Tight Trade-off in Contention Resolution without Collision Detection
aa r X i v : . [ c s . D C ] F e b Tight Trade-off in Contention Resolution without Collision Detection
HAIMIN CHEN,
Nanjing University, China
YONGGANG JIANG,
Nanjing University, China
CHAODONG ZHENG,
Nanjing University, China
In this paper, we consider contention resolution on a multiple-access communication channel. In this problem, a set of nodes arriveover time, each with a message it intends to send. In each time slot, each node may attempt to broadcast its message or remain idle. If asingle node broadcasts in a slot, the message is received by all nodes; otherwise, if multiple nodes broadcast simultaneously, a collisionoccurs and none succeeds. If collision detection is available, nodes can differentiate collision and silence (i.e., no nodes broadcast).Performance of contention resolution algorithms is often measured by throughput—the number of successful transmissions withina period of time; whereas robustness is often measured by jamming resistance—a jammed slot always generates a collision. Previouswork has shown, with collision detection, optimal constant throughput can be attained, even if a constant fraction of all slots arejammed. The situation when collision detection is not available, however, remains unclear.In a recent breakthrough paper [Bender et al., STOC ’20], a crucial case is resolved: constant throughput is possible withoutcollision detection, but only if there is no jamming. Nonetheless, the exact trade-off between the best possible throughput and theseverity of jamming remains unknown. In this paper, we address this open question. Specifically, for any level of jamming rangingfrom none to constant fraction, we prove an upper bound on the best possible throughput, along with an algorithm attaining thatbound. An immediate and interesting implication of our result is, when constant fraction of all slots are jammed, which is the worst-case scenario, there still exists an algorithm achieving a decent throughput: Θ ( 𝑡 / log 𝑡 ) messages could be successfully transmittedwithin 𝑡 slots. Contention resolution is a classical problem in parallel and distributed computing. In this problem, there are multipleplayers trying to access a shared resource; and the problem is solved once each player has successfully accessed theresource (at least) once. This problem is complicated by the requirement that accesses must be mutual exclusive : if twoor more players try to utilize the shared resource simultaneously, none would succeed. Contention resolution saw var-ious applications in computer science, some prominent examples include congestion control in computer networking(e.g., Ethernet and IEEE 802.11 wireless networks [16, 18]), concurrency control in database management systems andoperation systems (e.g., locking [20, 22]).In this paper, we consider the following more concrete setting, which is a standard model used in many papersstudying the problem (see, e.g., [6, 8]). The shared resource is a multiple-access communication channel , and time isdivided into discrete and synchronized slots . Each player (also called a node ) joins the system at the beginning of someslot with a single message (also called a packet ) it intends to send. We assume node arrival times are controlled by anadaptive adversary. In each slot, each node can either try to broadcast its message on the channel or remain idle. In casea single node broadcasts, the transmission succeeds and all participating nodes receive the unique message; otherwise,if multiple nodes broadcast simultaneously in a slot, a collision occurs and all transmission attempts in that slot fail.Upon collision, the exact channel feedback depends on the availability of a collision detection mechanism. Specifically,with collision detection, nodes can tell whether a wasted slot (i.e., a slot without a successful message transmission) isdue to silence (i.e., no node broadcasts in this slot) or collision (i.e., multiple nodes broadcast in this slot). By contrast,if collision detection is not available, nodes cannot differentiate silent slots and colliding slots.Achieving a high throughput is the primary goal of contention resolution algorithms. Though exact definition of aimin Chen, Yonggang Jiang, and Chaodong Zheng “throughput” differ in various papers, intuitively it measures how many messages an algorithm can successfully trans-mit within a period of time.Transmission attempts could fail even without collision, as the shared communication channel could suffer fromhardware/software errors (e.g., a shared printer may crash due to hardware/software bugs) or unintentional/intentionalexternal interference (e.g., a wireless link may be affected by electromagnetic noise). In the context of contentionresolution, such failures are often modeled by jamming (see, e.g., [4, 6, 10, 21]). Formally, if a slot is jammed, then acollision occurs on the channel in that slot, regardless of the actual number of broadcasting nodes.An important question in studying various contention resolution algorithms is to understand their robustnessagainst jamming. More fundamentally, in solving contention resolution, what is the inherent trade-off between throughput—measuring performance, and jamming-resistance—measuring fault-tolerance? As it turns out, the availability of col-lision detection makes a huge difference. More specifically, in case collision detection is available, even if a constantfraction of all slots could be jammed (which is the asymptotic worst-case scenario), constant throughput (which is theasymptotic optimal throughput) can be attained [6, 10]. By contrast, in the absence of collision detection, even withoutjamming, whether optimal throughput is possible for the vanilla contention resolution problem remains unknown fora long time. Last year, in a breakthrough paper [8], Bender et al. showed that such algorithm does exist. Moreover, theyhave also proved an impossibility result implying constant throughput is impossible when constant fraction of slotscould be jammed. However, an intriguing question remains open: in solving contention resolution, what is the exacttrade-off between throughput and jamming-resistance, when collision detection is not available? Main contribution.
In this paper, we answer the above question by providing a tight and complete characterizationof the trade-off between the best possible throughput and the severity of jamming. In particular, for any level ofjamming ranging from none to constant fraction, we have an upper bound on the best possible throughput, along witha corresponding algorithm attaining that bound. Together with previous work, this marks the complete understandingof the throughput versus jamming-resistance trade-off for the contention resolution problem.In the reminder of this section, we will discuss some additional model details, then proceed to a more careful intro-duction of our main results. We will conclude this section with a brief survey on related work.
Additional model details.
We assume each node has a single message to send, and a node will leave the systemimmediately once its message has been successfully transmitted. We assume node arrival times and jamming are bothcontrolled by an adaptive adversary. We often call this adversary Eve, and her adaptivity is reflected by the assumptionthat, in each slot, she could use past channel feedback to determine whether to jam the slot, and whether to inject (oneor more) new nodes. Notice, the multiple-access channel provide identical feedback to the nodes and the adversary,meaning Eve also does not posses the ability of collision detection.
Statement of results.
Throughout this paper, we call a slot active if there is at least one player in the system in thatslot. Following classical definition of throughput is adopted from [8]: let 𝑛 𝑡 be the number of players arriving thesystem in the first 𝑡 slots, and let 𝑎 𝑡 be the number of active slots among the first 𝑡 slots, the 𝑡ℎ𝑟𝑜𝑢𝑔ℎ𝑝𝑢𝑡 at slot 𝑡 isdefined to be 𝑛 𝑡 / 𝑎 𝑡 . An algorithm achieves a certain throughput 𝜆 if for all slots the throughput is lower bounded by 𝜆 .That is to say, for instance, if an algorithm achieves a constant throughput, then the number of active slots is at mostsome constant factor larger than the number of player arrivals.In this paper, we extend the above definition to better reflect the influence of jamming, see following. Definition 1.1 ( ( 𝑓 , 𝑔 ) -throughput). Let A be the algorithm each node runs after arriving. Denote the number of newlyarrived players and the number of jammed slots in the first 𝑡 slots as 𝑛 𝑡 and 𝑑 𝑡 , respectively. Let 𝑓 , 𝑔 : R + → R + be ight Trade-off in Contention Resolution without Collision Detection two functions. We say A achieves ( 𝑓 , 𝑔 ) -throughput if for any integer 𝑡 ≥ and any adaptive adversary strategy, thenumber of active slots in the first 𝑡 slots is at most 𝑛 𝑡 · 𝑓 ( 𝑡 ) + 𝑑 𝑡 · 𝑔 ( 𝑡 ) , with high probability in 𝑛 𝑡 . Notice that the above definitions of throughput (both the classical one and ( 𝑓 , 𝑔 ) -throughput) do not claim a boundon the number successful transmissions within a time interval. Nonetheless, as Bender et al. [8] have pointed out, theycould imply such results. To see this, consider an algorithm that achieves ( 𝑓 , 𝑔 ) -throughput and an interval of length 𝑡 .If the adversary does not jam too many slots or inject too many nodes in the sense that 𝑛 𝑡 · 𝑓 ( 𝑡 ) + 𝑑 𝑡 · 𝑔 ( 𝑡 ) < 𝜆𝑡 , thenone of the most recent 𝜆𝑡 slots is inactive, implying any node arriving before ( − 𝜆 ) 𝑡 has succeeded. We capture thisrelationship between ( 𝑓 , 𝑔 ) -throughput and the number of successes more precisely in Corollary 3.6 in Section 3.As mentioned earlier, our main result concerns with the trade-off between the best possible throughput and theseverity of jamming. Interestingly, this comes down to a trade-off between 𝑓 and 𝑔 . To see this intuitively, notice thatsuccessfully sending 𝑛 𝑡 messages in 𝑡 slots requires 𝑛 𝑡 · 𝑓 ( 𝑡 ) + 𝑑 𝑡 · 𝑔 ( 𝑡 ) < 𝑡 , which implies ( 𝑛 𝑡 / 𝑡 ) · 𝑓 ( 𝑡 ) + ( 𝑑 𝑡 / 𝑡 ) · 𝑔 ( 𝑡 ) ≤ .This suggests, if an algorithm wants to send 𝑛 𝑡 messages within 𝑡 slots, then 𝑛 𝑡 / 𝑡 —which roughly corresponds tothroughput—would be at most / 𝑓 , and 𝑑 𝑡 / 𝑡 —which corresponds to severity of jamming—would be at most / 𝑔 . Thishints the throughput versus jamming-resistance trade-off is in fact a trade-off between 𝑓 and 𝑔 .The following two theorems state the exact relationship between functions 𝑓 and 𝑔 : Theorem 1.2 (Algorithmic Result).
For any function 𝑔 such that log ( 𝑔 ) is sub-logarithmic, there exist a function 𝑓 where 𝑓 ( 𝑥 ) ∈ Θ (cid:16) log 𝑥 log 𝑔 ( 𝑥 ) (cid:17) and an algorithm achieving ( 𝑓 , 𝑔 ) -throughput. Theorem 1.3 (Impossibility Result).
For any functions 𝑓 and 𝑔 such that 𝑓 and log ( 𝑔 ) are both sub-logarithmic, if 𝑓 ( 𝑥 ) ∈ 𝑜 (cid:16) log 𝑥 log 𝑔 ( 𝑥 ) (cid:17) , then there does not exist any algorithm achieving ( 𝑓 , 𝑔 ) -throughput.Remark 1. Throughout this paper, we say a function 𝑓 : R + → R + is sub-logarithmic if: (1) 𝑓 ( 𝑥 ) ∈ 𝑂 ( log 𝑥 ) and isnon-decreasing; (2) for some large constant 𝑦 , there exists some constant 𝑥 such that 𝑓 ( 𝑥 ) ≥ 𝑦 when 𝑥 ≥ 𝑥 ; (3) forany constant 𝑐 > , there exists some constant 𝑐 such that for any 𝑛 ∈ N + , | 𝑓 ( 𝑐𝑛 ) − 𝑓 ( 𝑛 )| ≤ 𝑐 ; (4) for any constant 𝑐 > , 𝑓 ( 𝑥 𝑐 ) ∈ Θ ( 𝑓 ( 𝑥 )) . We define “sub-logarithmic” this way so as to avoid artificial pathological functions. Remark 2.
In Theorem 1.2, when log 𝑔 ( 𝑥 ) ∈ Θ ( p log 𝑥 ) , 𝑓 becomes a constant function and our algorithm couldachieve constant throughput, which is the best possible throughput.To illustrate the application of the above results, consider two interesting cases. Suppose 𝑔 is a constant function,meaning Eve can jam some constant fraction of all 𝑡 slots, then the best ratio of the number of player arrivals over 𝑡 —which roughly corresponds to throughput—is Θ ( / log 𝑡 ) . Moreover, we can devise an algorithm that attains thisthroughput. (This means in the absence of collision detection, even with constant fraction of jamming, we could send Θ ( 𝑡 / log 𝑡 ) messages in 𝑡 slots, achieving a decent—though sub-constant—throughput.) On the other hand, if we wouldlike 𝑓 to be some constant function, meaning sending Θ ( 𝑡 ) messages within 𝑡 slots, then the maximum number ofslots Eve can jam must be bounded by Θ (√ log 𝑡 ) . Again, we can devise an algorithm that tolerates such a jammingadversary. Related work.
One simple and standard algorithm to resolve contention is binary exponential backoff . In its classicalimplementation (e.g., in Ethernet [18]), each participating node waits a random time interval before trying to broadcastits message; if a transmission attempt failed, the node waits another randomly chosen time interval before retrying,and the expected length of the waiting interval doubles after each failure. An event happens with high probability (w.h.p.) in some parameter 𝜆 if it happens with probability at least − / 𝜆 𝑐 , for some constant 𝑐 ≥ .3 aimin Chen, Yonggang Jiang, and Chaodong Zheng Unfortunately, simple exponential backoff cannot provide optimal throughput, for both statistical arrival patterns [2,15] and batch/adversarial arrival patterns [5]. In view of this, numerous variations of the binary exponential backoffscheme have been proposed and analyzed (such as polynomial backoff and saw-tooth backoff), again for both statisticalarrival patterns (e.g., [9, 19]) and batch/adversarial arrival patterns (e.g., [5, 12]). This paper considers adversarial arrivalpattern, and the proposed algorithm utilizes two exponential backoff variants as key subroutines.Simple backoff algorithms usually do not depend on the availability of collision detection: nodes decrease sendingprobabilities (i.e., backoff) whenever an empty slot is observed. However, this behavior is not always correct: an emptyslot could also mean no node tries to broadcast; in such case, nodes should be more aggressive and increase theirsending probabilities (i.e., backon). Therefore, with collision detection, more clever backoff-backon algorithms can bedevised (e.g., [4, 6, 10, 21]). Such algorithms are especially helpful if external interference is present: often optimalperformance can be attained in spite of jamming. However, when collision detection is not available, the exact impactof jamming on solving contention resolution remains unclear. Our paper addresses this open question.Beside throughput, another important metric when evaluating contention resolution algorithms is the number ofchannel accesses a node has to make before successfully sending its message (some authors call this the energy com-plexity). Many existing algorithms (e.g., [6, 8]) have 𝑂 ( poly-log ( 𝑛 )) energy complexity, assuming there are 𝑛 nodes inthe system. Nonetheless, somewhat surprisingly, Bender et al. [7] show that 𝑂 ( log ( log ∗ 𝑛 )) channel accesses per nodeis enough for resolving contention.It is worth noting, if we only care about the first success (instead of requiring each node to succeed once), then theproblem essentially degrades to leader election—a classical symmetry breaking task. In a seminal work by Willard [23],a tight bound of Θ ( log log 𝑛 ) slots is proved, assuming 𝑛 nodes are activated simultaneously. In case nodes are injecteddynamically, the problem is known as the “wake-up problem” [11] or the “synchronization problem” [13].Lastly, we note that contention resolution is an extensively studied problem, and many interesting results are notcovered here. (E.g., there are papers focusing on deterministic algorithms [3, 17], and papers considering performancemetric other than throughput [1].) Interested readers are encouraged to find dedicated survey papers for more details. Paper outline.
In Section 2, we will first give an overview of the algorithm that achieves optimal throughput for anygiven level of jamming, including some key design ideas; and then provide a complete description of the algorithm. InSection 3, we will analyze the proposed algorithm and prove our algorithmic result—Theorem 1.2. We will also provea corollary connecting ( 𝑓 , 𝑔 ) -throughput and number of successful transmissions. Finally, in Section 4, we will provetwo impossibility results. The first one is Theorem 1.3, while the second one demonstrates a certain type of exponentialbackoff cannot achieve optimal throughput, justifying some decisions we made during the algorithm design process. Our algorithm has same high-level framework as [8], but with key adjustments made specifically for achieving thebest possible throughput against jamming.
Algorithm framework.
It is known that binary exponential backoff cannot provide constant throughput, even if 𝑛 nodes are activated simultaneously. Nevertheless, in such “batch” scenario, the first Θ ( 𝑛 ) slots of the process couldachieve a constant throughput. To see this, consider the following implementation of binary exponential backoff: eachnode broadcasts with probability / 𝑖 in slot 𝑖 . For this algorithm, around slot index 𝑛 , if Θ ( 𝑛 ) nodes have already suc-ceeded then we are done. Otherwise, at least Θ ( 𝑛 ) nodes remain, and the sum of their broadcasting probability—oftencalled the contention of a slot—is Θ ( ) . As the contention of a slot corresponds to the expected number of broadcast- ight Trade-off in Contention Resolution without Collision Detection ing nodes in that slot, a constant contention means a successful transmission will occur with constant probability.Therefore, starting from slot 𝑛 , after another Θ ( 𝑛 ) slots, there is a good chance that at least Θ ( 𝑛 ) successes will occur.However, such high throughput cannot be maintained in later portion of binary exponential backoff. Hence, we needa mechanism to stop the process once Θ ( 𝑛 ) slots are executed, and then restart. If nodes can access two independentchannels, then the following method would work. On one channel, called the “data channel”, nodes execute the abovestandard exponential backoff algorithm. On the other channel, called the “control channel”, nodes execute a modifiedbackoff algorithm. The goal of the modified backoff algorithm is to let the first success of the control channel occur inslot Θ ( 𝑛 ) , so that backoff on the data channel can stop at the right time. As it turns out, the modified backoff algorithmis pretty simple: each node broadcasts with probability ( log 𝑖 )/ 𝑖 in slot 𝑖 .In the vanilla contention resolution problem, nodes are injected dynamically over time, thus another mechanism isrequired to “synchronize” nodes, so that they can start a backoff process on the data channel in a batch manner. Again,with two independent channels, a simple solution exists. Specifically, a newly arrived node first runs exponentialbackoff on the control channel, until a success occurs on the control channel. (A new node cannot simply listen andwait for a success, as it might be the only node in the system.) At that point, all nodes in the system are synchronizedand can start an efficient new batch on the data channel.At this point, the only remaining issue is that the model only provides one channel. If nodes have access to a globalclock, then an easy solution would be: (1) groups odd slots together and call it “odd channel”, then assign odd channelto be control channel; and (2) groups even slots together and call it “even channel”, then assign even channel to be datachannel. Unfortunately, such global clock is also not available. Therefore, we need yet another mechanism to allownodes to make consensus on the role of slots. We defer the discussion of this mechanism to algorithm description. Achieving jamming resistance.
In the above algorithm framework, two types of backoff algorithms are used intwo different settings, with different purposes: (1) truncated exponential backoff in batch setting, with the goal ofachieving Θ ( 𝑛 ) successes in Θ ( 𝑛 ) slots, assuming 𝑛 nodes start simultaneously; (2) standard exponential backoff indynamic setting, with the goal of achieving a single success efficiently. It turns out that the truncated exponentialbackoff process is extremely robust against jamming. In particular, among the first Θ ( 𝑛 ) slots, even if a constantfraction is jammed, the procedure could still guarantee Θ ( 𝑛 ) successes, and halt at the correct time. On the other hand,however, standard backoff performs poorly against jamming in the dynamic arrival setting. Specifically, in the extremecase in which a single node executes standard exponential backoff, if the adversary jams early slots, then the node’ssending probability quickly decays to sub-optimal values, resulting it taking too much time to succeed.A natural fix to the above issue is to decrease nodes’ sending probabilities slower. But to what extent? After all, ifnodes’ sending probabilities remain high for too long, then in the other extreme case in which Eve injects a lot of nodeswithin a short period of time, contention among nodes themselves would prevent quick first success. This dilemmais exactly what we exploit in proving the impossibility results (see Section 4), and it also hints the optimal sendingprobabilities nodes should use when running backoff style algorithms in dynamic arrival setting. We first introduce two (parameterized) subroutines: the backoff subroutine and the batch subroutine. As the namesuggests, the backoff subroutine aims to achieve quick first success in dynamic arrival setting, whereas the batch subroutine aims to achieve good throughput in batch setting. Both subroutines are variants of the standard exponentialbackoff algorithm. (Careful readers might wonder why we use two different variants, the reason being: the batch aimin Chen, Yonggang Jiang, and Chaodong Zheng subroutine simplifies algorithm presentation and analysis; while the backoff subroutine is necessary for achievingoptimal throughput, see Theorem 4.2 in Section 4 for details.) Below are the definitions of the two subroutines. ℎ - backoff : Let ℎ : N + → N + be a function. We say a node runs ℎ - backoff starting from slot 𝑙 , if forany 𝑘 ∈ N , in slot interval I 𝑘 = [ 𝑙 − + 𝑘 , 𝑙 − + 𝑘 + ) , the node sends its message in slots { 𝑙 𝑖 | ≤ 𝑖 ≤ ℎ (|I 𝑘 |)} where each 𝑙 𝑖 is drawn uniformly at random (with replacement) from I 𝑘 . We call I 𝑘 as the 𝑘 -th stage of ℎ - backoff . ℎ - batch : Let ℎ : N + → R + be a function. We say a node runs ℎ - batch starting from slot 𝑙 , if for any 𝑘 ∈ N + , the node sends its message with probability min { , ℎ ( 𝑘 )} in slot 𝑙 − + 𝑘 .Our algorithm requires a function 𝑔 as an input parameter, where log 𝑔 ( 𝑥 ) = 𝑂 ( p log 𝑥 ) . This function signifies thelevel of jamming the algorithm cam tolerate. (Recall Definition 1.1 and discussions below it.) For instance, if Eve canjam some constant fraction of all slots, then 𝑔 ( 𝑥 ) should be a constant function; whereas if Eve can jam / log 𝑡 fractionof all slots over a time period of 𝑡 , then 𝑔 ( 𝑥 ) = log 𝑥 . Given 𝑔 ( 𝑥 ) , define function 𝑓 ( 𝑥 ) = 𝑎𝑐 log 𝑥 log ( 𝑔 ( 𝑥 )/ 𝑎 ) ; further definefunction ℎ 𝑐𝑡𝑟𝑙 ( 𝑥 ) = 𝑐 log 𝑥𝑥 and function ℎ 𝑑𝑎𝑡𝑎 ( 𝑥 ) = 𝑥 . Here, 𝑎, 𝑐 , 𝑐 are constant to be determined in later analysis.Conceptually, our algorithm uses two channels: the odd channel which contains odd slots, and the even channel which contains even slots. We use 𝛼 to represent one of the two channels, and use ¯ 𝛼 to represent the other channel.We are now ready to state the algorithm for a newly injected node 𝑢 . It contains three phases: Phase 1:
Suppose 𝑢 is injected at the beginning of slot 𝑙 . Node 𝑢 will run ( 𝑎 𝑓 ) - backoff on thechannel determined by the parity of 𝑙 , until hearing a success on either one of the twochannels. Node 𝑢 will then start Phase 2. Phase 2:
Suppose the first success 𝑢 observed during Phase 1 occurred in slot 𝑙 , and 𝑙 is in channel 𝛼 . Node 𝑢 will run ( 𝑎 𝑓 ) - backoff on channel ¯ 𝛼 starting from slot 𝑙 + , until hearing asuccess on channel ¯ 𝛼 in some slot 𝑙 . Node 𝑢 will then start Phase 3, with 𝑙 set to 𝑙 . Phase 3:
Starting from slot 𝑙 + node 𝑢 will run ℎ 𝑐𝑡𝑟𝑙 - batch on the channel determined by theparity of 𝑙 + , and starting from slot 𝑙 + node 𝑢 will run ℎ 𝑑𝑎𝑡𝑎 - batch on the channeldetermined by the parity of 𝑙 + . Let 𝛼 be the channel on which 𝑢 runs ℎ 𝑐𝑡𝑟𝑙 - batch . Oneexecution of Phase 3 ends when 𝑢 hears a successful transmission on channel 𝛼 in someslot 𝑙 ′ . By then, 𝑢 sets 𝑙 to 𝑙 ′ , and restarts Phase 3.Recall the algorithm framework introduced at the beginning of this section. Phase 1 allows a newly joined node 𝑢 andthe existing nodes to reach agreement on the role (data and control) of the two channels (odd channel and even channel).Specifically, the newly joined node 𝑢 treats the channel on which the success occurred as the data channel. (Notice,since it might be the case that there are only Phase 1 nodes in the system, nodes in Phase 1 cannot just passively waitfor successes. Instead, they run backoff to create the first success efficiently.) Once all nodes have reached agreementon the role of the two channels, in Phase 2, node 𝑢 runs ( 𝑎 𝑓 ) - backoff on the control channel and waits for a successto occur on the control channel. Once such a success occurs, node 𝑢 and the other nodes executing Phase 2 or Phase 3are synchronized, and can (re)start Phase 3, which contains an execution of the “truncated exponential backoff”. Lastly,one important detail worth noting is, whenever a node (re)starts Phase 3, it swaps its data channel and control channel.Finally, we note that a node halts once its message has been successfully transmitted, as specified by the model. Node 𝑢 does not need to know whether 𝑙 is in the odd channel or the even channel.6 ight Trade-off in Contention Resolution without Collision Detection backoff In this subsection, we will prove two key lemmas demonstrating the effectiveness and robustness of the backoff subroutine. They are used extensively in later analysis.We begin by stating a concentration inequality that will be used in proving these two lemmas.
Theorem 3.1 (McDiarmid’s Ineqality [14]).
Suppose 𝑓 ( 𝑥 , 𝑥 , · · · , 𝑥 𝑛 ) is a function satisfying: for any 𝑖 ∈ [ 𝑛 ] , itholds | 𝑓 ( ® 𝑥 ) − 𝑓 ( ® 𝑥 ′ )| < 𝑐 , where ® 𝑥 = ( 𝑥 , 𝑥 , · · · , 𝑥 𝑖 , · · · , 𝑥 𝑛 ) and ® 𝑥 ′ = ( 𝑥 , 𝑥 , · · · , 𝑥 ′ 𝑖 , · · · , 𝑥 𝑛 ) only differ in the 𝑖 -thcoordinate. Suppose { 𝑋 𝑖 } 𝑖 ∈[ 𝑛 ] are 𝑛 independent random variables. Then for any 𝛿 > : Pr [| 𝑓 ( 𝑋 , 𝑋 , ..., 𝑋 𝑛 ) − E [ 𝑓 ( 𝑋 , 𝑋 , ..., 𝑋 𝑛 )]| ≥ 𝛿 ] ≤ (cid:18) − 𝛿 𝑐 𝑛 (cid:19) The first key technical lemma concerns within the scenario where batch and backoff are being executed concur-rently. That is, in the system, a set of (synchronized) nodes are running batch , while some other (un-synchronized)nodes are running backoff . In such case, so long as the adversary does not inject too many new nodes or jam toomany slots, at least one success will occur sufficiently fast. One point worth noting is, conditioned on the state at thebeginning of the considered slot interval, prior to the first success, the power of an adaptive adversary and an obliviousadversary are identical (as all channel feedback is silence, an adaptive adversary has nothing to adapt to). Since ourapplications of the two lemmas only concern with first success, here we only consider an oblivious adversary.
Lemma 3.2.
Let 𝑡 and 𝑐 ′ be sufficiently large integers, 𝑓 be any sub-logarithmic function satisfying 𝑓 ( 𝑥 ) = 𝑂 ( log 𝑥 ) .Consider slot interval I = [ 𝐿, 𝑅 ] where |I| = 𝑡 , assume the following conditions hold:(1) For each slot 𝑖 ∈ [ 𝐿, 𝑅 ] , there are 𝑞 𝑖 (synchronized) nodes running some instance of batch , and each node’s sendingprobability is 𝑝 𝑖 . Moreover, it holds that log 𝑡𝑐 ′ ≥ 𝑞 𝑖 · 𝑝 𝑖 ≥ 𝑐 ′ log 𝑡𝑡 and 𝑝 𝑖 ≤ .(2) During slot interval [ , 𝑅 ] , aside from the nodes described in (1), an oblivious adversary injects at most 𝑡 𝑓 ( 𝑡 ) additional new nodes and jams at most 𝑡 slots.(3) Each node injected by the adversary runs 𝑓 - backoff after joining the system.Then there exists at least one success slot in I , with high probability in 𝑡 . Proof.
Let 𝑚𝑖𝑑 = ( 𝐿 + 𝑅 )/ . Call the nodes injected by the oblivious adversary as additional nodes. Let 𝑥 𝑖 be arandom variable denoting the number of additional nodes that send in slot 𝑖 . We consider two complement cases,depending on the value of Í 𝐿 ≤ 𝑖 < 𝑚𝑖𝑑 E [ 𝑥 𝑖 ] . Case 1.
Suppose Í 𝐿 ≤ 𝑖 < 𝑚𝑖𝑑 E [ 𝑥 𝑖 ] ≤ 𝑡 . . Call a slot occupied if at least one additional node sends in this slot, or if thisslot is jammed. Note that 𝑋 = Í 𝐿 ≤ 𝑖 < 𝑚𝑖𝑑 𝑥 𝑖 depends on at most 𝑡 log 𝑡 independent random variables: there are at most 𝑡 𝑓 ( 𝑡 ) additional nodes, each will broadcast in at most 𝑓 ( 𝑡 ) log 𝑡 slots during I ; the indices of the slots these nodesbroadcast are the random variables that determine 𝑋 . It is easy to see, changing the value of each such random variableaffects the value of 𝑋 by at most one. Since Í 𝐿 ≤ 𝑖 < 𝑚𝑖𝑑 E [ 𝑥 𝑖 ] ≤ 𝑡 . , according to Theorem 3.1, with high probability in 𝑡 we have Í 𝐿 ≤ 𝑖 < 𝑚𝑖𝑑 𝑥 𝑖 ≤ 𝑡 . . Thus, for sufficiently large 𝑡 , during [ 𝐿, 𝑚𝑖𝑑 ) , the number of occupied slots is at most 𝑡 / + 𝑡 . ≤ 𝑡 / . In each non-occupied slot, since there are 𝑞 𝑖 nodes sending messages each with probability 𝑝 𝑖 and log 𝑡𝑐 ′ ≥ 𝑞 · 𝑝 𝑖 ≥ 𝑐 ′ log 𝑡𝑡 , it is easy to verify a success occurs with probability Ω ( min { 𝑐 ′ log 𝑡𝑡 , 𝑡 − / 𝑐 ′ }) = Ω ( 𝑐 ′ log 𝑡𝑡 ) when 𝑐 ′ ≥ . Since there are at least 𝑡 / non-occupied slots during [ 𝐿, 𝑚𝑖𝑑 ) , by a Chernoff bound, we know for sufficientlylarge 𝑐 ′ , a success will occur in some non-occupied slot, with high probability in 𝑡 . aimin Chen, Yonggang Jiang, and Chaodong Zheng Case 2.
Suppose Í 𝐿 ≤ 𝑖 < 𝑚𝑖𝑑 E [ 𝑥 𝑖 ] > 𝑡 . . We now prove E [ 𝑥 𝑖 ] > 𝑡 − . for any 𝑖 ≥ 𝑚𝑖𝑑 . For each additional node 𝑢 ,denote 𝑎 𝑢𝑖 as the probability that 𝑢 sends in slot 𝑖 . Recall that 𝑓 -backoff will send at most 𝑓 ( 𝑗 ) times in the 𝑗 -th stageand the 𝑗 -th stage has length 𝑗 . If 𝑢 arrives before slot 𝐿 − 𝑡 , the stages that intersect with [ 𝐿, 𝑅 ] all have length at least 𝑡 . Thus, there are at most two stages intersecting [ 𝐿, 𝑅 ] , and 𝑎 𝑢𝑖 in each such stage differ by some constant factor. Thisimplies 𝑎 𝑢𝑘 is at least 𝑡 . Í 𝐿 ≤ 𝑖 < 𝑚𝑖𝑑 𝑎 𝑢𝑖 for 𝑘 ∈ [ 𝑚𝑖𝑑, 𝑅 ] . If, however, 𝑢 joins after slot 𝐿 − 𝑡 , we know when 𝑖 ∈ [ 𝑚𝑖𝑑, 𝑅 ] it must be the case 𝑎 𝑢𝑖 = Ω ( / 𝑡 ) , since the stages intersecting [ 𝑚𝑖𝑑, 𝑅 ] has length 𝑂 ( 𝑡 ) . Moreover, Í 𝐿 ≤ 𝑖 ≤ 𝑚𝑖𝑑 𝑎 𝑢𝑖 is atmost 𝑂 ( log 𝑡 ) since there are at most log 𝑡 stages in [ 𝐿, 𝑚𝑖𝑑 ] and each such stage contributes at most 𝑓 ( 𝑡 ) = 𝑂 ( log 𝑡 ) times of sending. Thus, 𝑎 𝑢𝑘 ≥ 𝑡 . Í 𝐿 ≤ 𝑖 ≤ 𝑚𝑖𝑑 𝑎 𝑢𝑖 for 𝑘 ∈ [ 𝑚𝑖𝑑, 𝑅 ] . As this point, we conclude: for any additional node 𝑢 and any 𝑘 ∈ [ 𝑚𝑖𝑑, 𝑅 ] , 𝑎 𝑢𝑘 ≥ 𝑡 . Í 𝐿 ≤ 𝑖 ≤ 𝑚𝑖𝑑 𝑎 𝑢𝑖 . Since E [ 𝑥 𝑖 ] = Í 𝑢 𝑎 𝑢𝑖 , we have E [ 𝑥 𝑖 ] > 𝑡 − . for any 𝑖 ∈ [ 𝑚𝑖𝑑, 𝑅 ] .Let 𝑥 ′ 𝑖 be the total (i.e., including nodes running batch as well as backoff ) number of nodes that send in slot 𝑖 .Since each additional node contributes at most 𝑓 ( 𝑡 ) log 𝑡 sending slots during [ 𝐿, 𝑅 ] , we know E [ Í 𝑚𝑖𝑑 ≤ 𝑖 ≤ 𝑅 𝑥 ′ 𝑖 ] ≤ 𝑡 𝑓 ( 𝑡 ) · 𝑓 ( 𝑡 ) log 𝑡 + 𝑡 · log 𝑡𝑐 ′ = 𝑡 log 𝑡 + 𝑡 log 𝑡 𝑐 ′ . For sufficiently large 𝑐 ′ , this means there are at most 𝑡 slots in [ 𝑚𝑖𝑑, 𝑅 ] with E [ 𝑥 ′ 𝑖 ] ≥ log 𝑡 , which further implies there are at least 𝑡 slots in [ 𝑚𝑖𝑑, 𝑅 ] with 𝑡 − . ≤ E [ 𝑥 ′ 𝑖 ] ≤ log 𝑡 . Call a slot 𝑖 good if: (1) 𝑡 − . ≤ E [ 𝑥 ′ 𝑖 ] ≤ log 𝑡 ; (2) no nodes are injected in 𝑖 ; and (3) the adversary does not jam 𝑖 . We know thereare at least 𝑡 good slots in [ 𝑚𝑖𝑑, 𝑅 ] . Moreover, it is easy to verify, a good slot has probability at least 𝑡 − . to generatea success. Now, let 𝑌 be a random variable denoting the number of successes in slots [ 𝑚𝑖𝑑, 𝑅 ] . We know E [ 𝑌 ] is atleast 𝑡 · 𝑡 − . = 𝑡 . . Note that 𝑌 is a function of at most 𝑡 log 𝑡 independent random variables each affecting thevalue of 𝑌 by at most one. Thus, according to Theorem 3.1, with high probability in 𝑡 we have at least one success in [ 𝑚𝑖𝑑, 𝑅 ] . (cid:3) The following second lemma focus on the scenario in which no instance of batch is running, or batch ends in themiddle of the considered interval. (Specifically, in the lemma statement, slot 𝑘 marks the end of the batch instance,and 𝑘 = means initially there is no instance of batch running). Once again, so long as the adversary does not injecttoo many new nodes or jam too many channels, successful transmission would soon occur. Lemma 3.3.
Let 𝑡, 𝑐 ′ , 𝑐 ′ be sufficiently large integers. Let 𝑔 be any function such that log 𝑔 ( 𝑥 ) is sub-logarithmic and log 𝑔 ( 𝑥 ) = 𝑂 ( p log 𝑥 ) . Consider slot interval [ , 𝑡 ] . There exists a sufficiently large constant 𝑐 ′ and a function 𝑓 ( 𝑥 ) = 𝑐 ′ log 𝑥 log 𝑔 ( 𝑥 ) such that, if the following conditions are satisfied:(1) There exists an integer 𝑘 ∈ [ , 𝑡 ] such that for each slot 𝑖 ∈ [ , 𝑘 ] , there are 𝑞 𝑖 (synchronized) nodes running someinstance of batch , and each node’s sending probability is 𝑝 𝑖 . Moreover, for each 𝑖 ∈ [ 𝑘 / , 𝑘 ] , it holds log 𝑘 ′ 𝑐 ′ ≥ 𝑞 𝑖 · 𝑝 𝑖 ≥ 𝑐 ′ log 𝑘 ′ 𝑘 ′ and 𝑝 𝑖 ≤ , where 𝑘 ′ = 𝑘 .(2) Aside from the nodes described in (1), an oblivious adversary injects at most 𝑡𝑐 ′ 𝑓 ( 𝑡 ) additional new nodes and jamat most 𝑡𝑐 ′ 𝑔 ( 𝑡 ) slots in [ , 𝑡 ] .(3) The adversary injects at least one additional node in the first max { 𝑘, } slots.(4) Each node injected by the adversary runs 𝑓 - backoff after joining the system.Then there exists at least one success in the first 𝑡 slots, with high probability in 𝑡 . Proof.
Let 𝑓 ( 𝑥 ) = 𝑐 ′ log 𝑥 log 𝑔 ( 𝑥 ) for some sufficiently large 𝑐 ′ to be specified latter. First consider the situation in whichthe adversary injects at least 𝑡 . nodes in the first 𝑡 / slots. Let random variable 𝑥 ′ 𝑖 be the number of nodes that sendin slot 𝑖 . Since each additional node arrived in the first 𝑡 / slots contributes at least / 𝑡 to E [ 𝑥 ′ 𝑖 ] when 𝑖 > 𝑡 / and 𝑐 ′ is sufficiently large, we have the lower bound E [ 𝑥 𝑖 ] ≥ 𝑡 − . for any 𝑖 > 𝑡 / . On the other hand, since each additionalnode contributes at most 𝑓 ( 𝑡 ) log 𝑡 sending slots in [ , 𝑡 ] , and since for existing nodes executing batch (if some batch ight Trade-off in Contention Resolution without Collision Detection is running) 𝑞 𝑖 · 𝑝 𝑖 is at most log 𝑡𝑐 ′ when 𝑖 > 𝑡 / , we have the upper bound E [ Í 𝑡 / < 𝑖 ≤ 𝑡 𝑥 ′ 𝑖 ] ≤ 𝑡𝑐 ′ 𝑓 ( 𝑡 ) · 𝑓 ( 𝑡 ) log 𝑡 + 𝑡 · log 𝑡𝑐 ′ .Then, apply same analysis as in Case 2 of the proof of Lemma 3.2, there is a success in [ 𝑡 / , 𝑡 ] with high probability in 𝑡 . In the reminder of the proof, we consider the situation in which the adversary injects at most 𝑡 . nodes in the first 𝑡 / slots. Specifically, we consider three cases depending on the value 𝑘 . Case 1:
Suppose 𝑘 ≥ 𝑡 / . For sufficiently large 𝑡, 𝑐 ′ , 𝑐 ′ , 𝑐 ′ , all the conditions in Lemma 3.2 are satisfied if we set 𝐿 = 𝑘 / and 𝑅 = 𝑘 . Thus, apply Lemma 3.2 and we know there is a success in [ 𝐿, 𝑅 ] with high probability in ( 𝑘 / ) > 𝑡 . Case 2:
Suppose 𝑡 / ≥ 𝑘 ≥ 𝑡 / 𝑔 ( 𝑡 ) . Recall that 𝑔 ( 𝑡 ) is at most 𝑂 (√ log 𝑡 ) . Also recall that at most 𝑡 . nodes areinjected before slot 𝑡 / , and in Lemma 3.2 we only require the fraction of jammed slots to be bounded by / . As aresult, for sufficient large 𝑐 ′ , all the conditions in Lemma 3.2 are satisfied if we set 𝐿 = 𝑘 / and 𝑅 = 𝑘 . Thus, there isat least one success in [ 𝐿, 𝑅 ] with high probability in ( 𝑘 ′ ) . Since 𝑘 is at least 𝑡𝑔 ( 𝑡 ) and 𝑔 ( 𝑡 ) is at most √ log 𝑛 , ( 𝑘 ′ ) isat least 𝑡 for sufficiently large 𝑡 . Case 3.
Suppose 𝑘 ≤ 𝑡 / 𝑔 ( 𝑡 ) . Suppose 𝑢 is a node arriving in slot 𝑡 𝑢 where 𝑡 𝑢 ≤ 𝑘 . (By lemma assumption, suchnode 𝑢 exists.) We will prove that there is at least one success for 𝑢 in the first 𝑡 / slots. We call a slot occupied ifadditional nodes other than 𝑢 send in this slot or this slot is jammed by the adversary. Since each additional nodecan send in at most 𝑓 ( 𝑡 ) log 𝑡 slots in [ , 𝑡 ] and there are at most 𝑡𝑐 ′ 𝑔 ( 𝑡 ) ≤ 𝑡𝑔 ( 𝑡 ) jammed slots, there are at most 𝑡 . · 𝑓 ( 𝑡 ) log 𝑡 + 𝑡𝑔 ( 𝑡 ) ≤ 𝑡𝑔 ( 𝑡 ) occupied slots for sufficiently large 𝑡 . (Recall that log 𝑔 ( 𝑡 ) = 𝑂 ( p log 𝑡 ) ).Define slot interval P 𝑗 = [ 𝑡 𝑢 + 𝑗 − , 𝑡 𝑢 + 𝑗 + − ) . Recall that 𝑢 will choose 𝑓 (|P 𝑗 |) slots uniformly at random(with replacement) to send in interval P 𝑗 . Suppose there are 𝑎 𝑗 occupied slots in interval P 𝑗 . Define ℓ as the largest 𝑗 such that P 𝑗 is entirely contained within interval [ , 𝑡 / ] . Recall that log 𝑔 is sub-logarithmic, which means 𝑔 is atleast for sufficiently large 𝑡 , hence 𝑘 ≤ 𝑡𝑔 ( 𝑡 ) ≤ 𝑡 . As a result, ℓ ∈ Θ ( log 𝑡 ) . Define ℓ ′ as the smallest 𝑗 such that 𝑡𝑔 ( 𝑡 ) ≤ 𝑗 . Clearly ℓ ′ > 𝑘 and ℓ ≥ ℓ ′ , meaning only occupied slots can jam the success of 𝑢 for backoff stages after ℓ ′ . Since there are at most 𝑡𝑔 ( 𝑡 ) occupied slots, the probability that there is no success slot for node 𝑢 in P ℓ ′ , · · · , P ℓ isat most: Ö ℓ ′ ≤ 𝑖 ≤ ℓ (cid:16) 𝑎 𝑖 𝑖 (cid:17) 𝑓 ( |P 𝑖 |) ≤ Ö ℓ ′ ≤ 𝑖 ≤ ℓ (cid:18) 𝑡 𝑖 𝑔 ( 𝑡 ) (cid:19) 𝑓 (cid:16) ℓ ′ (cid:17) ≤ (cid:18) 𝑔 ( 𝑡 ) (cid:19) · 𝑓 (cid:16) ℓ ′ (cid:17) · log ( 𝑔 ( 𝑡 )) Since ℓ ′ is the smallest 𝑗 such that 𝑡𝑔 ( 𝑡 ) ≤ 𝑗 and log 𝑔 ( 𝑡 ) = 𝑂 ( p log 𝑡 ) , ℓ ′ > 𝑡 . . Since 𝑓 ( 𝑡 𝑑 ) = Θ ( 𝑓 ( 𝑡 )) for any 𝑑 > , 𝑓 ( ℓ ′ ) = Θ ( 𝑓 ( 𝑡 )) . Recall 𝑓 ( 𝑡 ) = 𝑐 ′ log 𝑡 log 𝑔 ( 𝑡 ) , for sufficiently large 𝑐 ′ , the above probability is at most / 𝑡 . (cid:3) In this subsection, we present some additional notations so as to simplify the presentation of later analysis.We begin by introducing three special types of slots.
Beginning slot : Consider a slot 𝑡 , if there exists some active node in slot 𝑡 and all active nodes in thesystem in slot 𝑡 were injected at the beginning of slot 𝑡 , then slot 𝑡 is a beginning slot. Ending slot : Consider a slot 𝑡 , if there is only one active node in the system at the beginning ofslot 𝑡 , and it succeeds in slot 𝑡 , then slot 𝑡 is an ending slot. Transition slot : Consider a slot 𝑡 containing a success and denote the last beginning slot before 𝑡 (or 𝑡 itself if 𝑡 is a beginning slot) as 𝑏 . The success slot 𝑡 is called a transition slot if either:(1) there is no success within slots [ 𝑏, 𝑡 − ] ; or (2) 𝑡 − 𝑟 𝑡 is odd and there is no success aimin Chen, Yonggang Jiang, and Chaodong Zheng within slots { 𝑠 ∈ [ 𝑟 𝑡 + , 𝑡 − ] : ( 𝑡 − 𝑠 ) is a positive even integer } , where 𝑟 𝑡 is the lasttransition slot before 𝑡 .Intuitively, transition slots are the slots in which active nodes’ states change: from Phase 1 to Phase 2, or from Phase2 to Phase 3, or restart Phase 3.We then utilize above special slots to define complete intervals . Specifically, each of the following three kinds of timeintervals is a complete interval: • Interval [ 𝑏, 𝑟 ] from a beginning slot 𝑏 to the first transition slot 𝑟 after 𝑏 . • Interval [ 𝑟 + , 𝑟 ′ ] from slot 𝑟 + to the first transition slot 𝑟 ′ after 𝑟 , where slot 𝑟 is a transition slot. • Interval [ 𝑟 + , 𝑒 ] from slot 𝑟 + to the first ending slot 𝑒 after 𝑟 , where: (1) 𝑟 is a transition slot; (2) 𝑟 isnot an ending slot; and (3) there is no transition slot within [ 𝑟 + , 𝑒 − ] .Intuitively, for any interval, if we mark all the beginning-slots/ending-slots/transition-slots, then these special slotsdivide the interval into a set of “segments”. Each such segment that contains active node(s) is a complete interval.Now, consider an arbitrary interval I = [ 𝐿 I , 𝑅 I ] . If a node starts Phase 1 or Phase 2 of the main algorithm in I ,then the node is a new arrival of I . Notice that if some node arrives before slot 𝐿 I and begins Phase 2 in slot 𝐿 I , it isalso a new arrival of I by definition. Therefore, each node is a new arrival of at most two complete intervals.Lastly, inspired by [8], we define truncated length for complete intervals. This facilitates later amortized analysis.Consider an arbitrary complete interval I = [ 𝐿 I , 𝑅 I ] . The length of I is 𝑙 I = 𝑅 I − 𝐿 I + . The truncated length ¯ 𝑙 I is defined in the following way: (1) if the number of new arrivals of I is at most 𝑎𝑙 I /( 𝑐𝑐 𝑐 𝑓 ( 𝑙 I )) , and thenumber of jammed slots during I is at most 𝑎𝑙 I /( 𝑐𝑐 𝑐 𝑔 ( 𝑙 I )) , and the number of success during I is less than 𝑙 I /( 𝑐𝑐 ( 𝑡 + )) , then ¯ 𝑙 I = 𝑙 I ; (2) otherwise ¯ 𝑙 I = . Here, 𝑎, 𝑐, 𝑐 , 𝑐 , 𝑡 are constants to be specified in later analysis; 𝑓 and 𝑔 are the functions used by the algorithm, see Subsection 2.1. (Recall 𝑎 and 𝑐 are also used in describing thealgorithm.) The main goal of this subsection is to show the truncated length of any complete interval is likely to be small. Inparticular, for any sufficiently large integer 𝑡 , the probability that the truncated length of a complete interval reaching 𝑡 is at most / poly ( 𝑡 ) . Intuitively, this means each complete interval is able to maintain desirable throughput: eitherthe adversary jams a lot of slots or injects a lot of new nodes, or the algorithm generates sufficiently many successes.We first introduce a technical lemma bounding the sum of a set of dependent random variables. It will be usedseveral times in remaining analysis. We defer its proof to the appendix. Lemma 3.4.
Let 𝑡 be an arbitrary positive integer. Suppose 𝑋 , 𝑋 , · · · , 𝑋 𝑛 are 𝑛 (potentially dependent) random vari-ables. If Pr [ 𝑋 𝑖 = 𝑡 | 𝑋 = 𝑥 , 𝑋 = 𝑥 , · · · , 𝑋 𝑖 − = 𝑥 𝑖 − ] ≤ 𝑡 − holds for any integer 𝑡 ≥ 𝑡 , any ≤ 𝑖 ≤ 𝑛 , and anyvalues 𝑥 , 𝑥 , · · · , 𝑥 𝑖 − of 𝑋 , 𝑋 , · · · , 𝑋 𝑖 − , then with high probability in 𝑛 , we have Í 𝑛𝑖 = 𝑋 𝑖 ≤ ( 𝑡 + ) 𝑛 . We now proceed to bound the truncated length of complete intervals.
Lemma 3.5.
Consider a complete interval I that starts at the beginning of slot 𝐿 I , assume it ends at the end of slot 𝑅 I . 𝑅 I is a random variable. For any integer 𝑡 ≥ 𝑡 where 𝑡 is a sufficiently large constant, we have Pr [ ¯ 𝑙 I = 𝑡 ] ≤ / 𝑡 Ω ( ) ,regardless of the history before slot 𝐿 I . Proof.
By definition, it is easy to verify that in the beginning slot of a complete interval, each active node in ight Trade-off in Contention Resolution without Collision Detection the system will start some instance of backoff or batch from scratch. Thus, without loss of generality, we assumecomplete interval I begins at slot one. That is, 𝐿 I = .For any interval I ′ = [ 𝐿 I ′ , 𝑅 I ′ ] , let 𝑛 [ 𝐿 I′ ,𝑅 I′ ] be the number of new arrivals of I ′ , and 𝑑 [ 𝐿 I′ ,𝑅 I′ ] be the numberof jammed slots within I ′ . In this proof, define function 𝑓 ′ ( 𝑥 ) = 𝑓 ( 𝑥 )/ 𝑎 and 𝑔 ′ ( 𝑥 ) = 𝑔 ( 𝑥 )/ 𝑎 . For any 𝑡 ≥ 𝑡 , wehave 𝑛 [ ,𝑡 ] ≤ 𝑡 /( 𝑐𝑐 𝑐 𝑓 ′ ( 𝑡 )) and 𝑑 [ ,𝑡 ] ≤ 𝑡 /( 𝑐𝑐 𝑐 𝑔 ′ ( 𝑡 )) , otherwise the lemma trivially holds by definition oftruncated length. Therefore, for any interval I ′ = [ 𝐿 I ′ , 𝑅 I ′ ] ⊆ [ , 𝑡 ] whose length |I ′ | = 𝑅 I ′ − 𝐿 I ′ + is at least 𝑡 / 𝑐𝑐 , we have 𝑛 [ 𝐿 I′ ,𝑅 I′ ] ≤ 𝑛 [ ,𝑡 ] ≤ 𝑡 /( 𝑐𝑐 𝑐 · 𝑓 ′ ( 𝑡 )) ≤ |I ′ |/( 𝑐 · 𝑓 ′ (|I ′ |)) since |I ′ | ≤ 𝑡 ≤ 𝑐𝑐 |I ′ | ; similarly, 𝑑 [ 𝐿 I′ ,𝑅 I′ ] ≤ |I ′ |/( 𝑐 · 𝑔 ′ (|I ′ |)) .Call the nodes that start Phase 3 in slot 𝐿 I as the batch nodes of I . Let 𝑛 be the number of batch nodes of I .(Throughout this proof, we always use 𝑛 without subscript to refer to the number of batch nodes of I .)We first focus on the situation 𝑛 = . If the complete interval is from a beginning slot to a transition slot, we willapply Lemma 3.3 on the channel 𝛼 determined by the parity of 𝐿 I (i.e., the set of slots { 𝑡 ∈ [ 𝐿 I , 𝑅 I ] : 𝑡 − 𝐿 I + is odd } ).We argue the conditions of Lemma 3.3 are satisfied. Specifically, setting 𝑘 = satisfies the first condition; conditionsthree and four trivially satisfy. Denote the number of nodes that run backoff on channel 𝛼 in the first 𝑡 / slots of 𝛼 as 𝑛 𝛼 [ ,𝑡 / ] , then 𝑛 𝛼 [ ,𝑡 / ] ≤ 𝑛 [ ,𝑡 / ] ≤ 𝑛 [ ,𝑡 ] ≤ 𝑡 /( 𝑐 𝑓 ′ ( 𝑡 )) ≤ ( 𝑡 / )/(( 𝑐 / 𝑎 ) 𝑓 ( 𝑡 / )) . Similarly, the bound on the numberof jammed slots is also satisfied. Thus, the second condition of the lemma is satisfied. As a result, apply Lemma 3.3and we know, there is at least one success in the first 𝑡 / slots of channel 𝛼 , with high probability in 𝑡 / . This implies ¯ 𝑙 ≤ 𝑡 / < 𝑡 with high probability in 𝑡 , as desired. If 𝑛 = and the complete interval is from a transition slot to atransition slot or from a transition slot to an ending slot, then there must exist some node(s) running Phase 2 in 𝐿 I , sothese nodes can distinguish between the control channel and the data channel. Again we focus on the first 𝑡 / slots ofthe control channel and apply Lemma 3.3 with 𝑘 = in the first condition. Therefore, there is at least one success inthe first 𝑡 / slots of the control channel with high probability in 𝑡 / , implying ¯ 𝑙 ≤ 𝑡 / < 𝑡 with high probability in 𝑡 .Assume 𝑛 > in the reminder of the proof. Let 𝑙𝑎𝑠𝑡 be the slot that the last batch node successfully sends its messageamong the 𝑛 batch-nodes. (Let 𝑙𝑎𝑠𝑡 be 𝑡 if there are still batch nodes at the end of slot 𝑡 ). We consider three scenariosaccording to the value of 𝑡 : (1) 𝑡 > 𝑐𝑐 𝑛 ; (2) 𝑐 𝑛 < 𝑡 ≤ 𝑐𝑐 𝑛 ; and (3) 𝑡 ≤ 𝑐 𝑛 . Here, 𝑐 is a constant to be specifiedlater. The analysis for scenario two and three is very similar to the proof of Lemma 8 in [8], we defer them to theappendix to avoid redundancy. Here, we focus on scenario one: 𝑛 > and 𝑡 > 𝑐𝑐 𝑛 . We further divide this scenariointo four cases. Case 1: Suppose 𝑙𝑎𝑠𝑡 ≥ 𝑡 / . If 𝑙𝑎𝑠𝑡 ≥ 𝑡 / , we argue the conditions for applying Lemma 3.2 are satisfied, hence thereis a success on the control channel during time [ 𝑡 / , 𝑡 / ] with high probability in 𝑡 / , implying ¯ 𝑙 ≤ 𝑡 / < 𝑡 with highprobability in 𝑡 . Specifically, since 𝑛 · 𝑐 log ( 𝑡 / ) 𝑡 / ≤ log ( 𝑡 / ) 𝑐 ′ when 𝑡 ≥ 𝑐 ′ 𝑐 𝑛 , and since · 𝑐 log ( 𝑡 / ) 𝑡 / ≥ 𝑐 ′ log ( 𝑡 / ) 𝑡 / when 𝑐 ≥ 𝑐 ′ , the first condition is satisfied. Since 𝑛 [ ,𝑡 / ] ≤ 𝑛 [ ,𝑡 ] ≤ ( 𝑡 / )/(( 𝑐 / 𝑎 ) 𝑓 ( 𝑡 / )) , the bound on the number ofinjected nodes in the second condition is satisfied. Similarly, the bound on the number of jammed slots in the secondcondition is also satisfied. Case 2: Suppose 𝑙𝑎𝑠𝑡 < 𝑡 / , and all nodes never run backoff on control channel within time [ , 𝑙𝑎𝑠𝑡 ] . Then either thereis no node in the system at end of slot 𝑙𝑎𝑠𝑡 , or nodes only arrive on data channel after the slot in which the secondto last batch node succeeded. In the former case we have ¯ 𝑙 = 𝑙𝑎𝑠𝑡 < 𝑡 . As for the latter case, we apply Lemma 3.3 toshow there is a success within interval [ 𝑙𝑎𝑠𝑡 + , 𝑡 / ] with high probability in 𝑡 . Specifically, we set 𝑘 = for the firstcondition. The second condition is satisfied since the length of [ 𝑙𝑎𝑠𝑡 + , 𝑡 / ] is at least 𝑡 / . The third condition issatisfied since in time slot 𝑙𝑎𝑠𝑡 + all nodes in the system begin Phase 2. Case 3: Suppose 𝑙𝑎𝑠𝑡 < 𝑡 / , 𝑡 ≥ 𝑛 . , and there exists some node that runs backoff on control channel within time aimin Chen, Yonggang Jiang, and Chaodong Zheng [ , 𝑙𝑎𝑠𝑡 ] . We first introduce an additional type of “jammed” slots, and show the number of such “jammed” slots canstill be bounded by 𝑡 / 𝑐 𝑔 ′ ( 𝑡 / ) for later use. More specifically, call a control channel slot “interfered” if any batch nodesends in the slot. Define 𝜎 = 𝑡 𝑐 𝑔 ′ ( 𝑡 ) . The number of interfered control slots before time 𝜎 can be upper boundedby 𝜎 trivially. Next we upper bound the number interfered slots after time 𝜎 . Let 𝑥 𝑖 be the random number of batchnodes that send in the 𝑖 -th control slot, then for all 𝑖 ≥ 𝜎 , E [ 𝑥 𝑖 ] ≤ 𝑐 log 𝜎𝜎 · 𝑛 ≤ 𝑡 − . · 𝑛 ≤ 𝑡 − . · 𝑡 . = 𝑡 − . , wherethe last inequality is due to 𝑡 ≥ 𝑛 / . Let 𝑦 𝑖 be an indicator random variable taking value one when at least one batchnode sends in the 𝑖 -th slot of the control channel. By Markov’s inequality, Pr [ 𝑦 𝑖 ] = Pr [ 𝑥 𝑖 ≥ ] ≤ E [ 𝑥 𝑖 ] ≤ 𝑡 − . .Hence, the expected number of such interfered control slots within time interval ( 𝜎, 𝑡 ] is Í 𝜎 < 𝑖 ≤ 𝑡 / Pr [ 𝑦 𝑖 ] ≤ 𝑡 − . · 𝑡 / ≤ 𝑡 𝑐 𝑔 ′ ( 𝑡 ) . By a Chernoff bound, with high probability in 𝑡 , Í 𝜎 < 𝑖 ≤ 𝑡 / 𝑦 𝑖 is at most 𝑡 𝑐 𝑔 ′ ( 𝑡 ) . Therefore, if wesee interfered slots as another kind of jamming, the total number of jammed slots (i.e., interfered slots and jammingfrom the adversary) in [ , 𝑡 ] is bounded by 𝑡 𝑐 𝑔 ′ ( 𝑡 ) + 𝑡 𝑐 𝑔 ′ ( 𝑡 ) + 𝑡 𝑐 𝑔 ′ ( 𝑡 ) ≤ 𝑡 / 𝑐 𝑔 ′ ( 𝑡 / ) , where the third term is dueto 𝑑 [ ,𝑡 ] ≤ 𝑡 /( 𝑐 𝑔 ′ ( 𝑡 )) .Denote the first time that some node begins backoff on control channel as 𝑓 𝑖𝑟𝑠𝑡 (such slot exists by case assump-tion), then 𝑓 𝑖𝑟𝑠𝑡 ≤ 𝑙𝑎𝑠𝑡 < 𝑡 / . We now apply Lemma 3.3 on the control channel within interval [ 𝑓 𝑖𝑟𝑠𝑡, 𝑡 / ] , with 𝑘 = for the first condition, and consider both interfered slots and jamming from adversary for the second condi-tion. We can conclude, with high probability in 𝑡 / , there is a success (due to nodes running backoff , instead of thebatch nodes) on the control channel within interval [ 𝑓 𝑖𝑟𝑠𝑡, 𝑡 / ] . This further implies ¯ 𝑙 ≤ 𝑡 / < 𝑡 holds with highprobability in 𝑡 . Case 4: Suppose 𝑙𝑎𝑠𝑡 < 𝑡 / , 𝑡 < 𝑛 . , and there exists some node that runs backoff on control channel within time [ , 𝑙𝑎𝑠𝑡 ] . We first show with high probability in 𝑛 (thus also in 𝑡 as 𝑡 < 𝑛 . ), 𝑙𝑎𝑠𝑡 > 𝑐𝑐 𝑛 , by invoking the followingclaim with 𝑐 ′ set to 𝑐𝑐 . (Since we seek a lower bound on the number of slots that all 𝑛 batch nodes successfully sendstheir messages on the data channel, without loss of generality, assume there is no new arrival or jammed slots duringthe first 𝑐𝑐 𝑛 slots of the data channel.) Claim 3.5.1.
Assume there are 𝑛 nodes running ℎ 𝑑𝑎𝑡𝑎 - batch which begins at time 1 on a fixed channel, where ℎ 𝑑𝑎𝑡𝑎 ( 𝑥 ) = / 𝑥 . Then for any constant 𝑐 ′ , for sufficiently large 𝑛 , there is at least one node that has not succeeded during the first 𝑐 ′ 𝑛 slots of the channel, with high probability in 𝑛 . Proof.
Consider the first time slot ℓ when there are only . 𝑛 batch nodes in the system. Let ℓ 𝑖 be the first timeslot that 𝑖 of those nodes have left the system. Define ℓ = ℓ , and let 𝑠 𝑖 = ℓ 𝑖 + − ℓ 𝑖 . We will prove Pr [ Í ≤ 𝑖 ≤ . 𝑛 − 𝑠 𝑖 ≤ 𝑐 ′ 𝑛 ] < / 𝑛 , which implies that with high probability in 𝑛 , the batch runs for at least 𝑐 ′ 𝑛 slots.Since before time slot ℓ there are . 𝑛 times of successes, we have ℓ ≥ . 𝑛 , which means the probability thateach node sends after ℓ is at most . 𝑛 . Thus the probability that some of the . 𝑛 − 𝑖 nodes send in a slot between ℓ 𝑖 to ℓ 𝑖 + is at most ( . 𝑛 − 𝑖 ) · . 𝑛 . Since the occurrence of success implies one of them sends a message, we have Pr [ 𝑠 𝑖 ≥ 𝐿 ] ≥ ( − 𝑝 𝑖 ) 𝐿 where 𝑝 𝑖 = . 𝑛 − 𝑖 . 𝑛 for any positive integer 𝐿 .Define ¯ 𝑠 𝑖 as the truncated geometric distribution where Pr [ ¯ 𝑠 𝑖 = 𝐿 ] = 𝑝 𝑖 ( − 𝑝 𝑖 ) 𝐿 for ≤ 𝐿 < 𝑝 𝑖 , and Pr [ ¯ 𝑠 𝑖 = 𝑝 𝑖 ] = ( − 𝑝 𝑖 ) 𝑝𝑖 . It is easy to verify Pr [ 𝑠 𝑖 ≥ 𝐿 ] ≥ Pr [ ¯ 𝑠 𝑖 ≥ 𝐿 ] for any 𝐿 ≥ . Moreover, we have E [ ¯ 𝑠 𝑖 ] ≥ 𝑝 𝑖 . Write 𝑆 = Í ≤ 𝑖 ≤ . 𝑛 − ¯ 𝑠 𝑖 , we have E [ 𝑆 ] ≥ . 𝑛 ln ( . 𝑛 ) . By applying the Hoeffding’s inequality [14], we get Pr [ 𝑆 < 𝑐 ′ 𝑛 ] ≤ − ( . 𝑛 ln ( . 𝑛 ) − 𝑐 ′ 𝑛 ) Í ≤ 𝑖 ≤ . 𝑛 − (cid:0) 𝑛 . 𝑛 − 𝑖 (cid:1) ! = 𝑛 𝜔 ( ) (cid:3) ight Trade-off in Contention Resolution without Collision Detection Now assume 𝑙𝑎𝑠𝑡 ≥ 𝑐𝑐 𝑛 . We intend to apply Lemma 3.3 on the control channel within time interval [ , 𝑡 / ] .For the first condition, 𝑘 = 𝑙𝑎𝑠𝑡 / is a feasible value, since 𝑛 · 𝑐 log ( 𝑙𝑎𝑠𝑡 / ) 𝑙𝑎𝑠𝑡 / ≤ log ( 𝑘 / ) 𝑐 ′ when 𝑙𝑎𝑠𝑡 ≥ 𝑐 ′ 𝑐 𝑛 , and · 𝑐 log ( 𝑙𝑎𝑠𝑡 / ) 𝑙𝑎𝑠𝑡 / ≥ 𝑐 ′ log 𝑘𝑘 when 𝑐 ≥ 𝑐 ′ . The second condition is satisfied because the considered interval has length Θ ( 𝑡 ) . Recall that there is some node running backoff on control channel within time [ , 𝑙𝑎𝑠𝑡 ] by case assumption, sothe third condition of Lemma 3.3 is satisfied. Therefore, we conclude with high probability in 𝑡 , there is a success onthe control channel before time 𝑡 / , which implies ¯ 𝑙 ≤ 𝑡 / < 𝑡 . (cid:3) Remark.
Claim 3.5.1 could be of independent interest, in that it showcases ℎ 𝑑𝑎𝑡𝑎 - batch —a standard implementationof binary exponential backoff—cannot send all 𝑛 messages in 𝑂 ( 𝑛 ) slots, with high probability in 𝑛 . This holds even if 𝑛 nodes start simultaneously and there is no external interference. Nevertheless, assuming all nodes start simultaneously,with high probability in 𝑛 , ℎ 𝑑𝑎𝑡𝑎 - batch can send a constant fraction of all 𝑛 messages in 𝑂 ( 𝑛 ) slots, even if a constantfraction of all these slots are jammed. (See, e.g., the analysis for scenario two in the appendix.) In this subsection, we will first prove our main algorithmic result, and then prove a corollary clarifying the connectionbetween an algorithm’s ( 𝑓 , 𝑔 ) -throughput and the number of successful transmission it can guarantee. Proof of Theorem 1.2.
Recall the definition of ( 𝑓 , 𝑔 ) -throughput. We focus on the first 𝑡 slots and assume 𝑛 𝑡 nodesarrive within these slots. By definition of complete interval, the number of active slots in [ , 𝑡 ] is the summation of thelengths of the complete intervals that any of these 𝑛 𝑡 nodes involved in, excluding any active slots after 𝑡 . Denote thesecomplete intervals as I , I , · · · , I 𝑛 𝑡 , where some may be empty. (The number of such complete intervals is at most 𝑛 𝑡 since there is at least one success during each complete interval.) Specifically, if I 𝑙𝑎𝑠𝑡 is the last complete interval thatbegins in some slot in [ , 𝑡 ] , then all {I 𝑘 } 𝑘 > 𝑙𝑎𝑠𝑡 has length . Since we focus on the number of active slots in the first 𝑡 slots, we can assume nodes that are still active at the end of slot 𝑡 are allowed to continue the algorithm after slot 𝑡 while Eve does not inject new nodes or jam after slot 𝑡 , and bound the number of active slots in this setting instead.For each complete interval I 𝑗 , we use 𝑚 I 𝑗 to denote the number of successes occurred during I 𝑗 , use 𝑛 I 𝑗 to denotethe number of new arrivals during I 𝑗 , and use 𝑑 I 𝑗 to denote the number of jammed slots during I 𝑗 . Recall that 𝑙 I 𝑗 is the length of I 𝑗 , and ¯ 𝑙 I 𝑗 is the truncated length of I 𝑗 . If 𝑙 I 𝑗 > ¯ 𝑙 I 𝑗 occurs (i.e., ¯ 𝑙 I 𝑗 = ), then by the definition oftruncated length, at least one of following three conditions holds: (1) 𝑚 I 𝑗 ≥ 𝑙 I 𝑗 /( 𝑐𝑐 ( 𝑡 + )) , called the many-success condition ; (2) 𝑛 I 𝑗 > 𝑙 I 𝑗 /( 𝑐𝑐 𝑐 · 𝑓 ′ ( 𝑙 I 𝑗 )) , called the heavy-arriving condition ; (3) 𝑑 I 𝑗 > 𝑙 I 𝑗 /( 𝑐𝑐 𝑐 · 𝑔 ′ ( 𝑙 I 𝑗 )) ,called the heavy-jamming condition . (As in the proof of Lemma 3.5, define function 𝑓 ′ ( 𝑥 ) = 𝑓 ( 𝑥 )/ 𝑎 and 𝑔 ′ ( 𝑥 ) = 𝑔 ( 𝑥 )/ 𝑎 .)Let C (respectively, C or C ) be the set containing the complete intervals that satisfy the many-success condition(respectively, heavy-arriving condition or heavy-jamming condition).We intend to bound the total length of all intervals in C , C , and C . But before that, we first show any completeinterval that satisfies the heavy-arriving condition or the heavy-jamming condition will have length 𝑂 ( 𝑡 ) , otherwisethe theorem already holds. Recall that 𝑓 and log 𝑔 are sub-logarithmic, thus for any constant ˆ 𝑐 > , there exists someconstant 𝑐 such that for any 𝑥 ∈ N + , we have | 𝑓 ( ˆ 𝑐𝑥 ) − 𝑓 ( 𝑥 )| ≤ 𝑐 , implying 𝑓 ′ ( ˆ 𝑐𝑥 )/ 𝑓 ′ ( 𝑥 ) ≤ 𝑐 ; and | log 𝑔 ( ˆ 𝑐𝑥 ) − log 𝑔 ( 𝑥 )| ≤ 𝑐 , implying 𝑔 ′ ( ˆ 𝑐𝑥 )/ 𝑔 ′ ( 𝑥 ) ≤ 𝑐 . Let 𝛾 = 𝑐 · 𝑐𝑐 𝑐 . Now, if there exists some I 𝑗 ∈ C satisfying 𝑙 I 𝑗 ≥ 𝛾𝑡 for some 𝛾 > 𝛾 , then by the definition of the heavy-jamming condition, 𝑑 𝑙 I 𝑗 > 𝛾𝑡 /( 𝑐𝑐 𝑐 · 𝑔 ′ ( 𝛾𝑡 )) ≥ 𝑐 𝑡 /( 𝑐 · 𝑔 ′ ( 𝑡 )) ≥ 𝑡 / 𝑔 ′ ( 𝑡 ) , which further implies 𝑛 𝑡 𝑓 ( 𝑡 ) + 𝑑 𝑡 𝑔 ( 𝑡 ) ≥ 𝑎𝑑 𝑡 𝑔 ′ ( 𝑡 ) ≥ 𝑎𝑑 𝑙 I 𝑗 · 𝑔 ′ ( 𝑡 ) > 𝑎𝑡 ≥ 𝑡 , thus thenumber of active slots in [ , 𝑡 ] is trivially at most 𝑛 𝑡 𝑓 ( 𝑡 ) + 𝑑 𝑡 𝑔 ( 𝑡 ) , and the theorem is proved. Similarly, if there exists aimin Chen, Yonggang Jiang, and Chaodong Zheng some I 𝑗 ∈ C satisfying 𝑙 I 𝑗 ≥ 𝛾𝑡 for some 𝛾 > 𝛾 , then by the definition of the heavy-arriving condition, 𝑛 𝑙 I 𝑗 = 𝛾𝑡 /( 𝑐𝑐 𝑐 · 𝑓 ′ ( 𝛾𝑡 )) ≥ 𝑐 𝑡 /( 𝑐 · 𝑓 ′ ( 𝑡 )) ≥ 𝑡 / 𝑓 ′ ( 𝑡 ) , which further implies 𝑛 𝑡 𝑓 ( 𝑡 ) + 𝑑 𝑡 𝑔 ( 𝑡 ) > 𝑡 , thus the theoremtrivially holds. Therefore, from now on, we assume there is no I 𝑗 that satisfies the heavy-arriving condition or theheavy-jamming condition but with length larger than 𝛾𝑡 .We now bound the total length of all intervals in C , C , and C : (1) Complete intervals satisfying many-successcondition. Since Í I 𝑗 ∈C 𝑚 I 𝑗 ≤ 𝑛 𝑡 , we have Í I 𝑗 ∈C 𝑙 I 𝑗 ≤ Í I 𝑗 ∈C 𝑚 I 𝑗 · 𝑐𝑐 ( 𝑡 + ) ≤ 𝑐𝑐 ( 𝑡 + ) · 𝑛 𝑡 . (2) Completeintervals satisfying heavy-arriving condition. Since Í I 𝑗 ∈C 𝑛 I 𝑗 ≤ 𝑛 𝑡 (as each node is a new arrival for at most twodistinct complete intervals), we have Í I 𝑗 ∈C 𝑙 I 𝑗 ≤ Í I 𝑗 ∈C 𝑛 I 𝑗 · ( 𝑐𝑐 𝑐 ) · 𝑓 ′ ( 𝑙 I 𝑗 ) ≤ Í I 𝑗 ∈C 𝑛 I 𝑗 · ( 𝑐𝑐 𝑐 ) · 𝑓 ′ ( 𝛾 𝑡 ) ≤ 𝑐 · 𝑐𝑐 𝑐 · 𝑛 𝑡 · 𝑓 ′ ( 𝑡 ) . (3) Complete intervals satisfying heavy-jamming condition. Since Í I 𝑗 ∈C 𝑑 I 𝑗 ≤ 𝑑 𝑡 , we have Í I 𝑗 ∈C 𝑙 I 𝑗 ≤ Í I 𝑗 ∈C 𝑑 I 𝑗 · ( 𝑐𝑐 𝑐 ) · 𝑔 ′ (I 𝑗 ) ≤ Í I 𝑗 ∈C 𝑑 I 𝑗 · ( 𝑐𝑐 𝑐 ) · 𝑔 ′ ( 𝛾 𝑡 ) ≤ 𝑐 · 𝑐𝑐 𝑐 · 𝑑 𝑡 · 𝑔 ′ ( 𝑡 ) .As the final preparation for bounding total number of active slots, we show Í 𝑛 𝑡 𝑗 = ¯ 𝑙 I 𝑗 is 𝑂 ( 𝑛 𝑡 ) . For any 𝑗 ∈ [ 𝑛 ] , if I 𝑗 is not empty, then by Lemma 3.5, conditioned on any history up to the beginning of I 𝑗 , we have Pr [ ¯ 𝑙 I 𝑗 = 𝑡 ] ≤ 𝑡 − for 𝑡 ≥ 𝑡 ; and if I 𝑗 is empty (which implies ¯ 𝑙 I 𝑗 = ), then again we have Pr [ ¯ 𝑙 I 𝑗 = 𝑡 ] = ≤ 𝑡 − . Apply Lemma 3.4and we know, with high probability in 𝑛 𝑡 , Í 𝑛 𝑡 𝑗 = ¯ 𝑙 I 𝑗 ≤ ( 𝑡 + ) 𝑛 𝑡 .In conclusion, if we set 𝑎 = ( 𝑡 + ) + 𝑐𝑐 ( 𝑡 + ) + 𝑐𝑐 𝑐 · 𝑐 to be a sufficiently large constant, then wehave Í 𝑛 𝑡 𝑗 = 𝑙 I 𝑗 ≤ Í 𝑛 𝑡 𝑗 = ¯ 𝑙 I 𝑗 + Í I 𝑗 ∈C 𝑙 I 𝑗 + Í I 𝑗 ∈C 𝑙 I 𝑗 + Í I 𝑗 ∈C 𝑙 I 𝑗 ≤ 𝑎 · 𝑛 𝑡 · 𝑓 ′ ( 𝑛 𝑡 ) + 𝑎 · 𝑑 𝑡 · 𝑔 ′ ( 𝑡 ) . Recall 𝑓 ( 𝑥 ) = 𝑎𝑓 ′ ( 𝑥 ) , 𝑔 ( 𝑥 ) = 𝑎𝑔 ′ ( 𝑥 ) , so the number of active slots in first 𝑡 slots in at most 𝑛 𝑡 · 𝑓 ( 𝑡 ) + 𝑑 𝑡 · 𝑔 ( 𝑡 ) . (cid:3) Next, we state and prove the following corollary connecting an algorithm’s ( 𝑓 , 𝑔 ) -throughput and the number ofsuccessful transmission it can guarantee. Corollary 3.6.
Assume nodes run an algorithm that achieves ( 𝑓 , 𝑔 ) -throughput. For interval [ , 𝑡 ] , an adversarystrategy is called " smooth " if for any ≤ 𝑗 ≤ 𝑡 − , the number of nodes arrived in [ 𝑡 − 𝑗, 𝑡 ] is small enough in 𝑂 ( 𝑗 / 𝑓 ( 𝑗 )) and the number of jammed slots in [ 𝑡 − 𝑗, 𝑡 ] is small enough in 𝑂 ( 𝑗 / 𝑔 ( 𝑗 )) . Then under any "smooth" adversary strategy B , for any ≤ 𝑗 ≤ 𝑡 − , all nodes arrived before slot 𝑡 − 𝑗 will leave the system by the end of slot 𝑡 , with high probabilityin 𝑗 . Proof.
Fix an arbitrary integer 𝑗 ∈ [ , 𝑡 − ] . For any B , construct another adversary strategy B ′ in the followingway. Besides the same node arrivals and jammed slots as B , B ′ injects another 𝑗 . nodes in the last slot (i.e., slot 𝑡 )and jams the last slot. Notice that the execution under adversary strategy B and B ′ are identical until the end of slot 𝑡 − . Moreover, in B ′ nodes cannot succeed in slot 𝑡 since that slot is jammed. Thus, if we can prove the claim in B ′ then the claim also holds in B . Therefore, we focus on B ′ in the reminder of the proof.Suppose to the contrary, there exists some node arrived before slot 𝑡 − 𝑗 that leaves the system after slot 𝑡 . Then allslots in [ 𝑡 − 𝑗, 𝑡 ] are active. Therefore, there must exist some integer 𝑘 ∈ [ , 𝑡 − 𝑗 − ] such that all slots in [ 𝑘 + , 𝑡 ] are active and slot 𝑘 is inactive. (In case 𝑘 = then all slots in [ , 𝑡 ] are active.) Consider interval [ 𝑘 + , 𝑡 ] . Since theadversary strategy B is smooth, for B ′ the number of arrived nodes and jammed slots in [ 𝑘 + , 𝑡 ] is small enough in 𝑂 (( 𝑡 − 𝑘 )/ 𝑓 ( 𝑡 − 𝑘 )) + 𝑗 . = 𝑂 (( 𝑡 − 𝑘 )/ 𝑓 ( 𝑡 − 𝑘 )) (recall 𝑓 is sub-logarithmic and 𝑡 − 𝑘 > 𝑗 ) and 𝑂 (( 𝑡 − 𝑘 )/ 𝑔 ( 𝑡 − 𝑘 )) + = 𝑂 (( 𝑡 − 𝑘 )/ 𝑔 ( 𝑡 − 𝑘 )) , respectively. Therefore, according to Theorem 1.2 and the definition of ( 𝑓 , 𝑔 ) -throughput, (atleast) with high probability in 𝑗 . (thus also in 𝑗 ), the number of active slots in [ 𝑘 + , 𝑡 ] is (strictly) less than 𝑡 − 𝑘 , acontradiction. Summing over the distribution of 𝑘 we get with high probability in 𝑗 , all nodes arrived before slot 𝑡 − 𝑗 will leave the system by the end of slot 𝑡 . (cid:3) ight Trade-off in Contention Resolution without Collision Detection In this section, we will prove our impossibility result on the trade-off between the best possible throughput and theseverity of jamming, we will also prove another impossibility result demonstrating the necessity of backoff styleprocedures in attending optimal throughput when jamming is present.
Main impossibility result.
The impossibility result on the trade-off exploits the following dilemma that all contentionresolution algorithms must confront: on the one hand, when few nodes (e.g., only one node) are in the system, eachnode’s broadcasting probability should be sufficiently high, otherwise successes will not happen fast enough; on theother hand, however, when a lot of nodes are in the system, each node’s broadcasting probability should be sufficientlylow, otherwise contention among themselves would prevent successes from happening.The following lemma captures the case that when many nodes are in the system, each node’s broadcasting proba-bility cannot be too high for too long.
Lemma 4.1.
Let ℎ be a sub-logarithmic function and A be a contention resolution algorithm. Consider a node that runs A . Suppose in expectation, the node broadcasts 𝜔 ( ℎ ( 𝑡 ) log 𝑡 ) times in the first 𝑡 slots (since its activation) before it hearsthe first success, then A does not achieve ( ℎ,𝑔 ) -throughput for any 𝑔 . Proof.
Let 𝑥 𝑖 be the probability that A sends a messages in the 𝑖 -th slot since the execution of A starts, assumingno successes occur in slots to 𝑖 − . Consider an adversary strategy that injects ( 𝑡 )/ 𝑥 nodes in each of thefirst √ 𝑡 slots—call these nodes “batch-injected”, and injects another 𝑡 /( ℎ ( 𝑡 )) nodes in the first 𝑡 slots uniformly atrandom—call these nodes “random-injected”. We show no success occurs in the first 𝑡 slots, with high probability in 𝑡 .For each of the first √ 𝑡 slots, since each newly arrived node will send in the slot with probability 𝑥 and there are atleast ( 𝑡 )/ 𝑥 newly arrived nodes, the probability of a success occurring is at most / 𝑡 . (This is because, in eachsuch slot, at least ( 𝑡 )/ 𝑥 − newly arrived batch-injected nodes need to choose not send.)Consider a slot 𝑘 > √ 𝑡 and a random-injected node 𝑢 , the probability that 𝑢 sends in this slot is Í ≤ 𝑗 ≤ 𝑘 ( / 𝑡 ) · 𝑥 𝑘 − 𝑗 + = 𝐸 𝑘 / 𝑡 . (The probability that " 𝑢 arrives in the 𝑗 -th slot" times the probability that " 𝑢 sends in the ( 𝑘 − 𝑗 + ) -th slotsince its arrival".) Here, 𝐸 𝑘 = Í ≤ 𝑗 ≤ 𝑘 𝑥 𝑘 − 𝑗 + = Í ≤ 𝑗 ≤ 𝑘 𝑥 𝑗 . Due to lemma assumption, we know 𝐸 𝑘 = 𝜔 ( ℎ ( 𝑘 ) · log 𝑘 ) .Since ℎ is sub-logarithmic and 𝑘 > √ 𝑡 , we have ℎ ( 𝑘 ) = Ω ( ℎ ( 𝑡 )) , thus 𝐸 𝑘 = 𝜔 ( ℎ ( 𝑡 ) · log 𝑡 ) . As a result, the expectedcontention in slot 𝑘 is at least ( 𝑡 /( ℎ ( 𝑡 ))) · ( 𝐸 𝑘 / 𝑡 ) = 𝜔 ( log 𝑡 ) , implying the probability of success in slot 𝑘 is at most / 𝑡 .Apply a union bound, we know with high probability in 𝑡 there are no successes in the first 𝑡 slots.Assume 𝑡 is sufficiently large. Since ℎ is sub-logarithmic, ℎ ( 𝑡 ) = 𝑂 ( log 𝑡 ) and is at least . Hence, the number ofinjected nodes in the first 𝑡 slots is ( 𝑡 )/ 𝑥 · √ 𝑡 + 𝑡 /( ℎ ( 𝑡 )) ≤ 𝑡 /( . ℎ ( 𝑡 )) . This means 𝑛 𝑡 ≤ 𝑡 /( . ℎ ( 𝑡 )) ≤ 𝑡 and 𝑑 𝑡 = . Now, if algorithm A achieves ( ℎ,𝑔 ) -throughput, then the number of active slots in the first 𝑡 slots is boundedby 𝑡 /( . ℎ ( 𝑡 )) · ℎ ( 𝑛 𝑡 ) ≤ 𝑡 /( . ℎ ( 𝑡 )) · ℎ ( 𝑡 ) = 𝑡 / . . However, we know all the 𝑡 slots are active, a contradiction. (cid:3) To prove the impossibility result, what remains is to show that in case few nodes are in the system, each node mustgenerate enough contention to ensure a success can happen fast enough.
Proof of Theorem 1.3.
Suppose algorithm A achieves ( 𝑓 , 𝑔 ) -throughput and satisfies: 𝑓 , log ( 𝑔 ) are both sub-logarithmicand 𝑓 ( 𝑥 ) = 𝑜 (( log 𝑥 )/ log 𝑔 ( 𝑥 )) . We will show, in expectation, A broadcasts Ω (( log 𝑡 )/ log 𝑔 ( 𝑡 )) times in the first 𝑡 slots before any success occurs. Together with Lemma 4.1, the theorem is immediate.Consider an interval of 𝑡 slots. Consider an adversary strategy that injects one node 𝑢 in the first slot, and jams the aimin Chen, Yonggang Jiang, and Chaodong Zheng first 𝑡 /( 𝑔 ( 𝑡 )) slots as well as the last slot. The adversary also jams another 𝑡 /( 𝑔 ( 𝑡 )) slots which are chosen uniformlyat random from slot interval ( 𝑡 /( 𝑔 ( 𝑡 )) , 𝑡 ] . Since there are at most 𝑡 /( 𝑔 ( 𝑡 )) + jammed slots and 𝑡 /( 𝑓 ( 𝑡 )) + injected nodes, and since A achieves ( 𝑓 , 𝑔 ) -throughput, the number of active slots in the first 𝑡 slots is at most 𝑡 − with high probability in 𝑡 / 𝑓 ( 𝑡 ) , for sufficiently large 𝑡 . Therefore, there is at least one success in the first 𝑡 slots withhigh probability in 𝑡 / 𝑓 ( 𝑡 ) > √ 𝑡 , as 𝑓 ( 𝑡 ) = 𝑂 ( log 𝑡 ) .Denote 𝑘 ( 𝑡 ) as the number of times node 𝑢 broadcasts in slots ( 𝑡 /( 𝑔 ( 𝑡 )) , 𝑡 ] . Since in slot interval ( 𝑡 /( 𝑔 ( 𝑡 )) , 𝑡 ] the adversary randomly chooses 𝑡 /( 𝑔 ( 𝑡 )) slots to jam, the probability that no success occur in these slots is atleast Í 𝑘 ≤ 𝑡 𝑔 ( 𝑡 ) Pr [ 𝑘 ( 𝑡 ) = 𝑘 ]( 𝑔 ( 𝑡 )) 𝑘 . Since there must be a success in these slots with probability at least − /√ 𝑡 , we have Í 𝑘 ≤ 𝑡 𝑔 ( 𝑡 ) Pr [ 𝑘 ( 𝑡 ) = 𝑘 ]( 𝑔 ( 𝑡 )) 𝑘 ≤ √ 𝑡 . Since 𝑡 /( 𝑔 ( 𝑡 )) ≥ log 𝑔 ( 𝑡 ) √ 𝑡 for the family of 𝑔 we care, we have: Õ 𝑘 ≤ 𝑡 𝑔 ( 𝑡 ) Pr [ 𝑘 ( 𝑡 ) = 𝑘 ]( 𝑔 ( 𝑡 )) 𝑘 ≥ Õ 𝑘 ≤ log 𝑔 ( 𝑡 ) √ 𝑡 Pr [ 𝑘 ( 𝑡 ) = 𝑘 ]( 𝑔 ( 𝑡 )) log 𝑔 ( 𝑡 ) √ 𝑡 / = Õ 𝑘 ≤ log 𝑔 ( 𝑡 ) √ 𝑡 Pr [ 𝑘 ( 𝑡 ) = 𝑘 ]√ 𝑡 / = Pr h 𝑘 ( 𝑡 ) ≤ log 𝑔 ( 𝑡 ) (cid:16) √ 𝑡 / (cid:17) i √ 𝑡 / If Pr h 𝑘 ( 𝑡 ) ≤ log 𝑔 ( 𝑡 ) √ 𝑡 i > , then Í 𝑘 ≤ 𝑡 𝑔 ( 𝑡 ) Pr [ 𝑘 ( 𝑡 ) = 𝑘 ]( 𝑔 ( 𝑡 )) 𝑘 > / √ 𝑡 / = √ 𝑡 , which is a contradiction. As a result, we know Pr h 𝑘 ( 𝑡 ) ≤ log 𝑔 ( 𝑡 ) √ 𝑡 i ≤ , which means Pr h 𝑘 ( 𝑡 ) ≥ log 𝑔 ( 𝑡 ) √ 𝑡 i ≥ , implying E [ 𝑘 ( 𝑡 )] ≥ log 𝑔 ( 𝑡 ) √ 𝑡 .By an argument similar as above, for any 𝑖 ∈ [ 𝑙 ] where 𝑙 = log 𝑔 ( 𝑡 ) 𝑡 , in slot interval (cid:16) 𝑡 ( 𝑔 ( 𝑡 )) 𝑖 . (cid:16) 𝑔 (cid:16) 𝑡 ( 𝑔 ( 𝑡 )) 𝑖 (cid:17)(cid:17) , 𝑡 ( 𝑔 ( 𝑡 )) 𝑖 i ,the expected number of times node 𝑢 broadcasts is at least log 𝑔 ( 𝑡 /( 𝑔 ( 𝑡 )) 𝑖 ) √ 𝑡 /( 𝑔 ( 𝑡 )) 𝑖 . Since 𝑔 is non-decreasing, wehave 𝑡 ( 𝑔 ( 𝑡 )) 𝑖 / (cid:16) 𝑔 (cid:16) 𝑡 ( 𝑔 ( 𝑡 )) 𝑖 (cid:17)(cid:17) ≥ 𝑡 ( 𝑔 ( 𝑡 )) 𝑖 + , implying in expectation node 𝑢 must broadcast log 𝑔 ( 𝑡 /( 𝑔 ( 𝑡 )) 𝑖 ) √ 𝑡 /( 𝑔 ( 𝑡 )) 𝑖 times during slots (cid:16) 𝑡 ( 𝑔 ( 𝑡 )) 𝑖 + , 𝑡 ( 𝑔 ( 𝑡 )) 𝑖 i . Let 𝑏 𝑡 be the expected number of times 𝑢 broadcasts in the first 𝑡 slots, we have: 𝑏 𝑡 ≥ Õ ≤ 𝑖 ≤ 𝑙 / · log (cid:16)q 𝑡 ( 𝑔 ( 𝑡 )) 𝑖 . (cid:17) log (cid:16) 𝑔 (cid:16) 𝑡 ( 𝑔 ( 𝑡 )) 𝑖 (cid:17)(cid:17) ≥ 𝑙 · log (cid:18)q 𝑡 ( 𝑔 ( 𝑡 )) 𝑙 / (cid:30) (cid:19) log (cid:16) 𝑔 (cid:16) 𝑡 ( 𝑔 ( 𝑡 )) (cid:17)(cid:17) = Ω (cid:18) 𝑙 · log 𝑡 log 𝑔 ( 𝑡 ) (cid:19) = Ω (cid:18) log 𝑡 log 𝑔 ( 𝑡 ) (cid:19) Recall the first paragraph of this proof, the theorem is proved. (cid:3)
Remark.
Our proof critically relies on the absence of collision detection. To see this, for each node 𝑢 , denote 𝑥 𝑢𝑖 asthe probability that it broadcasts in the 𝑖 -th slot since its activation, assuming no success occurs in slots 1 to 𝑖 − .Without collision detection, for any two nodes 𝑣 and 𝑤 , for any 𝑖 ≥ , the distribution of 𝑥 𝑣𝑖 and 𝑥 𝑤𝑖 are identical, as 𝑣 and 𝑤 receive identical channel feedback. Thus the superscript is not necessary: we can use 𝑥 𝑖 to denote 𝑥 𝑢𝑖 . However,with collision detection, this no longer holds. For example, if 𝑣 hears silence in the first slot since its activation and 𝑤 hears collision in the first slot since its activation, then 𝑥 𝑣 and 𝑥 𝑤 might differ! More fundamentally, collision detectionbreaks the dilemma the impossibility result proof exploits: nodes can differentiate whether there are few nodes/lowcontention (hearing silence), or there are many nodes/high contention (hearing collision), and take different actions. Necessity of backoff . Recall that in our algorithm, two variants of the standard backoff procedure are used: (1) batch , which sends with probability / 𝑖 in slot 𝑖 ; and (2) backoff , which chooses several slots in a stage to send andthen increases stage length. A critical difference between these two procedures is that backoff is adaptive : in a slot,the sending probability of a node depends on the previous sending behavior of the node. The next theorem shows that backoff is necessary when jamming exists: non-adaptive sending pattern cannot provide optimal throughput. Theorem 4.2.
For algorithm A , if A will send with pre-defined probability 𝑎 𝑖 in the 𝑖 -th slot since it starts and before ight Trade-off in Contention Resolution without Collision Detection any success is heard, then for any function 𝑓 , 𝑔 such that 𝑓 , log ( 𝑔 ) are both sub-logarithmic and 𝑓 ( 𝑥 ) = 𝑜 (cid:16) log 𝑥 log 𝑔 ( 𝑥 ) (cid:17) ,algorithm A does not achieve ( 𝑓 , 𝑔 ) -throughput. Proof sketch.
Consider an interval of 𝑡 slots, we will show Í ≤ 𝑖 ≤ 𝑡 𝑎 𝑖 = Ω (cid:16) log 𝑡 log 𝑔 ( 𝑡 ) (cid:17) , and then apply Lemma 4.1to obtain our conclusion. The following part is similar to the proof of Theorem 1.3.For the sake of contradiction, assume A achieves ( 𝑓 , 𝑔 ) -throughput. Consider the adversary strategy that jams thefirst 𝑡 𝑔 ( 𝑡 ) slots as well as the last slot, and injects nodes in the first slot as well as 𝑡 𝑓 ( 𝑡 ) nodes in the last slot. Wecan show the probability that there is no success in the first 𝑡 slots is at most /√ 𝑡 .Denote 𝑝 𝑖 as the probability that the 𝑖 -th slot succeeds. Let interval 𝐼 = (cid:16) 𝑡 𝑔 ( 𝑡 ) , 𝑡 i . For every 𝑖 ∈ 𝐼 , we know 𝑝 𝑖 = 𝑎 𝑖 ( − 𝑎 𝑖 ) ≤ / . Due to the analysis in the last paragraph, we also have Î 𝑖 ∈ 𝐼 ( − 𝑝 𝑖 ) ≤ /√ 𝑡 . Notice that when ≤ 𝑝 𝑖 ≤ / , we have Î 𝑖 ∈ 𝐼 ( − 𝑝 𝑖 ) ≥ Î 𝑖 ∈ 𝐼 − 𝑝 𝑖 = − Í 𝑖 ∈ 𝐼 𝑝 𝑖 , hence − Í 𝑖 ∈ 𝐼 𝑝 𝑖 ≤ /√ 𝑡 , implying Í 𝑖 ∈ 𝐼 𝑝 𝑖 = Ω ( log 𝑡 ) .Since 𝑎 𝑖 ≥ 𝑝 𝑖 / , we have Í 𝑖 ∈ 𝐼 𝑎 𝑖 = Ω ( log 𝑡 ) . Recall the proof of Theorem 1.3, consider intervals (cid:16) 𝑡 ( 𝑔 ( 𝑡 )) 𝑖 + , 𝑡 ( 𝑔 ( 𝑡 )) 𝑖 i ,we have Í ≤ 𝑖 ≤ 𝑡 𝑎 𝑖 = Ω (cid:16) log 𝑡 log 𝑔 ( 𝑡 ) (cid:17) . Recall the first paragraph of this proof, the theorem is proved. (cid:3) ACKNOWLEDGMENTS
The authors would like to thank Prof. Yitong Yin for the comments and suggestions that greatly improve the overallquality of the paper.
REFERENCES [1] Kunal Agrawal, Michael Bender, Jeremy Fineman, Seth Gilbert, and Maxwell Young. 2020. Contention Resolution with Message Deadlines. In
Proceedings of the 32nd ACM Symposium on Parallelism in Algorithms and Architectures (SPAA ’20) . ACM, 23––35.[2] David Aldous. 1987. Ultimate Instability of Exponential Back-Off Protocol for Acknowledgment-Based Transmission Control of Random AccessCommunication Channels.
IEEE Transactions on Information Theory
33, 2 (1987), 219–223.[3] Lakshmi Anantharamu, Bogdan Chlebus, Dariusz Kowalski, and Mariusz Rokicki. 2019. Packet latency of deterministic broadcasting in adversarialmultiple access channels.
J. Comput. System Sci.
99 (2019), 27–52.[4] Baruch Awerbuch, Andrea Richa, and Christian Scheideler. 2008. A Jamming-Resistant MAC Protocol for Single-Hop Wireless Networks. In
Proceedings of the 27th ACM Symposium on Principles of Distributed Computing (PODC ’08) . ACM, 45–54.[5] Michael Bender, Martin Farach-Colton, Simai He, Bradley Kuszmaul, and Charles Leiserson. 2005. Adversarial Contention Resolution for SimpleChannels. In
Proceedings of the 17th Annual ACM Symposium on Parallelism in Algorithms and Architectures (SPAA ’05) . ACM, 325–332.[6] Michael Bender, Jeremy Fineman, Seth Gilbert, and Maxwell Young. 2018. Scaling Exponential Backoff: Constant Throughput, PolylogarithmicChannel-Access Attempts, and Robustness.
J. ACM
66, 1 (2018).[7] Michael Bender, Tsvi Kopelowitz, Seth Pettie, and Maxwell Young. 2016. Contention Resolution with Log-Logstar Channel Accesses. In
Proceedingsof the 48th Annual ACM Symposium on Theory of Computing (STOC ’16) . ACM, 499–508.[8] Michael A. Bender, Tsvi Kopelowitz, William Kuszmaul, and Seth Pettie. 2020. Contention Resolution without Collision Detection. In
Proceedingsof the 52nd Annual ACM SIGACT Symposium on Theory of Computing (STOC ’20) . ACM, 105–118.[9] John Capetanakis. 1979. Generalized TDMA: The Multi-Accessing Tree Protocol.
IEEE Transactions on Communications
27, 10 (1979), 1476–1484.[10] Yi-Jun Chang, Wenyu Jin, and Seth Pettie. 2018. Simple Contention Resolution via Multiplicative Weight Updates. In . Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, 16:1–16:16.[11] Bogdan Chlebus, Leszek Gąsieniec, Dariusz R. Kowalski, and Tomasz Radzik. 2005. On the Wake-Up Problem in Radio Networks. In
Proceedings ofthe 2005 International Colloquium on Automata, Languages, and Programming (ICALP ’05) . Springer Berlin Heidelberg, 347–359.[12] Bogdan Chlebus, Dariusz Kowalski, and Mariusz Rokicki. 2012. Adversarial Queuing on the Multiple Access Channel.
ACM Transactions onAlgorithms
8, 1 (2012).[13] Shlomi Dolev, Seth Gilbert, Rachid Guerraoui, Fabian Kuhn, and Calvin Newport. 2009. The Wireless Synchronization Problem. In
Proceedings ofthe 28th ACM Symposium on Principles of Distributed Computing (PODC ’09) . ACM, 190–199.[14] Devdatt P. Dubhashi. 2012.
Concentration of Measure for the Analysis of Randomized Algorithms . Cambridge University Press.[15] Johan Hastad, Tom Leighton, and Brian Rogoff. 1987. Analysis of Backoff Protocols for Multiple Access Channels. In
Proceedings of the 19th AnnualACM Symposium on Theory of Computing (STOC ’87) . ACM, 241–253.[16] James Kurose and Keith Ross. 2017.
Computer Networking: A Top-Down Approach, 7th Edition . Pearson.17 aimin Chen, Yonggang Jiang, and Chaodong Zheng [17] Gianluca De Marco, Dariusz Kowalski, and Grzegorz Stachowiak. 2019. Deterministic Contention Resolution on a Shared Channel. In
Proceedingsof the 39th International Conference on Distributed Computing Systems (ICDCS ’19) . IEEE, 472–482.[18] Robert Metcalfe and David Boggs. 1976. Ethernet: Distributed Packet Switching for Local Computer Networks.
Commun. ACM
19, 7 (1976),395–404.[19] Prabhakar Raghavan and Eli Upfal. 1995. Stochastic Contention Resolution with Short Delays. In
Proceedings of the 27th Annual ACM Symposiumon Theory of Computing (STOC ’95) . ACM, 229–237.[20] Raghu Ramakrishnan and Johannes Gehrke. 2002.
Database Management Systems, 3rd Edition . McGraw-Hill.[21] Andrea Richa, Christian Scheideler, Stefan Schmid, and Jin Zhang. 2010. A Jamming-Resistant MAC Protocol for Multi-Hop Wireless Networks. In
Proceedings of the 2010 International Symposium on Distributed Computing (DISC ’10) . Springer Berlin Heidelberg, 179–193.[22] Andrew Tanenbaum and Herbert Bos. 2014.
Modern Operating Systems, 4th Edition . Pearson.[23] Dan Willard. 1986. Log-Logarithmic Selection Resolution Protocols in a Multiple Access Channel.
SIAM J. Comput.
15, 2 (1986), 468–477.
APPENDIX
Proof of Lemma 3.4.
For any 𝑖 ∈ [ 𝑛 ] , for any values 𝑥 , · · · , 𝑥 𝑖 − of 𝑋 , · · · , 𝑋 𝑖 − , we have E [ 𝑋 𝑖 | 𝑋 = 𝑥 , · · · , 𝑋 𝑖 − = 𝑥 𝑖 − ] ≤ 𝑡 · Pr [ 𝑋 𝑖 ≤ 𝑡 | 𝑋 = 𝑥 , · · · , 𝑋 𝑖 − = 𝑥 𝑖 − ] + Í 𝑡 > 𝑡 𝑡 · Pr [ 𝑋 𝑖 = 𝑡 | 𝑋 = 𝑥 , · · · , 𝑋 𝑖 − = 𝑥 𝑖 − ] ≤ 𝑡 + Í 𝑡 > 𝑡 𝑡 · 𝑡 − ≤ 𝑡 + . Besides, Pr [ 𝑋 𝑖 ≥ 𝑛 . ] ≤ Í 𝑡 ≥ 𝑛 . 𝑡 − ≤ 𝑛 − / , for sufficiently large 𝑛 . Take aunion bound over the 𝑛 random variables, we know with high probability in 𝑛 , 𝑋 𝑖 ≤ 𝑛 . holds for each 𝑖 ∈ [ 𝑛 ] .Assume indeed 𝑋 𝑖 is at most 𝑛 . for each 𝑖 ∈ [ 𝑛 ] . Apply Lemma 3 from [8], we can show with high probability in 𝑛 , Í 𝑛𝑖 = 𝑋 𝑖 ≤ ( 𝑡 + ) 𝑛 . (cid:3) Missing parts in proof of Lemma 3.5.
Here we provide analysis for scenario two and scenario three, which is verysimilar to the proof of Lemma 8 in [8].
Scenario II: 𝑐 𝑛 < 𝑡 ≤ 𝑐𝑐 𝑛 . Let 𝜂 = 𝑐 ( 𝑡 + ) . We will prove with high probability in 𝑛 (thus also in 𝑡 ), there areat least 𝜂𝑛 successes on the data channel within interval [ , 𝑐 𝑛 ] . If this holds, in the case that the batch ends in thefirst 𝑐 𝑛 slots (i.e. 𝑅 I ≤ 𝐿 I + 𝑐 𝑛 − ), we have ¯ 𝑙 ≤ 𝑐 𝑛 < 𝑡 ; in the other case that the batch lasts for at least 𝑐 𝑛 slots, we have ¯ 𝑙 = since there are 𝜂𝑛 ≥ 𝑡 /( 𝑐𝑐 ( 𝑡 + )) successes, where the inequality is due to 𝑡 < 𝑐𝑐 𝑛 .To prove there are at least 𝜂𝑛 successes on the data channel within [ , 𝑐 𝑛 ] , we begin with some notations. Let 𝜏 = 𝑐 𝑛 . Let 𝑠 𝑖 be the number of slots (of the data channel) from 𝐿 I to the slot that the 𝑖 -th success of the data channeloccurs, and define 𝑠 = . Further we set 𝑠 𝑖 as 𝜏 if 𝑠 𝑖 exceeds 𝜏 . Let 𝑋 𝑖 be the interval from ( 𝑠 𝑖 − + ) -th slot to 𝑠 𝑖 -th slot of the data channel, thus 𝑠 𝑖 = Í 𝑖𝑗 = | 𝑋 𝑗 | . So what we need to prove is 𝑠 𝜂𝑛 ≤ 𝜏 . We also use 𝑛 𝑋 𝑖 and 𝑑 𝑋 𝑖 todenote the number of new arrivals and jammed slots in interval 𝑋 𝑖 , respectively. Recall we assume 𝑛 [ , 𝜏 ] ≤ 𝑛 [ ,𝑡 ] ≤ 𝑡 /( 𝑐𝑐 𝑐 𝑓 ′ ( 𝑡 )) ≤ 𝜏 /( 𝑐 𝑓 ′ ( 𝜏 )) and 𝑑 [ , 𝜏 ] ≤ 𝑑 [ ,𝑡 ] ≤ 𝑑 I ≤ 𝑡 /( 𝑐𝑐 𝑐 𝑔 ′ ( 𝑡 )) ≤ 𝜏 /( 𝑐 𝑔 ′ ( 𝜏 )) .We define three types of intervals, and for each type bound the total length of the intervals in { 𝑋 𝑖 } belonging to thattype. We call an interval 𝑋 heavy-arriving if 𝑛 𝑋 > | 𝑋 |/( 𝑐 𝑓 ′ (| 𝑋 |)) ; call 𝑋 heavy-jamming if 𝑑 𝑋 > | 𝑋 |/( 𝑐 𝑔 ′ (| 𝑋 |)) ; andcall 𝑋 light if 𝑋 is not heavy-arriving and not heavy-jamming. Let X 𝐻𝑒𝑎𝑣𝑦𝐴 (respectively, X 𝐻𝑒𝑎𝑣𝑦𝐽 or X 𝐿𝑖𝑔ℎ𝑡 ) be the setcontaining all heavy-arriving intervals (respectively, all heavy-jamming intervals or all light intervals). (These threesets are not necessarily disjoint.) We know Í 𝑋 𝑖 ∈X 𝐻𝑒𝑎𝑣𝑦𝐴 | 𝑋 𝑖 | ≤ Í 𝑋 𝑖 ∈X 𝐻𝑒𝑎𝑣𝑦𝐴 𝑛 𝑋 𝑖 · 𝑐 𝑓 ′ (| 𝑋 𝑖 |) ≤ 𝜏 𝑐 𝑓 ′ ( 𝜏 ) · 𝑐 𝑓 ′ (| 𝑋 𝑖 |) ≤ 𝜏 / since Í 𝑋 𝑖 ∈X 𝐻𝑒𝑎𝑣𝑦𝐴 𝑛 𝑋 𝑖 ≤ 𝑛 [ , 𝜏 ] ≤ 𝜏 /( 𝑐 𝑓 ′ ( 𝜏 )) . Similarly we can prove Í 𝑋 𝑖 ∈X 𝐻𝑒𝑎𝑣𝑦𝐽 | 𝑋 𝑖 | ≤ 𝜏 / .What remains it to bound the total length of light intervals. We further divide light intervals into three groups andbound the total length of intervals within each group. For light intervals that end by slot 𝑛 (of the data channel), wecan bound them trivially since Í 𝑋 𝑖 ∈X 𝐿𝑖𝑔ℎ𝑡 ∧ 𝑠 𝑖 ≤ 𝑛 | 𝑋 𝑖 | ≤ 𝑛 ≤ 𝜏 / when 𝑐 ≥ .The second group contains at most one interval 𝑋 𝑖 that begins before or at slot 𝑛 (which implies last success hap- In such case, it must be that the batch ends due to some success on the control channel, since by Claim 3.5.1 it take 𝜔 ( 𝑛 ) slots to generate 𝑛 successeson the data channel. So the end of batch also means the end of the complete interval the lemma is considering.18 ight Trade-off in Contention Resolution without Collision Detection pened before slot 𝑛 ), and ends after slot 𝑛 . We can show with high probability in 𝑛 this interval ends by 𝑛 + 𝜏 / ≤ 𝜏 / when 𝑐 ≥ , conditioned on any fixed value of 𝑖 and any fixed values of 𝑠 , 𝑠 , · · · , 𝑠 𝑖 − , by the following claim. Claim.
Fix any 𝑖 ≤ 𝜂𝑛 , and any values of 𝑠 , 𝑠 , · · · , 𝑠 𝑖 − satisfying 𝑠 𝑖 − < 𝑛 . Then with high probability in 𝑛 , there isat least a success in ( 𝑛, 𝑛 + 𝜏 / ] (thus also in interval ( 𝑛, 𝜏 / ] when 𝑐 ≥ ). Proof.
Notice interval [ , 𝑛 + 𝜏 / ] is light, since 𝑛 [ , 𝜏 ] ≤ 𝜏 / 𝑐 𝑓 ′ ( 𝜏 / ) and 𝑑 [ , 𝜏 ] ≤ 𝜏 / 𝑐 𝑔 ′ ( 𝜏 / ) . We intend to applyLemma 3.2 on interval [ 𝑛 + , 𝑛 + 𝜏 / ] and argue its conditions are satisfied. When 𝑛 is sufficiently large with respectto 𝑐 and 𝑐 , we have 𝑛 · 𝑛 ≤ log ( 𝜏 / ) 𝑐 ′ and ( − 𝜂 ) 𝑛 · 𝑛 + 𝜏 / ≥ . 𝑛𝜏 / ≥ 𝑐 ′ log ( 𝜏 / ) 𝜏 / , thus the first condition for applying thelemma is satisfied. Moreover, since interval [ , 𝑛 + 𝜏 / ] is light, it is easy to verify the second condition for applyingthe lemma is also satisfied. Therefore, by Lemma 3.2, with high probability in 𝑛 , there is a success in [ 𝑛 + , 𝑛 + 𝜏 / ] . (cid:3) The third group of light intervals consists of the intervals that begin after slot 𝑛 . We need the following claim andLemma 3.4 to show Í 𝑋 𝑖 ∈X 𝐿𝑖𝑔ℎ𝑡 ∧ 𝑠 𝑖 − > 𝑛 | 𝑋 𝑖 | ≤ 𝜂 · 𝑛 · ( 𝑡 + ) ≤ 𝑐 𝑛 / = 𝜏 / . Claim.
Fix any 𝑖 ≤ 𝜂𝑛 , and any values of 𝑠 , 𝑠 , · · · , 𝑠 𝑖 − satisfying 𝑠 𝑖 − ≥ 𝑛 . Recall each 𝑠 𝑗 is at most 𝑐 𝑛 . For any 𝑥 ≥ 𝑡 , with high probability in 𝑥 , Pr (| 𝑋 𝑖 | > 𝑥 ∧ 𝑋 𝑖 ∈ X 𝐿𝑖𝑔ℎ𝑡 ) ≤ / 𝑥 Ω ( ) , where 𝑡 is a sufficiently large constant. Proof.
Let 𝑥 ′ ≥ 𝑥 be the smallest integer such that ( 𝑠 𝑖 − , 𝑠 𝑖 − + 𝑥 ′ ] is light. It suffices to upper bound the probability Pr (cid:0) ( 𝑠 𝑖 − , 𝑠 𝑖 − + 𝑥 ′ ] is light ∧ ( 𝑠 𝑖 − , 𝑠 𝑖 − + 𝑥 ′ ) contains no success (cid:1) We intend to apply Lemma 3.2 on interval ( 𝑠 𝑖 − , 𝑠 𝑖 − + 𝑥 ′ ] and argue its conditions are satisfied. Specifically, since ( 𝑠 𝑖 − , 𝑠 𝑖 − + 𝑥 ′ ] is light, the second condition is satisfied. On the other hand, if we set 𝑡 ≥ 𝑐 ′ , then 𝑛 · 𝑛 ≤ log 𝑥 ′ 𝑐 ′ since 𝑥 ′ ≥ 𝑡 . Moreover, if we set 𝑡 ≥ ( 𝑐 ′ 𝑐 ) , then . 𝑛 · 𝑐 𝑛 ≥ . 𝑐 ≥ 𝑐 ′ log 𝑥 ′ 𝑥 ′ since 𝑛 − 𝜂𝑛 ≥ . 𝑛 and 𝑥 ′ ≥ 𝑡 . Hence,the first condition is also satisfied. Now, apply Lemma 3.2 on interval ( 𝑠 𝑖 − , 𝑠 𝑖 − + 𝑥 ′ ] , we conclude there is a successwithin interval ( 𝑠 𝑖 − , 𝑠 𝑖 − + 𝑥 ′ ] with high probability in 𝑥 ′ (thus also in 𝑥 ). (cid:3) We can now conclude the proof for scenario two. Specifically, 𝑠 𝜂𝑛 ≤ 𝜏 / + 𝜏 / + 𝜏 / + 𝜏 / + 𝜏 / ≤ 𝜏 by summing thetotal length of each type of interval. Therefore, there are at least 𝜂𝑛 successes in the first 𝜏 slots of the data channel,with high probability in 𝑛 (thus also in 𝑡 ), implying ¯ 𝑙 = . Scenario III: 𝑡 ≤ 𝑐 𝑛 . When 𝑐 ≥ 𝑐 , we have 𝑐 𝑛 ≤ 𝑐 𝑛 / , so assume 𝑡 ≤ 𝑐 𝑛 / . In the case that there areat least 𝑛 / = 𝑛𝑐 𝑐 ≥ 𝑡 𝑐 ≥ 𝑡 /( 𝑐𝑐 ( 𝑡 + )) successes within time [ , 𝑡 ] , we have ¯ 𝑙 = < 𝑡 . Otherwise, thereare at least 𝑛 / batch nodes within time [ , 𝑡 ] , and we argue there is no success in the first 𝑡 / slots on the controlchannel, with high probability in 𝑛 . This is because in each of the 𝑡 / slots, the contention of batch nodes is at least 𝑛 / · 𝑐 log ( 𝑐 𝑛 / ) 𝑐 𝑛 / ≥ ( 𝑐 𝑛 / ) ∈ Ω ( log 𝑛 ) . Therefore, in this case, with high probability in 𝑛 (thus also in 𝑡 ), 𝑅 I − 𝐿 I + > 𝑡 (since there is no success on control channel and there are active nodes in the system). In conclusion,for this scenario, with high probability in 𝑡 , the value of ¯ 𝑙 is either more than 𝑡 or . (cid:3)(cid:3)