Optimal Relay Probing in Millimeter Wave Cellular Systems with Device-to-Device Relaying
11 Optimal Relay Probing in Millimeter Wave CellularSystems with Device-to-Device Relaying
Ning Wei, Xingqin Lin, and Zhongpei Zhang
Abstract
Millimeter-wave (mmWave) cellular systems are power-limited and susceptible to blockages. As a result, mmWaveconnectivity will be likely to be intermittent. One promising approach to increasing mmWave connectivity and rangeis to use relays. Device-to-device (D2D) communications open the door to the vast opportunities of D2D and device-to-network relaying for mmWave cellular systems. In this correspondence, we study how to select a good relay fora given source-destination pair in a two-hop mmWave cellular system, where the mmWave links are subject torandom Bernoulli blockages. In such a system, probing more relays could potentially lead to the discovery of abetter relay but at the cost of more overhead. We find that the throughput-optimal relay probing strategy is a purethreshold policy: the system can stop relay probing once the achievable spectral efficiency of the currently probedtwo-hop link exceeds some threshold. In general, the spectral efficiency threshold can be obtained by solving a fixedpoint equation. For the special case with on/off mmWave links, we derive a closed-form solution for the threshold.Numerical results demonstrate that the threshold-based relay probing strategy can yield remarkable throughput gains.
I. I
NTRODUCTION
Recently, there has been a surge of interest in millimeter-wave (mmWave) cellular systems, operating in the 10-300GHz band [1]–[3]. Utilizing the large chunks of mmWave spectrum has the potential to enable the next-generationcellular systems to support multiple gigabit-per-second data rates and also can help mitigate the current spectrumcrunch [4]. Inspired by the great potential, much research work has been done to address the various aspects ofmmWave cellular system design, including propagation measurements and modeling, air-interface, beamforming,radio frequency components, and network architecture. We refer to [5] and references therein for a comprehensivereview of the up-to-date mmWave research.
N. Wei and Z. Zhang are with the National Key Laboratory of Science and Technology on Communications, University of ElectronicScience and Technology of China. (Email: { wn, zhangzp } @uestc.edu.cn.) X. Lin is with Ericsson Research, San Jose, CA, USA. (Email:[email protected].) a r X i v : . [ c s . I T ] O c t One major concern for mmWave communications is that their signals are quite susceptible to blockages [6], [7].In particular, mmWave signals cannot penetrate many solid materials and even human body can attenuate the signalsby as much as 20 to 35 dB [8]. This implies that mmWave connectivity is likely to be highly intermittent and thesystem needs to rapidly adaptable to the time-varying radio environment. One promising approach to overcomingthe drawbacks of the peculiar mmWave propagation characteristics is to use relays [5]. The intuition is that relayingcan help mmWave signals turn around the blockages and increase the chance to reach the destinations. Indeed, recentstudies have shown that multi-hop relaying can greatly increase mmWave connectivity [6], [9]. Moreover, relayingis essential if the system targets at providing outdoor-to-indoor mmWave coverage. Considering the importance ofrelaying in mmWave systems, it is of great interest to explore the relaying opportunities enabled by the emergingdevice-to-device (D2D) communications in cellular systems [10].Existing research on multi-hop cellular networks has been focused on networks operating on the spectrum below 5GHz [11]–[14]. Many different relaying schemes have been proposed: analog repeater, amply-and-forward, decode-and-forward (DF), compress-and-forward, and demodulate-and-forward [15]–[18]. Once a relaying scheme has beendetermined for a source-destination pair, the next key question is which relay(s) should be selected to assist thecommunication in the presence of multiple potential relays. Choosing multiple relays can potentially provide ahigher diversity gain but requires more overhead, which may not be desirable from a system perspective [19].An interesting result proved in [20] is that selecting the best relay can achieve the same diversity-multiplexingtradeoff obtained by using multiple relays. Therefore, studying how to select a single “best” relay is of interestfrom both theoretical and practical perspectives. The definition of the best relay, however, hinges on the adoptedperformance metrics such as transmission rate [21], network lifetime [22], and spatial reuse [23]. Relay selectionhas also been jointly studied with other problems including power control [24], channel allocation [25], and multi-antenna techniques [26]. Now multi-hop cellular networks have been/are being standardized and deployed in practice[27]. In addition to infrastructure-based relaying, D2D communications bring in new opportunities for D2D anddevice-network relaying [10].In contrast to the existing multi-hop cellular research focusing on communications using the spectrum below 5GHz, in this correspondence we focus on relay probing and selection in mmWave cellular systems. Specifically,we study a two-hop DF mmWave system. For a given source-destination pair, determining if a relaying deviceis good or not requires learning the channel qualities of source-relay and relay-destination channels. Note thatmmWave transmission requires beamforming to overcome the high pathloss as well as other losses due to rain andoxygen absorption and higher noise floor associated with larger bandwidth [28]. As a result, to estimate the channelquality of a mmWave link, the transmitter and the receiver need to steer their antenna beams towards each otherbefore carrying out the estimation. This beam searching and steering process results in additional non-negligible communication overhead. For example, the prototype in [3] requires ms for each adaptive beam searching andswitching. So there is a tradeoff in searching for good relaying devices for mmWave systems: Probing more relayingdevices increases the probability of finding a better relay but at the cost of more probing overhead. This tradeoffnaturally raises the central question studied in this correspondence: how many relaying devices should be probedin a mmWave system? In contrast, such non-negligible beamforming overhead in relay probing is not a concern innon-mmWave multi-hop cellular systems, and to the best of our knowledge has not been studied in the literature.In this correspondence, the mmWave links are subject to random Bernoulli blockages. Similar models have beenproposed by [7], [29] to study the coverage and capacity of mmWave systems. We also explicitly take into accountthe beamforing overhead involved in the relay probing process. For such a mmWave system, we are interested infinding the best relaying device that achieves the maximum throughput (bit/s). Using optimal stopping theory [30],we show that the throughput-optimal relay probing strategy is a pure threshold policy. Specifically, the system canstop relay probing once the achievable spectral efficiency (bps/Hz) of the currently probed two-hop DF link exceedssome threshold. Further, it is not necessary to recall any previously probed relays: just select the relay probed at thestopping stage. In general, the spectral efficiency threshold can be obtained by solving a fixed point equation. Forthe special case with on/off mmWave links, we derive a closed-form solution for the threshold; the derived solutionreveals the impact of key system parameters on the threshold and maximum throughput. We also numericallycompare the threshold-based relay probing strategy to several heuristic relay probing schemes. Numerical resultsdemonstrate that the threshold-based relay probing strategy can yield remarkable throughput gains.II. S YSTEM M ODEL AND P ROBLEM F ORMULATION
A. MmWave Transmission with I.I.D. Bernoulli Blocking
A distinct feature of mmWave transmission is its sensitivity to the blockage of the spatially distributed obstacles inthe radio environment. In this correspondence we consider random Bernoulli blockages, which has been proposed by[7], [29] to study the system level performance of mmWave systems. Specifically, denote by x and y the transmitterand the receiver of a typical mmWave link. Whether the link x → y is blocked or not is modeled by a Bernoullirandom variable χ x,y , which equals (with probability p ) if the link is not blocked and otherwise. To avoidtriviality, we assume p (cid:54) = 0 . The blocking events are assumed to be i.i.d. across the links. The received signal powerat receiver y from transmitter x is then modeled as P r ( x, y ) = χ x,y η x,y · P t G t G r PL ( (cid:107) y − x (cid:107) ) , (1)where η x,y models other random factors such as shadowing for the link x → y , P t denotes the transmit power, G t and G r denote the transmit and receive beamforming gains respectively, and PL ( · ) is a distance-dependent pathloss function. Further, { η x,y } are assumed to be i.i.d. across the links. B. Relaying Protocols
We assume a half-duplex DF relaying strategy, where the time resource for data transmission is equally dividedbetween source-relay transmission and relay-destination transmission; this work could be straightforwardly extendedto other relaying strategies. The spectral efficiency (bit/s/Hz) of the two-hop DF link x - z - y is given by R z = min (cid:18)
12 log(1 + 2
SNR x,z ) ,
12 log(1 + 2
SNR z,y ) (cid:19) , (2)where SNR x,y (cid:44) P r ( x, y ) / ( N W ) with N and W denoting the noise power spectral density and channel bandwidthrespectively. Denote by F R z ( r ) the cumulative distribution function (CDF) of the spectral efficiency R z of a typicaltwo-hop DF link, and assume that ≤ R z ≤ ¯ r , where ¯ r is the largest achievable spectral efficiency.We propose the following relay probing protocol for a typical communication pair: (i) the source-relay link is firstprobed, and (ii) if the source-relay link is blocked, the probing stops; otherwise, the relay-destination link is furtherprobed. The physical meaning of probing a mmWave link may be understood as measuring the correspondingreceived signal-to-noise ratio (SNR): the link is considered blocked if the SNR is below certain threshold and notblocked otherwise. To measure the received SNR, the transmitter and the receiver need to steer their antenna beamstowards each other. We assume that each beamforming process requires a time duration τ . The data communicationoccurs after the relay probing process and lasts for a time duration T , after which a new relay probing processstarts. Therefore, the communication process is periodic in time and each period consists of two steps: relay probingand data transmission.The relay probing is repeated independently across different periods. Note that the lengths of different periodsmay differ as different numbers of relays may be probed in different periods. Further, when probing a relay z ,whether the relay-destination z - y link is probed or not depends on the probing outcome of the source-relay x - z link. Thus, probing the relay z consumes a (random) time duration τ (1 + χ x,z ) , and for a sample period in which n relays are probed, the length of the period is given by T n = n (cid:88) i =1 τ (1 + χ x,z ) + T. (3)Figure 1 illustrates a sample realization of a period of relay probing and data transmission. ⌧ ⌧ ⌧ ⌧ T Phase I: Relay probing Phase II: Data transmission A probing needs 8me τ or 2 τ n T n Fig. 1. A sample realization of relay probing and data transmission in a communication period.
C. Problem Formulation
After probing n relays, the source-destination pair can select the best relay for the two-hop communication.Accordingly, the amount of bits that can be delivered equals U n = W T max( R z , ..., R z n ) . (4)If we repeatedly use a stopping rule across m different periods, the total number of bits that can be delivered equals U N + · · · + U N m , and the total amount of consumed time equals T N + · · · T N m , where N i denotes the number ofprobed relays in the i -th period. As a result, the average throughput (bit/s) equals ( U N + · · · + U N m ) / ( T N + · · · T N m ) .Dividing both the numerator and the denominator by m and letting m go to infinity, the last ratio converges to E [ U N ] / E [ T N ] by the law of large numbers.Our objective is to find an optimal stopping rule N (cid:63) which probes (in expectation) a finite number of relays andmaximizes the average throughput: µ (cid:63) = max N ∈C E [ U N ] E [ T N ] , (5)where C = { N ∈ N : E [ T N ] < ∞} denotes the set of admissible stopping rules, and by definition µ (cid:63) is themaximum throughput. III. T HROUGHPUT -O PTIMAL R ELAY P ROBING
A. The Associated Ordinary Optimal Stopping Problem
It turns out very challenging to directly solve the throughput maximization problem (5). Nevertheless, we maysolve (5) by considering the solution to the following ordinary optimal stopping problem: V ( µ ) (cid:44) max N ∈C E [ U N − µT N ] . (6)The following lemma proved in [30] can be used to establish the relation between the solution to the throughputmaximization problem (5) and the solution to the ordinary optimal stopping problem (6). Lemma 1. N (cid:63) is an optimal stopping rule that attains the maximum throughput µ (cid:63) in (5) if and only if N (cid:63) is anoptimal stopping rule for the ordinary optimal stopping problem (6) with µ = µ (cid:63) and V ( µ (cid:63) ) = 0 . Lemma 1 suggests the following way to solve the throughput maximization problem (5). First, for a given µ ,find the optimal stopping rule for the ordinary problem (6). Second, find µ (cid:63) such that V ( µ (cid:63) ) = 0 . Then the optimalstopping rule attaining V ( µ (cid:63) ) = 0 for the ordinary optimal stopping problem (6) is also an optimal stopping rulefor the original throughput maximization problem (5).For the ordinary optimal stopping problem (6), let us consider the following stopping rule: N ( µ ) = min { n ∈ N : U n − µT n ≥ E [ U n +1 − µT n +1 |F n ] } , (7)where the subscript and the parameter µ in N ( µ ) respectively indicate that the rule is only to look stage aheadand that the rule depends on µ , and {F n } is a filtration of the underlying probability space. The above 1-stagelook-ahead stopping rule is myopic: it calls for stopping as long as the current utility is not less than the expectedutility attained at the next stage. This myopic decision neglects the possibility that the expected utilities attainedbeyond the next stage may exceed the current utility, and thus in general is suboptimal. Somewhat surprisingly,this 1-stage look-ahead stopping rule turns out to be optimal for the problem in question, as stated in the followingLemma 2. Lemma 2.
Denote by M n = max( R z , ..., R z n ) . For any µ > , the 1-stage look-ahead stopping rule (7) is optimal.Further, it can be reduced to the following threshold rule: N ( µ ) = min { n ∈ N : M n ≥ ρ } , (8) where ρ is the unique root to the following equation: E [max( R z − ρ, µτW T (1 + p ) . (9)We omit the proof of Lemma 2 due to page constraints. B. Threshold Policy Achieves the Optimal Throughput
Lemma 2 gives the the optimal stopping rule for the ordinary problem (6) with an arbitrary µ > . As notedin Lemma 1, we are particularly interested in the µ (cid:63) such that V ( µ (cid:63) ) = 0 , because it would lead to the optimalstopping rule for the original throughput maximization problem (5). This µ (cid:63) can be found by invoking Lemma 2;the corresponding results are summarized in the following proposition. Proposition 1.
The optimal stopping rule for the throughput maximization problem (5) is given by N (cid:63) = min (cid:26) n ∈ N : R z n ≥ µ (cid:63) W (cid:27) , (10) where µ (cid:63) is the unique maximum throughput satisfying µ (cid:63) = E [max( W T R z − µ (cid:63) T, τ (1 + p ) . (11) Further, the optimal relay is the relay probed at stage N (cid:63) . Proof:
We first claim that the rule (8) is equivalent to the following: ˆ N ( µ ) = min { n ∈ N : R z n ≥ ρ } . Thiscan be shown by induction. Clearly, at stage the stopping rules N ( µ ) and ˆ N ( µ ) are the same because M = R .In particular, if M = R ≥ ρ , both N ( µ ) and ˆ N ( µ ) call for stopping. If M = R < ρ , both N ( µ ) and ˆ N ( µ ) call for continuing to stage . At stage , if M = max( R z , R z ) ≥ ρ , then M = R z because R z < ρ byinduction. It follows that both N ( µ ) and ˆ N ( µ ) call for stopping. If M = max( R z , R z ) < ρ , then R z < ρ .Thus, both N ( µ ) and ˆ N ( µ ) call for continuing to stage . Repeating this argument for stages , , ... , we can seethat N ( µ ) and ˆ N ( µ ) are indeed the same stopping rules.Now we have shown that the optimal relay selection rule is to select the first relay satisfying R z n ≥ ρ . Inparticular, we do not have to recall any of the previously probed relays R z m , m < n . Using this fact, we can seethat the problem is invariant in time: If a relay is probed and we stop, the utility is W T R z − µτ (1 + χ x,z ) − µT ;if we continue probing more relays, the utility is V ( µ ) − µτ (1 + χ x,z ) . Thus, the following optimality equationholds: V ( µ ) = E (cid:2) max (cid:0) W T R z − µτ (1 + χ x,z ) − µT, V ( µ ) − µτ (1 + χ x,z ) (cid:1)(cid:3) . (12)Since the optimal µ (cid:63) is the one such that V ( µ (cid:63) ) = 0 , the optimality equation reduces to E [max( W T R z − µ (cid:63) τ (1 + χ x,z ) − µ (cid:63) T, − µ (cid:63) τ (1 + χ x,z ))]= E [max( W T R z − µ (cid:63) T, − µ (cid:63) τ (1 + p ) , (13)from which we have E (cid:20) max( R z − µ (cid:63) W , (cid:21) = µ (cid:63) τ (1 + p ) W T . (14)Comparing (9) to (14), we conclude that ρ = µ (cid:63) W and complete the proof.Prop. 1 implies that the optimal stopping rule is a pure threshold policy: the relay probing process stops once theachievable spectral efficiency of the currently probed two-hop link exceeds µ (cid:63) /W . In particular, the stopping rule is based on the state of the currently examined relay only, and it is not necessary to recall any previously probedrelays: just select the relay probed at the stopping stage N (cid:63) . This is a desirable feature in practical systems. Inparticular, due to the timing varying radio environment, the beamforming patterns trained for an earlier relay mightbecome outdated or the two-hop link might enter deep fade. Choosing the most recently probed relay avoids suchnuisances.Note that it is challenging to derive a closed form solution for µ (cid:63) from the fixed point equation (11) for anon-trivial spectral efficiency distribution F R z ( r ) . Instead, numerical methods are usually needed to compute µ (cid:63) .One possible numerical iterative algorithm is given as follows: µ ( t + 1) = E [max( W T R z − µ ( t ) T, τ (1 + p ) , (15)where t is the iteration index. This iterative method is in essence a variation of Newton’s method with all iterationsusing a unit step size. As shown in [30], for any non-negative initial value µ (0) , the sequence { µ ( t ) } generated bythe iteration (15) converges quadratically to µ ∗ .IV. A PPLICATION : O N /O FF M M W AVE L INKS
To obtain some insights and intuitions, we apply the derived analytical results to a homogeneous network withon/off mmWave links. Specifically, for the communication pair ( x, y ) and any relay z , P r ( x, z ) = χ x,z P and P r ( z, y ) = χ z,y P , where P and P are some positive constants. Then the spectral efficiency of the two-hop link x - z - y is given by R z = ¯ r with probability p ;0 with probability − p . Plugging the above into (11) and solving for µ (cid:63) yields µ (cid:63) = W T p ¯ r (1 + p ) τ + p T . (16)Accordingly, the optimal stopping rule is given by N (cid:63) = min (cid:40) n ∈ N : R z n ≥ ¯ r (cid:18) pp τT (cid:19) − (cid:41) . (17)Several remarks are in order. Remark 1.
The numerator
W T p ¯ r in (16) is the expected number of bits that can be delivered in a communicationperiod, while the denominator (1 + p ) τ + p T in (16) is the expected duration of a communication period includingrelay probing and data transmission. Therefore, without a priori knowledge of the relays, the optimal stopping helpsthe system spend the right amount of resources on relay probing and yields the optimal throughput given in (16). Remark 2.
With a genie-aided relay selection, the system knows the right beamforming patterns and can directlyuse a two-hop link with spectral efficiency ¯ r and thus the achievable throughput is W T ¯ r/T = W ¯ r . The ratio ofthe optimally stopped throughput and the genie-aided throughput is µ (cid:63) W ¯ r = 11 + pp τT . (18)The gap between the optimally stopped throughput and the genie-aided throughput is solely determined by τT and p p . Intuitively, τT can be considered as normalized probing overhead, while p p , which is increasing in p ∈ (0 , ,can be seen as a measure of the channel condition. As expected, the smaller the probing overhead or the better thechannel condition, the smaller the probing overhead, the smaller the gap. In particular, if the mmWave links arealways available, i.e., p = 1 , the ratio is T / ( T + 2 τ ) , where the sole cost is τ time resources spent on finding theright beamforming patterns. Remark 3.
The optimal stopping rule (17) is a pure threshold policy: the relay probing process stops once R z n / ¯ r ≥ (cid:16) pp τT (cid:17) − , which interestingly equals the ratio (18) of the optimally stopped throughput and thegenie-aided throughput. Intuitively, the smaller the probing overhead or the better the channel condition, the higherthe threshold, i.e., the system is more willing to probe more relays. Remark 4.
Note that the threshold in (10) may not be unique, though the optimal throughput µ (cid:63) is unique. Ourcurrent application with on/off links is one such example. Specifically, the spectral efficiency of any two-hop link x - z - y is either or ¯ z . As a result, any value in (0 , ¯ r ] can be used as a threshold for the optimal stopping.V. S IMULATION R ESULTS
In this section, we provide simulation results to generate more insights into the derived theoretical results. ThemmWave simulation setup closely follows [2] and is described as follows. The source is a pico BS located at ( − m , and the destination is a device located at (250 m , . The potential relays are other devices randomlydistributed in the ball centered at (0 , with radius m. The transmit powers of the pico BS and a deviceare respectively dBm and dBm. The beamforming (either transmit or receive) gains of the pico BS and adevice are respectively dB and dB. The channel bandwidth is W = 500 MHz at a carrier frequency of GHz. The noise power spectral density is − dBm/Hz. The receiver noise figure is dB. The pathloss model is
10 log PL ( (cid:107) y − x (cid:107) ) = 141 . ( (cid:107) y − x (cid:107) / dB. Each link x → y is also subject to a random log-normalshadowing η x,y with dB deviation, and is available with probability p . The data transmission phase lasts T = 1 second. Note that the prototype in [3] requires ms for each beam searching and switching. Motivated by thisresult, in our simulation we consider two beamforming time overhead values: τ = 10 ms and ms, indicatinglow overhead and high overhead, respectively. Spectral Efficiency Threshold (bps/Hz) T h r oughpu t ( G bp s ) τ = 10 ms; p = 0.5 τ = 10 ms; p = 0.9 τ = 50 ms; p = 0.5 τ = 50 ms; p = 0.9 Fig. 2. Impact of probing overhead τ and link availability p on the optimal stopping threshold. In Figure 2, we study how the throughput performance varies with the stopping spectral efficiency threshold R z under different probing overheads and link availabilities. For each pair of ( τ, p ) , Figure 2 clearly shows that thereexists an optimal stopping threshold that achieves the maximum throughput. Further, the maximum throughputincreases if the probing overhead τ decreases and/or the link availability p increases, which is intuitive. On the onehand, for a fixed link availability p , the higher the probing overhead τ , the lower the optimal stopping threshold,i.e., less potential relaying devices are probed. On the other hand, for a fixed probing overhead τ , the higher thelink availability p , the higher the optimal stopping threshold, i.e., more potential relaying devices are explored.In Figure 3, we compare the throughput performance attained by the derived optimal stopping rule to theperformance of several heuristic relay probing strategies. In Figure 3, the probing overhead τ = 10 ms, whilesimilar observations hold for τ = 50 ms. The first heuristic scheme is termed myopic stopping, in which the relayprobing process is stopped at the first relay that can be used to establish both source-relay and relay-destinationlinks. The second heuristic scheme is to probe a fixed number β of relays and then selects the best one from theprobed relays. We consider two values for β in Figure 3: and . Figure 3 shows that, when the link availability isvery low (e.g., p = 0 . ), myopic stopping is nearly optimal and outperforms the two fixed probing heuristics. Thisimplies that, in a radio environment with many blockages, a mmWave system can stop probing relays immediatelyonce it finds a feasible relay but still achieves close-to-optimum throughput performance. This myopic stoppingscheme, however, performs poorly when the link availability p increases. In particular, with a higher link availability p , it is valuable to explore more potential relays to find a good source-relay-destination route. As a result, the twofixed probing heuristics outperform the myopic stopping once the link availability p becomes large enough. Thederived optimal stopping approach outperforms the three heuristic schemes in then entire range of link availability p , and the throughput gains of the optimal stopping are remarkably large. Probability of Link Availability: p T h r oughpu t ( G bp s ) Myopic probingFixed probing: β = 5Fixed probing: β = 10Optimal probing Fig. 3. Comparison of throughput performance under different relay probing strategies.
VI. C
ONCLUSIONS
In bandwidth-limited cellular systems, the gains of multi-hop relaying may be modest since the loss in the degreesof freedom (due to half-duplex constraints) eats into the gains. In contrast, mmWave systems have many degreesof freedom (in terms of bandwidth) but are power-limited and susceptible to blockages. Using infrastructure-based and/or D2D relaying can help increase the connectivity and range of mmWave cellular systems. In thiscorrespondence, we have taken some initial steps towards studying relaying probing and selection in a two-hop DFmmWave cellular system. We have found that a threshold-based policy can optimally balance the tradeoff betweenthe throughput gain from searching a better relay and the throughput loss due to higher relay probing overhead.Numerical results have demonstrated the remarkable throughput gains of the threshold-based relay probing policy(versus several heuristic schemes). Future work may consider a heterogeneous scenario where the mmWave linksare subject to non-i.i.d. Bernoulli blockages. It is also of interest to extend the current work to multiuser scenarioswhere many source-destination pairs exist. R
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