A statistical mechanics based framework to analyze ad hoc networks with random access.bak


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A statistical mechanics based framework to analyze ad hoc networks with random access.bak

  1. 1. 618 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 11, NO. 4, APRIL 2012 A Statistical Mechanics-Based Framework to Analyze Ad Hoc Networks with Random Access Sunil Srinivasa, Student Member, IEEE, and Martin Haenggi, Senior Member, IEEE Abstract—Characterizing the performance of ad hoc networks is one of the most intricate open challenges; conventional ideas based on information-theoretic techniques and inequalities have not yet been able to successfully tackle this problem in its generality. Motivated thus, we promote the totally asymmetric simple exclusion process (TASEP), a particle flow model in statistical mechanics, as a useful analytical tool to study ad hoc networks with random access. Employing the TASEP framework, we first investigate the average end-to-end delay and throughput performance of a linear multihop flow of packets. Additionally, we analytically derive the distribution of delays incurred by packets at each node, as well as the joint distributions of the delays across adjacent hops along the flow. We then consider more complex wireless network models comprising intersecting flows, and propose the partial mean-field approximation (PMFA), a method that helps tightly approximate the throughput performance of the system. We finally demonstrate via a simple example that the PMFA procedure is quite general in that it may be used to accurately evaluate the performance of ad hoc networks with arbitrary topologies. Index Terms—Ad hoc networks, throughput, end-to-end delay, statistical mechanics, network topology. Ç1 INTRODUCTION1.1 Motivation them using traditional methods such as information theory ad hoc network is formed by deploying nodes that becomes intractable and hence has yielded little in the wayA N possess self-organizing capabilities and typically con- of results [1]. This has motivated researchers to turn to othersists of several source-destination http://ieeexploreprojects.blogspot.comobtain ideas and methodologies that pairs communicating branches of study, towirelessly with each other in a decentralized fashion. In help better understand and characterize the dynamicalorder to conserve energy, yet efficiently deliver packets to behavior of multihop networks. Of late, statistical physicsdistant nodes, routing is often performed in a multihop has, in particular, captured the attention of the researchfashion, wherein relay nodes assist in the flow of packet community since it contains a rich collection of mathema-traffic from the sources to the destinations. Characteristi- tical tools and methodologies for studying interactingcally, the multihop nature of packet transmissions causes many-particle systems [2], [3], [4].interweaving of traffic flows, resulting in strong correlations, Along similar lines, we employ ideas from the totallyor interdependencies between the activities of the nodes. asymmetric simple exclusion process (TASEP) [5], a subfield in For instance, since a traffic flow is relayed across several statistical mechanics, to analyze multihop networks. Thehops, the packet arrival processes at the nodes (and hence, focus of this work is the investigation of the end-to-endthe departure processes) are coupled with each other. metrics, delay and throughput of ad hoc networks withThus, the end-to-end delay in multihop networks, deter- random access, taking into account the correlations in themined by the joint distribution of the successive delays of a system. Our main contributions are the following:packet traversing multiple nodes may hardly be expressedin a product form. Likewise, owing to the existence of 1. First, we consider a linear network model fed by arelay nodes that serve multiple packet flows, the through- single source and characterize its average delay andputs of the various flows in the network are correlated throughput performances. Additionally, we analyti-with each other. cally derive a) the probability mass functions (pmfs) On account of such intricate interactions, ad hoc net- of the delays incurred by packets at each node alongworks evade familiar link-based decompositions; studying the flow and b) the joint pmfs of the packet delays across adjacent nodes in the line network. 2. Second, we consider more complex ad hoc network. S. Srinivasa is with LSI Corporation, Milpitas, CA 95035. E-mail: ssunil@gmail.com. models comprising intersecting packet flows. We. M. Haenggi is with the Wireless Institute, Department of Electrical introduce an elegant technique, the partial mean-field Engineering, University of Notre Dame, Notre Dame, IN 46556. approximation (PMFA), which we employ to tightly E-mail: mhaenggi@nd.edu. approximate the throughput (and end-to-end delay)Manuscript received 5 Aug. 2010; revised 18 Jan. 2011; accepted 28 Jan. 2011; performance of such systems. We also demonstratepublished online 29 Apr. 2011. via a simple example on how to use the PMFAFor information on obtaining reprints of this article, please send e-mail to:tmc@computer.org, and reference IEEECS Log Number TMC-2010-08-0370. approach to accurately study networks with arbi-Digital Object Identifier no. 10.1109/TMC.2011.96. trary topologies. 1536-1233/12/$31.00 ß 2012 IEEE Published by the IEEE CS, CASS, ComSoc, IES, & SPS
  2. 2. SRINIVASA AND HAENGGI: A STATISTICAL MECHANICS-BASED FRAMEWORK TO ANALYZE AD HOC NETWORKS WITH RANDOM ACCESS 6191.2 Related Work also be scheduled for transmission in some time slots. This,Most earlier attempts at analyzing ad hoc networks have however, is equivalent to simply “stretching” the time axis.neglected the correlations in the system for tractability. An Also, note that the r-TDMA protocol does not entail spatialapproximation commonly used in this regard is Kleinrock’s reuse. However, in small networks (which we primarilyIndependence Assumption [6]. Accordingly, for a densely consider in this paper), spatial reuse is impractical, and theconnected network involving Poisson arrivals and uni- performance of the r-TDMA-based network is quite goodform loading among source-destination pairs, the queues (compared to other MAC schemes).at each link in the network behave independently Owing to the presence of (at most) a single transmitter inregardless of the interaction of traffic across different each slot, there is no interference in the system; thelinks. Kleinrock’s approximation has been used to probability of a successful transmission across a link,characterize the delay performance of wireless systems denoted by p , is dictated by the SNR model, i.e., s(see for example [7], [8]). Under general scenarios, ps ¼ IPðSNR > ÂÞ, for some received SNR threshold Â.however, this approximation may be very loose; thecorrelations in the system cannot be neglected. 2.2 Buffering Scheme Much of the prior work on the performance analysis of We consider the following buffering policy for each flow inmultihop networks has also focused on very small net- the network, which obeys the following two rules:works, (e.g., two-relay [9] or three-relay networks [10]).Their results, however, do not directly extend to larger 1. All the buffering in the network is performed at thenetworks. More recently, discrete-time queuing theory has source nodes, while each relay node has a bufferbeen applied to the study of end-to-end delay [11] and size of unity for each flow it is associated with. Thus,throughput [12] performances of ad hoc networks. The all the queuing occurs at the source, while relayauthors, however, focus specifically on a linear multihop nodes may hold at most one packet (per flow). Wenetwork model fed with a single flow, and do not consider also take that the source nodes are backlogged, i.e.,intersecting flows. To the best of our knowledge, this is the they always have packets to transmit.first attempt at studying the throughput performances of ad 2. Incoming transmissions are not accepted by relays ifhoc networks with arbitrary topologies. their buffer (corresponding to that flow) already The rest of the paper is organized as follows: Section 2 contains a packet.outlines the system and channel model. Section 3 providesan overview of the TASEP particle flow model, as well as These rules together mean that a successful transmissionthe matrix product ansatz (MPA), the analytical tools that may occur only when a node has a packet and its targetwe use extensively for our analysis. In Section 4, we node’s buffer is empty. http://ieeexploreprojects.blogspot.comconsider a wireless line network model, and characterize its Employing this transmission scheme has several benefitsdelay and throughput performances. Section 5 introduces such as keeping the in-network packet end-to-end delaythe PMFA framework, which we use to analyze the small and helping regulate traffic flow in a completelythroughput performances of more complex network topol- distributed manner. More details on the benefits of thisogies. Section 6 concludes the paper. “single packet buffering” policy are provided in our earlier work [13] (and references therein).2 SYSTEM MODELWe consider an ad hoc network comprising a set of source 3 PRELIMINARIESnodes intent to deliver packets to a set of destinations over We now review the totally asymmetric simple exclusionan infinite duration of time in a multihop fashion. We study process, a subfield in statistical mechanics that deals withseveral different network topologies in this paper; the the flow of particles across a lattice grid and studies theirspecifics of each topology will be described in its interactions. Later, we will use some results from thecorresponding analysis section. Time is slotted to the TASEP literature to characterize the delay and throughputduration of a packet, and packet transmissions occur at performances of ad hoc networks.slot boundaries. No power control is employed, and thetransmit power at each node is taken to be unity. 3.1 An Overview of TASEPs2.1 Channel Access Scheme The TASEP refers to a family of simple stochastic processesFor analytical tractability, we consider a modified version used to describe the dynamics of self-driven systems withof the traditional TDMA MAC scheme which we call several interacting particles and is a paradigm for none-randomized-TDMA (r-TDMA). In r-TDMA, the transmitting quilibrium systems [5]. The classical 1D TASEP model withnode in each time slot is simply chosen uniformly open boundaries is defined as follows: consider a systemrandomly from the set of all nodes in the network instead with N þ 1 sites, numbered 0 to N. Site 0 is taken to be theof being picked in an deterministic fashion (as in source that injects particles into the system. The model isconventional TDMA). said to have open boundaries, meaning that particles are The r-TDMA scheme may also be viewed as a time- injected into the system at the left boundary (site 1) and exitslotted version of the carrier sense multiple access (CSMA) the system on the right boundary (site N). The configuration ðNÞprotocol since in each time slot, only a single transmitter of site i, 1 i N at time t is denoted by i ½tŠ (or simplynode gains the right to access the wireless channel. The by i ½tŠ), which can only take values in f0; 1g, i.e., each siteonly difference between slotted CSMA and r-TDMA is that 1 i N may either be occupied (denoted as i ½tŠ ¼ 1) orin r-TDMA, nodes not having packets in their buffers may empty (denoted as i ½tŠ ¼ 0). The source, however, is taken to
  3. 3. 620 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 11, NO. 4, APRIL 2012be always occupied (0 ½tŠ 1; 8t 0). Also, at t ¼ 0, all sitesother than the source are empty (i ½0Š ¼ 0; 0 i N). In the discrete-time version of the TASEP, the move-ment of particles is defined to occur in time steps.Specifically, let ð1 ½tŠ; 2 ½tŠ; . . . ; N ½tŠÞ 2 f0; 1gN denote theconfiguration of the system in time slot t. In thesubsequent time slot t þ 1, a set of sites is chosen at first,depending on the updating procedure. Then, for every site Fig. 1. TASEP-equivalent network flow along with the (site dependent)chosen, if it contains a particle and the neighboring site on hopping probabilities p,
  4. 4. p, and p. The (backlogged) source node withits right has none, then the particle hops from that site to a large buffer connected to the TASEP particle flow model with N þ 1 sites, each with a buffer size of unity. Filled circles indicate occupiedits neighbor with a certain probability (which in general, is sites, and the others indicate holes. Jumping from site j to k is possiblesite dependent). only if the configuration ðj ; k Þ is (1, 0). In the above example, hopping is In this paper, we consider TASEPs with the random not possible between sites i and i þ 1.sequential update wherein a single site is uniformly ran-domly picked (with probability (w.p.) 1=ðN þ 1Þ) for the probability of finding the system in the configurationtransmission in each time step, and particle hopping is ¼ ð1 ; 2 ; . . . ; N Þ in time slot t. The master equationperformed as per the aforementioned rules. Formally, describes the evolution of the system with time and takessupposing that the ith site is picked in time slot t. Then, the formif 1 i N À 1, the particle on site i (if there is any) jumps X Ãto site i þ 1 (provided it is empty) w.p. p, i.e., ÁðP ð ; tÞÞ ¼ ð 0 ; ÞP ð 0 ; tÞ À ð ; 0 ÞP ð ; tÞ ; 0 IPði ½t þ 1Š ¼ 0Þ ¼ 1 À i ½tŠð1 À p þ piþ1 ½tŠÞ; where ÁðP ð ; tÞÞ ¼ P ð ; t þ 1Þ À P ð ; tÞ, and ð ; 0 Þ de- IPði ½t þ 1Š ¼ 1Þ ¼ i ½tŠð1 À p þ piþ1 ½tŠÞ; notes the rate of transition from to another configuration 0 . For further details on the master equation and itsand formulation, we refer the reader to [5], [16]. IPðiþ1 ½t þ 1Š ¼ 0Þ ¼ ð1 À iþ1 ½tŠÞð1 À pi ½tŠÞ; Interestingly, in the long time limit (t ! 1), the probability of finding the system in any configuration IPðiþ1 ½t þ 1Š ¼ 1Þ ¼ pi ½tŠ þ iþ1 ½tŠð1 À pi ½tŠÞ: becomes independent of t, i.e., limt!1 ÁðP ð ; tÞÞ ¼ 0 [5]. If i ¼ 0, site 1 remains occupied at time t þ 1 if it was The TASEP flow is then said to have reached a steady state. http://ieeexploreprojects.blogspot.comoccupied at time t and gets occupied w.p. p if it was Solving for the steady-state configuration probabilities is aempty. Accordingly, formidable task which may be accomplished by considering recursion-based techniques on the system size (see for, e.g., IPð1 ½t þ 1Š ¼ 0Þ ¼ ð1 À pÞð1 À 1 ½tŠÞ; [14], [15]). A more elegant and direct procedure, however, is IPð1 ½t þ 1Š ¼ 1Þ ¼ p þ ð1 À pÞ1 ½tŠ: to use a matrix product ansatz [16], wherein the probabilityIf i ¼ N, site N remains empty at t þ 1 if it was empty at of each configuration at steady state is decomposed into atime t, and gets emptied w.p.
  5. 5. p if it was occupied. Thus, product of matrices. According to the MPA formulation [16], the probability IPðN ½t þ 1Š ¼ 0Þ ¼ 1 À ð1 À
  6. 6. pÞN ½tŠ; of finding the TASEP system in the configuration ¼ IPðN ½t þ 1Š ¼ 1Þ ¼ ð1 À
  7. 7. pÞN ½tŠ: ð1 ; 2 ; . . . ; N Þ at steady state is independent of t and given byIn this manner, the particles are transported from site 0 Qthrough the system until their eventual exit at site N. The hW j N ði D þ ð1 À i ÞEÞjV i i¼1quantities ,
  8. 8. , and p may be regarded as the “influx,” and P ð Þ ¼ ; ð1Þ hW jC N jV i“outflux” rates and the hopping probability, respectively. It is apparent from the description of the TASEP model where D and E are square matrices that operate onthat it exhibits a similarity to a flow in an ad hoc network. occupied and empty sites, respectively, C ¼ D þ E, andThe sites can be taken to represent the relay nodes and the jV i and hW j are column and row vectors, respectivelyparticles of the packets. The hopping probability p is (represented here by the “ket” and “bra” notation). Inanalogous to the link reliability ps while the exclusion general, the matrices D; E and vectors V ; W in (1) are all 1principle models the unit buffer size at the relay nodes. infinite dimensional. A convenient choice of the matricesAlso, the random sequential update relates to the r-TDMA and vectors (assuming p 0) is [5]MAC scheme, and the condition 0 ½tŠ ¼ 1; 8t, models the fact 0 1 1=
  9. 9. 1 0 0 . . .that the source node is always backlogged. Fig. 1 depicts the B 0 1 1 0 ...CTASEP-equivalent network flow, wherein we assume that 1B 0 B 0 1 1 . . . C; Cthe source has a large buffer and regulates the packet flow D¼ B C p@ 0 0 0 1 ...Ainto a TASEP model. . . . . .. . . . . . . . . .3.2 The Matrix Product Ansatz FormulationThe starting point for studying the stochastic 1D TASEP 1. The only case for which the matrices are finite dimensional (in fact,model is to write down its master equation. Let P ð ; tÞ denote scalars) is when þ
  10. 10. ¼ 1 [16].
  12. 12. ¼ 1 and p ¼ ps , we may take 1 ¼ 1, and 2 ¼ 1 À ps in ð2Þ so that 0 1 1 1 0 0 ... B0 1 1 0 ...C 1B C 1 À ps T D ¼ B0 0 1 1 . . . C; E¼ D : ps B 0 0 0 @ 1 ...A C psFig. 2. A regular wireless line network. The backlogged source (node 0)attempts to deliver packets to the destination via N relays, each with . . . . . . . . .. . . . . .unit-sized buffer. The hopping probability across each node is ps . 0 1 For these forms of matrices D and E, and vectors W and V , ð1 À pÞ= 0 0 0 ... the following two properties hold: B 2 1Àp 0 0 ...C 1B B 0 1Àp 1Àp 0 C . . . C; C ¼ D þ E ¼ ps DE; ð4aÞ E¼ B C ð2Þ pB 0 0 1Àp 1Àp ...C @ A . . . . . . . . .. . . . . . ð2N þ 2Þ! pN hW jC N jV i :¼ ðNÞ ¼ s : ð4bÞ ðN þ 2Þ!ðN þ 1Þ!with While (4a) is straightforward to establish, (4b) is a hW j ¼ ð1; 0; 0; . . .Þ and jV i ¼ ð1; 0; 0; . . .ÞT ; consequence of [16, (80), (81)]. We use these results in thewhere ðÁÞT denotes transpose. Here, 1 and 2 may be remainder of this section.chosen so as to satisfy 4.1 Steady-State Probabilities and Occupancies 1 Using (1) along with the forms of matrices and vectors 1 2 ¼ ½1 À p À ð1 À pÞð1 À
  13. 13. pފ: discussed earlier, the steady-state probabilities can be
  14. 14. p computed in a straightforward manner, in particular for In conclusion, the MPA provides an analytical frame- small values of N. As examples, we have for N ¼ 1,work for describing the asymmetric exclusion process in acompletely algebraic manner. We will employ it extensively hW jEjV i hW jDjV ifor our analysis, in particular in the next section. P ð0Þ ¼ ¼ 1=2; and P ð1Þ ¼ ¼ 1=2: hW jCjV i hW jCjV i Likewise, one can show for N ¼ 2,4 http://ieeexploreprojects.blogspot.com THROUGHPUT AND DELAY ANALYSES OF A WIRELESS LINE NETWORK P ð0; 0Þ ¼ P ð0; 1Þ ¼ P ð1; 1Þ ¼ 1=5; and P ð1; 0Þ ¼ 2=5:We now use some existing results from the random Next, we compute the steady-state occupancy of eachsequential TASEP literature to study wireless networks. As node 0 i N, defined as the probability that it isa first step in this direction, we consider a simple linear occupied at steady state. Hereafter, we use the notation inetwork model running the r-TDMA MAC scheme and to denote the configuration of node i, 0 i N at steadyevaluate the steady-state throughput and the average end- state. From (1), we obtain the occupancy of node i to beto-end delay for a packet. Additionally, we use the MPAframework extensively to characterize the delay pmf hW jC iÀ1 DC NÀi jV iacross each hop in closed form and measure the IPði ¼ 1Þ ¼ ; 0 i N: hW jC N jV icorrelations between the delays experienced by packetsacross adjacent hops along the flow. From [14, (48)], this simplifies to The line network model considered comprises a singlesource node S intending to deliver packets to a destina- 1 1 ð2iÞ! ðN!Þ2 ð2N À 2i þ 2Þ! IEi ¼ þ ðN À 2i þ 1Þ: ð5Þtion node D in a multihop fashion via N relay nodes (see 2 4 ði!Þ2 ð2N þ 1Þ! ½ðN À i þ 1Þ!Š2Fig. 2). We take the arrangement of nodes to be a regularlattice with equal separation between adjacent nodes d. Note that since i can take values only in f0; 1g, IPði ¼The attenuation in the channel is modeled as the product 1Þ ¼ IEi and IPði ¼ 0Þ ¼ 1 À IEi .of a Rayleigh fading component and a large scale path Surprisingly, the node occupancies are independent ofloss component with exponent . Since the fading power ps . Also, notice the particle-hole symmetry,3 i.e., IEi ¼ PNis exponentially distributed, we obtain 1 À IENþ1Ài . Thus, i¼0 IEi ¼ 1 þ N=2. In a system with an odd number of relays, the middle relay has an ps ¼ Pr½SNR Š ¼ expðÀÂN0 d Þ; ð3Þ occupancy of exactly 1=2. The steady-state occupanciesfor each link in the system. This is equivalent to taking for an r-TDMA-based flow along N ¼ 5 relays in depictedp ¼ ps in the corresponding TASEP model. We also choose in Fig. 3.the (analytically tractable) operating point ¼
  15. 15. ¼ 1 forwhich the network accepts as many packets as it can (when 4.2 Steady-State Throughputthe first relay node’s buffer is empty), and also provides the We now derive the throughput of the r-TDMA-based linehighest possible service rate.2 network at steady state, defined as the average number of 2. Equivalently, the rate of packet flow across the r-TDMA-based 3. The movement of particles (packets) to the right is equivalent to thenetwork is maximized when ¼
  16. 16. ¼ 1. movement of holes (nodes with empty buffers) to the left.
  17. 17. 622 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 11, NO. 4, APRIL 2012 Corollary 4.2. For the wireless multihop network with N relays running the r-TDMA scheme, the average delay experienced by a packet at node i is 2ðN þ 1Þð2N þ 1ÞIEi IEDi ¼ ; 0 i N: ð9Þ ðN þ 2Þps Consequently, the average end-to-end delay is X N 2N 2 þ 3N þ 1 IEDe2e ¼ IEDi ¼ : ð10Þ i¼0 ps Proof. Recall that the rate of packet flow across each node is equal to T , and that the average number of packets at node i, 0 i N is IEi . From Little’s theorem [6], theFig. 3. The steady-state occupancy of nodes in an r-TDMA-based flowalong N ¼ 5 relays. Notice the particle-hole symmetry, i.e., average delay at node i is simply IEi =T . u tIEi ¼ 1 À IENþ1Ài . We see that the average end-to-end delay is proportionalpackets successfully delivered (to the destination) in a unit to the node occupancies and inversely proportional to thestep of time. We have the following result. link reliability. Also, it is interesting to note that theTheorem 4.1. For the r-TDMA-based line network with N nodes, product of throughput and delay is 1 þ N=2, which is the throughput at steady state is independent of ps . ps ðN þ 2Þ 4.4 Delay Distributions T¼ : ð6Þ In this section, we analytically derive the pmfs of the 2ðN þ 1Þð2N þ 1Þ steady-state delays incurred by packets at each node alongProof. At any instant of time, node N’s buffer contains a the linear flow, i.e., we evaluate IPðDi ¼ kÞ, k ! 1, 0 i N packet w.p. N ; furthermore, it is picked for transmission in closed form. w.p. 1=ðN þ 1Þ, and the transmission is successful w.p. To this end, suppose that a packet arrives at a node i in http://ieeexploreprojects.blogspot.com t (at steady state). The three events ps . Thus, the throughput of the line network is simply an arbitrary time slot that need to occur in the following order for the packet to be T ¼ ps IEN =ðN þ 1Þ: ð7Þ able to hop to node i þ 1 successfully are: Using (5) in (7), we obtain the desired result. u t 1. Node i þ 1 has an empty buffer. Since the network reliability is 100 percent, the through- 2. Node i is picked for transmission.put across each link is the same as specified by (7). Noting 3. Node i’s transmission is successful.that the probability that node i has a packet and node i þ 1 While (2) occurs w.p. 1=ðN þ 1Þ, (3) happens (indepen-none is IPði ¼ 1; iþ1 ¼ 1Þ ¼ IE½i ð1 À iþ1 ފ, T may also be dently of (2)) w.p. ps . Thus, at time t, if node i þ 1’s buffer isobtained using any of the N þ 1 equivalent expressions empty, the delay experienced by the packet at node i is simply geometrically distributed with mean ðN þ 1Þ=ps , i.e., T ¼ ps IE½i ð1 À iþ1 ފ=ðN þ 1Þ; ð8Þ kÀ1for any i 2 ½0; NŠ. The system throughput at steady state is ps ps IPðDi ¼ kÞ ¼ 1À :proportional to the link reliability and upper bounded by N þ1 N þ1ps =4, but decreases with increasing system size. If instead, there is another packet present in node i þ 1’s Note that instead of picking any of the N þ 1 nodes buffer, however, no packet at node i þ 2’s buffer, therandomly, if one only chooses among the nodes having a probability that the delay at the ith node is k time slots ispacket (as in CSMA), the throughput is improved by a equal to the probability that a single successful transmission Pfactor of N þ 1=ð N IEi Þ ¼ 2ðN þ 1Þ=ðN þ 2Þ, i.e., T ¼ (of the packet at node i þ 1) occurs within k À 1 slots, and i¼0ps =ð2N þ 1Þ. then the packet at node i hops in the kth time slot. Extending this argument, if j nodes adjacent to node i have4.3 Average End-to-End Delay at Steady State a packet, and the j þ 1th adjacent node has none, i.e., ifIn this paper, we are also interested in the in-network delay4measured by the number of time slots for by the packet at ðiþ1 ; . . . ; iþj ; iþjþ1 Þ ¼ ð 1; . . . ; 1 ; 0Þ; j ! 0; |fflfflfflffl{zfflfflfflffl}the head of the source’s queue to be delivered (to the j onesdestination). We first evaluate the average end-to-end delay then the packet at node i will successfully hop to nodeincurred by packets along each node in the network. Later, i þ 1 in exactly k time steps if j packets (those at nodeswe derive the complete distribution of the delays. i þ j; i þ j À 1; . . . ; i þ 1 in that order) hop within k À 1 time 4. There is no queueing delay at the source node since it is considered to slots, and then, the packet present at node i hops (in thebe always backlogged. kth time slot).
  18. 18. SRINIVASA AND HAENGGI: A STATISTICAL MECHANICS-BASED FRAMEWORK TO ANALYZE AD HOC NETWORKS WITH RANDOM ACCESS 623 Let ei;j denote the event that given a packet arrives atnode i (at some time t), j nodes adjacent to it are occupied. ðNÞWe now compute Ái;j :¼ IPðei;j Þ. We have À Á IP iþ1 ½tŠ ¼ 1; . . . ; iþj ½tŠ ¼ 1; iþjþ1 ½tŠ ¼ 0 j packet arrives at node i À Á IP iþ1 ½tŠ ¼ 1; . . . ; iþj ½tŠ ¼ 1; iþjþ1 ½tŠ ¼ 0;packet arrives at node i ¼ IP ðpacket arrives at node iÞ ¼ P ðiÀ1 ½t À 1Š ¼ 1; i ½t À 1Š ¼ 0; iþ1 ½t À 1Š ¼ 1; . . . ; iþj ½t À 1Š ¼ 1; iþjþ1 ½t À 1Š ¼ 0Þ 1  P ðiÀ1 ½t À 1Š ¼ 1; i ½t À 1Š ¼ 0Þ IPðthe packet at node i À 1 hops to node iÞ Â : IPðthe packet at node i À 1 hops to node iÞUsing the MPA formalism, we may write at steady state, Fig. 4. The pmf of the delay incurred by packets at various nodes in the line network with N ¼ 3. For this plot, all link reliabilities are taken to be ðNÞ hW jC iÀ2 DEDj EC NÀiÀjÀ1 jV i equal to ps ¼ 0:8. Ái;j ¼ hW jC iÀ2 DEC NÀi jV i iÀ1 jÀ1 NÀiÀj ð11Þ ðaÞ hW jC D C jV i The proof is presented in the appendix, available in the ¼ ; ps hW jC NÀ1 jV i online supplemental material.where in ðaÞ, we have used (4a) thrice (twice in the Theorem 4.5. The pmf of the packet delay at node i, 0 i N isnumerator term and once in the denominator term). given by ðNÞ The evaluation of Ái;j is relatively straightforward for X N Ài k À 1 jþ1small values of j. For instance, IPðDi ¼ kÞ ¼ ðNÞ Ái;j ð1 À ÞkÀ1Àj ; ð16Þ j¼0 j iÀ2 NÀiÀ1 hW jC DEEC ðNÞ jV i Ái;0 ¼ hW jC iÀ2 DEC NÀi jV i where ¼ ps =ðN þ 1Þ. hW jC iÀ1 EC NÀiÀ1 jV i ð12Þ Proof. The conditional probability IPðD ¼ k j e Þ is the ¼ hW jC NÀ1 jV http://ieeexploreprojects.blogspot.com i i;j i probability that j packets (present at nodes i þ 1; . . . i þ j) ðNÀ1Þ ¼ 1 À IEi ; hop out successfully in k À 1 time slots, and then, theand packet at node i is transmitted successfully only in the kth time slot. Hence, ðNÞ hW jC iÀ2 DEDEC NÀiÀ2 jV i Ái;1 ¼ kÀ1 j hW jC iÀ2 DEC NÀi jV i IPðDi ¼ k j ei;j Þ ¼ ð1 À ÞkÀ1Àj  : hW jC NÀ2 jV i ð13Þ j ¼ hW jC NÀ1 jV i Summing up the joint pmf IPðDi ¼ k; ei;j Þ over all the ¼ ðN À 2Þ=ðN À 1Þ: possible values of j (0 j N À i) yields the desired ðNÞ result, i.e., In order to compute Ái;j for higher values of j, we usethe following lemmas. X N ÀiLemma 4.3. The following relationship holds for j ! 2: IPðDi ¼ kÞ ¼ IPðei;j ÞIPðDi ¼ k j ei;j Þ; k 0; j¼0 ðNÞ ðNÞ ðNÀ1Þ Ái;j ¼ Ái;jÀ1 À Ái;jÀ2 ðN À 2Þ=ðN À 1Þ: ð14Þ which is equivalent to (16). u t The proof is presented in the appendix, which can be Fig. 4 plots the delay pmfs at each node in a line networkfound on the Computer Society Digital Library at http:// with N ¼ 3. Note that apart from the delay at the final relay,doi.ieeecomputersociety.org/10.1109/TMC.2011.96. none of the other delays are geometrically distributed, i.e.,Lemma 4.4. For j ! 2, we have they are not memoryless. Also note that D0 ! 2. This may be explained by the fact that whenever a packet hops out of bjÀ1c X 2 the source node (node 0), another packet arrives at the head ðNÞ ðN À k À 2Þ Ái;j ¼ ðÀ1Þk of the source. Thus, the packet at node 0 has to wait for at k¼0 ðN À 1Þ # ð15Þ least one time slot (for the packet at node 1 to hop out) jÀkÀ2 ðNÀkÀ2Þ jÀkÀ2 before attempting to hop.  IEi þ ; k kÀ1 4.5 Joint Delay Distributions ðNÞ where IEi denotes the occupancy of node i in the flow with Since the flow of packets in a wireless multihop network is N relays. relayed across multiple links, the delays experienced by a packet across hops are correlated. As mentioned earlier, the
  19. 19. 624 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 11, NO. 4, APRIL 2012study of delay correlations has often been neglected in priorwork; it is, however, crucial for the design of smarterretransmission and flow control algorithms. For instance, suppose it is known that the conditionaldelay probability IPðDiþ1 ¼ j j Di ¼ kÞ is high for somespecific value j ¼ ‘, i.e., given that a packet stayed atnode i for k slots, it is likely to be present in node i þ 1’sbuffer for ‘ slots. Node i þ 1 can then decide to hold backthe transmission of a packet for ‘ À 1 slots, thus reducingthe number of unnecessary transmissions. Knowing thespatial delay correlations also helps determine the varianceof the end-to-end delay. We begin by stating the following simple lemma.Lemma 4.6. In a multihop wireless network with N nodes, DN is independent of all the other hop delays. As a special case, when Fig. 5. The conditional delay pmfs IPðD1 ¼ ‘ j D0 ¼ kÞ for several values N ¼ 1, D0 and D1 are independent. of ‘ and k.Proof. Irrespective of the delay experienced by a packet at ðNÞ any arbitrary node, it can hop from node N to the We obtain closed-form expressions for i;j1 ;j2 considering destination (node N þ 1) if node N is picked, and its the following two cases: transmission is successful. Thus, DN follows a geometric Case 1: j1 ¼ 0. From (17), we obtain distribution with mean ðN þ 1Þ=ps and is independent of all other delays. u t ðNÞ hW jC iÀ1 EDj2 EC NÀiÀj2 À1 jV i i;0;j2 ¼ hW jC NÀ1 jV i We next compute IPðDiþ1 ¼ ‘; Di ¼ kÞ, i.e., the probabil- hW jC D EC NÀiÀj2 À1 jV i i j2ity that a packet will stay at nodes i and i þ 1 for k and ¼ ps hW jC NÀ1 jV i ð18Þ‘ slots, respectively, at steady state? The same procedure hW jC iÀ1 Dj2 þ1 EC NÀiÀj2 À1 jV imay be extended (with extra care) to evaluate the joint pmfs À ps hW jC NÀ1 jV iof the delays at nodes farther apart. ðNÞ ðNÞ http://ieeexploreprojects.blogspot.com 2 À Ái;j2 þ1 : To this end, let ei;j1 ;j2 denote the event that given a packet ¼ Áiþ1;jarrives at node i, we have Case 2: j1 0. Using the recursive equation (17), we obtain ðiþ1 ; . . . ; iþj1 ; iþj1 þ1 ; iþj1 þ2 . . . ; iþj1 þj2 þ1 ; iþj1 þj2 þ2 Þ ! 0 1 ðNÞ ðN À j1 À 1Þ ðNÀj1 Þ X ðNÀj À1þsÞ j1 1 i;j1 ;j2 ¼ Áiþ1;j1 þj2 þ Ái;sþj2 : ðN À 1Þ B C s¼2 ¼ @1; . . . ; 1 ; 0; 1; . . . ; 1; 0A: |fflfflfflffl{zfflfflfflffl} |fflfflfflffl{zfflfflfflffl} j1 ones j2 ones The following theorem establishes the joint pmfs between delays across adjacent hops in the network. ðNÞWe first evaluate i;j1 ;j2 :¼ IPðei;j1 ;j2 Þ. Using the same idea as Theorem 4.7. The joint pmf of the delays at nodes i and i þ 1 isin (11), we may write given by hW jC iÀ2 DEDj1 EDj2 EC NÀiÀj1 Àj2 À2 jV i X X 2j1 þj2 s1 s2 X ðNÞ i;j1 ;j2 ¼ ðNÞ kÀ1 hW jC iÀ2 DEC NÀi jV i IPðDiþ1 ¼ ‘; Di ¼ kÞ ¼ i;j1 ;j2 j1 ¼0 j2 ¼0 j¼j1 j hW jC iÀ1 Dj1 EDj2 À1 C NÀiÀj1 Àj2 À1 jV i ¼ : ‘À1 ps hW jC NÀ1 jV i  2j1 þj2 þ2 ð1 À Þkþ‘À2À2j1 Àj2 2j1 þ j2 À j ð19ÞSimplifying further, we get X 2j1 À1 ðNÞ k À 1 ‘ À 1 s3 X þ i;j1 ;j2 2j1 þ1 j ¼0 j¼j1 j 2j1 À 1 À j ðNÞ ðaÞ hW jC iÀ1 Dj1 À1 CDj2 À1 C NÀiÀj1 Àj2 À1 jV i 1 i;j1 ;j2 ¼  ð1 À Þkþ‘À1À2j1 ; p2 hW jC NÀ1 jV i s iÀ1 j1 þj2 À1 NÀiÀj1 Àj2 À1 where ¼ ps =ðN þ 1Þ, s1 ¼ minfk À 1; N À i À 1g, s2 ¼ ðbÞ ðN À 2Þ hW jC D C jV i ¼ ðN À 1Þ ps hW jC NÀ2 jV i minfk þ ‘ À 2j1 À 2; N À i À 1 À j1 g, and s3 ¼ minfk À 1; # ð17Þ ðk þ ‘ À 1Þ=2g. hW jC iÀ1 Dj1 À1 EDj2 À1 C NÀiÀj1 Àj2 À1 jV i þ ps hW jC NÀ2 jV i The proof is presented in the appendix, available in the ðN À 2Þ h ðNÀ1Þ ðNÀ1Þ i online supplemental material. ¼ Ái;j1 þj2 þ i;j1 À1;j2 : The conditional delay pmf may be obtained by using (19) ðN À 1Þ together with (16). Fig. 5 plots the conditional delay pmfsTo derive ðaÞ, we used (4a) for the term in the numerator, IPðD1 ¼ ‘ j D0 ¼ kÞ in a line network with N ¼ 3 andand to derive ðbÞ, we used the identity C ¼ D þ E. ps ¼ 0:8, for several values of ‘ and k.
  20. 20. SRINIVASA AND HAENGGI: A STATISTICAL MECHANICS-BASED FRAMEWORK TO ANALYZE AD HOC NETWORKS WITH RANDOM ACCESS 625 Fig. 7. The two flows S1 ! R ! D1 and S2 ! R ! D2 , each occurring via the relay node R are represented by solid and dashed arrows, respectively. When the relay node contains two packets, it routes either the packet meant for D1 w.p. q or the one for D2 w.p. 1 À q. The probability of a successful transmission across all links is ps . 5.1 Two Two-Hop Flows via a Common Relay We begin by considering the network model depicted inFig. 6. The correlation coefficients i;iþ1 , i;iþ2 , and i;iþ3 for ps ¼ 0:8 in amultihop r-TDMA-based system with N ¼ 10 relays. The delay correla- Fig. 7. It comprises two source nodes S1 and S2 (eachtions across nodes farther apart and closer to the destination are seen to numbered 0 with respect to (w.r.t) its corresponding flow)be relatively light. intending to deliver packets to destinations D1 and D2 (each numbered 2), respectively, each via a common relay4.6 Empirical Results node R (numbered 1). Here, we take that the relay node hasEvaluating the joint pmfs between delays across nodes a buffer size of two since it accommodates two flows.lying farther apart can be performed by essentially Furthermore, the r-TDMA dictates that in any time slot,following the aforementioned procedure, but it gets only one of the three nodes (S1 , S2 , or R) is (uniformly)computationally intensive and unwieldy. Instead, we resort randomly picked (w.p. 1=3) for transmission. Let ps denoteto simulation and present the behavior of the spatial delay the reliability of each link. Whenever R is picked, any onecorrelation coefficients. The correlation coefficient between of the following event occurs:delays at nodes i and j is defined as  À Áà If the relay’s buffer has no packet, it obviously does 1. IE ðDi À IEDi Þ Dj À IEDj not transmit anything. i;j ¼ ; Di Dj 2. If the relay’s buffer contains only one packet http://ieeexploreprojects.blogspot.com (intended for either of the destinations), that packetwhere Di and Dj represent the standard deviations of the is transmitted.delays at node i and j, respectively. 3. If the relay’s buffer has two packets (to be forwarded Fig. 6 plots the empirical values of correlation to both the destinations), it transmits either thecoefficients across one-, two- and three-hop neighbors in packet intended for D1 w.p. q or the packet meantan r-TDMA-based wireless network with N ¼ 10 relays for D2 w.p. 1 À q. Note that priority-based routingand ps ¼ 0:8. Observe that all the delay correlation may be modeled by setting q ¼ 1 (prioritizing thecoefficients are nonpositive. This can be explained by first flow) or q ¼ 0 (for the second flow). q ¼ 0:5noting that if the transmission of a packet is delayed at models having equal priorities for the flows.any node, the adjacent nodes’ buffers get emptied so that ½iŠ For notational convenience, let j represent the steady-the packet traverses faster across them. Likewise, if the state configurations for the buffers across the two flows,waiting time of a packet at any particular node is small, i ¼ f1; 2g, for each of the three nodes involved in each flow,the neighboring relay node buffers are still occupied and ½1Š ½2Š numbered j ¼ f0; 1; 2g. By definition, 0 ¼ 0 ¼ 1 andtherefore it takes longer for the packet to get transported ½1Š ½2Š 2 ¼ 2 ¼ 0. We shall now derive the steady-stateacross the system. Also, delays across hops closer to the throughput, T ½1Š , for the first flow; T ½2Š may simply bedestination, and delays across nodes farther apart arerelatively lightly correlated compared to the correspond- obtained by replacing q by 1 À q. Using the fact that for each flow, the throughput acrossing values near the source node. In fact, 8i, i;N ¼ 0, since each link is the same (8), we get from 1-3,DN is independent of all other delays (which is also aconsequence of Lemma 4.6).  ½1Š à  ½1Š À ½2Š Áà IE 1 À 1 ¼ IE 1 1 À ð1 À qÞ1 ð20Þ5 MORE COMPLEX TOPOLOGIES andSo far, we have only considered the linear wireless network  ½2Š à  ½2Š À ½1Š ÁÃmodel. In this section, we consider more complex ad hoc IE 1 À 1 ¼ IE 1 1 À q1 ð21Þnetworks comprising intersecting routes, i.e., networksconsisting of flows that travel through common relays. for the first and second flows, respectively. In order to solveWe also propose the partial mean-field approximation, a the above equations analytically, we can use the mean-fieldstatistical mechanics-based tool which may be used to approximation (MFA) [5], according to which all theapproximate the throughput performance of networks with correlations between the buffer occupancies are neglected.arbitrary topologies. Mathematically, the MFA takes that
  21. 21. 626 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 11, NO. 4, APRIL 2012 Fig. 9. The two three-hop flows S1 ! R1 ! R ! D1 and S2 ! R2 ! R ! D2 , each occurring via the relay R are represented by solid and dashed lines, respectively. In this case, the common relay is the node numbered 2. 5.2.1 Common Relay: Node 1 We first analyze the case wherein the common relay is the node numbered 1. Since the throughput across each link is the same (for each flow), we obtain at steady state,Fig. 8. Steady-state throughput across the first flow, T ½1Š versus q for  ½1Š À À Á ½2Š ÁÀseveral values of the link success probability ps . The results obtained ½1Š ½1Š Áà ½1Š 1 À IE1 ¼ IE 1 1 À 1 À q 1 1 À 2 ¼ IE2 ;numerically (dashed lines) closely approximate the empirical results(solid lines). and  ½jŠ ½lŠ à  ½2Š À ½jŠ ½lŠ IE i k ¼ IEi IEk ; ½2Š ½1Š ÁÀ ½2Š Áà ½2Š 1 À IE1 ¼ IE 1 1 À q1 1 À 2 ¼ IE2 :for all (valid) node pairs ði; kÞ and flow pairs ðj; lÞ. Evidently, when q ¼ 1, the second flow (the one without For this example in particular, we assume that the priority) does not affect the throughput across the first ½1Š ½2Š ½1Š ½2Š ½1Š ½1ŠIE½1 1 Š ¼ IE1 IE1 . Employing the MFA and the simpli- flow. Following (5), IE2 ¼ 2=5; T ½1Š ð1Þ ¼ ps IE2 =5 ¼ 0:08ps . ½1Š ½2Šfied notation IE1 ¼ x, IE1 ¼ y in (20) and (21), we obtain For general q, we may use the MFA to analytically ½1Š ½2Š evaluate the throughput. Indeed, setting IE1 ¼ x, IE1 ¼ y, 1 À x ¼ x À ð1 À qÞxy; ½1Š ½2Š IE2 ¼ u, and IE2 ¼ v, we obtain the following set of 1 À y ¼ y À qxy: four equations: http://ieeexploreprojects.blogspot.comSolving the above equations simultaneously, we obtain the 1 À x ¼ u;only meaningful solution as u ¼ xð1 À uÞð1 À ð1 À qÞyÞ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2q þ 3 À 4q2 À 4q þ 9 1 À y ¼ v; x¼ : 4q v ¼ yð1 À vÞð1 À qxÞ; Since the channel access probability for each node in the which may be solved numerically. It is easily seen thatsystem is 1=3, we see that the throughput for the flow when q ¼ 0, the first flow does not affect the throughput ½2ŠS1 ! R ! D1 is given by across the second flow. From (5), IE1 ¼ 3=5, so that pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi ½1Š 9 À 65 ps IE1 ps ð2q À 3 þ 4q2 À 4q þ 9Þ ½1Š IE1 ¼ % 0:234; T ½1Š ðqÞ ¼ ¼ : ð22Þ 4 3 12q ½1Š and T ½1Š ð0Þ ¼ IE1 ps =5 % 0:047ps .When q ¼ 1, i.e., when the first flow is always given priorityover the second flow, T ½1Š ð1Þ ¼ ps =6. On the other hand, 5.2.2 Common Relay: Node 2when q ¼ 0, we use the L’ Hopital rule to see that T ½1Š ð0Þ ¼ ˆ We now consider the case when the common relay is theps =9. Whenpffiffiffi both flows are prioritized equally, T ½1Š ðqÞ ¼ node numbered 2. For this scenario, we haveT ½2Š ðqÞ ¼ ps ð 2 À 1Þ=3. The achievable set of throughput for  ½1Š À ½1Š ½1Š Áà  ½1Š À ½2Š ÁÃthe first flow, T ½1Š is plotted in Fig. 8 for different values of 1 À IE1 ¼ IE 1 1 À 2 ¼ IE 2 1 À ð1 À qÞ2 ;ps . For comparison, we have also shown empirical results, andwhich match the theoretical ones (20) closely, in particularwhen ps is small. ½2Š  ½2Š À ½2Š Áà  ½2Š À ½1Š Áà 1 À IE1 ¼ IE 1 1 À 2 ¼ IE 2 1 À q2 :5.2 Two Three-Hop Flows via a Common Relay Using similar arguments as earlier, we obtain T ½1Š ð1Þ ¼We next consider the case where again, the two source ½2Š 0:08ps . When q ¼ 0, IE2pffiffiffiffiffi 3=5, and employing the MFA, ¼nodes S1 and S2 intend to deliver packets to different ½1Š we have T ð0Þ ¼ ð11 À 85Þ=30 % 0:06ps .destinations D1 and D2 , respectively. Here, however, we The above analysis suggests that the throughput acrosstake that in each flow, packets traverse two hops each, one the flow is higher when the bottleneck node is closer to theof which is the common relay. Evidently, the common relay destination (also see Fig. 10). This is explained by the factmay be the node numbered 1 or the node numbered 2 (see that the node occupancies monotonically decrease withFig. 9). The channel access probability for each node is 1=5. proximity to the destination.
  22. 22. SRINIVASA AND HAENGGI: A STATISTICAL MECHANICS-BASED FRAMEWORK TO ANALYZE AD HOC NETWORKS WITH RANDOM ACCESS 627 Fig. 12. Two multihop flows S1 ! D1 and S2 ! D2 across N1 and N2 nodes each occur via a common relay node R. The common relay is numbered n1 and n2 w.r.t. the first and second flows, respectively. Solving the above set of equations using the MFA yields ½kŠ IE1 ¼ k=ðk þ 1Þ, 1 k n. In this case, the channel accessFig. 10. Steady-state throughput across the first flow, T ½1Š versus q for probability for each node is 1=ðn þ 1Þ, so thatps ¼ 0:75 for different locations of the common relay node. The results ½kŠobtained numerically (dashed lines) closely approximate the empirical 1 À IE1 1results (solid lines). T ½kŠ ¼ ¼ ; 1 k n: ð23Þ nþ1 ðk þ 1Þðn þ 1Þ5.3 Multiple Flows via a Common Relay 5.4 The Partial Mean-Field ApproximationNext, we consider a network topology comprising multiple While the MFA tightly approximates the throughput(2) flows passing through a common relay (see Fig. 11). performance of networks comprising short flows, it canHere, the source nodes S1 ; S2 ; . . . ; Sn attempt to deliver get loose, in particular when the flows in the networkpackets to their corresponding destinations D1 ; D2 ; . . . ; Dn , traverse several nodes, since it neglects the correlationsthrough a common relay node R (that has a buffer size of n). between all the node occupancies. In this section, weWe also take that routing is priority based with packets present the partial mean-field approximation, which (as weintended for D1 having the highest priority and those meant shall see later) is more accurate than the MFA. Later, infor Dn the lowest. Thus, the relay node transmits the packet Section 5.5, we illustrate (via a simple example) how tomeant for node k, 1 k n, only when it does not have employ the PMFA framework to evaluate the throughput http://ieeexploreprojects.blogspot.comother packets corresponding to the destination nodes Dj , performance of a network with an arbitrary topology. We begin by considering a scenario where two generalj k in its buffer. Since the throughput of each flow is conserved, we multihop flows (of arbitrary lengths) both pass through a common relay node. Suppose that source node S1 deliversobtain the following set of equations: data to D1 in a multihop fashion via N1 nodes, while S2 ½1Š ½1Š forward packets to D2 via N2 relays, each via a common 1 À IE1 ¼ IE1 ; ½2Š À ½1Š Á ½2Š relay node R (see Fig. 12). We take that R is numbered 1 À IE1 ¼ 1 À IE1 IE1 ; 1 n1 N1 w.r.t. the first flow, and 1 n2 N2 w.r.t. the . . second flow. . In principle, the MFA may be used to compute the Y nÀ1 À ½iŠ Á ½nŠ 1 À IE1 ¼ 1 À IE1 IE1 : ½nŠ steady-state throughput of each flow. Indeed, we get for the i¼1 first flow ½1Š  ½1Š À ½1Š Áà 1 À IE1 ¼ IE 1 1 À 2 ¼ ÁÁÁ  ½1Š À ½1Š ÁÀ ½2Š Áà ¼ IE n1 1 À n1 þ1 1 À ð1 À qÞn2  ½1Š À ½1Š Áà ½1Š ¼ Á Á Á ¼ IE N1 À1 1 À N1 ¼ IEN1 ; and the second flow ½2Š  ½2Š À ½2Š Áà 1 À IE1 ¼ IE 1 1 À 2 ¼ ÁÁÁ  ½2Š À ½2Š ÁÀ ½1Š Áà ¼ IE n2 1 À n2 þ1 1 À qn1  ½2Š À ½2Š Áà ½2Š ¼ Á Á Á ¼ IE N2 À1 1 À N2 ¼ IEN2 : Employing the MFA, the above set of N1 þ N2 equations may be solved for the N1 þ N2 buffer occupancies, and consequently, the throughput of the networks at steady state for any 0 q 1. However, as aforementioned, theFig. 11. n flows S1 ! D1 , S2 ! D2 ; . . . ; Sn ! Dn passing through acommon relay node R. When routing, packets intended for D1 are taken MFA neglects all the correlations between the nodeto have the highest priority, and those meant for Dn , the lowest. occupancies.
  23. 23. 628 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 11, NO. 4, APRIL 2012 We now present a tighter approximation, which we termthe partial mean-field approximation, wherein the correla-tions between the occupancies of nodes involved inintersections alone are neglected.5 The basic idea behindPMFA is to “cut” the network flow into constituent linearflows, and to use the fact that the throughput across eachcut (or linear segment) in the flow is the same. To this end,we present the following lemma.Lemma 5.1. Consider an r-TDMA-based ad hoc network with N nodes (the channel access probability for each node is 1=ðN þ 1Þ). Let ps denote the packet success probability Fig. 13. A toy example consisting of two multihop flows S1 ! D1 and S2 ! D2 . The packet routing priorities at the common relay nodes R1 across each link in the network. The throughput across a cut and R5 are q1 and q2 , respectively. The dotted lines I and II represent two in the network comprising n nodes with influx and outflux cuts along the flow. rates and hopping probability ,
  24. 24. , and ps , respectively, (see Fig. 1) is given by One may then use (24) and (25) in conjunction with ½1Š 8 Lemma 5.1 to solve for the two unknowns IEn1 and ps =ðN þ 1Þ Â minf;
  25. 25. g n¼0 ½2Š IEn2 , and subsequently evaluate the throughputs across T ð;
  26. 26. ; nÞ ¼ Zð;
  27. 27. ; n À 1Þ : ps =ðN þ 1Þ Â n ! 1; the two flows. Zð;
  28. 28. ; nÞ 5.5 A Toy Example where Zð;
  29. 29. ; 0Þ ¼ 1 and We now describe how to use the PMFA to approximate the X ið2n À 1 À iÞ! ð1=
  30. 30. Þ À ð1=Þ n iþ1 iþ1 throughput performance of networks with arbitrary topol- Zð;
  31. 31. ; nÞ ¼ ; n ! 1: ogies. As intuitively expected, the PMFA method outper- i¼1 n!ðn À iÞ! 1=
  32. 32. À 1= forms the MFA method. For the purpose of illustration, weProof. Proving the case n ¼ 0 is straightforward; the rate of consider a simple example comprising two six-hop flows packet flow across the cut is the minimum of the influx across two common relays (see Fig. 13). The packet routing and outflux rates, multiplied by the channel access and priorities for the first flow S1 ! D1 at the common relay success probabilities (1=ðN þ 1Þ and ps , respectively). For nodes R1 and R5 are q1 and q2 , respectively. We evaluate the http://ieeexploreprojects.blogspot.com first flow; the computation of the the case n ! 1, the throughput across the flow is (7) throughput only for the throughput of the second flow is quite similar. ps IEn ps hW jC NÀ1 jV i The main idea to use is that for each flow, the throughput T ð;
  33. 33. ; nÞ ¼ ¼ : N þ 1 N þ 1 hW C N jV i into a common relay node equals the throughput out of it. Accordingly, we make some “cuts” along the multihop From [14, (39)], the lemma is established. u t flow, and equate the throughputs across the constituent We now show how to use the PMFA framework to linear flows. For the toy example shown in Fig. 13, we makeevaluate the throughput for the multihop network shown two cuts I and II along the flow. ½1Š ½1Šin Fig. 12. First, we cut each flow across S ! D at the Now, for notational convenience, let IE1 ¼ x, IE5 ¼ y, ½2Š ½2Šcommon relay node R to form two line network flows. IE1 ¼ z, and IE5 ¼ w. Since the rate of packet flow acrossThus, the flow S1 ! D1 is split into flows S1 ! Rn1 and each cut is the same, we obtain for the two flows, S1 ! D1Rn1 ! D 1 . Now, the flow S1 ! Rn1 may be modeled as a and S2 ! D2 ,line network flow across n1 À 1 relay nodes (considering T ð1; 1 À x; 1Þ ¼ T ðxð1 À ð1 À q1 ÞzÞ; 1 À y; 3Þ;Rn1 as the destination node for that flow); it has an influx ð1Þrate of 1 and an effective outflux rate of
  34. 34. ¼ 1 À IE . ½1Š T ð1; 1 À x; 1Þ ¼ T ðyð1 À ð1 À q2 ÞwÞ; 1; 1Þ; eff n1Likewise, for the latter flow spanning N1 À n1 relays and(through nodes Rn1 to D1 ), the effective influx rate is ð1Þ ½1Š ½2Š T ð1; 1 À z; 1Þ ¼ T ðzð1 À q1 xÞ; 1 À w; 3Þ;eff ¼ IE½n1 ð1 À ð1 À qÞn2 ފ, and the outflux rate is 1. Sincethe throughput across each cut is the same, we have T ð1; 1 À z; 1Þ ¼ T ðwð1 À q2 yÞ; 1; 1Þ; À ð1Þ Á À ð1Þ Á respectively. The above four equations may be solved to T 1;
  35. 35. eff ; n1 ¼ T eff ; 1; N1 À n1 : ð24Þ obtain the unknowns x, y, z, and w. The channel accessSimilarly, considering the second flow, we obtain probability for each node is 1=10; the steady-state through- À Á À ð2Þ Á put is T ½1Š ¼ ps ð1 À xÞ=10. ð2Þ T 1;
  36. 36. eff ; n2 ¼ T eff ; 1; N2 À n2 ; ð25Þ Alternatively, one may use the MFA to evaluate the ð2Þ ½2Š ð2Þ ½2Š ½1Šwhere
  37. 37. eff ¼ 1 À IEn2 and eff ¼ IE½n2 ð1 À qn1 ފ. throughput across the first flow at steady state. For simplicity ½1Š ½1Š ½1Š ½1Š of notation, let IE1 ¼ x1 , IE2 ¼ x2 , IE3 ¼ x3 , IE4 ¼ x4 , 5. The PMFA gives exact performance results in networks without ½1Š ½2Š ½2Š ½2Š ½2Š IE5 ¼ x5 , IE1 ¼ x6 , IE2 ¼ x7 , IE3 ¼ x8 , IE4 ¼ x9 , andintersections, i.e., for a linear flow of packets. The MFA, on the other hand, ½2Šis fairly inaccurate [14]. IE5 ¼ x10 . We obtain the following 10 equations: