Proposed ILP Parallelized sub-problem formulation

Separability. When analyzing the objective function described by (5.12a), the total PF metric corresponds to the sum of the individual PF metrics for all UEs. Furthermore, at each UE n ∈ N , it is assumed that the total instantaneous achievable data rate is equivalent to the linear combination of the decoupled achievable data rates per scheduled PRBs, as given by (5.12c). Therefore, it is possible to separate the CS with muting problem in (5.12), into L independent sub-problems corresponding to the scheduling decision of one PRB each1_{. By performing this parallelization, and by}

exploiting the advances in massively parallel processors [KmWH17], the computation time of the CS with muting problem is reduced without aecting the quality of the solution. In other words, the solution of the parallelized CS with muting problem remains optimal.

Reducibility. It is expected that some of the UEs connected to a common BS m ∈ M, share one or more cooperative interfering BSs. From a BS perspective, the set

Jm = ∪

n∈N cn,m=1

Jn ∀m ∈ M, (5.14)

contains all the unique muting indicator sets associated to its connected UEs. Similar to the set Jn of muting indicator sets for UE n, Jm is common to all PRBs in the

reporting period. The number of unique muting indicator sets for BS m, given by J0

m = |Jm|, depends on the number of UEs connected to BS m and the maximum

number J0 _{of reported interference scenarios per UE, as introduced in Section 5.2.1.}

Thus, J0 _{≤ J}0 m ≤

P

n∈Ncn,mJ0, where the lower bound corresponds to the case when

all connected UEs are interfered by the same set of cooperative interfering BSs, and the upper bound represents the case with all UEs having dierent cooperative interfering BSs. For the unique muting indicator set Jm,j0, where j0 ∈ J0

m and Jm0 ={1, . . . , Jm0 },

the set of indices of UEs, connected to BS m, with equal muting indicator set is dened as

Nm,j0 ={n ∈ N | c_n,m = 1, J_m,j0 ₍J_n, ∀m ∈ M, ∀j0∈ J0

m} . (5.15)

Based on the denitions in (5.14) and (5.15), the following Proposition 5.3 is given.

1_{It is worth mentioning that also non-linear combinations of the achievable data rates can be}

applied. One example is the Mutual Information based Exponential SNR Mapping (MIESM) [LKK12, WIN05]. However, in the case of non-linear combination, the formulation is no longer an ILP.

5.3 Centralized CS with Muting 57 Proposition 5.3. Assume BS m ∈ M, with unique muting indicator set index given by j0_{∈ J}0

m and the set Nm,j0 as introduced in (5.15). If |N_m,j0| > 1, then the optimal contribution of BS m on PRB l ∈ L to the total PF metric in (5.12a), corresponds to the PF metric of UE ˆn, with

ˆ n = arg max n∈Nm,j0 Ωn,l,j = arg max n∈Nm,j0 rn,l,j Rn ∀j ∈ J 0 _{| J} n,j =Jm,j0, (5.16) where Ωn,l,j is dened as in (3.11).

Proof. See Appendix C.

Based on Proposition 5.3, it is sucient that each BS m ∈ M forwards to the central controller on PRB l ∈ L, the CSIR-11 _{reports related to one UE per unique muting}

indicator set Jm,j0, ∀j0∈ J0

m, instead of the CSIR-11 reports from all connected UEs.

The set of indices of UEs connected to BS m that maximize the PF metric on PRB l, in at least one of the unique muting indicator sets indexed by j0 _{∈ J}0

m, is dened as Nm,l0 = ( ˆ n _{| ∃j : ˆn = arg max} n∈Nm,j0 Ωn,l,j, ∀j0∈ Jm0 , ∀j ∈ J0 | Jn,j =Jm,j0 ) . (5.17)

The cardinality of the set N0

m,l, is bounded as 1 ≤ Nm,l0

 ≤

P

n∈Ncn,m, where the lower

bound implies that only one UE provides the maximum PF metric on PRB l, among all unique muting indicator sets, and the upper bound corresponds to the case when each UE reports dierent muting indicator sets with respect to the other UEs connected to BS m.

Example 5.3. Assume a network with M BSs, where BS m serves a total of ˜N UEs. It is further assumed that all ˜N UEs served by BS m, have the same set of coopera- tive interfering BSs, such that Ic

n = I0, ∀n ∈ N | cn,m = 1, with |I0| = M0. Thus,

the number of unique muting indicator sets for BS m is J0

m = 2M 0

. If the reducibil- ity concept is not applied, BS m needs to forward to the central controller a total of

˜ N · J0

m CSIR-11 reports on PRB l ∈ L, corresponding to the interference scenarios of

all connected UEs. However, if BS m applies the reducibility concept, the number of forwarded CSIR-11 _{reports to the central controller decreases to J}0

m, with N_m,l0 

≤ J_m0 . Hence, the total number of connected UEs impacts marginally the size of the solution space.

At the central controller, all the achievable data rates rn,l,j, ∀n ∈ Nm,l0 , ∀m ∈ M,

∀l ∈ L, ∀j ∈ J0_{, are per denition set to zero for the interference scenarios where}

UE n does not provide the maximum PF metric among the UEs connected to the same BS m. The set N0

l = ∪m∈MNm,l0 is used to denote the indices of UEs to be consid-

ered in the reformulated ILP on PRB l. The cardinality of the set N0

l is described as

M _{≤ |N}0

l| ≤ N. In the special case of M0 = 0, all UEs report only one interference

scenario where no cooperative interfering BS is muted, and thus, |N0

58 Chapter 5: Centralized Coordinated Scheduling in Mobile Communication Networks

Lifting. In order to linearize the constraints in (5.12c), a variable transformation is introduced based on the lifting technique [Bal05]. A new coordinated decision variable is dened, denoted by the tensor S of dimensions N × L × J0_{. The new coordinated}

decision variable contains both, the scheduling and the muting decisions, and has ele- ments sn,l,j =      1 if PRB l ∈ L is scheduled to UE n ∈ N under interference scenario j ∈ J0

0 otherwise.

(5.18) The new decision variable, sn,l,j, is related to the muting and scheduling decisions in

(5.2) and (5.10), respectively, as

sn,l,j = 1⇔ ¯sn,l = 1∧ ¯αm,l= 1, ∀n ∈ N , ∀l ∈ L, ∀j ∈ J0, ∀m ∈ Jn,j, (5.19)

with ∧ denoting the logical and operator. Hence, the non-linear constraints in (5.12c) reduce to a linear combination of the achievable data rates, rn,l,j, and the new decision

variable, sn,l,j.

Problem Reformulation. Using the above described concepts of separability, reducibility and lifting, the CS with muting INLP formulation in (5.12) can be reformulated as an ILP, which can be eciently solved by commercial solvers as mentioned in Section 2.3. Hence, with the set N0

l as introduced above, and dening the binary decision variable

Sl to have dimensions |Nl0| × J0, the sub-problem formulation for PRB l ∈ L is

max {Sl} X n∈N0 l Ωn,l (5.20a) s.t. sn,l,j+ X u∈N0 l X i∈J0 cu,msu,l,i≤ 1 ∀n ∈ Nl0, ∀j ∈ J0, ∀m ∈ Jn,j, (5.20b) X n∈N0 l X j∈J0 cn,msn,l,j ≤ 1 ∀m ∈ M\ ∪n∈N0 l I c n, (5.20c) rn,l = X j∈J0 rn,l,jsn,l,j ∀n ∈ Nl0, (5.20d) sn,l,j = 0 ∀n ∈ Nl0, ∀j ∈ J0 | rn,l,j = 0, (5.20e) sn,l,j ∈ {0, 1} ∀n ∈ Nl0, ∀j ∈ J0, (5.20f)

where the objective in (5.20a) is to maximize the sum of the PF metric over all UEs on PRB l. The constraints in (5.20b) restrict the scheduling decisions of the cooperative interfering BSs of UE n ∈ N0

l, i.e., ∀m ∈ Jn,j, in order to agree with the muting state

considered in the interference scenario j ∈ J0_{. If PRB l is scheduled to UE n under the}

condition of muting the cooperative interfering BSs indexed by the set Jn,j ∈ Jn, then,

no other UE connected to the muted BSs can be simultaneously assigned to the same PRB l. Thus, if sn,l,j = 1 in (5.20b), the second term on the left-hand-side must be

equal to zero. Furthermore, in the case that sn,l,j = 0, the constraints in (5.20b) ensure

5.3 Centralized CS with Muting 59 serve a maximum of one UE per PRB, over all possible interference scenarios j ∈ J0_.

Since it is possible that specic BSs within the cooperation cluster do not belong to the set of cooperative interfering BSs of any UE, the constraints in (5.20c) complement the restriction on the single-user transmissions from (5.20b). Additionally, the total instantaneous achievable data rate on PRB l of UE n, denoted by rn,l, is calculated in

(5.20d) as the achievable data rate for the selected interference scenario j, as dened by the coordinated decision variable sn,l,j. It is worth noting that from the CSIR-11 reports,

rn,l,j is uniquely related to sn,l,j, thus, no lookup table function as used in (5.13) is

required. Furthermore, the constraints in (5.20e) are incorporated as a pre-processing step to ensure that no PRB is scheduled to UEs for which a maximum PF metric for the corresponding interference scenario j is not available. Finally, the elements of the coordinated decision matrix Sl are binary as described by the constraints in (5.20f).

Theorem 5.1. The LTE-Advanced CS with muting problem formulations in (5.12) and (5.20) are equivalent.

Proof. See Appendix D.

The proposed parallelized formulation in (5.20) reduces signicantly the CS with mut- ing problem complexity, enabling its application even for large-size networks as illus- trated in Section 5.4.