User-Association for Provisioned Backhaul Networks

We start our discussion for optimal user-association by first assuming that the backhaul network is provisioned. In other words the backhaul constraints are always satisfied and can be safely ignored when deriving the optimal association rules. Our focus will thus be on the radio access network performance: uplink/downlink and traffic-differentiation tradeoffs.

We remind to the reader that based on our system model, the association policy con-sists in finding appropriate values for the routing probabilitiesp^l,t_i (x), l∈ {D, U}, t∈ {b, d}, for DL and UL, best-effort and dedicated traffic, respectively (defined earlier in assumption B.5 and B.9). That is, for each locationx, we would like to optimally choose to which BSito route different flow types generated from (UL) or destined at (DL) users inx⁹. Our goal for this association problem is threefold: (i) ensure that the capacity of no BS is exceeded; (ii) achieve a good tradeoff between spectral efficiency, user QoS and load balancing, (iii) investigate howUL/DL association splitimpacts the optimal rule derivation and the overall performance benefits.

We define the feasible region for the aforementioned routing probabilities, by re-quiring that no BS capacity being exceeded.

Definition 6.3.1. (Feasible set): Letl∈ {U, D}, t∈ {b, d}, and letϵbe an arbitrarily

8Note that, since the routing paths on the backhaul network are given (see C.1), if a small cell uses links iandjto send its traffic to the macro cell, then all of its traffic will be send over both links.

9The use of a probabilistic association rule convexifies the problem at hand. As it will turn out, the optimal values will turn out to be integer anyway.

small positive constant. The setf^l,tof feasible BS loadsρ^l,t= (ρ^l,t₁ , ρ^l,t₂ , . . . , ρ^l,t_∥B∥)is f^l,t=

{

ρ^l,t |ρ^l,t_i =

∫

L

p^l,t_i (x)ρ^l,t_i (x)dx, 0≤ρ^l,t_i ≤1−ϵ,

∑

i∈B

p^l,t_i (x) = 1,

0≤p^l,t_i (x)≤1,∀i∈ B,∀x∈ L} .

(6.8)

Lemma 3. The feasible setsf^D,b, f^D,d, f^U,b, f^U,das well as the[f^D,b;f^D,d],[f^U,b;f^U,d], [f^D,b;f^U,b],[f^D,b;f^D,d;f^U,b;f^U,d],are convex.

Proof. The proof for the feasible setf^D,bis presented in [63]. It can be easily adapted for the other cases, too (e.g., see [239]).

Following [63] we extend the proposed objective to also include the DL dedicated traffic (see B.8-B.9). We introduce the parameter θ ∈ [0,1]that helps the operator weigh the importance of DL best effort versus DL dedicated traffic performance.α^D,b (α^D,d)≥0controls the amount of load balancing desired in the DL best-effort (dedi-cated) resources. Lets denoteα^D= [α^D,b;α^D,d]andρ^D= [ρ^D,b;ρ^D,d].

Definition 6.3.2. (Objective function for DL) Our objective is

ϕ_αD(ρ^D) =∑

i∈B

θ(1−ρ^D,b_i )^1−αD,b

α^D,b−1 + (1−θ)(1−ρ^D,d_i )^1−αD,d

α^D,d−1 , (6.9)

whenα^D,d, α^D,b ̸= 1. Ifα^D,b(or,α^D,d) is equal to1, the respective fraction must be replaced withlog(1−ρ^D,b_i )⁻¹(or,log(1−ρ^D,d_i )⁻¹).

As a final step, we want to further extend this objective to also capture the UL traffic performance and thus we introduceτ ∈[0,1]to trade it off with DL. Lets assume that α= [α^D,b;α^D,d;α^U,b;α^U,d]andρ= [ρ^D,b;ρ^D,d;ρ^U,b;ρ^U,d].

Definition 6.3.3. (Objective function for DL and UL) The objective that jointly con-siders DL and UL performance follows

ϕ_α(ρ) =τ·ϕ_αD(ρ^D) + (1−τ)·ϕ_αU(ρ^U). (6.10) Lemma 4. The objective functionϕα(ρ)is convex.

Proof. Sinceϕα(ρ)is a weighted non-negative sum of convex functions [63], convex-ity is preserved [240].

Definition 6.3.4. (Optimization Problem) Our problem is

minρ {ϕ_α(ρ)|ρ∈ F}. (6.11)

Lemma 5. This is a convex optimization problem.

Proof. This is a convex optimization problem since the objective is convex on the con-vex feasible setF.

Whileθ linearly weights the best effort versus dedicated flow performance (see Eq. 6.9), the impact ofα^b, α^d is not obvious. We now discuss their impact on the system performance and refer to [63], [71] for the respective proofs.

• Optimizing Spectral Efficiency:α^b = 0maximizes the average physical rate for best-effort flows (defined in B.3), whereasα^d = 0maximizes the number of ded-icated flows that can be served concurrently (defined in B.8). This is equivalent to maximizing the userSIN Rand thus the spectral efficiency.

• Optimizing flow-level QoS:if α^b = 1the corresponding optimal rule tends to maximize the average user throughput. If α^b = 2the per-flow delay is min-imized since the objective for best effort flows corresponds to the delay of an M/G/1/PS system. Ifα^d= 1the corresponding optimal rule becomes equivalent to the averageidlededicated servers in a k-loss system, and the actual blocking probability is minimized.

• Optimizing Load-Balancing:Asα^b→ ∞, the maximum BS utilization is min-imized. Hence, load-balancing has the highest priority. Similarly for α^d and balancing the loads in terms of dedicated traffic.

As a final note, observe that the association rules optimize the objective with respect to best effort traffic (e.g., maximally balanced loads) will usually be different than the association that optimizes for dedicated traffic. Since both dedicated and best effort traffic for each node must be served by the same BS, this creates a conflict that must be optimally resolved.

6.3.1 Optimal Split UL/DL User Association

In the Split UL/DL, since UL and DL traffic can be split at locationx, the problem of optimal DL and UL association decouples into two independent problems. Specifically, minρϕα(ρ) = min_ρDϕ_αD(ρ^D) + min_ρUϕ_αU(ρ^U).

Note that all DL best-effort and dedicated flows atxhave to be downloaded from the same BS, i.e.,p^D_i (x) =p^D,b_i (x) =p^D,d_i (x). Also, all UL best-effort and dedicated should be offloaded to the same BS, sop^U_i (x) = p^U,b_i (x) = p^U,d_i (x). Nevertheless, as explained in Split UL/DL scenariosp^D_i (x)andp^U_i (x)can take different values (see B.10). In that case, the optimal user association rules follow. In the remainder of this section, we focus on the downlink, and we omit the superscripts{D, U}to simplify notation. (We return to the Joint UL/DL association problem in the next subsection.) Theorem 3.1. Ifρ^∗= (ρ^∗₁, ρ^∗₂,· · · , ρ^∗_||B||)denotes the optimal load vector, the optimal association rule atxis

i(x) = arg max

i∈B

(1−ρ^∗_i^b)α^b

·(

1−ρ^∗_i^d)α^d

e^b(x)·(

1−ρ^∗_i^d)α^d

+e^d(x)·(

1−ρ^∗_i^b)α^b (6.12)

where e^b(x) = _µb^θz(x)c^Dz_i^b(x) and e^d(x) = ⁽¹_µd⁻(x)k^θ)zD_i(x)^zd, can be seen as weight for the corresponding individual rules, and together forming a (weighted) harmonic mean of these.

Proof. We prove that the above association rule (Eq. 6.12) indeed minimizes the ob-jective of Eq. (6.9). This is a convex optimization problem. Let ρ^∗ = [ρ^∗b;ρ^∗d]be the optimal solution of Problem (6.9). (We will relax this assumption in Section 6.3.3, as the optimal vectorρ^∗ is not necessarily known.) Hence, the first order optimality condition is that

⟨∇ϕ_α(ρ^∗), ρ−ρ^∗⟩ ≥0 (6.13)

for anyρ ∈ f. Letp(x)andp^∗(x)be the associated routing probability vectors for ρandρ^∗, respectively. Using the deterministic cell coverage generated by (6.12), the optimal association rule is given by:

p^∗_i(x) =1 Then the inner product in Eq. (6.13) can be written as:

⟨∇ϕ_α(ρ^∗),∆ρ^∗⟩= ∑

In the case of split UL/DL association, the above analysis can be appliedseparately on UL and DL traffic, and optimize UL and DL associations independently.

6.3.2 Optimal Joint UL/DL User Association

Current cellular networks (e.g. 3G/4G) suggest that a UE should be connected to a single BS for both UL and DL traffic [241], i.e.p^D_i (x) =p^U_i (x). In that case, the two problems are coupled and the (single) optimal association rule atxshall appropriately weigh DL and UL performance as it follows.

Theorem 3.2. Ifρ^∗ = (ρ^∗₁, ρ^∗₂,· · ·, ρ^∗_||B||)denotes the optimal load vector, and given

i(x)are the corresponding weight factors.

Proof. We refer the interested reader to [239].

Remark 1.The above optimal rule derived in Eq. (6.17) suggests that in theJoint UL/DL scenario associated with objectives that potentially conflict with each other (due to the different flow type performances), it is optimal to associate a user with the BS that maximizes a weighted version of theharmonic meanof the individual as-sociation rules when considering each objective alone. To better understand this, we focus on a simple scenario with only DL and UL best-effort traffic. And assume the following BS options for a user: (BS A) offers 50Mbps DL and only 1Mbps UL; (BS B) 200Mbps DL and 0.5Mbps UL; (BS C) 20Mbps DL and 5Mbps UL. If we care about UL and DL performance equally (i.e. τ = 0.5), one might assume that the BS that maximizes the arithmetic mean (or arithmetic sum) of rates would be a fair choice (i.e. BS B). However, this would lead to rather poor UL performance. Maximizing the harmonic mean would lead to choosing (BS C) instead¹⁰. Additionally, note that in the case ofsplit UL/DL, covered in Section 6.3.1, where each user is free to be associated with two different BSs for the DL and UL traffic offloading, DL traffic would be as-sociated with (BS B), and UL traffic with (BS C) by maximizing the arithmetic mean (or, sum) of their throughputs¹¹. These simple examples intuitively explain how split UL/DL impacts the user association policies, by allowing to independently optimize each objective. This also demonstrates why UL/DL split may perform considerably better than the joint association. We will further explore this in the simulation section.

Summarizing, we showed that our framework supports both best-effort and ded-icated (in both DL and UL direction) traffic demand and we analytically found the corresponding optimal rules in different scenarios. Nevertheless, we highlight that our derived formulas allow to add more dimensions in our setup andflexibly derive

10While this simple example captures the main principle, the actual rule is more complex, as it weighs each objective with the complex factore^l(x).

11The usage of harmonic mean and arithmetic mean/sum appears in a number of physical examples, such as in the calculation of the total resistance in circuits where all resistances are set in series or in parallel.

the optimal rules without any analytical calculations (e.g., by using the arithmetic or harmonic mean maximization). For instance, consider a more modern offloading tech-nique, where different downlink, or uplink, flow types are able to be offloaded to dif-ferent BSs (e.g., per flow/QCI offloading) with conflicting aims. Using our model we can consider an additional respectiveα-function for each flow type, and analytically optimize the complete objective as showed earlier.

6.3.3 Iterative Algorithm Achieves Optimality

The above derived association rules (see e.g., Eq. (6.12) or (6.17)) tell a UE at some locationx, where to associate given the optimal BS load vector. However, the loads might not be optimal at that time (i.e. equal toρ^∗, e.g. see proof of Theorem 3.1). In the following, we show that starting within a feasible pointρ⁽⁰⁾, iterative application of these rules, push the BS load vector to converge to the global optimal point. In such an iterative framework two parts are involved: the mobile device and the BS. Later, in our summarizing Remark 2, we show that our framework (in both over and under provisioned backhaul cases) (i) is applicable for distributed implementations, as well as (ii) satisfies various important properties (e.g., scalability).

Mobile Device: At the beginning of thek period, each new flow request coming from a user at locationxsimply selects the BSithat maximizes the derived rule (see e.g., Eq. (6.12)), using device-centric information (e.g. data rate) and BS broadcast control messages (e.g.currentBS load).

Base Station:At each iterationk, each BS measures its utilizationρ˜^(k)after some required amount of time (e.g., see Eq. (6.5)). Then, based on the previous BS loadsρ^(k) and the latest measurementρ(k), the new BS vectorρ˜^(k+1)needed for the broadcast message in the next iteration would be

˜

ρ^(k+1)=β^(k)·ρ^(k)+ (1−β^(k))·ρ˜^(k), (6.18) where β^(k) ∈ [0,1) is an exponential-averaging parameter. Note that, in the Split UL/DL scenario the UL and DL loads can be independently updated, whereas in the other cases the corresponding loads should be updated using the sameβ^(k).

This iterative algorithm converges to the global optimum, by requiring a simple modification of the proof found on the original algorithm [63]. The descent direction atx, improving the objective at the nextk+ 1iteration i.e. satisfying

⟨∇ϕ_α(ρ^(k)), ρ^(k+1)−ρ^(k)⟩<0, (6.19) is now provided from Eq. (6.12) under joint DL dedicated/DL best-effort association.

This formula appropriately projects the direction under the constraintp^b_i(x) =p^d_i(x);

as shown in the proof of Theorem 3.1. Similarly for the Joint UL/DL association, where the optimization problem does not decouple.

6.4 User-Association for Under-Provisioned Backhaul