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How delay scales with network size? Relation between delay and throughput? What are the tradeoffs? We developed queuing network models to analyze delay and throughput of multihop wireless ad hoc networks`qPfqPfiqPfiOutlineCIntroduction Queuing Network Model Main Results Simulation Results  DqPfDOutlineCIntroduction Queuing Network Model Main Results Simulation Results .DqPf 7#Delay in Multihop Wireless Networks$$(End-to-end delay is sum of queuing and transmission delays at intermediate nodes Queuing delay depends on Packet arrival process  how much traffic is handled by network? Node density  how many interferers are there? MAC protocol  how the channel is shared? Traffic pattern  how many times a packet is transmitted before it reaches destination Modeling all the factors is quite challengingjqPfqPf/qPfj+ # H/ (Throughput in Multihop Wireless Networks))(( Maximum achievable per node throughput of a network is the maximum rate at which the nodes of a network may generate traffic while keeping delay finite Maximum achievable throughput is inversely proportional to Average time a node takes to serve a packet Average number of flows served by a nodeXqPfUqPf&U  Related Work Gupta and Kumar  Capacity of Wireless Networks Under optimal scheduling, per node throughput scales as E.Gamal et al  Throughput Delay Trade-off in Wireless Networks D(n) = Q(n T(n)) Assuming that: Packet size scales with throughput Infinite backlog at source Centralized and deterministic scheduling Delay is proportional to number of hops traversed0qPfZ9qPfZ@qPfZ!qPfZjqPfZ3qPfZ09@ j3 OutlineCIntroduction Queuing Network Model Main Results Simulation Results <DqPf ! Network and Interference Model0 Network consists of n nodes that are distributed uniformly and independently distributed over a unit torus Transmission rate of each node = W bits/sec Interference Model: node i can successfully forward a packet to node j only if rij r(n) rjk > r(n) nodes k transmitting simultaneously with i qPfHqPf *    ,<      MAC Model ABefore transmitting a packet each node counts down a random timer The duration of the timer is exponentially distributed with mean 1/x Once the timer of a node expires it starts transmitting and at the same instant the timers of all interfering nodes is frozenNqPfAA~  Traffic ModelAEach node is source, destination and relay of traffic Size of each packet is fixed and equals L bits Each node generates packets at rate l packets/sec When a node receives a packet from its neighbor: The packet is absorbed by the node with probability p(n) (absorption probability) The packet is forwarded to a randomly chosen neighbor with probability 1-p(n) In other words, the fraction of packets received by a node that are destined to it equals p(n)qPfqPf`qPfC 3C4CC`CC[CC Queuing Network ModelIn order to characterize delay, ad hoc network modeled as G/G/1 queuing network Each node of the network is a station of queuing network Incorporate queuing delays at source and relay in delay analysis Diffusion approximation used to analyze the resulting queuing network *qPf   OutlineCIntroduction Queuing Network Model Main Results Simulation Results <DqPf#  Main Results A Mean service time ( ) Average time it takes for a node to serve a packet"MqPfMA$Interpreting the Service Time Result%% $Interpreting the Service Time Result%%  Main Results A Average end-to-end delay ( )  Average time in which packet reaches the destination after being generated at source"wqPfwA Main Results A) Maximum achievable throughput ( )"*qPf*A#Comparison with Kumar-Gupta Results$$C(X When parameters of our model are comparable to that of Kumar-Gupta model i.e. and 8PqPf qPfYAOutlineCIntroduction Queuing Network Model Main Results Simulation Results <DqPf0 Simulation ResultsA "qPfA!Conclusion and Future WorkADeveloped queuing network models for multihop wireless ad hoc networks Used diffusion approximation to evaluate average delay and maximum achievable per-node throughput Future work: extend analysis to many to one cases, taking deterministic routing into account <qPfA ASA%"Thanks!6 /#$&')+,- . / 0 1 23456789:;<=>  ` 33` Sf3f` 33g` f` www3PP` ZXdbmo` \ғ3y`Ӣ` 3f3ff` 3f3FKf` hk]wwwfܹ` ff>>\`Y{ff` R>&- {p_/̴>?" dd@,|?" dd@   " @ ` n?" dd@   @@``PR    @ ` ` p>> $(    6X  `}  T Click to edit Master title style! !  0  `  RClick to edit Master text styles Second level Third level Fourth level Fifth level!     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S  6, _P   P*    6/ _   R*  H  0޽h ? 3380___PPT10.b  0(  x  c $\0P  x  c $<"    H  0޽h ? 33___PPT10i.]+D=' = @B +  0(  x  c $`0   x  c $  H  0޽h ? ̙33___PPT10i.<+D=' = @B +  00(  x  c $0o`0   x  c $x  H  0޽h ? ̙33___PPT10i.<+D=' = @B +  ` 0(   x  c $PJ`0   x  c $$K  H  0޽h ? ̙33___PPT10i.<+D=' = @B +   00(  0x 0 c $pW`0   x 0 c $DX0P  H 0 0޽h ? ̙33___PPT10i.<+D=' = @B +   40(  4x 4 c $d`0   x 4 c $,h  H 4 0޽h ? ̙33___PPT10i.<+D=' = @B +   8l(  8x 8 c $x`0   x 8 c $x  4 8 ^Atxp_fig   SOURCE\documentclass{slides}\pagestyle{empty} \begin{document} $\Theta \bigg( \frac{W}{\sqrt{n \log{n}}} \bigg)$ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH348(BOXHEIGHT200"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH1066PICTUREFILESIZE7969H 8 0޽h ? ̙33___PPT10i.<+D=' = @B +  <0(  <x < c $@`0   x < c $  H < 0޽h ? ̙33___PPT10i.<+D=' = @B +C    Z R D (  Dx D c $`0   x D c $\0P   D ^Atxp_fig `` JB SOURCE\documentclass{slides}\pagestyle{empty} \begin{document} $\forall$ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH430(BOXHEIGHT298"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH134PICTUREFILESIZE905 D ^Atxp_figp w D< SOURCE\documentclass{slides}\pagestyle{empty} \begin{document} $\leq$ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH430(BOXHEIGHT298"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH134PICTUREFILESIZE831H D 0޽h ? ̙33___PPT10i.<+D=' = @B +    Ht(  Hl H C DA,interfering_neighbors``7  H <h)Yj V   : Neighbors of A  All nodes within distance r(n) of A + : Interfering neighbors of A  All nodes within distance 2r(n) of ARqPf qPf2Z`2 H s *3f" ` @ `2 H s *" ` 0 `2 H s *3f" ` `0 2 H 6L0Y" `@  ?A"0" H 6&Y"`P  H"0   H 007YP V sTransmission of A is guaranteed to be successful if none of the interfering neighbors of A transmit simultaneously"t0tH H 0޽h ? ̙33___PPT10i.<+D=' = @B +  LI(  Lx L c $J`0   x L c $? P  " L 6CY"` ` J$0G K L <DFY   oThe MAC model captures the collision avoidance mechanism of IEEE 802.11 and is still mathematically tractable To(qPf0oGH L 0޽h ? ̙33___PPT10i.<+D=' = @B +O  f^ P(  Px P c $`    x P c $  " P 6WY"`@ J$0G  P <4ZY@gNF p(n) characterizes the degree of locality of traffic  Low p(n) average hops between a source destination pair is large^y0G7GG:G;;H P 0޽h ? ̙33___PPT10i.<+D=' = @B +  0T0(  Tx T c $hY  Y x T c $Y Y H T 0޽h ? ̙33___PPT10i.<+D=' = @B +  @XE(  XN X C &Aqmodel X <Y   KThe queuing network0H X 0޽h ? ̙33___PPT10i.<+D=' = @B +  P\0(  \x \ c $Y`0   x \ c $$Y Y H \ 0޽h ? ̙33___PPT10i.<+D=' = @B +3  &&pdS&(  dx d c $HY`0  Y x d c $Y Y  d ^Atxp_figP f^ SOURCE\documentclass{slides}\pagestyle{empty} \begin{document} $\overline{X_{i}}$ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH348(BOXHEIGHT200"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH256PICTUREFILESIZE1798R d ^Atxp_fig`< SOURCE \documentclass{slides}\pagestyle{empty} \begin{document} $\overline{X_{i}} = \frac{1/\xi + L/W}{1 - 4nA(n)\lambda_{i}(L/W)}$ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH348(BOXHEIGHT200"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH2138PICTUREFILESIZE 15733VF P `a d   6 d ^Atxp_figP `  SOURCE\documentclass{slides}\pagestyle{empty} \begin{document} $A(n) = \pi r(n)^2$: Communication area of a node \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH348(BOXHEIGHT200"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH4668PICTUREFILESIZE 20660\ d ^A txp_figP a > SOURCE"\documentclass{slides}\pagestyle{empty} \begin{document} $\lambda_{i} = \frac{\lambda}{p(n)}$: Overall arrival rate at a node \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH348(BOXHEIGHT200"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH4018PICTUREFILESIZE 19562   d ^A txp_fig a SOURCE\documentclass{slides}\pagestyle{empty} \begin{document} $\frac{1}{\xi}$: Mean backoff duration \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH348(BOXHEIGHT200"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH2588PICTUREFILESIZE 126576  d ^A txp_figP s SOURCE\documentclass{slides}\pagestyle{empty} \begin{document} $\frac{L}{W}$: Time required to transmit a packet \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH348(BOXHEIGHT200"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH3988PICTUREFILESIZE 16916  d <|Yc  @Where,0C2  d 0Ԕ"`0` @P,$D 0  d <dY3"`@P,$@ 0 b(Service time in absence of interference)0)AB d@ <DoP`,$@ 0r d 6m"`P ,$@ 0 d <Y3"`P ,$D 0 c)Term introduced by interfering neighbors*0*AB d <DopP,$@ 0H d 0޽h ? ̙33  ___PPT10d .<+]DWD ' = @B D ' = @BA?%,( < +O%,( < +DY' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* d%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*d%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* d%(DY' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*d%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*d%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*d%(+p+0+ dY ++0+dY +   hU(  h+ h <Y" ` f  u Transmitter and receiver, in absence of interference Service time = Wait for timer to expire + transmission time =  vq̙v h jA txp_fig" p P@6 VN SOURCE\documentclass{slides}\pagestyle{empty} \begin{document} $1/\xi + L/W$ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH430(BOXHEIGHT298"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH1076PICTUREFILESIZE5076 h BDY `0  Y f2 h 0"`p P f2 h 0"`p jB h BD?P H h 0޽h ? 33___PPT10i.dⵧ+D=' = @B +  l&(  lf2 l 0"`p P f2 l 0"`p jB l BD?P f l <Y"    Now suppose there are k interferes, each with packet arrival rate a Fraction of time for which the channel is occupied by the interferers = The fraction of time the channel is available to the transmitter = In our model a = li and k = 4nA(n), therefore q̙+  l B$oY `0  Y f2 l 0"`@f2 l 0"` f2  l 0"``f2  l 0"`P f2  l 0"`0 f2  l 0"`pf2  l 0"`0 f2 l 0"`@f2 l 0"`p 0 f2 l 0"`Pf2 l 0"`f2 l 0"``@ l jA txp_fig"   XP SOURCE\documentclass{slides}\pagestyle{empty} \begin{document} $ k \alpha L/W$ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH430(BOXHEIGHT298"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH726PICTUREFILESIZE4521  l jAtxp_fig"  bbZ SOURCE\documentclass{slides}\pagestyle{empty} \begin{document} $ 1 - k \alpha L/W$ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH430(BOXHEIGHT298"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH1096PICTUREFILESIZE5131R l ^Atxp_fig< SOURCE \documentclass{slides}\pagestyle{empty} \begin{document} $\overline{X_{i}} = \frac{1/\xi + L/W}{1 - 4nA(n)\lambda_{i}(L/W)}$ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH348(BOXHEIGHT200"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH2138PICTUREFILESIZE 15733H l 0޽h ? 33___PPT10i.dⵧ+D=' = @B +y%  $$ p $(  px p c $6`0   x p c $f   p ^Atxp_figP` VN SOURCE\documentclass{slides}\pagestyle{empty} \begin{document} $\overline{D}$ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH348(BOXHEIGHT200"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH196PICTUREFILESIZE10126 p ^Atxp_figpP " SOURCE\documentclass{slides}\pagestyle{empty} \begin{document} $\overline{D} = \frac{\rho}{\lambda (1 - \hat{\rho})}$ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH348(BOXHEIGHT200"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH1106PICTUREFILESIZE5464 p <,v  @Where,0CF  ` p 0 P Z p ^Atxp_fig  < SOURCE \documentclass{slides}\pagestyle{empty} \begin{document} $\rho = \lambda_{i} \overline{X_{i}}$: Utilization factor of a node \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH348(BOXHEIGHT200"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH3778PICTUREFILESIZE 16020r  p ^Atxp_fig T SOURCE8\documentclass{slides}\pagestyle{empty} \begin{document} $\hat{\rho} = \exp \bigg( - \frac{2(1 - \rho)}{c_{A}^2\rho - c_{B}^{2}} \bigg)$ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH348(BOXHEIGHT200"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH1868PICTUREFILESIZE 12807d  p ^Atxp_fig F SOURCE*\documentclass{slides}\pagestyle{empty} \begin{document} $c_{A}^{2}$: Squared coefficient of variance (SCV) of inter-arrival time \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH348(BOXHEIGHT200"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH4688PICTUREFILESIZE 30209  p ^Atxp_fig0 `~v SOURCE\documentclass{slides}\pagestyle{empty} \begin{document} $c_{B}^{2}$: SCV of service time \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH348(BOXHEIGHT200"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH2438PICTUREFILESIZE 12302"  p 6I"`P J$0G   p <G  The value of end-to-end delay is governed by r and SCVs of service and inter-arrival times.L\0-G.GH p 0޽h ? ̙33___PPT10i.<+D=' = @B +5  LD t(  tx t c $ `0   x t c $    t ^Atxp_fig XP SOURCE\documentclass{slides}\pagestyle{empty} \begin{document} $\lambda_{max}$ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH348(BOXHEIGHT200"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH466PICTUREFILESIZE2892h t ^Atxp_fig R SOURCE6\documentclass{slides}\pagestyle{empty} \begin{document} $\lambda_{max} = \frac{p(n)}{\frac{1}{\xi} + \frac{L}{W} + 4nA(n)\frac{L}{W}}$ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH348(BOXHEIGHT200"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH2208PICTUREFILESIZE 15972 t <  9 m =or,0C  t <|@ 0  MWhere, *0CA t ^Atxp_fig  l SOURCEP\documentclass{slides}\pagestyle{empty} \begin{document} $\overline{s}$: Average number of hops traversed by a packet between source and destination \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH348(BOXHEIGHT200"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH4668PICTUREFILESIZE 38559"  t 6h"` ` J$0G   t <Բ @9& hAs expected, MAT varies inversely with mean path length, node density and communication radius of nodes$i0iGh  t jAtxp_fig" 00 !8 SOURCE\documentclass{slides}\pagestyle{empty} \begin{document} $\lambda_{max} = o\left( \frac{W}{\overline{s}(4nA(n))L} \right)$ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH348(BOXHEIGHT200"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH2098PICTUREFILESIZE 13488H t 0޽h ? ̙33___PPT10i.<+D=' = @B +L  c[ x(  xx x c $X`0   x x c $,   x ^Atxp_figwx SOURCE\documentclass{slides}\pagestyle{empty} \begin{document} $p(n) = \sqrt{\frac{\log(n)}{n}}$ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH348(BOXHEIGHT200"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH1478PICTUREFILESIZE 10102 x ^Atxp_figph SOURCE\documentclass{slides}\pagestyle{empty} \begin{document} $A(n) = \frac{\log(n)}{n}$ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH348(BOXHEIGHT200"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH1326PICTUREFILESIZE8028 x ^Atxp_fig   SOURCEt\documentclass{slides}\pagestyle{empty} \begin{document} $\lambda_{max} = \frac{\frac{1}{4 \pi} \frac{W}{\sqrt{n \log{n}}L}}{1 + \frac{1/\xi + L/W}{4\pi \log{n}L/W}}$ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH348(BOXHEIGHT200"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH1948PICTUREFILESIZE 19776 x <H@  <or0C B x ^Atxp_fig >4 , SOURCE\documentclass{slides}\pagestyle{empty} \begin{document} $\lambda_{max} = o \bigg( \frac{W}{\sqrt{n \log{n}}}\bigg)$ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH348(BOXHEIGHT200"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH1758PICTUREFILESIZE 11115"  x 6"` ` J$0G 1  x <Ⱥ & The bound is similar to Gupta-Kumar bound but is not achievable. This is expected as channel capacity is wasted due to random access.$0GH x 0޽h ? ̙33___PPT10i.<+D=' = @B +  @0(  x  c $0׵`0   x  c $׵  H  0޽h ? ̙33___PPT10i.<+D=' = @B +  `(  x  c $`0   x  c $`  F  C A+q1@ r F  C A,q2@ B   0<ϵ   l0Comparison of theoretical and simulation results 101A  <Hٵ `f s3Diffusion Approximation yields pretty good results.$404GH  0޽h ? ̙33___PPT10i.<+D=' = @B +  p0(  x  c $   x  c $d  H  0޽h ? ̙33___PPT10i.<+D=' = @B +   (  x  c $, ``    H  0޽h ? 3380___PPT10. _ 0 P((  ^  S      c $lG 0    H  0޽h ? 3380___PPT10.bp/ 0 p$((  $^ $ S     $ c $T 0    H $ 0޽h ? 3380___PPT10.bp/ 0 @((  @^ @ S     @ c $ 0    H @ 0޽h ? 3380___PPT10.bp/ 0 ``((  `^ ` S    Y ` c $dY 0  Y  H ` 0޽h ? 3380___PPT10.bp/ 0 P((  ^  S      c $$ 0    H  0޽h ? 3380___PPT10.bp/ 0 ((  ^  S      c $T 0    H  0޽h ? 3380___PPT10.bp/  0 (  X  C      S ( 0    H  0޽h ? 3380___PPT10. >T  0 (  X  C      S `b 0    H  0޽h ? 3380___PPT10. @T  0 (  X  C      S Xv 0    H  0޽h ? 3380___PPT10. @T  0 (  X  C      S H 0    H  0޽h ? 3380___PPT10. @T  0 (  X  C      S H!Y 0    H  0޽h ? 3380___PPT10. @T   0 (  X  C      S =Y 0    H  0޽h ? 3380___PPT10. @T   0 (  X  C      S PRY 0    H  0޽h ? 3380___PPT10. @T   0  (  X  C    Y  S Y 0  Y  H  0޽h ? 3380___PPT10. @T   0 0(  X  C    Y  S Y 0  Y  H  0޽h ? 3380___PPT10. @T   0 @(  X  C      S Y 0    H  0޽h ? 3380___PPT10. @T  0 P(  X  C    Y  S (Y 0  Y  H  0޽h ? 3380___PPT10. FCT  0 `(  X  C    Y  S Y 0  Y  H  0޽h ? 3380___PPT10. FCT  0 p(  X  C      S c 0    H  0޽h ? 3380___PPT10. FCT  0 (  X  C      S 0 0    H  0޽h ? 3380___PPT10. FCT  0 (  X  C      S  0    H  0޽h ? 3380___PPT10. FCT  0 (  X  C      S ͵ 0    H  0޽h ? 3380___PPT10. FCT  0 (  X  C      S , 0    H  0޽h ? 3380___PPT10. FCT  0 (  X  C      S @ 0    H  0޽h ? 3380___PPT10. FCT  0 (  X  C      S  0    H  0޽h ? 3380___PPT10. FCTr`=GMRhUWZ]dygs{C+ъbN  '``ACHjK"MBO& bQS)`UFPWY[ ^`2bFdZfnhjlnprtvy"{6}J? ^>Oh+'0P px    PowerPoint PresentationoweoweAAe4eMicrosoft PowerPointon@p|@@@@&J_ AG& g  9  -- @ !--'@Times New Roman-. 92 a!Queuing Network Models for Delay (('2(."System4-@Times New Roman-. 2 ? Analysis of '.-@Times New Roman-. 2 LMultihop2.-@Times New Roman-. 2 *Wireless Ad Hoc 4'( .-@Times New Roman-. 2 @Networks('.-@Times New Roman-. 2  NabhendraA .-@Times New Roman-. 2 `Bisnik  .-@Times New Roman-.  2 and  .-@Times New Roman-. 2  AlhusseinA  .-@Times New Roman-. 2 Abouzeid .-@Times New Roman-. 72  Rensselaer Polytechnic Institute        .-@Times New Roman-. 2 *Troy, NY  .-@Times New Roman-. 2 Xbisnin@rpi.edu  $  .--]]-@Times New Roman-.  2 X, .-@Times New Roman-. '2 Xabouzeid@ecse.rpi.edu $  .--]]3-՜.+,D՜.+,x4    On-screen Shown-sA ArialTimes New Roman WingdingsSymbolDefault DesignOQueuing Network Models for Delay Analysis of Multihop Wireless Ad Hoc NetworksImportant QuestionsOutlineOutline$Delay in Multihop Wireless Networks)Throughput in Multihop Wireless Networks Related WorkOutlineNetwork and Interference Model Slide 10 MAC ModelTraffic ModelQueuing Network Model Slide 14Outline Main Results%Interpreting the Service Time Result%Interpreting the Service Time Result Main Results Main Results$Comparison with Kumar-Gupta ResultsOutlineSimulation ResultsConclusion and Future WorkThanks!  Fonts UsedDesign Template Slide Titles<(PX4 _PID_HLINKSVersionA mailto:bisnin@rpi.edumailto:abouzeid@ecse.rpi.edu_NAAAA  !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnpqrstuvxyz{|}~Root EntrydO)Pictures0XCurrent UserSummaryInformation(oPowerPoint Document(rDocumentSummaryInformation8w