An Approximation to the QoS Aware Throughput Region of a Tree Network under IEEE CSMA/CA with Application to Wireless Sensor Network Design

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1 An Approxmaton to the QoS Aware Throughput Regon of a Tree Network under IEEE CSMA/CA wth Applcaton to Wreless Sensor Network Desgn Abht Bhattacharya a, Anurag Kumar a a Dept. of Electrcal Communcaton Engneerng Indan Insttute of Scence, Bangalore, , Inda Abstract In the context of wreless sensor networks, we are motvated by the desgn of a tree network spannng a set of source nodes that generate packets, a set of addtonal relay nodes that only forward packets from the sources, and a data snk. We assume that the paths from the sources to the snk have bounded hop count, that the nodes use the IEEE CSMA/CA for medum access control, and that there are no hdden termnals. In ths settng, startng wth a set of smple fxed pont equatons, we derve explct condtons on the packet generaton rates at the sources, so that the tree network approxmately provdes certan qualty of servce (QoS) such as end-to-end delvery probablty and mean delay. The structures of our condtons provde nsght on the dependence of the network performance on the arrval rate vector, and the topologcal propertes of the tree network. Our numercal experments suggest that our approxmatons are able to capture a sgnfcant part of the QoS aware throughput regon (of a tree network), that s adequate for many sensor network applcatons. Furthermore, for the specal case of equal arrval rates, default backoff parameters, and for a range of values of target QoS, we show that among all path-length-bounded trees (spannng a gven set of sources and the data snk) that meet the condtons derved n the paper, a shortest path tree acheves the maxmum throughput. Keywords: Throughput optmal network desgn, wreless sensor networks, QoS based desgn of wreless sensor networks, Throughput regon of CSMA/CA, desgn of multhop CSMA networks 1. Introducton Our work n ths paper s motvated by the followng broad problem of desgnng mult-hop ad hoc wreless networks that utlse IEEE CSMA/CA as the medum access control. Gven a network graph over a set of sensor nodes (also called sources), a set of potental relay locatons, and a data snk (also called base staton (BS)), where each lnk meets a certan target qualty requrement, the problem s to extract from ths graph, a hop length bounded 1 tree topology connectng the sensors to the BS, such that the resultng tree provdes certan qualty of Ths work was supported by the Department of Electroncs and Informaton Technology under the Automaton Systems Technology (ASTEC) program, and by the Department of Scence and Technology va a J.C. Bose Fellowshp. Emal addresses: abht@ece.sc.ernet.n (Abht Bhattacharya), anurag@ece.sc.ernet.n (Anurag Kumar) 1 For a dscusson of why such a hop count bound s needed, see [1, 2] Preprnt submtted to Elsever March 1, 2015

2 Fgure 1: A network graph over 10 sources (labelled S 1, S 2,, S 10) and 30 potental relay locatons (the unlabeled vertces); the base staton (BS) s at (0,0); each edge s assumed to have a packet error rate of no more than 2%. There are no hdden nodes. servce (QoS), typcally expressed n terms of a bound on the packet delvery probablty and/or on the mean packet delay, whle also achevng a large throughput regon. As an example, consder the network graph shown n Fgure 1 over 10 sensors, 30 potental relay locatons (the unlabeled vertces), and a BS at (0, 0); the lnks n the network have a worst case packet error rate of 2%. Suppose that the nodes use IEEE CSMA/CA, and that all the nodes are wthn carrer sense range of one another. The problem s to obtan from ths graph (by deployng relays at a subset of the potental locatons), a tree connectng the sensors to the snk, such that the hop count from each sensor to the snk s no more than, say 5, the packet dscard probablty on each lnk s no more than 2.08%, and the mean delay on each lnk s no more than 20 msec (these sngle hop QoS requrements translate to an end-to-end delvery probablty of 90%, and an end-to-end mean delay of 100 msec). In addton, among trees that meet these requrements, the resultng tree should acheve a large throughput regon. The followng are possble approaches for addressng such a problem. Va exhaustve search usng smulaton or an accurate performance analyss tool: One nave way of solvng the above mentoned network desgn problem s to consder all possble canddate tree topologes, and smulate each of them for a wde range of arrval rates to obtan ther QoS respectng throughput regons, and choose the one wth the largest throughput regon. Ths method s clearly neffcent as smulaton of each topology takes sgnfcant amount of tme, and there could be exponentally many canddate trees (see Fgure 1). An alternatve approach s to replace the smulaton step wth a network analyss tool (such as the one proposed n [3] for IEEE CSMA/CA networks) whch s consderably faster compared to smulatons; however, one stll requres to evaluate an exponental number of canddate trees for a wde range of arrval rates, and hence the method s stll neffcent. Va a characterzaton of the QoS respectng throughput regon: A more effcent way of solvng the network desgn problem would be to obtan an exact analytcal characterzaton of the QoS respectng throughput regon of a tree network under IEEE CSMA/CA n terms of the topologcal propertes of the network, and then derve network desgn rules from that characterzaton to maxmze the throughput regon. The dffculty wth ths approach s that for practcal CSMA/CA protocols such as IEEE , obtanng an explct exact characterzaton of the 2

3 Fgure 2: Two competng tree topologes obtaned from the network graph n Fgure 1. Left panel: a shortest path tree connectng the sources to the BS. Note that t uses two relays, namely R1 and R2. Rght panel: a tree obtaned usng an approxmaton algorthm for a certan Stener graph desgn problem (see the SPTRP algorthm [1]) to connect the sources to the snk. Note that t uses no relays, but has a hgher total hop count compared to the shortest path tree. QoS respectng throughput regon n terms of topologcal propertes s notorously hard. See Secton 1.1 for more detals. Therefore, some approxmate methodology s n order. Our strategy: Our approach n solvng ths problem s two-fold: 1. Obtan an explct approxmate nner bound to the QoS respectng throughput regon of a tree network n terms of the topologcal propertes of the network, and parameters of the CSMA/CA protocol. 2. Obtan a tree that maxmzes ths approxmate nner bound. Such an endeavour requres performance models of general multhop wreless networks under the CSMA/CA MAC, and the dervaton of desgn crtera from such models. In ths paper, we utlze our smplfcaton of a detaled fxed pont based analyss of a mult-hop tree network operatng under IEEE unslotted CSMA/CA [4], provded by Srvastava et al. [3], to develop certan explct desgn crtera for QoS respectng networks. As wll be explaned n a later secton (see Secton 6), t turns out that for default protocol parameters of IEEE CSMA/CA, and for a wde range of QoS targets, the resultng soluton s surprsngly smple: connect each sensor to the snk usng a shortest path (n terms of hop count). Although ths crteron s based on an approxmate nner bound to the QoS respectng throughput regon, we wll see n our numercal experments that ths actually acheves larger throughput than a wde range of competng topologes. Contnung the example of Fgure 1, Fgure 2 demonstrates two dfferent tree topologes connectng the sensors to the snk, that are both subgraphs of the example network graph of Fgure 1. The left panel shows a shortest path tree connectng the sources to the BS; ths tree requres two relays. The rght panel depcts a tree obtaned usng the SPTRP algorthm proposed n [1] for constructon of hop constraned Stener trees wth a small number of relays. In the context of ths example, our results n ths paper provde. explct formulas for nner boundng the set of arrval rates that can be carred by ether of these two topologes, whle respectng the QoS obectves, and. a bass for assertng that, for equal arrval rates from all sources, the shortest path tree topology n Fgure 2 wll acheve the larger QoS respectng arrval rate. 3

4 Indeed, for equal arrval rates from all the sensors, t turns out (by brute force search over a wde range of arrval rates usng the network analyss method presented n [3] for IEEE CSMA/CA) that the shortest path tree can handle up to 5 packets/sec from each source, whereas the other tree topology n Fgure 2 can handle 3.5 packets/sec from each source. The correspondng values from our explct nner bound formulas are obtaned as packets/sec and packets/sec respectvely. See Secton 6 for detals Related work Most of the prevous work on network desgn focuses on the lone-packet model, or a TDMA lke MAC, where there s no contenton n the network; see, for example, [5, 1, 6], and the references theren. On the other hand, most work on network performance modelng focuses on estmatng some measure of network performance under some smplfed form of CSMA/CA MAC protocol for a gven arrval rate vector, and a gven network topology; see, for example, [7, 8] and the references theren. Ths knd of work usually nvolves capturng the complex nteracton (due to contenton) among the nodes n the network va a set of fxed pont equatons, and then solvng these equatons usng an teratve scheme to obtan the quanttes of nterest (see [9, 10] for example). For practcal MAC protocols such as IEEE and IEEE , these fxed pont based schemes are too nvolved to gather any nsght on the general dependence of the network performance on the arrval rates and network topology; see, for example, [11, 12, 7]. The only papers (see [13, 14, 15]) that provde some nsght on the topology dependence of the network performance do so for smplfed network model and/or smplfed MAC protocols that are not mplemented n practce. In partcular, Marbach et al. [13] provde an approxmate nner bound on the stablty regon of a mult-hop network for a smplfed CSMA/CA MAC, and Bordenave et al. [14] provde an approxmate characterzaton of the network stablty regon for slotted ALOHA; however, nether work consders any QoS obectve. Srnvasa and Haengg [15] consder an nfnte network where the sources nodes as well as the relay nodes are dstrbuted accordng to a homogeneous Posson pont process, and analyze the throughput and end-to-end delay of a typcal flow n such a network under ALOHA, and a smplfed verson of CSMA. They provde system desgn nsgts such as the optmal densty of source nodes, and the optmal hop count n a typcal flow so as to maxmze the spatal throughput densty; however, ther results cannot be appled n practce snce practcal networks are of fnte sze, and they do not consder any practcal MAC protocol such as IEEE or IEEE CSMA/CA. Another lne of work studes the problem of ont rate control, routng and schedulng ([16, 17], and references theren) for throughput utlty optmzaton n a gven network graph. The schedulng algorthm proposed n [16] s centralzed, and therefore, hard to mplement n practce. In [17], the authors propose a dstrbuted rate control and schedulng algorthm that s guaranteed to acheve at least half the optmal throughput regon of a gven network. However, none of these papers consder any QoS obectve. There s related work that focuses on developng queue length aware CSMA/CA type dstrbuted algorthms that can acheve the optmal throughput regon, or some guaranteed fracton thereof, of a gven network, whle also yeldng low delay 2 ; see [18] and the references theren for a detaled account of the progress n ths lne of research. However, these queue length based dstrbuted algorthms are 2 Note that these schedulng algorthms essentally allow nfnte number of packet retransmssons to ensure 100% end-to-end delvery 4

5 not mplemented n any commercally avalable CSMA/CA (e.g., IEEE , IEEE ). Therefore, we shall not pursue ths lne of work any further n ths paper. Our current work s amed as a frst step towards brdgng the gap between network performance modelng, and network desgn for a practcal CSMA/CA MAC. In partcular, we are nterested n dervng, from network performance models, nsghts on how the network performance s affected by the topology and the arrval rate vector, and then convert that nsght nto useful crtera for network desgn Outlne of the paper In Secton 2, we descrbe the network model, the QoS obectves we consder n ths paper, and defne certan throughput regons; we then provde a summary of our contrbutons n the paper. Secton 3 brefly descrbes the CSMA/CA analyss n [3], and develops a set of smplfed fxed pont equatons startng from that analyss. In Sectons 4 and 5, we use the smplfed fxed pont equatons to derve an explct approxmate nner bound to the QoS aware throughput regon of a tree network. Secton 6 dscusses how we can use the approxmate crtera to derve a smple network desgn rule for a certan specal case. Detaled numercal results are provded n Secton 7, and we conclude n Secton 8. For lack of space, some of the proofs have been provded n a techncal report [2]. 2. The Model, some Notaton, and a Summary of the Contrbutons of ths Paper In ths secton we descrbe the network model, the QoS obectve, some notaton, and then we defne certan QoS respectng throughput regons. Ths then permts us to provde an overvew of our contrbutons, before we actually get nto the detals of the dervatons, and formal results Network model, QoS obectves, and throughput regons Network model In ths paper we focus on developng crtera for the desgn of tree topologes wth bounded path lengths. We consder a tree network T = (V, E T ) spannng a set of sensor nodes (henceforth, called sources ), Q (ncludng the base staton (BS)), wth Q = m + 1, and a set of relays R T such that V = Q R T. We assume that the edges n E T are such that the packet error probablty on any edge (for fxed packet lengths, non-adaptve modulaton, fxed bt rates, and fxed transmt powers) s no more than a target value, l; the set of relays R T may be needed to lmt lnk lengths n order to acheve such a bound on packet error probabltes. We assume that the source-snk path lengths n T obey a gven hop constrant, h max. As explaned n [2], the choce of such a hop constrant may be affected by several factors, ncludng the RF propagaton n the gven envronment, and the physcal characterstcs of the mote hardware. In ths work, we shall not concern ourselves wth detals of the partcular choce of h max. We further assume that the nodes use IEEE CSMA/CA for medum access. Moreover, all the nodes are assumed to be n the carrer sense range of one another. We shall refer to ths assumpton as the No Hdden Nodes (NH) assumpton. For a dscusson of why such an assumpton may not be too restrctve, see [2]. 5

6 QoS obectves In order to desgn mult-hop networks that provde QoS, or even to route connectons over such networks, gven the ntractablty of analyss of mult-hop networks (partcularly, CSMA/CA wreless networks), t has been the practce to adopt the approach of splttng the QoS over the hops along whch a flow s routed (see, for example, [19, Secton ]). In ths paper we adopt the approach that the end-to-end QoS obectves are splt equally over the lnks on each path. That s, to meet the target end-to-end delvery probablty obectve p del, we requre that the packet dscard probablty on each lnk s no more than δ 1 exp ( ) ln p del h max. Smlarly, to meet the target end-to-end mean delay obectve d max, we requre that the mean delay on each lnk s no more than d d max h max. Wth the values of h max, p del, d max beng gven, we shall assume n the rest of the paper that we are gven a target sngle-hop dscard probablty, δ, and a target sngle-hop mean delay, d Throughput regons Suppose source k generates traffc at rate λ k packets/sec, k = 1,..., m, accordng to some ergodc pont process. The throughput regon of a gven tree network, assumng that the nodes n the network use CSMA/CA for medum access, s defned as Λ(T) = {λ = {λ k } m k=1 : the network s stable} Informally speakng, the network s sad to be stable when the queue lengths do not grow unbounded wth tme. 3 Note, however, that ths noton of throughput regon does not consder any QoS obectve. Wth our QoS splttng assumptons stated earler, gven a tree network T, and assumng the nodes n the network use CSMA/CA for medum access, we defne Λ δ (T) = {λ : Under arrval rates λ, the dscard probablty on each lnk n the tree T s bounded by δ} Λ d (T) = {λ : Under arrval rates λ, the mean delay on each lnk n the tree T s bounded by d} Λ δ,d (T) = Λ δ (T) Λ d (T) Note that Λ δ,d Λ, for otherwse, at least one queue length would grow unbounded, and the mean delay constrant would not be met. For a tree T wth arrval rates λ k (packets/sec), 1 k m, at the sources, we defne ν (λ, T) : The packet rate nto node, f no packets are dscarded n any node. h k (T) : The hop count on the path from source k to the snk n tree T. For a gven tree T, we then defne the followng throughput regons n terms of () a detaled fxed pont analyss provded n [3], () a smplfed verson of ths fxed pont analyss that we provde n ths paper, and () numbers B(δ) and B (δ, d) for whch we wll provde explct formulas n terms of the parameters of the CSMA/CA protocol. 3 See [20] for dfferent notons of stablty. 6

7 ˆΛ δ (T) = {λ : Under arrval rates λ, the dscard probablty on each lnk s bounded by δ, accordng to the detaled analyss 4 presented n [3]} ˆΛ d (T) = {λ : Under arrval rates λ, the mean delay on each lnk s bounded by d, accordng to the detaled analyss presented n [3]} ˆΛ d,δ (T) = ˆΛ d (T) ˆΛ δ (T) ˆΛ δ (T) = {λ: Under arrval rates λ, the dscard probablty on each lnk s bounded by δ, accordng to the smplfed analyss presented n Secton 3.3} ˆΛ d (T) = {λ: Under arrval rates λ, the mean delay on each lnk s bounded by d, accordng to the smplfed analyss presented n Secton 3.3} ˆΛ d,δ (T) = ˆΛ d (T) ˆΛ δ (T) Λ δ (T) = {λ : m k=1 λ kh k (T) < B(δ)} Λ d,δ (T) = {λ : m k=1 λ kh k (T) < B(δ) and max 1 N ν (λ, T) B (δ, d)} 2.2. Contrbutons n ths paper Wth the above defntons, the man contrbutons of ths paper are summarzed as follows: 1. For the regme where packet dscard probablty s small, we obtan a smplfed set of fxed pont equatons n Secton 3.3. Unlke [3], we are able to show the unqueness of the fxed pont of our smplfed equatons n Secton Usng the smplfed fxed pont analyss, we provde expressons for B(δ) and B (δ, d) such that the followng set relatonshps hold: Λ d,δ (T) 1 ˆΛ d,δ (T) 2 ˆΛ d,δ (T) The set nequalty 1 has been establshed n Sectons 4 and 5 (see Theorems 1 and 2), where, n the process, we provde expressons for the functons B(δ), and B (δ, d). The tghtness of the set nequalty 1 has been evaluated through numercal experments (see Secton 7.2.2, and Table 5 n Secton 7.3.2; also see Secton VI- B3 n [2]). Approxmaton 2 has been verfed through extensve numercal experments n Secton for the regme where our smplfed fxed pont equatons have a unque soluton. These results are related to the throughput regon of the orgnal system, Λ δ,d (T), through the followng approxmaton whch was shown to be very accurate (well wthn 10% n the regme where packet dscard probablty on a lnk s about the same as the PER on the lnk) n [3] by extensve comparson aganst smulatons: ˆΛ d,δ (T) Λ δ,d (T) 4 Ths means that f we analyze the tree T under the gven arrval rate vector λ usng the detaled analyss, then the QoS values obtaned from the analyss satsfy the target requrements. 7

8 Remark: It follows from the above set relatonshps that the explctly defned set Λ d,δ (T) can be taken as an approxmate nner bound to the orgnal throughput regon, Λ δ,d (T). Furthermore, our numercal experments suggest that Λ d,δ (T) captures a sgnfcant part of the throughput regon ˆΛ d,δ (T) (whch, n turn, s a good approxmaton for Λ δ,d (T)). In partcular, when the QoS targets are n the range δ , and d 20 msec, for the tested network topologes, t follows from our approxmate nner bound that an arrval rate of at least 2-3 packets/sec from each source can be handled wthout volatng QoS (see Table 5 n Secton 7, and the dscusson thereafter). Ths arrval rate s more than enough for many sensor networkng applcatons ncludng those of ndustral telemetry, and non-crtcal montorng and control applcatons [21, 22]. 3. Fnally, for the specal case of equal arrval rates at all the sources, default backoff parameters of IEEE CSMA/CA, and a range of target values δ and d, we have shown n Secton 6 that Λ δ (T) = Λ d,δ (T) (1) Furthermore, let us defne, for any tree T, λ(t) max λ λ:λ1 Λ d,δ (T) where, 1 s the vector of all 1 s havng the same length as the number of sources. Let T be any shortest path tree spannng the sources and rooted at the snk. Then, for any other tree T spannng the sources and rooted at the snk, we argue n Secton 6 that λ(t) λ(t ) (2) 3. Modelng CSMA/CA for a Network wth No Hdden Nodes In ths secton we frst brefly revew the IEEE CSMA/CA mechansms, then we descrbe the overall modelng approach used by Srvastava et al. n [7] and [3]. Ths modelng approach yelds a system of fxed pont equatons from whch we derve a set of very smple fxed pont equatons on whch the work n ths paper s based The beaconless IEEE CSMA/CA protocol We provde a bref sketch. When a node has data to send, t ntates a random back-off [4] at the end of whch t performs a CCA (Clear Channel Assessment) to determne whether the channel s dle. We note that, unlke the IEEE CSMA/CA, the back-off tmer of a node s not frozen durng the transmssons of other nodes. If the CCA succeeds, the node does a Rx-to-Tx turnaround, whch s 12 symbol tmes, and starts transmttng on the channel. CCA falure starts a new back-off process wth the back-off exponent rased by one. After a prespecfed maxmum number of successve CCA falures for the same packet, the packet s dscarded at the MAC layer. Upon successful packet recepton, the recever sends a fxed sze ACK packet. When a transmtted packet colldes or s corrupted by the PHY 8

9 layer nose, the ACK packet s not generated, whch s nterpreted by the transmtter as falure n delvery. The node retransmts the same packet for a prespecfed maxmum number of tmes before dscardng t at the MAC layer The overall modelng approach n [7, 3] Although there has been consderable research on analytcal modelng of CSMA/CA, the work reported n [7], [3], n the context of beacon-less IEEE CSMA/CA, appears to be the most comprehensve (capturng aspects such as mult-hoppng, hop-by-hop queueng, presence of hdden termnals, etc.), and also very accurate. For ease of reference, we brefly descrbe the modelng phlosophy of ther analyss here. In [7, 3], akn to the approach n [9] and [10], a decouplng approxmaton s made, whereby each node s modeled separately, ncorporatng the nfluence of the other nodes n the network by ther average statstcs, and as f these nodes were ndependent of the tagged node. In the lterature, such an approach has also been called a mean feld approxmaton, and formal ustfcaton has been provded n, for example, [14]. Modelng the actvty of a tagged node, say : All packets enterng node are assumed to have the same fxed length, and hence the same fxed transmsson tme over the medum (denoted by T tx ). In a network wth no hdden nodes, the channel actvty perceved by node s due to all the other nodes n the network. Each node alternates between perods durng whch ts transmtter queue s empty and those durng whch the queue s non-empty. Durng perods n whch ts transmtter queue s non-empty, when the node s not transmttng t s performng repeated backoffs for the head-of-the-lne (HOL) packet n ts queue. Each backoff ends n a CCA attempt. Fgure 3 s a depcton of ths alternaton n node behavor durng the perods when ts queue s nonempty. We employ the decouplng approxmaton to analyse ths process. W (k 1) W (k) W (k+1) X (k 1) X (k) X (k+1) X (k+2) Total backoff duraton wth CCAs Transmsson duraton T tx Fgure 3: (Taken from [3]) The alternatng (contenton-transmsson) process obtaned by observng the process at a node after removng all the dle tme perods at Node. The transmsson completon epochs are denoted by {X (k) } and the contenton-transmsson cycle lengths by {W (k) }. Focusng on duratons durng whch Node s queue s nonempty (see Fgure 3), consder the completon of a transmsson by Node ; ths could have been a success or a collson. We note that packet collsons can occur even n the absence of hdden nodes, snce transmssons from two transmtters placed wthn the CS range of each other can overlap (known as Smultaneous Channel Sensng), as depcted n Fgure 4. Due to the small propagaton delay, and the no hdden nodes assumpton, we neglect the up to 12 addtonal symbol duratons for whch other smultaneous transmssons could last. After the completon of ts transmsson, Node contends for the channel by executng successve backoffs; the nodes ether have non-empty queues and are also contendng, or have empty queues and are not contendng. Wth these observatons n mnd, the CCA attempt process at Node condtoned on beng n backoff perods s modeled as a Posson process of rate β. For each node 9

10 12 T s Backoff CCA DATA Process evolvng at node Ω Process evolvng at node t t Fgure 4: (Taken from [3]) Node fnshes ts backoff, performs a CCA, fnds the channel dle and starts transmttng the DATA packet. Node fnshes ts backoff anywhere n the shown 12 T s duraton and as there s no other ongong transmsson n the network, ts CCA succeeds and t enters the transmsson duraton. As a result, the DATA packets may collde at recever of or that of., the CCA attempt process, condtoned on s queue beng non-empty and beng n backoff, or beng empty, s modeled by an ndependent Posson process wth rate τ (),. Ths s the rate of attempts of Node as perceved by Node durng Node s back-off perods. Now let us return to the process depcted n Fgure 3,.e., the alternatng contenton and transmsson perods of Node when ts queue s nonempty. As a result of the assumpton of ndependent Posson attempt processes at the nodes, t can be observed that the nstants X (k) are renewal nstants, and the ntervals W (k) form an..d. 5 sequence. Wth the above observatons and defntons, the detaled evoluton of the attempt processes at the nodes can be analyzed usng renewal theoretc arguments, thus leadng to a set of fxed pont equatons nvolvng quanttes such as the collson probablty, the CCA falure probablty, the rates of CCA attempts, the packet dscard probablty at a node, queue non-empty probablty of a node, and the probablty that a node s n back-off condtoned on ts queue beng non-empty. The resultng fxed pont equatons, for the NH case, are shown n [2, Secton II-B2]. The analyss of packet delay between a source node and the snk node utlzes the approxmaton technques n Whtt s Queueng Network Analyzer (QNA, [23]). The arrval processes at the source nodes are modeled as ndependent Posson processes. 6 Each node s modeled by a GI/GI/1 queue. Usng the soluton to the fxed pont equatons obtaned earler, the frst two moments of the servce tme at each queue can be computed. The arrval and departure processes of the nodes are approxmated by renewal processes, whose frst two moments are computed recursvely, startng from the leaf nodes, and usng the soluton to the fxed pont equatons obtaned earler. Gven the frst two moments of the nterarrval tme and of the servce tme, QNA uses a standard GI/GI/1 mean delay approxmaton for the mean delay at a queue. In ths manner, startng wth the model of the nput processes at the source nodes, the end-to-end mean delay can be approxmated. In [3], the authors fnd that the analyss s accurate to wthn 10% n the regme where the dscard probablty on a lnk s about the same as the PER on that lnk. It s evdent from the complex structure of the fxed pont equatons dsplayed n [2] that even for the no hdden nodes case, t s dffcult to use ths analyss to extract nsght about the relaton between network topology and the 5 Independently and dentcally dstrbuted 6 We note that the methodology does not rely on ths Posson arrvals assumpton. Other renewal arrval processes can be assumed, ncludng perodc arrvals. 10

11 QoS measures. We therefore, n the next secton, resort to smplfyng these detaled equatons n a certan regme of network operaton Smplfed fxed pont equatons n the low dscard regme Our pont of departure from the analyss n [3] s to show that, n the low dscard regme 7, the fxed pont equatons can be substantally smplfed. We have obtaned ths smplfcaton by ntroducng the followng approxmatons that are approxmately vald n the low dscard regme. (A1) The dscard probablty at any node, δ 0 so that the total arrval rate nto node s ν = m k=1 z k, λ k, where z k, = 1 ff node s n the path of source k, and z k, = 0 otherwse. (A2) The probablty, r, that the head-of-lne (HOL) packet at node encounters a transmsson falure s very small,.e., r 1. (A3) The fracton of tme, q, that the queue of node s non-empty s very small,.e., q 1. (A4) The probablty, c, that node fnshes ts backoff wthn 12 symbol tmes after another node has fnshed backoff s neglgbly small,.e., c 0. (A5) The probablty, γ, that a transmtted packet from node encounters a collson or lnk error s very small,.e., γ 1, so that (1 γ ) 1. Wth these approxmatons, t easly turns out (by smple substtuton of the approxmatons) that the detaled fxed pont equatons n [3] smplfy to the followng vector fxed pont equatons (see [2] for detals): ( m ) τ = z k, λ k (1 + α α n c 1 ) = 1,..., N (3) α = k=1 T txτ 1 + T tx τ = 1,..., N (4) where, τ s the total CCA attempt rate seen by node, and α s the CCA falure probablty at node. Although these equatons have been derved by ntroducng the above approxmatons n the detaled equatons of [3], t s easy to provde a drect ntuton for them as follows. Consder the term correspondng to node n the summaton n (3). Note that the frst factor n ths term s the total arrval rate nto a node, and the second factor s the mean number of CCA attempts of a packet at node. Thus, each term nsde the summaton s the total CCA attempt rate due to a node. Hence the summaton (whch does not nclude node ) s ndeed the total CCA attempt rate seen by node. Here, of course, we are nvokng the NH property of the network. 7 Note that a sngle-hop dscard probablty of even 3% results n an end-to-end delvery probablty less than 90% on a path wth 4 or more hops; hence, for most practcal purposes, we would be nterested n operatng the network n the low dscard regme. 11

12 Also, let us defne N CCA (t) and N CCA f (t) to be the number of CCA attempts and number of CCA falures respectvely at node up to tme t. Then, α = lm N CCA f (t) t N CCA, whch by usng the renewal model dscussed n Secton 3.2, and (t) elementary ergodc results, gves α = NCCA f N CCA (5) where N CCA s the mean number of total CCA attempts by node n a renewal cycle, and N CCA f of faled CCA attempts by node n a renewal cycle. s the mean number Note that when node s backoff fnshes frst among all contendng nodes (ths happens wth probablty β β +τ ), ts frst CCA succeeds, and the cycle ends after node s transmsson. When some other contendng node s backoff fnshes frst (ths happens wth probablty τ β +τ ), that node starts transmttng on the medum, and node contnues to perform CCAs at rate β for the entre transmsson duraton T tx,.e., a total of β T tx CCAs, all of whch fal (note that we are usng (A4) n makng ths asserton); at the end of the transmsson duraton, all the nodes agan enter contenton, and the cycle contnues. Usng these observatons, N CCA Ths, after smplfcaton, yelds, N CCA = β β + τ 1 + can be wrtten recursvely as τ β + τ (β T tx + N CCA ) N CCA = 1 + τ T tx Usng smlar arguments, N CCA f can be wrtten recursvely as N CCA f = β β + τ 0 + τ β + τ (β T tx + N CCA f ) Ths, after smplfcaton, yelds, N CCA f = τ T tx Fnally, pluggng these back nto (5), we have T txτ α = 1 + T tx τ We now wrte down certan other quanttes of nterest n terms of the fxed pont varables. Usng (A4), the probablty of a packet collson (or, equvalently, the probablty of smultaneous channel sensng) s easly seen to be (recall the modelng phlosophy from Secton 3.2) (1 exp( 12τ )). Hence, the transmsson falure probablty, γ at node can 12

13 be wrtten as γ = l + (1 l)(1 exp( 12τ )) (6) where we have assumed the worst case PER l on a lnk. The packet dscard probablty at node, denoted δ, s gven by δ = α n c (1 + r + r r n t 1 ) + r n t (7) where, r = γ (1 α n c ) s the probablty that the HOL packet at node was transmtted, and t encountered a transmsson falure (ether due to collson or due to lnk error). n c and n t denote respectvely, the maxmum allowed number of successve CCA falures, and the maxmum allowed number of transmsson falures after whch the packet s dscarded. Fnally, the CCA attempt rate, β, condtoned on beng n back-off perods, can be wrtten as β = 1 + α + α α n c 1 (8) B Ths can be nterpreted as a consequence of the Renewal Reward Theorem (RRT; see, for example, [24]), where the numerator s the mean number of CCA attempts for the HOL packet at node, and the denomnator, B, s the mean tme spent n backoff by the HOL packet at node. For default back-off parameters of IEEE CSMA/CA, B = α + 318(α 2 + α α n c 1 ) (9) It s easy to see (usng Lemma 5.1 n [25]) that β s non-ncreasng n α. Suppose L s the set of nodes on the path from source to the snk. Then, assumng packet dscards are ndependent across nodes, the end-to-end delvery probablty of source s gven by L (1 δ ), where δ s the packet dscard probablty at node as defned earler. Smplfcatons for delay approxmaton We have lsted the detaled equatons for delay approxmaton n Secton II-B2 of [2]. We wrte down here, the smplfcatons we obtan from those equatons under approxmatons (A1)-(A5) (see [2] for detals). The end-to-end mean packet delay for a source node, provded that the set of nodes along the path from ths node to the BS s L, s gven by = L (10) 13

14 where, s the mean soourn tme at a node, and s gven by (also see Eqn. 19 n [2], and the dscusson leadng to Eqn. 24 n [2]) = ρ E(S )(1 + c 2 S ) 2(1 ρ ) + E(S ) (11) Here, ρ = ν E(S ) denotes the traffc load at node wth ν = m k=1 z k,λ k beng the total arrval rate nto node, S denotes the servce tme at node, and c 2 S = Var(S ) E 2 (S ) s the squared coeffcent of varance of servce tme at node. Note that the above expresson s precsely the Pollaczek-Khntchne formula (see, for example, [24]) for the mean delay of an M/G/1 queue. Thus, n the low dscard regme, the delay approxmaton (see Eqn. 19 n [2]) reduces to modelng each node n the network by an M/G/1 queue. Furthermore, the MGF M S (z) of S can be expressed as (see [3]), M S (z) = β (1 α )(1 γ )e zt tx z + β (1 α )(1 γ e zt tx ). Startng wth ths MGF, and takng dervatve w.r.t z, straghtforward calculatons yeld, E(S ) = d dz M S (z) = 1 + β (1 α )T tx z=0 β (1 α )(1 γ ) (12) where we recall from Secton 3.2 that β s the mean CCA attempt rate of node condtoned on beng n backoff perods. β s a functon of α and the protocol parameters (see (8)). Smlarly, by straghtforward calculatons, E(S 2 ) = d2 dz M 2 S (z) = T 2 2T tx tx + z=0 β (1 α )(1 γ ) + 3β (1 α )γ Ttx 2 β (1 α )(1 γ ) Fnally, after some algebrac manpulatons, we have + 2(1 + β (1 α )γ T tx ) 2 (β (1 α )(1 γ )) 2 (13) c 2 S = Var(S ) E 2 (S ) where, n Eqn. (14), we have used (A5). = γ + 1 γ (1 + β (1 α )T tx ) γ (14) (1 + β (1 α )T tx ) 2 Remark: In our numercal experments wth default backoff parameters, the above smplfcatons yeld delay values that are accurate to wthn 6% w.r.t the delay values obtaned from the detaled analyss, whch themselves were accurate to well wthn 10%. In what follows, we shall use ths smplfed fxed pont analyss to obtan suffcent condtons for the membershp of ˆΛ δ and ˆΛ d. We start by establshng a condton for the unqueness of the soluton to the smplfed vector fxed pont equatons (3) and (4). 14

15 3.4. Exstence and unqueness of the smplfed vector fxed pont Let us defne α max δ 1/n c, and a α max T tx (1 α max ), where δ s the target dscard probablty (at a node) as defned earler. Observe from Equaton (4) that α α max τ a. Moreover, note that n our regme of nterest,.e., the regme where max 1 N δ δ, we have α α max for all = 1,..., N, or equvalently, τ a for all = 1,..., N. Then, we have the followng proposton. Proposton 1. If m λ k h k < mn { k=1 a 1 + α max +... α n c 1 max, } 1 B T tx (1 + 2α max (n c 1)α n 1 (δ) (15) c 2 max ) then the smplfed fxed pont equatons defned by (3) and (4) have a unque soluton {τ } N =1 n [0, a]n. Proof. The proof proceeds by establshng that the smplfed fxed pont equatons defne a contracton mappng on [0, a] N. See Appendx 9.2 for detals. We call the regme defned by Equaton (15) the unqueness regme. Our numercal experments suggest that n ths regme, the smplfed fxed pont equatons well-approxmate the detaled fxed pont equatons (to wthn 10% for τ ),.e., we have ˆΛ d,δ ˆΛ d,δ, where ˆΛ d,δ and ˆΛ d,δ are as defned n Secton See Secton 7 for detals. In lght of above dscusson, we shall focus on obtanng suffcent condtons for membershp of ˆΛ δ and ˆΛ d nstead of ˆΛ δ and ˆΛ d. 4. Dervaton of A Suffcent Condton for Membershp of ˆΛ δ In ths secton, we shall derve a suffcent condton for the membershp of the set ˆΛ δ (T) (defned n Secton 2.1.3) for a gven tree T. In other words, we shall derve the structure of the set Λ δ (T) ntroduced n Secton A control on the maxmum packet dscard probablty From equaton (7), we can derve the followng Lemma. Lemma 1. For α n c 1/n t, and γ n t α and γ,.e., δ,1 δ,2 whenever α,1 α,2 and γ,1 γ,2. 1/n t, the packet dscard probablty at node, δ, s monotoncally ncreasng n Proof. The proof proceeds by showng that δ (α, γ ) s ncreasng n each coordnate when the other coordnate s held fxed. See [2] for detals. From the smplfed fxed pont equatons (3), (4), and (6), we have the followng lemma. Snce the proof s short, we present the proof here tself. Lemma 2. α and γ are monotoncally ncreasng n τ. Proof. From (6), t s clear that γ s monotoncally ncreasng n τ. To see that α s monotoncally ncreasng n τ, observe that the dervatve of the R.H.S of (4) w.r.t τ s non-negatve. 15

16 Combnng Lemmas 1 and 2, we have Proposton 2. δ s monotoncally ncreasng n τ. It follows from Proposton 2 that to control the maxmum packet dscard probablty, we need to control max =1,...,N τ,.e., where τ max s the upper bound on τ that ensures δ δ. max =1,...,N δ δ max =1,...,N τ τ max 4.2. A scalar fxed pont Before proceedng further, we take a slght detour, and ntroduce a further smplfcaton to the fxed pont equatons descrbed n Secton 3.3. We shall obtan a scalar fxed pont equaton whch wll be exploted later on to extract nformaton about topologcal dependences. We start wth the followng observaton from our numercal experments wth the detaled analyss (see Secton 7.2.2). Observaton 1. For all = 1,..., N, τ τ.e., the total attempt rate seen by a node s roughly equal for all nodes, and s approxmately equal to the total attempt rate, denoted τ, n the network. An analytcal ntuton behnd ths observaton for large N s also provded n [2]. From the above observaton, and (4), t follows that the CCA falure probabltes of the nodes are roughly equal, and are equal to α = T txτ 1+T tx τ. Moreover, the total CCA attempt rate, τ, n the network can be computed as τ = ( m k=1 λ kh k )(1 + α α n c 1 ), where the frst factor on the R.H.S s the total packet arrval rate nto the network, and the second factor s the mean number of CCA attempts due to each packet. Thus, we have the followng scalar fxed pont equaton n τ, the total attempt rate n the network: m τ = ( λ k h k )(1 + α α nc 1 ) (16) α = k=1 T txτ 1 + T tx τ Note that (17) can also be wrtten as α = T tx 1, whch can be nterpreted as the fracton of tme that the medum s τ +T tx occuped by a packet (consequently, all CCA attempts durng ths tme fal). (17) 16

17 4.3. Exstence and unqueness of the scalar fxed pont Let us, as before, defne a have the followng proposton. α max T tx (1 α max ), where α max = δ 1/n c. Observe from (17) that α α max τ a. Then, we Proposton 3. If the condton defned by (15) holds, then the scalar fxed pont equaton defned by (16),(17) has a unque soluton n [0, a]. Proof. The proof proceeds by showng that the scalar fxed pont equaton defnes a contracton mappng on [0, a]. See Appendx 9.3 for detals. Note that (15) ensures unque fxed pont for both the vector and the scalar fxed pont equatons A tght upper bound on max 1 N τ We now come back to the man thread of our dscusson. Recall the scalar fxed pont equatons (16), (17) derved n Secton 4.2. We had promsed that these equatons wll fnd use n dervng topologcal propertes that control the network performance. To ths end, we start wth the followng proposton. Proposton 4. In the unqueness regme (defned by (15)), τ max 1 N τ (18) where τ s the unque soluton to the scalar fxed pont equatons (16),(17), and {τ } N =1 s the unque soluton to the smplfed vector fxed pont equatons (3),(4). Proof. The proof s va nducton. See Appendx 9.4 for detals. Remarks: (a) The ntuton behnd the above proposton s clear. Each component of the smplfed vector fxed pont approxmates the total attempt rate seen by the correspondng node n the network, whch should clearly be upper bounded by the total attempt rate of all the nodes n the network, approxmated by the scalar fxed pont. (b) Observaton 1 (from our numercal experments) suggests that the above upper bound on max 1 N τ s tght. Further numercal experments verfyng the tghtness of the bound are provded n Secton VI-B3 of [2]. From Proposton 4, t follows that to ensure max 1 N τ τ max, t suffces that we ensure τ τ max,.e., to control the maxmum packet dscard probablty, max 1 N δ (or, equvalently, max 1 N τ ; recall Proposton 2, and the dscusson thereafter), t suffces to control the approxmate total attempt rate, τ. We, therefore, next nvestgate the dependence of τ on the network topology and arrval rate vector to come up wth a suffcent condton for the membershp of ˆΛ δ. 17

18 4.5. A suffcent condton for membershp of ˆΛ δ The followng proposton s an easy consequence of Part 2 of Lemma 5 stated n Appendx 9.3, and hence we omt the proof. Proposton 5. τ, the soluton to the scalar fxed pont equatons (16),(17), s monotoncally ncreasng n m k=1 λ kh k. It follows from Proposton 5 that gven δ (or, equvalently, τ max ), there exsts B 2 (δ) such that m k=1 λ kh k B 2 (δ) τ τ max. Hence, as long as the vector fxed pont equatons (3),(4) and the scalar fxed pont equatons (16),(17) have unque solutons, the followng sequence of mplcatons hold: m k=1 λ k h k B 2 (δ) τ τ max max 1 N τ τ max max 1 N δ δ However, (15) gves us a suffcent condton for a unque soluton of the fxed pont equatons. Thus, we have the followng theorem: Theorem 1. Let B(δ) mn{b 1 (δ), B 2 (δ)}, where B 1 (δ) s as defned n (15), and B 2 (δ) s as defned n the above dscusson. Suppose, for a gven tree network topology, an arrval rate vector λ satsfes m λ k h k < B(δ) (19) k=1 Then, λ ˆΛ δ. Note that the above theorem characterzes the set Λ δ (T) ntroduced n Secton Dervaton of a Suffcent Condton for the Membershp of ˆΛ d,δ In ths secton, we shall derve a suffcent condton for the membershp of ˆΛ d,δ, thereby defnng the set Λ d,δ (T) for a gven tree T Dependence of E(S ) and c 2 S on τ We make the followng clam. Lemma 3. E(S ) and c 2 S are monotoncally ncreasng n τ. Proof. See [2] A bound on Recall from Proposton 2 that the constrant on the packet dscard probablty, namely max 1 N δ δ, translates to a constrant on τ, namely max 1 N τ τ max. Ths, together wth Lemma 3, mples a bound on E(S ) and on c 2 S, namely that, for all, E(S ) S, and c 2 S c 2 S. Notng that ρ 1 ρ s monotoncally ncreasng n ρ, Ths yelds a bound on for fxed ν, namely, ν S 2(1 ν S ) S (1 + c2 S ) + S 18

19 5.3. A suffcent condton for membershp of ˆΛ d,δ ν In order that d for all, t suffces that for all, S S (1 + 2(1 ν S ) c2 S ) + S d. After some straghtforward algebrac manpulatons, ths yelds an upper bound, desgnated by B (δ, d), on ν for all. Thus, we have the followng theorem: Theorem 2. If an arrval rate vector λ = {λ k } m k=1 satsfes Equaton (19), and the followng holds: max 1 N ν B (δ, d) (20) where ν = m k=1 z k,λ, and B (δ, d) s as defned n the above dscusson, then, λ ˆΛ d,δ. Note that the above theorem characterzes the set Λ d,δ (T) ntroduced n Secton From the defntons of Λ δ (T), and Λ d,δ (T), t also follows that Λ δ (T) Λ d,δ (T). 6. Specal Case: Equal Arrval Rates at All Sources In ths secton, we shall focus on the noton of equal throughput,.e., the scenaro where the external arrval rates at all the sources are equal, say, λ. Ths leads to the followng nterestng consequences. 1. Condton (20) reduces to λ max 1 N m B, whereas, condton (19) reduces to λ N =1 m B, where m s the number of sources whose paths to the snk use node. Now, for default backoff parameters, and for δ =.0208, d = 20 msec (correspondng to, for example, p del = 90% and d max = 100 msec for h max = 5), t turns out that B(δ) = packets/sec, and B (δ, d) = packets/sec,.e., B(δ) < B (δ, d) so that for all λ satsfyng the dscard probablty obectve (19), λ max 1 N m λ N =1 m < B(δ) < B (δ, d),.e., the mean delay obectve (20) s also satsfed. Hence, t follows that for equal arrval rates, and default backoff parameters, Λ d,δ = Λ δ. Furthermore, t was observed that ths concluson contnues to hold for a range of QoS targets such that.015 δ 0.04 and 20 msec d 45 msec. 2. A smple network desgn crteron: Suppose we are gven a graph G = (V, E) wth V = Q R, where R s a set of potental locatons where one can place relays, and E s the set of admssble edges wth PER at most l. Our obectve s to desgn a tree network spannng the sources and the BS, possbly usng a few relays (placed at a subset of the potental locatons), such that the resultng network meets a gven hop constrant h max on each source-snk path, and meets gven per-hop dscard probablty and mean delay targets, whle achevng a large throughput regon. In lght of the dscusson n Item 1 above, for default backoff parameters and reasonable QoS targets, an approxmate lower bound to the throughput of a gven tree network topology s B(δ) mk=1 h k (usng Λ d,δ as an approxmate nner bound to Λ δ,d ; recall the set relatonshps from Secton 2.2). Ths, n turn, gves a smple crteron for approxmately throughput optmal network desgn for the case of equal arrval rates, namely, obtan a tree that mnmzes the total hop count from the sources to the snk, m k=1 h k; ths s nothng but the shortest path tree over graph G (wth hop count as cost). 8 Note that ths s smply a re-wordng of Equaton (2) stated n Secton Note that f the hop constrant, h max, s feasble, the shortest path tree wll also meet the hop constrant. 19

20 Observe that the same crteron holds for the noton of max mn throughput (see, for example, [2] for a formal defnton), snce the max-mn s acheved along the drecton of equal arrval rates. Further dscusson of the example ntroduced n Fgures 1 and 2: Recall the example ntroduced n Fgures 1 and 2 n Secton 1. Observe that our expermental observaton there that the shortest path tree acheves a better throughput than the other topology matches wth our theoretcal observaton n Item 2 above. Moreover, for the gven QoS requrements, and default backoff parameters, t follows from the dscusson above that the shortest path tree (left panel of Fgure 2) can handle an arrval rate of at least B(δ) mk=1 h k = = packets/sec from each source, whereas the topology obtaned from SPTRP algorthm [1] (rght panel of Fgure 2) can handle an arrval rate of at least B(δ) mk=1 h k = = packets/sec from each source. Note that these lower bounds are obtaned usng our explct formula for the set Λ d,δ as opposed to a brute force search over all possble arrval rates, and hence are much smpler to compute. As we shall see n our numercal experments (Secton 7), although the shortest path tree crteron s based on an approxmaton to Λ δ,d, t, n fact, acheves better throughput than a wde range of competng topologes. 7. Numercal Results 7.1. The settng We conducted all our numercal experments usng the default protocol parameters of IEEE CSMA/CA. In partcular, we assumed n c = 5, n t = 4, and default back-off parameters ([4]) n all the experments. We chose the worst case packet error rate (PER) on each lnk to be l = 2%, and packet length T = 131 bytes 9. The target packet dscard probablty, δ = , and the target sngle-hop mean delay d = 20 msec (whch correspond, for example, to a target end-to-end delvery probablty p del = 90%, and target end-to-end mean delay d max = 100 msec for a hop constrant h max = 5). We assumed equal arrval rates at all the sources for the experments. All the experments were conducted on 5 random network topologes, each of whch was generated as follows: 10 sources and 30 potental relay locatons were selected unformly at random over a m 2 area. The snk was chosen at a corner pont of the area. A Stener tree was then formed connectng the sources to the snk, usng no more than 4 relays 10, such that the hop count from each source to the snk was at most 5. The algorthm used to form ths Stener tree s a varaton of the SPTRP algorthm ([1]), and s descrbed n the Appendx Model valdaton Accuracy of the smplfed vector fxed pont equatons To verfy the accuracy of the smplfed vector fxed pont equatons (3) n the unqueness regme (descrbed by Eqn. (15)) w.r.t the detaled fxed pont equatons n [3], we analyzed each of the 5 test networks usng both the orgnal equatons, and the smplfed equatons for several dfferent arrval rates wthn the unqueness regme for that 9 Ths ncludes 70 bytes of data, 8 bytes of UDP header, 20 bytes IP header, 27 bytes MAC header, and 6 bytes Phy header. 10 Ths restrcton was mposed snce, n practce, the network planner would lke to budget the use of addtonal relays. 20

21 topology. 11 For each topology, we computed the worst case 12 percentage error n the smplfed fxed pont {τ } N =1 w.r.t ther values obtaned from the detaled analyss. We also computed the worst case error n max 1 N δ, the maxmum packet dscard probablty over all the nodes n the network. Note that when max 1 N δ 0.002, even an absolute error of would result n a percentage error of 25%. We, therefore, adopt the followng conventon for reportng the errors n max 1 N δ. For arrval rates at whch max 1 N δ 0.002, we report the worst case absolute error; for arrval rates at whch max 1 N δ > 0.002, we report the worst case percentage error. Fnally, we also computed the worst case percentage error n the end-to-end probablty of delvery. Table 1 summarzes the results. Table 1: Worst case (over all nodes and all arrval rates) errors of the smplfed vector fxed pont scheme w.r.t the orgnal fxed pont scheme n the unqueness regme Topology Node Total hop % error Absolute error % error % error % error count count n τ n max 1 N δ n max 1 N δ n end-to-end n end-to-end for max N for max > N delvery probablty delay Observatons: 1. The error n τ never exceeded 10.05%. 2. The error n max 1 N δ never exceeded 6% n the regme where max 1 N δ > The error n end-to-end delay never exceeded 6%. 4. The error n end-to-end delvery probablty was neglgbly small over the entre unqueness regme, never exceedng 0.11%,.e., the smplfed fxed pont equatons predcted the end-to-end delvery probablty extremely accurately. Remark: In ths paper, we have chosen to compare the results obtaned from our smplfed fxed pont equatons aganst those obtaned from the detaled fxed pont equatons n [3] as opposed to comparng aganst smulatons of the orgnal system, whch would be more tme consumng. The detaled fxed pont equatons were shown to be very accurate (well wthn 10% compared to smulatons) n the regme where the dscard probablty on a lnk s close to the lnk PER. Thus, n ths low dscard regme, an error of 10% w.r.t the detaled equatons translates to an error of at most 19%-21% w.r.t the orgnal system Valdty of the equalty assumpton The numercal experments leadng to Observaton 1 were conducted as follows: we analyzed each of the 5 test networks usng the detaled analyss n [3](also descrbed n Secton II-B2 of [2]) to obtan the vector {τ } N =1 for several dfferent arrval rates rangng from pkts/sec to 4 pkts/sec. Then, we computed Jan s farness ndex ([26]) for each of these vectors. The Jan s farness ndex, J(x), s a measure of farness (equalty) among the components n a 11 Snce we are consderng equal arrval rates at all the sources, the unqueness regme for a partcular network s gven by an upper bound on the arrval rate λ determned by the R.H.S of Eqn. (15), and the total hop count of the concerned network. 12 By worst case, we mean, over all nodes and all arrval rates 21

22 gven vector x = {x 1,..., x N }. In partcular, closer the value of J(x) to 1, better s the farness among the components. As t turned out, n all our experments, J({τ } N N 1 =1 }), and wthn 1.5% of 1, ndcatng that the components were N ndeed roughly equal. The results are summarzed n Table 2, where, for compact representaton, we have reported, for each topology, the least value of the farness ndex over all the arrval rates. Table 2: Valdaton of Observaton 1, usng Jan s Farness Index; farness ndex of 1 mples exact equalty Topology Node count Worst case farness ndex Dscusson on the valdty of assumptons (A1)-(A5) Note that assumptons (A1)-(A5) are dealzatons made prmarly for mathematcal convenence. These assumptons are never satsfed exactly n practce, and ths s reflected n the fact that the smplfed fxed pont equatons obtaned usng these assumptons ncur addtonal error wth respect to the orgnal detaled fxed pont equatons. However, as our numercal experments n Secton suggest, the error ncurred by the smplfed fxed pont equatons wth respect to the detaled fxed pont equatons s no more than 10% n the unqueness regme; ths ndcates that the assumptons (A1)-(A5) cannot be wldly naccurate n ths regme. In ths secton, we further nvestgate, to what extent the dealzatons n assumptons (A1)-(A5) are volated n the soluton to the detaled fxed pont equatons n the unqueness regme. To do ths, we proceeded n the same manner as n Secton 7.2.1, and analyzed each of the fve test networks usng the detaled fxed pont analyss for several dfferent arrval rates wthn the unqueness regme. For each topology, we computed the worst case (over all nodes, and all arrval rates) values of the followng quanttes: 1 δ, 1 γ, 1 q, and 1 c. Note that f assumptons (A1)-(A5) were exactly true, the values of each of these quanttes would be exactly 1. Also observe that the error n 1 r s always less than that n 1 γ. Hence, we do not report the values of 1 r separately. Table 3 summarzes our fndngs. Table 3: Worst case (over all nodes and all arrval rates) values of the quanttes n assumptons (A1)-(A5) n the orgnal fxed pont scheme n the unqueness regme; deally, these values would be 1 Topology 1 δ 1 γ 1 q 1 c Observatons: 1. Assumpton (A1) (.e., δ 0) s accurate wthn 0.17%. 2. Assumpton (A5) (.e., 1 γ 1), and hence (A2) (.e., r 1) are accurate wthn 5.6%. 3. Assumpton (A3) (.e., q 1) s accurate wthn 9.8%. 4. Assumpton (A4) (.e., c 0) s accurate wthn 10%. 5. Fnally, recall from Secton that the overall loss of accuracy n the smplfed fxed pont equatons w.r.t the detaled fxed pont equatons due to volaton of these dealzatons s wthn 10%. 22

23 7.3. Throughput optmalty of the shortest path tree To verfy the throughput performance of shortest path trees, we generated 60 random nstances, each wth 10 sources, and 30 potental relay locatons deployed unformly over a m 2 area. For brevty, we report detaled results from 5 nstances here; the observatons from the other nstances were smlar Comparson aganst competng topologes We are lookng for a desgn that uses a small number of nodes and has a large throughput regon for the gven target QoS. Snce an exhaustve search for the optmal throughput over all possble Stener trees s computatonally mpractcal, we proceeded as follows to compute an estmate of the optmal throughput for each nstance. Intutvely, two topologcal propertes can affect the throughput of a gven network, namely, the total hop count (.e., the total load n the network), and the number of nodes n the network. Snce the SPT gves the least total hop count, clearly there s no need to look at desgns that use more relays than an SPT. Hence, for each nstance, we frst constructed an arbtrary shortest path tree, T SPT, connectng the sources to the snk. Let n SPT be the number of relays n ths SPT. We then generated many other canddate Stener trees as follows: for each n N such that 0 n n SPT, we consdered all possble combnatons of n relays, and wth each of these combnatons, we constructed a shortest path tree connectng the sources to the snk. If the resultng tree volated the hop constrant h max = 5, t was dscarded; otherwse, t was accepted as a canddate soluton. For the chosen QoS targets (see Secton 7.1), we analyzed each of these canddate trees usng the detaled analyss (Secton II-B2 n [2]) for ncreasng arrval rates, startng from pkts/sec, and obtaned the maxmum arrval rate up to whch the target dscard probablty and mean delay requrements were met. The maxmum value of ths arrval rate among all the canddate trees was taken as the estmate for the optmal throughput for that nstance. Let us denote ths as ˆλ. Ths was compared aganst the throughput acheved by the ntally constructed SPT, T SPT, obtaned usng the detaled analyss n the same manner as descrbed above, and denoted by ˆλ SPT. For all the 5 nstances, t turned out that ˆλ SPT = ˆλ. The results are summarzed n Table 4. Table 4: Verfcaton of throughput optmalty of shortest path tree Scenaro Maxmum throughput among canddate trees (ncludng T SPT ) Maxmum throughput of T SPT ˆλ (pkts/sec) ˆλ SPT (pkts/sec) As an llustraton, we depct n Fgure 5, the network layout of Scenaro 2 (left panel of Fgure 5), the shortest path tree constructed from ths network graph to connect the sources to the snk (mddle panel of Fgure 5), and also one of the other canddate tree topologes that connects the sources to the snk usng ust one relay (rght panel of Fgure 5). It turns out from the numercal experments that the shortest path tree acheves a throughput of 4 packets/sec from each source (as already mentoned n Table 4), whle the other canddate topology depcted n Fgure 5 acheves a throughput of 3 packets/sec from each source. The correspondng values from our approxmate nner bound formula can be calculated to be 2.8 packets/sec and packets/sec respectvely. 23

24 Fgure 5: Depcton of the network graph, and two canddate topologes for Scenaro 2. Left panel: The network layout, consstng of 10 sources, and 30 potental relay locatons. The BS s at (0,0). Each edge has a PER of at most 2%. Mddle panel: A shortest path tree connectng the sources to the BS. Note that t uses three relays (R1, R2, and R3). Rght panel: a competng tree topology connectng the sources to the snk usng ust one relay (namely, R1). Note that t has a hgher total hop count compared to the shortest path tree Comparson aganst the outcome of the SPTRP algorthm [1] For each nstance, we also computed a hop count feasble Stener tree, T SPTRP usng the SPTRP algorthm proposed n [1] (T SPT was used as the ntal feasble soluton n runnng the SPTRP algorthm). The dea s to check how much we gan n terms of throughput by usng a shortest path desgn (wthout any constrant on the number of relays used) nstead of the SPTRP desgn whch uses a nearly mnmum number of relays. Hence, for ths Stener tree also, we computed ts throughput usng the detaled analyss n the same manner as descrbed above. Furthermore, for the chosen QoS targets, we computed the nner bound on the throughput (maxmum arrval rate) as predcted by our approxmate formula for both T SPT and T SPTRP. The results are summarzed n Table 5. From Table 5, we observe the followng: Table 5: Throughput comparson of the shortest path tree and the SPTRP desgn Scenaro Predcted λ max (pkts/sec) from formula λ max (pkts/sec) from detaled analyss T SPTRP T SPT T SPTRP T SPT As we had predcted, the shortest path tree always acheved better throughput than the outcome of the SPTRP algorthm. 2. However, even the SPTRP desgn can operate at a sgnfcant postve load, whle possbly savng on the number of relays used. Hence, when relays are costly, the desgn based on the SPTRP algorthm can be used nstead of the shortest path tree wthout much loss n throughput. 3. Fnally, comparng column 2 aganst column 4, and column 3 aganst column 5, we observe that the nner bound on the throughput regon ( Λ δ,d ) predcted by our formula are wthn 30% of the throughput regon obtaned usng the detaled analyss ( ˆΛ δ,d ). Remarks: 1. We have also performed numercal experments to see what happens to the QoS performance of the shortest path tree when the arrval rates at the sources are ndependently subected to small devatons from the maxmum 24