and T off denote the average duratons of the two perods, respectvely. Under the above assumptons, the slot-level loss rate for the foreground traffc should be equal to the fracton of tme that the background traffc s n ON perods. That s: T on T on + T off = l asyn mn (7) Assumng T on = T µ, the above equaton yelds T off = lasyn mn l asyn T µ. mn A packet s successfully receved f t starts n a background OFF perod and the rest of ths background OFF perod lasts at least the packet transmsson tme. Wth exponentally dstrbuted background OFF perods, we have: L asyn mn = [ T off exp T µ T on + T off = ( l asyn mn ) exp ] T off [ lasyn mn l asyn mn ] (8) (9) where the frst term on the rght hand sde of Equaton (8) s the probablty that the packet transmsson starts n a background OFF perod and the second term s the probablty that the rest of ths background OFF perod lasts for at least T µ. 5. UNICAST TRAFFIC In ths secton, we extend our broadcast models to handle uncast traffc. There are two key dfferences between uncast and broadcast transmssons. On the sender sde, the transton matrx M s dfferent under uncast due to addtonal ACK overhead and exponental backoff. On the recever sde, there are addtonal losses due to ACKs colldng wth both data and other ACKs. We present the sender sde extensons followed by the recever sde extensons. 5. Extensons to Sender Model The transton matrx for the sender, n partcular, CW(m), OH(m) and Q(m) n Equaton (2) are dfferent for uncast traffc. Computng CW(m): We derve CW(m) from packet loss rate L mn across all recevers n as follows. Let H(L) be the average contenton wndow under packet loss rate L, and RMAX be the maxmum number of retransmssons. Then we have: RMAX H(L) = k= mn{(cw mn + )2 k,cw max } L k (2) 2 A sender may transmt to more than one recever, each wth a dfferent loss rate. We estmate CW(m) as the weghted average over all recevers, where the weghts are based on the total transmssons to the recevers. For smplcty, we approxmate the G weght as mn d mn r G mr d mr, where G mn denotes the expected number of transmssons (ncludng the frst transmsson) for each data packet sent from m to n. Assumng ndependent packet losses, G mn = RMAX k= Lmn. k Therefore: [ CW(m) = H(L mn ) G ] mn d mn (2) n r G mr d mr Computng OH(m): Uncast nvolves addtonal overhead from SIFS and ACK. Let T SIFS and T ACK denote the duraton for SIFS and ACK. The average overhead for uncast transmssons from m to n s: OH(m,n) = T DIFS+T SIFS +( L mn )T ACK T slot. We then compute OH(m) as the the weghted average of OH(m,n) over all n: [ OH(m) = OH(m,n) G ] mn d mn (22) n r G mr d mr

6 Computng Q(m): For saturated uncast demands, Q(m) =. For unsaturated uncast demands, we can update Q(m) n the same way as n Secton The only adjustment we need s to account for retransmsson by m. Ths can be acheved by smply changng the rght hand sde of Equaton () from d m to r G mr d mr. Computng throughput t mn : Wth the new CW(m), OH(m), and Q(m), we can compute the transton matrx M and statonary probabltes π for uncast traffc. We can then compute the throughput from m to n as t mn = t m G mn d mn r G mr d mr, where t m = :m S π. 5.2 Extensons to Recever Model Consder node m sendng data to node n. Smlar to broadcast, we decompose uncast packet loss rate L mn nto three components: (a) Lmn rss losses due to low RSS, (b) Lmn syn losses due to collson of synchronzed transmssons, and (c) Lmn asyn losses due to collson wth asynchronous transmssons. Assumng ndependence among them, we have: L mn = ( Lmn) rss ( Lmn syn ) ( Lmn asyn ). The key extensons that we make are: () extend Lmn rss to nclude RSS nduced losses for both data and ACK packets, and () extend l mn S to nclude SINR nduced slot-level loss due to collson between ACK/data, data/ack, ACK/ACK (n addton to data/data), whch can then be used to compute Lmn syn and Lmn asyn n the same way as n Secton Below we descrbe these extensons n detal. Estmatng Lmn: rss Let l rss mn = Pr{R mn < γ n } and l rss nm = Pr{R nm < γ m }. The combned RSS-nduced loss on data and ACK s then L rss mn = ( l rss mn) T µ(m)/t slot ( l rss nm) T ACK(n)/T slot where T ACK (n) s the duraton of an ACK sent by n. Estmatng l mn S : We consder the followng three cases of low SINR nduced slot-level loss: C: Data loss due to collson wth other data. Data sent from m to n can get lost due to collson wth data sent by other nodes n S \ {m}. Ths s dentcal to the broadcast case and the slot-level loss rate s l C mn S = Pr{ R mn Imn S C W n + B n + t S \{m} R tn. < δ n }, where I C mn S = C2: Data loss due to collson wth other ACKs and data. In ths case, a synchronzaton group G (m G) exts the transmsson mode whle all nodes n S \ G contnues transmttng. The ACKs generated by recevers of senders n G could collde wth data from m to n. To quantfy such effects, let random varable R ack (m,n S ) denote the nterference at n that s caused by ACKs sent by the recever of sender m after m stops transmttng n state (whch we wll analyze below). The slot-level loss rate caused by other ACKs and data s thus: l C2 mn S (G) = Pr{ R mn I C2 mn S < δ n }, where I C2 mn S = W n + B n + t S \(G {m}) R tn + t G R ack (t,n S ) s the total nterference at n when m s transmttng data to n. C3: ACK loss due to collson wth other ACKs and data. In ths case, the synchronzaton group that m belongs to (denoted by G m ) exts the transmsson mode whle all nodes n S \ G m contnue transmttng. ACKs sent by recevers of senders n G m \ {m} combned wth data sent by nodes n S \ G m could collde wth ACKs sent from n to m. The resulted slot-level loss rate s l C3 mn S = Pr{ R nm I C3 nm S < δ n }, where I C3 nm S = W m +B m + t S \G m R tm + t Gm \{m} R ack (t,m S ) s the total nterference at sender m when recever n s transmttng an ACK to m. Among the above three cases, C2 (wth dfferent G) and C3 are mutually exclusve because only one G can be the frst synchronzaton group that stops transmttng. Note that wth our prunng strateges descrbed n Secton 4..3, we need not consder havng two groups stop transmttng at the same tme (because the transton probablty would become too small). For a gven G, the probablty for G to be the frst group that stops transmttng n state s M(, (G)) j M(, j) = M(, (G)) M(,), where (G) s the resulted state after G stops transmsson n state. The combned loss rate for C2 and C3 can then be computed as: l C2+C3 mn S = [ G:m G M(, (G)) M(,) lc2 mn S (G) ] + M(, (G m )) M(,) lc3 mn S (23) Assumng ndependence between C and C2/C3, the combned slot-level loss rate n state can be computed as: l mn S = ( l C mn S ) ( l C2+C3 mn S ) (24) The only remanng task s to analyze R ack (m,n S ). The man challenge s that m may have multple recevers and RSS from dfferent recevers are dfferent. To address ths, for each sender we compute the weghted average of nterference that ts recevers generate, where the weghts are based on the traffc demands and delvery probabltes to the recevers. Specfcally, we approxmate the dstrbuton of R ack (m,n S ) usng a log-normal dstrbuton, whose mean R ack (m,n S ) and varance R var ack (m,n S ) are computed as: [ ] dmr R ack (m,n S ) = p ack r r d mr S mr R rn (25) [ R var ack (m,n S dmr ) = p ack r r d mr S mr R var rn ] (26) where p ack mr S denotes the probablty for a transmsson from sender m to successfully trgger an ACK from ts recever r n state. To smplfy the analyss of p ack mr S, we gnore data loss due to collson wth other ACK and only consder data loss caused by ether low RSS or collson wth other data. Wth such smplfcaton, we can approxmate p ack mr S ( l C mn S ) ( l rss mn) T µ(m)/t slot. Fnally, once all the loss rates L mn are avalable and t mn has been computed, we can compute the goodput as g mn = η t mn Lmn RMAX+ G mn, where G mn s the average number of transmssons per data packet, ( Lmn RMAX+ ) gves the packet delvery rate (after the ntal transmsson and RMAX retransmssons), and η s used to exclude the header and preamble overhead. 5.3 Puttng It Together Unlke for broadcast traffc, there s a tght couplng between the sender model and recever model for uncast traffc. Specfcally, n order to compute CW(m) and OH(m) before dervng the transton matrx M and the statonary probabltes π, the sender model requres the knowledge of packet loss rates L mn (as descrbed n Secton 5.). Meanwhle, the recever model needs to know π n advance n order to convert slot-level loss rates nto packet loss rates L mn (as descrbed n Secton 5.2 and Secton 4.2.2). To break such nter-dependency, we apply an teratve framework to progressvely refne our model. As summarzed n Fgure, we ntally set L mn =. Durng each teraton, we frst apply the sender model to update CW(m), OH(m), M and π based on L mn from the prevous teraton. We then use the updated π to compute the new L mn. For unsaturated uncast demands, we also teratvely update Q(m) (whch are ntalzed to ). For quck convergence, we apply the relaxaton procedure n Secton 4..2 to update L mn and Q(m). Convergence s reached when the relatve changes n L mn and Q(m) become small enough. In our evaluaton, we fnd that the model always converges quckly wthn teratons.

7 ntalzaton: L mn = ( m n); Q(m) = ( m); converged = false 2 whle (not converged) // sender model: see Secton 5. and Secton 4. 3 compute CW(m), OH(m) usng L mn 4 derve transton matrx M from CW(m), OH(m) and Q(m) 5 compute statonary probabltes π usng M 6 compute Q new (m) based on π and prevous Q(m) // recever model: see Secton 5.2 and Secton compute packet loss rates Lmn new from slot-level loss rates usng π // relaxaton for quck convergence (currently we set α =.9) 8 L mn = α Lmn new +( α) L mn 9 Q(m) = α Q new (m)+( α) Q(m) // test for convergence converged = true f changes n L mn and Q(m) are small enough end Fgure : An teratve framework for modelng uncast traffc. 6. OBTAINING MODEL INPUTS In ths secton, we descrbe how we obtan the varous nputs to our model. To estmate parwse RSS and the external nterference at each node, namely R mn and B n, we measure RSSI at n when only m s transmttng. We only requre O(N) measurements because wreless s broadcast medum and all recevers can measure RSSI when a node transmts. From Res et al. [24], RSSI mn = log ( R mn+b n W n ). For smplcty, we assume B n =. (Ths holds when nterference from external transmtters s neglgble.) We then estmate R mn by fndng a log-normal dstrbuton that best fts the measured RSSI. Let R mn and R var mn denote the mean and varance of the best fttng log-normal dstrbuton. The fnal RSS dstrbuton s estmated as a log normal dstrbuton wth mean of R mn and varance of R var mn T preamble T slot. We estmate RSS varance as R var mn T preamble T slot because we are nterested n RSS varaton n the tme scale of slots whle RSSI s measured as an average over the preamble perod and R var T mn s slot T preamble of the slot-level RSS varance. As mentoned n Secton 4.2., when the RSS dstrbuton s avalable, we can estmate Pr{R mn < γ n } mmedately from the dstrbuton. In practce, because RSSI measurements are only avalable on receved packets, estmatng the true RSS dstrbuton s hard. To get around the problem, we can estmate Pr{R mn < γ n } by drectly computng the loss rate (.e., the fracton of packets that are lost) usng the RSSI measurement data. We fnd that when the delvery rate s too low (e.g., below %), computng the mean and varance of RSS based on RSSI measurements yelds sgnfcant bas because RSSI measurements are only avalable on receved packets. Accurately estmatng the true R mn under such cases s an nterestng subject on ts own, and we leave t as future work. In our current testbed evaluaton, we consder only the sender groups such that every node par m and n wthn the sender group has ether L mn 9% or L mn = %. For far comparson wth the UW model, n all 2-sender evaluaton we do not apply the above flterng, and compare the estmated and actual values over all sender groups. Our model also requres the values of a few rado-dependant constants. For testbed experments, based on our hardware, we use -95 dbm as thermal nose, 2.5 db as SINR threshold, and -85 dbm as CCA threshold. For smulaton experments, we use the default values n Qualnet, where the thermal nose s dbm n 82.a and dbm n 82.b, SINR threshold s 2.5 db, and CCA threshold s -85 dbm n 82.a and -93 dbm n 82.b. 7. SIMULATOR-BASED EVALUATION We evaluate the accuracy of our model n both smulaton and testbed settngs. These two evaluaton methodologes are complementary. Testbed experments allow us to quantfy accuracy n more realstc scenaros whch are subject to fluctuaton n the RF envronment, measurement errors, and varatons across real hardware. Smulaton offers a more controlled envronment and allows us to more comprehensvely assess the accuracy of ndvdual components n our model. Many smplfyng assumptons n our model relate to the nteracton of the MAC protocol, and any naccuracy due to these assumptons mpacts the smulator results as well. 7. Qualnet Modfcatons We use Qualnet for our evaluaton. It has been shown to provde a relatvely accurate and realstc smulaton envronment [27]. We make the followng modfcatons to the Qualnet. Correct desynchronzaton problem: The IEEE 82. standard states that when the medum s busy at any tme durng a backoff slot, the backoff procedure must be suspended wthout decreasng the value of the backoff tmer. However n Qualnet, the backoff tmer s decremented by propagaton delay and causes tme desynchronzaton. Such desynchronzaton results n an unrealstcally low collson rato, as reported n [3] and confrmed by our results. We fx the problem by ensurng that the backoff tmer s not decremented when the medum s busy at any nstant wthn a tme slot. Dsable EIFS: Accordng to the IEEE 82. standard, n DCF a frame transmsson must use EIFS whenever a frame transmsson begns but does not result n the correct recepton of a complete MAC frame. However several research papers [9, 3] report that EIFS results n unfarness, and suggests dsablng EIFS by settng EIFS duraton to the same value as DIFS. Exstng chpsets such as Atheros also have a confgurable EIFS duraton. We use the above method to dsable EIFS, and leave modelng EIFS for future work. Modfy capture effects: In Qualnet, a recever accepts frames wth stronger sgnals only when they arrve earler than recepton of other frames. Recently, Kochut et al. [3] report that real wreless cards accept frames wth stronger sgnals even f they arrve after recepton has started. Therefore, we modfy Qualnet to accept frames wth stronger sgnals regardless of whether they arrve earler or later than recepton of other frames. In contrast to modfcatons used n [3], we also accept frames that arrve after preamble of the frame beng receved. Ths smplfes our model, and we plan to explore a detaled model of capture effects n our future work. Support SINR model: The 82. mplementaton n Qualnet uses a Bt-Error-Rate (BER) model, where t computes SINR of the current packet, uses the SINR to determne the BER, and then converts the BER to the packet loss rate. To match Qualnet smulaton, our model needs the same BER table as n Qualnet. However Qualnet source code does not reveal the BER table t uses for 82.. To ensure consstency between our model and Qualnet, we mplement the commonly used SINR model n both Qualnet and our model. If the BER table becomes avalable, our model can mmedately support BER model by usng BER table to map SINR to loss rate. 7.2 Evaluaton Methodology We evaluate our model for both broadcast and uncast by varyng the number of smultaneous senders, the frequency band, and the network topologes. We consder both saturated demands and unsaturated demands. The demand s normalzed by the physcal layer data rate, and a sender wth saturated demand has demand. Throughout the evaluaton, we use 25-node topologes. Senders generate 24-byte UDP packets at a constant bt rate (CBR). The actual sendng rate to the ar may not be constant, however, due to varable contenton delay. We use the lowest MAC data rates,.e., 6Mbps n 82.a and Mbps n 82.b. The communcaton ranges of 82.a and 82.b wth the lowest data rates are 69 m and 348 m, respectvely. For each scenaro, we conduct random runs, where each run randomly selects the senders and recevers and the demands. We quantfy the accuracy of our model by comparng wth the actual

8 Ours (RMSE=.28) UW (RMSE=.45) Ours (RMSE=.5) UW (RMSE=.664) Sender-Recever Par ID Fgure 2: 2 saturated broadcast senders usng 82.a n a 5 5 grd topology over an 3m 3m area Ours (RMSE=.46) Ours (RMSE=.89) Sender-Recever Par ID Fgure 3: saturated broadcast senders usng 82.a n a 5 5 grd topology over an 3m 3m area. normalzed throughput and goodput (defned n Secton 3) and computng the root mean square error (RMSE). RMSE s defned as (est actual ) 2 P, where P s total number of predctons. We also study the accuracy n detal usng scatter plots of actual and estmated values. For clarty, n the scatter plots the data ponts are plotted n the ncreasng order of actual values. We consder the followng scenaros below: () 2 broadcast senders wth saturated demands; () N broadcast senders wth saturated demands; () N broadcast senders wth unsaturated demands; (v) N uncast senders wth saturated demands; and (v) N uncast senders wth unsaturated demands. For the frst scenaro, we compare our model wth both Qualnet smulaton and UW model [24]. The UW model predcts the mpact of nterference n the presence of two broadcast senders wth saturated demands. It s seeded usng O(N) measurements smlar to ours each node takes turn to broadcast packets and other nodes log RSSIs and packet delvery rate. Each node obtans ts RSSI versus delvery rate profle usng these measurements. To predct the mpact of two senders tryng to send smultaneously, t frst estmates the probablty wth whch senders defers based on the RSSIs they receve from each other. To estmate a recever s goodput from a sender, t uses the standard SINR model, whle treatng transmssons from the second sender as addtonal nterference at the recever. Snce there are no exstng models for the other scenaros, we compare our model only wth the actual values obtaned n Qualnet. 7.3 Broadcast Traffc We begn our evaluaton by studyng broadcast traffc, startng wth the smple case of two senders wth saturated demands Two Saturated Senders Fgure 2 shows the accuracy of throughput and goodput estmates of our model and the UW model. The graphs plot the actual values obtaned n Qualnet and the predctons of the two models. The legend contans the RMSE values for the two models. We see that both models perform well overall, though our model s more accurate. The RMSE n our model s wthn.5 whle that of the UW model s.45 or more. The UW model also tends to have hghly naccurate predctons for a few cases Ours (RMSE=.33) Ours (RMSE=.7) Sender-Recever Par ID Fgure 4: saturated broadcast senders usng 82.b n a 5 5 grd topology n an 5m 5m area Ours (RMSE=.52) ; Ours (RMSE=.69) Sender-Recever Par ID ; Fgure 5: saturated broadcast senders usng 82.a n a 5 5 grd topology n an 5m 5m area. The error n UW model s manly because t assumes that a packet can be receved as long as ts SINR exceeds the threshold. It gnores the other condton that RSS should exceed the rado senstvty for a packet to be receved. For example, when there s only thermal nose (-95 dbm) and RSS s above -92.5, SINR would be above the 2.5 db threshold. The UW model would thus predct % packet delvery. In realty, however, when RSS s between dbm and -85 dbm (the rado senstvty value for 82.a n Qualnet), the delvery rate s n fact. Unfortunately, there s no smple extenson to the UW model to accommodate the rado senstvty constrant because the model bulds RF profle drectly based on delvery rate. Wth the rado senstvty constrant, there s no drect translaton between R mn and delvery rate snce ther relatonshp changes from Pr{ R mn I n +W n δ n } to Pr{R mn γ n R mn I n +W n δ n }. For a gven delvery rate, R mn s no longer unque N Saturated Senders Next we consder the case of N broadcast senders. Each sender has saturated demand, as before. We evaluate our model by varyng the frequency band, network topology, and the number of senders. Dfferent frequency bands (82.a and 82.b): Fgures 3 and 4 show the scatter plots of actual and predcted throughput and goodput under 82.a and 82.b. In each case, there are broadcast senders wth nfnte demands. We see that our model s hghly accurate n both cases, wth less than.5 RMSE. The goodput error s lower than the throughput error because many recevers have no connectvty to one or more senders, and t s easer to predct the exact goodput for such recevers. Dfferent network topologes (grd and random): Fgure 5 and 6 show the results for broadcast senders usng 82.a n a 5m 5m grd topology and 3m 3m random topology, respectvely. In each case, the model closely tracks the actual values and the error s around or below.5. Dfferent number of senders (2-): Fgure 7 plots throughput and goodput RMSE as a functon of the number of broadcast senders. We see that the error tends to ncrease slghtly wth the number of senders due to more complex nteractons. Yet under all numbers of senders, the model can keep RMSE wthn.7 for throughput estmaton and wthn.25 for goodput estmaton.