A Novel Description approach based on sorted rectangles for scheduling information bearing in OFDMA systems

Computer Networks 115 (2017) 82–99

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A Novel Description approach based on sorted rectangles for scheduling information bearing in OFDMA systems Tsung-Yu Tsai a,∗, Yi-Hsueh Tsai b, Zsehong Tsai a, Shiann-Tsong Sheu c a

GICE, National Taiwan University, Taipei, Taiwan Institute for Information Industry, Taipei, Taiwan c Dept. of Communication Engineering, National Central University, Taoyuan, Taiwan b

a r t i c l e

i n f o

Article history: Received 6 September 2015 Revised 7 January 2017 Accepted 25 January 2017 Available online 3 February 2017 Keywords: OFDMA Scheduling information Signaling overhead Resource allocation

a b s t r a c t Dynamic time/frequency resource allocation has been widely employed in wireless OFDMA systems. In such mechanisms, sophisticated scheduling decisions need to be conveyed to users and may lead to considerable signaling overhead. In this paper, a new efficient approach, namely, Sorted-Rectangle Description (SRD) is proposed for describing the scheduling decision in an OFDMA frame. SRD first partitions a frame into minimum number of rectangular polygons, namely, rectangular bursts (RB), in which each RB contains the resource units belonging to the same assignment. We show that each RB-partitioned frame is sufficient to be fully reconstructed by the top-left coordinates of each RB in a specific sorted order. Sorting and reconstructing algorithms with linear time complexity in the total number of RBs are also proposed. Simulations based on a multi-user OFDMA system with full-buffer or periodic small-data traffic model are conducted to evaluate the performance of the proposed SRD method under two representative scenarios. The simulation results show that, for the full-buffer traffic model with a proportional fair (PF) scheduler, the proposed description approach can achieve significant reduction of the overhead ratio compared to that of the existing systems and is expected to lead to considerable increase of system throughput. Meanwhile, for the periodic small-data traffic model, it is shown that the proposed description approach can greatly reduce the rate of packet loss due to timeout or buffer overflow and thus provides significant improvement of the system capacity. © 2017 Elsevier B.V. All rights reserved.

1. Introduction 1.1. Background Orthogonal frequency-division multiple access (OFDMA) is a promising technique for broadband wireless access systems and has been widely adopted by several existing and next generation communication standards such as 3rd Generation Partnership Project (3GPP) Long Term Evolution (LTE) [1], LTE Advanced [2] and IEEE 802.16e/m [3,4]. One of the main advantage of OFDMA is that it allows flexible two-dimensional resource allocations with which users could be assigned different frequency units (e.g. subcarriers or subchannels) and time units (e.g. OFDM symbols) at the same time. The additional scheduling flexibility provides a finer allocation granularity and is beneficial to improve the system utilization. Moreover,

∗

Corresponding author. E-mail address: [email protected] (T.-Y. Tsai).

http://dx.doi.org/10.1016/j.comnet.2017.01.017 1389-1286/© 2017 Elsevier B.V. All rights reserved.

it also allows the system to schedule time/frequency resource units (RUs) to those users with better channel gains at these resources and achieves a significantly higher spectrum efficiency. However, due to the fact that the channel status can generally vary relatively fast in mobile wireless environments and the traffic characteristic of many services are bursty, the resource scheduling decisions should be done frequently enough to adapt to the variation of channel condition and queueing status of each user. In most existing systems, the scheduling cycle is usually done in a per frame basis within several milliseconds. The resource scheduling decisions made by the system are usually encapsulated into control messages called scheduling information (SI) messages and then be encoded and sent to the users. A typical SI message contains the information the scheduled users need for data transmission or reception. For example, allocated RUs, user identities (ID) of the scheduled users, transmission format including the modulation and coding schemes (MCS), hybrid auto repeat request (HARQ) related information, and multipleinput-multiple-output (MIMO) configurations adopted for the data transmission or reception.

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Since the SI message carries essential control information, it should be transmitted at a sufficiently robust MCS level to meet certain error rate requirement, which is in general much more strict than the requirement for data channels. Thus, although the SI message is in general shorter than data plane packets, it may still occupy a considerable amount of radio resources and introduce non-ignorable signaling overhead. There are mainly two transmission strategies to encode and transmit SI messages, and both have been adopted by existing systems: 1) Joint Encoding and Broadcasting (JEB): In this strategy, the SI of all scheduled users is encapsulated into an SI message, encoded to a single codeword, and broadcast to all scheduled users. To let the users to have the necessary information to decode the SI message, there may be a common header which has a fixed size and is transmitted in pre-determined MCS to carry the length and MCS information of the SI message. 2) Separately Encoding and Transmission (SET): In this strategy, the SI of different scheduled users are encapsulated into separate SI messages which are encoded separately according to the channel conditions of each scheduled users and then sent individually. To let each scheduled user to identify the SI message destined for it, each SI message is attached a user-specific cyclic redundancy check (CRC) code and is encoded by the MCS selected from a finite set of pre-defined MCS settings. User-specific CRC is obtained by masking the conventional CRC calculated by the SI message with the user’s ID. So the scheduled user could recognize its SI message by trying different MCS candidates and perform CRC check after de-masking the CRC portion of the decoded message with its ID until the CRC check is passed. This mechanism is refered as blind decoding and is adopted in several systems such as 3GPP LTE/LTE-A [2] and IEEE 802.16m [4]. The main drawback of JEB is that the SI message should be encoded with a robust enough MCS level such that all the scheduled users could decode it under a given message error probability. Thus, the transmission of the SI message may not be very efficient since its coding rate should adapt to the user with the worst channel status. On the other hand, SET has the advantage that separate SI messages could be encoded adaptive to the channel statuses of different users and thus has a better transmission efficiency. However, large number of searches in the blind decoding procedure may introduce power consumption issues to the user devices. To reduce the number of search steps, practical systems employ some additional mechanisms, which will degrade the efficiency of the usage of the control channel. For example, the search space defined in 3GPP LTE systems [2] are adopted to reduce the number of necessary searches for blind decoding with the cost of reducing the utilization of control channel resource. Moreover, as addressed in [7], the probability that at least one user fails to receive the SI message of SET is in general higher than that of JEB. 1.2. Related works The overhead of SI messages is an important issue when designing practical systems. Recently, several research works have highlighted this problem and some analytical results of the performance degradation caused are provided. In [5], Gross et al. analyzed the impact of signaling overhead to the system throughput in OFDMA systems with dynamic resource allocation and conducted simulations under different setting of system parameters. In [6], So proposed a queueing theoretic analytical model to evaluate the performance of VoIP services in IEEE 802.16e OFDMA system in consideration of the signaling overhead. The results indicated that the signaling overhead ranges from 11% to 53% for IEEE 802.16e

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system dominated by VoIP users. A comprehensive comparison of JEB and SET schemes is provided in [7] by Moosavi et al. These two strategies were evaluated in terms of overall spectrum efficiency and system throughput under different channel models, scheduling algorithms and resource granularities to catch a variety of practical environments. Moreover, the authors also presented the idea of using run-length encoding to compress the SI messages in both strategies to reduce the signaling overhead. There are also literatures aiming at the method to reduce the signaling overhead of SI messages, which is also the focus of this paper. Basically, there are three main methodologies: overheadaware resource scheduling, improving the transmission efficiency of the SI messages, and condensing information bits needed to convey to each scheduled users. Overhead-aware resource scheduling is a technique which schedules the pending packets and allocates the RUs in an OFDMA frame with consideration of the resultant overhead (include SI messages and preamble signals) and achieve a balance between overhead reduction and Quality of Service (QoS) requirement. In [8], Cohen and Katzir proposed a two-tier scheduling scheme, i.e. macro scheduling and micro scheduling. Their macro scheduling determines the set of pending packets to be transmitted in the OFDMA frame based on the profit value and the set of feasible MCS types of each packet. The additional bits needed to be carried in the SI message is proportional to the number of MCS types adopted in a frame. On the other hand, the micro scheduling is performed to fit these scheduled packets with the same MCS type into a collection of rectangular allocations in the OFDMA frame, where larger number of rectangular allocation could avoid the leftover RUs but also introduce larger overhead penalty of SI signaling. The authors proposed efficient algorithms to perform macro and micro scheduling separately, with a guaranteed approximation to maximize the total profit. Similar issue is also investigated in [9,10]. In [9], Ben-Shimol et. al. proposed a variety of algorithms to efficiently encapsulate the scheduled bursts into rectangular allocations in an OFDMA frame to achieve higher resource efficiency while obeying the priority constraint of scheduled bursts. Israeli et. al. [10] also addressed the similar problem of fitting the scheduled bursts with strict priority order to rectangular allocations in an OFDMA frame table, where hardness result and approximation algorithms were both provided. Moreover, in [11], Chen et. al. also proposed heuristic algorithms for mapping scheduled data to reduce the number of rectangular allocations in an OFDMA frame and thus reduced the resulting SI overhead. Improving the transmission efficiency of SI is typically achieved by utilizing the variation of channel conditions of different users and adjusting the modulation or coding rate according to the channel conditions of different users. In [12], Kwon and Cho suggested to use adaptive modulation and coding (AMC) technique for the transmission of SI message and their results showed that it could reduce the SI overhead particularly in systems with short-sized packet services such as Voice over Internet Protocol (VoIP). Based on a similar concept, Sternad [13] and Yeom [14] further proposed to group users with similar channel gains and transmit the SI messages in a per-group basis with the modulations and coding rates according to the channel qualities of each group. Moreover, Yeom also proposed a mechanism to dynamically adjust the MCS level of each user based on acknowledgement signals from the users to further reduce the overhead of channel status feedback. In general, the effectiveness of the overhead reduction of this methodology highly depends on the given realization of users’ channel conditions. When the ratio of users with good channel conditions is relatively small, the overhead reduction will be limited. From a different perspective, several literatures also investigated the methods to reduce the amount of bits of SI mes-

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sages. Moosavi [7] and Nguyen [15] both presented the idea to use run-length coding to compress the SI messages. Nguyen further suggested a message compression scheme which contains a run-length encoder followed by a universal-variable length code (UVLC). Another technique is proposed in [16] by Moosavi et al., which utilizes the fact that given a radio resource is allocated to a user, then the radio resource will not be assigned to other users. Thus the base station can encode the SI of each user differentially and reduce the amount of overall information bits transmitted. The information theoretic analysis of this differential compression approach is also provided in [17]. Further, Moosavi and Larson [18] also proposed a user-assisted SI compression scheme, which is operated under the assumption that the side information of users’ channel conditions are known in advance by the base station and users. The above approaches reduce the amount of overall information bit by applying data compression techniques. To the best of our knowledge, there is relatively few work investigating the fundamental question that if there is more efficient description to sufficiently identify any given resource allocations. In this paper, we propose a novel approach, namely, SortedRectangle Description (SRD), to describe the resource allocation in OFDMA systems. SRD can be implemented over JEB schemes. It treats the frame and RUs as rectangular polygons in Cartesian plane and can employ a polynomial time algorithm to partition the frame into minimum number of smaller rectangular polygons, called rectangular bursts (RBs). Where each RB contains the RUs belonging to the same assignment. Then, we prove such an RBpartitioned frame can be identified by sorted top-left coordinates of the RBs in it. Hence, only one corner (e.g. left-top) coordinate of each RB is needed to be carried in the SI message and other information is embedded implicitly in the order of the appearance of the coordinate information in the SI message. Sorting and reconstructing algorithms with linear time complexity are also developed. The rest of this paper is organized as follows. In Section 2, we introduce the system model. An overview of approaches for describing scheduling information in existing systems is also provided. Next, the proposed SRD method and the sorting/reconstructing algorithms are presented in Section 3. Performance evaluation via computer simulation are presented and shown in Section 4. Section 5 concludes this paper.

2. System model and preliminaries 2.1. System model Consider a conventional OFDMA point-to-multipoint (PMP) downlink transmission system, which consists of a base station surrounded by a number of users. All data transmissions are performed in frames. It is assumed that the users have obtained both time and frequency synchronizations to the base station. In this paper, we suppose SI messages are transmitted by the base station in each downlink frame via JEB or SET strategies. We further assume that in each frame, all subcarriers are divided into C non-overlap subsets, namely, subchannels, and all OFDM symbols are grouped into S non-overlap time slots. Let the frequencydomain be the horizontal axis and time-domain be the vertical axis in Cartesian plane. We can then regard a frame as the following rectangular polygon with length S and width C:

A := {(x, y ) : 0 ≤ x < C, 0 < y ≤ S}.

(1)

C Θ0 Θ1

Vertical axis (time slots)

84

Θ2 Θ3

S

Θ4

A

Horizontal axis (subchannels) Resource Unit (RU) Fig. 1. An example of resource allocation of a frame A with five users allocated.

Further, A is the union of CS non-overlap rectangular polygons,  and can be expressed as A = Rc,s , where c,s

Rc,s : = {(x, y ) : c ≤ x < c + 1, s ≤ y < s + 1} c ∈ {0, · · · , C − 1}, s ∈ {0, · · · , S − 1}. Rc, s is refered as a Resource Unit (RU), which is the minimal resource entity to be allocated to a user for the transmissions in data channels. Assume there are Nsche RUs which can be scheduled to a user for data transmission and are denoted as Rd0 , · · · , RdN −1 , resche

spectively. Nsche is a positive integer which is smaller than or equal to CS. The resource allocation of data transmission is in general dynamic in a per frame basis. Let there be Nu users scheduled in a given frame A, which are indexed as 0, · · · , Nu − 1 and each RU can be allocated to at most one user. Thus, we can denote 1u ⊆ {0, · · · , Nsche − 1} as the set of RU indices which are allocated to scheduled user u, u ∈ {0, · · · , Nu − 1}. The region corresponding to the resource allocated to an scheduled user u is denoted as u :=  Rd and data region Ad is defined as the union of RUs which k∈1u k  are allocated for data transmission. That is Ad := u∈{0,··· ,Nu −1} u . Additionally, we define the union of RUs that are not allocated for data transmission as empty region Ae := AAd . An example of resource allocation layout of a frame A is depicted in Fig. 1. In this paper, SI messages are sent to convey 1u to each scheduled user u in a frame via JEB or SET strategies. To do this, the scheduled users are first divided into one or more subsets and the SI of each user subset are signaled by individual SI messages. JEB and SET strategies can be regarded as two extreme examples to determine user subsets, in which the first treats all scheduled user as a set and the latter treats every single scheduled user as a subset. For an SI message P, the RUs allocated to each user signaled by P may be further divided into one or more non-overlapped subsets, namely, bursts. The SI message contains the description of the bursts such that a signaled user can reconstruct its assignment from the description. One of the representative examples is the IEEE 802.16e system, where the RUs allocated to each user is partitioned into rectangular-shape portions. Let the allocated RUs of users signaled by P be divided into totally NB bursts B0 , · · · , BNB −1 ,

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2) Rectangular Partition Description (RPD): In this approach, u is first partitioned into bursts with rectangular shape. We refer the specific burst with rectangular shape as rectangular burst (RB).

which satisfy

Bk ⊆ Ad , for each k ∈ {0, · · · , NB − 1} Bk ⊆ u , for some u ∈ {0, · · · , Nu − 1} Bk1 ∩ Bk2 = ∅, if k1 = k2

(2)

where ∅ denotes the empty set. Denote ⊆ {0, · · · , NB − 1} as the set of the indices of the bursts allocated to a scheduled user u signaled by P. Furthermore, define Dk ⊆ {0, · · · , Nsche − 1} as the set of indices of the RUs forming Bk . The region corresponding to the resource allocated to scheduled user u can then be expressed as:

2u

u =



85

Bk .

(3)

k∈2u

We refer the set of bursts satisfying (2) and (3) as a burst partition for user u. Note that the burst partition for a given u is in general not unique. Different burst partitions may lead to different SI message overhead depending on the method used to describe the  burst information. Moreover, it is clear that 1u = k∈2 Dk . u

The method for describing the bursts of the destined scheduled user in P is referred as burst description method. Representive burst description methods adopted in existing systems are introduced in Section 2.3. Moreover, in this paper, we refer a combination of transmission strategy (i.e. JEB or SET) and the burst description method as an SI bearing scheme. 2.2. Model of SI message overhead The size in bits of an SI message P, denoted as LSI , is composed by three portions:

LSI = Md + Mu + LCRC .

(4)

where Md is the length in bits of the resource description portion, Mu is the length in bits of the user information portion, and LCRC is the length in bits of the CRC portion of an SI message1 . Md represents the number of bits for describing the set of the bursts associated to the scheduled users indicated in P, namely,   BP := Bk , where UP is the set of the indices of the scheduled u∈UP k∈2 u

users indicated in P. If JEB strategy is used, UP = {0, · · · , Nu − 1}. When SET strategy is adopted and P is destined to scheduled user u, then UP = {u}. Note that Md typically depends on the burst description method adopted. Finally, Mu is the number of bits for describing the necessary information for users in UP , including the transmission format adopted, and the mapping relation of each burst in BP to the user IDs of each user in UP . 2.3. Burst description methods adopted in existing systems There are two main approaches to describe a given burst set in existing systems: 1) Bitmap Description (BMD): In this approach, u is treated as one burst, thus |2u | = 1 for u ∈ {0, · · · , Nu − 1}, where | · | denotes the cardinality. The bursts associated to the scheduled users u are described by using a bitmap bu = {buk : k ∈

{0, · · · , Nsche − 1}} where buk = 1 when Rdk ⊆ u and buk = 0 otherwise. BMD is in general employed by combining SET strategy, and the length of the resource allocation portion can be expressed as Md = MBMD = |bn | = Nsche . BMD with SET strategy is the SI bearing scheme used in 3GPP LTE/LTE-A and IEEE 802.16m systems.

1 When SET strategy is used, the user ID is masked in the CRC portion (we suppose the length of user ID is not larger than LCRC ). Then the user can determine whether the SI message is destined to it by blind decoding technique. The detailed procedure of blind decoding will be addressed in Section 4.1.6.

˜ k is a rectangular polygon ⊆A which can be Definition 1. An RB B expressed as:

˜ k := {(Xk + t1 , Yk − t2 ) : 0 ≤ t1 < Wk , 0 ≤ t2 < Hk } B

(5)

where Xk ∈ {0, · · · , C − 1}, Yk ∈ {1, , S}, Wk ∈ {1, · · · , C − Xk }, Hk ∈ {1, , Yk } are refered as subchannel offset, time offset, width, ˜ k , respectively. and height of B It can be seen from (5) that the 4-tuple (Xk , Yk , Wk , Hk ) is sufficient to identify each RB, and can be described by log2 C , log2 S , log2 C and log2 S bits, respectively. Thus, the number of bits used to identify an RB MRPD is 2( log2 C + log2 S ). RPD is in general employed by combining JEB strategy, it is also the SI bearing scheme used by IEEE 802.16e systems. When combining with JEB strategy, the length of the resource allocation portion is given by Md = NB MRPD = 2NB ( log2 C + log2 S ), where  u −1 2 NB = N |u | is the total number of RBs. In Section 3.4, we will u=0 discuss the optimal partition algorithm to minimize NB and hence minimize the overhead. 3. The Sorted-Rectangle method for efficient description of resource allocation In this section, we present Sorted-Rectangle Description (SRD) method for efficient description of the resource allocation of a frame. In Section 3.1, we first consider a simplified resource allocation model in which a frame is regarded as a rectangular polygon partitioned by a set of non-overlapped RBs. Then, we prove such an RB-partitioned rectangular polygon is sufficient to be described by one-corner coordinates (e.g. the top-left corner) of each RB in a specific sorted order. In Section 3.2, we further establish efficient sorting and reconstruction algorithms with linear complexity in the total number of RBs to determine the order of RBs and reconstruct the RB-partitioned frame from the sorted coordinates of each RB. At last, we discuss the strategies to partition a resource allocation in a frame into an RB-partitioned rectangular polygon in the presence of empty regions (i.e. Ae = ∅) and non-rectangular resource allocation in Sections 3.3 and 3.4, respectively. 3.1. Condensed decription of a RB-partitioned frame using sorted coordinates Consider a polygon A as defined in (1) which represents a given frame, and an SI message containing the assignment information of NB bursts 0 := {B0 , · · · , BNB −1 }, where each burst belongs to the region of the RUs of the same assignment in the given frame. For convenience of illustration, we first assume the following assumptions: A1: Empty region is not present in A, thus A is covered by the data region. That is A = Ad . A2: Assume the bursts are non-overlapped RBs as defined in Definition 1 and partition Ad (thus partition A due to A1). N B −1 That is, Bk = Ad = A and Bk1 ∩ Bk2 = ∅ for k1 = k2 . k=0

We define an RB set 0 which satisfies A1 and A2 as a Complete RB Partition (CRP), and the set of all CRPs as . Moreover, in the following, we denote the number of RBs of a CRP 0 as NCRP , and denote the RBs in 0 as B0 , · · · , BNCRP −1 . To further reduce the information needed for describing a CRP, a natural question is: whether one can reconstruct a CRP with only

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T.-Y. Tsai et al. / Computer Networks 115 (2017) 82–99

Fig. 2. Two examples of relation U in CRPs. (a) For the CRP with 4 RBs in it, U = { (B1 , B3 ), (B2 , B3 ), (B3 , B1 ), (B3 , B2 )}. (b) For the CRP with 6 RBs in its, U = ∅.

the knowledge of the coordinate of one of the four corners of each RB? To answer this question, we develop a procedure to obtain the RBs’ lengths and widths from the coordinates of only one corner of each RB. After investigations, it can be found that one cannot always reconstruct a CRP with only the knowledge of the coordinate information unless the CRP satisfies certain specific conditions. Without loss of generality, for a CRP 0 , we assume the topleft coordinate (Xn , Yn ) of the RB Bn in 0 is known in advance (n ∈ {0, · · · , NCRP − 1}). Then, we have the following observations. Lemma 1.1. For an RB Bi ∈ 0 , if there is another RB Bj , j = i satisfies j = arg min{Xk : Xk > Xi , Yk ≥ Yi }, then Wi ≥ X j − Xi . If no such Bj k

exists, then Wi = C − Xi Lemma 1.2. For an RB Bi ∈ 0 , if there is another RB Bj , i = j, satisfies Xj > Xi and Y j = Yi . Then Wi ≤ X j − Xi . Lemma 1.3. For an RB Bi ∈ 0 , if there is another RB B j ∈ 0 , j = i, satisfies j = arg min{Xk : Xk > Xi , Yk ≥ Yi } and Y j = Yi , then Xi + Wi = k

Xj.

Lemma 3. For an RB Bi in a CRP 0 , its height and width can be obtained from the top-left coordinates of all RBs ∈ 0 if there is no RB ∈ 0 which Bi is U-related to. Proof. When |0 | = 1, the result is clear to stand. When |0 | > 1, if there is no RB which Bi is U-related to, the RBs ∈ 0 Bi will fall into one of the following five sets T0 to T4 : T0 : An RB Bj ∈ T0 if Xj > Xi and Y j = Yi . T1 : An RB Bj ∈ T1 if X j = Xi , and Yj < Yi . T2 : An RB Bj ∈ T1 if Xj > Xi and Yj < Yi . T3 : An RB Bj ∈ T2 if Xj ≤ Xi and Yj > Yi . T4 : An RB Bj ∈ T3 if Xj < Xi and Y j = Yi If there is no RB∈ T0 , then combining the facts that there is no RB which Bi is U-related to and 0 is a CRP, it can be seen that there is no RB Bk satisfying Xk > Xi and Yk ≥ Yi . In this case, Wi = C − Xi according to Lemma 1.1. Otherwise, due to the fact that there is no RB which Bi is U-related to, an RB Bk satisfying Xk > Xi and Yk ≥ Yi also has the property that Yk = Yi . And the width of Bi can then be obtained by Wi = X j − Xi according to Lemma 1.3, where j = arg min{Xk : Bk ∈ T0 }. k

Proof. By Lemma 1.1, Wi ≥ X j − Xi , and by Lemma 1.2, Wi ≤ X j − Xi . Consequently, Xi + Wi = X j .  In a similar fashion, we can also obtain similar inequalities and equalities on the height of an RB ∈ 0 . Lemma 2.1. For an RB Bi ∈ 0 , if there is an RB B j ∈ 0 , i = j, j = arg max{Yk : Yk < Yi , Xk ≤ Xi }, then Yi − Hi ≤ Y j . If no such Bj exists, k

then Yi − Hi = 0. Lemma 2.2. For an RB Bi ∈ 0 , if there is an RB B j ∈ 0 , j = i satisfies Yj < Yi and X j = Xi . Then Hi ≤ Yi − Y j . Lemma 2.3. For an RB Bi ∈ 0 , if there is an RB B j ∈ 0 , j = i satisfies j = arg max{Yk : Yk < Yi , Xk ≤ Xi } and X j = Xi , then Yi − Hi = Y j . k

From Lemma 1.1 to 2.3, we can obtain a sufficient condition that a CRP 0 can be reconstructed with only the knowledge of the topleft coordinates of each RB in it. To show this, we start with the following definition. Definition 2. Let U be a symmetric binary relation on two RBs in a CRP. For two RBs Bi and Bj in a CRP 0 , we say the ordered pair (Bi , Bj ) ∈ U if either 1) Xj > Xi and Yj > Yi , or 2) (Bj , Bi ) ∈ U. When (Bi , Bj ) ∈ U, we also say Bi is U-related to Bj . Fig. 2 shows the ordered pair sets U of two distinct CRPs. Note that U can be empty set in some CRPs, such as the one depicted in Fig. 2(b).

On the other hand, if there is no RB∈ T1 , then combining the fact that there is no RB which Bi is U-related to and 0 is a CRP, it can be seen that there is no RB Bk satisfying Xk ≤ Xi and Yk < Yi . In this case, Hi = Yi according to Lemma 2.1. Otherwise, due to the fact that there is no RB which Bi is U-related to, an RB Bk satisfying Yk < Yi and Xk ≤ Xi also has the property that Xk = Xi . And the height of Bi can then be obtained by Hi = Yi − Y j , according to Lemma 2.3, where j = arg max{Yk : Bk ∈ T1 }. k

From above, it has been proved that there exists a procedure to determine the width and height of Bi , which is sufficient to lead to the desired result.  Theorem 1. A CRP 0 can be reconstructed with only the knowledge of the top-left coordinates of all its RB if U = ∅. Proof. Since U = ∅, the width and height of each RB∈ 0 can be obtained by the procedure illustrated in the proof of Lemma 3. Thus, 0 can be reconstructed.  An example of the reconstruction procedures illustrated above is depicted in Fig. 3. Fig. 3(a) is a CRP which satisfies the condition in Theorem 1. Suppose only the top-left coordinate of each RB is known initially, as shown in Fig. 3(b). Fig. 3(c) and (d) illustrate the procedure of reconstructing the widths and the heights of each RB, respectively. Let a be a subset of  which represents the set of CRP with no RB ordered pair ∈ U. And let Ca be the set formed by the collection of top-left coordinates of the RBs in a CRP ∈ a . Theorem 1 implies that the function Fa : a → Ca which maps a CRP ∈ a to the collection of the top-left coordinates of the RBs in

T.-Y. Tsai et al. / Computer Networks 115 (2017) 82–99

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Fig. 3. An example of reconstruction a CRP which satisfies the condition in Theorem 1 with the top-left coordinates of each RB in it. (a) a CRP satisfying the condition in Theorem 1 (i.e. U = ∅). (b) The top-left coordinates of all RBs in it. (c) Obtaining the RB widths of this CRP based on the knowledge of the top-left coordinates as illustrtated in the proof of Lemma 3. (d) Obtaining the heights of each RB of the CRP as illustrated in the proof of Lemma 3.

it is bijective, and since Fa is obviously onto by definition and is also one-to-one due to the invertibility according to Theorem 1. Theorem 1 states a sufficient condition for a CRP 0 to be fully reconstructed with only the knowledge of the top-left coordinates of only one corner of each RB in it is 0 ∈ a . Conversely, the necessary condition that an ambiguity occurs is that 0 ∈  \ a . That is, there exist two RBs Bi , Bj ∈ 0 such that (Bi , Bj ) ∈ U. Let Cb be the set formed by the collection of the top-left coordinates of the RBs in a CRP ∈ a . In such a case, the function Fb :  \ a → Cb that maps the CRP ∈ a to the top-left coordinates of the RBs is no longer one-to-one and more than one CRP ∈ a may map to exactly the same collection of top-left coordinates. Fig. 4 is an example that a CRP fails to be reconstructed by the top-left coordinates of the RBs in it. Fig. 4(a) is a CRP with four RBs B0 to B3 . Note that B0 and B1 are not U-related to other RBs in this CRP, and their widths and heights can be determined by the procedure described in the proof of Lemma 3. However, B2 and B3 are U-related to one another and their widths and heights can not be reconstructed since more than one possible CRPs exist and cause ambiguities. Fig. 4(c) and (d) show two CRPs with the same collection of top-left coordinates and different widths and heights of B2 and B3 . Since Fa and Fb are onto, the above results also implies that |a | = |Ca | and |a | > |Cb |. Thus, to uniquely reconstruct a CRP, we need to introduce information besides the top-left coordinates of the RBs in it. To do this, we need to define another binary relation on the RBs in a CRP. Definition 3. Let G be a binary relation on two RBs in a set of RBs. For two RBs Bi and Bj , we say the ordered pair (Bi , Bj ) ∈ G if 1) Xi + Wi = X j . 2) Yi − Hi < Y j ≤ Yi or Y j − H j < Yi < Y j .

When (Bi , Bj ) ∈ G, we also say Bi is G-related to Bj . Moreover,

we say (Bi , B j ) ∈ G if Bi is G-related to Bj and Y j − H j < Yi ≤ Y j .



When (Bi , B j ) ∈ G , we also say Bi is G -related to Bj . Definition 4. Let  := {B0 , · · · , BN−1 } be an RB set and X be a binary relation on the RBs in . Let i ∈ {0, · · · , N − 1}, LiX, is defined as

LiX, := {B j : (Bi , B j ) ∈ X}.

Fig. 5 is an illustrative example of G and G relations in an RB set. It can be proved that a CRP can be identified by the top-left

coordinates of each RB and its associated relation G . Theorem 2. Let  denote the set formed by the collection of the top left coordinates of each RBs and its associated relation G of all CRPs. The function FG :  →  which maps a CRP to the left-top coordi

nates of its RBs and the associated relation G is bijective. Proof. See Appendix A.



Theorem 2 reveals that each CRP can be identified by the top left coordinates of its RBs and the associated relationship G . Now,

the next question is how to convey G with the smallest amount of bits to minimize the overall overhead. Here, we must prove the

relationship G of a CRP can be one-to-one mapped to a specific linear order relation which implies that it is sufficient to describe an arbitrary CRP by carrying the top-left coordinates of the RBs in a specific sorted order in the SI message. Definition 5.  is a specific collection of RB set. An RB set with K RBs {B0 , · · · , BK−1 } is in  if it satisfies the following properties

88

T.-Y. Tsai et al. / Computer Networks 115 (2017) 82–99

Fig. 4. An example that a CRP fails to be reconstructed by the top-left coordinates of the RBs in it (a) a CRP which does not satisfy the condition of Theorem 1 (i.e. U = ∅). (b) The top-left coordiantes of the RBs in (a). (c) One possible CRP which has the set of top-left coordinates in (b). (d) Another possible CRP which has the set of top-left coordinates.



Fig. 5. An illustrative example of G and G relations. For the RB set {B0 , , B7 },

the RBs which B0 is G-related to is B2 , B3 and B4 . Additionally, B0 is also G -related to B2 . On the other hand, B1 is G-related to B6 and B2 is both G-related to and

G -related to B7 . For this RB set, G = { (B0 , B2 ), (B0 , B3 ), (B0 , B4 ), (B1 , B6 ), (B2 , B7 )},

G = { ( B0 , B2 ), ( B2 , B7 )}.

1) Bi



B j = ∅, for i = j. 2) Let point p = (X, Y ) ∈ A. If p ∈ /

 Bk , k

then so does the line { p + (0, −t )|t ∈ [0, Y )}.

Fig. 6. An illustrative example of M and M relations. For the RB set {B0 ,, B5 },

the RBs which B2 is M-related to is B1 and B0 , and B2 is also M -related to B1 . For this RB set, M = { (B2 , B1 ), (B2 , B0 ), (B1 , B0 ), (B5 , B4 ), (B5 , B3 ), (B4 , B3 )}, and

M = { (B2 , B1 ), (B1 , B0 ), (B5 , B4 ), (B4 , B3 )}. Moreover, B0 is the only isolated RB in the RB set.

Lemma 4. There is at least one isolated RB in an RB set  := {B0 , · · · , BK−1 } ∈ . Proof. See Appendix B.



Definition 6. M is a binary relation on two RBs Bi and Bj ∈ A. We say the ordered pair (Bi , Bj ) ∈ M if X j ≤ Xi + Wi , Yj < Yi , and X j + W j > Xi .

Lemma 5. Given an RB set  := {B0 , · · · , BK−1 } ∈ . When K > 1, and Bm is isolated in , then Bm ∈ .

Moreover, we say (Bi , B j ) ∈ M if Y j = Yi − Hi and X j ≤ Xi < X j + Wj.

Proof. See Appendix C.

Definition 7. Let  := {B0 , · · · , BK−1 } ∈ , an RB Bk ∈  is said to be isolated in  if LkG, = ∅ and LkM, = ∅.

Fig. 6 is an illustrative example of M and M relations and isolated RB in an RB set.



Theorem 3. Let 0 := {B0 , · · · , BNCRP −1 } be a CRP. There is an ordered sequence o0 , · · · , oNCRP −1 where ok ∈ {0, · · · , N − 1} and ok1 = o

ok2 , for k1 = k2 , with which LGk, ⊆ {Bok+1 , · · · , BoN −1 }, for k < CRP 0 o o NCRP . Moreover, for an RB Bok ∈ 0 , if LGk, = ∅, then |LGk , | = 1 and

0

o

0

LGk , = {Bm : m ∈ {ok+1 , · · · , oNCRP −1 }, Xm = αk , Ym = βk } 0

T.-Y. Tsai et al. / Computer Networks 115 (2017) 82–99

where

Combining the above two lemmas, we have the following theorem:

αk = min{Xn : Bn ∈ {Bok+1 , · · · , BoNCRP −1 }, Xn > Xk }

Theorem 4. For a CRP 0 ∈ {B0 , · · · , BNCRP −1 } and l ∈ {0, · · · , NCRP − 1}, the following statements are true

βk = min{Yn : Bn ∈ {Bok+1 , · · · , BoNCRP −1 }, Xn = αn , Yn ≥ Yok }. On

the

other

hand,

o LGk, 0

if

= ∅,

{Xn : Bn ∈

then

{Bok+1 , · · · , BoNCRP −1 }, Yn ≥ Yok , Xn > Xok } = ∅. Proof. See Appendix D.

l

l1 ≤ l. 2. For an RB Bk ∈ 0 , LkM, = ∅ if and only if Lk

of the RBs (X0 , Y0 ), · · · , (XNCRP −1 , YNCRP −1 ) and the order sequence o0 , · · · , oNCRP −1 can uniquely identify G . Thus, the top-left coordinates and the order sequence is sufficient to reconstruct 0 from Theorem 2. As a result, encapsulating the coordinate information in the order (Xo0 , Yo0 ), · · · , (XoN −1 , YoN −1 ) in the CRP

CRP

SI message is enough to describe 0 . Since the heights and widths information are not needed, the corresponding length of the resource description portion when using the SRD method is Md = NCRP MSRD , where MSRD = log2 C + log2 S . Compared to the RPD method illustrated in Section 2.3, the bits for describing the resource allocation is saved by 50%. Moreover, the event that part of coordinates are missing or in errors during transmissions should not need to be concerned in the cases we considered in the paper, since all the ordered coordinates are encapsulated in the SI message, which is attached by CRC for error detection. 3.2. Sorting/reconstruction algorithms

Consider a CRP 0 := {B0 , · · · , BNCRP −1 }. In this subsection, we investigate efficient algorithms for sorting the RBs in 0 to obtain the order sequence o0 , · · · , oNCRP −1 as described in Theorem 3 and reconstructing 0 from the sorted top-left coordinates (Xo0 , Yo0 ), · · · , (XoN −1 , YoN −1 ), respectively. It was proved CRP

CRP

that, with the aid of appropriate data structures, the proposed sorting and reconstruction algorithms both have linear time complexity O (NCRP ). In the following illustration, we adopt the notations in the proof of Theorem 3. Where 0 := {B0 , · · · , BNCRP −1 } is a CRP, and l denotes the RB set after the lth excluding procedure. 3.2.1. Sorting algorithm The sorting algorithm is based on implementing the sequential excluding procedure which find an isolated RB in RB set l , l ∈ {0, · · · , NCRP − 1} and excluding it to obtain a new RB set l+1 as described in the proof of Theorem 3. To further improve the efficiency of the excluding procedure, we have the following lemmas. Lemma 6. For an RB Bk ∈ 0 , if |LkG, | ≥ 2, then there exist two RBs 0

Bm1 and Bm2 , where Bm1 ∈ LkG , and Bm2 ∈ LkG, \ LkG , , moreover, 0

0

∃l1 ∈ {0, · · · , N − 2} such that Bm1 ∈ l1 , and Bm2 ∈/ l1

0

l

Since Ym1 ≥ Yk > Ym2 , and Xm1 = Xm2 = Xk + Wk , Consequently, in the excluding procedure in the proof of Theorem 3, Bm1 can not be excluded before Bm2 is. Hence, there exists l1 ∈ {0, · · · , N − 2} such that Bm1 ∈ l1 and Bm2 ∈ / l 1 .  Lemma 7. For an RB Bk ∈ 0 , if |LkM, | ≥ 2, then there exist two RBs 0

Bm1 and Bm2 , where Bm1 ∈ LkM , and Bm2 ∈ LkM, \ LkM , , more0

0

0

over, ∃l2 ∈ {0, · · · , N − 2} such that Bm1 ∈ l2 , and Bm2 ∈ / l 2

= ∅, for each

2

l2 ≤ l. 3. The necessary and sufficient condition that an RB Bk is isolated in

l is Lk = ∅ and Lk = ∅ for each l1 , l2 ∈ {0, · · · , l }. G , l

M , l

1

Proof. See Appendix E

2



The results of Theorem 4 could provide an efficient way to obtain an isolated RB in each excluding procedure. The basic concept is maintaining two sets S k and S k and two binary indicators G

M



I1k and I2k for each RB Bk , where S k := {k : (Bk , Bk ) ∈ G } and



G

S k := {k : (Bk , Bk ) ∈ M }. If Bk is excluded in the lth excluding M

procedure (l ∈ {1, · · · , NCRP − 1}), then we know from the state ment 1 in Theorem 4 that all the RBs which Bk , k ∈ S k are G

G-related to have been excluded. And we mark the indicators



I1k , k ∈ S k as 1. Similarly, the indicators I2k , k ∈ S k are also G

M

marked as 1 to indicate that all the RB Bk , k ∈ S k are M-related M

to have been excluded according to statement 2 of Theorem 4. Finally, for an RB that the associated indicators are both marked as 1, it means the RB has been isolated and can be selected and excluded in the sequential excluding procedures. The index of such RB is then stored in a set I. S k , S k , and the initial values of I1k , I2k , and I can be obtained G

M

by Algorithm 1, with the aid of a data structure allocation maAlgorithm 1 Algorithm to obtain the initial value. Input: Allocation matrix Z, and RB information (Xk , Yk ), (Wk , Hk ), k ∈ {0, · · · , NCRP − 1}. Assume the RBs form a CRP of frame A. Output: S k , S k , and the initial value of I1k , I2k and I, k ∈ G

M

{0, · · · , NCRP − 1}.

for k:=0 to NCRP − 1 do if Xk + Wk = C then I1k ← 1. 3: 4: else I1k ← 0. 5:

1:

2:

6: 7: 8: 9: 10: 11: 12:

|LkG | ≥ 2, there are two RBs Bm1 ∈ LkG and

1

M ,

CRP

Bm2 ∈ LkG \ LkG . ( Bm1 , Bm2 ) ∈ M.

1. For an RB Bk ∈ 0 , LkG, = ∅ if and only if Lk = ∅, for each G , l

l



According to Theorem 3, for any CRP 0 := {B0 , · · · , BNCRP −1 }, there exists a bijective mapping J : {B0 , · · · , BNCRP −1 } → {0, · · · , NCRP − 1} such that one can let J (Bo0 ) = 0, J (Bo1 ) = 1, · · · , J (BoN −1 ) = NCRP − 1. In addition, the top-left coordinates

Proof. Assume

89

Add k into S

zX +W +1,Y k k k .

G

end if if Yk − Hk = 0 then I2k ← 1. else I2k ← 0 Add k into S

zX +1,Y −H k k k .

M

end if if I1k == 1 and I2k == 1 then Add k into I. 15: end if 16: 17: end for

13: 14:

trix Z, where Z = {zi, j }, i ∈ {1, · · · , S}, j ∈ {1, · · · , C }, and zi, j = k if (i − 1, j ) ∈ Bk . Then, the order sequence o0 , · · · , oNCRP −1 is calculated by Algorithm 2, which is based on the basic concept illustrated above. The variables of Algorithms 1 and 2 are listed as follows

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T.-Y. Tsai et al. / Computer Networks 115 (2017) 82–99

Algorithm 2 o0 , · · · , oNCRP −1 .

Algorithm

to

obtain

the

order

Step 1: Set all elements in Q to 0. Step 2: Set variable k to NCRP − 1 Step 3: Set two variables w and h to 1. Step 4: Mark the matrix element corresponding to the lefttop corner of Bok to 1, that is, set qXo +1,Yo to 1.

sequence

Input: I1k , I2k , S k , S k , and I, k ∈ {0, · · · , NCRP − 1}. G

M

Output: The ordered sequence o0 , · · · , oNCRP −1 defined in Theorem 3. 1: for k:=0 to NCRP − 1 do Remove the first element m from I. 2: oNCRP −k−1 ← m 3: for each k ∈ S m do 4:

Step 5:

6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 16:

k

k

+w+1,Yo

k

k

is equal to 1, go to Step

6. Otherwise, set qXo +w+1,Yo to 1 and set w to w + 1. Then, k k repeat Step 5. Step 6: If Yok − h = 0 or qXo +1,Yo −h is equal to 1, go to Step

G

I1k ← 1 if I1k == 1 and I2k == 1 then Add k into I. end if end for for each k ∈ S m do

5:

If Xok + w = C or qXo

7. Otherwise, set qXo

k

k

+1,Yo −h k

k

to 1 and set h to h + 1. Then,

repeat Step 6. Step 7: Set Wok to w, and set Hok to h. If k is equal to 0, terminate the procedure. Otherwise, set k to k − 1 and go to Step 3. The main operations in this reconstruction algorithm are in the check and mark operations of Step 5 and Step 6. One can observe that the number of operations of Step 5 and Step 6 are proporNCRP −1 NCRP −1 tional to k=0 Wk and k=0 Hk , respectively. By the facts 1 ≤ Hk ≤ S and 1 ≤ Wk ≤ C, we have

M

I2k ← 1 if I1k == 1 and I2k == 1 then Add k into I. end if end for end for

NCRP −1

2NCRP ≤



(Wk + Hk ) ≤ (S + C )NCRP .

k=0

Z: An allocation matrix. Z = {zi, j }, i ∈ {1, · · · , S}, j ∈ {1, · · · , C }, where zi, j = k if (i − 1, j ) ∈ Bk . I1k (k ∈ {0, · · · , NCRP − 1} ): A binary indicator which is 1 if Lk = ∅ (and thus LkG, = ∅ from Lemma 6) and is 0 othG ,l

l

as that of I1k .

S k (k ∈ {0, · · · , NCRP − 1} ): The set of the RB indices k satisfying

( Bk , Bk ) ∈ G .

k S (k ∈ {0, · · · , NCRP − 1} ): The set of the RB indices k satisfyM

ing (Bk , Bk ) ∈ M . I: The set of the indices of the isolated RBs. Note that the number of iteratives in Algorithms 1 and 2 are NCRP −1 proportional to NCRP and (|S k | + |S k | ), respectively. It k=0 M NCRP −1 kG  can be proved that neither k=0 |S | nor N−1 |S k | be larger k=0 G

M

than the total amount of RB NCRP (see Theorem 5), thus the entire encoding procedure consisting initializing the data structures (Algorithm 1) and calculating the order sequence (Algorithm 2) has a time complexity O (NCRP ). NCRP −1 k  CRP −1 k Theorem 5. Neither |S | nor Nk=0 |S | are not larger k=0 G

than NCRP . Proof. See Appendix F.

M



3.2.2. Reconstruction algorithm Based on the conceptual recovering procedure with the top left coordinates and the associated G relation in the proof of Theorem 2 and the result of Theorem 3, we propose an algorithm to reconstruct the CRP 0 upon a user receives the sorted top-left coordinate (Xo0 , Yo0 ), · · · , (XoN −1 , YoN −1 ). The reCRP

CRP

construction algorithm calculates the widths and heights of RBs in BoN −1 , · · · , Bo0 sequentially, with the aid of a binary matrix Q := CRP

3.3. Strategy to partition the empty region

l

erwise. l ∈ {0, · · · , NCRP − 1} is the index of excluding procedure which is illustrated in the proof of Theorem 3. I2k (k ∈ {0, · · · , NCRP − 1} ): A binary indicator which is 1 if LkM, = ∅ and is 0 otherwise. The definition of l is the same

G

Thus, the reconstruction algorithm also has a time complexity O (NCRP ).

{qi, j , i ∈ {1, , S}, j ∈ {1, , C}}, where qi, j is corresponding to the lattice point (i − 1, j ) ∈ A. The reconstruction algorithm is done by the following steps:

In Section 3.2, we have developed an approach to identify a CRP with the sorted top-left coordinate of its RBs. Now we consider a more general case which describes an non-CRP RB set  := {B0 , · · · , BNB −1 }, where Ad ⊂ A, and Ae = A \ Ad = ∅ in order to incorporate the presence of the empty region. In general, the empty region Ae is not necessarily a rectangular polygon. However, since it is a union of RUs, it can be partitioned into finite number of non-overlap virtual RBs, denoted as NVB . The virtual RB also obeys the definition of RB as described in Definition 1. In other words, nonNV B −1 overlapped virtual RBs BNB , · · · , BNB +NV B −1 satisfy Ae = BNB +k k=0

and BNB +k1 ∩ BNB +k2 = ∅, k1 = k2 . Therefore, the extended RB set

 := {B0 , · · · , BNB −1 , BNB , · · · , BNB +NV B −1 } forms a CRP which is applicable for the proposed SRD method. One could also sort the RBs by Algorithms 1 and 2 in Section 3.2. Note that the number of information bits used to describe the CRP in the proposed SRD method is monotone increasing with the number of RBs in the CRP. Thus, Ae is prefered to be partitioned into the smallest number of virtual RBs (i.e. minimize NVB ) to minimize the overhead. Now we can make use of the following observation. Since Ae is the union of RUs, it can be uniquely decomposed as polygons such that each polygon intersects with each other with at most one point. Moreover, the decomposed polygons are rectilinear, that is, the edges of the polygons are either parallel to the horizontal axis or the vertical axis. It is clear that Ae is partitioned into minimal number of virtual RBs if and only if each rectilinear polygon is partitioned into minimal number of rectangular polygons. The problem to partition a rectilinear polygon into minimal number of rectangular polygons has been investigated and solved in literatures [21–23]. In Appendix G, we illustrate the approach proposed in [21], which translated the rectilinear partition problem to a minimum independent set searching problem in a biparite graph, and can be efficiently solved by several existing algorithms with a polynomial time complexity in the number of vertices of

T.-Y. Tsai et al. / Computer Networks 115 (2017) 82–99

91

Table 1 Key parameter settings for simulations. Parameter

Value

Description

Nint Nslot Lo LID LCRC Ri

10 0 0 24,0 0 0 10 12 16 50 15,0 0 0 600 1,2,4,6 12 12 4 63 24 0.8 0.2 300 0.05 200 500 100 500 20,0 0 0

Number of iterations to average each simulation result Number of time slots simulated in each iteration Length in bits of the transmission format information Length in bits of user ID Length in bits of CRC The inner radius in meters of a cell Subcarrier spacing in hertz Number of used subcarriers Frame duration, number of time slots a frame occupies Number of subcarriers per subchannel Number of OFDM symbols in a time slot Path loss exponent The average SNR in dB for the user at cell inner cycle Number of REs in an CCE Penalty factor for data channels Penalty factor for control channels Number of users in a cell in usage scenario 1 Probability for a user to be active in usage scenario 1 Moving averaging window size for PF scheduler The TB size in bits in usage scenario 2 The period in ms of packet arrival in usage scenario 2 The tolerant delay in ms of each packet in usage scenario 2 The buffer size at the base station in usage scenario 2

f

Nc Ns Nsc Nsymb

α

SNR0 NCCE

δd δc

Nuser pa Tw LM2M TM2M DM2M BM2M

the rectilinear polygon. Also note that, the additional operations for partitioning the rectilinear polygons is only needed at the base station.

In the following simulations, Ri and the transmission power of the base station are fixed and Ro is adjusted to evaluate the performance in different cell sizes (and thus different cell-edge SNR). We will illustrate the detailed setting of Ro later.

3.4. Strategy to partition allocated resources to RBs In this subsection, we address the issue that partition the resource allocated to each scheduled user to a set of RBs. As illustrated in Section 3.3, the resource allocated to each scheduled user is a union of RUs ∈ {Rd0 , · · · , RdN −1 }. Thus, similar to the empty sche

region Ae , the resource allocated to each scheduled user can also be divided into disconnected rectilinear polygons as illustrated in Section 3.3. Consequently, the algorithm for solving the rectilinear partition can also be invoked to partition the resource allocated to each scheduled user into minimal number of RBs with which the overhead of the SI message is also minimized. 4. Simulation results In this section, via Monte Carlo simulations, we evaluate the performance of different SI bearing schemes under different traffic models, each of which corresponds to a usage scenario. The SI bearing scheme is the combination of transmission strategy (i.e. JEB or SET) and the burst description methods, such as BMD and RPD in Section 2.3 and the SRD method proposed in Section 3. Each simulation result is obtained by averaging over Nint independent iterations in which Nslot time slots are simulated. The setting of key simulation parameters are summarized in Table 1. 4.1. Simulation parameters 4.1.1. System-level deployment The considered OFDMA PMP downlink system consists of one base station surrounded by a number of users. The users are are indexed as 0, 1,  and uniformly distributed in a donut shaped cell layout with inner and outer radiuses Ri and Ro , respectively. The distance between user u and the base station is denoted as ru and has the following probability density function (PDF):



f r u (x ) =

2x , R2o − R2i 0,

Ri ≤ x ≤ Ro otherwise.

(6)

4.1.2. Physical-layer parameter setting The OFDMA system in the simulation has a subcarrier spacing f 15 kHz and totally Nc = 600 data subcarriers around the DC subcarrier, which are indexed as 0, · · · , Nc − 1 according to the ascending order of the baseband frequencies Nc  Nc  − 2 f , · · · , − f ,  f , · · · , 2 f . The duration of an OFDM symbol can be expressed as Ts = TCP + Td , where Td = −1 is the duf

ration of the signal portion and TCP = 0.125Td is that of the cyclic prefix (CP) portion of an OFDM symbol, respectively. Moreover, we define a time slot is a time-domain unit comprising Nsymb consecutive OFDM symbols, where Nsymb = 12, and subchannel is a frequency-domain unit comprising Nsc = 12 consecutive subcarriers. The time slots are indexed as 0, 1,  and so on. In each time slot, there are NNscc subchannels and are indexed as 0, · · · , NNscc − 1 from the lower frequency to the higher frequency. Also note that the setting of a time slot is equivalent to the duration of 1 ms. We define the time/frequency resource comprised by a subcarrier and an OFDM symbol as a resource element (RE). In a given time slot, we use the ordered pair (m, n) to identify the RE corresponding to subcarrier m and the nth OFDM symbol in it (m ∈ {0, · · · , Nc − 1} and n ∈ {0, · · · , Nsymb − 1}). Finally, every consecutive Ns time slot is grouped as a frame. Thus, there are Ns Nsymb OFDM symbols in a frame and indexed as 0, · · · , Ns Nsymb − 1. 4.1.3. Channel propogation models The modeling of the physical wireless channel between a base station and a user includes large-scale and small-scale fading models. The large-scale fading model contains a path loss factor which reflects the distance dependent attenuation of the transmitted signals. For user u, the signal attenuation multiplicative factor is modu eled by log-distance function which is given by lo ( r )−α , where α Ro denotes the path loss exponent and lo is a positive constant. On the other hand, the small-scale fading effect is caused by the multi-path factor of the channel impulse response (CIR) which can be charaterized by a tapped-delayed line model. Combining these

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T.-Y. Tsai et al. / Computer Networks 115 (2017) 82–99

two fading models, the baseband CIR from the base station to user u at time t can be given by

CIR (τ , t ) = u



r u −α  l0

aui (t )δ (τ − τiu (t ))e j2πψi (t )t

R0

u

(7)

i

√ where j = −1, and aui (t ), τiu (t ), and ψiu (t ) are the tap gain, delay, and Doppler frequency shift of the ith path at time t, respectively. Moreover, the Doppler spread is defined as the supremum of the absolute value of ψiu (t ), which is proportional to the user’s velocity [24]. In the simulations, we assume all users are fixed and the environment is stationary in the simulated duration. Therefore, the variable t in (7) can be temporarily omitted and each wireless channel can be regarded as a linear time-invariant system with channel frequency response (CFR)

H (f) = u



r u −α  l0

aui e− j2π f τi .

R0

u

i

In a given time slot, the CFR of RE (m, n) for user u is defined as hum,n := Hu ((− Nc + m ) f ). The tap gains aui are assumed to be 2 independent zero-mean complex Gaussian random variables with  the average power sum normalized to unity. That is, E(aui aui ) = 1 i

and





E hum,n hum,n = l0 (

r u −α ) Ri

(8)

where (· ) is the conjugate of the argument, and E(· ) denotes the expectation. Moreover, the delay and mean power of each tap are set according to the Extended Typical Urban (ETU) model defined by International Telecommunication Union (ITU) [27]. The received SNR at RE (m, n) for user u in a given time slot is defined as u γm,n :=

(hum,n hum,n )Pm,n N0  f

where N0 is the power spectrum density of the noise and Pm, n is the average power of the modulated signal allocated in RE (m, n). Assume that equal power allocation is adopted in the downlink channel. That is, when RE (m, n) is allocated for data transmission, the average power of the modulated signal is P/Nc , where P is the total power of the base station. In fact, equal power allocation should be close to optimal if the scheduler selects the users with SNR higher than a reasonable threshold [7]. Combining (6) and (8), the cumulative distribution function of the average SNR per RE can be expressed as 0.1SNR

u P r (E(γm,n ) ≤ ζ) =

R2i ( 10 ζ 0 ) α R2o − R2i

where SNR0 = 10 log10

l0 N0  f

4.1.5. Resource element drawing for transmitting SI messages In a frame, one or more (depending on the adopted strategy, JEB or SET) SI messages are sent to convey the scheduling information to the users scheduled in it. Since the SI message is smaller than most data packets, logical resource with a size smaller than that of RU, namely control channel elements (CCE), is used for its transmissions. An CCE contains NCCE REs which are drawn from the interlaced RUs at the same time slot with a period of N C CC E subchannels. CCEs are indexed from 0 and an CCE with index k is composed by drawing an RE from the RUs occupying subchannels (kmod N C ) + NnC , n ∈ {0, · · · , NCCE − 1}. In each frame, the CCEs CC E CC E are assigned from the latest time slot to the earlier time slots in accordance to the indices. We assume each SI message is encoded with coding rate ranged from 18 to 78 and modulated by Quadrature Phase-Shift Keying (QPSK). The combinations of the available coding rates and modulations correspond to spectrum efficiencies 0.25 to 1.75 bits per degree of freedom. When an SI message is transmitted, it is encoded with coding rate matching to the smallest amount of CCEs such that the corresponding spectrum efficiency is smaller than the estimated control channel capacity of the destined user with the weakest channel status. Where the definition of the estimated control channel capacity will be described later. Moreover, the CCEs are allocated in the ascending order of their indices until every SI message is carried. When an CCE is assigned to transmit an SI message, the REs in the CCE will not be used for data transmission in avoidance of collisions. That is, when an RU is allocated for data transmission, the number of available REs is Nsc Nsymb minus the number of REs which are assigned to CCEs for transmitting SI messages. Since the subcarriers of CCEs are spreaded over the available subcarriers, the estimated control channel capacity of a user is obtained by calculating the sample average of the estimated capacity of all the REs in a frame. That is, for user u, its estimated control channel capacity Cˆu is decided by

Cˆu =

1 Nc Ns Nsymb



u log2 (1 + δc γm,n )

m∈{0,··· ,Nc −1},n∈{0,··· ,Ns Nsymb −1}

where δ c is the penalty factor for control channels to include the performance gap between the practical channel coding and the optimal channel capacity [25]. 4.1.6. SI bearing schemes Three SI bearing schemes are evaluated and compared.

1

(9)

is the average SNR per RE in dB for

the user at the inner cycle of the cell. Following the assumption in 3GPP, we set SNR0 = 63 dB, which is corresponding to P = 49 dBm [26], N0 = −174 dBm/Hz, and l0 = 83 dB. Finally, we also define the cell-edge SNR as the 5-percentile worst SNR in the cell. 4.1.4. Resource unit for data transmission RU is the time/frequency resource allocation unit which is constituted by a time slot and a subchannel. The setting results in an S × C time/frequency resource table with height S = Ns and width C = NNscc = 50 for each frame. To observe the performance under different scheduling cycle, we will evaluate different numbers of time slots a frame occupies (i.e. different Ns ). Moreover, in a given time slot n, the RUs in it are indexed as 0, · · · , NNscc − 1. In this time slot, RU k occupies the radio resource of subchannel k and contains the RE set Rk = {(m, n ) : m ∈ {kNsc , · · · , (k + 1 )Nsc − 1}, n ∈ {0, · · · , Nsymb − 1}}. The relationship of RE, RU, and frame is shown in Fig. 7.

1. BMD-SET: SET transmission strategy is adopted, and each SI message describes the resource allocation with BMD method illustrated in Section 2.2. It is an alternative scheme in LTE [1], LTE-A [2] and IEEE 802.16m [4] 2. RPD-JEB: JEB transmission strategy is adopted and each SI message describes the resource allocation with RPD method illstrated in Section 2.2. It is also the scheme adopted by IEEE 802.16e [3]. 3. SRD-JEB: JEB transmission strategy is adopted and the SI message describes the resource allocation with proposed SRD method. In the BMD-SET scheme, the resource allocation of each scheduled user is described with BMD method. Separate SI messages are sent to individual scheduled users with the message size (in bits)

LBMD−SET = MBMD + Lo + LCRC bits, where the resource description portion occupies MBMD bits, and the user information portion occupies Lo bits. MBMD = Nsche = SC is the length of the bitmap for BMD method. Lo is the length in bits of the transmission format information for the destined user

T.-Y. Tsai et al. / Computer Networks 115 (2017) 82–99

93

Fig. 7. The relationship of resource element (RE), resource unit (RU), and frame.

of the SI message. LCRC denotes the length in bits of CRC portion of the SI messages. When SET strategy is adopted, blind decoding capability is assumed for each user. The user ID of each destined user is masked on the CRC portion of each SI message and does not appear in the data portion. For RPD-JEB scheme, the resource allocation u of each scheduled user u is partitioned to minimum number of non-overlapped RBs using the algorithm in Section 3.4. Then, the partitioned RBs of each scheduled user are described using RPD method. The scheduling information of all scheduled users are finally encapsulated into a single SI message which has the message size (in bits)

LRPD−JEB = LB + NB (MRPD + LID ) + Nu Lo + LCRC where the resource description portion occupies NB MRPD bits, and the user information portion occupies LB + NB LID + Nu Lo bits. LB = log2 SC denotes the number of bits used to describe the total number of RBs NB (NB ≤ SC). LID is the length of user ID. Nu is the number of the users scheduled in the frame. Moreover, a fixedsized common header is sent to indicate the information for the users to decode the SI message, which contains the number of CCEs the SI message occupies and the used coding rate. It is supposed to be decoded by all users and thus is always encoded with the most robust coding rate (i.e. 18 ) and transmitted by the CCEs with the smallest indices. The size of common header is set to be 14 bits to accommodate 7 possible coding rates and at most 1800 CCEs in a frame (corresponds to the evaluated frame duration Ns = 6). After encoded, common header is transmitted by CCE 0,1,2 in each frame. SRD-JEB scheme is similar to RPD-JEB scheme except the proposed SRD method is adopted to describe the resource allocation instead of using RPD method. Since the SRD method requires the entire frame to be partitioned by RBs to obtain a CRP, the empty region, if present, should also be partitioned into virtual RBs as described in Section 3.3. Moreover, the virtual RBs are regarded as the allocation of a virtual user which is also assigned an user ID. Let the total number of partitioned RBs (including the RBs associated to data region and empty region) be NCRP . The length of an SI message in bits of SRD-JEB scheme is given by

4.2. Numerical results Two usage scenarios with different traffic models, namely, the full buffer traffic model and the periodic small-data traffic model are employed to evaluate the performance under two particular usage cases. 4.2.1. Usage scenario 1 (the full-buffer traffic test) The full-buffer traffic model is typically used to model elastic or best-effort services [28]. In each iteration, there are Nuser users indexed as 0, · · · , Nuser − 1, and each user is marked as active users with a probability pa . When an user is active, it is assumed there always is downlink data to transmit. As in many studies, for this scenario, a proportional fair (PF) scheduler [19,20] is adopted to allocate RUs to each active user, which is known capable of promising system throughput while meeting stringent fairness criteria [20]. RU k of time slot n is allocated to the user with index arg max Duk /T u (n − 1 ), where A is u∈A

the set of the indices of active users, Dku is the instantaneous data rate user u can get if RU k is scheduled to it, and T u (n − 1 ) is the historical average data rate of user u before time slot n. The  u ), instantaneous data rate is given by Dku = log2 (1 + δd γm,n (m,n )∈Rk

where 0 < δ d < 1 is the penalty factor for data channel. Since control channels in general require higher robustness than that of data channels, we assume δ c ≤ δ d . Historical data rates T u (n ) are updated per time slot n as

T u (n ) = (1 − Tw−1 )T u (n − 1 ) + Tw−1



Dku

k∈S u (n )

LSRD−JEB = LB + NCRP (MSRD + LID ) + Nu Lo + LCRC

where S u (n ) denotes the index set of the RUs allocated to user u at time slot n, and Tw ∈ N is the moving averaging window size. At the beginning of each iteration, the historical average data rate of each active user is set to 0. Three metrics are used for performance comparison under different cell-edge SNR in this scenario. The first is average overhead Nˆ tot NCC E ratio , which is the averaged ratio of the number of REs Nc Nsymb Ns for transmitting SI messages in a frame. Nˆ tot is the averaged num-

where the resource description portion occupies NCRP MSRD bits, and the user information portion occupies LB + NCRP LID + Nu Lo + LCRC bits. Finally, common header is also needed to be send as in RPD-JEB scheme.

ber of used CCEs in a frame. The second metric is effective spectrum efficiency, which is defined as averaged amount of the transmitted data in bits divided by the total number of REs in a frame. The third metric is Jain’s fairness index [29] for measuring the extent of

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Fig. 8. Average overhead ratio versus cell-edge SNR (5-percentile worst SNR), by comparing BMD-SET, RPD-JEB, and SRD-JEB.

fairness under different SI bearing schemes. In every iteration, the 

( u∈A ϒu )2  Jain’s fairness index is defined as , where Yu denotes the |A| u∈A ϒu2

amount of data in bits transmitted to user u. In Figs. 8 and 9, it can be observed that, for all evaluated frame duration (i.e. different Ns ), the schemes adopting SRD-JEB have the lowest average overhead as well as the highest effective spectrum efficiency among all schemes at each cell-edge SNR. Comparing to RPD-JEB, we can see SRD-JEB provides about 18.9% to 29% considerable reduction of average overhead. Note that larger overhead reduction ratio is achieved when frame duration Ns goes large. The main reason of this trend is due to the fact that in general there are more RBs to be described when frame size is larger, which leads to much better efficiency for the proposed description method. On the other hand, when Ns = 1, SRD-JEB provides 37.5% and 12.3% overhead reduction comparing to BMD-SET at −4 and 2 dB cell-edge SNRs, respectively. When Ns = 6, about 56.7% and 36.8% overhead reductions are achieved. Similar to the comparison with RPD-JEB, the overhead reduction ratio in comparison with BMDSET tends to be larger when frame duration Ns goes large. The main reason is the number of bits of the resource description portion for BMD-SET MSET grows linearly in Ns , and that of SRD-JEB grows in a logarithmic manner. Moreover, it can also be observed that the overhead reduction ratio obtained by replacing BMD-SET with SRD-JEB becomes smaller when cell-edge SNR goes large. It is because SET transmission strategy has relatively better transmission efficiency since it could adjust the coding rate of SI messages according to the individual control channel capacity of each user. However, from the simulation results, SRD-JEB still has 12.3% to

25% considerable reduction of average overhead ratio comparing to BMD-SET at high cell-edge SNR under different frame duration Ns . From Fig. 9, we can see when frame duration Ns = 1, SRDJEB could provide about 12.5% and 40.6% improvement of effective spectrum efficiency comparing to that of RPD-JEB and BMDSET at −4 dB cell-edge SNR, and about 8.7% and 4.9% improvement at 2 dB cell-edge SNR. When frame duration Ns = 6, SRDJEB could provide about 6.8% and 29.2% improvement of effective spectrum efficiency comparing to that of RPD-JEB and BMD-SET at −4 dB cell-edge SNR, and about 6% and 7.3% improvement at 2 dB cell-edge SNR. The improvement of effective spectrum efficiency is caused directly by the additional resource gained from overhead reduction, thus the improvement of effective spectrum efficiency becomes more significant at lower cell-edge SNR and smaller frame duration. Since the ratio of the REs (i.e. number of CCEs) for transmitting SI messages is tend to be larger when celledge SNR is low or frame duration Ns is smaller. Finally, Table 2 shows the Jain’s fairness index under different SI bearing schemes with frame duration Ns = 4. It can be observed that the differences of fairness among different SI bearing schemes are almost negligible at the same cell-edge SNR. Also note that fairness increases as the cell-edge SNR goes high. It is not a surprise since there are more active users with relatively good channel conditions (and higher data rates) in the high cell-edge SNR case. In summary, in full-buffer traffic scenarios, regardless of the frame duration, RPD-JEB and BMD-SET have better performance than that of each other at lower and higher cell-edge SNRs, respectively. When cell-edge SNR is low, RPD-JEB performs better than BMD-SET since RPD in general describes resource allocation more efficiently than BMD does. On the other hand, when cell-edge SNR

T.-Y. Tsai et al. / Computer Networks 115 (2017) 82–99

95

Fig. 9. Effective spectrum efficiency versus cell-edge SNR (5-percentile worst SNR), by comparing BMD-SET, RPD-JEB, and SRD-JEB. Table 2 Jain’s fairness index versus cell-edge SNR (5-percentile worst SNR) when Ns = 4, by comparing BMD-SET, RPD-JEB, and SRD-JEB. SI bearing scheme

Cell-edge SNR (dB) −4 dB

−3 dB

−2 dB

−1 dB

0 dB

1 dB

2 dB

BMD-SET RPD-JEB SRD-JEB

0.6656 0.6661 0.6661

0.6877 0.6874 0.6876

0.6937 0.6938 0.6938

0.7211 0.7211 0.7212

0.7355 0.7355 0.7355

0.7467 0.7467 0.7467

0.7801 0.7801 0.7802

is high, the performance of BMD-SET becomes better than that of RPD-JEB since the benefit of transmission efficiency of SET strategy becomes more significant. However, the proposed SRD description method describes the resource allocation in a more efficient way. And hence SRD-JEB could provide superior performance to that of both RPD-JEB and BMD-SET under all evaluated cell-edge SNRs. 4.2.2. Usage scenario 2 (the small-data transmission test) In contrary to the full-buffer traffic model in usage scenario 1, some practical applications are dominated by short messages (small-data transmissions) associated to large number of users. A typical example is Machine-to-Machine Communications (M2M) in which massive number of M2M users such as sensors or meters are attached to the base station at the same time, and downlink channel is usually used to transport short and low-rate control information from some centralized controllers for the purposes such as pinging, status report request, etc [30]. To model the scenario of the environment dominant by smalldata transmissions, we consider a cell with −4 dB cell-edge SNR and frame duration Ns = 4. To evaluate the performance, different numbers of M2M users are dropped around the base station inde-

pendently according to the distance distribution in (6). Each M2M user has a data session with periodically arrived downlink packets with period TM2M ms and size LM2M bits. The start time of each data session is uniformly distributed in the range from 0 to TM2M ms. In each frame, the base station transmits the arrived packets in a First-Come-First-Served (FCFS) manner in which the first-come packet is transmitted by the RUs with lower indices in the time slot with the lowest index. Moreover, each packet has a tolerant delay DM2M ms and a finite buffer size BM2M = 20 0 0 0 is assumed at the base station. If a packet is not transmitted within the tolerant delay or arrives when the buffer is full, it will be dropped. Here, FCFS discipline could ensure each packet in the buffer to be scheduled for transmission in accordance to its deadline. In this usage scenario, packet loss rate is used to evaluate the performance under different numbers of M2M users, which is defined as the ratio of the number of dropped packets and the number of arrived packets. In Fig. 10, we can obviously see that BMD-SET has the highest packet drop ratio among all schemes. The result is reasonable since the small-data dominant environment is more resource-inefficient for SET transmission strategy comparing to the full-buffer scenario,

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large coverage or large number of users. Hence, we recommend SRD as an excellent scheduling information bearing choice to be employed in the next-generation mobile communication systems. Appendix A. Proof of Theorem 2 FG is onto by definition, thus we focus on the verification of the one-to-one property. Let 0 := {B0 , · · · , BNCRP −1 } be a CRP, the width of an RB Bn ∈ 0 can be determined by the following procedure. If Ln = ∅, G ,0

it implies Xn + Wn is larger than the subchannel offsets of all the RBs ∈ 0 . The only possibility is Wn = C − Xn . On the other hand, if Ln = ∅, let Bm be an arbitrary RB ∈ Ln . Then the width of G ,0

Fig. 10. Packet loss rate versus number of M2M users, by comparing BMD-SET, RPD-JEB, and SRD-JEB.

since aggregation of small data packets results in large number of SI messages in a frame and each SI message should be encoded and transmitted by CCEs. In addition, the small-data dominant environment also causes the increase of scheduling assignment in a frame, which further emphasizes the efficiency of description method of resource allocation and saves more resources for SI signaling. Thus, we can see SRD-JEB has a significantly low packet loss rate, when compared with both BMD-SET and RPD-JEB. Besides, the saved resources could be used to transmit more short messages and increase the system capacity. According to the simulation results, given a tolerant packet loss rate 2.5% as an example, SRD-JEB can accommodate 50 to 60 more M2M users than that of RPD-JEB. 5. Conclusion In this paper, an efficient Sorted-Rectangle Description (SRD) method for describing the resource allocation in OFDMA systems is presented in details. This SRD method first partitions a scheduled OFDMA frame into rectangular polygons such that all RUs in the same rectangle belong to the same assignment. We proved the sufficiency of reconstructing such partitioned frame with the onecorner (e.g. top-left) coordinates of the rectangles in a specific order. Thus, only a series of sorted coordinates, each corresponding to a single rectangle is needed to be conveyed to the users. Moreover, efficient sorting and reconstruction algorithms are also developed and they are proved to be linear in the number of rectangular polygons partitioning the frame. The full-buffer scenario simulation shows that the proposed SRD description method outperforms the schemes adopted in existing systems such as BMD-SET and RPD-JEB in terms of effective overhead reduction as well as considerable improvement of effective spectrum efficiency at different cell-edge SNRs. The performance enhancement is especially substantial when cell-edge SNR is low (i.e. with larger cell coverage), and in such cases, large amount of resources for transmitting SI messages can be saved and large improvement of effective spectrum efficiency can also be achieved. For the periodic small-data usage scenario, it can be found that the proposed SRD method can provide significantly lower packet drop rate than that of other evaluated schemes under the same number of users. It shows that SRD-JEB is an out-performing choice to be employed in scenarios with numerous users and small packet sizes, such as M2M. In summary, the proposed SRD method could effectively improve the service quality as well as achieve higher system capacity by significantly reducing the scheduling information overhead. The gain is expected to be even more considerable in systems with

G ,0

Bn can be simply obtained by Xm − Xn according to the definition

of relation G . After we obtain the RB widths W0 , · · · , WNCRP −1 , now the RB heights can also be determined by the fact that the bottom-left coordinate of an RB in a CRP always intercepts either one of the top boundaries of other RBs or the bottom boundary of A. From above, it is sufficient to prove the desired result. Appendix B. Proof of Lemma 4 When K = 1, the result clearly stands. In the cases that K > 1, due to the fact that A is bounded and the RBs B0 , · · · , BK−1 are included in A with the coordinates of each corner are at lattice points (according to Definition 1), there is at least one RB Bk ∈  with which LkG, = ∅.

Let Bk be an RB which satisfies LkG, = ∅ and Yk − Hk = min{Yn − Hn : LnG, = ∅}. Assume LkM, = ∅, from the definition of , there must be an RB Bk such that Yk = Yk − Hk , Xk < Xk + Wk < Xk +

Wk and LkG, = ∅. Since Yk − Hk < Yk = Yk − Hk , it contradicts the assumption that Yk − Hk = min{Yn − Hn : LnG, = ∅}. Thus, LkM, = ∅ and Bk is isolated in . It is sufficient to lead to the desired result. Appendix C. Proof of Lemma 5 Consider a point p := (X, Y ) ∈ /



Bk . If p lays in Bm , the line  {(X, Y − t ) : 0 ≤ t < Y } does not intersect with Bk , since Bm is k=m

k=m

isolated in and thus LkM, = ∅. If p does not lay in Bm , then p ∈ /   Bk and {(X, Y − t ) : 0 ≤ t < Y } also does not intersect with Bk k=m

k

since  ∈ . It is sufficient to argue that Bk also ∈  when || > 1. Appendix D. Proof of Theorem 3 When NCRP ≤ 2, the result clearly stands. Thus, we focus on the cases that NCRP > 2. It can be simply verified that a CRP is in . Thus, according to Lemma 4, one can find an RB BoN −1 ∈ 0 which is isolated in 0 . CRP

Exclude BoN

CRP −1

from 0 and obtain a new RB set:

1 := 0 \ BoNCRP −1 . From Lemma 5, 1 is also in , thus there is also an RB BoN −2 ∈ 1 isolated in 1 . Then, one can obtain 2 := 1 \ CRP

BoN

CRP −2

similarly. By performing the above process repeatedly

NCRP − 1 times, we finally extract NCRP − 1 RBs BoN

CRP −1

, · · · , Bo1 in

the order of extraction. Moreover, we further define Bo0 := 0 \ {Bo1 , · · · , BoN −1 }. CRP

For an RB Bok , with which k ∈ {1, · · · , NCRP − 1}, we can observe that there is no RB in {Bo0 , · · · , Bok−1 } which Bok is G-related to,

T.-Y. Tsai et al. / Computer Networks 115 (2017) 82–99

since Bok is isolated in NCRP −1−k = 0 \ {Bok−1 , · · · , BoN

CRP −1

fact implies that

LGk, ⊆ {Bok+1 , · · · , BoNCRP −1 }. o

97

}. This (10)

0

Then, there are two possible cases: o

o

Case 1 (LGk, = ∅): If LGk, = ∅, it can be further observed that 0 0 |Lok | = 1, since there can be only one RB with which G ,0

the time offset is larger than or equal to Yok and Bok is G-related to. Let RB Bm ∈ L

ok ,

G ,0

it is clear that Xm > Xok

and Ym ≥ Yok by the definition of relation G . Moreover, for each RB Bn ∈ {Bok +1 , · · · , BoN −1 } such that Xn > Xok and CRP

Yn ≥ Yok , Xn must be larger than or equal to Xok + Wok since if Xn < Xok + Wok , then (Bn , Bok ) ∈ M and Bn can not be selected until Bok is excluded in the (NCRP − k )th excluding procedure, which contradicts the assumption that Bn ∈ {Bok+1 , · · · , BoN−1 }. Combine the above analysis and (10), we can conclude that

Xm = Xok + Wok = min{Xn : Bn ∈ {Bok+1 , · · · , BoNCRP −1 }, Xn > Xok , Yn ≥ Yok } : In addition, from the definition of relation G , it is not difficult to verify that Bm must have the smallest time offset among the RBs in {Bok+1 , . . . BoN −1 } with time offset larger

Fig. 11. An example of obtaining an MRP of a rectilinear polygon (a) a given rectilinear polygon F. (b) The red segements is one of the maximum-cardinality independent chord set of F. (c) The yellow segments have one of the end containing the remained concave vertices and do not intersect with the chords drawn in (b). The red and yellow segments partition F into the smallest number of rectangular polygons. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Ym = min{Yn : Bn ∈ {Bok+1 , · · · , BoNCRP −1 }, Xn = αk , Yn ≤ Yok } (12)

Appendix G. An example for obtaining minimum-cardinality rectilinear partition

=

αk

(11)

CRP

than or equal to Yok and frequency offset α k . That is

Combining (11) and (12), we {ok+1 , · · · , oNCRP −1 }, Xn = αk , Yn = βn }. o

have

o

LGk = {Bn : n ∈

o

Case 2(LGk, = ∅): If LGk = ∅, the necessary condition is Xok + 0

Wok = C. For each RB Bn with Xn > Xok and Yn ≥ Yok , Xn must be smaller than Xok + Wok , that is, (Bok , Bn ) ∈ M. Thus, Bn ∈ / {Bok+1 , · · · , BoN−1 }. Then, we can conclude that {Xn : Bn ∈ {Bok+1 , · · · , BoN−1 }, Yn ≥ Yok , Xn > Xok } = ∅.

The analysis of the above two cases leads to the desired result. Appendix E. Proof of Theorem 4

In this appendix, we illustrate the algorithm of decomposing a rectilinear polygon into a minimum number of rectangles with which the interiors are not overlapped by presenting an example. A rectlinear polygon is a polygon in which each edge is parallel to the axes of Cartesian plane. More formally, we have the following definition: Definition G.1. Let D be the set of rectangular polygons with which the edge is parallel to the axes of Cartesian plane. Consider a bounded rectangular polygon B ∈ D and a set of rectangular polygon H ⊂ D such that for each rectangular polygon R ∈ H has R⊆B.  A rectilinear polygon F = B, H is defined as F = B \ R. R∈H

The necessity of the first statement is clear. From Lemma 6, we know Bm1 ∈ Lk can not be excluded until all other RBs in G ,0

LkG, are excluded. Thus, Lk

G ,

0

l

= ∅ is also the sufficient condition

R∈L

1

of LkG, = ∅ for each l1 ≤ l. The proof of the second statement is l similar. By definition, the necessary and sufficient condition for an RB

Bk to be isolated in l is LkG, = ∅ and LkM, = ∅ for each l1 , l2 ∈ l

1

l

2

{0, · · · , l }. Combining this fact and the first and the second statements leads to the third statement. Appendix F. Proof of Theorem 5



Due to the fact that |Lk | ≤ 1, if k ∈ S k , then k ∈ / G ,0 G N−1 k m S , for m = k. Thus, |S | is not larger than the total numk=0 G G N−1 k ber of RBs NCRP . The proof of the inequality |S | ≤ NCRP is k=0 similar.

Definition G.2. A rectilinear partition L of a rectilinear polygon F is a set of rectangular polygons ∈ D with which the interiors are  not overlapped such that F = R.

M

Our goal is to find a minimum-cardinality rectilinear partition (MRP) L∗ with contains fewest rectangular polygons among all possible rectilinear partitions of F. Here, we adopt the algorithm originally proposed by Ohtuski [21] which can find an MRP of a given rectilinear polygon with polynomial time complexity. In the following, we adopt the definition of the necessary terms from [22], such as concave verex, convex vertex, cohorizontal, and chord. Fig. 11(a) depicts an example of rectilinear polygon (the red region), with which angle A ∼ G are concave vertices and angle a ∼ k are convex vertices. {B, D} and {E, G} are cohorizontal vertex pairs and {A, F} and {C, D} are covertical vertex pairs. Moreover, {BD, EG}, {BD, AF}, {CD, EG}, and {AF, CD} are independent chord pairs. On the other hand, chords AF and GE are not independent since they intersect. Ohtuski’s algorithm for obtaining an MRP of rectilinear polygon F has the following steps [21]:

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T.-Y. Tsai et al. / Computer Networks 115 (2017) 82–99

Step 1: Find a independent set of chords with maximum cardinality. Step 2: Draw the chords in the independent set obtained in Step 1. Note that it partition F into smaller rectilinear polygons with no independent chord set. Step 3: From each of the concave vertices which a chord was not drawn in Step 2, draw a vertical or horizontal line wholly within the smaller rectilinear polygons obtained in Step 2 with which one of the end contains the concave vertex. It can be proved, the rectangular polygons formed by the polygon edges, chord of step 2, and lines of step 3 form an MRP (see, e.g. [22]). Moreover, the computation complexity of the above algorithm is dominant by Step 1, which is equivalent to finding a maximum independent set in a bipartite graph. There have been several polynomial-time algorithms discovered to find a maximum independent set for a given bipartite tree. Thus, Ohtuski’s algorithm can also have a polynomial-time complexity. For example, if the algorithm proposed by Imai and Asano [23] is adopted, the resulting time complexity of Ohtuski’s algorithm is O (Nv3/2 log Nv ), where Nv denotes the number of vertices of F. Fig. 11 is an example of obtaining an MRP of a rectilinear polygon using the above algorithm. The red segments in Fig. 11(b) is one of the independent chord set with maximum cardinality. The red segments partition the rectilinear polygon into four smaller rectilinear polygons. The yellow segments in Fig. 11(c) finally partition the rectilinear polygon into six rectangles. References [1] 3GPP TS 36.213 v9.3.0, Physical Layer Procedures, Oct. 2010. [2] 3GPP TS 36.213 v11.11.0, Physical Layer Procedures, July 2015. [3] IEEE Standard for Local and Metropolitan Area Networks - Part 16: Air Interface for Fixed and Mobile Broadband Wireless Access Systems, Feb. 2006. [4] IEEE Standard for Local and Metropolitan Area Networks - Part 16: Air Interface for Broadband Wireless Access Systems Amendment 3: Advanced Air Interface, 2011. [5] J. Gross, H. Geerdes, H. Karl, A. Wolisz, Performance analysis of dynamic OFDMA systems with inband signaling, IEEE J. Sel. Areas Commun. 3 (3(March)) (2006). [6] J. So, Performance analysis of voIP services in the IEEE 802.16e OFDMA system with inband signaling, IEEE Trans. Vehicular Technol. 57 (3(May)) (2008). [7] R. Moosavi, et al., Comparison of strategies for signaling of scheduling assignments in wireless OFDMA, IEEE Trans. Vehicular Technol. 59 (9(Nov.)) (2010).

[8] R. Cohen, L. Katzir, Computational analysis and efficient algorithms for micro and macro OFDMA downlink scheduling, IEEE/ACM Trans. Netw. 18 (1(Feb.)) (2010). [9] Y. Ben-Shimol, I. Kitroser, Y. Dinitz, Two-dimensional mapping for wireless OFDMA systems, IEEE Trans. Broadcasting 52 (3(Sept.)) (2006). [10] A. Israeli, D. Rawitz, O. Sharon, On the complexity of sequential rectangle placement in IEEE 802.16/wimax systems, in: Proc. ESA, Perth, Australia, 2007. [11] Y. Chen, C. Chen, Y. Lin, Cross-layer design for radio resource allocation based on priority scheduling in OFDMA wireless access network, EURASIP J. Wirel. Commun. Netw. (2011). [12] T. Kwon, D. Cho, Adaptive-modulation-and-coding-based transmission of control messages for resource allocation in mobile communication systems, IEEE Trans. Vehicular Technology Vol. 58 (No. 6) (July 2009). [13] M. Sternad, T. Svensson, M. Dottling, Resource allocation and control signaling in the WINNER flexible MAC concept, in: Proc. IEEE VTC 2008-Fall, Sep. 2008. [14] J. Yeom, Y. Lee, Efficient transmission of multicast SIs in IEEE 802.16e, IEICE Trans. Commun. E91-B (10(Oct.)) (2008). [15] H. Nguyen, et al., Compression of associated signaling for adaptive multi-carrier systems, in: Proc. IEEE VTC 2004-Spring, May 2004. [16] R. Moosavi, J. Eriksson, E.G. Larsson, Differential signaling of scheduling information in wireless multiple access systems, in: Proc. IEEE GLOBECOM 2010, Dec. 2010. [17] R. Moosavi, E.G. Larsson, Optimized encoding of scheduling assignments using finite blocklength coding bounds, IEEE Wirel. Commun. Lett. 3 (3(June)) (2014). [18] R. Moosavi, E.G. Larsson, Reducing physical layer control signaling using mobile-assisted scheduling, IEEE Trans. Wirel. Commun. 12 (1(Jan.)) (2013). [19] A. Jalali, R. Padovani, R. Pankaj, Data throughput of CDMA-HDR a high efficienct-high data rate personal communication wireless system, in: in Proc. IEEE VTC 20 0 0-Spring, 20 0 0. [20] K.C. Beh, S. Armour, A. Doufexi, Joint time-frequency domain proportional fair scheduler with HARQ for 3GPP LTE systems, in: Proc. VTC 2008-Fall, Sept. 2008. [21] T. Ohtsuki, Minimum dissection of rectilinear regions, in: Proc. ISCAS, Rome, Italy, May 1982. [22] R. Chadha, D.C.S. Allison, Decomposing rectilinear figures into rectangles, in: Proc. CSC’88, Feb. 1988. [23] H. Imai, T. Asano, Efficient algorithms for geometric graph search problems, SIAM J. Comput. 15 (2(May)) (1986). [24] D. Tse, P. Viswanath, Fundamentals of Wireless Communication, Cambridge University Press, 2005. [25] X. Qiu, K. Chawla, On the performance of adaptive modulation in cellular systems, IEEE Trans. Commun. 47 (6(June)) (1999). [26] 3GPP TR 36.814 v9.0.0, further advancements for e-UTRA physical layer aspects, March 2010. [27] 3GPP TS 36.521 v11.2.0, User Equipment (User) Conformance Specification Radio Transmission and Reception Part 1: Conformance Testing, Sep. 2013. [28] B. Sadiq, R. Madan, A. Sampath, Downlink scheduling for multiclass traffic in LTE, EURASIP J. Wirel. Communi. Netw. (2009). [29] R. Jain, D. Chiu, W. Hawe, A quantitative measure of fairness and discrimination for resource allocation in shared systems, DEC TR-301, Littleton, Digital Equipment Corporation, MA, 1984. [30] EU FP7 project LOLA (Achieving Low-Latency in Wireless Communications), Project Contract number 248993, d4.5 scheduling policies for M2M and gaming traffic, v2.0, January 2012.

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Tsung-Yu Tsai received the B.S. degree in electrical engineering from the National Cheng Kung University (NCKU), Tainan, Taiwan, in 2005 and the M.S. degree in communication engineering from the National Taiwan University (NTU), Taipei, Taiwan, in 2007. He is currently working toward the Ph.D. degree in communication engineering in NTU. From 2007 to 2014, he was with Institute for Information Industry (III) and worked on the standards development in IEEE 802.16m/p and 3GPP LTE/LTE-A. His current research interests include wireless communications and networkings.

Yi-Hsueh Tsai received the B. S. degree in electrical engineering from National Taiwan University of Science and Technology in 1998 and the M. S. and Ph.D degrees in electrical engineering from National Taiwan University in 20 0 0 and 2005. He started working in IEEE 802.16 standardization in 2006 and received Certificates of Appreciation for contributing to IEEE Std 802.16j-2009 in July 2009. He started working in 3GPP standardization in 2011. He is currently a senior engineer in Institute for Information Industry and involved in wireless standards development in both WiMAX and LTE.

Zsehong Tsai received the B.S. degree in electrical engineering from National Taiwan University (NTU), Taipei, in 1983, and the M.S. and Ph.D. degrees from the University of California, Los Angeles, in 1985 and 1988, respectively. During 1988–1990, he worked as a Member of Technical Staff at AT&T Bell Laboratories, where he investigated performance aspects of network management systems. Since 1990, he has been with the Department of Electrical Engineering and Graduate Instate of Communication Engineering of NTU, where he is currently a professor. Dr. Tsai has been active in Telecommunication deregulations and related research since Taiwan started the liberalization process of its telecomm market. For many years, he was a member of Telecommunications Advisory Board (TAB) of Ministry of Transportation and Communications (MOTC), Taiwan, R.O.C. In 1998-2004, he joined National Telecommunication Program Office (NTPO) of National Science Council (NSC), R.O.C. as the leader of the Broadband Internet Research Group. In 20 0 0, he served as the co-chair of the 3G Study Group for DGT, the telecommunication regulator in Taiwan. During 20 04-20 06, he was the Deputy Executive Secretary of STAG (Science and Technology Advisory Group) of the Executive Yuan. Currently, he is the Deputy Executive Officer of the Networked Communications Program of NSC in Taiwan. He is also an independent director of Chunghwa Telecom. Co. Dr. Tsai’s research interests include broadband Internet, next generation wireless network, telecomm regulations, and digital convergence policy. Dr. Tsai is a receipt of the CIE (Chinese Institute of Engineers) Technical Paper Award in 1997. He is also a member of IEEE. Shiann-Tsong Sheu received his B.S. degree in applied mathematics from National Chung Hsing University in 1990, and received his Ph.D. degree in computer science from National Tsing Hua University in 1995. From 1995 to 2002, he was an associate professor at the Department of Electrical Engineering, Tamkang University. Since Feb. 2002, he became a professor at the Department of Electrical Engineering, Tamkang University. In Aug. 2005, he joined to the faculty of Department of Communication Engineering, National Central University. Dr. Sheu received the outstanding young researcher award from the IEEE Communication Society Asia Pacific Board in 2002. His research interests include next-generation wireless communication, optical networks, protocol designs and intelligent control algorithms.