IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 46, NO. 3, MAY 2000
Capacity of the Wireless Packet Collision Channel Without Feedback George Thomas, Member, IEEE
Abstract—It is desirable in random-access protocols for wireless mobile communications to avoid the need for feedback and retransmissions. Pure Aloha protocol can use erasure correction coding in place of retransmissions to maintain a throughput of 0 184. We show in this note that if we use error-correction coding (instead of erasure correction coding) throughput levels not less than 0 322 can be realized for asynchronous random access without feedback or retransmissions. Index Terms—Error-control coding, multiuser communications, random access, wireless networks.
I. INTRODUCTION Random-access packet communication protocols have been studied extensively in the context of wireless as well as other media. With ternary feedback on channel state (idle, success, collision), the best known stable collision resolution algorithm attains a throughput of 0:487 [1]. The simpler (but potentially unstable) Aloha protocol has the well-known maximal throughput of e01 = 0:368. These algorithms require that all transmitters have synchronized clocks which demarcate a common time axis into slots equal in size to packet length. Unslotted (“pure”) Aloha relaxes this timing requirement but consequently has a capacity one half that of slotted Aloha, i.e., 0:184. The inherent feedback-and-retransmit approach of the Aloha protocols is beset with two major drawbacks. First, the feedback causes instability problems. Second, managing the retransmissions adds to the complexity of the protocol. The latter is particularly significant in next-generation advanced wideband wireless data networks where high-speed cell transmission (e.g., “wireless ATM”) is employed. The old paradigm that propagation delay is a small fraction of the packet length is reversed, and feedback-based strategies become inefficient or even infeasible. In mobile wireless packet data communications, a number of mobile users may pass through a given cell at different times transmitting packet data to a base station which must relay the same to their appropriate destinations. Considering the mobility of the users, it would be desirable if the packets did not need to be acknowledged to or retransmitted by the sender who may presently have moved on to a different cell or region. This motivates the study of packet collision channels without feedback. One way to dispense with feedback altogether is to employ some form of forward error-control (FEC) coding which can provide a degree of resilience to channel collisions. In feedback-free random access, one essentially replaces retransmissions by collision recovery coding. Reed–Solomon (RS) codes can be used quite effectively here. For instance, in slotted Aloha operating at an offered traffic load of packets per slot, the probability of collision for a transmitted packet is 1 0 e0 . Out of a long sequence of successive packet transmissions from a user, almost exactly a fraction 1 0 e0 may be expected to be erased by
Manuscript received January 11, 1999; revised July 16, 1999. The material in theis correspondence was presented at the 33rd Annual Conference on Information Sciences and Systems, Johns Hopkins University, Baltimore, MD 21218, March 1999. The author is with the Department of Electrical and Computer Engineering, University of Louisiana, Lafayette, LA 70504-3890 USA (e-mail:
[email protected]). Communicated by M.L. Honig, Associate Editor for Communication. Publisher Item Identifier S 0018-9448(00)03104-7.
1141
channel collisions, invoking the law of large numbers. If each user encodes its successive packets as code symbols in a long RS code with rate R = e0 , and if the decoder uses channel erasure as side information, then for long code sequences, the probability of successful decoding is nearly unity and the effective throughput is e0 which attains the maximum value 0:368 when = 1. To make this scheme work in practice, we would need to add to each individual packet (i.e., RS code symbol) a header that contains the transmitter identification and the sequence number of the packet (code symbol) in the codeword. The receiver will need to store packets received from different transmitters separately and in respective sequential order, and infer the “erasures” from the missing packet numbers. We do not pursue the details here but merely note that RS coding holds the potential for attaining a throughput e01 equal to that of retransmission Aloha. In practice, it is advantageous to relax the slotted-time assumption even if there is a throughput penalty factor of 2. It turns out that RS coding over long sequences of successive packet transmissions will work for unslotted Aloha as well. By the same arguments as before, we can deduce a maximal throughput of 0:184 for the feedback-free unslotted random-access channel. In mobile networks, the avoidance of the need for network-wide time synchronization among mobile nodes is often an overwhelmingly powerful advantage by itself, outweighing the disadvantage of reduced throughput. The challenge then is whether we can retain the extreme simplicity of unslotted Aloha without feedback and retransmissions, and yet hope to improve the throughput beyond the limit of 0:184. In this note we show that this is indeed possible. In fact, we show that throughputs as high as 0:322, not too far below the 0:386 capacity of the ideal unslotted Aloha channel [2], are achievable. The key to this accomplishment will be seen to consist of discarding the traditional erasure-correction approach and instead adopting an error-correction approach. Historically, the most celebrated result in erasure correction coding for collision channels without feedback is due to Massey et al. [2] who used a (non-RS) erasure-correction coding to demonstrate a maximal channel utilization (or information theoretic capacity) of e01 = 0:368 for asynchronous random access. A key assumption in [2] is that individual packets involved in collisions are totally and irretrievably lost. No attempt is made toward making use of salvageable parts of packets in collision. But by marking the collided packets as total erasures and resorting to an overall erasure correction code spread over several packets, they develop an ingenious erasure recovery scheme for packet sequences. They use a set of protocol sequences which cleverly schedule successive packet transmissions (as a superpacket) such that, for all code sequences and for all relative phase shifts, there are enough collision-free slots for the erasure correction scheme to work. Basically, thus, the technique in [2] is one of erasure correction over several packets rather than of bit-level error correction within each packet. Hui [3] observed that the techniques used in Massey’s proof of the e01 capacity might be difficult to use in practice because the set of simultaneous users could be large and varying over time. Besides, Massey’s model assumes that if enough fragments of the superpacket survive a collision, the entire superpacket may be reconstructed by virtue of the coding regardless of where the collision damages occurred. Very commonly in practical data communications, vital packet protocol information (including the “preamble” or “sync” bits that facilitate carrier, bit, and word timing acquisition) are located at the front end of the packet. A partial collision at the front end of the packet thus leads to loss of this data and renders the rest of the packet useless even if received physically intact. Hui thus set up a more practical erasure-correction model by stipulating that packets are possibly recoverable only from partial tail-end collisions with other
0018–9448/00$10.00 © 2000 IEEE
1142
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 46, NO. 3, MAY 2000
packets. The resulting practically attainable capacity of the collision channel can then be expected to be possibly less than 0:368. By means of erasure-correction coding arguments, Hui showed that a capacity of about 0:295 is attainable. Notice that Hui’s model is in essence the feedback-free asynchronous (or unslotted) random-access channel. It has an inherently low capacity of 0:184 without any coding. But by using RS coding for recovery from tail-end collisions, Hui improved the attainable capacity to 0:295. In this note we consider the same unslotted, feedback-free collision channel model as above but employ bit-error correction instead of packet-erasure correction. We show that if we consider the mutual interference in the collision channel as a noise process leading to bit errors in the collided packets, and then use error-correction coding (rather than erasure-correction coding) and try to recover from such errors, a capacity much higher than 0:184 can be realized. In particular, for totally asynchronous access, we demonstrate a maximal throughput in excess of about 0:322. This exceeds the previously known capacity bound of 0:295 in [3] and indeed moves us closer to the benchmark value of 0:368 established by Massey and Mathys [2]. Whether the unslotted, feedback-free random-access channel with the added restriction on front-end collisions can still attain the ultimate throughput of e01 remains unknown. II. ANALYSIS A. System Model Consider an infinite population of packet transmitters which together generate Poisson packet arrivals into a common channel at the rate of packets per second. All packets are of equal length, and each consists of M header bits followed by N data bits, N M . Bit duration is T seconds. The time axis is marked in intervals of T seconds, the bit duration, and all transmitter clocks are assumed to be synchronizable at the bit level. This assumption facilitates the development of our analysis, and it will be shown later that such synchronizability is, in fact, unnecessary. Packets are generated asynchronously. Each packet is assumed to “arrive” at one of the transmitters at a specific instant in time. All packets with “arrival points” in the bit interval [(i 0 1)T; iT ) are transmitted into the channel beginning at the instant iT . Each transmitted packet (or multipacket sequence, as in [2] and [3]) could have been encoded as an error-correction codeword. However, in a slot-asynchronous access channel, the individual bits in a given packet see varying and mutually dependent levels of noise and the analysis of the performance of the code becomes difficult. To circumvent this complication, we apply coding individually to each bit position, over a number of successive packet transmissions from each user. We then develop upper bounds to these bit-error probabilities and hence lower bounds to the corresponding bit channel capacities, in terms of their corresponding worst case Gaussian channels. (A key to our result is the recent demonstration by Shamai and Verdú [4] that for low signal-to-noise ratios, the noise distribution with the worst bit error rate on a binary-input, unquantized-output channel is Gaussian.) The sum of these individual bit position capacities, averaged over the arrival statistics, then yields a lower bound to the channel capacity. We carry out this procedure for the slot-wise asynchronous multiple-access channel and derive a lower bound for throughput. Forward error-correction (FEC) codes are used as in Fig. 1. For each transmitter, and for each ith bit position in the data portion in its packet, i = 1; 1 1 1 ; N , the n consecutive ith bits over n successive packets form a codeword in an (n; ki ) error-correcting code. Alternatively, after transmitting ki information bits in the ith bit position, the transmitter adds n 0 ki parity-check bits in the same bit position in succeeding packets such that the sequence of n successive bits at the ith
Fig. 1. Packet-encoding model. The ith bits of form an (n; k ) codeword.
n
successive
N
-bit packets
bit positions from any one given transmitter is an (n; ki ) codeword, for each i; i = 1; 1 1 1 ; N . This means that the effective information arrival rate for the entire system is Ri packets per second, where Ri = ki =n is the code rate. These rates vary with i because, as we explain below, each bit in a packet sees a different level of multiple-access interference. The packets in the channel may experience partial or total overlap with other packets. If a packet encounters interference in its first M (header) bit positions, the packet is deemed irretrievably lost. Otherwise, if the M header bits are received free of interference from other packets, the packet is deemed to have been acquired by the receiver and the remainder of the packet (N bits of data) may be decodable in presence of further interference in those bit positions. (We use this terminology of “acquisition” to emphasize that upon successful demodulation of the header, the receiver has synchronized itself with the packet transmission parameters and is thus cognizant of the existence of the particular packet but that the bits in the rest of the packet may or may not be sufficiently noise-free. The decoding gets done, of course, only after all of a set of n successive packets are received.) Assumptions about partial packet overlap similar to the foregoing have been used in other studies on asynchronous multiple access by Hui [3] and others. Given a packet that begins transmission into the channel at a nominal time t = 0, it is clear that the first M bit positions may be interfered with if and only if there are other Poisson arrival points in the interval from t = 0(M + N )T to +(M 0 1)T , i.e., a total of 2M + N 0 1 bit intervals. The probability of a packet loss due to interference to the header bits is, then
PL () = 1 0 exp(0(2M + N 0 1)T ) 1 0 exp(0NT ) where the approximation holds for the practical case where N M , i.e., the relative length of the header is negligible. We shall henceforth use this approximation (M = 0), so that a packet of length NT seconds, starting at t = 0, will be deemed to have successfully been “acquired” by the receiver if there were no other transmissions into the channel in the preceding (N 0 1)T second interval (0NT; 0T ]. For a packet which is thus successfully acquired by the receiver, with probability 1 0 PL (), the successive bits within that packet will see a monotonically nondecreasing interference level (Fig. 2) over the length of that packet, due to possible subsequent arrivals from other
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 46, NO. 3, MAY 2000
1143
Ci () are capacities based on the marginal distribution of noise in the individual bit positions while C () is based on the joint distribution of noise in all bit channels and hence accounts for the statistical dependencies among them. As is well known, noise correlation increases capacity in vector channels. C. Gaussian Lower Bounds for Low Signal-to-Noise Ratios
Fig. 2. Successive bits in a packet see nondecreasing levels of multiuser interference.
transmitters which will all outlast the present packet’s duration. For a given bit position i in a given packet, let m be the number of other packets which have one of their bits interfering into the same bit interval [(i01)T; iT ). These interfering packets are precisely those which have their Poisson arrival points belonging to the interval [0T; (i 0 1)T ), i.e., in the previous i bit intervals. (Any arrivals in the NT second interval earlier to that would have resulted in total loss of the packet under consideration.) The probability i (m; ) of m such interfering packets being present in the ith slot of the given packet is
i (m; ) = exp(0iT )
The capacities Ci () above may appear not amenable to easy calculation. A given user’s transmitted symbol X = 61 sees a discrete noise channel whose output is Y = X + Z where Z is the noise variable belonging to the set f0; 61; 62; 1 1 1 ; 6mg when there are m other bit-synchronous multiuser interferers present. The channels are clearly non-Gaussian, their noise probability mass functions being distributed on a countably infinite set of points on the real axis, i.e., the set of positive and negative integers. Such discrete noise distributions on a lattice arise in the study of worst noise distributions for the class of binary input channels with constrained noise power. For such channels, Shamai and Verdú [4] have shown that the worst case capacity is indeed the same as the capacity of the Gaussian channel with the same power constraint, if the signal-to-noise ratio is less than unity. In our present case, the signal-to-noise ratio is in fact less than unity by the nature of the multiple-access problem (Eb =N0 = 1=m and m > 0). Hence, calculating C (m) using the Gaussian noise model must give us an exact value of Ci () and not merely a bound to it (see [4, Fig. 2]). Thus C (m) is equal to the capacity of a binary-input, Gaussian-noise channel with unquantized output, such that Eb =N0 = 1=m, which is well-known to be (see, e.g., [5, p. 153])
C (m) = 0
iT )m : m!
(
1 2
e) 0
log2 (2
1
01
gm (y) log2 [gm (y)] dy
where B. Interference Noise Modeling The ith bit transmitted by a given user sees a random number m of other users’ transmissions as interference noise. Each of these m signals contributes an amount of energy Eb as “noise” into the ith-bit position of the desired packet. Recall that thus far we are assuming bit level synchronization among the transmitters. The signal energy per bit is Eb . The total “noise” power mEb =T and the signal power Eb =T , together yield an “Eb =No ” value of 1=m, which is the key parameter for determination of the error probability in the ith-bit position. For the sake of specificity in estimating bit-error probabilities or channel capacities, we will assume antipodal signaling with ideal matched-filter detection. Given m, then, we can find in principle the capacity C (m) of the binary-input channel with m equal energy interferers (see Section II-C below). For each bit position i in a given user’s N -bit packet, the average capacity Ci () of the channel seen by the bits sent in the ith position is
Ci () = exp (0NT )
1
m=0
C (m)i (m; )
where the factor exp(0NT ) accounts for the packet loss due to interference to the header portion. Averaging over all N bit positions, we can then find the average capacity of any bit-position channel, for a given and N , as follows:
CN ()
1
N
C (): N i=1 i
The inequality above results because the noise processes are not independent across the bit channels; an interference packet affecting bit position i also affects all bit positions i + 1 through N (Fig. 2). The
gm (y) =
1 2
+
p
1
1 2
exp
0 y0
p
exp
2
2
2
2
1
2
m
0
y+
2
m
2
2
:
Thus we can compute CN () which gives the information-theoretic capacity (in “bits” of information per transmitted binary digit) for the average multiple-access channel seen by a transmitted bit. There are N such transmitted bits per packet and T packets per second on the average. Thus the throughput of the system is NCN () bits of information per packet, or CN () packets worth of information per transmitted packet. This translates to NT CN () packets worth of information per second, i.e., N CN () packets of information per packet length. The lower bound on throughput (in packets per packet length) can then be found by letting N ! 1 and by searching for the supremum of N CN () over all , as follows:
C
N !1 lim
sup
N CN () :
This lower bound to throughput is numerically determined to be about 0:322 (Fig. 3). Thus we can guarantee that the multiple-access channel without feedback must have a capacity of at least 0:322, when the packets are transmitted in a bit-synchronous fashion. D. Relaxation of the Bit Synchronization Assumption We have shown that the ideal, bit-synchronous but packet-asynchronous, error-correcting collision channel without feedback, has a capacity in excess of about 0:322. Our argument assumed that there was bit-level synchronization among the users. However, this
1144
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 46, NO. 3, MAY 2000
Fig. 3. Throughput as a function of the packet arrival rate normalized with respect to packet length.
assumption is not at all necessary and we can now relax the requirement as follows. If two packets overlap totally asynchronously (i.e., without bit-level synchronization), then typically each bit gets partially interfered by two adjacent bits from each interfering packet. Note that the total interference energy per bit is still Eb . However, now there is additional interbit correlation in the interference noise variables. But this correlation serves only to increase the capacity of the packet channel, the worst noise being the one with no bit-to-bit correlation. Also, the first or last bits of packets may see only partial interference from one bit (i.e., energy
