Power Allocation and Coverage for a Relay-Assisted Downlink with Voice Users Junjik Bae, Randall Berry, and Michael L. Honig Department of Electrical Engineering and Computer Science Northwestern University, Evanston, IL 68 USA junjik@northwestern.edu,{rberry,mh}@ece.northwestern.edu Abstract We study the downlink coverage of a base station terminal (T), which has access to a relay node. Continuing a previous study in which the T is assumed to provide a variable-rate data service, here we assume that each active user requires a target data rate, corresponding to a voice type of service. The relay is assumed to serve a separate set of (noncellular) users, corresponding to a WiFi Access Point (). A one-dimensional model is considered in which cellular and noncellular users are uniformly distributed along a line. The T and jointly allocate available power across users and the T- link to imize the total number of users served. We characterize the optimized set of active cellular users served by the T directly and the relay, and the non-cellular users served by the. We also give a closed-form upper bound on the increase in the total number of users provided by the relay as a function of user densities and path loss exponents. Our results show that depending on the distance between the T and the, the addition of a relay gives a modest increase in the total number of active users. I. INTRODUCTION Adding fixed relays to a cellular network can potentially increase the network capacity and extend network coverage [] [5], [7]. This has motivated the introduction of relays in emerging standards such as 8.6j [8] and 4G mobile systems [9]. Rather than deploying new cellular relays, in some cases, it may be possible for a cellular service provider to use an existing wireless LAN (WLAN) access point () as a relay for cellular traffic [6]. In that case, although the potentially brings additional resources to the cellular network, those resources must be allocated across both WLAN and cellular user requests. Previous studies on relay-assisted cellular systems have generally assumed that the relays are dedicated to serving only cellular users (e.g., see [3] [5], and [6], where relays are proposed to relieve congestion in a particular cell by siphoning off traffic to neighboring cells). This paper is continuation of previous work [7] in which we evaluate the increase in downlink capacity (sum data rate over all users) associated with the addition of a relay to a single cell in a cellular network. Whereas [7] evaluates a sum data rate, which corresponds to a data type of service, here we assume a voice type of service in which each user requires a target data rate. The system objective is then to imize the total number of users served by both the cellular system and the. This work was supported by the Motorola-Northwestern Center for Telecommunications. As in [7], we consider a single one-dimensional cell with two relays ( nodes) symmetrically placed on either side of the Base STation (T), and a static user population. Both the cellular users and non-cellular users, served by the, are assumed to be continuously and uniformly distributed along the line with given densities. This corresponds to the large system model presented in []. All users (both cellular and ) are assumed to be orthogonal in time, frequency, and/or signature space, and do not interfere. Furthermore, the is assumed to use a different band from the cellular band, so that cellular users do not interfere with non-cellular users. Also, here we consider only the power allocation problem subject to a total power constraint. We assume that each user receives a single unit of bandwidth (i.e., channel, time slot, or signature), and that the total number of users does not exceed a bandwidth constraint. Finally, in the model considered here path loss is determined only by distance. Random propagation effects, such as shadowing, are not explicitly modeled, but can be incorporated in the model by changing the distance metric. Our problem is to allocate available T and power across both cellular and users, as well as to the link from the T to the, to imize the total number of active users. We solve this problem for the following two scenarios: (i) the information flows to the cellular users served by the relay are jointly encoded and transmitted from the T to the ; and (ii) the preceding information flows are transmitted in parallel (separately encoded) from the T to the. Joint encoding requires less power to transmit the data destined for relayed users from the T to the, but requires that the received data at the be demultiplexed. We also give an upper bound on the increase in total number of users provided by the, based on the scenario in which the T and have a wireline connection, so that the T does not expend any power to communicate with the. The optimization of power across users determines intervals of cellular users served by the T directly, relayed cellular users, and non-cellular users. The relative position of these intervals depends on the T and power constraints, the path loss exponents associated with the cellular, non-cellular, and T- links, and the user densities. We give numerical examples, which show that the total number of active users can This change in distance metric effectively amounts to changing the distribution of users along the line.
Fig.. γ P T for Total flow conservation P B (x) for Individual Flow Conservation P B (r) P A (x) r Distance d P A (x) One-dimensional model of a cell with relay nodes (s). be increased by more than 5% if the position of the relay is optimized. If the has no users to serve, then the increase in the number of the cellular users can be more than 35%. Jointly encoding the information from the T to the is shown to result in only a small performance gain. II. SYSTEM MODEL The one-dimensional cellular model is illustrated in Fig.. We assume a symmetric cell in which two s are placed at the same distance d from the T. As in [7], [], we assume a static set of users, which are uniformly and continuously distributed along the line. The density of the cellular users, served by the T, is ρ B, and the density of the users is ρ A. This corresponds to a large system limit in which the number of users in the system tends to infinity in proportion with the available bandwidth (i.e., fixed users per Hz). Given a finite total power constraint, the T and each serve a finite number of users. In what follows, we assume that each active user receives a fixed unit of bandwidth, and that the corresponding number of available channels (and/or time slots and/or signatures) exceeds the total number of active users. All users are therefore assumed to be orthogonal, i.e., they do not interfere. We first consider the situation in which the T does not use the as a relay. From symmetry, we need only consider the power allocation and set of active users on one side of the T. Because each active user receives a target rate, our objective is to imize the number of active users for a given power constraint. Assuming a path loss of /r a, where a is the path-loss exponent, and optimal coding, the power needed to transmit at rate R B to a cellular user at distance r is P B (r) = ( R B W B e W B ) r a Br a, where is the noise density and W B is the cellular bandwidth per user. The imum number of cellular users that the T can support with power P is then ρ B C, where C is the length of the interval containing x the active users and satisfies C ρ B P B (r)dr = P, i.e., ( a + C = ρ B B P. () Similarly, assuming non-cellular and cellular users have the same path-loss exponents, the minimum power needed by the to transmit at rate R A to a user at distance x is P A (x) = ( R A W A e W A ) x a Ax a, where W A is the bandwidth per user. The imum number of non-cellular users that the can support with power P is therefore ρ A D, where D = ( P ρ AA is the one-sided coverage (i.e., for users at distance d + x, with x ). 3 Therefore, without relaying, the total number of active cellular and non-cellular users (on one side of the T) is ρ B C + ρ A D. Now we consider the case where the T uses the to relay voice traffic to cellular users. We assume that a cellular user receives data either from the T directly or through, but not simultaneously. 4 The presence of the relay adds a flow conservation constraint, namely, the total rate that the provides to cellular (relayed) users must be the same as the total rate it receives from the T. With target rates R B and R A for T and users, respectively, the power allocation problem is then to imize the total number of T and users, namely, ρ B C + ρ A D, () {C,D} subject to the power and flow conservation constraints. Here, C and D are the regions of active cellular and one-sided non-cellular users, respectively, and denotes the size (i.e. Lebesgue measure) of the corresponding region. It can be easily shown that D is always a single connected interval. However, since each cellular user can receive data from either the T or, the cellular coverage set C can consist of two or more disjoint intervals C d and C r, where C d = i C d,i is the union of non-overlapping intervals corresponding to the direct (non-relayed) cellular users and similarly, C r corresponds to the relayed users. We assume that C r is one (connected) segment. In what follows, we consider two different techniques for coding the streams transmitted from the T to the for the relayed users. Joint coding or total flow rate conservation assumes that the T jointly encodes all of the data destined for relayed users, and transmits the resulting stream to the. The then decodes and demultiplexes the individual data flows. In that case, the total data rate from the T to the is equal to the sum rate of the flows to relayed users. In the second technique, individual flow rate conservation, data flows, which are to be relayed by the, are transmitted in parallel from the T to the. The then decodes the The total T power for users on both sides of the T is then P. 3 Note we are assuming that the coverage region is symmetric about the location. This will only be the case if D d, which implies P ρ AA d. Otherwise the coverage would be bounded on the left at zero (since the other users would be covered by the to the left of the T). 4 This is in contrast to the model for data service considered in [7] where the T and can simultaneously transmit to a user.
packets for each data flow separately. Hence the code rate used by the T to transfer the data to the is the same as that used by the to transmit to the relayed user. Total flow rate conservation generally requires less power for the T- link than individual flow rate conservation, but is more complex, since the relayed users must be multiplexed at the T and demultiplexed at the. III. TOTAL FLOW RATE CONSERVATION The optimization problem in this case is to imize () subject to ρ B P B (r) dr + γp C d ρ A P A (x) dx + D W BA log ( + P, (3) ρ B P A(x) dx P, (4) C r ) γp W BA d b = R B ρ B C r, (5) C = C r C d, C r C d =, γ [, ] (6) where W BA is the bandwidth between the T and the, and γ is a variable to be optimized over indicating the fraction of T power allocated to the relay channel. The power allocation for a relayed cellular user at distance x from the is P A (x) = N ( R B W A e W A ) x a = A x a. The pathloss exponent between the T and the is b, which can differ from the direct path-loss exponent a. That is, through proper placement of the, we may have b < a. Constraints (3) and (4) are the total power constraints for the T and, respectively, and (5) is the total flow rate conservation. To solve this optimization problem, we first assume that the allocates power αp for relayed cellular users, where α [, ], and subsequently optimize over α. With this assumption the preceding problem decomposes into two independent optimization problems for the T users and the users. Given α, the optimization problem is to imize the number of active non-cellular users. The solution ( α)p is ρ A D(α), where D(α) = ( ρ AA. The T optimization problem is to imize the coverage of the cellular users C(α). To solve the T optimization problem, we first write constraints on the interval containing relayed users, which follow from the power allocation αp and the T power allocation γp. Denote the interval of relayed users by [d+x, d+ x], where x d. Given x, from (4) x x A (x), where x A (x) is the value of x that satisfies x x ρ B P A (x) dx = αp. (7) Likewise, from (5), we have x x B (x), where x B (x) = x + W BA log ( + R B ρ B Combining these, we must have that γp W BA d b ). (8) x x(x) min{ x A (x), x B (x)}. (9) Given α and γ, to imize the total coverage this bound will be met with equality. The T allocates ( γ)p for the direct-path cellular users. Consider the two possible inequalities d+x ρ B P B (r) dr ( γ)p. () The left side of the inequality is the T power required to activate users in the interval (, d + x). This can be rewritten as ( a + x x = ( γ)p d, () ρ B B where [, d + x] is the largest interval of users which can be served directly by the T given the available power. For a given interval of relayed users [d + x, d + x], if x > x, then there is a gap between the cellular users served directly by the T and the cellular users served by the. The total cellular user coverage is then C(γ, α, x) = d + x + x x. () If x x, then there is no gap in the coverage of cellular users. That is, the T power needed to activate users in [, d+x] is less than ( γ)p. Hence, there will be a second interval of cellular users served directly by the T given by [d + x, C(γ, α, x)], where C(γ, α, x) = [ (d + x) + (d + x) (d + x) ] (3) is the total cellular coverage. Given α, the T optimization problem reduces to optimizing x, which determines the interval of relayed users, and γ, i.e., C(α) = C(α, γ, x). (4) γ x d Finally, the imum total number of users served by both the T and is given by ρ B C(α) + ρ A D(α). (5) α Although we are unable to obtain a closed-form solution to the preceding optimization problem, we can follow the approach in [7] to derive a closed-form upper bound on the total number of users served by the T and. Namely, we assume that the connection between the T and is free, i.e., does not require any expenditure of power. This could correspond to the situation in which there is a wired connection between the T and the. In that case, the intervals of users served by the T and have lengths ( a + C W L, = ρ B B P ( ρb A /a + + ( a + P ρ A A /a ) ρ B A /a, (6)
( a + D W L, = P ( ρb A + /a ρ ) A A /a A /a (7) provided that the distance d between the T and the is large enough so that there is no coverage overlap between the users directly served by the T and the relayed users. The relative increase in the total number of users due to the addition of the relay is then ρ B C W L, + ρ A D W L, ρ B C + ρ A D + = ( P P ( B /a A /a ( ) a ( + ρa B P ρ B A ) + ρa B /a a ρ B A /a P. (8) If the does not have a separate set of non-cellular users to serve (ρ A = ), then this ratio becomes ρ B C W L, + ρ A D W L, ( B P = + ρ B C + ρ A D A. P (9) Even with a wired connection between the T and, the preceding bound is not tight unless the distance d between the T and is large enough so that the intervals covered by T and do not overlap. With small d this bound can be improved by optimizing the total number of users served over x and α as before, assuming that the T- link requires no power consumption (since the T- link requires no power, we no longer have to optimize over γ). IV. INDIVIDUAL FLOW RATE CONSERVATION Next we consider individual flow rate conservation. The T individually encodes all data destined to the relayed users and transmits these individual streams in parallel to the. to the relayed users, we can again decompose the optimization problem () into two separate problems. The problem is still the same as the one for total flow rate conservation. The T problem for given Assuming the allocates power αp user density ρ B now becomes imizing the cellular user coverage C = C d + C r subject to ρ B P B (r) dr + ρ B P B (d) dx P, () C d C r C r ρ B P A(x) dx αp. () Note in this case, () combines the T power constraint and the total flow constraint; namely, P B (d) = Bd a is the power required from the T to deliver each relayed flow to the. As before, C d and C r are also constrained to be disjoint sets. To simplify our discussion, first we consider the case where b = a, i.e., the path-loss between the T and the is the same as the path-loss between the T and each cellular user. Then for a cellular user at distance r < d, the T will not receive any power savings by using the relay. Therefore, if the T power P is less than ρbb d, the T will not use the relay even when α >. In this case, the total coverage is C(α) = C = ( ρ P BB < d. However, if P > ρbb d, using the relay reduces the power needed by the T. Again, denote the interval of relayed users by [d + x, d + x]. Given x and α, x is again constrained by x x A (x), where x A (x) is the value of x that satisfies (7). The following proposition shows that when the relay is used, the optimal relay user interval starts at x =. Proposition : Given power αp and b = a, if > ρbb d, then the optimal relay user interval is P given by [d, d + x], where x x A (). We omit the proof to save space. Note the constraint x x A () may not be tight if the T does not have enough power to relay sufficient traffic to utilize all the allocated power. In this case, x can be calculated from (). Next, we consider the case where the path-loss between the T and the is /r b, where b < a. In this case, the power required by the T to transmit directly to a user at distance r at rate R B is P B (r) = Br a and the power required by the T to transmit a relayed packet to the for this user is P B (d) = Bd b. Hence, the T can save the power by using the relay if r > d = d b/a. When the T power is small enough so that it can only serve users at distance less than d, it will not usethe relay. As its power increases, it will begin to serve the relayedusers. Initially, the set of relayed users will lie in a symmetric interval around the. Given power, let [d, d + ] be the largest symmetric interval of relyed users that can be supported around the, i.e., = ( αp ρ BA. As the T power further increases, the resulting coverage will follow two possible evolutions, depending on whether or not d d. Two examples of this evolution are shown in Figure. On the left is the case where d d. In this case the relay always serves a symmetric set of cellular users, and as the T power increases, it will eventually serve some users to the right of the relayed set. On the right is the case where d < d. In this case, when there is sufficient relayed traffic, the relayed set is not symmetric, but extends further in the positive direction. This is because the users to the left are served directly by the T. For the former case (d d ), the coverage as a function of the T power is given in table I; we have derived a similar table for the latter case, but omit it due to space considerations. αp In Table I, P (critical) represents the minimum power such that when P > P (critical), the T serves users to the right of the relayed interval. In this case, the cellular coverage as a function of x becomes [ a + C(α, x) = ρ B B P + (d + x A (x)) (a + )d b (x A (x) x) (d + x) ], () where x is constrained to be no smaller than min{, d d}. Therefore, the optimal cellular coverage for a given α is C(α) = C(α, min{x, ẋ}), (3)
TABLE I CELLULAR COVERAGE AS A FUNCTION OF P WHEN b < a AND d d. A B C D P T Power P < P ρ BB d P ρ BB d C(α) = C = T coverage C(α) ρ B B P + ρ B Bd b C(α) = d + P ρ B B ρ B Bd b (critical) P d n `P ρ B B ρ BBd b o + = ρ BB (d ) + ρ B Bd b C(α) = P > P (critical) C(α) See eq. () and (3) Fig.. Cellular user coverage as a function of P. On the left is the case where d d and on the right the case where d < d. where x and ẋ are obtained by solving C(α,x) x x=x = and C(α, ẋ) = d + x A (ẋ), respectively. As in Section III, given C(α), the optimal number of cellular and users can then be found by searching for the optimal α [, ]. V. NUMERICAL RESULTS We next give some numerical results to illustrate the gains from relaying under the two flow rate conservation constraints. First, we consider total flow rate conservation. Figure 3 shows the ratio of the total number of users (T+) with relaying to those without relaying as a function of distance d between the T and the. Two different path loss scenarios are considered: a = 4, b = 3 and a = b = 4, with the remaining parameters indicated in the caption. For each case the ratio of the total number of cellular users is also shown (T only). Note there is clearly an optimal location for imizing each of these ratios. When the path-loss exponent b between the T and the is the same as the path-loss exponent a between the T and each user, the increase in the total number of active users with the relay is very small (less than 7% when a = b = 4). As the path-loss between the T and the improves, however, the ratio of the total number of users increases more significantly. The relay achieve more than a 7.8% increase in total number of users served when a = 4 and b = 3. In this case the cellular users served increase by more than 34.9%. (This is offset by a decrease in the number of users.) It follows that the increase in the total number of cellular user is larger than 34.9% when the only serves as a relay (i.e., ρ A = ). The upper bound we obtained by assuming the wired connection between the T and the for d = 4.5 is. (.3 for d = ), and the loss due to the power consumption on the back-haul link between the T and the is small in this example. This is partially because we only consider one. In a two dimensional model that has more s, the back-haul wireless communication loss should be larger. Fig. 4 shows the ratio of total number of users as a when a = 4 and b = 3, assuming the is located d = 4.5 from the T. It can be seen that there is an optimal power which imizes this gain; in this case this is approximately the power used in Figure 3. Figures 5 and 6 show the analogous results under the individual flow conservation constraint. These figures are very similar to those with total flow conservation. Interestingly, the imum gains occur at similar power levels and locations in the two cases. In this case, the gains are slightly lower due to the increased power needed for the backhaul link. For example, in the case of a = 4, b = 3, the imum percentage increase in the total number of users served is about 6.6% (v.s. 7.8% in the total flow case). In these examples the gain from joint encoding is very small, and may not warrant the additional implementation complexity required. function of P VI. CONCLUSION We have studied the total number of downlink voice users in a one dimensional model of a single cell site when two s located symmetrically around a T are available as relays. In addition to relay traffic, we assumed that the has its own customers to serve. We studied two different flow conservation schemes which correspond to whether or not the relay traffic is multiplexed when sent to the. Our results show that when the sets of active users are optimized, the total number of cellular users increases significantly under both flow rate conservation schemes, but at the cost of reducing the users. This corresponds to a coverage extension of the cellular network. Although the relay schemes considered in this paper increase the coverage of the T, it is only possible when the is cooperative. Relay cooperation can be achieved either
.4.35 T+ (a=4,b=3) T only (a=4,b=3) T+ (a=4,b=4) T only (a=4,b=3).4.35 T+ (a=4,b=3) T only (a=4,b=3).3.3.5..5.5. upper bound when P =5..5.5..5 3 3.5 4 4.5 5 5.5 6 Distance d.5 3 35 4 45 5 55 6 65 7 T power Fig. 3. Ratio of the total number of users served under total flow rate conservation as a function of distance d. Here P = 5, P = 5, W B = W A =, W BA = 3, R B = R A =, ρ B = ρ A =. Fig. 6. Ratio of the total number of users served under individual flow conservation as a function of P P. Here, = 5, W N B = W A =, R B = R A =, ρ B = ρ A =, d = 4.4..35.3.5..5 upper bound when P =5 T+ (a=4,b=3) T only (a=4,b=3). 3 35 4 45 5 55 6 65 7 T power Fig. 4. Ratio of the total number of users served under total flow rate conservation as a function of P. Here P = 5, W N B = W A =, W BA = 3, R B = R A =, ρ B = ρ A =, d = 4.5..4.35.3.5..5..5 T+ (a=4,b=3) T only (a=4,b=3) data3 (a=4,b=4) data4 (a=4,b=4) when the T owns the or when the T and have an agreement on usage of resources, for example, through a bargaining process. Bargaining between two non-cooperative agents, such as the T and the, is an interesting topic for future work. REFERENCES [] R. Pabst et al, Relay-Based Deployment Concepts for Wireless and Mobile Broadband Radio, IEEE Commun. Mag., vol. 4, pp. 8-89, Sept. 4. [] Z. Dawy, S. Davidovič, and I. Oikonomidis, Coverage and Capacity Enhancement of CDMA Cellular System via Multihop Transmission, Proc. IEEE GLOBECOM 3, pp. 47-5, Dec. 3. [3] S. Mukherjee and H. Viswanathan, Resource Allocation Strategies for Linear Symmetric Wireless Networks with Relays, Proc. IEEE ICC, pp. 366-37, April. [4] E. Yanmaz and O. K. Tonguz, Dynamic Load Balancing and Sharing Performance of Integrated Wireless Networks, IEEE J. Select. Areas Commun., vol., pp. 86-87, June 4. [5] H. Wu, C. Qiao, S. De, and O. K. Tonguz, Integrated Cellular and Ad-Hoc Relay Systems: icar, IEEE J. Select. Areas Commun.,vol. 9, pp. 5-5, Oct.. [6] H. Wei and R. D. Gitlin, WWAN/WLAN Two-hop-relay Architecture for Capacity Enhancement, Proc. IEEE WCNC 4, vol., pp. 5-3, March 4. [7] J. Bae, R. Berry, and M. L. Honig, Power Allocation, Rate, and Coverage for Relay-Assisted Downlink Data Transmission, Proc. IEEE ICC 6, June. 6, to appear. [8] P8.6j: PAR developed by 8.6 s Working Group Study Group on Mobile Multihop Relay, http://ieee8.org/6/docs/sg/mmr/86mmr- 6.pdf, Jan. 6. [9] Wireless World Initiative New Radio (WINNER), see, e.g., https://www.ist-winner.org. [] C. Zhou, P. Zhang, M. L. Honig, and S. Jordan, Two-Cell Power Allocation for Downlink CDMA, IEEE Transactions on Wireless Communications, vol. 3, no. 6, pp. 56-66, Nov. 4..5 3 3.5 4 4.5 5 5.5 6 Distance d Fig. 5. Ratio of the total number of users served under individual flow conservation as a function of distance d. Here P = 5, P = 5, W B = W A =, R B = R A =, ρ B = ρ A =.