LTE uplink interference aware resource allocation

Computer Communications 66 (2015) 45–53

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Computer Communications journal homepage: www.elsevier.com/locate/comcom

LTE uplink interference aware resource allocation Kai Yanga,b,∗, Steven Martinb, Tara Ali Yahiyab a b

School of Information and Electronics, Beijing Institute of Technology, Beijing, China Laboratoire de Recherche en Informatique (LRI), University of Paris-Sud – CNRS, 91405 Orsay, France

a r t i c l e

i n f o

Article history: Received 6 March 2014 Revised 5 April 2015 Accepted 7 April 2015 Available online 15 April 2015 Keywords: LTE Resource allocation Inter-cell interference Interference graph

a b s t r a c t In this paper, we investigate a multi-cell Long Term Evolution (LTE) uplink resource allocation problem by mitigating the inter-cell interference based on interference graph, in which the vertices represent the user equipments (UEs) and the edges represent interference relations between the UEs. We derive two different interference graphs based on the cellular channel statistics, namely, the enhanced interference graph (EIG) and simplified interference graph (SIG). For EIG, the edge between two UEs is associated with the weight value related to the interference state between them; whereas for SIG, the edge between two UEs exists only when they interfere with each other. Given that the LTE uplink resource allocation is NP-hard, heuristic algorithms are proposed based on EIG and SIG, respectively, to give solutions with reasonable complexity. The system-level simulation results show that all the algorithms can mitigate the inter-cell interference significantly. EIG-based algorithm has the better fairness relative to SIG-based algorithms, however SIG-based algorithms are preferred in terms of throughput, complexity, and overhead. © 2015 Elsevier B.V. All rights reserved.

1. Introduction With the increasing demand for better quality of service (QoS) and the emergence of bandwidth consuming multimedia applications, the Long Term Evolution (LTE) is developed by Third Generation Partnership Project (3GPP) [1,2]. The selected multiple access techniques for LTE are orthogonal frequency-division multiple access (OFDMA) for downlink and single-carrier frequency-division multiple access (SC-FDMA) for uplink. Different from conventional downlink OFDMA resource allocation [3,4], SC-FDMA in LTE uplink requires that all resource blocks (RBs) allocated to the same user must be contiguous in frequency domain within each time slot, which makes the uplink resource allocation problem much harder to solve [5,6]. Consequently, the techniques developed for OFDMA cannot be directly applied to SC-FDMA resource allocation problem, primarily due to the limited allocation flexibility resulted from this frequency adjacency restriction. LTE uplink resource allocation is achieved by assigning different RBs to different user equipments (UEs) based on UEs’ metrics on RBs. The uplink resource allocation problem is NP-hard [5]. Authors in [5] explored the LTE SC-FDMA uplink scheduling problem by adopting the conventional time-domain proportional fairness algorithm to maximize its objective in the frequency-domain ∗ Corresponding author at: School of Information and Electronics, Beijing Institute of Technology, Beijing, China. E-mail address: [email protected] (K. Yang).

http://dx.doi.org/10.1016/j.comcom.2015.04.002 0140-3664/© 2015 Elsevier B.V. All rights reserved.

setting and presented a set of practical algorithms fine tuned to this problem. This problem was reformulated as a pure binaryinteger program called the set partitioning problem in [6], and a greedy heuristic algorithm that approached the optimal performance in cases of practical interest was presented. The binary-integer programming-based solution in [6] was extended to different problems in [7], and the simpler greedy algorithms for these problems were also proposed. In [8], the structure of the underlying optimization problem was exploited to obtain an interior point method for optimal resource allocation. Although [5–8] give practical algorithms to the uplink resource allocation, they only consider the cellular network with a single evolved NodeB (eNB) and multiple UEs. In other words, they did not take the inter-cell interference (ICI) into account. ICI arises as a result of RB reuse by UEs located in neighboring cells, and the ICI is the most important interference source in LTE systems, especially for the cell-edge capacity. It is pointed out that the loss in system and celledge capacity due to a collision can be compensated by utilizing advanced link and system state adaptive resource management methods [9]. Some interference coordination schemes, e.g., soft frequency reuse (SFR), are proposed for 3GPP LTE. In [10], a decentralized adaptive uplink SFR scheme with two essential features that consist of RB reuse avoidance and cell-edge bandwidth breathing was proposed. By exploiting the flexibility of frequency selective scheduling and rate adaptation while dynamically limiting the interference experienced by the neighboring cells to a predefined limit, an SFR-based uplink ICI

46

K. Yang et al. / Computer Communications 66 (2015) 45–53

coordination mechanism was proposed in [11]. It shall be pointed out that SFR has low spectrum efficiency because the cell-edge UEs can only use a fraction of the entire frequency band [12]. Besides the SFR-based algorithm, graph theory is also used for the reduction of interference in LTE network [13–15]. The interference graph, in which vertices model UEs, is constructed by evaluating the interference between each two UEs. In [15], the uplink resource allocation problem was modeled as weighted fractional coloring problem (WFCP) in terms of graph theory, and a heuristic algorithm was proposed to obtain a solution in reasonable time. Unfortunately, the uplink frequency adjacent restriction is not taken into account in [15]. In this paper, we use the graph theory to investigate the LTE uplink resource allocation problem. Based on the cellular channel statistics, we derive two different interference graphs. The first one is called enhanced interference graph (EIG), where each edge contains a weight that characterizes the potential interference between two UEs. Due to the NP-hardness of the uplink resource allocation problem, we propose a centralized heuristic algorithm to give solutions in reasonable time based on EIG. The EIG-based heuristic algorithm imposes a restriction on the sum of the interference from neighboring cells to constrain the amount of ICI at each eNB. As EIG-based heuristic algorithm requests the massive interference weight and cannot be performed in a distributed manner due to the constraint on the sum of interference, we derive the second interference graph, called simplified interference graph (SIG), by taking only the strongest interference from neighboring cells into account. In SIG, each edge indicates that the interference between the two linked UEs is intolerable. Based on SIG, both the centralized and distributed heuristic algorithms are proposed to give solutions with reasonable complexity. The centralized algorithm gives better performance than the distributed algorithm at the cost of higher computation complexity and larger overhead. Compared with the improved riding peaks (IRP) algorithm [16], simulation results show that all the proposed algorithms can mitigate the ICI significantly and give better performance in terms of throughput and fairness. The rest of the paper is organized as follows. The system model and EIG are presented in Section 2, which is followed by the EIGbased problem formulation and related centralized heuristic algorithm in Section 3. In Section 4, the SIG is derived. In Section 5, the uplink resource allocation problem is formalized based on SIG, and related centralized and distributed heuristic algorithms are proposed. Simulation results that evaluate the performance of our heuristic algorithms are presented in Section 6, which is followed by the conclusion in Section 7.

2. System model and enhanced interference graph In this paper, we consider a multi-cell network, as shown in Fig. 1, where the antenna of each eNB provides 25 dBi gain with a 70-degree beamwidth and X2 interface is used by eNBs to communicate with each other [17]. In the multi-cell network, there are NeNB eNBs, each of which has NRB RBs, and i active UEs associated with eNB i, 1 ≤ i ≤ NeNB . Here, there are NUE i = {1, 2, . . . , N i }, and  we use eNB = {1, 2, . . . , NeNB }, UE RB = UE {1, 2, . . . , NRB } to denote the eNB, UE, and RB sets respectively, and use tuple (i, j ) to represent the UE j of cell i. UEs (i, j ), (i , j ), and (i , j ) in Fig. 1 are associated with eNBs i, i , and i , respectively. The interference graph G(V, E ), where the vertex set V stands for all UEs and the edge set E contains the weight that characterizes the interference between two UEs, is constructed by evaluating the interference between each two UEs.

Fig. 1. Illustration of LTE uplink inter-cell interference. (The green and red arrows represent the intended and interference signals, respectively.) (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

In order to determine the weights related to the edges, the interference state between each two UEs is required. In LTE, the sounding reference signals (SRS) are transmitted on the uplink and allow the network to estimate the channel state information at different frequencies [18]. The information provided by the estimations can be used to decide whether two UEs in neighboring cells would interfere with each other and to obtain the interference graph [15]. It is noticed that obtaining the interference graph through SRS could decrease the periodicity of SRS transmission and hence increase the system overhead. Here, we propose to utilize the cellular channel statistics to obtain the interference weight to construct the interference graph. In the multi-cell environment, one UE can be interfered by any UE that occupies the same RB in the adjacent cells. For each interfering UE, there exists one signal-to-interference ratio (SIR), which is defined as the ratio of the desired power from interfered UE to the interference power from interfering UE. For example, at eNB i, the SIR between UEs (i, j ) and (i , j ) is given by1



γ j, j =

2

h  P(i, j ) L−1 i,(i, j ) i,(i, j ) 

2

h    P(i , j ) L−1 i,(i , j ) i,(i , j )

(1)

where P(α ,β ) , (α , β ) ∈ {(i, j ), (i , j )}, is the uplink transmit power of UE (α , β ), and Lα ,(α ,β ) and hα ,(α ,β ) denote the large-scale fading coefficient and Rayleigh fading channel coefficient between eNB α and UE (α , β ), respectively. Here we assume that hi,(i, j ) and hi,(i , j ) are independent and identically distributed, and |hi,(i, j ) |2 and |hi,(i , j ) |2 are exponentially distributed with unit mean. To constrain the interference from UE (i , j ) on the signal of UE (i, j ), the SIR γ j, j shall be no less than a predefined threshold, γth , which is usually related to the QoS requirement of UE (i, j ). Letting γ j, j ≥ γth and taking the expectation on (1) yields

P(i , j ) L−1 ≤ i,(i , j )

P(i, j ) L−1 i,(i, j )

γth

.

(2)

In (2), we only take the large-scale fading, i.e., path-loss and shadowing, into account, and assume that the large-scale fading is static. In the following, we consider both the small-scale and large-scale fading statistics. From (1), the probability of γ j, j being no less than γth is

1 We assume that UEs (i, j ) and (i , j ) transmit their signals on the same RB to eNBs i and i , respectively. In such a case, the uplink signal from UE (i, j ) at eNB i is interfered by the signal from UE (i , j ).

K. Yang et al. / Computer Communications 66 (2015) 45–53



Pr γ j, j ≥ γth





2 P(i , j ) L−1 2  γ  i,(i , j ) th  = Pr hi,(i, j )  ≥ hi,(i , j )  −1  =

∞

Pr

P(i, j ) Li,(i, j )

  hi,(i, j ) 2

≥

N0

0

P(i , j ) L−1 γ i,(i , j ) th P(i, j ) L−1 i,(i, j )



higher the weight value. It is noticed that if two UEs are in the same cell, ω ((i, j ), (i, j ), k ) = ∞, j = j, ∀k.

 x f|hi,(i , j ) |2 /N0 (x )dx

(3) where f|hi,(i , j ) |2 /N0 (x ) is the probability density function (PDF) of |hi,(i , j ) |2 /N0 , and N0 is the variance of additive white Gaussian noise (AWGN) with zero mean at eNB i. For Rayleigh distributed h, the PDF of |h|2 /N0 is f |h|2 (x ) = N0

1

γ

 exp −

x

(4)

2





Pr γ j, j ≥ γth =

γ

P(i , j ) L−1 i,(i , j ) th P(i, j ) L−1 i,(i, j )

1 P(i, j ) L−1 i,(i, j ) γth

x(i, j,k )







i∈eNB

i j∈UE

k∈RB

η

x(i, j, k ) ∈ {0, 1},

−1 +1

(5)

.

x(i, j, k ) p(i, j, k )

(10)

i ∀i ∈ eNB , ∀ j ∈ UE , ∀k ∈ RB

ω ((i , j ), (i, j ), k )x(i , j , k )x(i, j, k ) ≤ 

i i ∈eNB , j ∈UE

i ∀i ∈ eNB , ∀ j ∈ UE , ∀k ∈ RB

Letting Pr(γ j, j ≥ γth ) ≥ η, we have

P(i , j ) L−1 ≤ i,(i , j )

In Section 2, we obtained the enhanced interference graph based on the cellular channel statistics and used the reciprocal of the SIR to characterize the strength of the interference. In this section, we will formalize the uplink resource allocation problem based on the EIG, and then give the EIG-based heuristic algorithm. The uplink resource allocation problem can be formalized as follows:

subject to

where γ = E[|h| ]/N0 , and E[x] denotes the expectation of x. Substituting (4) into (3) and performing some algebraic manipulations yield



3. EIG-based heuristic algorithm

max

γ

47





−1 .

(6)

x(i, j, k ) ≤ di, j ,

i ∀i ∈ eNB , ∀ j ∈ UE

(11)

1

γth

,

(12)

(13)

k∈RB 

In (6), the implicit assumption is that |hi,(i, j ) |2 /N0 and |hi,(i , j ) |2 /N0 are independent and identically distributed exponential random variables. If they are non-identically distributed variables, we can rewrite (6) as

P(i , j ) L−1 ≤ i,(i , j )

γ 1 P(i, j ) L−1 i,(i, j ) 1 γth γ 2

η

−1

 (7)

where γ 1 = E[|hi,(i, j ) | ]/N0 and γ 2 = E[|hi,(i , j ) | ]/N0 . Eqs. (2) and (6) give different criteria to check whether the interference from UE (i , j ) on the signal of UE (i, j ) is tolerable or not at eNB i based on the SIR requirement. It is observed that (6) is equivalent to (2) when η = 0.5. This is from the assumption that the fast fading coefficients hi,(i, j ) and hi,(i , j ) are independent with each other. Based on (2) or (6), we can construct the enhanced interference graph as follows. 2

2

1. If the constraint of (2) or (6) is satisfied, the interference weight associated with the edge between UEs (i, j ) and (i , j ) is

ω ((i , j ), (i, j ), k ) =

P(i , j ) L−1 i,(i , j ) P(i, j ) L−1 i,(i, j )

(8)

where ω ((i , j ), (i, j ), k ) indicates the interference effect of UE (i , j ) to UE (i, j ) on RB k. Obviously, ω ((i , j ), (i, j ), k ) is usually unequal to ω ((i, j ), (i , j ), k ). 2. Otherwise, the interference weight for the edge is

ω ((i , j ), (i, j ), k ) = ∞

(9)

which means that the interference from UE (i , j ) on UE (i, j ) is intolerable and they cannot coexist with each other. In EIG, the edge weight value is related to the reciprocal of the SIR. Obviously, the stronger the interference between two UEs, the

k

x(i, j, k ) = k − k + 1,

i ∀i ∈ eNB , ∀ j ∈ UE ,

k=k

if x(i, j, k ) = x(i, j, k ) = 1, k ≤ k

(14)

where 1/γth is the bound for the sum of the interference, p(i, j, k ) denotes the metric of allocating RB k to UE (i, j ) in cell i, x(i, j, k ) is a binary variable, and x(i, j, k ) = 1 means that cell i allocates RB k to UE (i, j ), otherwise x(i, j, k ) = 0. In proportional fairness algorithms, the metric for RB allocating, p(i, j, k ), is usually defined as

p(i, j, k ) =

c (i, j, k ) c (i, j )

(15)

where c (i, j, k ) is the channel capacity between eNB i and UE (i, j ) on RB k, and c (i, j ) is the average throughput of UE (i, j ). Constraint (12) imposes the restriction on the sum of interference power. We can take the reciprocals of both sides of (12) to get it in a more familiar form, namely, the total SIR of the signal from UE (i, j ) at eNB i is no less than γth . Constraint (13) indicates that there exists a maximum number of RBs, i.e., di, j , that can be allocated to UE (i, j ). The value of di, j is related to the maximum transmit power of UE (i, j ) or the amount of information UE (i, j ) needs to transmit. Specifically, the value of di, j can be determined according to the ratio of allowed maximum transmit power of UE and actual transmit power per RB or the ratio of amount of information and RB capacity.2 For the same allowed maximum transmit power, the farther the UE to the serving eNB, the higher the transmit power per RB needed to compensate for the path-loss, and hence the lower the value of di, j . On the other hand, for the same amount of information, the farther the UE to the serving eNB, the lower the modulation and coding scheme (MCS) adopted by UE to provide higher resiliency and robustness to errors, and hence the higher the value of di, j . Constraint (14) enforces the contiguous RB 2 The RB capacity is the amount of data (expressed in bits) that one RB can carry according to the modulation and coding scheme adopted by the UE for uplink transmission.

48

K. Yang et al. / Computer Communications 66 (2015) 45–53

allocation in frequency domain, which is in line with the uplink resource allocation in LTE. Now we need to maximize the objective function (10) under the constraints of (11)–(14). However, the constraints, especially the contiguous RB allocation, make the uplink resource allocation problem extremely hard to be solved. In the following, we will outline an EIGbased centralized heuristic algorithm to give sub-optimal solution with low complexity. The centralized heuristic algorithm needs to be performed in a cooperative fashion by a central allocating unit for a set of cooperating eNBs, and the central allocating unit may be incorporated in one of the cooperating eNBs and shall be connected with all the cooperating eNBs via a fast high-capacity low-delay backhaul network. In order to derive the global interference graph, each eNB determines whether the interference between two UEs is tolerable or not based on (2) or (6) and then derives the edge weight based on (8) and (9) to determine the local interference graph, which is sent to central allocating unit to construct the global interference graph. The global interference graph needs to be updated periodically according to the UEs’ movements, which affect the transmit powers and path-losses of UEs. With the global interference graph and metric set  p , we can perform the centralized heuristic algorithm, as shown in Algorithm 1. In each while loop, we select the UE-RB pair with the highest metric, which is similar to the concept of riding peaks (RP) algorithm [19]. If allocating the RB according to the highest metric UE-RB pair is not in line with the constraints, this highest metric UE-RB pair would be removed from the metric set  p . We then choose the highest metric UE-RB pair in the updated metric set, which is the second highest metric UE-RB pair in the previous metric set. Through exhaustively searching available UE-RB pairs, we try to maximize the throughput through assigning the RBs to preferred UEs while minimizing the ICI. It is observed that the pseudo-code on Line 11 is actually covered by the pseudo-code of the condition statement on Line 6. The reason of including the pseudo-code on Line 11 in the algorithm is that it is much easier to check whether the edge weight is infinite than to check whether the constraint (12) is satisfied.

Algorithm 1. EIG-based centralized heuristic algorithm

i 1: Initiate x(i, j, k ) = 0, ∀i ∈ ψeNB , ∀ j ∈ ψUE , ∀k ∈ ψRB 2: Construct the metric set i  p , p(i, j, k ) ∈  p , ∀i ∈ eNB , ∀ j ∈ UE , ∀k ∈ RB 3: while  p = φ 4: {i∗ , j∗ , k∗ } = argmax p(i, j, k ) p(i, j,k )∈ p ∗ ∗

Remove p(i , j , k ) from  p if x(i∗ , j∗ , k∗ ) = 1 violates the constraints (12)–(14) Continue else x(i∗ , j∗ , k∗ ) = 1 end if Remove p(i, j, k∗ ) from  p , if ω ((i, j ), (i∗ , j∗ ), k∗ ) = ∞, i or ω ((i∗ , j∗ ), (i, j ), k∗ ) = ∞, ∀i ∈ eNB , ∀ j ∈ UE 12: end while 5: 6: 7: 8: 9: 10: 11:

∗

which means cell i is in serious ICI condition, cell i may blank partial RBs to achieve an interference aware resource allocation. i |, Although the initial size of metric set  p is |RB | i∈eNB |UE the while loop would iterate about |eNB ||RB | times due to the removal operation on Line 11. Within the while loop, the computation complexity decreases with the decrease of metric set, and the asympi | ), which is totic complexity of the operations is O(|RB | i∈eNB |UE dominantly determined by the maximum value finding operation on Line 4. Hence, the total complexity of the centralized algorithm is 2 i | ). It is noteworthy that the same maxO(|eNB ||RB | i∈eNB |UE imum value finding operation exists in each while loop. We can sort the entries in the metric set in a descending order before the while loop to avoid repeating the maximum metric value finding operation within the while loop and to further decrease the complexity of the centralized algorithm.

4. Simplified interference graph In the previous sections, we derived the enhanced interference graph and the EIG-based heuristic algorithm. One major disadvantage of the EIG-based algorithm is that it cannot be implemented in a distributed manner due to the restriction on the sum of interference. In this section, we derive the simplified interference graph by ignoring the non-dominant interference to simplify the uplink resource allocation. In SIG, the edge set E is to characterize the interference between two UEs in a straightforward manner, where there exists one edge between UEs (i, j ) and (i , j ) if they interfere with each other, i.e., ((i, j ), (i , j ), k ) ∈ E. Obviously, if two UEs in the same cell use the same RB, they would interfere with each other, i.e., ((i, j ), (i, j ), k ) ∈ E, j = j , ∀k. In the multi-cell environment, each UE is interfered by multiple UEs from the adjacent cells, and the total signal-to-interferenceplus-noise-ratio (SINR) is defined as the ratio of the desired power to the sum of interference powers from interfering UEs and noise power. For UE (i, j ), the total SINR at eNB i is given by



γ(i, j ) =

−1

(α  ,β  )∈ P(α  ,β  ) Li,(α  ,β  )



≈

2

  hi,(α ,β  ) 2 + N0

h  P(i, j ) L−1 i,(i, j ) i,(i, j ) 

2

h    + N0 P(i , j ) L−1 i,(i , j ) i,(i , j )

(16)

where  is the interfering UE set corresponding to UE (i, j ). The approximation in (16) is due to the fact that interferences from the other nodes can be ignored compared with the dominating interference coming from the interfering node with the nearest distance and/or lowest path-loss [20]. In (16), we assume that the uplink signal of UE (i , j ) is the dominant interference on the uplink signal of UE (i, j ) at eNB i. To guarantee the QoS of UE (i, j ), the SINR γ(i, j ) usually needs to be no less than a required threshold. Here, we define γth as the SINR threshold. From (16) we have



2

h  P(i, j ) L−1 i,(i, j ) i,(i, j ) 

It is noticed that there may exist unallocated RBs in some cells after completing the resource allocation in the extreme serious ICI case. For example, if most of the active UEs in cell i are at cell edge, and cell i is surrounded by cell-edge UEs of neighboring cells,

2

h  P(i, j ) L−1 i,(i, j ) i,(i, j )

2

h    + N0 P(i , j ) L−1 i,(i , j ) i,(i , j )

≥ γth .

(17)

Taking the expectation on (17) and ignoring the noise produce the same result as shown in (2).

K. Yang et al. / Computer Communications 66 (2015) 45–53

In (17), only the large-scale fading is considered. In the following, we take both the small-scale and large-scale fading into account. From (17), the probability of γ(i, j ) being no less than γth is 

Pr γ(i, j ) ≥ γth



= Pr  =

  hi,(i, j ) 2



≥

N0

P(i, j ) L−1 i,(i, j )

  hi,(i, j ) 2

∞

Pr

≥

N0

0



2 γ  P(i , j ) L−1 i,(i , j ) th hi,(i , j ) N0

γ P(i , j ) L−1 i,(i , j ) th P(i, j ) L−1 i,(i, j )

x+

+



γth P(i, j ) L−1 i,(i, j )



γth

f 

2  h     i,(i , j ) 

P(i, j ) L−1 i,(i, j )

(x )dx.

N0

(18) Substituting (4) into (18) yields







Pr γ(i, j ) ≥ γth =

P(i , j ) L−1 γ i,(i , j ) th P(i, j ) L−1 i,(i, j )

−1 +1

 exp −

P(i, j ) L−1 i,(i, j )

≤



γth



1

exp −

η

P(i, j ) L−1 γ i,(i, j )

 . (19)



γth

P(i, j ) L−1 γ i,(i, j )

 −1 .

(20)

If we consider non-identically distributed small-scale fading variables, (20) is rewritten as P(i , j ) L−1 i,(i , j )

P(i, j ) L−1 γ i,(i, j ) 1

≤





1

γth γ 2

η

exp −



γth



−1 .

P(i, j ) L−1 γ i,(i, j ) 1

(21)

In (20), the denominator term of the exponent, γ , is the average SNR of signal from UE (i, j ) at eNB i. It is noticed that (20) is th meaningless if η > exp(− P γL−1 ). The reason is that the probability γ (i, j ) i,(i, j )

th ), namely, of SNR of UE (i, j ) no less than γth is exp(− P γL−1 (i, j ) i,(i, j ) γ

Pr





2

h  P(i, j ) L−1 i,(i, j ) i,(i, j ) N0

≥ γth



= exp −

γth



P(i, j ) L−1 γ i,(i, j )

.

(22)

Eqs. (2) and (20) give different criteria to check whether the signal from UE (i , j ) is the interference to signal from UE (i, j ) at eNB i. If (2) or (20) is unsatisfied, ((i, j ), (i , j ), k ) ∈ E. Hence, we can obtain the interference graph accordingly. 5. SIG-based heuristic algorithms In this section, we will use the SIG to formulate the uplink resource allocation problem, and then give both the centralized and distributed heuristic algorithms. Based on SIG, the uplink resource allocation problem can be formalized as follows:

max

x(i, j,k )







i∈eNB

i j∈UE

k∈RB

x(i, j, k ) p(i, j, k )

(23)

subject to

x(i, j, k ) ∈ {0, 1},

i ∀i ∈ eNB , ∀ j ∈ UE , ∀k ∈ RB

x(i, j, k ) + x(i , j , k ) ≤ 1,

x(i, j, k ) ≤ di, j ,

∀k ∈ RB ,if((i, j ), (i , j ), k ) ∈ E

i ∀i ∈ eNB , ∀ j ∈ UE

Algorithm 2. SIG-based centralized heuristic algorithm i 1: Initiate x(i, j, k ) = 0, ∀i ∈ ψeNB , ∀ j ∈ ψUE , ∀k ∈ ψRB 2: Construct the metric set i  p , p(i, j, k ) ∈  p , ∀i ∈ eNB , ∀ j ∈ UE , ∀k ∈ RB 3: while  p = φ 4: {i∗ , j∗ , k∗ } = argmax p(i, j, k ) p(i, j,k )∈ p

P(i, j ) L−1 i,(i, j )



Constraints (24), (26), and (27) are the same with Constraints (11), (13), and (14) in the EIG-based problem formulation. Constraint (25) states that the same RB cannot be allocated to two UEs simultaneously if they interfere with each other. Obviously, this constraint includes the case that two UEs in the same cell cannot occupy the same RB simultaneously. Compared with (12), we exclude each interference power beyond the threshold instead of imposing restriction on the sum of interference power. In the following we propose two heuristic algorithms. The first one is centralized, which makes full use of the global interference graph. And the second one is distributed, which makes use of the local interference graph at eNB to mitigate the ICI.

5.1. Centralized heuristic algorithm

Letting Pr(γ(i, j ) ≥ γth ) ≥ η, we have P(i , j ) L−1 i,(i , j )

γth

49

(24)

(25)

(26)

k∈RB

Remove p(i∗ , j∗ , k∗ ) from  p if x(i∗ , j∗ , k∗ ) = 1 violates the constraints (26) and (27) Continue else x(i∗ , j∗ , k∗ ) = 1 end if Remove p(i, j, k∗ ) from  p , if i ((i, j ), (i∗ , j∗ ), k∗ ) ∈ E, ∀i ∈ eNB , ∀ j ∈ UE 12: end while

5: 6: 7: 8: 9: 10: 11:

With the global interference graph and metric set  p at the central allocating unit, we can perform the centralized heuristic algorithm, as shown in Algorithm 2. For each chosen UE-RB pair, besides removing the corresponding UE-RB metric from metric set, the metrics of adjacent UEs3 on the same chosen RB are also removed, which is to avoid allocating the same RB to adjacent UEs and hence mitigate the ICI. As the meanings of ((i, j ), (i∗ , j∗ ), k∗ ) ∈ E in SIG and ω ((i, j ), (i∗ , j∗ ), k∗ ) = ∞ in EIG are the same, the pseudo code of Algorithm 2 on Line 11 is equivalent to that of Algorithm 1 on Line 11. Actually, the only difference between Algorithms 1 and 2 exists in the condition statement on Line 6. Hence, we can conclude that the total complexity of the SIG-based centralized algorithm is 2 i | ). O(|eNB ||RB | i∈eNB |UE However, the complexity of SIG-based centralized algorithm is indeed lower than that of EIG-based centralized algorithm. Firstly, the while loop times of EIG-based algorithm is more than that of SIG-based algorithm. By removing the adjacent UEs’ metrics, it is impossible to choose a UE that would interfere with the already scheduled UEs in SIG-based algorithm. In EIG-based algorithm, the chosen UE-RB pair may still unsatisfy the amount of interference constraint (12), which would result in extra while loops. Secondly, the complexity of EIG-based algorithm within the while loop is higher than that of SIG-based algorithm within the while



k

i x(i, j, k ) = k − k + 1, ∀i ∈ eNB , ∀ j ∈ UE ,if x(i, j, k ) = x(i, j, k ) = 1, k ≤ k .

k=k

(27)

3 Two UEs that interfere with each other are adjacent UEs in the simplified interference graph.

50

K. Yang et al. / Computer Communications 66 (2015) 45–53

loop. This is due to the facts that EIG-based algorithm needs to check 3 constraints on Line 6 compared to the 2 constraints in SIG-based algorithm and that the amount of interference constraint involves all the scheduled UEs occupying the chosen RB.

5.2. Distributed heuristic algorithm

Algorithm 3. SIG-based distributed heuristic algorithm i 1: Initiate x(i, j, k ) = 0, ∀ j ∈ ψUE , ∀k ∈ ψRB for cell i for cell i i 2: Initiate the RB set RB = RB , for cell i 3: Construct the metric set i i  pi , p(i, j, k ) ∈  pi , ∀ j ∈ UE , ∀k ∈ RB , for cell i 4: Update the metric set  pi : if i i , ∀k ∈ RB , violates the constraint x(i, j, k ) = 1, ∀ j ∈ UE (25) according to local interference graph, remove p(i, j, k ) from  pi . 5: while  pi = φ 6: { j∗ , k∗ } = argmax p(i, j, k ) p(i, j,k )∈ pi

7: Remove p(i, j∗ , k∗ ) from  pi 8: if x(i, j∗ , k∗ ) = 1 violates the constraints (26) and (27) 9: Continue 10: else 11: x(i, j∗ , k∗ ) = 1 12: end if i 13: Remove p(i, j, k∗ ) from  pi , ∀ j ∈ UE 14: end while

The centralized heuristic algorithm requires a central unit to coordinate the resource allocation among the cooperating eNBs and then distribute the resource allocation results to them. In the following, we will propose a distributed heuristic algorithm, which allows to implement the interference aware resource allocation at each eNB once it has the resource allocation results of its neighboring cells and its local interference graph. The straightforward way to derive the distributed algorithm is to set eNB = 1 in the above-mentioned SIG-based centralized algorithm and to constrain the metric set into a single cell, as shown in Algorithm 3. In the distributed algorithm, the ICI is mitigated firstly by taking the resource allocation results of neighboring cells4 into account based on the local interference graph. Hence, the local interference graph and neighboring cells’ allocation results are the priori-information for the distributed algorithm. For distributed algorithm, eNBs may also need to exchange their local interference graph with each other via the X2 interface to improve the quality of their local interference graph. The reason can be found from the following example. In Fig. 1, we assume the channels between eNB i and UE (i, j ) and between eNB i and UE (i , j ) are none-line-of-sight and line-of-sight, respectively. Due to the large path-loss between eNB i and UE (i, j ), Li ,(i, j ) , UEs (i, j ) and (i , j ) can occupy the same RB from the point of view of eNB i . However, if the channels between eNB i and UE (i, j ) and between eNB i and UE (i , j ) are line-of-sight, UEs (i, j ) and (i , j ) cannot occupy the same RB with high probability from the point of view of eNB i.

4 The cells that have already completed the resource allocation need to inform their allocation results to neighboring cells in order to signal the neighboring cells to avoid ICI on certain RBs based on local interference graph.

The local interference graph at an eNB may only have the interference information around itself or have the entire interference information of the network. The more accurate the local interference graph, the better performance of the distributed algorithm. Obviously, the entire interference information means information exchanging among eNBs via X2 interface. For cell i, the while loop needs to iterate about |RB | times and the asymptotic complexity of the operations within the while loop is i || | ). Hence, the complexity of the distributed algorithm O(|UE RB i | ), and the total complexity of the disfor cell i is O(|RB | |UE 2

i | ). Similar to centralized tributed algorithm is O(|RB | i∈eNB |UE algorithm, we could also use the descending order sorting on metric set before the while loop to take place the repeated maximum metric value finding within while loop to further decrease the complexity of the distributed algorithm. The complexity of the distributed heuristic algorithm is only one out of |eNB | of that of the centralized heuristic algorithm. Obviously, lower complexity leads to worse performance relative to centralized heuristic algorithm, as shown in the following section. 2

6. Simulation results The system performance of the centralized and the distributed heuristic algorithms are thoroughly evaluated by means of extensive system-level simulations based on the 3GPP methodology [17], where the wrap around model is adopted in order to avoid any border effects [21]. The system parameters are given in Table 1. In the simulation, 25 UEs are dropped following uniform distribution within each cell, and the speed of UE is set as 3 kmph, where each UE randomly chooses one direction and moves from time to time with this direction. As the speed is only 3 kmph, which means that the movement of one UE is less than 1 m during 1 s (the period of 1000 subframes), the handover is not considered in the simulations. Since RP algorithm outperforms other proportional fairness algorithms [19] and IRP [16] is the improved version of RP, we compare our proposed algorithms with IRP algorithm and the conventional proportional fairness algorithm in terms of UE’s throughput and fairness in this section. In our system-level simulation platform, the uplink fractional power control is used [23], and the EIG and SIG are obtained based on cellular channel statistics as described in Sections 2 and 4, respectively. The adaptive MCS, which is based on the uplink channel quality indicator information, ensures a block error ratio (BLER) value smaller than 10% [24]. Based on the received SINR and the adopted MCS, the error probability of a received data block is derived according to the AWGN BLER curves [25]. Based on the block error probability, we can determine whether the uplink transmission is successful or not by throwing the dice. Finally, dividing the total volume of the data transmitted from each UE, which are carried on the successfully transmitted data blocks, with the simulation time yields the throughput of each UE. Tables 2 and 3 show the cell-average and cell-edge (5% worst) throughputs of different algorithms respectively, and Tables 4 and 5 show the 10% worst and 20% worst throughputs of different algorithms respectively. Here, the cell-edge (5% worst) throughput is defined as 5% point of the cumulative distribution function (CDF) of the user throughput [26], and 10% worst and 20% worst throughputs are defined as 10% and 20% points of the CDF of the user throughput, respectively. The cell-average, celledge, 10% worst, and 20% worst throughputs of IRP algorithm are 389.71 kbps, 110.89 kbps, 153.65 kbps, and 227.44 kbps, respectively. The cell-average, cell-edge, 10% worst, and 20% worst throughputs of conventional proportional fairness algorithm are 374.07 kbps,

K. Yang et al. / Computer Communications 66 (2015) 45–53

55.35 kbps, 91.87 kbps, and 185.71 kbps, respectively. The items “EIG-Cen” and “SIG-Cen” stand for the EIG-based and SIG-based centralized heuristic algorithms respectively, and the items “SIGDis1” and “SIG-Dis2” stand for the SIG-based distributed heuristic algorithms with completed and partial interference graphs

respectively. The completed interference graph at eNB means a lot of ICI state information exchanging between eNBs through X2 interface, and the partial interference graph means that each eNB is only aware of the ICI state information estimated by itself. From Table 2, it is observed that both the SIG-based centralized algorithm and SIG-based distributed algorithm with completed interference graph give better cell-average throughput than SIG-based distributed algorithm with partial interference graph. This results are expected as the algorithms could minimize the effect of ICI to a great extent with completed interference graph. The cell-average throughput of EIG-based centralized algorithm is almost the lowest across the whole range of γth . This can be explained by the fact that the imposed restriction on the sum of interference degrades the resource allocation flexibility, resulting in lower frequency diversity and frequency reuse factor. From Table 3, it is shown that EIG-based centralized algorithm gives better cell-edge throughput with low γth and SIG-based centralized algorithm gives better cell-edge throughput with high γth . The same phenomenon exists for the 10% worst and 20% worst throughputs in Tables 4 and 5, respectively. This is related to the constraints on interference of different algorithms. It also reveals that enforcing the single interference is not enough in the low γth regime and enforcing the sum of interference is too rigorous in the high γth regime. With proper γth , the maximum cell-edge throughput of the EIG-based and SIG-based centralized algorithms is almost the same, both of which outperform the SIG-based distributed algorithm. It should be pointed out that there exist breaking points in Tables 2–5, which demonstrate the effect of γth on the performance of different algorithms.

Table 1 System-level simulation parameters. Parameter

Setting

Deployment scenario Network deployment

3GPP case 1 [17] 7 sites with 3 sectors per site Yes [21] 500 m 2.0 GHz 10 MHz 48 PRBs (PUSCH) 2 PRBs (PUCCH) 0.1 MMSE [17] 3GPP Spatial Channel Model (SCM) [22] 8 dB log-normal 1/2 Omnidirectional ϕ3dB = 70◦ , Am = 25 dB >= 35 m >= 3 m 23 dBm (200 mW) 3 kmph (quasi-static) 25

Wrap around model Inter-site distance (ISD) Carrier frequency Bandwidth Total number of PRBs Target BLER Receiver type Channel model Shadowing UE/eNB antenna number UE antenna pattern eNB antenna pattern Minimum distance between UE and eNB Minimum distance between UEs UE power class UE mobility Average UE number per sector

51

Table 2 Cell-average throughput of different algorithms (kbps).

γth (dB)

3

5

6

7

8

9

EIG-Cen SIG-Cen SIG-Dis1 SIG-Dis2

435.82 439.17 440.44 433.00

432.58 446.69 450.57 436.47

427.83 450.41 454.74 439.22

422.65 450.76 456.35 439.36

414.17 450.95 454.81 439.10

401.74 448.48 449.54 435.92

10 388.39 442.59 441.41 431.64

11 373.05 435.71 429.25 425.08

12 357.24 425.85 415.09 416.32

15 305.67 381.48 364.42 385.60

Table 3 Cell-edge throughput of different algorithms (kbps).

γth (dB)

3

5

6

7

8

9

EIG-Cen SIG-Cen SIG-Dis1 SIG-Dis2

175.09 131.45 122.35 106.52

226.57 160.51 148.44 112.34

250.56 178.97 166.89 116.18

263.64 206.67 188.90 124.07

256.49 231.48 210.21 132.24

250.36 254.58 225.63 142.45

10 239.33 264.56 223.54 152.22

11 228.54 263.20 210.37 164.31

12 215.68 252.27 194.09 175.22

15 179.64 219.13 156.05 175.32

Table 4 10% worst throughput of different algorithms (kbps).

γth (dB)

3

5

6

7

8

9

EIG-Cen SIG-Cen SIG-Dis1 SIG-Dis2

227.69 179.31 173.29 154.84

273.43 212.21 204.25 160.73

285.08 236.60 227.70 164.61

287.25 255.16 244.67 174.25

282.37 277.29 263.45 183.95

268.80 289.08 262.17 194.63

10 260.76 290.04 254.67 205.17

11 247.24 286.00 240.45 217.95

12 236.21 276.98 225.50 225.18

15 198.53 239.60 176.70 205.53

Table 5 20% Worst Throughput of Different Algorithms (kbps)

γth (dB)

3

5

6

7

8

9

EIG-Cen SIG-Cen SIG-Dis1 SIG-Dis2

308.17 266.89 265.82 243.28

326.27 298.53 300.23 251.88

324.01 314.39 313.90 260.77

318.77 323.45 319.94 267.80

311.09 329.18 318.71 276.49

299.07 328.66 308.11 283.29

10 287.18 323.57 293.41 285.56

11 273.29 313.63 277.63 287.12

12 260.36 305.36 261.96 281.70

15 218.65 265.94 210.31 244.04

K. Yang et al. / Computer Communications 66 (2015) 45–53

Figs. 2 and 3 plot the CDFs of UEs’ average received SINRs and throughputs for different algorithms with certain γth , respectively. From Fig. 2, we observe that the average received SINRs of our proposed algorithms improve significantly compared with that of IRP. Although Fig. 2 shows that with proper γth the proposed algorithms have similar performance in terms of average received SINR, Fig. 3 shows that the proposed algorithms have different performance in terms of throughput. With proper γth , the maximum celledge throughput enhancements of EIG-based and SIG-based centralized algorithms are greater than 130% compared to IRP algorithm and 370% compared to conventional proportional fairness algorithm respectively, and the maximum cell-average throughput enhancements of both algorithms are about 12% and 16% compared to IRP algorithm and 16% and 20% compared to conventional proportional fairness algorithm respectively. Fig. 3 shows that, with proper γth , SIG-based centralized algorithm outperforms EIG-based centralized algorithm in terms of cell-average throughput while maintaining almost the same cell-edge throughput with EIG-based centralized algorithm. Fig. 4 shows the fairness of different algorithms with varying γth . Here, the fairness factor is defined as 2 i |

| i |

|UE

i |· ( i∈eNB n=1UE cn ) /( i∈eNB |UE cn2 ) [27], i∈eNB n=1

1

EIG−Cen, γ =7dB th

SIG−Cen, γth=10dB

0.8

SIG−Dis1, γth=9dB SIG−Dis2, γth=12dB

0.6

CDF

IRP

0.92 0.9 0.88

Fairness Factor

52

0.86 0.84 0.82

EIG−Cen SIG−Cen SIG−Dis1 SIG−Dis2 IRP Proportional Fairness

0.8 0.78 0.76

4

6

8

γth

10

12

Fig. 4. Fairness of different algorithms with varying γth .

where cn is the throughput of UE n. The fairness factors of IRP and conventional proportional fairness algorithms are 0.827 and 0.771, respectively. EIG-based centralized algorithm has the highest fairness factor of 0.906 when γth = 7 dB. The fairness factors of SIG-based centralized and distributed algorithms approach 0.89 with γth = 9 dB and 0.88 with γth = 8 dB, respectively. However, with the increase of γth , the fairness factors of the algorithms would decrease eventually due to the detrimental impact of interference graph on the flexibility of resource allocation. From the above simulation results, it is observed that the value of γth is crucial to the performance. SIG-based algorithms achieve a better tradeoff between the performance and complexity, and SIG-based centralized algorithm outperforms SIG-based distributed algorithm in terms of throughput and fairness at the price of complexity.

0.4

7. Conclusions

0.2

0 −2

0

2

4

6

8

10

12

Average Received SINR of UE (dB) Fig. 2. Cumulative distribution function of UEs’ average received SINRs for different algorithms with certain γth .

1

0.8

In this paper, we derived the interference graphs based on the cellular channel statistics, and proposed heuristic algorithms for the LTE uplink resource allocation problem. All the algorithms can mitigate the ICI evidently and give better performance in terms of throughput and fairness compared with IRP algorithm. It is shown that the proposed algorithms can improve the cell-average and cell-edge throughput up to 16% and 130% relative to that of IRP algorithm, respectively. It is noticed that the interference graph is critical to the proposed algorithms, since it has negative impact on frequency diversity and frequency reuse factor. It is obvious that there is a tradeoff between resource allocation flexibility and ICI mitigation, and we need to decide the value of γth carefully to obtain a proper interference graph.

CDF

0.6

References

EIG−Cen, γth=7dB SIG−Cen, γ =10dB

0.4

th

SIG−Dis1, γth=9dB SIG−Dis2, γth=12dB

0.2

IRP Proportional Fairness 0

0

200

400

600

800

Throughput of UE (kbps) Fig. 3. Cumulative distribution function of UEs’ throughputs for different algorithms with certain γth .

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