Hop capacity balancing in OFDMA relay networks

Computer Networks 72 (2014) 33–44

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Hop capacity balancing in OFDMA relay networks Dong Geun Jeong a,⇑, Jeong Ae Han b,1, Wha Sook Jeon b,1 a b

Department of Electronics Engineering, Hankuk University of Foreign Studies, Yongin-si, Kyonggi-do 449-791, Republic of Korea Department of Computer Science and Engineering, Seoul National University, Seoul 151-742, Republic of Korea

a r t i c l e

i n f o

Article history: Received 2 November 2013 Received in revised form 18 June 2014 Accepted 13 July 2014 Available online 22 July 2014 Keywords: Relay networks OFDMA Scheduling Hop capacity balancing

a b s t r a c t A novel scheduling scheme for the orthogonal frequency division multiple access (OFDMA) relay systems is proposed. In the relay systems, downlink frame is divided into two slots for two-hop transmission, corresponding to the base station (BS)–relay station (RS) and the RS–mobile station (MS) transmissions, respectively. If the capacities of two hops are not equal, cell throughput can be degraded by the bottleneck hop and radio resource can be underutilized on the non-bottleneck hop. The proposed scheme aims at increasing cell throughput while guaranteeing the minimum rate requirements of MSs, by balancing the capacities of two hops. We formulate a throughput maximization problem which satisfies the minimum rate requirements of MSs, by adjusting the lengths of both slots and allocating the subchannels appropriately. To alleviate the high computational complexity of the problem, we suggest a practical scheduling scheme that uses linear programming relaxation and postprocessing algorithm. The simulation results show that the proposed scheme guarantees the minimum rate requirements of real-time MSs more properly and provides higher cell throughput compared to the existing hop capacity balancing schemes for OFDMA relay systems. Ó 2014 Elsevier B.V. All rights reserved.

1. Introduction Orthogonal frequency division multiple access (OFDMA) system has been considered as one of the most promising technologies to support the increasing demands for wireless and mobile communication services since it can offer high transmission rate and high scheduling flexibility on time and frequency dimensions [1,2]. However, even with the advanced OFDMA technology, a cellular system often fails to guarantee the quality-of-service (QoS) of a mobile station (MS) when the MS is located at the cell edge and its channel from the base station (BS) is of inferior quality. Although this problem can be solved by installing additional BSs at the areas, the BS installation ⇑ Corresponding author. Tel.: +82 31 330 4169; fax: +82 31 330 4120. E-mail addresses: [email protected] (D.G. Jeong), [email protected] (W.S. Jeon). 1 Tel.: +82 2 880 1839; fax: +82 2 880 1841. http://dx.doi.org/10.1016/j.comnet.2014.07.002 1389-1286/Ó 2014 Elsevier B.V. All rights reserved.

requires high cost and the site survey and location is not easy in practice. Among the other techniques that can compensate this poor channel quality, we focus on the relaying [3,4], since the relaying system is usually very small, inexpensive, and easy to implement. On the downlink, a relay station (RS) receives data from BS and forwards it to the destination MS. By using relaying, since the long link between BS and MS is split into two short links, the channel quality can be heightened. Accordingly, the QoS of MS can be improved and the cell throughput is able to be increased with an appropriate resource management. Various downlink scheduling algorithms for OFDMA relay networks have been proposed recently [5–11]. In these schemes, for guaranteeing RSs to finish the data reception before they forward data to MSs, the downlink OFDMA frame is divided into two slots, where the BS–RS and RS–MS transmissions are carried out at the first slot and at the second slot, respectively. When using this two-hop transmission, the amount of data that an MS is capable of

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D.G. Jeong et al. / Computer Networks 72 (2014) 33–44

receiving for a frame is restricted to the minimum of capacities of the first (BS–RS) hop and the second (RS–MS) hop. Thus, the bottleneck hop may cause cell throughput degradation and this, in turn, incurs underutilization of the scarce radio resource on the non-bottleneck hop. This type of throughput loss can be minimized when two hops have equal capacity, i.e., when the hop capacities are balanced. In this paper, we investigate the ‘‘hop capacity balancing’’. The hop capacity balancing can be achieved in three approaches: (a) efficient subchannel assignment between two hops (frequency-domain approach); (b) suitable transmission power allocation among the transmitters of two hops (power-domain approach); and/or (c) appropriate slot length (i.e., transmission time) differentiation between the first and second slots in a frame (time-domain approach). The scheduling schemes in [5,6] take the first approach. For example, the scheduling scheme proposed in [5] searches the best subchannel pairs for BS–RS and RS–MS transmission. However, since this scheme is unable to guarantee the equal capacity for both hops, some wireless resources may still be wasted. In fact, under the constraint that the amount of data transmitted at the second hop is restricted by that transmitted at the first hop, the hop capacity balancing is capable of being hardly accomplished by using just the subchannel assignment. This method, however, can be exploited adjunctively when other methods are used. The scheduling schemes proposed in [7,8] allocate different transmission power for each of two hops in order to balance the capacities of two hops. If the condition of RS–MS channel is much poorer than that of the BS–RS channel, the RS uses considerably large transmission power to compensate the low channel quality of RS–MS hop. However, since RSs are usually located close to the cell edge, the increasing RS transmission power in this case will cause high interference to the neighbor cells. This interference may disturb the transmission in the neighbor cells and, in turn, cause the degradation of throughput in whole cellular system. The scheduling schemes suggested in [8–11], adjust the transmission times of the first and second slots. Although [9] has suggested a good analytic model for time-domain approach, it considers the relay system with only one MS, while a number of MSs are served at the same time in practical OFDMA systems. In [8,10], the first and second slots are exclusively used by BS and RSs, respectively. However, since the BS transmits data not only to RSs but also to MSs in a direct manner, the efficiency of radio resource can be improved by heightening the flexibility of BS in utilizing slots. In [11], we have designed a transmission time adjustment scheme that permits BS to transmit data in the second slot as well as in the first slot, which increases the flexibility of BS in resource usage. Nevertheless, Ref. [11] uses only a heuristic design method and the resulting scheduling algorithm can be improved highly by using another elaborate design approach. The hop capacity balancing in time-domain does not have the shortcomings of other approaches. Moreover, this approach can harmonize well with the time division duplex (TDD) mode of OFDMA operation since TDD is also a time-domain resource management method (we do not

treat TDD issue here). In this paper, we investigate the hop capacity balancing with a time-domain approach, together with a frequency-domain approach adjunctively. We design a scheduling scheme such that the BS is a unique transmitter to RSs and MSs in the first slot, while the BS and RSs are the transmitters to the respective MSs in the second slot. Specifically, we propose a scheduling problem that decides the lengths of two slots for given frame length and allocates subchannels to MSs on a per OFDM symbol basis, for maximizing the cell throughput while guaranteeing QoS and taking multiple MSs into account. The remainder of the paper is organized as follows. The system model under consideration is described in Section 2. The proposed scheduling scheme is presented in Section 3. In Section 4, we discuss the performance of the proposed scheme with simulation results. The paper is concluded with Section 5. 2. System model We consider the downlink of an OFDMA-TDD system which consists of a BS, K MSs, and N RSs, as shown in Fig. 1. The total bandwidth B is divided into M subchannels with equal bandwidth Bs :¼ B=M (see Fig. 2). The length of a downlink frame is denoted by T f and each subchannel has L OFDM symbols in a downlink frame. The resource allocation unit in time can be one or more fixed number of OFDM symbol times. In this paper, for the convenience in description, we assume that an OFDM symbol time is the resource allocation unit in time. The downlink frame is again divided into two slots for supporting the twohop transmission where l denotes the number of OFDM symbols in the first slot. Both BS and RSs are transmitters. The RSs transmit data only at the second slot, whereas the BS can transmit data at the second slot as well as at the first slot. We denote the BS as the 0th transmitter and the RS n as the nth transmitter (n ¼ 1; . . . ; N). Each MS in the cell is served by either BS or RS according to the channel condition. An MS served by BS is referred to as a ‘‘direct’’ MS, whereas an MS whose data is relayed by RS is regarded as an ‘‘indirect’’ MS. We assume that the serving transmitter for each MS is determined a priori, by using an appropriate transmitter selection scheme which we do not treat in this paper, as in [12]. For the relay selection problems, one can refer to other literatures (e.g., [13,14]). We consider the minimum

RS3 Direct MS RS2 BS

Indirect MS

RS1

Fig. 1. Architecture of OFDMA relay system.

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3. Proposed scheduling scheme We use an optimization technique to find out the most ðiÞ appropriate l and aj;m ’s. We first formulate a throughput maximization problem, which is a non-linear integer programming (LP) problem, as will be described below. To alleviate the high computational complexity of this problem, we suggest a practical scheduling scheme that uses LP relaxation and postprocessing algorithm. 3.1. Scheduling problem Fig. 2. Downlink frame structure of OFDMA relay system under consideration.

rate requirement of each link for reflecting various QoS requirements of MSs. Note that the delay constraint of real time traffic can be converted into a minimum rate requirement [15]. Fairness for best effort service can be also improved by guaranteeing a minimum rate. At the first slot, the BS sends the RS n the aggregated data which is destined to the indirect MSs served by the RS n. After receiving the data, the RS n classifies the data according to the destination and forwards each of them to the corresponding indirect MS at the second slot. As shown from Fig. 2, the direct MSs can receive data from BS at any slot. Since RSs receive data from the BS, there can be totally K þ N wireless links in the cell. We index the wireless links as follows. At first, the link between MS k and its serving transmitter is indexed by 1; 2; . . . ; K, and the link from the BS to the RS n is indexed by K þ 1; K þ 2; . . . ; K þ N. In addition, let Sn denote the set of the links where the transmitter n sends data, and let uj denote the transmitter on the link j. We consider a system where the transmission powers of BS and each RS on a subchannel are predetermined. It is noted that the transmission power of BS is usually fixed in the practical cellular systems such as IEEE 802.16 and the adaptive modulation and coding (AMC) scheme without power control is used to cope with the channel variation. Let Rj;m denote the achievable transmission rate of the link j when using the subchannel m. In the practical OFDMA system, Rj;m can be estimated and reported (to BS) by the corresponding MS. Note that the Rj;m ’s of indirect MSs are first collected to their serving RSs and then are at once forwarded to the BS by RSs. Since the channel between BS and RS is usually much better than that between BS and MS or RS and MS, this forwarding overhead may not be a problem. The number of OFDMA symbols in the first slot (i.e., the length of the first slot), l, is adjusted for balancing the capacity of each slot. Since each slot consists of an integer multiple of OFDM symbols, l has an integer value between ðiÞ

1 and L  1. Let aj;m indicate the portion of the subchannel m allocated to the link j at the ith slot ði ¼ 1; 2Þ. Let V ðiÞ denote ðiÞ

the set of feasible values that aj;m can have. Then, V ð1Þ ¼  1   1  ; 1 and V ð2Þ ¼ 0; Ll ; . . . ; Ll1 ; 1 . Now, our 0; l ; . . . ; l1 l Ll ðiÞ

scheduling problem is to decide l and aj;m ’s every frame.

To maximize the cell throughput while guaranteeing the minimum rate requirements of users, the proposed scheme determines the length of the first slot l and the subchannel ðiÞ allocation variables aj;m of each slot. Since the RSs do not ð2Þ receive but transmit data at the second slot, aj;m ¼ 0, for ð1Þ K þ 1 6 j 6 K þ N; 8m. Moreover, aj;m ¼ 0; 8j R S0 ; 8m, since the only transmitter at the first slot is the BS. When considering that each OFDM symbol is allocated to at most P PK ð2Þ ð1Þ one link, KþN j¼1 aj;m 6 1 and j¼1 aj;m 6 1; 8m. The proposed scheme aims at guaranteeing the minimum rate requirement of each MS. Let Dj denote the minimum rate requirement of the link j where the MS j is the receiver of the link. If Dj is not satisfied at a certain frame, it is said that an outage occurs. To avoid the outage, the achievable transmission rate of the link j with its allocated resources should be greater than or equal to Dj . That is, M  X l m¼1

L

ð1Þ

aj;m þ

 L  l ð2Þ aj;m Rj;m P Dj ; L

1 6 j 6 K:

ð1Þ

An RS without data buffering cannot forward data more than it received from the BS. Thus,

Kn :¼

M X l

L m¼1

ð1Þ

aKþn;m RKþn;m 

M XX Ll j2Sn m¼1

L

ð2Þ

aj;m Rj;m P 0; 1 6 n 6 N; ð2Þ

where Kn is the data rate difference between receiving and forwarding of RS n, which should be minimized for maximizing the channel utilization. To minimize this difference, ðiÞ both the subchannel allocation aj;m and the length of the first slot l should be determined appropriately since they directly affect the achievable throughput of RSs and MSs. ðiÞ To find the optimal l and aj;m , we formulate the following throughput maximization problem with above constraints.

 L  l ð2Þ aj;m Rj;m ; L L j¼1 m¼1  M  X l ð1Þ L  l ð2Þ aj;m þ aj;m Rj;m P Dj ; s:t: L L m¼1

max

K X M  X l

ð1Þ

aj;m þ

Kn P 0;

1 6 n 6 N;

ð1Þ aj;m ð2Þ aj;m

¼ 0;

8j R S0 ;

¼ 0;

K þ 1 6 j 6 K þ N;

KþN X ð1Þ aj;m 6 1;

8m;

j¼1 K X ð2Þ aj;m 6 1; j¼1

8m:

1 6 j 6 K;

8m;

ð3Þ l8m;

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The objective function in (3) is composed of a multipliðiÞ cation of l and aj;m , which is a non-linear function. Also, l ðiÞ has an integer value in ½1; L  1 and aj;m should be in V ðiÞ which is a finite set. Therefore, the maximization problem (3) can be translated into a non-linear integer programming problem whose complexity increases intractably as the number of MSs and subchannels increases [16, Chapter 1.4]. To reduce the complexity of this optimization probðiÞ lem, we relax the constraint on aj;m so that it can have any real value between 0 and 1. Hereafter, we call the optimization problem with this relaxation the ‘‘relaxed problem’’. The relaxed problem is still a non-linear integer programming problem because of l. However, if the value of l is given, the above maximization problem can be translated into an LP problem that efficiently determines the optimal ðiÞ (under relaxation) aj;m with the polynomial complexity [17]. Since there are several widely known algorithms to solve this LP problem [17], we do not explain the specific procedure for solving the relaxed problem. To find out the optimal l, an iteration method can be used. That is, by repeatedly computing (3) with candidate l’s, two optimizaðiÞ tion variables l and aj;m can be jointly optimized. More specifically, the proposed scheme conducts the following steps for the resource allocation. First, the proposed scheme gets the solution of the relaxed problem for optimal l with Algorithm 1 in Section 3.2. The optimal l obtained from Algorithm 1, which is based on the relaxaðiÞ tion of the constraint on aj;m , cannot be directly applicable to the practical OFDMA system. Thus, in Section 3.3, we will suggest a postprocessing algorithm for obtaining a practical solution. From now on, we describe each step of the proposed scheme in detail.

condition within a cell, i.e., the number of MSs, the minimum rate requirements of MSs, and the serving transmitters  of MSs, does not largely vary over frame, it is expected that lG of current frame is not greatly different from that of the previous frame. In Algorithm 1, if U G ðlÞ is greater than or equal to  U G ðl  1Þ and U G ðl þ 1Þ, this l becomes lG which can achieve  the maximum throughput U G with given G. Otherwise, i.e., if either U G ðl  1Þ or U G ðl þ 1Þ is higher than U G ðlÞ, further search should be proceeded in the direction which can provide the higher throughput. That is, the above test is carried out for l  1 if U G ðl  1Þ > U G ðlÞ; and for l þ 1 if U G ðl þ 1Þ > U G ðlÞ. On the other hand, when all of U G ðlÞ; U G ðl  1Þ, and U G ðl þ 1Þ are 1, the further search is proceeded to l  1 until l reaches to 1. If U G ðlÞ’s for all l 6 ^l  are 1, the search for finding lG is continued from ^l þ 1 to L  1. For simplicity in algorithm description, we set U G ð0Þ ¼ U G ðLÞ ¼ 1 as boundary condition. The detailed  procedure for finding lG and U G is presented in Algorithm 1. 

Algorithm 1. Searching for lG

3.2. Fast search for slot length Let G be a set of MSs that participate in the scheduling. In usual underload condition, G is the set of all MSs. Let U G ðlÞ be the upper-bound throughput obtained by solving the relaxed problem (3) for given l and G. And, let   lG : argmax U G ðlÞ and U G : U G ðlG Þ. 16l6L1

For given l, the relaxed problem may not have a feasible solution depending on the channel quality of each MS. For example, when the length of the second slot (L  l) is too short to satisfy Dj of an indirect link j with poor channel quality even though all subchannels at the second slot are allocated to the link j, the relaxed problem of (3) does not have a feasible solution for given l. In this case, U G ðlÞ is set to 1.  To search lG more quickly, we exploit the following property on U G ðlÞ.

If a feasible scheduling solution is found by Algorithm 1  (i.e. U G > 1), then U G and lG respectively become U  and  l that are the final solution of the relaxed problem. The ðiÞ subchannel allocation variables aj;m are also determined together with U G in Algorithm 1.



Lemma 1. U G ðlÞ monotonically increases for l 6 lG , and  monotonically decreases for lG < l. Proof. See Appendix A.

h

Based on the observation of Lemma 1, we design  Algorithm 1 that searches U G and lG with the reduced computational time. The algorithm starts from ^l, which is  lG of the previous frame. When considering that the traffic

Remark. Note that a solution does not always exist. For example, there can be no solution when the channel qualities of some links are too bad to satisfy the minimum rate requirements of all MSs in G with any feasible l. In this  case, two policies are recommended: one is to use l of previous frame; the other is to exclude the MS with the worst channel quality and then try to find the solution again. When the second policy is adopted, the channel quality of

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D.G. Jeong et al. / Computer Networks 72 (2014) 33–44

each MS should be examined for finding the worst quality MS. The following is one example for evaluating the channel quality of MS k. When C k denotes the maximum achievable throughput of MS k when all subchannels are assigned to the links for the MS k, according to [11],

Ck ¼

8 M X > > > Rk;m ; <

if uk ¼ 0;   PM  P M > RKþu ;m  R > k > : Pm¼1 PMm¼1 k;m ; otherwise: M

ðiÞ

ð4Þ

m¼1 

R þ m¼1 Kþuk ;m

3.3. Postprocessing of the proposed scheduling scheme ðiÞ

Recall that we have relaxed the constraint that aj;m 2 V ðiÞ for reducing the complexity of the optimization problem (3). Thus, the result of Algorithm 1 cannot be directly applicable to a practical OFDMA system and a postprocessðiÞ ing on aj;m is required. Now, we suggest a postprocessing algorithm (Algorithm 2) that adjusts the allocation variable ðiÞ aj;m to be in the set V ðiÞ . The postprocessing is carried out for each slot separately. We apply it to the second slot first, since the minimum rate requirement of an indirect MS can be examined individually for the corresponding RS–MS link at the second slot, whereas the minimum rate requirements of indirect MSs are together aggregated to that of the link for their serving RS at the first slot. Let jðiÞ denote the ratio of the ith slot  length to the downlink frame length. That is, jð1Þ ¼ lL and 

jð2Þ ¼ LlL . Let fðiÞ j be the minimum rate requirement of the link j on the ith slot. Since indirect MSs are scheduled only ð2Þ

for the indirect MS is equal to Dj . In

addition, since each RS should support minimum rates of P ð1Þ its all subordinate MSs, fKþn of the RS n is set to j2Sn \G Dj . On the other hand, when assuming that each direct MS requires the minimum rate evenly for the whole downlink frame, the minimum rate requirement of direct MS j at the second slot can be set to jð2Þ Dj . It is noted that the rate allocated to the direct MS j at the second slot according to P ð2Þ ð2Þ Algorithm 1 is M m¼1 j aj;m Rj;m . Since a direct MS is served not only at the second slot but also at the first slot, if the allocated rate of direct MS j at the second slot is smaller than jð2Þ Dj , we adjust its minimum rate requirement at the secP ð2Þ ð2Þ ond slot to M m¼1 j aj;m Rj;m , for simplifying postprocessing. Accordingly, before starting the second-slot postprocessing,  P ð2Þ ð2Þ ð2Þ we set fj ¼ min jð2Þ Dj ; M m¼1 j aj;m Rj;m . After postprocessing, the allocated rate of direct MS j at the second slot PM ð2Þ ð2Þ ð2Þ is finalized as m¼1 j aj;m Rj;m , using the adjusted aj;m ’s. Thus, the minimum rate requirement of the direct MS j at  P ð1Þ ð2Þ ð2Þ the first slot is set as fj ¼ max 0; Dj  M m¼1 j aj;m Rj;m . The proposed postprocessing algorithm is composed of ðiÞ two steps. First, each aj;m is adjusted to one of the values in ðiÞ V . However, after this adjustment, the minimum rate

ðiÞ

than or equal to aj;m . It is reasonable to adjust aj;m to   ðiÞ ðiÞ v ðiÞ aðiÞ aj;m is not smaller than j;m . However, because v ðiÞ

R m¼1 k;m

The MS k with the lowest C k is selected as the worst quality MS. After excluding the worst quality MS from G, the BS  tries to find l again by using Algorithm 1. If there still exists no solution with the reduced G, the BS can again choose one of two policies in consideration of computation time.

in the second slot, fj

constraints of some MSs may not be satisfied. Accordingly, the second step of postprocessing tries to re-allocate some subchannels for compensating the resource loss of some MSs. ðiÞ At the first step, aj;m is adjusted to one of the values in V ðiÞ .  ðiÞ Let v ðiÞ aj;m be the smallest value in V ðiÞ which is greater

aj;m , some links of which adjustment is scheduled later may be allocated less resources compared to their requireðiÞ ments fj and this may result in additional outages. For the PM ðiÞ ðiÞ ðiÞ ðiÞ link j at the ith slot, let F j :¼ m¼1 j aj;m Rj;m  fj . And, let zm be the remaining resource of the subchannel m. In order to reduce the occurrence of these additional outage events, the first step is applied in order of the links with ðiÞ ðiÞ smaller F j . For the selected link ej, a is adjusted to ej;m     ðiÞ ðiÞ min v ðiÞ a ; zm since a cannot exceed zm . ej;m ej;m Algorithm 2. Postprocessing for the ith slot

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D.G. Jeong et al. / Computer Networks 72 (2014) 33–44

For reducing the additional outage events further, as the second step, the postprocessing scheme reallocates resource to the links of which minimum rate requirements have not been satisfied still, even though the cell throughput ðiÞ

can be lowered due to this reallocation. Note that F j is the ðiÞ

rate requirement subtracted from the allocated rate. F j

smaller than zero means that the minimum rate requirement of link j has been not satisfied yet. Thus, the link ej with ðiÞ

ðiÞ

the smallest F is selected. If F < 0, in order to meet the ej ej ðiÞ minimum rate requirement f of the link ej, each subchannel ej e in decreasing order of the channel quality Re is checked m e j; m whether any other link can yield its allocated resource to the ðiÞ link ej. Among the links with surplus resource (i.e. F > 0), j

the link being able to yield most resource on subchannel e is selected as a donor (the link h in line 20 of m Algorithm 2). It is noted that the maximum resource e is that any link j can yield on the subchannel m    ðiÞ ðiÞ ðiÞ ðiÞ min a ; x F j =Rj; m e , where x ðxÞ means the biggest e j; m value in V ðiÞ which is smaller than or equal to x. Then, the amount of reallocating resource is determined taking into account both the amount of additional resource required by the link ej and the maximum resource that the donor can give (see line 21 of Algorithm 2). ðiÞ

This resource reallocation is proceeded until fj ’s of all links are satisfied or there remains no resource that can be reallocated. Note that only the links having surplus resource can become a donor. As a result, it is obvious that the number of links that suffer from outage is decreased through the second post processing step.

4. Performance 4.1. Simulation model We evaluate the performance of the proposed scheme by using simulation. We consider both the single circular cell model with radius of 500 m and the multicell model being composed of seven hexagonal cells with side-length of 500 m. It is assumed that there are one BS and three RSs in a cell. The BS is located at cell center. The RSs are spaced equi-angularly and their distances from BS are 250 m. The downlink frame length is set to T f ¼ 5 ms and the number of OFDM symbols within a downlink frame is L ¼ 24. In addition, the total bandwidth is B ¼ 10 MHz and the number of subchannels is set to M ¼ 10. Thus, Bs ¼ 1 MHz. It is noted that these parameter values are chosen nominally and do not reflect a specific OFDMA system. Nevertheless, since the number of resource units that should be allocated for each scheduling is comparable with those in practical systems (i.e., several hundreds units per 5 ms), the simulation results can demonstrate well the advantages and disadvantages of the proposed scheme. At the start of simulation run, the MSs are distributed uniformly over its serving cell. The moving speed of an MS is determined by a uniform distribution with mean of

60 km/h and standard deviation of 10 km/h and its moving direction is determined between 0 and 2p uniformly. The moving speed and direction of an MS are updated with a random interval distributed uniformly between 0 and 10 s. To maintain the fixed number of MSs in a cell during a simulation run, it is assumed that, when an MS moves out of its serving cell, it moves back into the cell from the opposite side (i.e., wrapping around). Any serving transmitter selection (i.e., mode selection between direct and indirect transmissions) scheme can be applied to the proposed scheme. However, it is assumed for the simplicity that, for MS k, the transmitter that can provide the highest maximum achievable throughput C k is selected as the serving transmitter. Since an MS can gather the channel quality information from BS and RSs by listening the downlink and also that from BS to RSs by overhearing the channel quality reports of each RS, C k for each transmitter n can be easily measured by each MS. Therefore, the serving transmitter may be selected by MS without severe overhead. The MSs are categorized into two groups according to their service classes. The half of MSs are for the real-time (RT) services. Unless noted otherwise, the minimum rate requirement of RT MSs is set to 200 kb/s. The other half of MSs are non-real-time (NRT) MSs not having strict minimum rate requirement. Thus, the outage probability is measured only for RT MSs, which is defined as the average ratio of the number of outage MSs to the total number of RT MSs. Note that an outage MS means the RT MS whose minimum rate requirement is not satisfied. In addition, the BS is assumed to have an infinite backlog for each MS. The transmission powers of BS and RS per subchannel are set to P 0;m ¼ 100 mW and Pn;m ¼ 50 mW for all m, respectively, and the noise spectral density is N 0 ¼ 163 dBm/Hz. We consider path loss, log-normal shadowing, and Rayleigh fading for a channel model. When the distance between two stations is denoted by d, the path loss is modeled as 20log10 ð4pf c =cÞ þ 10elog10 d (dB), where f c is the center frequency of the channel and is set to 2.4 GHz in simulation, c is the speed of light, and e is the path loss exponent [18, Chapter 2.6]. Since RSs do not have mobility and are usually located where the line-of-sight from the BS is kept, for BS–RS link, shadowing is not considered and the path loss exponent of 3 is used, like in [5]. On the other hand, for BS–MS and RS–MS links, lognormal shadowing with standard deviation 8 dB and the path loss with exponent 4 are applied. In this paper, the achievable transmission rate of MSs is assumed as Shannon capacity. When g j;m denotes the channel gain of the link j on the subchannel m, with the single cell model,  g j;m Pu ;m Rj;m ¼ Bs log 1 þ N0 Bsj . Rj;m in multi-cell model will be explained in the Section 4.3. 4.2. Schemes for performance comparison To demonstrate the performance of the proposed scheme, we compare some variants of the proposed scheme with other existing schemes. Hereafter, we call the proposed scheme the ‘‘adaptive slot length (ASL) scheme.’’

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D.G. Jeong et al. / Computer Networks 72 (2014) 33–44 30

Throughput (Mb/s)

In order to assess the efficiency of the proposed postprocessing algorithm, we compare the performance of ASL scheme using postprocessing (i.e., the proposed scheme) with that of an ideal but unrealistic ASL scheme where the subchannel allocation variables are allowed to have any real value between 0 and 1, thus the postprocessing is not needed. Therefore, this ideal ASL scheme can be used for providing an upper-bound of the performance of the standard integer programming approach using (3). In addition, to evaluate the performance improvement achieved by determining the slot length adaptively, the proposed ASL scheme is compared with the scheme using a fixed slot length (FSL), called the ‘‘FSL scheme.’’ Another  variant is the ‘‘periodic ASL scheme,’’ which determines l not every frame but periodically. The periodic ASL scheme is an extension of the original ASL scheme, since the latter can be interpreted as the former with the period of just one frame. In the simulation, we set the period of the periodic ASL scheme to 20 frames. The periodic ASL scheme with period of several frames can be used for reducing computational overhead. Note that the proposed postprocessing algorithm is also applied to the FSL and periodic ASL schemes. We also compare the performance of the proposed ASL scheme with those of the following two existing schemes, of which explicit design objectives is hop balancing in relay networks. One is the transmission ‘‘power adaptation’’ scheme with equal slot length (i.e., l ¼ 12 in simulation), suggested in [7]. This scheme adjusts transmission power of each transmitter, with the maximum power constraint on the total transmission powers of BS and RSs. For the fair comparison, the maximum transmission power of BS in this scheme is set to 2 W, which is equal to the total transmission power of the proposed scheme when the BS transmits on all subchannels and at both slots, and the total transmission power of RS is 500 mW. Since the power adaptation scheme is designed to guarantee the throughput fairness among all MSs, it is slightly modified so as to support the minimum rate requirements of RT MSs and to maximize the cell throughput as in the proposed scheme. The scheme proposed in our earlier work [11] is also adopted for comparison, which adjusts l on a heuristic basis. This scheme is referred to as the ‘‘heuristic ASL’’ scheme.

25

20

15 1

4

8

12

16

Number of OFDM symbols in the first slot, l Fig. 3. Cell throughput according to l when K ¼ 10 and L ¼ 24.

of simulation runs, although we do not present the pdf in   this paper. Even though L ¼ 24; Pr½lG 6 11 ’ 0:79 and lG is most probably 11 in the pdf. Thus, the value of fixed l in the FSL scheme is also set to 11. Figs. 4 and 5 respectively show the outage probability and the cell throughput of the comparison schemes, according to the minimum rate requirement of RT MS. We first investigate the efficiency of the proposed postprocessing algorithm by comparing the performances of the ASL scheme with postprocessing (i.e., the proposed scheme) and the ideal ASL scheme. It is obvious that the ASL scheme achieves better performance under the unrealistic situation where postprocessing is not necessary than the practical situation where the postprocessing is required, since more flexible resource allocation is possible. However, as shown in Fig. 4, the outage probability difference of the ASL with postprocessing and the ideal ASL without postprocessing is very small. This is because the resource reallocation for reducing the outage events is performed at the second step of postprocessing. On the other hand, since the postprocessing basically pursues minimum adjustment, as we can observe in Fig. 5, the throughput difference between two situations is also not greatly

0.08

4.3. Simulation results

0.06

Outage probability

To demonstrate visibly the concavity of U G ðlÞ on l by Lemma 1, we illustrate in Fig. 3 an example of throughput (U G ðlÞ) curves versus l, when K ¼ 10 and L ¼ 24. It is observed from Fig. 3 that U G ðlÞ increases monotonically  until l reaches lG ð¼ 6Þ and decreases monotonically when   l > lG . By using this property, Algorithm 1 can find lG efficiently. Moreover, the throughput is higher with l ¼ 6 than with l ¼ 12 which means two equal-length slots and is the most commonly used slot partition in many relay networks. This is because the channel quality of BS–RS link is usually better than that of RS–MS link. For balancing the capacities of two hops, the first slot should be shorter than the second slot in usual. This tendency is also con firmed with the empirical pdf of lG from several thousands

Proposed ASL (with postprocrssing) Ideal ASL (without postprocessing) FSL Periodic ASL Power adaptation Heuristic ASL

0.07

0.05 0.04 0.03 0.02 0.01 0.00 100

150

200

250

300

350

400

Minimum rate requirement of RT MS (kb/s) Fig. 4. Outage probability according to the minimum rate requirements of RT MSs when K ¼ 10.

40

D.G. Jeong et al. / Computer Networks 72 (2014) 33–44 45 40

Throughput (Mb/s)

35 30 25 20 15

Proposed ASL (with postprocrssing) Ideal ASL (without postprocessing) FSL Periodic ASL Power adaptation Heuristic ASL

10 5 0 100

150

200

250

300

350

400

Minimum rate requirement of RT MS (kb/s) Fig. 5. Cell throughput according to the minimum rate requirements of RT MSs when K ¼ 10.

noticeable. Hereafter, we will call the ASL with postprocessing just the proposed ASL scheme shortly, if there is no confusion. Next, we compare the proposed ASL scheme and the FSL scheme. As shown in Fig. 4, the outage probability of the FSL scheme increases rapidly as the minimum rate requirement of RT MS increases, since the scheme cannot balance efficiently the capacities of two hops. That is, in order to compensate the capacity gap between two hops for an indirect MS, more subchannels are often required to be allocated to the MS even when the MS experiences poor quality on the subchannels. Due to this inefficient resource allocation, with the FSL scheme, both of outage probability and throughput performance are deteriorated, as shown in Fig. 5. From the above observation, we can see that the slot boundary within a frame needs to be adaptively adjusted as in the proposed ASL scheme, not only for the better RT MS support but also for the higher cell throughput. The periodic ASL scheme adjusts l not every frame but with longer interval. We can observe in Figs. 4 and 5 that, even when a long period such as twenty frames is applied, the periodic ASL scheme achieves better performance than the FSL scheme, especially in outage probability. When considering that the proposed ASL scheme is a special case of the periodic ASL scheme having the period of one frame, the periodic ASL is a good alternative of the proposed ASL that can reduce computational complexity at the cost of a little performance degradation. Next, let us compare the proposed ASL scheme with the heuristic ASL scheme. The heuristic ASL scheme also adjusts the slot length adaptively, like the proposed ASL scheme. However, the heuristic ASL selects l while taking account of only the capacity difference between BS–RS and RS–MS hops; whereas the proposed ASL determines  l that maximizes cell throughput in consideration of the rate requirements for RT MSs. Thus, both outage probability and throughput of the proposed ASL scheme are remarkably superior to those of heuristic ASL scheme. Lastly, we examine the performance of the power adaptation scheme. As shown in Fig. 4, the power adaption scheme provides lower outage probability than the proposed ASL. This is possible since the power adaptation

scheme tunes the transmission power for each MS in a fine granule whereas the proposed ASL only adjusts the slot boundary for all of MSs without power control. Thus, the power adaptation scheme can satisfy the minimum rate requirements of RT MSs more easily. However, note that the achievable capacity for a link is a linear function of the allocated time but it is a log function of the allocated power. Accordingly, the power adaptation scheme should spend lots of power to compensate the poor channel gain between RS and MS, whereas the proposed ASL scheme can achieve comparative outage probability by adjusting the slot boundary and allocating the resource in a unit of symbol, while providing much higher throughput as shown in Fig. 5. Moreover, in the power adaptation scheme, the adjusted power allocation can cause unexpected other-cell interference (OCI) since the transmission power is concentrated on a few of subchannels, which may incur the higher outage probability in neighbor cells. To investigate the effect of OCI and the performance of ASL scheme in practice, we extend the simulation model to the multi-cell environment. In simulation, we consider a cellular system composed of seven hexagonal cells. The seven cells form two tiers of rings, i.e., a cell in the center and the second tier with six cells. The cell structure is assumed to be wrapped around so that any cell is enclosed by the other six cells. With wrapping-around model, an MS in each cell can move to one of six neighbor cells and can suffer from OCI from six neighbor cells. We assume that, when a transmitter determines its transmission rate in a downlink frame, it uses the OCI measured at the previous frame as the estimate of OCI. Let Oj;m denote the average OCI received at the MS of link j on the subchannel m during the previous downlink frame. Then, the transmission rate  g j;m P uj ;m , where g is a Rj;m is assumed to be Bs log2 1 þ N0 Bs þOj;m þg parameter introduced for compensating the prediction error on the actual OCI due to, for example, channel fading. In the simulation, g is set to N 0 Bs =10. In Figs. 6 and 7, we assess the outage probability and the throughput of the proposed ASL, the heuristic ASL, and the power adaptation scheme, under multi-cell environment. Note that the realized transmission rate of a link can be lower than its expected rate since the actually received OCI at the current frame may be higher than ðOj;m þ gÞ. Thus, RT MSs may not receive data enough to satisfy their minimum rate requirements due to high OCI, and additional outage events can occur. Owing to this effect, the outage probability of the proposed ASL scheme in Fig. 6 is higher than that in Fig. 4. As shown in Figs. 6 and 7, the proposed ASL scheme achieves the lower outage probability and the higher throughput compared to the power adaptation scheme. This is because the power adaptation scheme should consume too much transmission power for compensating the lower quality hop without adjusting the transmission time of both slots. In addition, the power adaptation scheme pours the transmission power to increase the transmission rate of the cell-edge RT MSs. This can cause two revers effects: less transmission power would be left for the good-quality MSs; the MSs in other cells would suffer from unexpectedly high OCI. Thus, in the power adaptation

D.G. Jeong et al. / Computer Networks 72 (2014) 33–44

the proposed ASL scheme can achieve high throughput by flexibly allocating a part of a subchannel after satisfying the minimum rate requirements of RT MSs. Therefore, the throughput of the proposed ASL scheme can be growing as the number of MSs is increased, in contrast to the other two schemes.

0.15

Outage probability

Proposed ASL Power adaptation Heuristic ASL

0.10

5. Conclusion 0.05

0.00 8

10

12

14

16

18

20

Number of MSs, K Fig. 6. Outage probability according to the number of MSs.

60 50 Throughput (Mb/s)

41

40 30 20 Proposed ASL Power adaptation Heuristic ASL

10

Acknowledgments

0 8

10

12

We have proposed a downlink scheduling scheme for the OFDMA relay system, which aims at maximizing cell throughput, while guaranteing rate requirements of RT MSs. The proposed ASL scheme not only performs subchannel allocation for MSs and RSs but also adjusts the length of each slot for balancing hop capacities, considering unbalanced channel quality of two hops. Since the formulated scheduling problem is a non-linear integer programming whose complexity is very high, we have relaxed it into an LP problem that offers an upper-bound solution with a low computational complexity. The proposed ASL scheme with postprocessing can be implemented with minor performance degradation in comparison with an ideal but unrealistic scheme. The performance comparison with other schemes shows the effectiveness of the adaptive control of slot length in hop balancing and the competitiveness of the proposed ASL scheme. In summary, the proposed scheme can be a good solution to the hop balancing problem in relay networks.

14

16

18

20

Number of MSs, K Fig. 7. Cell throughput according to the number of MSs.

The work of D.G. Jeong was supported by the Hankuk University of Foreign Studies Research Fund. The authors thank the editor and the anonymous reviewers for their valuable comments. Appendix A. Proof of Lemma 1

scheme, the cell throughput decreases while the outage probability gets higher, as the number of MSs increases. Also, Fig. 6 shows that the RT MSs experience outage events more often in the heuristic ASL scheme than the proposed ASL scheme. Moreover, the proposed ASL scheme achieves much higher throughput than the heuristic ASL scheme (see Fig. 7). These remarkable differences in outage probability and throughput reveal the efficiency of the proposed scheme in determining l, both on the minimum rate guarantee and on the cell throughput enhancement. On the other hand, the throughput difference between the proposed ASL scheme and the other two schemes is further expanded due to the inflexible subchannel allocation of the latter schemes. Since a subchannel is allocated to only one link in both power adaptation scheme and heuristic ASL scheme, most of subchannels should be allocated to the links for RT MSs to prevent an outage. Thus, when there are many RT MSs in a cell, subchannels are hardly allocated to the links for NRT MSs even when each of them has good channel quality. As a result, the cell throughput is hardly increased or is slightly decreased. In the meantime,

The proof of Lemma 1 is composed of five steps. First, we introduce a simple one-slot scheduling problem (A.1) that maximizes the weighted throughput sum of all links while satisfying the given minimum rate requirements of the links. As the second step, we split the proposed twoslot scheduling problem into two one-slot (sub) problems. Next, we convert the second-slot subproblem into the form of (A.1) and, by using Lemma 2 in Appendix B, we proves that the objective function of the second-slot problem decreases concavely as l increases. As the fourth step, the first-slot subproblem is converted into the form of (A.1) and it is proven that the objective function of the first-slot problem increases concavely as l increases, based on Lemmas 2 and 3 in Appendix C. Finally, Lemma 1 is proven by using the results from the above third and fourth steps. Let us examine each steps. We formulate the one-slot scheduling problem (A.1) as follows. The total number of links is j and the minimum rate requirement of link j is denoted by Y j . Let Y :¼ fY 1 ; . . . ; Y J g.

42

D.G. Jeong et al. / Computer Networks 72 (2014) 33–44

WðYÞ :¼ max

J X M X

J ¼ jS0 j þ N, where jAj denotes the cardinality of a set A.

wj aj;m Rj;m ;

ð1Þ

Let wj

j¼1 m¼1

s:t:

Y ð1Þ n

M X aj;m Rj;m P Y j ;

8j;

ðA:1Þ

m¼1 J X

8m;

aj;m 6 1;

j¼1

where wj denotes the weight for the throughput of link j and wj P 0. According to Lemma 2, WðYÞ is a non-increasing affine and concave function of Y. Next, we partition the problem (3) into two subproblems for the first and second slots. Without loss of generality, it is noted that the minimum rate requirement of each direct MS can be divided into two parts so that the minimum rate requirement of a direct MS j is set to aj Dj at the first slot and ð1  aj ÞDj at the second slot, where 0 6 aj 6 1. Then, the throughput maximization problem at the second slot can be reformulated as

max s:t:

K X M X L  l ð2Þ a Rj;m ; L j;m j¼1 m¼1 M X L  l ð2Þ a Rj;m P ð1  aj ÞDj ; L j;m m¼1 M X L  l ð2Þ a Rj;m P Dj ; L j;m m¼1 K X

ð2Þ

aj;m 6 1;

j 2 S0 ; ðA:2Þ

j 2 [16n6N Sn ;

8m:

ð2Þ

of MSs, J ¼ K. Let wj

ð2Þ

¼ Ll for all j, and Y j L

ð2Þ Yj

¼

L D Ll j

¼

for j 2 [16n6N Sn . Then,

the problem (A.2) is converted to the problem (A.1). Note that when l increases, Y ð2Þ also increases. According to Lemma 2, the throughput at the second slot is a nonincreasing and concave function of l. Next, the throughput maximization problem at the first slot can be expressed as

max

¼ Cn for each BS–RS link n. Then, the problem (A.3) is also converted to the problem (A.1). According to Lemma 3 in Appendix C, the throughput of the indirect MS link j at the second slot  PM Ll ð2Þ ð2Þ i:e:; is equal to Y j or a non-increasing m¼1 L aj;m Rj;m function of Y ð2Þ . Remind that Y ð2Þ increases as l increases.  P P Ll ð2Þ Thus, Cn ¼ j2Sn M m¼1 L aj;m Rj;m is a non-increasing funcð1Þ

tion of l. As described above, since Y j Y ð1Þ n

¼ Cn for a BS–RS link n, Y ð1Þ decreases MS link j and as l increases. According to Lemma 2, this means that the total throughput of the first slot increases concavely when l increases. When l increases, since the throughput at first slot concavely increases and the throughput at the second slot concavely decreases, the sum of transmission rates on both slots, U G ðlÞ, is also a concave function of l under the assumption that l is a real number between 0 and L. However, l is actually a positive integer smaller than L. Thus, we cannot say U G ðlÞ is a concave function but U G ðlÞ monotonically increases as l increases until U G ðlÞ achieves the highest throughput and then it monotonically  decreases as l increases beyond lG . h

Proof. We denote Y :¼ ðY 1 ; . . . ; Y J Þ, where Y j is a rate requirement of link j and J is the number of links. Let Dj;m ðYÞ be the part of Y j achieved on the subchannel m P and M m¼1 Dj;m ðYÞ ¼ Y j . Note that, the required time for realizing Dj;m ðYÞ on the subchannel m is

s:t:

m¼1

D

X j2S0

KþN X

ð1Þ

aj;m þ

ð1Þ

aj;m 6 1;

j;m

DðYÞ

J X

wj

j¼1;j–h

! ) J X Dj;m ðYÞ Dj;m ðYÞ Rh;m ; Rj;m þ wh 1  Rj;m Rj;m j¼1;j–h ðB:1Þ

j 2 S0 ; ðA:3Þ

where DðYÞ :¼ ½Dj;m ðYÞJM . When wm Rm ¼ max16j6J wj Rj;m , WðYÞ can be reexpressed

1 6 n 6 N; as

8m;

j¼Kþ1

P

ðYÞ

That is, WðYÞ ¼ maxsumM m¼1

M X l ð1Þ aKþn;m RKþn;m P Cn ; L m¼1

which corre-

to each link j – h and the time of ing the time of j;m R  j;m PJ Dj;m ðYÞ to the link h, for each subchannel m. 1  j¼1;j–h R

0

l ð1Þ a Rj;m P aj Dj ; L j;m

Dj;m ðYÞ , Rj;m

sponds to aj;m . Let h ¼ argmax16j6J wj Rj;m . The maximum weighted throughput sum WðYÞ can be achieved by assign-

(

M XX l ð1Þ aj;m Rj;m ; L j2S m¼1 M X

¼ Ll aj Dj for a direct

L l

Lemma 2. The objective function of (A.1), WðYÞ, is a nonincreasing affine and concave function of Y.

Let us convert the problem (A.2) into the form of (A.1). For explicitly presenting the slot number, we denote Y and w at the ith slot by Y ðiÞ and wðiÞ , respectively. Since the number of links at the second slot is equal to the number  aj ÞDj for j 2 S0 and

ð1Þ

¼ Ll aj Dj for j 2 S0 , and wn ¼ 0 and

Appendix B. Lemma 2

j¼1

L ð1 Ll

ð1Þ

¼ Ll and Y j

L l

PM Ll ð2Þ where Cn :¼ m¼1 L aj;m Rj;m , which is the total j2Sn transmission rate of RS n for serving its indirect MSs at the second slot. Note that the number of links at the first slot is the number of direct MSs and RSs. That is,

( !) J X J M M X X X Dj;m ðYÞ   WðYÞ ¼ max wj Dj;m ðYÞ þ wm Rm 1  DðYÞ Rj;m m¼1 j¼1 m¼1 j¼1 ! J J X M M X X X wm Rm ¼ max wj Y j  Dj;m ðYÞ þ wm Rm : DðYÞ Rj;m m¼1 j¼1 j¼1 m¼1 ðB:2Þ

43

D.G. Jeong et al. / Computer Networks 72 (2014) 33–44

As shown in (B.2), the throughput loss paid for satisfying the rate D of the link j on the subchannel m is wm Rm Rj;m

D  wj D. Thus, the link j prefers a subchannel m1 –m2

when

wm Rm

1

1

6

Rj;m

1

j1 to j2 when

wm Rm

2

2

R

j;m2 wm Rm

6

Rj

1 ;m

. Also, the subchannel m prefers a link wm Rm . Rj ;m

; . . . ; D‘;J ðY 0 ÞÞ

Based on this preference, we will prove that WðYÞ is a non-increasing piecewise affine function of Y with decreasing gradient so that it is a concave function of Y. Let Dj;m ðYÞ denote the value of Dj;m ðYÞ for satisfying Y while maximizing WðYÞ. In addition, without loss of generality, we assume that the subchannels are sorted in wm Rm Rj ;m

R‘;m

link j , where  ¼ d Rj ;m1 . Note that 1

WðY 0 Þ ¼

þ

WðY 0 Þ ¼

wj Y j þ wj d 

j¼1 m¼1

Rj;m

j¼1



wm1 Rm1

R‘;m1

ðB:3Þ wm Rm 1 1 Rj ;m 1

The gradient of link j , which is wj  , is always less than or equal to 0, according to the definition of wm1 Rm1 . Thus, WðYÞ is a non-increasing affine function of Y. Moreover, when d cannot be supported in the subchannel m1 , the link j should be assigned the subchannel m2 (> m1 ). In this case, WðY 0 Þ can be defined as follows: J J X M X X wm Rm  WðY Þ ¼ wj Y j þ wj d  D ðYÞ Rj;m j;m j¼1 j¼1 m¼1 0

Rj ;m2

wm Rm

2 2 Rj ;m 2

R‘;m2



wm3 Rm3



R‘;m3

wm Rm

     wm1 Rm1 wm2 Rm2 wm3 Rm3 R‘;m1 d wj  þ  Rj ;m1 R‘;m2 R‘;m3 Rj ;m1



wm Rm

ðB:5Þ



60 and discussed above, wj  wm Rm 3 3 6 0. Thus, the gradient of WðYÞ in this  R‘;m 1 1 Rj ;m 1

3

Appendix C. Lemma 3 Lemma 3. For given rate requirements Y, the weighted throughput of the link j is either Y j or a non-increasing function of Y. Proof. Let wj ðYÞ be the weighted throughput of the link j .

X

wj Rj ;m 1 

J X Dj;m ðYÞ j¼1

!

;

Rj;m

ðC:1Þ

j;m

m¼1

ðB:4Þ

2

6

wm2 Rm2

n o P D ðYÞ where Mj ¼ mjj ¼ argmax16j6J wj Rj;m ; Jj¼1 j;m < 1 . If R

This also shows that WðYÞ is a non-increasing affine  wm Rm wj  Rj2;m 2 6 0. Furthermore, 1 1 Rj ;m 1

M X

dþ

m2Mj

function of Y owing to wm Rm

R‘;m1

wj ðYÞ ¼ wj Y j þ

M X dþ wm Rm

  wm2 Rm2 d þ WðYÞ: ¼ wj  Rj ;m2

since

wm1 Rm1

case is not only negative but also is less than or equal to the gradient of the first case. Accordingly, WðYÞ is a nonincreasing piecewise affine and concave function of Y. h





from the

þ WðYÞ:

 m1 Rm1

M X dþ wm Rm

Rj ;m1 m¼1   wm1 Rm1 d þ WðYÞ: ¼ wj  Rj ;m1

wm2 Rm2

¼

As w

Dj;m ðYÞ

wm Rm 1 1 R‘;m1

m¼1

sufficiently small d P 0 and for any link j , we consider the case that Y is increased to Y0 ¼ ðY 1 ; Y 2 . . . ; Y j þ d; . . . ; Y J Þ. First, let us assume that only the Dj ;m1 ðY 0 Þ is increased by d and all other Dj;m ðY 0 Þ are equal to Dj;m ðYÞ. Then,

wm Rm

P

J J X M X X wm Rm  wj Y j þ wj d  D ðYÞ Rj;m j;m j¼1 j¼1 m¼1



subchannel m1 where Dj ;m ðYÞ ¼ 0; 8m > m1 and  Dj ;m1 ðYÞ > 0 due to the subchannel preference. For a

J X M X

wm Rm 3 3 R‘;m3

subchannel preference. Then,

for any link j . Then, there is a

J X

in order to support the additional d for the



2

increasing order of

treated with the same way by splitting d into smaller values.  That is, D‘;1 ðYÞ; . . . ; D‘;m2 ðYÞ; . . . ; D‘;m3 ðYÞ; . . . ; D‘;J ðYÞ is  D‘;1 ðY 0 Þ; . . . ; D‘;m2 ðY 0 Þ  ; . . . ; D‘;m3 ðY 0 Þþ changed to

, we have wj 

wm Rm 1

Rj ;m

1

1

P wj 

wm Rm

2

2

Rj ;m

.

2

Thus, it is shown that the gradient of WðYÞ decreases as Y j increases for any link j , which means that WðYÞ is a concave function of Y when the increase of minimum rate of one link j does not affect to Dj;m of any other link j. Next, we examine the case that, because of the increased Y j þ d, not only Dj ;m1 ðY 0 Þ is increased by d, but also two D‘;m2 ðYÞ and D‘;m3 ðYÞ of another link ‘ are affected. Note that we do not consider that several links are affected by the increased Y j þ d since we assume that the d is sufficiently small. The cases that Dj;m of multiple links are affected can be

Mj ¼ ;, it is obvious that wj ðYÞ is equal to wj Y j , regardless of Y. Let us examine the case that Mj – ;. Suppose that Y j is increased by a sufficiently small d and Y 0 denotes the increased minimum rate requirements of links. When any subchannel m0 in Mj is chosen for satisfying the additional d, the weighted throughput of the link j will be changed to

wj ðY 0 Þ ¼ wj ðY j þ dÞ þ wj Rj ;m0 1 

J X Dj;m0 ðYÞ j¼1

þ

X

w R j

m2Mj ;m–m0

¼ wj Y j þ

X m2Mj

¼ wj ðYÞ:

j ;m

Rj;m0 1



d

!

Rj ;m0

J X Dj;m ðYÞ j¼1

wj Rj ;m 1 

!

Rj;m

J X Dj;m ðYÞ j¼1

!

Rj;m ðC:2Þ

44

D.G. Jeong et al. / Computer Networks 72 (2014) 33–44 0

In addition, if Y j0 of other link j is increased by 0 of some subchannels would be increased for supporting the additional , but it is certain that the increase of Dj0 ;m ðY 0 Þ’s does not increase wj ðYÞ. Thus, wj ðYÞ is a non-increasing function of Y when Mj – ;. Furthermore, from (C.2), wj ðYÞ is equal to Y j when Mj ¼ ;. h

; Dj ;m ðYÞ’s

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Dong Geun Jeong received the BS, MS, and PhD degrees from Seoul National University, Korea, in 1983, 1985, and 1993, respectively. From 1986 to 1990, he was a researcher with the R&D Center of DACOM, Korea. During 1994–1997, he was with the R&D Center of Shinsegi Telecomm, Inc., Korea, where he conducted and led research on advanced cellular mobile networks. In 1997, he joined the faculty at the Hankuk University of Foreign Studies, Korea, where he is currently a professor in the School of Electronics and Information Engineering. His research interests include resource management for wireless and mobile networks, mobile communications systems, communication protocols, and network performance evaluation. He served as the program committee vice chair for IEEE VTC 2003-Spring. From 2002 to 2007, he served on the editorial board of the Journal of Communications and Networks (JCN). He is a senior member of the IEEE.

Jeong Ae Han received the B.S., M.S., and Ph.D. degrees in computer science and engineering from Seoul National University, Seoul, Korea, in 2003, 2005, and 2011 respectively. She is currently a researcher with the R&D Center of SAP, Korea. Her research interests include cognitive radio systems, resource management for wireless and mobile networks, and relay enhanced communication systems.

Wha Sook Jeon received the BS, MS, and PhD degrees in computer engineering in 1983, 1985, and 1989, respectively, from Seoul National University, Korea, where she joined the faculty in 1999, and is currently a professor in the School of Electrical Engineering and Computer Science. From 1989 to 1999, she was with the Department of Computer Engineering, Hansung University, Korea. Her research interests include resource management for wireless and mobile networks, mobile communications systems, high-speed networks, communication protocols, and network performance evaluation. She currently serves on the editorial board of the Journal of Communications and Networks (JCN). She is a senior member of the IEEE.