Packet Latency of Deterministic Broadcasting in Adversarial Multiple Access Channels
Lakshmi Anantharamu, Bogdan S. Chlebus, Dariusz R. Kowalski, Mariusz A. Rokicki
aa r X i v : . [ c s . D C ] M a r Packet Latency of Deterministic Broadcastingin Adversarial Multiple Access Channels ∗ Lakshmi Anantharamu † Bogdan S. Chlebus † Dariusz R. Kowalski ‡ Mariusz A. Rokicki ‡ Abstract
We study broadcasting in multiple access channels with dynamic packet arrivals and jam-ming. Communication environments are represented by adversarial models that specify con-straints on packet arrivals and jamming. We consider deterministic distributed broadcast al-gorithms and give upper bounds on the worst-case packet latency and the number of queuedpackets in relation to the parameters defining adversaries. Packet arrivals are determined bya rate of injections and a number of packets that can be generated in one round. Jamming isconstrained by a rate with which an adversary can jam rounds and by a number of consecutiverounds that can be jammed.
Keywords: multiple access channel, adversarial queuing, jamming, distributed algorithm, de-terministic algorithm, packet latency, queues size. ∗ The results of this paper appeared in a preliminary form in [7] and [8]. † Department of Computer Science and Engineering, University of Colorado Denver, Denver, Colorado 80217, USA.The work of this author was supported by the National Science Foundation under Grant No. 1016847. ‡ Department of Computer Science, University of Liverpool, Liverpool L69 3BX, United Kingdom.
Introduction
We study broadcasting in multiple access channels by deterministic distributed algorithms. Thecommunication medium may experience a mild form of jamming. We evaluate the performanceof communication algorithms by upper bounds on their packet latency (delay) and the number ofpackets queued at stations (queues size). The performance metrics are understood in their worst-case sense and are considered in adversarial frameworks of packet injection and jamming. Thereare no statistical components in either algorithms or traffic generation.The traditional approach to distributed broadcasting in multiple access channels uses random-ization to arbitrate for access to a shared medium. Typical examples of randomized broadcastalgorithms include backoff ones, like the binary exponential backoff employed in the Ethernet. Theenduring effectiveness of the Ethernet, as a real-world implementation of local area networks [38],is a compelling evidence that randomized broadcasting can perform well in practice.Using randomization in algorithms, intended as practical solutions to broadcasting, may appearto be inevitable in order to cope with bursty traffic. Among the main challenges that broadcastingon a shared channel faces is resolving conflicts for access to the communication medium. In real-world applications, most stations stay idle for most of the time, so that periods of inactivity areinterspersed with unexpected bursts of activity by groups of stations configured unpredictably.Randomness appears to be a most natural way to break symmetry in attempts to access a channel.Since traffic demands are typically assumed to be unpredictable, the methodological underpinningsof key performance metrics of broadcasting, like queue sizes and packet delay, have traditionallybeen studied with stochastic assumptions in mind. In a matching manner, simulations have beengeared towards models of packet generation defined by stochastic constraints. All these factorshave historically contributed to a popular perception that randomness and stochastic assumptionsare inevitable aspects of broadcasting in multiple access channels.This paper addresses the efficiency of deterministic broadcast algorithms for dynamic traffic de-mands. Performance of algorithms is measured by packet delay and the number of queued packetspending transmission, while packet injection is constrained by formal adversarial models. Studyingalgorithmic paradigms useful for deterministic distributed broadcasting, for dynamic packet injec-tion, is a topic interesting in its own sake. We do this in a model of continuous packet injectionwithout any stochastic assumptions about how packets are generated and where and when they areinjected. This model, known as adversarial queueing, is an alternative to representing packet gen-eration by stochastic constraints. Adversarial queuing has proved useful in providing frameworksto study dynamic communication while imposing only minimal constraints on traffic generation. Itis an important benefit of adversarial queuing to provide a methodology to assess the performanceof deterministic algorithms by worst-case bounds, with respect to suitable metrics.Jamming in wireless networks can be understood as either malicious disruptions of communica-tion medium or inadvertent effects occurring on the physical layer. The former is an effect of foreignmessages sent deliberately to hinder the flow of information by creating interferences of legitimatesignals with such external disrupting transmissions. An example of jamming in this sense is adegradation-of-service attack that produces dummy packets that interfere with legitimate packets.The latter interpretation of jamming is about the physical layer affected by external factors, such asthe supply of energy, weather, or crowded bandwidth. A closely related motivation is to interpretjamming as inadvertent collision of signals with concurrent foreign communication. This occurs1hen groups of stations pursue their independent communication tasks, and so for each group aninterference caused by foreign transmissions is logically equivalent to jamming. To make our picturesimple, jamming is understood in this paper as purely logical, in that this is a symptom we have totake into account without deliberating its causes. There are no assumptions made to justify whya transmitted message is not heard on the channel, including any references to the physical layer,while a message should be heard since only one station transmits in the round. A jammed round hasthe same effect as one with multiple simultaneous transmissions of stations attached to a channel,in that stations cannot distinguish a jammed round from a round with multiple transmissions.
A summary of the methodology and results.
We investigate deterministic broadcast al-gorithms for dynamic packet injection. No randomization is used in algorithms nor there is anystochastic component that affects packet injection in the considered communication environments.The studied communication algorithms are distributed in that they are executed with no central-ized control. The two performance metrics are the queues size (maximum total number of packetssimultaneously stored in the queues at stations while pending transmission) and packet latency(maximum number of rounds spent by a packet in a queue from injection until a successful trans-mission).A set of stations attached to a channel is fixed and their number n is known, in that it can beused in codes of algorithms. Stations are equipped with private queues, in which they can storepackets until they are transmitted successfully.We use the slotted model of synchrony, in which an execution of a communication algorithm ispartitioned into rounds, so that a transmission of a message with one packet takes one round. Allthe stations attached to the channel are activated in the same initial round, each with an emptyqueue.It is the assumed synchrony that allows to define the rate of injecting packets and the rate ofjamming rounds. A round comprises a short atomic duration of time during which some eventshappening in the system can be considered as occurring simultaneously. For example, the burstinessof traffic is understood as the maximum number of packets that can be injected simultaneously,meaning in one round. The related concept of burstiness of jamming is understood as the maximumnumber of contiguous rounds that are unavailable for successful transmissions because of continuousjamming. Similarly, it takes a full round to transmit a message.We consider broadcasting against adversaries that control both injections of packets into stationsand jamming of the communication medium. Packet injection is limited only by the rate of injectingnew packets and the number of packets that can be injected simultaneously. Jamming is limitedby the rate of jamming different rounds and by how many consecutive rounds can be jammed.All the considered algorithms have bounded packet latency for each fixed injection rate ρ andjamming rate λ subject only to the necessary constraint that ρ + λ <
1. The obtained upper boundson packet latency and queue sizes of broadcast algorithms are understood in the worst-case sense.Here “queue size” means the maximum number of packets stored in the queues at the same time,as a function of ρ , for a given number n of stations, and packet latency is the maximum possiblenumber of rounds spent by a packet in a queue waiting to be heard on the channel.The upper bounds on queue size and packet latency of the algorithms studied in this paper aresummarized in Tables 1 and 2. All the algorithms we consider are reviewed in detail in Section 3.We consider non-adaptive algorithms for channels without jamming when either collision detec-2lgorithm Queues Latency Injection Proved OF-RRW ρ − ρ · n + β − ρ · n + β (1 + ρ ) ρ < RRW [25] ρ − ρ · n + β − ρ (1 − ρ ) · n + β − ρ ρ < OF-SRR ρ − ρ · n + β − ρ · n + β (1 + ρ ) ρ < OF-SRR β β (2 + lg n ) ρ ≤ n Thm 3 Sec 4
SRR [25] ρ − ρ · n + β − ρ (1 − ρ ) · n + β − ρ ρ < SRR [25] 2 β β (2 + lg n ) ρ ≤ n Thm 4 Sec 4
MBTF [24] ρ (1 + ρ ) · n + β ρ − ρ − ρ · n + β − ρ ρ < n stations, executed against an adversary of injection rate ρ < β ≥
1. Algorithm
MBTF is adaptive, and the remaining four algorithmsare non-adaptive.tion is not available (algorithms
OF-RRW and
RRW ) or when it is available (algorithms
OF-SRR and
SRR ). These algorithms have a property that queue sizes grow unbounded with injection rate ρ approaching 1, for a fixed n . We conjecture that this is a general phenomenon. Conjecture 1
Each non-adaptive algorithm for channels without jamming that provides boundedqueues, for injection rate ρ < , has its queue bound grow arbitrarily large as a function of injectionrate ρ , if ρ approaches , for all sufficiently large and fixed numbers of stations n . Adaptive algorithm
MBTF for channels without jamming has bounded queues even when ρ = 1,but its packet latency grows unbounded when ρ < Conjecture 2
Each broadcast algorithm for channels without jamming that provides bounded packetlatency, for injection rate ρ < , has its packet-latency bound grow arbitrarily large as a functionof injection rate ρ , if ρ approaches , for all sufficiently large and fixed numbers of stations n . We show that a non-adaptive algorithm for channels with jamming achieves bounded packetlatency for ρ + λ < Conjecture 3
Each non-adaptive broadcast algorithm for channels with jamming can be madeunstable by some adversaries with injection rates ρ and jamming rates λ satisfying ρ + λ < , forall sufficiently large and fixed numbers of stations n . Adaptive algorithm
C-MBTF for channels with jamming has bounded queues when ρ + λ = 1but its packet latency increases unbounded when ρ + λ < n ; see the3lgorithm Queues Latency Proved OF-JRRW( J ) β +1)1 − ρ − λ · n + β β +1)(1 − λ )(1 − ρ − λ ) · n + β (1+ ρ − λ )(1 − λ ) Thm 6 Sec 6
JRRW ( J ) β +1)1 − ρ − λ · n + β β +1)(1 − λ )(1 − ρ − λ ) · n + β (1 − λ )1 − ρ − λ Thm 7 Sec 6
OFC-RRW ρ − ρ − λ · n + β − ρ − λ · n + β (1+ ρ − λ )(1 − λ ) Thm 8 Sec 7
C-RRW ρ − ρ − λ · n + β − λ )(1 − ρ − λ ) · n + β (1 − λ )1 − ρ − λ Thm 9 Sec 7
C-MBTF ρ (1 − λ )+ ρ (1 − λ ) · n + β ρ − λ − ρ − ρλ (1 − λ )(1 − ρ − λ ) · n + β (1 − λ )1 − ρ − λ Thm 10 Sec 7Table 2: Upper bounds on queue size and packet latency for a channel with jammingwith n stations, when the injection and jamming rates satisfy ρ + λ < β ≥
1. The jamming burstiness is assumed to be at most J for algorithms OF-JRRW( J ) and JRRW( J ) , where J is part of their codes. Algorithms OF-JRRW( J ) and JRRW( J ) are non-adaptive, and the remaining three algorithms areadaptive.discussion following the proof of Theorem 10 in Section 7 for details. We conjecture that this is ageneral phenomenon. Conjecture 4
Each broadcast algorithm for channels with jamming that provides bounded packetlatency, for injection rate ρ and jamming rate λ such that ρ + λ < , has its packet-latency boundgrow arbitrarily large as a function of injection rate ρ and jamming rate λ , if ρ + λ approaches ,for all sufficiently large and fixed numbers of stations n . Previous work on adversarial multiple access channels.
Now we review previous work onbroadcasting in multiple-access channels in the framework of adversarial queuing. The first suchwork, by Bender et al. [15], concerned the throughput of randomized backoff for multiple-accesschannels, considered in the queue-free model. Deterministic distributed broadcast algorithms formultiple-access channels, in the model of stations with queues, were first considered by Chlebuset al. [25]; that paper specified the classes of acknowledgment based and full sensing deterministicdistributed algorithms, along the lines of the respective randomized protocols [22].The maximum throughput, defined to mean the maximum rate for which stability is achievable,was studied by Chlebus et al. [24]. Their model was of a fixed set of stations with queues, whosenumber n is known. They developed a stable deterministic distributed broadcast algorithm withqueues of sizes that are O ( n +burstiness) against leaky-bucket adversaries of injection rate 1. Thatwork demonstrated that throughput 1 was achievable in the model of a fixed set of stations whosenumber n is known. The paper [24] also showed some restrictions on traffic with throughput 1;in particular, communication algorithms have to be adaptive (may use control bits in messages),achieving bounded packet latency is impossible, and queues have to be of Ω( n + burstiness) sizes.Anantharamu et al. [9] extended work on throughput 1 in adversarial settings by studying theimpact of limiting window-type adversaries by assigning individual rates of injecting data for eachstation. That paper [9] gave a non-adaptive algorithm for channels without collision detection of O ( n + w ) queue size and O ( nw ) packet latency, where w is the window size; this is in contrast4ith general adversaries, against whom bounded packet latency for injection rate 1 is impossibleto achieve.Bie´nkowski et al. [19] studied online broadcasting against adversaries that are unbounded in thesense that they can inject packets into arbitrary stations with no constraints on their numbers norrates of injection. Paper [19] gave a deterministic algorithm optimal with respect to competitiveperformance, when measuring either the total number of packets in the system or the maximumqueue size. This algorithm was also shown in [19] to be stochastically optimal for any expectedinjection rate smaller than or equal to 1.Anantharamu and Chlebus [6] considered an ad-hoc multiple access channel, which has anunbounded supply of anonymous stations attached but only the stations activated with injectedpackets participate in broadcasting. They studied deterministic distributed broadcast algorithmsagainst adversaries that are restricted to be able to activate at most one station per round. Thealgorithms given in [6] can provide bounded packet latency for injection rates up to 1 /
2, withspecific rates depending on additional features of algorithms. It was also shown in [6] that noinjection rate greater than can be handled with bounded packet latency on such ad-hoc channelsby deterministic algorithms. Related work.
A natural basic communication problem in multiple access channels concernscollision resolution: there is a group of active stations, being a subset of all stations connectedto the channel, and we want to have either some station in the group or all of them transmitsuccessfully at least once. For the recent work on this topic, see the papers by Kowalski [36],Fernandez Anta et al. [27], and De Marco and Kowalski [26].Most related work on broadcasting in multiple access channels has been carried out with ran-domization playing an integral part; see the survey [22]. Randomness can affect the behavior ofprotocols either directly, by being a part of the mechanism of a communication algorithm, or in-directly, when packets are generated subject to stochastic constraints. With randomness affectingcommunication in either way, the communication environment can be represented as a Markovchain with stability understood ultimately as ergodicity. Stability of randomized communicationalgorithms can be considered in the queue-free model, in which a packet gets associated with anew station at the time of injection, and the station dies after the packet has been heard on thechannel. Full sensing protocols were shown to fare well in this model; some protocols stable forinjection rate slightly below 1 / Structure of the document.
We review the model of multiple-access channels and summarizethe classes of adversaries and deterministic broadcast algorithms in Section 2. Section 3 containsa description of all the deterministic broadcast algorithms we consider, both old and new. Theanalysis of performance of broadcast algorithms is given in subsequent sections. These are Section 4about non-adaptive algorithms for channels without jamming, Section 5 about adaptive algorithmsfor channels without jamming, Section 6 about non-adaptive algorithms for channels with jamming,and Section 7 about adaptive algorithms for channels with jamming. The final Section 8 includesa concluding discussion.
In this section, we review the model of multiple access channels and adversarial packet injection.The considered communication environments allow to develop efficient deterministic distributedbroadcast algorithms.A communication medium is called a channel . There are a number of communicating unitsattached to such a channel, which are called stations .We use the slotted model of synchrony, in which time is partitioned into rounds . The stationshave access to a global clock measuring rounds, starting from round zero. An execution of acommunication algorithm starts with all the stations activated in this round zero.The stations receive packets continuously and their goal is to have each of them eventuallybroadcast. Each station is equipped with a private buffer space to store packets pending trans-mission. Such a buffer is considered to have unbounded capacity, in that it can accommodate anarbitrary finite number of packets. The buffer memory of a station typically operates under a fixedqueuing discipline and is referred to as a queue of this station.A message transmitted by a station on the channel may include at most one packet and it mayinclude auxiliary control bits to coordinate actions of the stations. The size of messages and theduration of rounds are calibrated such that a transmission of a message takes one round; this meansthat a station can transmit at most one message in a round. Two messages transmitted by differentstations in the same round overlap in time and are said to be transmitted simultaneously .A successful transmission of a message on the channel means that the message gets broadcastto all the stations. If a message is delivered to a station then we say that that the message is heard by the station. If a message is heard by one station then it is also heard by all the stations. Around when no message is heard on the channel is called void .A round may be jammed , which disrupts the communication functionality of the channel in thisround; a round that is not jammed is called clear . A jammed round is always void but a clearround merely makes it possible to hear a message on the channel.A communication environment we consider operates as a broadcast network consisting of “ac-7ive” stations, which execute communication algorithms in a distributed manner, and a “passive”channel available for each station. The “external world” uses such a communication environmentby providing packets, which are injected individually into the stations, and it also determines whichround is jammed.
Multiple access channels.
Broadcast networks we consider allow for jamming in general, but wealso consider the case when no round can be jammed. A broadcast network is said to be a multiple-access channel without jamming when no round is ever jammed and a message transmitted by astation is heard if and only if it is the only message transmitted in the round. A broadcast networkis said to be a multiple-access channel with jamming when some rounds may be jammed and amessage transmitted by a station is heard if and only if it is the only message transmitted in theround and the round is not jammed.In every round, all the stations receive feedback from the channel. The feedback in a round isthe same for each station; in particular, we do not differentiate between stations that transmit in around and those that do not. If a message is heard on the channel, then the message itself is sucha feedback. A round with no transmissions is said to be silent ; in such a round, all the stationsreceive from the channel the feedback we call silence . Multiple transmissions in the same roundresult in conflict for access to the channel, which is called a collision . If a round is jammed thenall the stations receive in this round the same feedback from the channel as in a round of collision.Now we recapitulate all the possible reasons a round is void, that is, no message is heard. Onepossibility is that the round is silent, in that there is no transmission. The round may be jammed,then it does not matter whether there is any transmission in the round or not. Finally, there maybe a collision caused by multiple simultaneous transmissions. Stations cannot distinguish betweena round of collision, caused by multiple simultaneous transmissions, from a round in which thechannel is jammed, in that the channel is sensed in exactly the same manner in both cases.We say that collision detection is available when stations can distinguish between silence andcollision/jamming in a round by the feedback they receive from the channel in the round. If such adiscerning mechanism is not available then the channel is without collision detection . Next we specifythe four possible kinds of channels, determined by jamming or lack thereof, and, independently, bycollision detection or lack thereof, which determine how stations perceive rounds by the obtainedfeedback from the channel.A channel without jamming and without collision detection: a void round is caused by either si-lence or collision; a specific cause of voidness of a round is not perceivable.A channel without jamming and with collision detection: a void round is caused by either silenceor collision; a specific cause of voidness of a round is identifiable.A channel with jamming and without collision detection: a void round is caused by either silenceor collision or jamming; a specific cause of voidness of a round is not perceivable nor any canbe excluded.A channel with jamming and with collision detection: a void round is caused by either silence orcollision or jamming; silence can be perceived distinctly from the other two possible causes ofvoidness, but collision and jamming cannot be distinguished from each other.A communication algorithm for channels without jamming can be executed on channels with8amming, without any changes in its code. This is because a channel with jamming does notproduce any special “interference” signal indicating that a round is jammed, and stations obtaineither a silence or collision as feedback from the channel when a round is void.
An adversarial model of packet injection without jamming.
We use a leaky-bucket ad-versarial model of packet injection, when a channel cannot be jammed, similarly as considered in[10, 24]. An adversary is determined by its maximum rate of injecting packets and a burstinessof traffic it can generate. Let a real number ρ and integer β satisfy the inequalities 0 < ρ ≤ β ≥
1; the leaky-bucket adversary of type ( ρ, β ) may inject at most ρt + β packets into anarbitrary set of stations in each contiguous segment of t > ρ, β )is said to have injection rate ρ and burstiness component β . The burstiness of an adversary meansthe maximum number of packets that can be injected in one round. An adversary of type ( ρ, β )has burstiness ⌊ ρ + β ⌋ , so if ρ < β is the adversary’s burstiness.In some broadcast algorithms, in which the place and time of injection of packets determines theorder of their future transmissions, a prescribed quantity k of rounds that occur allows the adversaryto inject ρk packets, which then take ρk rounds to be transmitted, thus delaying transmissions ofolder packets. If this pattern can be iterated, then this creates a combined delay of the followingduration: k + ρk + ρ k + · · · ≤ k − ρ . We say that the quantity k − ρ is obtained from k by stretching-by-injecting . An adversarial model of packet injection and jamming.
For channels with jamming, weconsider adversaries that control both packet injections and jamming. Given real numbers ρ and λ in the interval (0 ,
1] and integer β ≥
1, the leaky-bucket jamming adversary of type ( ρ, λ, β ) caninject at most ρt + β packets and, independently, it can jam at most λt + β rounds, in each contiguoussegment of t > ρ as the injection rate , to λ as the jamming rate , and to β as the burstiness component . We can observe that a non-jamming adversaryof type ( ρ, β ) is formally the same as a jamming adversary of type ( ρ, , β ). The number of packetsthat a jamming adversary can inject in one round is called its injection burstiness, similarly as fora non-jamming leaky-bucket adversary. This parameter equals ⌊ ρ + β ⌋ . If λ = 1 then every roundcould be jammed, making the channel dysfunctional. Therefore, we always assume that a jammingrate λ satisfies λ < k non-jammed rounds, possibly in-terspersed with x additional jammed rounds. If the adversary wants to stretch k + x as much aspossible by maximizing x , then the inequality λ ( k + x ) + β ≥ x has to hold. If this is appliedrepeatedly and the adversary jams at full power then the burstiness component β can be appliedonly once. Disregarding the burstiness component β in the inequality λ ( k + x ) + β ≥ x is the sameas setting β = 0, so we have the inequality λ ( k + x ) ≥ x , which gives x ≤ λ − λ · k . We obtain thefollowing estimate k + x ≤ k + k · λ − λ = k − λ . We say that the quantity k − λ is obtained from k by stretching-by-jamming .If the adversary injects with injection rate ρ during these k non-jammed rounds extended byinserted jammed rounds, then the number of injected packets in the whole interval that includes9ammed rounds is at most the quantity ρ − λ · k , which is the same as if ρ got expanded to a virtual injection rate ρ − λ by an effect similar tostretching-by-jamming. The quantity ρ − λ can indeed be interpreted as injection rate because ρ − λ <
1, as ρ < − λ . If the adversary applies this virtual injection rate, already obtained bystretching-by-jamming, by creating a stretching-by-inserting effect, an interval of k clear roundsgets extended to the following number of rounds k (cid:0) ρ − λ + (cid:0) ρ − λ (cid:1) + . . . (cid:1) = k − ρ − λ = k (1 − λ )1 − ρ − λ . We say that the quantity k (1 − λ )1 − ρ − λ is obtained from k by combined stretching .A maximum continuous number of rounds that an adversary can jam is called its jammingburstiness . We can find what is the jamming burstiness for a leaky-bucket jamming adversary oftype ( ρ, λ, β ) as follows. Let x be a number of rounds that make a contiguous interval and are alljammed. The inequality λx + β ≥ x needs to hold, as otherwise x rounds within an interval of x rounds could not be jammed. We conclude by algebra that the adversary can jam at most β − λ consecutive rounds, which is an instance of stretching-by-jamming. Deterministic distributed broadcast algorithms.
Broadcast algorithms control timings oftransmissions by individual stations in a deterministic manner, starting from round zero when allthe stations are activated simultaneously. All the algorithms we consider are work-preserving inthat if a station is scheduled to transmit and it has pending packets then a transmitted messageincludes a packet.A state of a station is determined by the values of the private variables occurring in the codeof an algorithm and by the number of outstanding packets in its queue that still need to be trans-mitted. The local queues of packets at stations operate under the first-in-first-out discipline, whichminimizes packet latency. A station obtains a packet to broadcast by removing the first packetfrom the queue. If a station transmits a packet that is not heard then the station will transmit thesame packet in the immediately following round in which a transmission is scheduled. A packet isnever dropped by a station before it is heard on the channel.A state transition is a change in a state of a station in one round, which depends on the state atthe end of the previous round, the feedback from the channel in this round, and the packets injectedin this round. A state transition of a station in a round consists of the following actions in order.If packets are injected into the station in this round then they are immediately enqueued into thelocal queue. If the station broadcasted successfully in the previous round, then the transmittedpacket is discarded. If a new packet to transmit is needed and the local queue is nonempty then apacket is obtained by dequeuing the queue. Finally, a message for the next round is prepared, ifany will be transmitted.An event in a round comprises the following four actions by each station in the given order:(a) a station either transmits a message or pauses, accordingly to its state, (b) a station receivesa feedback from the channel, in the form of either hearing a message or collision signal or silence,(c) new packets are injected into a station, if any, and finally, (d) the suitable state transition occursat a station. An execution of an algorithm is a sequence of events occurring in consecutive rounds.10e categorize broadcast algorithms according to the terminology used in [24, 25]. All thealgorithms considered in this paper are full sensing, in that nontrivial state transitions can occur ata station in any round, even when the station does not have pending packets to transmit. This maybe interpreted as if the attached stations “sense the channel” in all rounds. Algorithms that usecontrol bits piggybacked on packets or can send messages comprised of only control bits, when astation does not have a packet to transmit, are called adaptive , and otherwise they are non-adaptive . Performance of broadcast algorithms.
The basic quality for a communication algorithm ina given adversarial environment is stability , understood to mean that the number of packets in thequeues at stations stays uniformly bounded at all times. For a stable algorithm in a communicationenvironment, an upper bound on the number of packets waiting in queues is a natural performancemetric, see [24, 25].We may observe that stability is not achievable by a jamming adversary with injection rate ρ and a jamming rate λ satisfying ρ + λ >
1. To see this, observe that it is equivalent to ρ > − λ ,so when the adversary is jamming with the maximum power, then the bandwidth remaining fortransmissions is 1 − λ , while the injection rate is greater than 1 − λ .A sharper performance metric is that of packet latency ; it denotes an upper bound on the timespent by a packet waiting in a queue, counting from the round of injection through the roundwhen the packet is heard on the channel. It is possible to achieve stability in the case ρ + λ = 1,by adapting the approach for ρ = 1 (and λ = 0) in [24], but packet latency is then inherentlyunbounded.An algorithm for an environment without jamming is universal when it is stable for any injectionrate smaller than 1. This can be extended to jamming by having stability for each case of ρ + λ < n and thetype ( ρ, λ, β ) of a leaky-bucket (jamming) adversary, subject only to the restriction ρ + λ < Knowledge.
A property of a system is said to be known when it can be referred to explicitlyin codes of algorithms. We assume throughout that the number of stations n is known to thestations. Each station has a unique integer name in [0 , n − J on the jamming burstiness of an adversary as part of its code; this algorithm attainsthe claimed packet latency when the adversary’s jamming burstiness happens to be at most J . We summarize the specifications of deterministic distributed broadcast algorithms whose packetlatency is analyzed in the following Sections.
Three broadcast algorithms.
We start with a summary of three deterministic distributedalgorithms for channels without jamming that are already known in the literature. These are thealgorithms
RRW , SRR and
MBTF , which can be described as follows.11lgorithm
Round-Robin-Withholding ( RRW ) is a non-adaptive algorithm for channelswithout collision detection. It operates in a round-robin fashion, in that the stations gain ac-cess to the channel in the cyclic order of their names. A station with the right to transmit is said tohold a conceptual token . Once a station receives the token then it withholds the channel to unloadall the packets in its queue. A silent round is a signal for the next station, in the cyclic order ofnames, to take over the token. Algorithm
RRW was introduced in [25] and showed to be universal,that is, stable for injection rates smaller than 1.Algorithm
Search-Round-Robin ( SRR ) is a non-adaptive algorithm for channels with colli-sion detection. Its execution proceeds as a systematic continuous search for the next station withpackets to transmit, under the cyclic ordering of stations by their names. The search is interpretedas binary one and is implemented by using a virtual distributed stack. If a station with pendingpackets is identified by the search, the search is suspended while the station withholds the channelto transmit all its packets. After all the packets held by a station have been unloaded, a silentround follows, which triggers the search to be resumed. A basic step in searching is to verify ifthere is a station with pending packets whose name is in a given interval of integers. Such a step isaccomplished by all the stations in the interval transmitting their packets. Every station receivesthe same feedback from the channel, whether it transmitted or not, so all the stations know if theinterval is empty (silence), or it contains a single station (packet heard), or it contains multiplestations (collision). A search for the next station is completed by a packet heard. A silence indicatesthat no station in the tested segment has packets and the interval is discarded. A collision results inhaving the interval partitioned into two halves of equal sizes, with one part processed immediatelynext while the other one is pushed on a stack to wait. If a processed interval becomes empty or itis verified by silence that there is no station with packets in it, then a new interval is obtained bypopping the stack. One instance of a full sweep through all the stations is called a phase . A phasestarts with the interval [0 , n −
1] representing all the stations placed on the stack, and it ends withthe stack becoming empty. Once a phase is completed, the next similar phase begins immediately.Algorithm
SRR was introduced in [25] and showed to be universal.Algorithm
Move-Big-To-Front ( MBTF ) is an adaptive algorithm that can be executed onchannels without collision detection. Each station maintains a dynamic list of all the stations in itsprivate memory. Such a list is initialized in each station to have all the names of stations arrangedin the increasing order: 0 , , . . . , n −
1. The lists are manipulated in the same way by all thestations so they are identical copies of each other. The algorithm schedules exactly one station totransmit in a round, so that collisions never occur. This is implemented by having a conceptualtoken travel through the stations, which is initially assigned to the first station in the list. A stationwith the token broadcasts a packet, if it has any, otherwise the round is silent. A station considersitself big in a round when it has at least n packets; such a station attaches a control bit to everypacket it transmits to indicate this status. A big station is moved to the front of the list and ittakes the token with it. If a station that is not big transmits in a round, or when it pauses dueto a lack of packets while holding the token so the round is silent, then the token is passed in thisround to the next station in the list ordered in a cyclic fashion. Algorithm MBTF was introducedin [24] and showed to be stable for injection rate 1.
The “old-go-first” approach.
We obtain new algorithms by modifying
RRW and
SRR so thatpackets are categorized into “old” and “new.” Intuitively, packets categorized as “new” becomeeligible for transmissions only after all the packets categorized as “old” have been heard. Formally,an execution is structured as a sequence of conceptual phases, which are contiguous segments of12ounds of dynamic length, and then the notions of old versus new packets are defined with respectto them.A phase is defined as a full cycle made by the conceptual token visiting the stations. Noadditional communication is needed to mark a transition to a new phase as all the stations candetect this by monitoring the position of the virtual token. A token leaves a station holding itafter the station has transmitted all its old packets while new packets may remain waiting for thenext token’s visit. In a given phase, packets are old when they had been injected in the previousphase, and packets injected in the current phase are considered new for the duration of the phase.If a new phase begins, the old packets have already been heard on the channel and the new onesimmediately graduate to becoming old. This means that the “old-go-first” principle is implementedby having packets injected in a given phase transmitted only in the next phase. In particular, thefirst phase does not include any transmissions of packets, as all the packets, if any, are new.Specifically, algorithm
Old-First-Round-Robin-Withholding ( OF-RRW ) operates by ma-nipulating the token similarly as algorithm
RRW does, except that when a station gets access tothe channel by transmitting successfully, then the station unloads all the old packets, while newpackets stay in the queue when the token is passed to the next station. Algorithm
Old-First-Search-Round-Robin ( OF-SRR ) performs search similarly as algorithm
SRR does, except thatsearching is for old packets only while new ones are ignored for the duration of a phase. Thisapproach is also applied to algorithm
JRRW ( J ) for channels with jamming, as explained next.The approach to modify a token algorithm by making old packets go first makes packet latencysmaller than in the original version but queue bounds remain the same, as reflected by the boundssummarized in Tables 1 and 2. The difference in packet latency is such that a “regular” versionof an algorithm for channel without jamming, which is either RRW or SRR , has an additionalfactor of − ρ present in its bound on packet latency as compared to their versions with old-go-firstspecification, and the bound for algorithm JRRW ( J ) has an extra factor of − ρ − λ present, ascompared to the bound on packet latency for algorithm OF-JRRW ( J ). This might be counter-intuitive, as an old-go-first version of broadcasting is a “lazy” implementation, in the sense that apossible immediate transmission of a packet is delayed for later when the packet happens to be stillnew. This can be explained intuitively as follows. Consider a regular version of a given broadcastalgorithm, like RRW . An injected packet may be transmitted either in the current phase or in thenext phase, depending on how the station that the packet is injected into is located in the cycle ofstations with respect to the station holding the token at the round of injection. We may say thatinjecting “behind the token” results in transmitting in the next phase and injecting “ahead of thetoken” results in transmitting in the current phase. If the adversary consistently injects “behindthe token” so that packets are transmitted as already old then a execution is indistinguishable fromthat of the old-go-first version of the algorithm. There is a possibility of an effect of stretching-by-injecting occurring in executions of the old-go-first version and this is reflected in the factor of − ρ in the bound on packet latency. If the adversary exercises the option to inject “ahead-of-the-token,”for the regular version of the algorithm, then this creates an additional possibility of enforcingstretching-by-injecting, and so adds another factor of − ρ . Non-adaptive algorithms for channels with jamming.
We introduce a non-adaptive broad-cast algorithm
Jamming-Round-Robin-Withholding ( J ), abbreviated JRRW ( J ), for channelswith jamming. The design of the algorithm is similar to that of RRW , the difference is in how thetoken is transferred from a station to the next one, in the cyclic order among the stations. Just13ne void round should not trigger a transfer of the token, as it is the case in
RRW , because nothearing a message may be caused by jamming.The algorithm has a parameter J interpreted as an upper bound on jamming burstiness ofthe adversary. This parameter is used to facilitate transfer of control from a station to the nextone by way of forwarding the token. The token is moved after precisely J + 1 contiguous voidrounds, counting from either hearing a packet or moving the token; the former indicates that thetransmitting station exhausted its queue, while the latter indicates that the queue was empty. Moreprecisely, every station maintains a private counter of void rounds. The counters show the samevalue across the system, as they are updated in exactly the same way determined only by thefeedback from the channel. A void round results in incrementing the counter by 1. The token ismoved to the next station when the counter reaches J + 1. If either a packet is heard or the tokenis moved then the counter is zeroed.Algorithm Old-First-Jamming-Round-Robin-Withholding ( J ), abbreviated OF-JRRW ( J ),is obtained from JRRW ( J ) similarly as OF-RRW is obtained from
RRW . An execution is struc-tured as consisting of consecutive phases, and packets are categorized into old and new, with thesame rule to graduate packets from new to old. If a token visits a station, then only the old packetsare transmitted while the new ones will be transmitted during the next visit by the token.
Structural properties of algorithms.
We say that a communication algorithm designed for achannel without jamming is a token one if it uses a virtual token to determine a station that gainsthe right to transmit successfully. All the algorithms discussed in this paper could be consideredas token ones. This is clearly the case for algorithms
RRW , OF-RRW , JRRW , OF-JRRW , and
MBTF , as their design specifies how a token is handled. Algorithms
SRR and
OF-SRR can alsobe interpreted as token ones, even though they make collisions possible to happen. A station thattransmits a packet successfully can be considered as holding the token, in that it can safely withholdthe channel, and the right to transmit was acquired by the virtue of being the next station withpackets after the previously transmitting one, in the cyclic ordering of stations.A token algorithm for channels without collision detection and without jamming can be modifiedto the model with jamming, but still without collision detection. This can be done in the followingmanner. If a station has the right to transmit a packet in the original algorithm, then the modifiedalgorithm has the station transmit a packet as well, otherwise the station transmits a control bit.A round in which only a control bit is transmitted by a modified token algorithm is called a controlround otherwise it is a packet round . The effect of sending control bits in control rounds is that ifa round is not jammed then a message is heard in this round; this message is either just a controlbit or it includes a packet. This approach to replace silent rounds by rounds with messages withcontrol bits allows for jamming detection: when a void round occurs then this round has to bejammed, as otherwise a message would be heard. Once a communication algorithm can identifyjammed rounds, we may ignore their impact on the flow of control and repeat the performed actionsin the next round, exactly as they were performed in the immediately preceding jammed ones. Theresulting algorithm is clearly adaptive. This method cannot be applied to algorithms relying oncollision detection, like
SRR and
OF-SRR .We will apply this method of modifying token algorithms to the non-adaptive algorithms
RRW and
OF-RRW , denoting the modified versions by
C-RRW and
OFC-RRW , respectively. Similarly,we modify algorithm
MBTF such that a station with a token sends a control message even if thestation does not have a packet; the modified algorithm is denoted by
C-MBTF . The letter C is a14nemonic to indicate using control rounds for jamming detection.Algorithms with executions structured into phases, so that each station with packets has oneopportunity to transmit its packets in a phase, are referred to as phase algorithms . Among thealgorithms considered in this paper, all are phase ones except for
MBTF and
C-MBTF . Thephase algorithms consist of
RRW , OF-RRW , C-RRW , OFC-RRW , SRR , OF-SRR , JRRW and
OF-JRRW . If the old-go-first approach is used in a phase algorithm then it is an old-go-first version of the algorithm, otherwise it is a regular version of the algorithm. In particular,
RRW , C-RRW , SRR and
JRRW are all regular phase algorithms, while
OF-RRW , OFC-RRW , OF-SRR and
OF-JRRW are all old-go-first phase algorithms.Let us consider an execution of a token algorithm. If a packet is injected into a station whosenumber is smaller than that of the current token’s holder then we say that the packet is injected behind the token , and otherwise it is injected ahead of the token . If the considered token algorithmis a regular one, like
RRW , then packets injected behind the token are transmitted in the nextphase, and those injected ahead of the token are transmitted in the current phase.
In this Section, we consider deterministic distributed non-adaptive algorithms for channels withoutjamming for injection rates ρ <
1. For each of these algorithms, we give upper bounds for thequeue size and packet latency as functions of the number of stations n and the type ( ρ, b ) of aleaky-bucket adversary. We begin with algorithms
OF-RRW and
RRW for channels without collision detection. Each ofthem is a token algorithm. The token is advanced to the next station when a station holding thetoken at the moment pauses, which results in a silent round.
Theorem 1
If algorithm
OF-RRW is executed by n stations against an adversary of type ( ρ, β ) then the number of packets simultaneously queued in the stations is at most ρ − ρ · n + β (1) and packet latency is at most − ρ · n + β (1 + ρ ) . (2) Proof:
Let T i denote the duration of phase i , where T = n . Let Q i denote the number of oldpackets in the beginning of phase i , where Q = 0. The sequences ( Q i ) i ≥ and ( T i ) i ≥ satisfy thefollowing recursive dependencies, where we disregard the effect of burstiness: Q i +1 ≤ ρ · T i and T i +1 ≤ n + Q i +1 ,
15y the algorithm’s design and the constraints imposed on the adversary. Iterating these recurrencesproduces the following bound T on the duration of a phase: T i +1 ≤ n + ρ · T i ≤ n + ρn + ρT i − ≤ n (1 + ρ + ρ + . . . ) ≤ n − ρ = T . (3)A packet waits to be transmitted through at most two consecutive phases, each taking at most T rounds. A bound for T given in (3) disregards the effect of burstiness. We can account for theeffect of burstiness as follows. Let the adversary inject additional β packets in a round of a phase.This instantaneously increases the number of packet queued in the current phase but extends theduration of the next phase, which is the phase when these packets are transmitted as old. Thesetransmissions in turn allow the adversary to inject ρβ additional packets, which extends the durationof the next phase by ρβ rounds.We conclude with the following estimates. The maximum number of queued packets is obtainedby combining at most ρT old packets with at most ρT new packets, along with at most β packetsinjected in a burst, which together give (1) as a bound. The maximum number of rounds spent bya packet waiting to be heard on the channel is obtained by adding twice the upper bound T on aduration of a phase (3), incremented by β extra rounds in a phase immediately following one of abursty injection, along with ρβ rounds of the next phase, which together give (2). (cid:3) The bounds of Theorems 1 are asymptotically tight. We give a strategy of the adversary tomake queue sizes and packet latency close to these for algorithm
OF-RRW . When a phase beginsthen the adversary injects its first packet into station n −
1, to make it wait almost two phases.The adversary injects at full power, that is, as soon as a packet can be injected while satisfyingthe restriction that the number of packets injected is at most ρt within the first t rounds of anexecution, then a packet is injected. The first phase takes exactly n rounds, and the adversaryinjects ρn packets during this phase, but all of them will be transmitted in the next phase. Sowhen the second phase begins, there are already ρn packets queued. The duration of phases keepsincreasing such that when one takes r rounds then the next one takes r + ρr rounds, starting from n ,so that it gets arbitrarily close to n − ρ . The number of old packets is ρ times the duration of aphase. Burstiness allows to add β to the number of queued packets and extend two consecutivephases by β (1 + ρ ) rounds.Next we estimate the performance of algorithm RRW . Theorem 2
If algorithm
RRW is executed by n stations against an adversary of type ( ρ, β ) thenthe number of packets simultaneously queued in the stations is at most ρ − ρ · n + β (4) and packet latency is at most − ρ (1 − ρ ) · n + β − ρ . (5) Proof:
First consider the queue sizes. Packets injected behind the token are transmitted inthe next phase, which is consistent with the design of
OF-RRW and so with its bound. Packetsinjected ahead of the token are transmitted in the current phase, which slows down the phasecompared to
OF-RRW . If a phase is longer then more packets can be injected in it, but each extra16ound is spent on a transmission, because this is the reason a phase is longer, while not each extraround has to have a new packet injected in it. This means that the upper bound on the number ofpackets stored in the queues (1) derived for
OF-RRW also applies to
RRW , so we make it equalto (4).Next we estimate packet latency. Packets injected behind the token and ahead of the tokenare considered separately. If packets are injected only behind the token then the bound (3) on thelength of a phase for
OF-RRW applies, in that each phase takes at most T = n − ρ rounds. Suchlength of a phase is determined by the packets that are already queued when a phase begins. Now,consider the effect of injections only ahead of the token while the old packets are already queued.The duration of a phase is obtained from a duration T of a phase of OF-RRW slowed down asmuch as possible by injecting packets in front of the token. The upper bound on the duration ofsuch a phase becomes T (1 + q + q + · · · ) = T − ρ ≤ n (1 − ρ ) . (6)Packet latency is upper bounded by the duration of two consecutive phases. The lengths of twoconsecutive phases are at most a sum of the lengths given by (3) and (6): n − ρ + n (1 − ρ ) = 2 − ρ (1 − ρ ) · n , because injecting only in front of the token prevents creating old packets to be transmitted in thenext phase, and the following phase starts with empty queues. The second of these two phases maybe additionally extended by at most β − ρ , due to the stretching-by-injecting effect, which gives theultimate bound (5). (cid:3) The bounds of Theorems 2 are asymptotically tight, which can be demonstrated by giving aspecific adversary’s strategy. Let the adversary first keep injecting just after the token. Thesepackets are transmitted in the next phase, which simulates the behavior of
OF-RRW . Eventuallythe phase lengths gets arbitrarily close to n − ρ . Then, at the beginning of a new phase, the adversarystarts injecting just ahead of the token. The duration of this one phase gets extended by anadditional factor of − ρ due to stretching-by-injecting.The tightness of the bounds implies that the advantage of the old-go-first mechanism appliedin algorithm OF-RRW , as compared to
RRW , is the speedup of packet latency by the followingfactor 2 − ρ (1 − ρ ) · − ρ · − ρ − ρ > − ρ ) , which is measured having an adversary fixed and n growing unbounded. We consider algorithms
Old-First-Search-Round-Robin ( OF-SRR ) and
Search-Round-Robin ( SRR ), both of which use collision detection. Executions are partitioned into phases. A phase de-notes one full sweep of search through all the names of stations.We begin with a technical estimate that will be used in proving bounds on packet latency. Letlg x denote ⌈ log x ⌉ . 17 emma 1 If there are already x packets in the system when a phase of algorithm OF-SRR begins,then the phase takes at most min [ x (2 + lg n ) , x + 2 n − rounds. Proof:
We argue that there are at most 1 + lg n void rounds between two packets are heard onthe channel. This is because of two reasons. First, when a station finishes its transmissions, thenone silent round either triggers the next search or completes the phase. Second, when a new searchto identify a station with a packet begins, it takes at most lg n collisions to identify a single stationwith pending packets. There are also x rounds spent to hear the x packets.Next we give the following alternative estimate. A phase can be represented by a binary searchtree in which each interval on a stack corresponds to a node. In particular, a station with pendingpackets is in an interval that is a leaf, and an interval that creates a collision corresponds to aninternal node. Observe that we may associate one void round with each node on such a tree. Theassociation depends on the kind of node. First, if a node represents a station with packets, whichis a leaf, then there is a silent round following all the transmissions by the station, which can beassociated with the node. Second, if this is an internal node, then it is associated with a collision.It follows that the total number of nodes in the tree and the number of void rounds in a phase areequal. There are at most 2 n − n leaves. The void roundsin the phase are added to the x rounds used to hear the x packets. (cid:3) Now we give the performance bounds for the algorithm
OF-SRR . Theorem 3
If algorithm
OF-SRR is executed by n stations against an adversary of type ( ρ, β ) then the number of packets simultaneously queued in the stations is at most ρ − ρ · n + β (7) and packet latency is at most − ρ · n + β (1 + ρ ) . (8) If ρ ≤ n then the number of packets simultaneously queued in the stations is at most β andpacket latency is at most β (2 + lg n ) . Proof:
Let T i denote the duration of phase i , where T = 1. Let Q i denote the number of oldpackets in the beginning of phase i , where Q = 0. Let Q be an upper bound on the number ofqueued old packets and T an upper bound on the duration of a phase.First, we consider the case of ρ ≤ n . The inequality Q ≤ β holds, including the effect ofburstiness, so that T ≤ β (2 + lg n ). Then again Q ≤ ρ · β (2 + lg n ) ≤ β . The pattern repeats,so the invariants Q i ≤ β and T i ≤ β (2 + lg n ) are maintained. This allows to set Q = β and T = β (2 + lg n ). The queues size is at most the number of old and new packets together, whichis 2 Q = 2 β , and packet latency is at most twice the duration of a phase T , which is at most2 T = 2 β (2 + lg n ).Next, we consider the general case. The sequences ( Q i ) i ≥ and ( T i ) i ≥ satisfy the followingrecursive dependencies, by Lemma 1, where we disregard the effect of burstiness: Q i +1 ≤ ρ · T i T i +1 ≤ n + Q i +1 . Iterating these recurrences produces the following bound T on the duration of a phase: T i +1 ≤ n + ρ · T i ≤ n + ρ n + ρT i − ≤ n (1 + ρ + ρ + . . . ) ≤ n − ρ = T . (9)A packet spends at most two consecutive phases waiting to be heard, each phase taking at most T rounds. A bound for T given in (9) disregards the effect of burstiness, which can be accounted foras follows. If the adversary injects β packets in one round then this increases the number of packetqueued in the current phase. This injection extends the duration of the next phase rather then thecurrent one, because this will be the phase when these packets are transmitted as old. These extratransmissions make it possible for the adversary to inject ρβ packets, which extends the durationof the next phase by ρβ rounds.Here are the concluding estimates. The maximum number of queued packets is at most ρT old packets added to at most ρT new packets, and at most β packets injected in a burst, whichgives (7). The maximum number of rounds spent by a packet waiting to be heard on the channelis twice the upper bound T on a duration of a phase (9), incremented by β extra rounds in a phaseof a bursty injection along with ρβ rounds of the next phase, which gives (8). (cid:3) The bounds of Theorems 3 are asymptotically tight, which can be shown as follows. Thereare two bounds on queues and latency, and tightness of a bound occurs when the adversary’s typesatisfies additional conditions. First, the case of small ρ , say, ρ =
12 lg n . Queues size is tight asthe bound is proportional to the burstiness component. Let the adversary inject packets in pairsinto two adjacent stations, a packet per station, such that they are at some point together in aninterval on the stack that is of a constant-size. There are β/ n/β apart. For the burstiness component β such that lg β = o (lg n ), ittakes Ω( β log n ) to transmit β packets injected simultaneously, so the phase duration is also tight.Next, the case of large injection rates ρ , in particular, when ρ > . Let the adversary keep injectinginto pairs of stations, a packet per station, that belong together to intervals of a constant lengththat are on the stack at some point in time. The time spent waiting to hear a new packets isΩ(log n ) initially, while the adversary injects at a rate larger than . Eventually, the rate of hearingconsecutive packets becomes Ω(1), but at that point the number of packets queued becomes Ω( ρn ).The adversary continues injecting at full power to extend a phase’s length close to Ω( n − ρ ), by thestretching-by-injecting effect. The adversary may add β packets in a burst and next extend thetwo following phases by about β + ρβ rounds.Next, we consider algorithm SRR . We begin with a preliminary fact.
Lemma 2
Let us consider the beginning of a phase of algorithm
SRR . If the number of packetsthat are either already queued or they are injected during the phase into stations that belong to someintervals on the stack is y then the phase takes at most min( y (2 + lg n ) , y + 2 n − rounds. Proof:
A proof is similar to that of Lemma 1, with a difference regarding which packets gettransmitted in a current phase. While in algorithm
OF-SRR these are the packets already queuedwhen the phase starts, algorithm
SRR has all the available packets transmitted, including thosealready present when the phase begins but also newly injected ones. Each station that holds packets19ompetes for access to the channel in a phase, unless its name is no longer on an interval on thestack. A round of the first transmission by such a station occurs when the interval including thestation’s name is removed from the top of the stack and the station is the only one in the intervalthat holds packets pending transmission. (cid:3)
Now we give the performance bounds for the algorithm
SRR . Theorem 4
If algorithm
SRR is executed by n stations against an adversary of type ( ρ, β ) thenthe number of packets simultaneously queued is at most ρ − ρ · n + β (10) and packet latency is at most − ρ (1 − ρ ) · n + β − ρ . (11) If ρ ≤ n then the number of packets simultaneously queued is at most β and packet latency isat most β (2 + lg n ) . Proof:
Packets that are injected into stations that do not belong to the intervals on the stackare transmitted in the next phase. The way the algorithm handles these packets is consistent withthe design of
OF-SRR , so their packet latency conforms to the bound on packet latency for
OF-SRR . Packets injected into stations that belong to the intervals on the stack are transmitted inthe current phase, which may slow down the phase as compared to
OF-SRR . The extra roundsare either spent on transmissions or they produce collisions while a next station with packets isidentified. Each round spent on transmissions decreases the number of packets in the queues, butnot each of these rounds is used by the adversary to inject new packets and so increase the numberof packets queued. Regarding the rounds producing collisions, they are estimated as overheads ofeither 2 + lg n per packet or 2 n − OF-SRR includes both the old and new packets, but accounted for separately. Sinceaccounting for transmission of old and new packets together is consistent with accounting for themseparately, the upper bounds on the size of queues of
OF-SRR also applies to algorithm
SRR . Weconclude that the bound (10) on queues size can be made identical to (10), along with the boundof 2 β for the suitably small injection rates.Next we estimate packet latency. There are two cases, the general one and a special one ofsuitably small injection rates. Let T i denote the duration of phase i , and T an upper bound on theduration of a phase.First the case of ρ ≤ n . If the adversary injects only into stations that are not on the stackthen these packets are old, in the sense that they will be heard in the next phase, so the bound T = β (2 + lg n ) for algorithm OF-SRR applies. If the adversary injects only into stations that arestill on the stack, then this allows to extend a phase’s duration by a factor of 1 / (1 − ρ ). A packetcan be delayed at most two consecutive phases, which is the following, for sufficiently large n : T (cid:16) − ρ (cid:17) = T · − ρ − ρ ≤ T · − n = 2 T · n n ≤ T .
OF-SRR applies, in that a phase takes at most T = n − ρ rounds. Each such a duration sufficesto hear the packets that are already queued when a phase begins. Now, consider the effect ofinjections only into stations that belong to intervals on the stack at the round of injection, whilethe old packets are already queued. The duration of a phase is obtained from the duration T of aphase of OF-SRR slowed down as much as possible by injecting packets into stations whose namesare in the intervals on the stack. An upper bound on the duration of such a phase is obtained bythe stretching-by-injecting effect to be at most the following: T (1 + ρ + ρ + · · · ) = T − ρ ≤ n (1 − ρ ) . (12)The maximum of a sum of lengths of two consecutive phases is obtained as a sum of the lengthsgiven by (9) and (12), because injecting only into stations on the stack results in not creating anyold packets to be heard in the next phase. The obtained bound is as follows:2 n − ρ + 2 n (1 − ρ ) = 4 − ρ (1 − ρ ) · n . The second of these two phases may be additionally extended by at most β − ρ , due to the stretching-by-injecting effect combined with burstiness, which gives (11). (cid:3) The bounds of Theorem 4 are asymptotically tight, which can be shown by finding a specificadversary’s strategy. The case of small injection rate is similar as for algorithm
OF-SRR , sincealgorithm
SRR has its performance bounds differ from those for
OF-SRR by constant multiplica-tive factors when injection rates are smaller than 1 / (2 + lg n ). Next we discuss the general case.Let the adversary first keep injecting into stations whose names are not in the intervals on thestack, similarly as in the case of algorithm OF-SRR . These packets are transmitted in the nextphase, which is consistent with the behavior of
OF-SRR , so that eventually the phase lengthsgets arbitrarily close to n − ρ . Then, at the beginning of a new phase, the adversary starts injectinginto stations that are still on the stack. The duration of this one phase can get extended by anadditional factor of − ρ due to stretching-by-injecting. This same phase can be further extendedby β − ρ by burstiness amplified by stretching-by-injecting.The tightness of the bounds implies that the advantage of the old-go-first mechanism appliedin algorithm OF-SRR , as compared to
SRR , is the speedup of packet latency by a factor that isgrater than − ρ ) , similarly as in the case of algorithm OF-RRW compared to
RRW . Algorithm
Move-Big-To-Front ( MBTF ) is an adaptive one for channels without collision de-tection. This algorithm is stable even when injection rate is 1, but for this rate packet latency isunbounded, in that even an eventual hearing of a packet is not guaranteed [24].Algorithm
MBTF works with stations arranged in a dynamic list, and we refer to the stationsnot by their names but by their positions on this list. There are n positions: 1 , , , . . . , n , withstation 1 at the front of the list and station n at the end.21he list of stations is traversed by a token that gives the right to transmit. Let a traversal ofthe token, which starts at the front of the list and ends by reaching again the front station of thelist, be called a pass of the token. A pass is concluded by either discovering a new big station ortraversing the list to its end.We monitor the number of packets in the queues at the end of a pass, to see how the passcontributed to the number of packets stored in the queues. If the number of queued packets at theend of a pass is smaller than at the end of the previous pass, then such a pass is called decreasing ,otherwise it is non-decreasing .We partition passes into two categories, depending on whether a big station is discovered in apass or not. If a big station is discovered in a pass then such a pass is called big and otherwiseit is called small . A discovery of a big station results in moving this big station to the front ofthe list, which concludes the pass. The next pass begins by a transmission of the newly discoveredbig station, just after it is moved up to the front position in the list. We begin the analysis ofperformance of algorithm MBTF by investigating how many packets can be accumulated in thequeues when small passes occur.
Lemma 3
If algorithm
MBTF is executed by n stations against an adversary of type ( ρ, β ) , insuch a manner that all the passes have been small up to a given round, then the number of packetsstored in the queues in this round is at most ρn + β . Proof:
If the adversary injects packets at the rate as close to injection rate as possible thenburstiness component can be applied only once, and we will conclude with its contribution, whileinitially we disregard it. A small pass takes n rounds. The adversary can inject ρn packets duringa time segment of these many rounds. This number ρn is also an upper bound on the number ofstations with packets during a non-decreasing small pass, because if there were more such stations,then each of them would transmit a packet during a pass.Each station with packets has at most n − ρn · ( n − ρn + β packets in the course of any of thesepasses. We conclude that the number of packets is at most ρn + β in a round by which only smallpasses have occurred. (cid:3) The adversary may use big passes to accumulate packets in queues and delay packets at theend of the list of stations by preventing the token to reach the tail of this list. The accumulationof packets is largest when the token traverses as many stations with empty queues as possiblebefore discovering a big station. During such passes, the adversary can inject at the rate of ρ whilestriving to make the ratio of the number of rounds with messages heard on the channel smallerthan ρ , which results in the number of queued packets growing. Theorem 5
If algorithm
MBTF is executed by n stations against an adversary of type ( ρ, β ) thenthe number of packets stored in the queues in any round is at most ρ (1 + ρ ) n + β (13) and packet latency is at most ρ − ρ − ρ · n + β − ρ . (14)22 roof: We will disregard the burstiness component through the initial stages of the analysis, toapply it at the end of the process of accounting for time and injected packets.By Lemma 3, if no big station has been discovered yet then there are at most ρn packets intotal. We explore now how much the queues can increase when big passes occur. If there are atmost ρn stations with packets then the sum of the lengths of big passes is maximized when thefollowing is the case: (1) stations holding packets are located at the end of the list, and (2) eachtime the token reaches one of these stations, for the first time since big passes started to occur,then the station is discovered to be big. Therefore, the sum of the lengths of big passes is at mostthe following: ρn X i =1 (cid:0) n − ρn + i (cid:1) ≤ ρn , for sufficiently large n . During these big passes, at most ρ · ρn new packets are injected. The totalnumber of packets at this point is at most ρn + ρ n = ρ (1 + ρ ) · n . Injecting β packets in one round can be increase the total number of packets to at most (13).Next we estimate packet latency. Let us consider some packet p and we argue about its delayby building a worst-case scenario. We may assume that p gets injected when the configuration ofpackets is already as in Lemma 3, which is such that at most ρn packets are located in the ρn stations located at the end of the list, each holding at most n packets, but possibly fewer. Letpacket p be injected into the last station, which takes the longest for the token to reach whenstarting from the front. Additionally, if the last station is never discovered to be big, which is thecase when the total number of packets in this station is at most n − p , then the tokenwill never discover the station to be big before a packet that is at the bottom when p is injectedis ready to be transmitted. Packet p may be at the bottom of its queue just after it is injected,and we may assume it is preceded by n − n − p when it is already ready to be transmitted. Eachsuch a traversal of the whole list makes a small pass. In the meantime, the token may be delayedby discovering big stations, what makes the token return back to the front station without reachingthe station holding p .We estimate how much time may pass before the token finally visits the p ’s station, when p is already at the top of the queue ready to be transmitted, by accounting for the following threegroups of rounds contributing to p ’s waiting time:(1) a delay due to discovering big stations,(2) a delay due to small passes and packets injected during such passes,(3) the effect of burstiness.We begin with the effect of discovering big stations. Starting from the p ’s injection, the adversarymay inject packets into the trailing ρn stations to make each of them big, with the exception ofthe last one. The discoveries of up to ρn big stations at the end of the list provide delays of up tothese many rounds: ρn − X i =1 (cid:0) n − ρn + i (cid:1) ≤ ρn . ρ − ρ · n . Next, we consider the effect of small passes. It has two components. There are n − p when at the top of its queue, each pass contributing n rounds,for the total of n ( n − ≤ n rounds, which is the first component.During small passes, packets can be injected to introduce additional delay, possibly throughdiscovering big stations. Suppose some x such packets are injected. If they are located in bigstations that are discovered big for the first time then there are at most xn such stations, eachcontributing a delay of at most n rounds for the total of at most xn · n = x rounds of delay.Otherwise, if some new packets are injected into a station that has already been discovered big andis at position i in the list, then this station has at most n − i packets inherited from the time it wasdiscovered big and moved to the front, so at least i packets are needed to make it big again, andthese i packets contribute to delay i by making the station big. Any excess of y packets beyond n injected into a big station will contribute to a delay of y when the station is moved to the front ofthe list and starts transmitting. So overall, the delay is upper bounded by the number of packetsinjected. There are at most ρ · n packets injected during small passes. The resulting delay is atmost such, which is the second component.Finally, burstiness allows to inject β packets into a big station, which can be extended to β − ρ by stretching-by-injecting.We have assessed the three contributions to packet delay. Adding them together gives a totalof at most these many rounds: ρ − ρ · n + (1 + ρ ) · n + β − ρ = ρ + (1 + ρ )(1 − ρ )1 − ρ · n + β − ρ = 1 + ρ − ρ − ρ · n + β − ρ , which is the claimed upper bound on packet latency (14). (cid:3) The bounds given in Theorem 5 are asymptotically tight. The factors 1 + ρ and 1 + ρ − ρ inthe upper bounds (13) and (14) are Θ(1) because 1 < ρ − ρ <
2. It is sufficient to show howto construct a configuration with Ω( ρn + β ) queued packets and a packet whose delay is Ω (cid:0) n + β − ρ (cid:1) .Let the adversary build queues of n − ρn stations. This occurs in the courseof small passes during which the adversary injects two packets into each of some fixed ρn stations,so each of them grows in a pass. After n − n − ρn + β packets so that the numberof queued packets is at least ρn + β .Next we consider packet latency. Let the adversary build queues of n − ρn ≤ n stations, while the first − ρ · n ≥ n stations have empty queues. A packet p is injectedinto the last station as its last packet at the bottom. Let the adversary make each of the stationswith packets big by inserting one extra packet, starting with the station in the smallest position,but skipping the last station. After that, the adversary keeps injecting at full power into the stationthat is last but one, which also includes injecting β packets in one round.We consider two cases. The first case is when ρ ≤ , which implies n + β − ρ ≤ n + 2 β . The bigpasses contribute at least β rounds and the small passes that follow contribute at least n ( n − ρ > . The number of void rounds in big passes is at least P n i =1 (cid:0) n − n + i (cid:1) ≥ n . When the last-by-one station is discovered big, the adversary injectsadditional β packets into it. The number of rounds n + β can be extended by stretching-by-injecting to at least · n + β − ρ rounds. All these rounds contribute to the delay of packet p . Thisquantity grows unbounded if injection rate ρ converges to 1. We show that non-adaptive algorithms may have bounded worst-case packet latency on channelswith jamming. The caveat is that they are correct only against adversaries whose jamming bursti-ness is bounded from above by a parameter we denote J . This parameter J is part of code, andto emphasize this, is included as part of the names of algorithms OF-JRRW( J ) and JRRW( J ) .The value of J does not occur in the upper bounds on packet latency we derive, as the jammingburstiness of a jamming adversary of type ( ρ, λ, β ) is at most β/ (1 − λ ). Lemma 4
If there are x old packets in the queues when a phase of algorithm OF-JRRW( J ) executed by n stations begins, against an adversary of type ( ρ, λ, β ) whose jamming burstiness is atmost J , then the phase takes at most these many rounds: x + n ( J + 1) + β − λ . Proof:
It takes x rounds to transmit the x old packets. It takes n intervals, of J + 1 void roundseach, for the token to make a full cycle and so visit every station with old packets. Therefore, atmost n ( J + 1) + x clear rounds are needed to hear the x old packets. Consider a contiguous timesegment of z rounds in which some x packets are heard. At most zλ + β of these z rounds can bejammed. Therefore, the following inequality needs to hold: z ≤ n ( J + 1) + x + zλ + β . Solving for z , we obtain the following bound z ≤ x + n ( J + 1) + β − λ on a length of a contiguous time interval in which at least x packets are heard. (cid:3) Lemma 4 could be explained by referring to the stretching-by-jamming effect directly: thereare x rounds to successfully transmit the old packets, there are n ( J + 1) rounds to get the tokenaround, and there is the burstiness component β , each of them stretched by the factor 1 / (1 − λ ). Aphase takes close to the upper bound in Lemma 4 when the adversary does not jam the n intervalsof J + 1 void rounds, each used to advance the token once. In what follows, similar facts are arguedabout by referring directly to the stretching-by-jamming effect.During analyses of algorithms, if rounds are counted in disjoint intervals and the adversary jamsat full power then the burstiness component can be applied only once. So Lemma 4 may be usedfor one phase as formulated above, and in the remaining ones the bound is restricted to a smallerquantity ( x + n ( J + 1)) / (1 − λ ). 25 heorem 6 If algorithm
OF-JRRW( J ) is executed by n stations against a jamming adversary oftype ( ρ, λ, β ) such that its jamming burstiness at most J then the number of packets queued in anyround is at most β + 1)1 − ρ − λ · n + β (15) and packet latency is at most β + 1)(1 − λ )(1 − ρ − λ ) · n + β (1 + ρ − λ )(1 − λ ) . (16) Proof:
Let T i be the duration of phase i and Q i be the number of old packets in the beginningof phase i , for i ≥
1. The following two estimates lead to a recurrence for the numbers T i , in whichwe disregard the burstiness component. One estimate reads Q i +1 ≤ ρT i , (17)by the definitions of old packets and of type ( ρ, λ, β ) of the adversary, and the other estimate is T i +1 ≤ n ( J + 1) + Q i +1 − λ , (18)by Lemma 4. Let us denote n ( J + 1) = a . Substitute (17) into (18) to obtain T i +1 ≤ a + Q i +1 − λ ≤ a − λ + ρ − λ · T i ≤ c + dT i , for c = a − λ and d = ρ − λ . Note that d <
1, as ρ < − λ . An upper bound on the duration of aphase is found by iterating the recurrence T i +1 ≤ c + dT i to obtain a bound on the duration T ofa phase: c + dc + d c + . . . d i c ≤ c − d = T . (19)After substituting c = a − λ and d = ρ − λ into (19), we obtain the following estimate: T = a − λ · − ρ − λ = a − ρ − λ . (20)Replacing a by n ( J + 1) in (20) expands T to the following quantity: T = n ( J + 1)1 − ρ − λ . (21)We apply the estimate J ≤ β/ (1 − λ ) to (21) to obtain the following upper bound on T : T ≤ n ( β − λ + 1)1 − ρ − λ = n ( β + 1 − λ )(1 − λ )(1 − ρ − λ ) ≤ n ( β + 1)(1 − λ )(1 − ρ − λ ) . (22)A packet waits to be transmitted through at most two consecutive phases, each taking at most T rounds, where a bound for T given in (22) does not account for burstiness. Let the adversary injectextra β packets in a round of a phase. This increases the number of packets in the current phasebut extends the duration of the next phase by β − λ , which is the phase when these packets aretransmitted as old. These transmissions in turn allow the adversary to inject ρ · β − λ additional26ackets, which extends the duration of the immediately following phase by ρβ (1 − λ ) rounds by thestretching-by-jamming effect.We conclude with the following estimates. The maximum number of queued packets is obtainedby adding at most ρT old packets to at most ρT new packets, along with at most β packets injectedin a burst, which together makes the following bound:2 ρn ( β + 1)(1 − λ )(1 − ρ − λ ) + β ≤ n ( β + 1)1 − ρ − λ + β , where we used ρ < − λ . This yields (15). The maximum number of rounds spent by a packetwaiting to be heard on the channel is obtained by adding twice the upper bound T on a duration ofa phase (22), incremented by β − λ extra rounds in the phase immediately following one of a burstyinjection, along with ρβ (1 − λ ) rounds of the following phase. This gives the following amount:2 · n ( β + 1)(1 − λ )(1 − ρ − λ ) + β − λ + ρβ (1 − λ ) = 2 n ( β + 1)(1 − λ )(1 − ρ − λ ) + β (1 + ρ − λ )(1 − λ ) , where we used ρ < − λ . This yields (16). (cid:3) The bound of Theorem 6 is tight, by the following scenario. A phase includes n ( J + 1) voidrounds to advance the token around, which the adversary does not jam. If the adversary injectsat full power, and at the same time jams at full power the rounds during which some station triesto transmit, then this is equivalent to injections with rate ρ − λ . Eventually phases get arbitrarilyclose to the following magnitude, by combined stretching: n ( J + 1)1 − λ · − ρ − λ = n ( J + 1)1 − ρ − λ . If the adversary is such that J = β − λ then a phase takes close to nβ (1 − λ )(1 − ρ − λ ) rounds. The numberof packets injected during a phase of such duration can be made close to ρ · nβ (1 − λ )(1 − ρ − λ ) , which canbe made asymptotic to nβ − ρ − λ , if ρ = Θ(1 − λ ).Next, we analyze algorithm JRRW ( J ). Theorem 7
If algorithm
JRRW( J ) is executed by n stations against a jamming adversary of type ( ρ, λ, β ) such that its jamming burstiness at most J then the number of packets stored in the queuesin any round is at most β + 1)1 − ρ − λ · n + β (23) and packet latency is at most β + 1)(1 − λ )(1 − ρ − λ ) · n + β (1 − λ )1 − ρ − λ . (24) Proof:
Packets injected by the adversary may be transmitted in the current phase or in the nextone, depending one how the station into which they are injected is related to the station with atoken. We consider separately the impact of such injections to extend phases, by first estimating thephase length when packets are transmitted in the next phase and then when they are transmittedin the current phase. 27ackets injected at stations behind the one that holds the token at the moment are transmittedin the next phase. These new packets will be visited by the token only after they become old. Itfollows that the adversary can make algorithm
JRRW ( J ) behave as OF-JRRW ( J ) by choosingstations to inject packets into in this very manner. If all packets are injected this way, an upperbound on the duration of a phase is given by (22), which we denote by T = n ( β +1)(1 − λ )(1 − ρ − λ ) .Next, we estimate the contribution of packets injected at stations ahead of the station thatholds the token at the moment, and which are transmitted in the current phase, compounded withpackets already at the stations, which were injected behind the station holding the token. Thepackets get injected with the rate extended by stretching-by-jamming effect. The total number ofrounds in such a phase is at most T + T · ρ − λ + T · (cid:16) ρ − λ (cid:17) + . . . = T − ρ − λ = T · − λ − ρ − λ . (25)Substituting T = n ( β +1)(1 − λ )(1 − ρ − λ ) into (25) results in the following bound n ( β + 1)(1 − λ )(1 − ρ − λ ) · − λ − ρ − λ = n ( β + 1)(1 − ρ − λ ) , (26)which is the maximum possible length of a single phase, if we disregard the effects of burstiness.To account for burstiness, the adversary can inject β packets in front of the token, and then byiterating stretching-by-jamming by injecting at full power, the resulting extra β rounds get extendedto β · − λ − ρ − λ . The duration of two consecutive phases is bounded from above by a sum of (22), whichwe denote by T = n ( β +1)(1 − λ )(1 − ρ − λ ) , of (26), and of a one-time extension of a phase due to burstiness,which we calculated to be β · − λ − ρ − λ . They together make the following bound: n ( β + 1)(1 − λ )(1 − ρ − λ ) + n ( β + 1)(1 − ρ − λ ) + β · − λ − ρ − λ ≤ n ( β + 1)(1 − λ )(1 − ρ − λ ) + β (1 − λ )1 − ρ − λ , which is the upper bound (24). (cid:3) The upper bound given in Theorem 7 is asymptotically tight, which can be justified by thefollowing scenario. Let the adversary initially inject behind the token, which results in all injectedpackets transmitted in the next phase. The accompanying pattern of jamming is such as to makequeues and packet latency get asymptotic to the bounds given in Theorem 6. This gives thetightness of queue bounds, as they are identical in Theorems 6 and 7. At this point, a phase takesclose to nβ (1 − λ )(1 − ρ − λ ) rounds. Now, the adversary switches to injecting just before the token, tomake the old packets injected in the previous phase and the currently injected packets transmittedin the current phase, so there are no outstanding packets when the phase is over. Injecting andjamming at full power has the effect of stretching injection rate to ρ − λ , which eventually makes aphase take close to the following amount: nβ (1 − λ )(1 − ρ − λ ) · − λ − ρ − λ = nβ (1 − ρ − λ ) , by the estimate as in (25), which is asymptotic to (24), if λ = Θ(1).The upper bound on packet latency given in Theorem 7 differs by the factor − ρ − λ > ρ + λ gets suitably close to 1.This difference between the two bounds reflects the benefit of the approach “old-go-first” applied inthe design of algorithm OF-JRRW ( J ), as compared to algorithm JRRW ( J ).28 Adaptive Algorithms with Jamming
We give worst-case upper bounds on queues size and packet latency against jamming adversariesfor the following three adaptive algorithms:
OFC-RRW , C-RRW , and
C-MBTF . Each of thesealgorithms is stable for any jamming burstiness, unlike the non-adaptive algorithms we consideredin Section 6, which include in their codes a bound on jamming burstiness which they can withstandin a stable manner.First, we estimate the worst-case performance of
OFC-RRW , which combines adaptivity withthe old-go-first approach, on top of the round-robin-withholding way to use a token.
Lemma 5
If there are x old packets in the queues, when a phase of algorithm OFC-RRW executedby n stations begins, against a type ( ρ, λ, β ) adversary, then the phase takes at most the followingnumber of rounds: x + n − λ . Proof:
It takes up to n control rounds for the token to pass through all n stations. It takes x rounds to hear the x packets. These x + n rounds can be extended to x + n − λ by the stretching-by-jamming effect. (cid:3) Now we give performance bounds for algorithm
OFC-RRW . Theorem 8
If algorithm
OFC-RRW is executed by n stations against a jamming adversary oftype ( ρ, λ, β ) then the number of packets queued in any round is at most ρ − ρ − λ · n + β (27) and packet latency is at most − ρ − λ · n + β (1 + ρ − λ )(1 − λ ) . (28) Proof:
Let T i denote an upper bound on the duration of phase i , for i ≥
1, where T = n − λ , asit consists of n rounds possibly stretched by jamming. Let Q i be the number of old packets in thebeginning of phase i , for i ≥
1. We use the following two estimates to derive a recurrence for thenumbers T i . One is Q i +1 ≤ ρT i , (29)which follows from the definition of old packets and the adversary of type ( ρ, λ, b ). The other is T i +1 ≤ Q i +1 + n − λ , (30)which follows from Lemma 5. Using the abbreviations c = n − λ and d = ρ − λ , we substitute (29)into (30) to obtain T i +1 ≤ n − λ + ρ − λ T i ≤ c + dT i . To find an upper bound T on the duration of a phase, we iterate the recurrence T i +1 ≤ c + dT i ,which produces c + dc + d c + . . . d i c ≤ c − d = T . (31)29fter substituting c = n − λ and d = ρ − λ into (31), we obtain the following estimate of the durationof a phase T = n − λ · − ρ − λ = n − λ · − λ − ρ − λ = n − ρ − λ . (32)A packet spends at most two consecutive phases waiting to be heard. A phase takes at most T rounds, where a bound for T is given in (32). This bound does not include effects due to burstiness.To extend it, we can argue as follows. Let the adversary inject extra β packets in a round of aphase. This extends the duration of the next phase by β − λ , because the injected packets will beold then. Next, the adversary injects extra ρβ − λ additional packets, which are transmitted in thenext phase to extend its duration by ρβ (1 − λ ) rounds, by stretching-by-jamming.Now we can estimate packet latency as 2 T incremented by the effects of jamming, to obtain2 n − ρ − λ + β − λ + ρβ (1 − λ ) = 2 n − ρ − λ + β (1 + ρ − λ )(1 − λ ) , which is the bound (28). The number of packets in the stations’ queues equals the sum of thenumbers of the new and old packets, which is at most 2 T ρ , increased by burstiness to 2
T ρ + β ,which combined with (32) yields the following value2 nρ − ρ − λ + β as a bound on the queue size (27). (cid:3) The bounds of Theorem 8 are tight, as can be argued as follows. Let the adversary inject atfull power. The first phase takes exactly n clear rounds, which are extended to n − λ rounds byjamming at full power. During this time, the adversary injects ρn − λ packets. So the combinedeffect of jamming and injecting at full power is injecting with rate ρ − λ . The duration of phaseskeeps increasing such that when one takes r rounds then the next one takes r (1 + ρ − λ ) rounds.Eventually, the duration of a phase gets arbitrarily close to n − λ (cid:0) ρ − λ + (cid:0) ρ − λ (cid:1) + . . . (cid:1) = n − λ · − ρ − λ = n − ρ − λ . If a phase lasts close to this number of rounds, the adversary injects about ρn − ρ − λ packets. In oneround, the adversary injects β packets, which increases the number of packets in the queues toabout ρn − ρ − λ + β . These extra β packets then allow the adversary to extend the duration of thenext two phases by close to β − λ + ρβ (1 − λ ) = β (1 − λ + ρ )(1 − λ ) many rounds.Next, we estimate the performance of algorithm C-RRW . Theorem 9
If algorithm
C-RRW is executed by n stations against a jamming adversary of type ( ρ, λ, β ) then the number of packets queued in any round is at most ρ − ρ − λ · n + β (33) and packet latency is at most − λ )(1 − ρ − λ ) · n + β (1 − λ )1 − ρ − λ . (34)30 roof: Packets injected behind the token are transmitted in the next phase, which is consistentwith the behavior of
OF-RRW and so with its bound on the queue size. Packets injected ahead ofthe token are transmitted in the phase of injection, which slows down the phase compared to phasesof
OF-RRW . Each such an extra round is spent on a transmission, because this why a phase islonger, while it is not necessary to have a packet injected in each extra round. Therefore the upperbound on the number of packets stored in the queues (27) derived for algorithm
OFC-RRW alsoapplies to
C-RRW , so it is made equal to (33).We estimate how injected packets contribute to extending phases by separately consideringpackets that are transmitted after a phase of injection and those that are transmitted in a phase ofinjection. Packets injected behind the token are transmitted in the next phase, which means thatthis manner of injecting packets can increase a phase length to be at most as long as the boundof (32), obtained for algorithm
OFC-RRW . Let us denote this bound by T = n − ρ − λ .Packets injected ahead of the token get transmitted in the phase of injection. We may assumewithout loss of generality that when a phase begins the stations store so many old packets that thephase would last T rounds without any additional injections. If the adversary switches to injectingahead of the token then combined stretching can extend a phase to at most the following duration: T (cid:16) ρ − λ + (cid:0) ρ − λ (cid:1) + . . . (cid:17) = T − ρ − λ = T · − λ − ρ − λ . (35)Substituting T = n − ρ − λ into (35) produces the following bound on the duration of a phase: n − ρ − λ · − λ − ρ − λ ≤ n (1 − λ )(1 − ρ − λ ) . (36)When such a phase ends, then there are no old packets to be transmitted in the following phase.Therefore the lengths of two consecutive phases is at most a sum of (32) and (36), when disregardingthe effect of burstiness. One phase may be further extended by double stretching combined withburstiness by β (1 − λ )1 − ρ − λ rounds. We can conclude with the following bound on packet latency n − ρ − λ + n (1 − λ )(1 − ρ − λ ) + β (1 − λ )1 − ρ − λ ≤ n (1 − λ )(1 − ρ − λ ) + β (1 − λ )1 − ρ − λ , (37)which is the bound (34). (cid:3) The tightness of the bounds given in Theorem 9 can be established as follows. The queue-sizebound is the same as for algorithm
OFC-RRW , and the adversary can make the number of packetsget close to the bound by making
C-RRW behave like
OFC-RRW by injecting just behind thetoken. In a similar manner, a phase’s duration can become close to (31). Then the adversaryswitches to injecting at full power ahead of the token to create one phase of a length close to (36).When the second of these two consecutive phases gets extended by burstiness combined with doublestretching, the two consecutive phases take time close to the bound in the derivation (37).Finally, we estimate the queue sizes and packet latency of algorithm
C-MBTF against jammingadversaries.The relevant terminology we use follows the one developed for executions of algorithm
MBTF discussed in Section 5, in particular, the vocabulary related to kinds of passes. Similarly, we refer tothe stations by their positions on the list of all the stations, numbered 1 , , , . . . , n , with station 1at the front of the list, and station n at the end.31 emma 6 If algorithm
C-MBTF is executed by n stations against a jamming adversary of type ( ρ, λ, β ) such that all the passes have been small since the beginning of the execution, then thenumber of packets stored in the queues in a round of a still small pass is at most ρn − λ + β . (38) Proof:
A small pass takes n clear rounds, which can be extended to take n − λ rounds by thestretching-by-jamming argument. During a time segment of these many rounds the adversary caninject ρn − λ packets.We may assume that a pass under consideration is non-decreasing, as otherwise the last sucha pass would witness more packets in the queues. The quantity ρn − λ is an upper bound on thenumber of stations with packets during a non-decreasing small pass, because if there were moresuch stations, then each of them would transmit a packet during a pass and more packets wouldbe transmitted than injected.Each station with packets has at most n − ρn − λ · ( n −
1) + ρn − λ = ρn − λ when the pass is over. The adversary can inject extra β packets in any round of the passes used tocollect these many packets in queues, which justifies the bound (38). (cid:3) A limit that small passes impose on the adversary to build queues is captured by Lemma 6.Big passes may contribute to delaying specific packets by preventing the token to reach the tail ofthe list of stations over an extended period of rounds. This may be amplified by the token passingthrough many stations with empty queues before discovering a big station, while the fraction ofvoid rounds in a pass could be greater than − ρ − λ . During a cascade of such passes, the adversarymay maintain a fraction ρ − λ of the number of rounds with injections, while the fraction of thenumber of rounds when messages are heard on the channel is continuously smaller than ρ − λ , thuscontributing to accumulation of packets in the queues. These insights are employed in the proof ofTheorem 10. Theorem 10
If algorithm
C-MBTF is executed by n stations against a jamming adversary oftype ( ρ, λ, b ) then the number of packets queued in any round is at most ρ (1 − λ ) + ρ (1 − λ ) · n + β (39) and packet latency is at most ρ − λ − ρ − ρλ (1 − λ )(1 − ρ − λ ) · n + β (1 − λ )1 − ρ − λ . (40) Proof:
We initially disregard the effect of burstiness in the course of an execution. If no bigstation has been discovered yet then there are at most ρn − λ packets in queues, by Lemma 3. We32xplore now how much the queues can increase when big passes occur. If there are at most ρn − λ stations with packets then the sum of lengths of big passes is maximized when the stations withpackets are located at the end of the list and when each time the token reaches one of these stations,for the first time since big passes started to occur, then the station is discovered to be big. Thesum of lengths of big passes can be estimated to be at most the following: ρn − λ X i =1 (cid:0) n − ρn − λ + i (cid:1) ≤ ρn − λ , which is the number of clear rounds only. These big passes can be extended by stretching-by-jamming to last at most − λ · ρ − λ · n rounds during which at most ( ρ − λ ) · n new packets areinjected. The total number of packets at this point is at most ρ − λ · n + ρ (1 − λ ) · n = ρ (1 − λ ) + ρ (1 − λ ) · n . This can be increased to at most (39) by injecting β packets in one round.Next we estimate packet latency. Let us consider a round when some packet p is injected andcount the rounds until it is heard. Burstiness-related effects are disregarded through the initialstages of the analysis, to be accounted for at the end. We may assume that p gets injected whenthe token is at the first station and the configuration of packets is as in Lemma 6, which is suchthat at most ρn − λ packets are located in the last ρn stations on the list, each holding at most n packets. The last station stores the packet p . If there are n − p in its queuethen p will be transmitted when the token visits its station n − p ’s injection. Then thetoken will need to cover the whole length of the list n − p . Each such a traversalof the whole list makes a small pass, since the last station in the list stores fewer than n packets.We estimate how much time may pass before the token visits the p ’s station when p is at the topof the queue ready to be transmitted, by accounting for the following groups of rounds contributingto p ’s waiting time:(1) a delay due to discovering big stations,(2) a delay due to small passes and packets injected during these passes,(3) the effect of burstiness.We begin with the effect of discovering big stations by identifying a worst-case scenario. Startingfrom the injection of p , the adversary may inject packets into the trailing ρn − λ stations to makeeach of them but the last one big. The discoveries of up to ρn − λ big stations at the end of the listprovide delays of up to these many clear rounds: ρn − λ X i =1 (cid:0) n − ρn − λ + i (cid:1) ≤ ρn − λ . During these big passes, and until the last big station becomes small after moved to the front, theworst case occurs when the duration of waiting time is obtained from the number of clear roundsby combined stretching to at most these many rounds in total: ρn − λ (cid:0) ρ − λ + ρ (1 − λ ) + . . . (cid:1) = ρn − λ · − ρ − λ = ρ − ρ − λ · n . (41)33ext, we consider the effect of small passes. There are n − p , each requiring n clear rounds, for the total of n ( n −
1) clear rounds, which can be extendedby stretching-by-jamming to at most n − λ . There is also a delay caused by packets injected duringsmall passes. It is upper bounded by the number of packets injected during small passes, as shownin the proof of Theorem 5. The number of packets injected during small passes is at most ρ · n − λ ,which makes the total delay incurred by small passes to be at most the following: n − λ + ρn − λ = 1 + ρ − λ · n . (42)Burstiness amplified by combined stretching adds at most these many rounds to a big pass: β (cid:0) ρ − λ + ρ (1 − λ ) + . . . (cid:1) = β − ρ − λ = β (1 − λ )1 − ρ − λ . (43)We have identified the three components (41), (42) and (43) of packet delay. Adding them togethergives a total of at most these many rounds: ρ − ρ − λ · n + 1 + ρ − λ · n + β (1 − λ )1 − ρ − λ = ρ (1 − λ ) + (1 + ρ )(1 − ρ − λ )(1 − λ )(1 − ρ − λ ) · n + β (1 − λ )1 − ρ − λ (44)= 1 + ρ − λ − ρ − ρλ (1 − λ )(1 − ρ − λ ) · n + β (1 − λ )1 − ρ − λ , which is the bound (40). (cid:3) The bounds given in Theorem 10 are asymptotically tight. We verify this by giving a specificstrategy of an adversary of type ( ρ, λ, β ) to make queues grow suitably big and a packet delayedby a suitable number of rounds.First, we consider the queues. The factor ρ (1 − λ )+ ρ (1 − λ ) in the upper bound (39) is Θ( ρ − λ ) because1 < ρ − λ <
2. It is sufficient to show how to construct a configuration with Ω( ρ − λ · n + β )queued packets. Let the adversary begin by building and maintaining queues of n − ρ − λ · n selected stations. This is accomplished during a sequence of n − n clearrounds each, which can be extended to n − λ rounds a pass by the stretching-by-jamming effect.During each such a pass, the adversary injects ρ − λ · n packets. Packets are injected two per station,for as long as there are fewer than n − ρ − λ · n . This quantity can be increased tothe claimed magnitude by injecting β packets in one round.Next we consider packet latency. The factor ρ − λ − ρ − ρλ − λ in the upper bound (40) is Θ(1)because of the following estimates1 < ρ + (1 + ρ ) (cid:0) ρ − λ (cid:1) = 1 + ρ − λ − ρ − ρλ − λ < − λ + ρ − λ < , where we used ρ < − λ . It is sufficient to show how to create a packet whose delay isΩ (cid:16) n + β (1 − λ )1 − ρ − λ (cid:17) = Ω (cid:18) n − λ + β − ρ − λ (cid:19) . n − ρ − λ · n stations in the courseof a series of small passes. In each such a pass, the number of packets in a station with packetsgrows by one, while all the remaining stations have empty queues. It takes at least n clear roundsto complete these passes, which can be extended to n − λ ) by stretching-by-jamming. A packet p is injected into the last station at the bottom of its queue. Let the adversary make each of thestations with packets big by inserting one extra packet, starting with the station in the smallestposition, but skipping the last station. After that, the adversary keeps injecting at full power intothe station that is last but one.We consider two cases. The first case is when ρ − λ ≤ . Then the following upper bound holds n − λ + β − ρ − λ ≤ n − λ + 2 β . The big passes contribute at least β rounds and the small passes that followcontribute at least n − λ ) rounds. The second case is when ρ − λ > . The big passes generate atleast P n i =1 (cid:0) n − n + i (cid:1) ≥ n void rounds in total. This amount can be extended to at least · n − λ rounds by stretching-by-jamming. When the last-but-one station is discovered big, the adversaryinjects β packets into it, to contribute β more rounds needed to transmit these packets. The numberof rounds · n − λ + β can be extended by double stretching to at least these many rounds18 · n + β (1 − λ )1 − λ · − λ − ρ − λ = 18 · n + β (1 − λ )1 − ρ − λ . This quantity grows unbounded if ρ − λ converges to 1. We present a comprehensive study of distributed deterministic broadcast algorithms in adversarialmultiple-access channels, in which an adversary controls packet injection and, optionally, jamming.The model assumes a fixed set of n stations attached to a shared channel, with the stations equippedwith unique names in the interval [0 , n − n using the channel.Algorithms are categorized as either adaptive or non-adaptive, channels may either have collisiondetection mechanism or not, and an adversary may either be able to jam the channel or not. Thecase of channels with both jamming and collision detection is omitted from the considerations, inthat we do not consider algorithms designed specifically for such an environment. There are tworeasons for this apparent omission. One is that this model does not allow to distinguish a round withcollision form a jammed one, so it is not clear how to make use of collision detection. Secondly, suchenvironments allow to execute any algorithm for a channel without collision detection and withoutjamming that avoids collisions, adapted only by the stipulation that jammed rounds, which aredetected by a collision-detection mechanism, are to be ignored.Algorithms OF-SRR and
SRR for channels without jamming but with collision detection havebounds on running times about twice as large as algorithms
OF-RRW and
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