Analytical Study of Incremental Approach for Information Dissemination in Wireless Networks
Andrey Belogaev, Evgeny Khorov, Artem Krasilon, Andrey Lyakhov
AAnalytical Study of Incremental Approach forInformation Dissemination in Wireless Networks
Andrey Belogaev ∗ , Evgeny Khorov ∗† , Artem Krasilov ∗† , Andrey Lyakhov ∗∗ Institute for Information Transmission Problems, Russian Academy of Sciences, Moscow, Russia † National Research University Higher School of Economics, Moscow, RussiaE-mail: { belogaev, khorov, krasilov, lyakhov } @iitp.ru ©2018 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, includingreprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, orreuse of any copyrighted component of this work in other works. Abstract —In many scenarios, control information dissemina-tion becomes a bottleneck, which limits the scalability and theperformance of wireless networks. Such a problem is especiallycrucial in mobile ad hoc networks, dense networks, networksof vehicles and drones, sensor networks. In other words, thisproblem occurs in any scenario with frequent changes in topologyor interference level on one side and with strong requirementson delay, reliability, power consumption, or capacity on theother side. If the control information changes partially, it maybe worth sending only differential updates instead of messagescontaining full information to reduce overhead. However, suchan approach needs accurate tuning of dissemination parameters,since it is necessary to guarantee information relevance in error-prone wireless networks. In the paper, we provide a deep studyof two approaches for generating differential updates — namely,incremental and cumulative — and compare their efficiency. Weshow that the incremental approach allows significantly reducingthe amount of generated control information compared to thecumulative one, while providing the same level of informationrelevance. We develop an analytical model for the incrementalapproach and propose an algorithm which allows tuning itsparameters, depending on the number of nodes in the network,their mobility, and wireless channel quality. Using the developedanalytical model, we show that the incremental approach is veryuseful for static dense network deployments and networks withlow and medium mobility, since it allows us to significantly reducethe amount of control information compared to the classical fulldump approach.
Index Terms —Wireless multihop networks, dense deployment,control information dissemination, full dump, differential update.
I. I
NTRODUCTION
In many existing and emerging scenarios, it is highlynecessary for the networking devices to have relevant infor-mation from their neighbors. For example, various routingprotocols for mobile ad hoc networks (MANET) exchangeinformation about links and their quality [1] to distributivelybuild a graph representing network topology. Outdated routinginformation may lead to loops and user data losses. Anotherexample is dense deployment, where access points may reduceinterference by coordinating transmissions in their networkswith a sort of time division [2], [3], dynamic sensitivity control[4] or adaptive transmission power control [5] enabled by thenovel IEEE 802.11ax amendment. Information disseminationis especially crucial for vehicular networks [6].
The research was done at IITP RAS and supported by the Russian ScienceFoundation (agreement No 14-50-00150).
Often, both user and control information share the samechannel resources. Thus, efficient control information dissem-ination increases capacity for user traffic, in addition to im-proving scalability in terms of both the number of nodes whichgenerate information updates and the rate of such updates.This makes the problem of control information disseminationcrucial for narrow-band networks (see [7]) and networks withhighly mobile devices.Typically, in ad hoc networks, control information is broad-cast without being acknowledged. Because of error-pronenature of the wireless channel, broadcast messages can belost. That is why in many protocols (e.g. OLSR [1], RR-ALOHA [8]), such broadcast messages contain full dumps of information and are sent periodically, even if no changesoccur in the network. The period of such messages is chosenas a trade-off between information relevance and channel timeconsumption. Despite the simplicity of the full dump approach,it is robust to packet losses since the neighboring nodes willrecover lost information with the next successfully receivedmessage. Moreover, when a new node appears in the network,it receives all necessary information in the first receivedmessage. The cost for these benefits is a huge overhead (e.g.see [9]).Another approach referred to as group-based one is pro-posed in the IEEE 802.11-2012 standard [10] for the determin-istic channel access protocol called Mesh coordination func-tion Controlled Channel Access (MCCA, for details see [11]).According to this approach, a node divides various piecesof information (information elements) into a relatively smallnumber of groups and periodically sends information onlyabout those groups that have been changed. Specifically, withMCCA protocol, a station periodically sends information abouttime intervals reserved for transmissions [12]. In [13], [14]various group management algorithms are studied and com-pared in terms of the amount of control information, using thedeveloped analytical models and simulations. However, thesepapers do not take into account the relevance of informationat neighboring nodes, since it is assumed that all controlmessages are transmitted reliably.The popular approach to reduce control information is tointerleave full dump messages with short differential updates,which contain only modified information. This approach canbe implemented in two ways: with cumulative and incre-mental updates. The cumulative differential updates contain a r X i v : . [ c s . N I] A ug ll information elements modified since the last full dumpmessage. Such an approach is used by the DSDV [15] routingprotocol. In contrast, incremental differential updates containonly information elements modified since the previously trans-mitted message (a full dump or an incremental message). Thisapproach is adopted by the OSPF-MDR [16] and the PSR [17]routing protocols. Obviously, when all control messages aretransmitted reliably, the cumulative approach produces morecontrol information than the incremental one. However, afailure in differential update transmission for the incrementalapproach may lead to irrelevant information, while, for the cu-mulative approach, any successfully received control messagemakes the information relevant, except for the case when afull dump message has been lost.In this paper, we compare the two approaches and showthat the incremental approach produce less control informationcompared to the cumulative one at the same information rele-vance level. We develop an analytical model of the incrementalapproach, which allows finding its optimal parameters in termsof the minimal amount of control information subject to somepredefined probability of information relevance. Using thedeveloped analytical model, we show that the incrementalapproach is very useful for static dense network deploymentsand networks with low and medium mobility, since it allowssignificantly reduce the amount of control information com-pared to the classical full dump approach.The rest of the paper is organized as follows. In Section II,we specify the considered network scenario. Section III com-pares the cumulative and incremental approaches by means ofsimulation. In Section IV, we develop an analytical model ofincremental approach which allows estimating the amount ofgenerated control information and the probability of informa-tion relevance. Based on this model, we provide an algorithmfor selecting the optimal parameters. In Section V, we validateour model by simulation and evaluate the performance of theincremental approach and the developed algorithm using thismodel. Section VI concludes the paper.II. S CENARIO
In this paper, we consider the following scenario. Each nodeof a wireless network periodically broadcasts to its neighborscontrol messages of two types: full dump and differentialupdate messages. We refer to the period of control messagetransmission as a slot . We assume that a node transmits acontrol message at the beginning of each slot. To reducethe amount of sent control information, the node interleavesdifferential update messages and the full dump ones. Let N be the transmission period for the full dump messages, i.e., anode transmits full dumps at the beginning of every N -th slot,while all other messages are differential updates.The size of control messages and their loss probabilitiesdepend on the rate of appearing new information elements(e.g., appearing of new scheduling information for channelaccess protocols or new links and metric updates for routingprotocols) and their lifetimes. In this paper, we assume thatthe number of information elements appearing at a node in a time slot has a Poisson distribution with rate λ . The lifetimeof each information element has an exponential distributionwith mean /µ slots. The size of each information element isconstant and equals V bits. Similar to our previous works [13],[14], we assume that the average lifetime of an informationelement is much longer than one slot, i.e. /µ (cid:29) . Also,we limit the number of information elements tracked by eachnode with threshold R .Let us consider a node (referred to as node A) having M neighbors. We assume that the wireless channel is errorprone and the probability that neighbor i cannot decodea control message from node A containing s informationelements equals p ( i ) err ( s ) . For definiteness, we estimate p ( i ) err ( s ) as follows: p ( i ) err ( s ) = 1 − (1 − ber ( i ) ) sV , (1)where ber ( i ) is the Bit Error Rate (BER) at node i . However,any other dependency can be considered as well.We assume that the network topology is dynamic, i.e., thenodes are mobile. Hence, node A loses connections with itsneighbors when they move far away and establishes newconnections with the nodes appearing in its coverage area.In this paper, we assume that the duration of the connectedphase for each neighbor (i.e. the time interval during which itis connected to node A) has the exponential distribution withmean /γ .When a neighbor fails to receive an update from node A, theinformation at this neighbor becomes irrelevant. To obtain up-to-date information, the neighbor needs to successfully receivea full dump message from node A in case of the incrementalapproach or any type of control message in case of the cu-mulative one. Let us define information relevance probabilityas the probability that at any arbitrarily selected time slot all connected neighbors have relevant information generated bynode A. To provide correct operation of a protocol whichdisseminates control information, we should guarantee that therelevance probability of that control information is higher thansome threshold p thresh .To increase the relevance probability, nodes use unsolicitedretries scheme. It means that nodes broadcast messages sev-eral times in row. Let us denote n f and n d the numberof transmission attempts for the full dump and differentialupdate messages, respectively. These two variables togetherwith period N of full dump messages are considered asthe parameters and can be tuned in order to minimize theamount of generated control information and to keep therelevance probability above the threshold p thresh . We estimatethe amount of control information as the average number ofinformation elements which are sent at the beginning of a slot.III. P RELIMINARY ANALYSIS
In this section, we compare incremental and cumulativeapproaches for generating differential updates. For that, weuse an event-driven custom simulation program, run exper-iments in the scenario described in Section II, and averagethe obtained results over simulation runs. We consider ig. 1. Comparison of incremental and cumulative approaches. transmission of control information via a single link (i.e. M = 1 ). For higher values of M , we obtain similar re-sults. Unless otherwise stated, further we set: R = 1000 , γ = 0 . , V = 2 bytes. BER on the considered link isset to ber = 6 . · − , which means that the probabilityof incorrect reception of a message containing R informationelements equals p err ( R ) ≈ .For each of two approaches, we find via the exhaustivesearch such triple ( N ∗ , n ∗ d , n ∗ f ) that minimizes the amount ofcontrol information subject to the relevance probability beinghigher than p thresh = 0 . . Fig. 1 shows the average amountof control information for optimal triple (for each λ and µ )as a function of the load defined as λµR . We can see that forthe load higher than . , the amount of control informationdoes not change since the number of information elementstracked by node A is close to R and cannot further increase.Note that the curves corresponding to the cumulative approachexperience a significant growth at the load of . .. . . Atthis point, the cumulative approach — which is more robustto packet losses by design — needs to increase the numberof retries for full dumps and/or differential updates in orderto satisfy the given requirement on the information relevanceprobability. Certainly, the incremental approach also increasesthe number of retries with the load. However this leads only toa slight increase of the amount of control information, sincethe size of differential update messages for the incrementalapproach is much lower than for that of the cumulative one.From the presented above results, we can conclude thatthe incremental approach outperforms the cumulative one:for all the considered loads and µ values, the incrementalapproach generates three times less control information. So,further in this paper, we focus on the incremental approachand develop an analytical model which allows selecting itsoptimal parameters (i.e. N , n d and n f ).IV. A NALYTICAL MODEL
Let us develop an analytical model of the incrementalapproach which allows estimating the average amount ofgenerated control information and the information relevance probability. Based on this model, we propose an algorithm tooptimize its performance in terms of minimizing the amountof control information subject to a given requirement oninformation relevance.
A. Estimation of the average amount of control information
Let us consider node A and estimate the average amount ofthe generated control information. For that we need to findstationary probabilities π r that the node has r informationelements. In our previous paper [13], we have developed ananalytical model that allows estimating the average amount ofgenerated control information for the full dump approach (i.e.for the case when N = 1 ). The model is based on a discretetime Markov chain with state r and the time unit equal to theslot. For that chain, we have found the stationary probabilities π r , which we further use in this paper.Following [13], let us introduce the following notation: d is the number of information elements deleted in the currentslot; n is the number of new information elements added inthe current slot.Since the lifetime of each information element has expo-nential distribution, the probability of deleting a particularinformation element during a slot equals ˜ p = 1 − e − µ .As lifetimes of different information elements are mutuallyindependent random variables, the probability of deleting d out of r information elements equals p d | r = C dr ˜ p d (1 − ˜ p ) r − d . (2)Using π r and (2), we can find the average numbers E [ r ] and E [ d ] of the total and deleted information elements, re-spectively: E [ r ] = R (cid:88) r =0 rπ r ,E [ d ] = R (cid:88) d =0 d R (cid:88) r = d π r · p d | r . At the steady state, the average number of informationelements E [ n ] added during a slot equals the average numberof information elements E [ d ] deleted during a slot. Since everydifferential update message contains information elements,which are added and deleted during the slot, the average size ofa differential update message equals E [ d ] . Since a full dumpmessage contains all information elements, its average sizeequals E [ r ] . Taking into account that each full dump messageand each differential update message are repeated n f and n d times, respectively, we can estimate the average amount ofcontrol information as follows: E [ V ] = 1 N n f E [ r ] + 2 N − N n d E [ d ] . (3) B. Information relevance probability
Now let us estimate the information relevance probability.First, we consider a pair of nodes (e.g. node A transmitscontrol messages to node B) and estimate the probability p rel that the control information obtained at node B is relevant.After that, we generalize the result for the case when node Ahas several neighbors.et us calculate the information relevance probability withinthe connected phase of node B (see Fig. 2). For that, weconsider N consecutive slots (the full dump cycle) and assumethat at the beginning of the first slot, node A sends a fulldump message. We can consider different full dump cyclesindependently since only the full dump message sent at thebeginning of each full dump cycle can recover informationrelevance at node B. Hence, we only need to estimate therelevance probability ˆ p rel within one full dump cycle.From [13], the conditional probability of adding n newinformation elements for given r and d values equals: p n | r,d = p k ( n ) , n < R + d − r, − n − (cid:80) k =0 p k ( k ) , < n = R + d − r, , n = R + d − r, (4)where p k ( k ) = λ k k ! e − λ .Using stationary probabilities π r and equations (2) and (4),we can estimate loss probability p f after n f retries of a fulldump message and loss probability p d after n d retries of adifferential update message as follows: p f = (cid:34) R (cid:88) r =0 π r · p err ( r ) (cid:35) n f ,p d = R (cid:88) r =0 π r r (cid:88) d =0 p d | r R − ( r − d ) (cid:88) n =0 p n | r,d · p err ( d + n ) n d , where p err ( r ) is calculated according to equation (1) (we omitindex i here).If a full dump message is lost, the information at nodeB is irrelevant for the whole full dump cycle, i.e. N slots.Otherwise, if the i -th differential update (out of N − differential updates) is the first lost control message withinthe full dump cycle, the information at node B is relevant for i time slots and irrelevant for the other N − i slots. Hence, therelevance probability within one full dump cycle equals ˆ p rel = p f · − p f ) · (cid:20) N − (cid:80) i =1 (1 − p d ) i − p d · iN + (1 − p d ) N − · (cid:21) .After summing up, we obtain: ˆ p rel = (1 − p f )(1 − (1 − p d ) N ) N p d . (5)According to the considered scenario, node B can be in twostates, connected and disconnected. The average duration ofconnected phase equals γ . Since the duration of the full dumpcycle N should be small enough compared to the duration ofconnected phase to meet information relevance requirement p rel ≥ p thresh (note that p thresh is close to ), we canwrite γ >> N . Hence, the probability that node B hasrelevant information subject to the condition that node B isin connected state and node A has already sent full dumpmessage equals ˆ p rel calculated according to equation (5). Thestartup period, i.e. the period between the time when node Bconnects to node A and the time of first full dump message tt... F FD ...
D Dconnects to A startup full dump cycle
F F F F F ... ... ... ... ... ... connected phase node A:node B: disconnects from A
Fig. 2. Connected phase. transmitted by node A (see Fig. 2) has uniform distributionover the interval [0 , N ] with the mean value N . Thus, theinformation relevance probability at node B — when it is inthe connected state — can be calculated as follows: p rel = ˆ p rel (cid:16) γ − N (cid:17) / γ . (6)Now let us consider the case when node A disseminates con-trol information to M neighbors. Assuming that all neighborsprocess messages and move independently, we can calculatethe probability that the control information is relevant at all neighbors as follows: p ( all ) rel = M (cid:89) i =1 p ( i ) rel , where p ( i ) rel is calculated according to equation (6). C. Asymptotic analysis
Let us study the asymptotic case when λ → ∞ . In this case,the total number (and the average number E [ r ] ) of informationelements r in every slot is constant and equals R .When in a particular slot d information elements are deleted,the same number of information elements are added. Hence,for the loss probabilities p f and p d and the average numberof deleted information elements per slot E [ d ] we have: p f = [ p err ( R )] n f , p d = (cid:34) R (cid:88) d =0 p d | R · p err (2 d ) (cid:35) n d ,E [ d ] = (1 − e − µ ) R. Therefore, the average amount of control information canbe recalculated as follows: E [ V ] = 1 N n f R + 2 N − N n d R (1 − e − µ ) . Thus, for the considered asymptotic case, we obtain closed-form expressions which allow estimating the average amountof control information and the relevance probability withoutneed in solving the Markov chain and finding stationaryprobabilities π r , which significantly reduces computationalcomplexity. D. Tuning parameters
The performance of the incremental approach depends onits parameters (i.e. N , n f and n d ). These parameters shouldbe selected in order to minimize the amount of controlinformation V subject to p ( all ) rel ≥ p thresh . The latter inequalityimposes a restriction on N , since for each node the probabilitythat information is irrelevant cannot be less than the ratio of ig. 3. Validation of the developed analytical model. the duration of startup period and the duration of connectedphase. Hence, the following condition for N should hold: (1 − N/ /γ ) M ≥ p thresh . Thus, the maximal value N max forparameter N , for which the condition p ( all ) rel ≥ p thresh holds,can be found as follows: N max = (cid:22) − M √ p thresh ) γ (cid:23) . (7)We propose the following algorithm for selecting the opti-mal parameters for the incremental approach:1) Find the maximal value N max according to (7).2) Find triples ( N, n f , n d ) , for which condition p ( all ) rel ≥ p thresh holds, e.g., using the exhaustive search ( N = { ..N max } , n f and n d within reasonable limits, e.g. n f , n d = { .. } ),3) Choose from the triples found at step 2 the triple ( N ∗ , n ∗ f , n ∗ d ) providing the minimal amount of controlinformation V according to (3).In Section V, we also consider triple ( ˜ N , ˜ n f , ˜ n d ) , which isfound with the same algorithm for the case λ → ∞ , and showthat this triple provides close to optimal results.V. N UMERICAL RESULTS
A. Model validation
Let us estimate the accuracy of the analytical model de-veloped in Section IV. For that, we vary µ and load, andcompare the average amount of control information sent perslot obtained with the analytical model and simulations. Weconsider the optimal parameter values ( N ∗ , n ∗ f , n ∗ d ) , whichare chosen according to the algorithm in Section IV. Wealso consider the optimal parameter values ( ˜ N , ˜ n f , ˜ n d ) forthe asymptotic case λ → ∞ . As in the preliminary analysispresented in Section III, we validate the analytical model in asingle link scenario. We set threshold p thresh = 0 . , mobility γ = 0 . and BER corresponding to p err ( R ) = 10% .Fig. 3 shows that the curves obtained with analytical andsimulation models almost coincide. Since the algorithm ofselection ( N ∗ , n ∗ f , n ∗ d ) requires an accurate estimation forboth information relevance probability p rel and the amount Fig. 4. Performance of the incremental approach depending on the scenarioparameters. of control information V , we can conclude that the developedanalytical model provides high accuracy.We can see that the amount of sent control informationincreases with the load and reaches its maximal value at highload (load > . ), where the estimations provided by theanalytical model (solid line) and asymptotic analysis (dashedline) coincide. Note that parameter configuration ( ˜ N , ˜ n f , ˜ n d ) provides results very close to optimal ones for medium andhigh load values. Specifically, for λµ · R > = 0 . the amount ofcontrol information for both configurations is almost the same.Moreover, closed-form expressions obtained in Section IVallow tuning parameters online. Hence, further we investigatethe incremental approach in the asymptotic case λ → ∞ . B. Performance evaluation
Now let us evaluate the performance of the incrementalapproach combined with the algorithm for tuning its param-eters, based on the analytical model. The performance of theapproach depends on the scenario parameters, such as mobility γ , number of neighboring nodes M , BER at each link ber ( i ) and information elements generation parameters λ and µ . Dueto the lack of space, in all the experiments below we set µ = 0 . . However, our experiments show that for other valuesof µ the results are quite similar.Fig. 4 shows the dependencies of the amount of controlinformation V and optimal full dump cycle duration ˜ N onobility γ for M = { , , } . For all nodes, we set BER tothe same value, which corresponds to p err ( R ) = { , } .Specifically, in Fig. 4(a) we consider the ratio of the amountof control information V for the incremental approach withparameter configuration ( ˜ N , ˜ n f , ˜ n d ) to the amount of controlinformation V full dump for the full dump approach. Here, thefull dump approach means that the node broadcasts full dumpmessages in every slot and does the minimal number of retriesin order to satisfy requirement p ( all ) rel ≥ p thresh . Note that thefull dump approach is the special case of the incremental onewith the full dump cycle set to . We can see that V rises with γ as higher mobility requires more frequent full dump messagetransmissions. Besides, we can see that V also rises with BERsince the node has to increase the number of retries n f and n d for the messages. At a particular level of mobility, the amountof control information generated by the node increases withthe number of neighboring nodes, because it needs to maintaina given information relevance level at all neighbors. Fig. 4(b)shows that the full dump cycle duration ˜ N decreases whenmobility, BER and the number of stations increases.It should be noted that N max calculated according to (7) canreach , i.e., at some critical level of mobility, a node cannotfurther tune parameters to satisfy condition p ( all ) rel ≥ p thresh .Consequently, we can see that curves have no points in highmobility area γ > γ critical . Specifically, the less the numberof neighboring stations, the higher is the critical level γ critical .So, we can conclude that the incremental approach allowssignificantly reducing the amount of generated control infor-mation compared to the classical full dump approach, whileproviding a high level of information relevance for densenetwork deployments with low and medium mobility.VI. C ONCLUSION