Reliability Constrained Packet-sizing for Linear Multi-hop Wireless Networks

Transcription

1 Reliability Constrained acket-sizing for inear Multi-hop Wireless Networks Ning Wen, and Randall A. Berry Departent of Electrical Engineering and Coputer Science Northwestern University, Evanston, Illinois 628 Eail: Abstract We consider optiizing the packet-sizes and the reuse factor to iniize the delay required to send a essage between two nodes in a linear ulti-hop wireless network subject to a reliability constraint. Initially assuing no re-use, we give a bound on the required delay. Next, in a infinite syste with re-use, we analyze the rate of growth of the delay as a function of the essage size. Two cases are considered: one in which packets are decoded/re-encoded on each hop and one in which this is concatenated with an end-to-end outer code. The later is shown to result in lower delays. I. INTRODUCTION Consider a wire-line network in which a essage is to be divided into several packets of equal size and sent over a sequence of error-free links with transission rate R. Furtherore, suppose that an entire packet ust be received over one link before it ay be sent over the next, and that the network is lightly loaded so that whenever this condition is true the packet ay be sent. Ignoring any overhead per packet, it is then well-known that the end-to-end delay of the essage is iniized by aking the packet-size as sall as possible so as to benefit fro pipelining. If overhead is not ignored, there is an optial packet-size which balances this pipelining effect with the aortization of overhead given by using larger packets. In this paper, we consider a siilar question in the context of a linear ulti-hop wireless network. Here, several new issues arise. First we assue that all nodes transit in a coon frequency-band and so ultiple links interfere with each other. Also, nodes are precluded fro sending and receiving at the sae tie i.e. they ust satisfy a half-duplex constraint. Finally, we assue that links are not error free and that a node has a constraint on the reliability i.e. probability of decoding error at which a essage ust be obtained. Given this odel we consider optiizing the packet-size as well as the schedule of transissions with the objective of iniizing the total delay for sending a packet fro a given source node to a given destination. In this setting for a given transission schedule packet sizes ust now balance the pipe-lining gains with both aortization of overhead as well as eeting the reliability constraint. We forulate a siple linear odel for a ulti-hop network in which all transissions transissions are sent hop-by-hop and interfering transissions are treated as noise. To odel the reliability constraint we use an error exponent odel derived fro the rando coding bound for a Gaussian channel. We initially consider a syste without spatial re-use, i.e. each link is scheduled in its own tie-slot, and derive an upper bound on the end-to-end delay. We then turn to a odel with spatial reuse consider the packet size and re-use distance which either axiize the throughput or iniize the total delay. In the later case, to gain insight we focus on the asyptotic growth of the total delay as the essage size increases to infinity. In this regie, we characterize the optial nuber of packets, for two different coding schees. In the first, each hop is coded individually. In the second, a concatenated coding schee is used to add end-to-end coding. In the first case, the optial nuber of packets satisfies 2 log = Θ, while in the second it satisfies 2 = Θ. In other words, with the concatenated coding schee we use saller packets. The delay under both schees grows at the sae first order rate, but the second order growth is saller with the concatenated schee. Finally we conclude with soe nuerical exaples. Related work includes [], which considers reliability bounds for ulti-hop networks but does not account for spatial reuse and thus pipe-lining as we do here, and [2], which addresses the re-use but focuses on throughput as opposed to delay. II. MODE We consider a one-diensional odel, where all nodes are regularly placed on a one-diensional line and labeled by an integer. One node x is assued to have nats of data to send to another node y. et D be the distance between node x and y, and let H be the nuber of nodes between x and y, each of which is assued to be a relay for the essage i.e. the nuber of hops is H. To siplify the analysis, we assue the queuing delay on each hop is zero. This is reasonable assuing that the given flow has higher priority over other flows in the network. All nodes are assued to transit in the sae frequency band with noralized bandwidth of and treat any interfering transission as noise. The channel between any pair of nodes is odeled by a distance dependent path-loss plus additive Gaussian noise. Furtherore, we assue that the nodes eploy a regular TDM-schedule of length so that In order to siplify the notations, we use nat as the unit for inforation in this paper.

2 in tie-slot t, the nodes n + t od are allowed to transit, for n =...,,,,.... Figure illustrates this space-tie reuse odel. Tie slot Tie slot 2 Tie slot + Node Node 2 Node + Node Node 2 Node +... Node Node 2 Node + Fig.. Space-tie reuse schee. To begin consider a siple odel where all the nodes have the sae transission rate. Assue the nats of inforation are transitted in equal-size packets, and there are h additional nats of overhead in each packet. et RH, denote the transission rate in ters of nats per channel use under certain H and and assue that all transissions are reliable. It follows that +h RH, is the length of the tie-slot for one packet. The end-to-end delay in channel uses is then DH, = + h + H + h RH, RH,, where the first ter is the tie for the source to send all the nats over the first hops, and the second ter is the tie required for the last packet to traverse the reaining H hops. Essentially, this is a pipe-lining calculation as described in the introduction, only now a packet ust traverse hops before a new packet ay be transitted. If H and are fixed, and we ignore the integer constraint on, then the delay is iniized if takes the value: H H, =, 2 h and the optial delay is: 2 + H h D H, = RH,. 3 Notice that if the overhead h is negligible, then the optial / goes to infinity, and so the optial packet size of RH, channel uses goes to zero. We are also interested in the case where each packet ust also eet a reliability constraint, in addition to overhead considerations, this will also prevent the use of arbitrarily sall packets. We next extend our odel to account for this. Specifically we assue that the end-to-end essage error probability for delivering the nats ust be no greater than a constant η. Given this constraint, we then want to iniize the end-to-end delay. Assuing that each node uses a rando Gaussian code, then fro [3], [4], the block error probability b of code with a block-length of N b channel uses is bounded by: b exp N b E, SINR, 4 for any [, ], where b is the nuber of inforation nats contained in one block, SIN R is the Signal-to-Noise-plus- Interference ratio at receiver, and E, SINR is the error exponent deterined by and SINR. For a coplex Gaussian channel with unit bandwidth, a siple expression for the error exponent E, SINR is given by: E, SINR = log + SINR. 5 Given an upper bound η hb on the block error probability b for a single hop, the iniu N hb satisfying N hb log + SINR b log η 6 is the iniu delay for sending a block of b nats over that hop for which we can use 5 to guarantee that the reliability constraint is et. In the rest of the paper, we this value of N hb as the iniu delay for each hop. III. DEAY WITHOUT SATIA REUSE IN FINITE ENGTH SYSTEM Now we return to proble of sending nats of data between two nodes x and y over H equal length hops, but include the reliability constraint. We assue that the distance D between the source and destination is noralized to. 2. Furtherore, there is no other node beyond x and y, and no other interference present. Here we consider the siplest case where only one transission is possible in each tie slot. et N denote the noise power spectru density. Notice that there is exactly one active transitter in each tie slot, we assue the transission power is for each transitter. We consider what are the optial nuber of hops H and what is the nuber of blocks containing the original nats. Before going into the details, we introduce following basic result. ea : In an optiized syste, the delay for each hop is the sae for each packet. This follows directly fro the assuptions that each packet has the sae size and that the channel for each hop is the sae and thus so is its SINR. Thus we assue that the delay is the sae for each hop and we denote this by N hb. et SINR = N H α be the SINR for a hop, where α is the path-loss factor 3. Then, 6 can be rewritten as: N N hb log + H α + h log η hb, 7 2 Changing this distance is equivalent to changing the transission power. Since our results are suitable for any power, we use the noralized distance to siplify our notation. 3 Here, /N is also noralized

3 where η hb is the per hop per block error constraint. Assuing no retransissions, and that coding is done independently over each hop then η hb and the end-to-end error constraint η will satisfy η = η hb H. 8 Considering that η < and N, we have η H H η, 9 which eans that setting η hb = H guarantees the end-to-end essage error probability is less than η. et N = N hb H denote the total delay, then the iniu N satisfying η N H + Hh H log η H log + N H α can be guaranteed to satisfy the reliability constraint. roposition : Without an integer constraint on H, the total delay is upper bounded by H + H h H log η H, log + N H α where H = zα /N α, and zα is the solution to + z log + z = α. roof: Notice that H and are positive integers. The right-hand-side of is onotonically increases with. Thus any given H, the that iniizes is. Note that H + Hh H log η < H + Hh H log η H. 2 log + N H α log + N H α Miniizing, the left-hand side of 2 with respect to H yields the desired result. IV. INFINITE ENGTH SYSTEM Now we consider the situation where siultaneous transissions are enabled so that the SINR includes the effect of interference. To calculate the interference, the knowledge of the sets of transitters in each tie slot is necessary, which coplicated the proble. To siplify this we assue there are infinitely any nodes regularly placed in a one-diensional line that each node in the line always has traffic to send. Thus, for a given transission schedule each node will see the sae SINR. In this section, we noralize the distance between adjacent nodes to be 4 and assue the nuber of hops H between the source node x and the destination node y is a constant. The nodes follow the space-tie reuse schedule as in Fig., so that for a schedule of length the distance between two adjacent transitters is hops. Each user has an average 4 Notice this is a different assuption than the one in previous sections. power constraint of 5 and so transits with power when scheduled. We are still assuing that the essage fro x can be sent to y via ulti-hop transission without queuing delay and are priarily interested in the case where the essage size is large. A. Throughput Optiization Before considering the optial delay proble, we address the related proble of optiizing the end-to-end throughput given a fixed block size of b nats and an end-to-end error constraint of η. et N hb be the one-hop iniu delay for nats of inforation which eets the corresponding per-hop reliability constraint in 6 The average throughput can then be written as R = b. 3 N hb We next consider axiizing this over b and. et γ, be the received SINR at each node, where γ N, = N + i= i + α +.4 j= j α Now, the reliability constraint 6 can be written as γ N hb log +, b log η H. 5 Substituting 5 into 3 yields log + γ N R log η H,. 6 The right-hand side of 6 can be decoposed into two parts, denoted by R and R respectively, Assuing a fixed value of, the first part R = only depends on, and log η H N γ, log+ the second part R = only depends on. It is clear that R is axiized if takes the largest possible value, and a finite axiizes R. To gain soe insight, we consider the optial when takes extree values. In the low SINR regie, where goes to, γ, goes to and optial is the sallest possible. On the other hand, if goes to, then the syste is in interference liited region and the optial is a bounded constant deterined by the syste paraeters. B. Optial Delay The analysis in Section IV-A shows that longer block lengths are preferred to axiize throughput. However, this prohibits pipe-lining and introduces longer delay. Now we return to the proble of iniizing the delay to send a essage of nats of inforation over H hops. As before this essage can be divided blocks of equal size b = +h, h 5 Notice this is also a slightly different noralization than in the previous section.

4 denotes the additional overhead needed per packet. The delay D of sending one block over one hop ust now satisfy D + h log η H, 7 log + γ N, to ensure the reliability constraint is et. Note that D is also the length of one tie-slot. Fro the discussion in Section II, the nuber of tie slots fro x sending out the first block until y receives the last block is H +. Thus the total delay satisfies should satisfy [ ] H + + h log η H D. 8 log + γ N, To gain insight into the paraeters which optiize this we next consider an asyptotic regie, where goes to. V. ASYMTOTIC ANAYSIS In this section, we consider study the asyptotic behavior of the optial paraeters, and as the total inforation goes to. A. Optial orders of and The right-hand side of 8 can be rewritten as D, = log + γ N, + b + log + h + b 2 + b log + b h + b 2, 9 where b = H, and b 2 = log η H. We next consider the behavior of this as for a fixed. roposition 2: et and iniize 9 over and. If, then and satisfy,, and 2 log = Θ. roof: On the right-hand-side of 9, note that + γ N, is bounded, no atter what value is used. The highest order ter is, and all the other ters are of saller orders than. This iplies that, and log = o. The coefficient of the highest order ter,, is + γ N,. It requires to iniize the delay. Therefore,,, and log = o. Now the candidates for the second highest order ters are b and log. These two ters should be asyptotically equivalent to iniize the total delay. Thus which yields the desired result. b log, 2 roposition 2 shows that 2 log increases linearly with, if is fixed. Suppose that there is no reliability constraint, i.e. the transission rate is deterined by SIN R and is independent of. In this case, fro 2, it follows that should increase at the order of Θ 2. This iplies that grows faster without a reliability constraint than when one is present. In other words, with the reliability constraint, an order larger block-size is preferred. B. Optial Fro the proof of roposition 2, the highest order-ter of D, as inf is D, = log + γ N, Furtherore, is shown to go to in roposition 2. Thus the optial is given by log+γ N = arg in k,. log 2 + γ,. as. Thus the optial is a bounded constant. Notice that 2 is the inverse of R in Section IV-A, and so the sae results are valid for the extree values of in high and low SINR regies. VI. CONCATENATED CODING SCHEME reviously, we considered that a essage containing nats is divided into blocks, and the essage is successfully received when there is no error for every block on every hop. Next we relax this assuption and consider a concatenated coding schee as in [5], in which an outer code is also used to correct issing packets which do not arrive at the destination. Assue that the inner code length is N i channel uses, and the outer code length is N out. Assue the diensional rate of the outer code is r, then N out = r N i. 22 Now, consider the error probability per block per hop, it is given by bh exp + h γ N i log +,. 23 Then the end-to-end block error, which is also the inner code error, can be upper bounded as b H bh exp{ N i E i }, 24 where E i = log + γ N, N i h N i log H N i can be interpreted as the error exponent of the inner code. Based on [5], the error exponent of the outer code can be expressed as E o R o = ax re i R i, 25 rr i=r o

5 where R i and R o are the rate for the inner and outer codes, respectively. And R i = N i and R o = N out. roposition 3: et and iniizes the total delay over and. With the concatenated coding schee, if, then the optial and satisfy,, and 2 = Θ. roof: The total essage error can be expressed as 6 e { [ γ exp ax r log +, rr i=r o h log H ] } N out N i N i N i = exp γ Nout log +, } 2 + h + log H, 26 where the optial r in the final step is r + h + log H =. 27 N out log + γ N, To satisfy the end-to-end error requireent η, we have γ Nout log +, + h + log H + log η. 28 Using the sae arguent as the previous section, the total end-to-end delay is D, = H + rn out. 29 lugging 27 and 28 into 29, and using the sae technique as in roposition 2, yield the result. Notice that the total end-to-end delay still grows linearly with. However, the second highest order ters now grow at a slower rate than the schee without concatenated coding, which iplies that this schee will have a order saller endto-end delay. VII. NUMERICA RESUT To illustrate the asyptotic, soe nuerical results are provided in this section. Figure 2 shows the growth order of the inial delays for both the schees with and without concatenated coding. It is clear that the highest order ter doinates the total delay, when the aount of inforation goes to infinity. It also illustrates the changes of optial according to different. Figure 3 are the optial and. The results shown in these two figures are consistent with our analysis. In all of above figures, α = 3, η =., N / =, h =, and H =. Delay Optial Optial Optial 8 6 Optial Delay w/o CC 4 First Order w/o CC Optial Delay with CC First Order with CC Fig. 2. Optial w/o CC Optial with CC Miniu delays and optial s Optial w/o CC Optial with CC Fig. 3. Optial s and s. REFERENCES Optial w/o CC Optial with CC [] O. Oyan, Reliability bounds for delay-constrained ulti-hop networks, in 44th Annual Allerton Conference, Illinois, Sept. 26. [2] M. Sikora, J. N. anean, M. Haenggi, D. J. Costello, and T. Fuja, On the Optiu Nuber of Hops in inear Ad Hoc Networks, in IEEE Inforation Theory Workshop ITW 4, San Antonio, TX, Oct 24. [3] R. G. Gallager, Inforation Theory and Reliable Counication. New York, NY, USA: John Wiley & Sons, Inc., 968. [4] I. E. Telatar and R. G. Gallager, Cobining queueing theory with inforation theory for ultiaccess, IEEE Journal on Selected Areas in Counications, vol. 3, no. 6, pp , August 995. [5] G. D. Forney, Concatenated Codes. M.I.T. ress, Notice E o is the actual error exponent when r is close to. We use E o to calculate the error probability since r indeed approaches when is large.