Windowed Backoff Algorithms for WiFi: Theory and Performance under Batched Arrivals
WWindowed Backoff Algorithms for WiFi
Theory and Performance under Batched Arrivals
William C. Anderton · Trisha Chakraborty · Maxwell YoungAbstract
Binary exponential backoff (BEB) is a decades-old algorithm for coor-dinating access to a shared channel. In modern networks, BEB plays an importantrole in WiFi (IEEE 802.11) and other wireless communication standards.Despite this track record, well-known theoretical results indicate that underbursty traffic BEB yields poor makespan, and superior algorithms are possible. Todate, the degree to which these findings impact performance in wireless networkshas not been examined.To address this issue, we investigate one of the strongest cases against BEB: asingle burst ( batch ) of packets that simultaneously contend for access to a wirelesschannel. Using Network Simulator 3, we incorporate into IEEE 802.11g severalnewer algorithms that, while inspired by BEB, possess makespan guarantees thatare theoretically superior. Surprisingly, we discover that these newer algorithmsunderperform BEB.Investigating further, we identify as the culprit a common abstraction regard-ing the cost of collisions. Our experimental results are complemented by analyticalarguments that the number of collisions—and not solely makespan—is an impor-tant metric to optimize. We argue that these findings have implications for thedesign of backoff algorithms in wireless networks.
Keywords
Backoff algorithms · contention resolution · WiFi
This research is supported by the National Science Foundation grant CNS-1816076 and theU.S. National Institute of Justice (NIJ) Grant 2018-75-CX-K002.W. C. AndertonE-mail: [email protected]. ChakrabortyDepartment of Computer Science and Engineering, Mississippi State University, MS, USAE-mail: [email protected]. Young (contact author)Department of Computer Science and Engineering, Mississippi State University, MS, USAE-mail: [email protected] a r X i v : . [ c s . D C ] N ov William C. Anderton · Trisha Chakraborty · Maxwell Young
Randomized binary exponential backoff ( BEB ) plays an important role incoordinating access by multiple devices to a shared resource. Originally designedfor use in old Ethernet systems decades ago [1], BEB has since found applicationin a range of domains such as transactional memory [2,3], concurrent memoryaccess [4,5,6], and congestion control [7]. However, arguably, the most prominentapplication of BEB today is in IEEE 802.11 (WiFi) networks, where the sharedresource is a wireless communication channel.Given its importance, BEB has been studied at length and is known to yieldgood throughput under well-behaved traffic [8,9,10,11,12,13,14,15,16]. In con-trast, when traffic is bursty, BEB is suspected to perform sub-optimally. Thisis true, even for batched arrivals , where n packets simultaneously arrive in atime-slotted system, begin contending for the channel, and succeed in being trans-mitted before the next batch arrives. Here, Bender et al. [17] prove that BEBhas Θ ( n log n ) makespan , which is the number of time slots until all packets aresuccessfully transmitted. In other words, the throughput is O (1 / log n ) and thustends asymptotically to zero.In light of this shortcoming, there has been significant interest in developingalgorithms with improved makespan. Under batched arrivals, Bender et al. [17]derive upper and lower bounds for several variants of backoff that are asymptot-ically superior to BEB in this regard. Furthermore, again under batched arrivals,sawtooth backoff [18] and truncated sawtooth backoff [19] algorithms achieve theasymptotically optimal O ( n ) makespan.This growing body of results provokes an obvious question: How do neweralgorithms compare to BEB in practice?
Here, we make progress towards an answerby restricting ourselves to bursty wireless traffic. In particular, we examine the caseof a single burst ( batch ) of packets. This is a prominent case in the theoreticalliterature where BEB is anticipated to do poorly, and it should be possible toidentify which of the following situations is true: (1) A newer contention-resolutionalgorithm outperforms BEB, or (2) BEB outperforms newer contention-resolutionalgorithms.Interestingly, neither of these outcomes is very palatable. In one form or an-other, BEB has operated in networks for over four decades and it remains anessential ingredient in several wireless standards. Bursty traffic can arise in prac-tice [20,21] and its impact has been examined [22,23,24,25]. If (1) holds, thenBEB is potentially in need of revision and the ramifications of this are hard tooverstate.Conversely, if (2) holds, then theory is not translating into improved perfor-mance in a prominent application domain. At best, this is a matter of asymptotics.At worst, this indicates a problem with the abstract model upon which newer re-sults are based. In this latter case, it is important to understand what assumptionsare faulty so that the abstract model may be revised. n ≥ contention reso-lution addresses the number of slots until any one of the stations transmits alone.A natural consideration is the time until a subset of k stations each transmits indowed Backoff Algorithms for WiFi 3 alone; this often falls under the same label, but is also sometimes referred to as k -selection in the literature (for example, see [26]).Here, we focus on the case of k = n and examine the performance of vari-ous backoff algorithms in solving the contention resolution problem. Much of thealgorithmic work shares an abstract model. Three common assumptions are: – A0. Each slot has length that can accommodate a packet. – A1.
If a single packet transmits in a slot, the packet succeeds , but failure occurs if two or more packets transmit simultaneously due to a collision . – A2. A collision incurs a delay of a single slot in which the failure occurred.Assumption A0 is near-universal, but technically inaccurate for reasons dis-cussed in Section 2. To summarize, under BEB, each station selects a random slotfrom a set of 2 r consecutive slots, for some integer r ≥
0; this is equivalent tosetting a counter randomly to a value in { , , ..., r − } , which is decremented ateach slot, and transmission occurs when the counter reaches 0. Thus, from an al-gorithmic perspective, stations decrement their respective counters as they marchthrough these slots uninterrupted. However, in practice, transmission of the fullpacket may occur past the slot, while all other stations pause their execution (notdecrementing their respective counters) until the transmission ends. Given thispausing behavior, this assumption is sufficiently close to reality that we shouldnot expect performance to deviate greatly as a result.Assumption A1 is prevalent in the literature (see [26,27,28,29,19,30,31]), al-though variations exist. An alternative is the signal-to-noise-plus-interference (SINR)model [32,33], which is less strict about failure in the event of simultaneous trans-missions. Another model that has received attention is the affectance model [34].Nevertheless, these all share the assumption that simultaneous transmissions maylower the probability of success.Assumption A2 is also widely adopted (see the same examples for A1) andimplicitly addresses two quantities that affect performance: the time to transmit apacket, and the time to receive any channel feedback on success or failure. Assign-ing a negligible delay to these quantities admits a simplified model, but ignoresthe associated performance impact. For example, the functionality for obtainingchannel feedback is provided by a medium access control (MAC) protocol, of whicha backoff algorithm is only one component.1.2 Our Main MessageOur main message is that A2 is flawed in the WiFi setting. In particular, the costof a collision is more significant than acknowledged by the abstract model. Thisis not a matter of minor adjustments to the assumption, or an artifact of hiddenconstants in the algorithms examined. Rather, in WiFi networks, the way in whichcollisions are detected requires a revision to the problem of contention resolution if we aim to design algorithms for many wireless settings.Several corollaries follow from our main claim, which we illustrate using de-tailed simulations of the well-known IEEE 802.11g standard. Additionally, we pro-vide analytical arguments that support our experimental findings (see Section 5).Our belief is that these findings generalize to several wireless settings—not just William C. Anderton · Trisha Chakraborty · Maxwell Young
WiFi networks—implying that contention-resolution algorithms that ignore thecost of collisions will likely not perform as advertised in these cases too (see Sec-tion 6.3).Finally, we emphasize that our findings are aimed at complementing priortheoretical results whose importance to many application domains is evident (seeSection 7). However, arguably, a major application of backoff algorithms is inwireless networks, and this is what motivates our investigation. To understand our findings, it is helpful to summarize IEEE 802.11g and howBEB operates within it. However, outside of this section and the description ofour experimental setup, discussion of such aspects and terminology is kept to aminimum.Throughout, we will often use interchangeably the terms packet and station depending on the context; the two uses are equivalent given that in the batchedsetting, each station seeks to transmit a single packet. Also, to be explicit, whilewe use the general term packet , we are specifically referring to a frame .Exponential backoff [1] is a widely deployed algorithm for distributed multipleaccess. Informally, a backoff algorithm operates over a contention window (CW) wherein each station makes a single randomly-timed access attempt. In the eventof two or more simultaneous attempts, the result is a collision and none of thestations succeed. Backoff seeks to avoid collisions by dynamically increasing thecontention-window size such that stations succeed.IEEE 802.11 handles contention resolution via the distributed coordinationfunction (DCF) which employs BEB; as the name suggests, successive CWsdouble in size under BEB. The operation of DCF is summarized as follows. Priorto transmitting data, a station first senses the channel for a period of time knownas a distributed inter-frame space (DIFS) . If the channel is not in use overthe DIFS, the station transmits its data; otherwise, it waits until the currenttransmission finishes and then initiates BEB.For a contention window of size w , a timer value is selected uniformly at randomfrom [0 , w − paused for the duration of the current transmission ,and then resumed (not restarted) after another DIFS.After a station transmits, it awaits an acknowledgement (ACK) from thereceiver. If the transmission was successful, then the receiver waits for a shortamount of time known as a short inter-frame space (SIFS) —of shorter durationthan a DIFS—before sending the ACK. Upon receiving an ACK, the station learnsthat its transmission was successful. Otherwise, the station waits for an ACK-timeout duration before concluding that a collision occurred. This series of actionsis referred to as collision detection ; the cost of which lies at the heart of ourargument. If a collision is detected, then the station must attempt a retransmission via the same process with its CW doubled in size.Note that both the transmission of data and the acknowledgement processoccur “outside” of the backoff component of DCF; Figure 1 highlights this. Incontrast, the focus of many algorithmic results is solely on the slots of this backoffcomponent. indowed Backoff Algorithms for WiFi 5
Busy ACKPacket
SIFS … DIFS
Contention Window
Fig. 1: An overview of DCF.Finally, RTS/CTS (request-to-send and clear-to-send) is an optional mecha-nism. Informally, a station will send an RTS message and await an CTS messagefrom the receiver prior to transmitting its data. Due to increased overhead, thereis some debate on whether RTS/CTS is worthwhile and, if so, when it should beenabled [35,36,37]; typically, it is not. Here, we focus on the case where RTS/CTSis disabled, although our experiments show that our findings continue to hold whenthis mechanism is used (see Section 4.2).
We employ Network Simulator 3 (NS3, version 3.25) [38] which is a widely usednetwork simulation tool in the research community [39]. Our simulation code,MATLAB scripts, data, and a description of these files will be made availableonline at . In order to motivate our designchoices and for the purposes of reproducibility, our experimental setup is describedin this section.Our reasons for using NS3 are twofold. First, wireless communication is difficultto model and employing NS3 helps allay concerns that our findings are an arti-fact of poorly-modeled wireless effects. Second, given the assumptions upon whichcontention-resolution algorithms are based, NS3 can reveal whether we are beingled astray by an assumption that appears reasonable, but results in an importantdiscrepancy between theory and practice.Table 1 provides our experimental parameters. Path-loss models with defaultparameters are known to be faithful [40] and, therefore, our experiments employthe log-distance propagation loss model in NS3. For transmission and reception ofpackets, we use the YANS [41] module.At the MAC layer, we make use of IEEE 802.11g in our experiments. IEEE802.11g provides a data rate in the tens of megabits per second (Mbit/s) , operatesin the 2.4 GHz band, and remains in use today. Although IEEE 802.11ac and IEEE802.11n are more recent members of the WiFi family, IEEE 802.11g is sufficientlyrepresentative to establish our main thesis.We implement changes to the behavior of the contention window based on thealgorithms we investigate. All experiments use IPv4 and UDP. Our investigationemploys UDP instead of TCP to reduce the impact of potential transport-layer effects that may complicate the interpretation of our results. Ultimately, given theexplanation for our findings, this choice does not alter our final conclusions. While our results focus on the more common case of RTS/CTS being disabled, we dobriefly report on the impact of RTS/CTS in Section 4.2. The standard offers a theoretical maximum of 54 Mbit/s. William C. Anderton · Trisha Chakraborty · Maxwell Young
Parameter Value
Wireless specification 802.11gData rate 54 Mbits/secSlot duration 9 µ sSIFS 16 µ sDIFS 34 µ sACK timeout 75 µ sPreamble 20 µ sTransport layer protocol UDPPacket overhead 64 bytesContention-window size min. 1Contention-window size max. 1024RTS/CTS Off Table 1: Parameter values used in our experiments.The amount of overhead for each packet is 64-bytes: 8 bytes for UDP, 20 bytesfor IPv4, 8 bytes for an LLC/SNAP header, and 28 bytes of additional overheadat the MAC layer. Fragmentation is disabled in our experiments.The duration of an acknowledgement (ACK) timeout is specified by the mostrecent IEEE 802.11 standard to be roughly the sum of a SIFS (16 µ s), standardslot time (9 µ s), and preamble (20 µ s); a total of 45 µ s. However, in practice, this issubject to tuning. In our experiments, an ACK-timeout below 55 µ s gave markedlypoor performance; there is insufficient time for the ACK before the sender decidesto retransmit. We use the default value of 75 µ s in NS3 since this yielded goodperformance and is still the same order as suggested in the standard.In our experiments, n stations are placed in a 40 meter ×
40 meter grid, andthey are laid out at regular spacing, starting at the south-west corner of the gridmoving left to right by 2-meter (m) increments, and then up when the current rowis filled. A wireless access point (AP) is located (roughly) at the center of thegrid. We do not simulate additional terrain or environmental phenomena, sinceour goal is to test the performance under ideal conditions without complicatingfactors.Throughout, the following common approach is used to identify outliers in ourdata. Let ∆ be the distance between the first and third quartiles, Q and Q ,respectively. Any data point smaller than Q − . ∆ or larger than Q + 1 . ∆ isdeclared an outlier and discarded. We examine a single batch of n packets that simultaneously begin their contentionfor the channel. As competitors with BEB , we experiment with the following al- This timeout period is specified in Section 10.3.2.9, page 1317 of [42]. We have also run experiments with 1 and 3 meter increments and the qualitative behaviorremains the same; therefore, we omit those results.indowed Backoff Algorithms for WiFi 7
Monotonic Windowed Backoff Algorithm w ← initial window size f ( w ) ← window-scaling functionA station with a packet to transmit does the following until successful: – Attempt to transmit in a slot chosen uniformly at random from w . – If the transmission failed, then wait until the end of the window and set w ← (cid:100) (1 + f ( w )) w (cid:101) .Fig. 2: A generic backoff algorithm with monotically-increasing windows. In par-ticular, LLB, LB, and BEB may use an initial window size of at least 4 , , and 1,respectively, and f ( w ) = 1 / lg lg w, / lg w , and 1, respectively.gorithms: Log-Backoff (LB),
LogLog-Backoff (LLB) [17] and
Sawtooth-Backoff (STB) [43,44]. We also investigate the performance of
TruncatedSawtooth-Backoff (TSTB) [19], although we defer this to Section 6.1.Both LLB and LB are closely related to
BEB in that they execute using a CWthat increases in size monotonically. Here, each window increases by a (1 + f ( w ))-factor in size over the previous window, where f ( w ) is a function that controlsthe amount by which the window size grows. For LLB, LB, and BEB, f ( w ) =1 / lg lg w, / lg w, and 1, respectively, and the initial window sizes are 4 , , and 1,respectively, in our experiments. The pseudocode is presented in Figure 2.In contrast, STB is non-monotonic and executes over a doubly-nested loop. Theouter loop sets the current window size w to be double that used in the precedingouter loop; this is like BEB. Additionally, for each such window, the inner loopexecutes over lg w windows of decreasing size: w, w/ , w/ , ...,
1. For each suchwindow, a slot is chosen uniformly at random for the packet to transmit; this isthe “backon” component of STB.Finally, TSTB executes identically to STB except that, for some constant c > c lg w ) windows of decreasing size: w, w/ , w/ , ..., max {(cid:98) w/c lg w (cid:99) , } . Our Metrics.
For a single batch of n packets, prior algorithmic results addressthe number of slots required to complete all n packets. These slots correspond onlyto those belonging to contention windows, even though many results refer to thisas makespan. To avoid confusion, we will refer to this metric more explicitly as contention-window slots (CW slots) .Table 2 summarizes the known with-high-probability (w.h.p.) guarantees onCW slots. Note that the Θ -notation implies an upper and lower bound on thenumber of CW slots until all n packets succeed.For CW slots, LB, LLB, STB, and TSTB have superior guarantees over BEB. In particular, both STB achieve and TSTB achieve Θ ( n ) CW slots, which is asymp-totically optimal. Despite this similarity with STB, TSTB is worth evaluating given For our later asymptotic analysis, the constants used for the initial window sizes areflexible, and we revise these in Section 5 when deriving bounds on the number of collisions. With probability at least 1 − /n c for a tunable constant c >
1. William C. Anderton · Trisha Chakraborty · Maxwell Young
Algorithm Contention-Window Slots
BEB Θ ( n log n )LB Θ (cid:16) n log n log log n (cid:17) LLB Θ (cid:16) n log log n log log log n (cid:17) STB, TSTB Θ ( n ) Table 2: Prior w.h.p. results on CW slots for a batch of n packets under BEB, LB,LLB [17] and STB [43,44], and TSTB [45,19].that its smaller number of windows may offer superior performance in practice,although we defer its examination until Section 6.1.We also examine a second metric. As described in Section 2, events occuroutside of contention windows, such as SIFS, DIFS, full packet transmission, ACKtimeouts. To denote the total duration—including the time spent in contentionwindows—between when the single batch of packets arrives and when the lastpacket successfully transmits (including its ACK timeout), we refer to total time . n = 150. Percentage increases or decreases are calculatedby the standard formula: 100 × ( A − B ) /B , where B is always the value for BEB(the “old” algorithm) and A corresponds to a value for one of LLB, LB, or STB(the “new” algorithms). We provide results from our NS3 experiments using relatively small packets, witha 64-byte (B) payload, and larger packets, with a 1024B payload.Figures 3(A) and 4(B) illustrate our experimental findings with respect to CWslots. The behavior generally agrees with theoretical predictions that each of LLB,LB, and STB should outperform BEB.Interestingly, LLB incurs a greater number of CW slots than LB despite theformer’s better asymptotic guarantees. We suspect this is an artifact of hiddenconstants/scaling, and evidence of this is presented later in Section 6.2.Nevertheless, in agreement with theory, LLB, LB, and STB demonstrate im-provements over BEB, giving a respective decrease of 40 . . . . . . For comparison, Figure 3(C) depicts CW slots derived from a simple Javasimulation that implements only the assumptions of the abstract model; that is,the simple simulation ignores physical-layer effects, processing performed at layers We exclude the time required for a station to associate with the access point and ARPrequests/replies, as this will be the same regardless of the algorithm being evaluated.indowed Backoff Algorithms for WiFi 9
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Number of Packets (64 Bytes) C on t en t i on - W i ndo w S l o t s BEBLBLLBSTB (A)
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Number of Packets (1024 Bytes) C on t en t i on - W i ndo w S l o t s BEBLBLLBSTB (B)
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Number of Packets C on t en t i on - W i ndo w S l o t s BEBLogLLBSTB (C)
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Number of Packets (64 Bytes) C on t en t i on - W i ndo w S l o t s f o r n / P a ck e t s BEBLBLLBSTB (D)
Fig. 3: Median values are reported: (A) and (B) CW slots from NS3 experimentswith 30 trials for each value of n with 64B and 1024B payloads, (C) CW slots viasimple Java simulation with 50 trials for each value of n , (D) number of CW slotsrequired to finish n/ n . Bars represent 95% confidence intervals.of the protocol stack, etc. Reassuringly, our NS3 results also roughly agree withthis data in terms of magnitude of values and the separation of BEB from theother algorithms; albeit, the performances of LLB, LB, and STB do not separateas cleanly in this data.Despite BEB’s inferior performance with respect to CW slots, is it possible thatmany packets finish quickly—perhaps faster than under the newer algorithms—and only a few packets account for the remaining makespan? We examine this inFigure 3(D) using the 64B payload, where we present the number of CW slotsrequired for half of the packets to succeed. A few observations can be made. First,BEB does not seem to strongly exhibit such behavior. Second, for all algorithms,the remaining n/ backon behavior of STB. Result 1.
Experiments confirm theoretical predictions that LLB, LB, andSTB outperform BEB with respect to CW slots. · Trisha Chakraborty · Maxwell Young
It is tempting to consider the single-batch scenario settled. However, if we focus onthe total time for both the 64B and 1024B payload sizes, then a different pictureemerges.
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Number of Packets (64 Bytes) T o t a l T i m e BEBLBLLBSTB (A)
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Number of Packets (1024 Bytes) T o t a l T i m e BEBLBLLBSTB (B)
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Number of Packets (64 Bytes) T o t a l T i m e f o r n / P a ck e t s BEBLBLLBSTB (C)
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Number of Packets (1024 Bytes) T o t a l T i m e f o r n / P a ck e t s BEBLBLLBSTB (D)
Fig. 4: NS3 results with the median reported from 30 trials for each value of n atdistance 2m and time measured in µs : (A) and (B) give the total time for 64B and1024B payloads, (C) and (D) time required to complete n/ BEB is erased, as de-picted in Figures 5(A) and 5(B). In fact, the order of performance is reversed withtotal time ordered from least to greatest as BEB, LLB, LB, STB. Specifically, for64B payloads, LLB, LB, and STB suffer an increase of 12 . . . . . . n/ newer algorithms do better for the bulk of packets, but suffer from a few stragglers?Interestingly, Figures 5(C) and 5(D) suggest that this is not the case. Indeed, fora 64B payload, BEB performs even better over LLB, LB, and STB with the latterexhibiting an increase of 36 . . . . . . indowed Backoff Algorithms for WiFi 11
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Number of Packets (64 Bytes) M a x N u m be r o f A C K T i m eou t s BEBLBLLBSTB
Fig. 5: The maximum number of ACK timeouts per station over all stations witha 64B payload from NS3 experiments. The median from 30 trials is reported foreach value of n . Result 2.
In comparison to BEB, the total time for each of LLB, LB, andSTB is significantly worse.
These findings are troubling since, arguably, total time is a more importantperformance metric in practice than CW slots. Critically, we note that this behav-ior is detected only through the use of NS3; it is not apparent from the simplerJava simulation. What is the cause of this phenomenon?4.2 The Cost of CollisionsThe number of ACK timeouts per station provides an important hint. As Figure 5shows, relative to BEB, the newer algorithms are incurring substantially more ACKtimeouts. This evidence points to collisions as the main culprit, since each ACKtimeout may be considered to arise from a collision; we elaborate on this below inSection 4.3. In particular, the way in which collision detection is performed meansthat each collision is costly in terms of time. In support of our claim, we performsome back-of-the-envelope accounting of the delay caused by collisions. We useBEB with n = 150 and a payload size of 64 bytes as an example throughout our discussion below. Back-of-the-Envelope (BOTE) Calculations . When a collision occurs, the cor-responding station waits for an ACK timeout and then retransmits its packet; here,we try to estimate the resulting delay. Recall from Section 3 that the durationof an ACK timeout is 75 µ s. To estimate the costs of transmissions, we assume · Trisha Chakraborty · Maxwell Young that a packet of size 128B (64B payload plus 64B overhead) requires roughly bits Mbits/s ≈ µ s plus the associated 20 µ s preamble. We break our BOTE calculation into two pieces: (a) collisions that occur priorto a window of size n , and (b) collisions that occur in windows of size n and larger.For (a), we note that most slots in windows with size up to and including n/ n − n/ · (19 µs + 20 µs + 75 µs ) =8 , µs .For (b), we reason in the following way. The maximum number of ACK time-outs for BEB—and, thus, the number of collisions—experienced by an unluckystation is 9 according to Figure 5(E), and the median value is 7. Many of thesecollisions will involve several stations, and these occur when the windows are smalland we have already incorporated these in our part (a) calculations. However, asmall number of these collisions should occur in larger windows. Why? Recall fromFigure 4(D) that the last n/ n/ n/ n/ · · · · (19 µs + 20 µs + 75 µs ) = 17 , µs .Finally, we can add to this the time required for the (final) successful trans-mission for each of the 150 stations, where we treat the time to receive the ACKas negligible; this yields an addition time of 150 · (19 µs + 20 µs ) = 5 , µs , and sowe have a total of 8 , µs + 17 , µs + 5 , µs = 31 , µs .In comparison, in Figure 3(A), BEB incurs 1326 CW slots for n = 150, eachof duration 9 µ s, spent in CWs which yields a total of 9(1326) = 11 , µs . Thissuggests that collisions have a much larger cost than CW slots.
As a sanity check, summing up the the time due to collisions and CW slotsyields roughly 43 , µs . This is right order of magnitude if we compare againstthe measured median total-time value for BEB of 53 , µs in Figure 4(A). Theshortfall likely arises in part from assuming the full data rate of 54 Mbits/s, ig-noring collisions involving more than two stations in larger windows, and the factthat we did not account for the SIFS and DIFS.Finally, this analysis has implications for packet size. For the 1024B payloadplus the 64B of overhead, the transmission delay grows to 161 µs . Using the sameBOTE analysis for BEB with a 1024B payload, the cost due to (a) and (b) is57 , µs . By comparison, using Figure 3(B), the median number of CW slotsremains essentially unchanged at 1 , , µs = 11 , µs . Therefore, transmission time dominates, and the degree to which it dominatesincreases with packet size; we discuss this further in Section 6 In practice, 54 Mbits/s is not achieved and so the true transmission time is larger. There-fore, we are being conservative; the time for a (re)transmission is likely higher.indowed Backoff Algorithms for WiFi 13
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Number of Packets (64 Bytes) M ed i an T o t a l N u m be r o f A L O I n s t an c e s BEBLBLLBSTB
Fig. 6: The median total ALO instances for BEB, LB, LLB, and STB.
Discussion.
Do the simulation results bear out these BOTE calculations, or isthis a “just-so” story? We answer this question by examining the simulation datafrom a slightly different angle. From our simulations with n = 150, we consider thetotal interval of time for a single trial; that is, until all n packets succeed. Whenone or more stations is sending its packet, we count this as a at-least-one (ALO)instance . We then look at the number of ALO instances over the trial and thentake the median over all trials. This is useful since it tells us which algorithmresults in the most disjoint transmissions by stations.This data is plotted in Figure 6. We expect that the number of transmissionsto be lowest for BEB, followed by LLB, LB, and STB in increasing order, aligningwith Figures 5, and indeed this is what we see.As plotted in Figure 6, BEB has a median of 345 ALO instances. Informally,we may consider this to be composed of 150 final successful transmissions and195 retransmissions that resulted from collisions. This implies a total time of150(19 µs + 20 µs ) + 195(19 µs + 20 µs + 75 µs ) = 28 , µs .By comparison, in our BOTE calculations for BEB above, we estimated 75collisions(in part (a)) plus 150 collisions (in part (b)) for a total of 225 colli-sions involving two or more stations, plus the 150 successful transmissions; thisyielded 31 , µs , which is slightly higher, but arguably a reasonable estimate. Us-ing BEB’s median of 345 ALO instances, we can add the time of 11 , µs due to1326 CW slots to arrive at a total cost of 40 , µs which again aligns reasonablywell with BEB for n = 150 in Figure 4(A).These experimental results provide us with strong evidence that assumption A2 is not accurate with regards to the cost of collisions.
Result 3.
The amount of time incurred by collision detection is not prop-erly accounted for by A2, and this cost dominates the time incurred bycontention window slots. · Trisha Chakraborty · Maxwell Young
ACK Timeout ≈ Collision.
It is true that not all ACK timeouts necessarilyimply that the corresponding packet suffered a collision. For example, an ACKmight be lost due to signal attenuation, even if the packet was transmitted withoutany collision. Note that, in such a case, the sending station still diagnoses a failure,and so the same costs described in (I)-(III) hold.However, for our simple NS3 setup, virtually all ACK failures result from acollision. This is evident from Figures 7(A) and 7(B) which illustrate a trial with n = 20 under BEB and STB, using a 64B payload. Collisions occur only when twoor more stations transmit (duration of transmission denoted by a thick blue line)at the same time and the result is an ACK timeout event (indicated by a thin redline); in all other cases, the transmission is successful and the corresponding ACKis received.Another interesting feature of these plots is that they again illustrate the dif-ference in number of collisions experienced by BEB versus STB. Figures 7(C)and 7(D) condense the executions in Figures 7(A) and 7(B) by depicting the timeover which: there are no transmissions, collisions occur, there is a successful trans-mission (and collision detection). For BEB, roughly 28% of the execution time isincurred by collisions and collision detection, while for STB this number is 39%. Total Time and Number of Collisions.
We observe that the total time doesnot grow linearly with the maximum number of ACK timeouts experienced by astation. Under LLB, an unlucky station suffers roughly 1 . × the number of ACKtimeouts, but the total time of LLB is not 1 . × that of BEB.Why? Consider n stations where each collision involves only two stations. Then,there are n/ n stations transmit in the same slot andcollide. Then, there is a single collision which adds only a single failed transmissiontime to the total time.The number of stations involved in a single collision is larger for algorithmswhose CWs grow more slowly than under BEB, such as LB and LLB. Here, thewindows are smaller for a longer period of time, so a collision typically involvesmany packets. For STB, a similar phenomenon is at work; the backon componentyields collisions involving many stations. That is, LB, LLB, and STB are closerto the second case above. In contrast, BEB is closer to the first as it grows its windows the fastest and does not have a backon component. Therefore, LB canhave roughly 1 . × as many ACK timeouts as BEB, but the total time for LB isnot 1 . × that of BEB. RTS/CTS.
Although it is not examined in detail in our work, we remark on theuse of RTS/CTS. When enabled, stations can experience collisions among the RTS indowed Backoff Algorithms for WiFi 15(A) (B)(C) (D)
Fig. 7: Execution of BEB (A) and STB (B) with 20 stations for 64B payload anddistance 2 m . The thick orange lines correspond to the time spent transmitting thepacket, and the thin blue lines indicate an ACK timeout; note that there is nosuch timeout in the case where the transmission is successful. For BEB and STB,respectively, plots (C) and (D) depict in red the time spent on transmitting thepacket and waiting for the ACK timeout (i.e., the time spent sending plus the timespend performing collision detection), and the green illustrates the time where asuccessful transmission is performed. For BEB, roughly 28% of the execution timeis incurred by collisions and collision detection, while for STB this number is 39%.frames (rather than among packets, for the most part). These are smaller in size(20B), but the remainder of the total-time calculation remains the same, and ad-ditional time is incurred due to additional inter-frame spaces and the transmissionof CTS frames.Ultimately, we witness the same qualitative behavior when RTS/CTS is en-abled. For example, without RTS/CTS, recall from Section 4.1.2 that the totaltime for LLB (BEB’s closest competitor) increases by 12 .
9% and 19 .
6% for the64B and 1024B, respectively, over BEB. With RTS/CTS, the respective increasesare 14 .
5% and 9 . Consider the following two cases. In Case 1, we have small data packets. Con-sequently, the impact of RTS collisions is significant to the total time becauseresending an RTS takes time on the same order as the transmission time for thedata frame. Since LLB has more RTS collisions, and RTS collisions are significant,LLB will do much worse than BEB in this case. · Trisha Chakraborty · Maxwell Young
Conversely, in Case 2, we have large data packets. Here the impact of RTS isnegligible compared to the time to send the data frame (which is never involvedin a collision). While LLB has more RTS collisions, each one is costs very littletime compared to sending the data frame (which never collides). Thus, in contrastto Case 1, the degree to which LLB underperforms BEB will be less. This alignswith the notion that the RTS/CTS option can be helpful when packets are large;however, this does not make the other algorithms competitive with BEB.
The reason for the discrepancy between theory and experiment is now apparent.LLB increases each successive contention window by a smaller amount than BEB;in other words, LLB is backing off more slowly ; the same is true of LB. Informally,this slower-backoff behavior is the reason behind the superior number of CW slotsfor LB and LLB since they linger in CWs where the contention is “just right” fora significant fraction of the packets to succeed. However, backing off slowly alsoinflicts a greater number of collisions.Note that BEB backs off faster, jumping away from such favorable contentionwindows and thus incurring many empty slots. This is undesirable from the per-spective of optimizing the number of CW slots. However, the result is fewer col-lisions. Given the empirical results, this appears to be a favorable tradeoff. Weexplicitly note that LLB backs off faster than LB. In this way, LLB is closer toBEB and, therefore, is not outperformed as badly in terms of total time as illus-trated in Figures 4(A) and 4(B).We highlight this observation below:
Result 4.
For algorithm design, optimizing CW slots at the expense ofincreased collisions is a poor design choice. A , denoted by T A , isapproximated as: T A = C A · ( P + ρ ) + W A · s where C A is the number of slots with a collision, P is the transmission time for apacket, ρ is the preamble duration, W A is the corresponding number of CW slots,and s is the duration of a slot.Abstracting further, we may treat ρ and s as constants to get: T A = Θ ( C A · P + W A ) . In other words, total time depends on the number of slots with collisions, eachof which incurs a delay that depends on P , and the number of CW slots.How does P behave? We assume it increases with packet size, which can bebroken into two parts: the amount of payload and the amount of control informa-tion (for example, the header of a packet). Focusing on the control information, indowed Backoff Algorithms for WiFi 17 we consider how it might be influenced by n ; in particular, the packet header mustcontain enough bits to uniquely address all n stations. Thus, as n scales, the num-ber of bits required to address devices must also increase, and we may assume P scales as the logarithm of n .With this in mind, we look to characterize T A asymptotically for BEB, LB,LLB, and STB. Previous results have already established W A , so the parameterof interest is C A , which we investigate next. In order to provide additional support for our empirical findings, we derive asymp-totic bounds on C A . Our arguments are couched in terms of packets and slots, butwhat follows is a balls-into-bins analysis, where balls correspond to packets, andslots correspond to bins. To bound C A , we are interested in the number of bins—where bins correspond to the slots in a CW—that contain two or more balls, sincethis is the equivalent of a collision. Throughout our analysis, we use the terminol-ogy of slots and bins interchangeably. Finally, all of our results hold given that n is sufficiently large.Our arguments make use of the following well-known inequalities. Fact 1
For any ≤ x < , e − x/ (1 − x ) ≤ − x . Fact 2
For any x , − x ≤ e − x . Throughout our analysis, we make use of the following previously-known con-centration result:
Theorem 1 (Method of Bounded Differences, Corollary 5.2 in [47]) Let f be afunction of independent variables X , ..., X N such that for any b, b (cid:48) it holds that | f ( X , ..., X i = b, ..., X N ) − f ( X , ..., X i = b (cid:48) , ..., X N ) | ≤ c i for i = 1 , ..., N . Then,the following holds: P r ( f > E [ f ] + t ) ≤ e − t / ( (cid:80) i c i ) P r ( f < E [ f ] − t ) ≤ e − t / ( (cid:80) i c i ) . Note that this concentration result holds for random variables that may be depen-dent. The following result follows directly, and we factor it out here since we useit in several of our arguments:
Corollary 1
Let Z be the number of collisions in a window of size w . Then: P r ( Z > E [ Z ] + t ) = e − t /w P r ( Z < E [ Z ] − t ) = e − t /w . Proof
Let Z denote the number of collisions in window i of a windowed backoff al-gorithm. We may view Z as a function Z ( X , ..., X n ), where for each k ∈ { , ..., n } , X k is a random variable with a value in { , ..., w } corresponding to the bin in whichthe k th ball lands. Note that Z is certainly a function of these X k variables, al-though we do not specify the function itself, nor do we need to in order to useTheorem 1. For example, these inequalities are established in Lemma 3.3 by Richa et al. [46].8 William C. Anderton · Trisha Chakraborty · Maxwell Young
The X k variables are independent, since the bin in which ball k lands is inde-pendent of the bin in which any other ball lands. We observe that X k variablessatisfy: | Z ( X , ..., X i = b, ..., X w ) − Z ( X , ..., X i = b (cid:48) , ..., X w ) | ≤ Z is tightly bounded to its expectation: P r ( Z > E [ Z ] + t ) ≤ e − t / ( (cid:80) w = e − t /w and: P r ( Z < E [ Z ] − t ) ≤ e − t / ( (cid:80) w = e − t /w which completes the argument. (cid:117)(cid:116) A similar result holds for the number of successes in a window; equivalently,the number of bins that contain a single ball.
Corollary 2
Let Z be the number of successes in a window of size w . Then: P r ( Z > E [ Z ] + t ) = e − t /w P r ( Z < E [ Z ] − t ) = e − t /w . The proof for Corollary 2 is nearly identical to that of Corollary 1, and so weomit it here.5.1 Upper Bounding Collisions in BEB
Theorem 2
For a single batch of n packets, with high probability the number ofcollisions for BEB is O ( n ) .Proof In the execution of
BEB , consider a contention window of size n i for aninteger i ≥
1; let the windows be indexed by i . Note that over all windows up toand including size n , we have O ( n ) collisions since there are O ( n ) slots by the sumof a geometric series.Let the indicator random variable Z j = 1 if slot j in window i contains a collision; Z j = 0 otherwise. Considering the corresponding balls-into-bins problem,the probability of a collision is equal to the probability that 2 or more balls aredropped into this bin. Pessimistically, assume exactly n balls are dropped in eachconsecutive window i , since this only increases the probability of a collision; inactuality, packets finish over these windows and reduce the probability of collisions. indowed Backoff Algorithms for WiFi 19 P r ( Z j = 1) = n (cid:88) (cid:96) =2 (cid:32) n(cid:96) (cid:33) (cid:18) n i (cid:19) (cid:96) (cid:18) − n i (cid:19) n − (cid:96) = n (cid:88) (cid:96) =2 O (cid:18) i(cid:96) (cid:19) by Fact 2= O (cid:18) i (cid:19) Let Z i = (cid:80) n i j =1 Z j be the number of collisions in window i . By linearity ofexpectation: E [ Z i ] = n i (cid:88) j =1 E [ Z j ] = n i · O (cid:18) i (cid:19) = O (cid:16) n i (cid:17) There are dependencies between the Z j variables. To handle this complication, weemploy Corollary 1 to argue that Z i is tightly bounded to its expectation; that is,for some constant d >
0, we have:
P r (cid:18) Z i > dn i + t (cid:19) ≤ e − t / ( n i ) By [17,48], w.h.p.
BEB finishes within O (lg lg n ) windows subsequent to a windowof size n ; therefore, i ≤ d (cid:48) lg lg n for some constant d (cid:48) >
0. Setting t = n/ i , andplugging into the above: P r (cid:18) Z i > dn i + n i (cid:19) ≤ e − n/ i ≤ e − n/ lg d (cid:48) n Taking a union bound over i ≤ d (cid:48) lg lg n windows, it follows that w.h.p. the numberof collisions under BEB is (cid:80) d (cid:48) lg lg ni =0 Z i = (cid:80) d (cid:48) lg lg ni =1 O (cid:0) n i (cid:1) = O ( n ). (cid:117)(cid:116) w i is used consecutively(1 /
2) lg lg( w i ) times, and then w i +1 = 2 w i for i ≥ Again, we defer discussionof the size of the initial window until after Lemma 1.We argue that the i th contention window under LLB* is larger than the i th contention window under LLB, assuming that we set the initial window size foreach to be a sufficiently large constant. This implies that a lower bound on the We highlight that the constant 1 / · Trisha Chakraborty · Maxwell Young number of collisions experienced by LLB* is a lower bound on the number ofcollisions experienced by LLB. The remainder of our analysis is devoted to showinga lower bound on the number of collisions experienced by LLB*.
Lemma 1
The window size doubles in fewer windows under LLB* than underLLB for a sufficiently large constant-sized window.Proof
Under LLB, consider any window W i of size w i for i ≥
0. How many windowsare required until the window size becomes at least 2 w i ? It holds that: w i + k = k − (cid:89) j =0 (cid:18) w i + j (cid:19) w i by specification of LLB ≤ (cid:18) w i (cid:19) k w i since w i is the smallest window size ≤ e k/ lg lg w i w i by Fact 2Solving for e k/ lg lg w i ≥ k ≥ ln(2) lg lg w i . That is, at least: (cid:98) ln(2) lg lg w i (cid:99) ≥ ln(2) lg lg w i − (cid:100) (1 /
2) lg lg w i (cid:101) ≤ (1 /
2) lg lg w i + 1 (2)windows to double. Solving:(1 /
2) lg lg( w i ) + 1 < ln(2) lg lg( w i ) − w i > / (ln(2) − / = O (1), the upper bound in Equation 2 is lessthan the lower bound in Equation 1. That is, after this window size is reached,LLB* will be doubling its window size in fewer windows than LLB. Setting Initial Window Size.
By Lemma 1, beyond a window size of w i ∗ > / (ln(2) − / , LLB* requires fewer windows to double its window size compared tothe window of the same index under LLB. We set LLB’s initial window size tobe a constant at least as large as w i ∗ . Then, we set LLB*’s initial window size tobe at least twice LLB’s initial window size. This ensures that LLB*’s window willalways be larger than the window of the same index under LLB. For our analysis of LLB*, we focus on a window of size cn/ lg lg lg n , for some suffi- ciently small constant c >
0. Our argument proceeds as follows. Given (cid:15)n packets,for any positive constant (cid:15) ≤
1, we first demonstrate in Lemma 2 an upper boundof O ( n/ (lg lg n ) d ) successes in any execution over a window of size cn/ lg lg lg n ,where d > (cid:15) and c . Next, in Lemma 3, we showthat Θ ( n ) packets survive until the critical window size is reached. Lemmas 2 and 3 indowed Backoff Algorithms for WiFi 21 imply that there are Θ ( n ) packets remaining for Ω (lg lg n ) executions of a windowof size cn/ lg lg lg n . Finally, in Theorem 3, we prove that for each such execution, Ω ( n/ lg lg lg n ) collisions occur, and this yields a total of Ω ( n lg lg n/ lg lg lg n ) col-lisions. Lemma 2
For any constant (cid:15) ≤ , consider (cid:15)n packets executing LLB* overa window of size cn/ lg lg lg n for a positive constant c < (cid:15) lg( e ) / . With highprobability, O ( n/ (lg lg n ) d ) packets succeed in the window for a constant d > depending on (cid:15) and c .Proof Let Y j = 1 if slot j contains a single packet; otherwise Y j = 0. We have: P r ( Y j = 1) = (cid:32) (cid:15)n (cid:33) (cid:18) lg lg lg ncn (cid:19)(cid:18) − lg lg lg ncn (cid:19) (cid:15)n − (3) ≤ (cid:15) lg lg lg nc e − ( (cid:15)n −
1) lg lg lg ncn (4) ≤ (cid:15) lg lg lg nc e − (1 / (cid:15)n lg lg lg ncn (5) ≤ (cid:15) lg lg lg nc e − (cid:15) lg lg lg n c (6) ≤ (cid:15) lg lg lg nc (lg lg n ) (cid:15) lg( e )2 c (7)= O (cid:18) n ) d (cid:19) for a constant d > (cid:15)n ≥ n , and Equation 8 follows from noting that (cid:15) lg( e ) / (2 c ) > (cid:15) ≤ c < (cid:15) lg( e ) / Y = (cid:80) cn/ lg lg lg nj =1 Y j be the number of successes over the entire window.From the above, we have: E [ Y ] = O (cid:18) n (lg lg n ) d lg lg lg n (cid:19) By Corollary 2:
P r ( Y > E [ Y ] + t ) ≤ e − t ncn and letting t = √ n ln n is sufficient to prove that the number of successes in thewindow is O (cid:16) n (lg lg n ) d lg lg lg n (cid:17) with high probability. Noting that:(lg lg n ) d lg lg lg n = (lg lg n ) d + lg lg lg lg n lg lg lg n = (lg lg n ) d + o (1) completes the proof. (cid:117)(cid:116) We now argue that, for LLB*, very few packets succeed prior to a window ofsize cn/ lg lg lg n . Lemma 3
Consider a single batch of n packets executing LLB*. With high prob-ability, o ( n ) packets succeed prior to the first window of size cn/ lg lg lg n for asufficiently small constant c > . · Trisha Chakraborty · Maxwell Young
Proof
To begin, note that w.h.p. no packet finishes in any slot up until the endof all windows of size n/ (4 ln n ). To see this, note that for any such slot, theprobability that it contains a single packet is at most: (cid:32) n (cid:33) (cid:18) nn (cid:19) (cid:18) − nn (cid:19) n − ≤ ne n −
1) ln n/n (9)= 4 ln nn n − /n (10) ≤ nn (11) ≤ n (12)where the first line follows from Fact 2, and the third line follows from n ≥ O ( n ) slots up to this point yields a probability thatany packet succeeds is at most O (1 /n ) . Now, we consider how many packets can succeed between those windows withsize n/ (4 ln n ) and cn/ lg lg lg n . Starting with a window of size n/ (4 ln n ), howmany intervening unique-sized windows exist before reaching the first window ofsize cn/ lg lg lg n ? This is given by solving for i in the following:2 i n n ≤ cn lg lg lg n yielding i ≤ lg(4 c ln n ) − lg lg lg lg( n ) = k lg lg n for some positive constant k .Pessimistically assume each such intervening window has size cn/ lg lg lg n (thiscan only reduce the number of collisions). By the above argument, we have (cid:15)n packets for some constant (cid:15) >
0, and for c sufficiently small, and so Lemma 2guarantees w.h.p. that each window results in O ( n/ (lg lg n ) d ) successful packetsfor some constant d > O (lg lg n ) times. There-fore, the total number of packets finished over these windows is: O (cid:18) n (lg lg n ) d − (cid:19) = o ( n )since d >
1, and this completes the argument. (cid:117)(cid:116)
Theorem 3
For a single batch of n packets, with high probability LLB* experi-ences Ω (cid:16) n lg lg n lg lg lg n (cid:17) collisions.Proof We focus on a window of size cn/ lg lg lg n for a sufficiently small constant c >
0. Conservatively, we do not count collisions prior to this window (countingthese can only improve our result).
By Lemma 3, w.h.p. the number of packets that succeed prior to reaching thefirst window of size cn/ lg lg lg n is o ( n ). Therefore, at the start of this window, wehave (cid:15)n packets some positive constant (cid:15) . Define an indicator random variable X j such that X j = 1 if slot j contains a collision; otherwise, X j = 0. Then, we have P r ( X j = 1): indowed Backoff Algorithms for WiFi 23 = 1 − (cid:88) k =0 (cid:32) (cid:15)nk (cid:33) (cid:18) lg lg lg ncn (cid:19) k (cid:18) − lg lg lg ncn (cid:19) (cid:15)n − k (13) ≥ − (cid:18) − lg lg lg ncn (cid:19) (cid:15)n − (cid:15) lg lg lg nc (cid:18) − lg lg lg ncn (cid:19) (cid:15)n − (14) ≥ − (cid:18) − lg lg lg ncn (cid:19) (cid:15)n (cid:18) (cid:15) lg lg lg n/c − lg lg lg n/ ( cn ) (cid:19) (15) ≥ − (cid:18) − lg lg lg ncn (cid:19) (cid:15)n (cid:18) (cid:15) lg lg lg nc (cid:19) (16) ≥ − O (cid:32) (cid:15) lg lg lg nc (lg lg n ) (cid:15) lg( e ) /c (cid:33) (17)= Ω (1) (18)where Equation 16 follows from Fact 2.Let X = (cid:80) cn/ lg lg lg nj =1 X j . By linearity of expectation, the expected number ofcollisions over the contention window W is: E [ X ] = (cid:88) j E [ X j ] = Ω (cid:18) n lg lg lg n (cid:19) By Corollary 1:
P r ( X < E [ X ] − t ) = e − t ncn and letting t = √ n ln n is sufficient to prove that the number of collisions in thewindow is Ω (cid:16) n lg lg lg n (cid:17) with high probability.By specification of LLB*, the window w is executed (1 /
2) lg lg( cn/ lg lg lg n ) = Ω (lg lg n ) times. By Lemma 2, O (cid:16) n (log log n ) d (cid:17) packets are succeeding in each suchexecution for some constant d >
1, assuming the constant c > Ω ( n ) packets remaining after each of the Ω (lg lg n ) executions of this window size. This means we can apply the abovelower bound on the number of collisions per window of size cn/ lg lg lg n , and sow.h.p. there are a total of Ω ( n/ lg lg lg n ) · Ω (lg lg n ) collisions, as desired. (cid:117)(cid:116) It may be helpful to note that, since O ( n/ (log log n ) d ) packets are succeeding ineach such execution of the window of size cn/ lg lg lg n , it may appear that provinga stronger lower bound is possible by proceeding to asymptotically larger windowsin the execution of LLB*. However, we highlight the dependency between d , c , and (cid:15) ; that is, for d > c < (cid:15) lg( e ) / < . (cid:15) ≤ .
73, and sothe argument does not hold for larger windows.We also note that, under LLB, the window size of cn/ lg lg lg( n ) is criticalto showing that all packets succeed within Ω (cid:16) n lg lg n lg lg lg n (cid:17) slots [17]; therefore, ourbound on the number of collisions derived in Theorem 3 is asymptotically tight. · Trisha Chakraborty · Maxwell Young
Finally, we turn to the analysis of LB. Here, we define the algorithm LB* wherea contention window of size w i is used consecutively (1 /
2) lg( w i ) times, and then w i +1 = 2 w i for i ≥
0. An analog to Lemma 1 implies that a lower bound on thenumber of collisions experienced by LB* provides a lower bound on the numberof collisions for LB; we omit this, although we give the important details of theremaining lower-bound argument below.
Theorem 4
For a single batch of n packets, with high probability Log-Backoff experiences Ω (cid:16) n lg n lg lg n (cid:17) collisions.Proof The argument is similar to that of LLB. Our focus is on a window of size cn/ lg lg n , and we state the three main steps of the argument here for complete-ness: (1) First, the analog of Lemma 2 holds:
For any constant (cid:15) ≤ , consider (cid:15)n packets and a CW of size cn/ lg lg n fora sufficiently small constant c > . With high probability, O ( n/ (lg n ) d ) packetssucceed in the CW for a constant d > depending on (cid:15) and c . This is proved in the same way, where the number of iterated logarithms isreduced by one in Equations 3-8, and Corollary 1 applies again to give the w.h.p.result. (2)
Second, the analog of Lemma 3 holds:
Consider a single batch of n packets executing Log-Backoff . With high prob-ability, o ( n ) packets succeed prior to the first window of size cn/ lg lg n for asufficiently-small constant c > . The argument remains essentially unchanged. Equations 9 to 12 in the proofof Lemma 3 hold. Next, we ask: how many intervening unique-sized windows existbefore reaching the first window of size cn/ lg lg n ? This is given by solving for i in the following: 2 i n n ≤ cn lg lg n which yields (again) i ≤ k lg lg n for some positive constant k .Pessimistically assume each such intervening window has size cn/ lg lg n (thiscan only reduce the number of collisions). By the above argument, we have (cid:15)n packets for some constant (cid:15) >
0, and for c is sufficiently small, the equivalent ofLemma 2 above guarantees w.h.p. that each window results in O ( n/ (lg n ) d lg lg n )successful packets for some d >
1. Each such intervening window executes O (lg n )times. Therefore, the total number of packets finished over these windows is O (cid:16) n (lg n ) d − lg lg n (cid:17) = o ( n ). (3) Third, the equivalent analysis in Equations 13-17 holds, showing that theprobability of a collision in this window is constant:= 1 − (cid:88) k =0 (cid:32) (cid:15)nk (cid:33) (cid:18) lg lg ncn (cid:19) k (cid:18) − lg lg ncn (cid:19) (cid:15)n − k (19)= Ω (1) (20) indowed Backoff Algorithms for WiFi 25 By Corollary 1, the expected number of collisions is tightly bounded:
P r ( Z < E [ Y ] − t ) = e − t ncn and letting t = √ n ln n is sufficient to prove that the number of collisions in thewindow is Ω (cid:16) n lg lg n (cid:17) with high probability.By specification of LB, the window w is executed lg( cn/ lg lg n ) = Ω (lg n )times. By Step 2, O ( n/ (lg n ) d lg lg n ) = o ( n ) packets are succeeding in each suchexecution for some constant d > c > (cid:15) (cid:48) n packets remaining in each execution of this window sizefor some sufficiently small constant (cid:15) (cid:48) >
0. This means we can apply the abovelower bound on the number of collisions per window of size cn/ lg lg n , and sow.h.p. there are a total of Ω ( n/ lg lg n ) · Ω (lg n ) collisions, as desired. (cid:117)(cid:116) n packets, it is known that w.h.p. STB has O ( n ) makespan,and this is a trivial upper bound on the number of collisions. Here, we derive alower bound of Ω ( n ) on the number of collisions, thus implying that w.h.p. STBhas Θ ( n ) collisions.Recall from Section 4 that STB is non-monotonic and executes over a doubly-nested loop. The outer loop sets the current window size w to be double that usedin the preceding outer loop. For each such window in the outer loop, the innerloop executes over lg w windows of decreasing size: w, w/ , w/ , ...,
1, and we referto this sequence of windows as a run . Theorem 5
For a single batch of n packets, with high probability Sawtooth-Backoff experiences Ω ( n ) collisions.Proof Consider a run that starts with a window of size n/
8. The total number ofslots up to the end of this window (including all corresponding runs) is less than n/
2; therefore, more than n/ W of size n/ j . Let the indicator random variable Z j = 1 ifslot j in this window has a collision; Z j = 0 otherwise. P r ( Z j = 1) ≥ n/ (cid:88) (cid:96) =2 (cid:32) n(cid:96) (cid:33) (cid:18) n/ (cid:19) (cid:96) (cid:18) − n/ (cid:19) ( n/ − (cid:96) ≥ n/ (cid:88) (cid:96) =2 (cid:16) n(cid:96) (cid:17) (cid:96) (cid:18) n (cid:19) (cid:96) e − (4 /n )(( n/ − (cid:96) )1 − /n = n/ (cid:88) (cid:96) =2 (cid:16) n(cid:96) (cid:17) (cid:96) (cid:18) n (cid:19) (cid:96) e − (8 /n )(( n/ − (cid:96) ) ≥ n/ (cid:88) (cid:96) =2 (cid:16) n (cid:96) (cid:17) (cid:96) (cid:18) n (cid:19) (cid:96) (cid:32) e (cid:96)/n e (cid:33) ≥ n/ (cid:88) (cid:96) =2 (cid:18) (cid:96) (cid:19) (cid:96) (cid:18) e (cid:19) = Ω (1) · Trisha Chakraborty · Maxwell Young
Algorithm A Num. of Collisions C A Total Time T A BEB O ( n ) O ( n · P + n log n )LB Θ (cid:16) n log n log log n (cid:17) Ω (cid:16) n log n log log n · P (cid:17) LLB Θ (cid:16) n log log n log log log n (cid:17) Ω (cid:16) n log log n log log log n · P (cid:17) STB Θ ( n ) Θ ( n · P ) Table 3: Asymptotic bounds on collisions and total time.where the second line follows from Fact 1 and a lower bound on binomial coefficient.Therefore, by linearity of expectation, over the n/ W , theexpected number of collisions is Ω ( n ).In order to obtain a bound with high probability, we use Corollary 1. Letting Z denote the number of collisions in w , then: P r ( Z < E [ Z ] − t ) ≤ e − t / ( n/ . Letting t = (cid:112) dn (ln n ) / d ≥ P r ( Z < E [ Z ] − t ) ≤ e − d ln n = 1 /n d and yields the result with high probability. (cid:117)(cid:116) T A = Θ ( C A · P + W A )yields the third column Table 3.For large values of n , one may argue that P ought to be treated as a slowlygrowing function of n , such as Ω (log n ), given that lg n bits are required to uniquelylabel n machines. In this case, we note that both T LB and T LLB exceed T BEB asymptotically. In fact, even a smaller bound P = ω (lg n lg lg lg n/ lg lg n ) is suffi-cient to yield this asymptotic behavior. This analysis offers additional support forour conjecture that the number of collisions is an important metric—more so thanthe number of CW slots—when it comes to the design of contention-resolutionalgorithms.Recall that in examining total time (Section 4.1.2), an increase in packet sizefavored BEB over LLB, the latter being the closest competitor to BEB in ourexperiments. This aligns with the above discussion on the impact of packet size.Furthermore, as empirical support for our claim, we use NS3 to examine the rela-tive performance of these two algorithms as packet size increases in Figure 7. As the packet size grows, LLB performs increasingly worse than BEB. We fit a linearregression model of the value LLB - BEB on the number of packets. This fittingmodel implies that when the payload size increases by 100B, the average increasein total time for LLB is roughly 700 µs more than the increase experienced byBEB. The increase rate is statistically significant ( p -value less than 0 . indowed Backoff Algorithms for WiFi 27 llllllllllllllllllllllllllllll llllllllllllllllllllllllllllll llllllllllllllllllllllllllllll llllllllllllllllllllllllllllll llllllllllllllllllllllllllllll llllllllllllllllllllllllllllll llllllllllllllllllllllllllllll llllllllllllllllllllllllllllll llllllllllllllllllllllllllllll llllllllllllllllllllllllllllll − Packet Size (payload in bytes) LL B − BEB ( m s )
100 200 300 400 500 600 700 800 900 1000
Fig. 8: The difference in total time between LLB and BEB for n = 150 as packetsize increases (30 trials per size). Result 5.
Theoretical bounds on total time imply that LLB and LB shouldunderperform BEB for sufficiently large n and P . The elephant in the room is STB since, for P = Ω (log n ), BEB and STB areasymptotically equal in terms of total time. Despite this, STB underperforms BEBin our experiments, We explore this issue in Section 6.2. We have presented our evidence for why assumption A2 is flawed. In this section,we discuss a few issues that we have left unresolved until this point. We start byexamining TSTB (from Section 4). Next, we discuss the legitimacy of our findingsin the context of other protocols/networks.6.1 Performance of TSTBRecall from Section 4, that TSTB proceeds like STB, except that its runs that aretruncated. We implemented TSTB in order to compare its performance againstthe other algorithms.To begin our discussion, we consider two extreme behaviors that depend on the value c . When c is sufficiently small, the run is truncated after the first window,and TSTB behaves identically to BEB. Conversely, when c is sufficiently large, therun is not truncated at all, and TSTB behaves identically to STB. These two casesare illustrated in Figures 9 (A) and (B), respectively. Given this view of TSTB,it is not surprising that it cannot outcompete BEB in terms of total time over · Trisha Chakraborty · Maxwell Young
10 20 30 40 50 60 70 80 90 100 110 120 130 140 150
Number of Packets (64 Bytes) T o t a l T i m e BEBLBLLBSTBTSTB (A)
10 20 30 40 50 60 70 80 90 100 110 120 130 140 150
Number of Packets (1024 Bytes) T o t a l T i m e BEBLBLLBSTBTSTB (B)
0 10000 20000 30000 40000 50000 60000 70000 80000 90000 100000
Number of Packets C on t en t i on - W i ndo w S l o t s STBTSTB (C)
Number of Packets C on t en t i on - W i ndo w S l o t s STBTSTB (D)
Fig. 9: Median values are reported: (A) Total time from NS3 experiments with 30trials for each value of n with 64B payload with 2 m distance with truncation point X = 1; (B) Total time from NS3 experiments with 30 trials for each value of n with 1024B payload with 2 m distance with truncation point X = 5; (C) CW slotsvia simple Java simulation with 100 trials for each value of n ranging from 400 to100000; (D) A zoomed-in portion of plot (C) illustrating the minor improvementof TSTB over STB in terms of CW slots. Bars represent 95% confidence intervals.the range of n -values investigated. We explored a variety of values for c , and nonegave an improvement.Of course, in terms of CW slots, when TSTB behaves closely to STB, it willoutperform BEB. Indeed, over large values of n , the number of CW slots usedby TSTB is smaller, although not by a significant amount; this is illustrated inFigures 9 (C) and (D). While we do not include this analysis, it is easy to see thatTSTB will also incur Θ ( n ) collisions.These findings—especially those regarding total time—are the reason for post-poning discussion of TSTB until this point. indowed Backoff Algorithms for WiFi 29 n = 10 to 150. What is thelong-term behavior?As we discussed previously in Section 3, NS3 is valuable in revealing flawedassumptions via the extraordinary level of detail it provides; however, this alsoprevents experimentation with NS3 at larger scales. We attempt to shed light on(i) - (iii) by examining larger values of n in order to see if our predictions aremet, and we employ our simpler Java simulation for this task. While this is notan accurate simulation if we are interested in total time, we are only interested incollisions; for this purpose, the simple Java simulation is adequate.To address (i), we look at n ≤ as plotted in Figure 10(A). Now, in terms ofCW slots, we see that LLB is indeed outperforming LB for large values of n . Thissupports prior theoretical results for the number of CW slots given a sufficientlylarge value for n .In regard to (ii), we again take n ≤ and plot the ratio of collisions: LB vsSTB and LLB vs STB. Figure 10(B) demonstrates that the number of collisions forLB quickly exceeds STB. The tougher case is LLB which only begins to evidencea greater number of collisions at approximately n = 30 , n . Note that the plot ofBEB/STB is (roughly) flat, as expected from our asymptotic analysis of collisions.This suggests that for even larger values of n , BEB will exhibit superior total time.6.3 Scope of Our FindingsIn this section, we consider to what extent our findings are an artifact of IEEE802.11g, and whether LB, LLB, STB might do better inside other wireless proto-cols. IEEE 802.11g uses a truncated BEB, is this significant?
In our experi-ments, the maximum congestion-window size is 1024 which differs from the ab-stract model where no such upper bound exists. However, even for n = 150, thismaximum is rarely reached during an execution of BEB and this does not seem tohave any noticeable impact on the trend observed in Figures 4(A) and 4(B). · Trisha Chakraborty · Maxwell Young
Number of Packets C on t en t i on - W i ndo w S l o t s BEB L B ( a b o v e ) a n d L L B ( b e l o w ) STB (A)
Number of Packets R a t i o o f C o lli s i on s LB/STBLLB/STBBEB/STB (B)
Fig. 10: Results from Java simulation results with 200 trials per n ≤ in in-crements of 400: (A) CW slots with median values plotted, (B) ratio of mediannumber of collisions for BEB, LLB, and LB versus STB. What if smaller packets are used?
During a collision, the time lost to trans-mitting would be reduced. In an extreme case, if the transmission of a packet fitwithin a slot, this would align more closely with A2.Due to overhead, packet size has a lower bound in IEEE 802.11. Additionally,in NS3, there is a 12-byte payload minimum which translates into a minimum totalpacket size of 76 bytes for our experiments. For this packet size, we witnessedthe same qualitative behavior is observed in terms of CW slots and total time.Alternatives to 802.11 might see more significant decreases. However, there isa tradeoff for any protocol. A smaller packet implies a reduced payload given theneed for control information (for routing, error-detection, etc.) and this meansthat throughput is degraded.
What if the ACK-timeout duration is reduced or acknowledgements areremoved altogether?
This would also bring us closer to A2. In our experiments,the ACK-timeout is 75 µ s (recall Section 3) and values below this threshold willlead a station to consider its packet lost before the ACK can be received. Thisresults in unnecessary retransmissions and, ultimately, poor throughput.Totally removing acknowledgements (or some form of feedback) is difficultin many settings since, arguably, they are critical to any protocol that providesreliability; more so when transmissions are subject to disruption by other stationsover a shared channel. To what extent do these findings generalize to other protocols?
We do not claim that our findings hold for all protocols. If (a) sufficiently small packets are feasible and (b) reliability is not paramount, performance should align betterwith theoretical guarantees derived from using assumption A2.We do claim that the performance of how collision detection is performed—andwhich is ignored under A2—seems common to several other protocols. Examples This is set within the
UdpClient class of NS3.indowed Backoff Algorithms for WiFi 31 include members of the IEEE 802.11 family, IEEE 802.15.4 (for low-rate wirelessnetworks), and IEEE 802.16 (WiMax); these employ some form of backoff, incuroverhead from control information, and use feedback via acknowledgements or atimeout to determine success or failure. This is a significant slice of current wirelessstandards.
Result 6.
Designing contention-resolution algorithms using assumptionA2 seems likely to translate into poor performance in practice for a rangeof wireless protocols.
A setting where the abstract model may be valid is networks of multi-antennadevices. If a collision can be detected more efficiently, perhaps by a separate an-tenna, the delay due to transmission time can be reduced. Canceling the signalat the sending device so that other transmissions (that would cause the colli-sion) can be detected is challenging. However, this is possible (for an interestingapplication, see [49]) and such schemes have been proposed using multiple-inputmultiple-output (MIMO) antenna technology [50,51].Finally, we note that future standards may satisfy (a) and (b). A possiblesetting is the Internet-of-Things (IoT); for example, [52] characterizes IoT trans-missions as “small” and “intermittent, delay-sensitive, and short-lived”. To reducedelay, the authors argue for removing much of the control messaging used by tra-ditional MAC protocols. Therefore, this setting seems more closely aligned withA2. However, using this same logic, [52] also argues for the removal of any backoff-like contention-resolution mechanism. So, these standards are in flux and we mayindeed see protocols that avoid the issues we identify here.
A preliminary version of this work appeared previously [53]. Here, we have revisedand expanded on much of the material. Experiments regarding CW slots and totaltime for all algorithms have been redone using different distances between nodes inorder verify that the results hold. The superior performance of BEB with respectto total time is explored further via additional experiments measuring the numberof collisions (Section 4.2). We also investigated a new algorithm, TSTB, in orderto measure its performance (Section 6.1). Finally, our analytical arguments arepresented in full (Section 5).
Theoretical Results.
There is a vast body of theoretical literature on the prob-lem of contention resolution. Exponential backoff has been studied under the case where arrival times of packets are stochastic (see [8,9,10,11,28]). Guarantees onstability are known [12,13,14], and under saturated conditions [15].Moving away from stochastic arrival times, batched arrivals have received sig-nificant attention [17,48]. Makespan results are also known for batched arrivalswhen packets can have different sizes [29]. In this case, BEB is again found to offer · Trisha Chakraborty · Maxwell Young poor makespan, although the model deviates significantly from wireless communi-cation protocols. Energy efficiency is important to multiple access in many low-power wirelessnetworks [54,55,19,45]. The communication channel may be subject to adversarialdisruption (for examples, see [56,57,58,59,60,61,62,63,64]), several results addressthe challenge of multiple access [65,66,67,46,30,68,69,70,71,72,73,30,74].Regarding the time required for a single successful transmission, a lower-boundof Ω (log log n ) is known [75]. Additionally, under different channel-feedback modelshave been investigated [31,76,77].Deterministic contention-resolution algorithms have been considered [78,79].Additionally, deterministic protocols for the closely-related problem of broadcaston a multiple-access channel have also received significant attention [80,81,82].In contrast, the performance of adaptive algorithms—where packets make use ofchannel feedback—and the power of randomized versus deterministic approacheshas been examined. When players have access to a global clock deterministic, non-adaptive contention-resolution has been investigated [83]. Conversely, without aglobal clock, both adaptive and non-adaptive results are known [84]. The wake-upproblem is another closely-related to contention resolution, and it addresses howlong it takes for a collection of devices to receive a wake-up transmission [85,86,87,88,89].Several results incorporate issues of scheduling or time-constrained access to ashared resource [90,91,92,93].Finally, several approaches involve first estimating the number of devices con-tending for the channel in order to start with a larger initial CW [94,95,96,97,98,99,19,45]. Preliminary results suggest that such size-estimation algorithms maybe promising [53]. We omit this content here given that the results presentedare already substantial, and the performance of size-estimation results deserve anin-depth investigation that we believe falls outside the scope of the current work. Performance under 802.11.
In terms of the performance of backoff algorithmsin practice, the performance of IEEE 802.11 has received significant attention [100,101,102,103,104,105,106]. There are several results that focus specifically on theperformance of BEB within IEEE 802.11, and we summarize those that are mostclosely related.Under continuous traffic, windowed backoff schemes are examined in [107] witha focus on the tradeoff between throughput and fairness. The authors focus onpolynomial backoff and demonstrate via analysis and NS2 (the predecessor toNS3) simulations that quadratic backoff is a good candidate with respect to bothmetrics.Work in [108] addresses saturated throughput (each station always has a packetready to be transmitted) of exponential backoff; roughly, this is the maximumthroughput under stable packet arrival rates. Custom simulations are used to con-firm these findings.In [109], the authors propose backoff algorithms where the size of the contention window is modified by a small constant factor based on the number of successfultransmissions observed. NS2 simulations are used to demonstrate improvementsover BEB within 802.11 for a steady stream of packets (i.e. non-bursty traffic). In particular, large packets being transmitted can be interrupted at any point by otherpackets that access the channel without performing carrier-sense multiple access.indowed Backoff Algorithms for WiFi 33
Lastly, in [110], the authors examine a variation on backoff where the contentionwindow increases multiplicatively by the logarithm of the current window size(confusingly, also referred to as “logarithmic backoff”). NS2 simulations imply anadvantage to their variant over BEB within IEEE 802.11, again for non-burstytraffic.
We have offered evidence that a commonly-used model for designing contention-resolution algorithms is not adequately accounting for the cost of collisions inthe domain of WiFi and related wireless communication standards. Interestingquestions remain. In terms of analytical work, we have argued for why collisionshave an impact on performance, but what is the optimal tradeoff between collisionsand CW slots? Assuming that this tradeoff is known, can we design algorithms thatleverage this information? Can we analyze the more general case where packets donot arrive in batches, but may instead arrive at arbitrary times?
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