Device-to-Device Load Balancing for Cellular Networks

10/07/2017 ∙ by Lei Deng, et al. ∙ Zhejiang University University of Calgary Peking University The Chinese University of Hong Kong 0

Small-cell architecture is widely adopted by cellular network operators to increase network capacity. By reducing the size of cells, operators can pack more (low-power) base stations in an area to better serve the growing demands, without causing extra interference. However, this approach suffers from low spectrum temporal efficiency. When a cell becomes smaller and covers fewer users, its total traffic fluctuates significantly due to insufficient traffic aggregation and exhibiting a large "peak-to-mean" ratio. As operators customarily provision spectrum for peak traffic, large traffic temporal fluctuation inevitably leads to low spectrum temporal efficiency. In this paper, we advocate device-to-device (D2D) load-balancing as a useful mechanism to address the fundamental drawback of small-cell architecture. The idea is to shift traffic from a congested cell to its adjacent under-utilized cells by leveraging inter-cell D2D communication, so that the traffic can be served without using extra spectrum, effectively improving the spectrum temporal efficiency. We provide theoretical modeling and analysis to characterize the benefit of D2D load balancing, in terms of total spectrum requirements of all individual cells. We also derive the corresponding cost, in terms of incurred D2D traffic overhead. We carry out empirical evaluations based on real-world 4G data traces to gauge the benefit and cost of D2D load balancing under practical settings. The results show that D2D load balancing can reduce the spectrum requirement by 25 balancing, at the expense of negligible 0.7

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I Introduction

The drastic growth in mobile devices and applications has triggered an explosion in cellular data traffic. According to Cisco [2], global cellular data traffic reached exabytes per month in 2016 and will further witness a 7-fold increase in 2016-2021. Meanwhile, radio frequency remains a scarce resource for cellular communication. Supporting the fast-growing data traffic demands has become a central concern of cellular network operators.

There are mainly two lines of efforts to address this concern. The first is to serve cellular traffic by exploring additional spectrum, including offloading cellular traffic to WiFi [3] and the recent 60GHz millimeter-wave communication endeavor [4]. The second is to improve spectrum spatial efficiency. A common approach is to adopt a small-cell architecture, such as micro/pico-cell[5]. By reducing cell size, operators can pack more (low-power) base stations in an area and reuse radio frequencies more efficiently to increase network capacity.

While the small-cell architecture improves the spectrum spatial efficiency, it comes at a price of degrading the spectrum temporal efficiency. When a cell becomes smaller and covers fewer users, there is less traffic aggregation. Consequently, the total traffic of a cell fluctuates significantly, exhibiting a large “peak-to-mean” ratio. As operators customarily provision spectrum to a cell based on peak traffic, high temporal fluctuation in traffic volumes inevitably leads to low spectrum temporal efficiency.

To see this concretely, we carry out a case-study based on 4G cell-traffic traces from Smartone [6] (this complements the study in our conference version [1], which was based on 3G data traces), a major cellular network operator in Hong Kong, a highly-populated metropolis. The detailed analysis and description can be found in Appendix A

. Based on this case study, we observe that the average cell-capacity utilization is very low and the peak traffic of many pairs of adjacent BSs occurs at different time epochs. This confirms that small-cell architecture indeed causes very low spectrum temporal utilization, and it suggests ample room to do traffic load balancing to improve temporal utilization.

Motivated by the above observations, we advocate device-to-device (D2D) load-balancing as a useful mechanism to improve spectrum temporal efficiency. D2D communication [7] [8] is a promising paradigm for improving system performance in next generation cellular networks that enables direct communication between user devices using cellular frequency. It is conceivable to relay traffic from congested cells to adjacent underutilized cells via inter-cell D2D communication, enabling load-balancing across cells at the expense of incurred inter-cell D2D traffic.

We remark that an idea of this kind was also studied by Liu et al. in their recent work [9]. They focus on important aspects of examining the technical feasibility of D2D load balancing and practical algorithm design in three-tier LTE-Advanced networks. This work is complement to their study and focuses on the following two important questions:

  • How much spectrum reduction can D2D load balancing bring to a cellular network?

  • What is the corresponding D2D traffic overhead for achieving the benefit?

Answers to these questions provide fundamental understanding of the viability of D2D load balancing in cellular networks. In this paper, we answer the questions via both theoretical analysis and empirical evaluations based on real-world traces. We make the following contributions.

In Sec. III, using perhaps the simplest possible example, we illustrate the concept of D2D load balancing and show that it can reduce peak traffic for two adjacent cells by 33%. We also compute the associated D2D traffic overhead.

For general settings beyond the example, we provide tractable models to analyze the performance of D2D load balancing in Sec. IV. We also exploit the optimal solutions without and with D2D load balancing in Sec. V and Sec. VI, respectively.

Theoretically, for arbitrary settings, we derive an upper bound for the benefit of D2D load balancing, in terms of sum peak traffic reduction in Sec. VII-B. We show that the bound is asymptotically tight for a specified network scenario, where we further derive the corresponding overhead, in terms of incurred D2D traffic. Our bound and analysis reveal the insight behind the effectiveness of D2D load balancing: by aggregating traffic among adjacent cells via inter-cell D2D communication, we can leverage statistical multiplexing gains to better serve the overall traffic without requiring extra network capacity.

Empirically, in Sec. X, we use real-world 4G data traces to verify our theoretical analysis and reveal that D2D load balancing can reduce sum peak traffic of individual cells by 25%, at the cost of 0.7% D2D traffic overhead. This implies significant spectrum saving at a negligible system overhead.

Throughout this paper, we assume that time is slotted into intervals of unit length, and each wireless hop incurs one-slot delay. We focus on uplink communication scenarios, while our analysis is also applicable to the downlink communication. In addition, in the rest of this paper, for any two positive integers with , we use notation to denote set , i.e., . When , we further simplify notation to be , i.e., .

Ii Related Work

In this paper, we use a dataset from Smartone to show that the peak traffic of different adjacent BSs occurs at different time epochs. Similar observation is also obtained from the measurement studies in [10] and [11]. The authors in [10] analyze the 3G cellular traffic of three major cities in China during 2010 and 2013 and a city in a Southeast Asian country in 2013. They show that the correlation coefficient of the traffic profiles of different BSs is small (between 0.16 and 0.33). The authors in [11] analyze the 3G/4G cellular traffic of 9600 BSs in Shanghai, China in 2014. They show that different areas (residential area, business district, transport, entertainment, and comprehensive area) have different traffic patterns, which have different peak epochs. All these traffic measurements motivate us to do load balancing among different BSs so as to reduce the peak demand (spectrum requirement).

In this paper, we propose the D2D load balancing scheme to reduce the peak demand (spectrum requirement) of BSs. There are other load balancing schemes to achieve the goal, including smart user association [12, 13] and mobile offloading [14].

Smart user association [12, 13] dynamically associates users to the BSs so as to balance the traffic demand of all BSs. However, (i) smart user association schemes normally should be operated on large timescale to overcome the large overhead incurred by frequently switching from one BS to another BS (a.k.a., handover) [12]; thus it is not designed for balancing traffic across BSs on small timescale, and (ii) smart user association scheme in [13], where cellular operators globally associate every user to a BS in a centralized manner, incurs high overhead and complexity. Other smart user association schemes through cell breathing [15] or power control methods, where every user locally connects to the BS with strongest signal in a distributed manner, will change the interference levels significantly and thus they may need for spectral re-allocations across the whole networks. Instead, D2D scheme can do load balancing on short timescale since D2D communications often occur locally within short distances and low power and thus D2D scheme has limited impact to the cellular network. Although D2D load balancing may need to switch between the BS mode (connecting to the BS) and the D2D mode (connecting to the device), such a switch happens locally and it is more lightweight than the global handover between different BSs. Therefore, though D2D load balancing scheme will incur some overhead during D2D communications, it has some unique advantages over smart user association schemes. Meanwhile, we also remark that D2D scheme and smart user association schemes are complementary for load balancing in the sense that we might simultaneously use smart user association schemes on large timescale and use D2D scheme on small timescale. Thus, in this paper we advocate the D2D load balancing scheme.

Mobile offloading [14, 16, 17, 3] is another scheme to reduce the cellular traffic demand. It mainly uses WiFi infrastructure. However, mobile offloading and D2D load balancing are technically different schemes: mobile offloading aims to exploit outband spectrum, but our D2D load balancing scheme targets to increase inband cellular temporal spectrum efficiency. Furthermore in D2D load balancing, the cellular operation can ubiquitously control everything, including both D2D and user-to-BS transmissions. However, mobile offloading usually outsources a portion of traffic to a thirdparty entity, imposing unpleasant unreliability for transmissions. Therefore, our proposed D2D load balancing scheme can ensure better QoS than mobile offloading. Again, our D2D load balancing scheme are orthogonal to the mobile offloading scheme in the sense that the operators can simultaneously use them to reduce the cellular spectrum requirement.

In addition to those traffic load balancing schemes, spectrum reallocation is another effective approach to reduce the spectrum requirement. Instead of moving traffic among different cells, spectrum reallocation dynamically allocate the spectrum among different cells to better match the time-varying traffic demands [18, 19, 20, 21]. However, spectrum allocation incurs high complexity. The state-of-the-art spectrum allocation solution is proposed in [21], which can obtain near-optimal performance for a network with up to 1000 APs and 2500 active users. Furthermore, spectrum reallocation again is operated on large timescale. Hence, the cellular operator can simultaneously do spectrum reallocation on large timescale based on aggregated traffic information [19] and use our proposed D2D load balancing scheme on small timescale based on the fine-grained traffic information to reduce the spectrum requirement.

We further remark that there are some existing works on D2D load balancing. For the three-tier LTE-Advanced heterogenous networks, [9] examines the technical feasibility and designs practical algorithm for D2D load balancing; [22, 23, 24] propose research allocation strategies to achieve load balancing goal via D2D transmission. In [25], an auction-based mechanism is proposed to incentivize the mobile users to participate in D2D load balancing. However, all existing works do not directly answer the two important questions proposed in Sec. I.

Iii An Illustrating Example

We consider a simple scenario shown in Fig. 1(a), where 4 users are each aiming at transmitting 3 packets to two base stations (BS) subject to a deadline constraint. We compare the peak traffic of both BSs for the case without D2D load balancing (Fig. 1(b)) and for the case with D2D load balancing (Fig. 1(c)). We illustrate the concept of D2D load balancing and show that it can reduce the peak traffic for two adjacent cells by 33%.

(a) Cellular network topology and traffic demands.
(b) Conventional cellular approach without D2D.
(c) Our approach with D2D load balancing.
Fig. 1: A simple example for demonstrating the concept of D2D load balancing, and that it can reduce the peak traffic for both cells by 33% (both from 3 to 2) at the cost of 4 extra inter-cell D2D transmissions.

Specifically, we consider a cellular network of two adjacent cells served by BS and BS , and four users , , , . BS (resp. ) can directly communicate with only users and (resp. users and ). BS and BS use orthogonal frequency bands. Due to proximity, users and can communicate with each other using frequency band of either BS or , creating inter-cell D2D links. Both user and user generate 3 packets at the beginning of slot 1, and both user and user generate 3 packets at the beginning of slot 3. All packets have the same size and a delay constraint of 2 slots, i.e., a packet must reach BS or within 2 slots from its generation time. We assume that a packet is successfully delivered as long as it reaches any BS, since BSs today are connected by a high-speed optical backbone, supported by power clusters, and can coordinate to jointly process/forward packets for users.

In the conventional approach without D2D load balancing, a user only communicates with its own BS. It is straightforward to verify that the minimum peak traffic of both BS and BS is 3 (unit: packets), and can be achieved by the scheme in Fig. 1(b). For instance, the minimum peak traffic for BS is achieved by user (resp. user ) transmitting all its 3 packets to BS in slot 1 (resp. slot 2).

With D2D load balancing, we can exploit the inter-cell D2D links between users and to perform load balancing and reduce the peak traffic for both BS and BS .

  • In slot 1, user transmits two packets and to BS , and user transmits two packets and to user using the orthogonal frequency band of BS . The traffic is 2 for both cells. In slot 2, users and transmit their remaining packets and to BS , and user relays the two packets it received in slot 1, i.e., and , to BS . The traffic is again 2 for both cells. By the end of slot 2, we deliver 6 packets for users and to BSs.

  • In slots 3 and 4, note that users and have the same traffic pattern as users and , but offset by 2 slots. Thus we can also deliver 3 packets for both users and in two slots. The traffic of both BSs is 2 per slot.

Overall, with D2D load balancing, we can serve all traffic demands with peak traffic of 2 for both BSs, which is 33% reduced as compared to the case without D2D load balancing.

The intuition behind this example is that the peak traffic for the two cells occurs at different time instances. When users and transmit data to BS in the first two slots, BS is idle. Meanwhile, BS is idle when users and transmit data to BS in the last two slots. Therefore, D2D communication can help load balance traffic from the busy BS to the other idle BS, reducing the peak traffic for both BSs. However, D2D load balancing also comes with cost, since it requires transmissions over the inter-cell D2D links. In the example, the total traffic is packets and the D2D traffic is packets, yielding an overhead traffic ratio of . Such D2D traffic is the overhead that we pay in return for peak traffic reduction.

Iv System Model

In this section, we present the system model for a general network topology and a general traffic demand model beyond the simple example expounded in the previous section. Such models will be used to analyze the benefit of D2D load balancing in general settings, in terms of spectrum reduction ratio, and the cost in terms of D2D traffic overhead ratio.

Iv-a Cellular Network Topology

Consider an uplink wireless cellular network with multiple cells and multiple mobile users. We assume that each cell has one BS and each user is associated with one BS111We say that user is associated with BS if user is in the cellular cell covered by BS . When a user is covered by multiple BSs, we assume that this user has been associated with one of them, e.g., the one with the strongest signal-to-noise ratio. In the rest of this paper, we will also use the terminology, cell , to represent the cell covered by BS .. Define as the set of all BSs, as the set of users belonging to BS , and as the set of all users in the cellular network. Let denote the cell (or BS) with which user is associated. We model the uplink cellular network topology as a directed graph with vertex set and edge set where if there is a wireless link from vertex (user) to vertex (BS or user) .

Iv-B Traffic Model

We consider a time-slotted system with slots in total, indexed from 1 to . Each user can generate a delay-constrained traffic demand at the beginning of any slot. We denote as the demand set. Each demand is characterized by the tuple where

  • is the user that generates demand ;

  • is the starting time/slot of demand ;

  • is the ending time/slot (deadline) of demand ;

  • is the volume of demand with unit of bits.

Namely, demand is generated by user at the beginning of slot with the volume of bits and it must be delivered to BSs before/on the end of slot , implying a delay requirement . We also call interval the lifetime of the demand . We further denote as the set of demands that are generated by the users in BS , i.e., Demand is delivered in time if every bit of demand reaches a BS before/on the end of slot . Note that different bits in demand could reach different BSs. Thus, every user can transmit a bit either to its own BS directly in a single hop or to another user via the D2D link between them such that the bit can reach another BS in multiple hops.

Iv-C Wireless Channel/Spectrum Model

For each link , we denote its link rate as (units: bits per slot per Hz), which is the number of bits that can be transmitted in one unit (slot) of time resource and with one unit (Hz) of spectrum resource. Then if we allocate (unit: Hz) spectrum to link at slot , this link can transmit bits of data from node to node in slot . Note that we simplify the channel model by assuming a linear relationship between the allocated spectrum and the transmitted data. This assumption is reasonable for the high-SNR scenario when we use Shannon capacity as the link rate [26].

Iv-D Performance Metrics

In this paper, we aim at minimizing the total (amount of) spectrum to deliver all demands in in time. In particular, we need to obtain the minimum spectrum/frequency to serve all demands in time without D2D (resp. with D2D), denoted by (resp. ). To evaluate the impact of D2D load balancing, we characterize both the benefit and the cost for D2D load balancing. The benefit is in terms of spectrum reduction ratio,

(1)

The cost is in terms of (D2D traffic) overhead ratio,

(2)

where is the volume of all D2D traffic and is the volume of all traffic directly sent by cellular users to BSs.

The spectrum reduction ratio evaluates how much spectrum we can save if we apply D2D load balancing. The overhead ratio evaluates the percentage of D2D traffic among all traffic. D2D traffic incurs cost in the sense that any traffic going through D2D links will consume spectrum and energy of user devices but do not immediately reach any BS. Overall, the spectrum reduction ratio captures the benefit of D2D load balancing and hence larger means larger benefit; the overhead ratio captures the cost of D2D load balancing and hence smaller means smaller cost. In the following, we will discuss how to obtain in Sec. V and in Sec. VI. Then we will show the theoretical upper bounds for and in Sec. VII.

V Optimal Solution without D2D

In this section, we describe how to compute the minimum spectrum without D2D, i.e., . Since there are no D2D links, we can calculate the required minimum spectrum for each BS separately. Let us denote as the minimum spectrum of BS to deliver all its own traffic demands, i.e., . Then the total minimum spectrum without D2D is222Here for simplicity, we assume that all BSs use orthogonal spectrum. We discuss how to extend our results to the practical case of spectrum reuse in Sec. IX.

V-a Problem Formulation

For each BS , we formulate the problem of minimizing the spectrum to deliver all demands in cell without D2D, named as ,

(3a)
s.t. (3b)
(3c)
(3d)
(3e)

where is the allocated spectrum (unit: Hz) for transmitting demand from user to BS at slot , the auxiliary variable is the total used spectrum from users to BS at slot , and is the allocated (peak) spectrum to BS ,

Our objective is to minimize the total allocated spectrum of BS , as shown in (3a). Without D2D, users can only be served by its own BS. Equation (3b) shows the volume requirement for any traffic demand , i.e., the total traffic volume needs to be delivered from user to BS during its lifetime. Equation (3c) depicts the total needed spectrum of cell (i.e., ) in slot , which is the summation of allocated spectrum for all active jobs in slot . Inequality (3d) shows that the total needed spectrum of cell in any slot cannot exceed the total allocated spectrum of BS . Finally, inequality (3e) means that the allocated spectrum for a job in any slot is non-negative.

Let us denote as the maximum delay among all demands. Then the number of variables in is and the number of constraints in is also .

V-B Characterizing the Optimal Solution

To solve

, we can use standard linear programming (LP) solvers. However, LP solvers cannot exploit the structure of this problem. We next propose a combinatorial algorithm that exploits the problem structure and achieves lower complexity than general LP algorithms.

We note that resembles a uniprocessor scheduling problem for preemptive tasks with hard deadlines [27]. Indeed, we can attach each task with an arrival time and a hard deadline and the requested service time . Then for a given amount of allocated spectrum (which resembles the maximum speed of the processor), we can use the earliest-deadline-first (EDF) scheduling algorithm [28] to check its feasibility. Since we can easily get an upper bound for the minimum spectrum, we can use binary search to find the minimum spectrum , supported by the EDF feasibility-check subroutine.

More interestingly, we can even get a semi-closed form for , inspired by [29, Theorem 1]. Specifically, let us define the intensity [29] of an interval to be

(4)

where is the set of all active traffic demands whose lifetime is within the interval . Then we have the following theorem.

Theorem 1

.

Proof:

Since the proof of Theorem 1 was omitted in [29] and the theorem is not directly mapped to the minimum spectrum problem, we give a proof in Appendix C for completeness.

Theorem 1 shows that is the maximum intensity over all intervals. To obtain the interval with maximum intensity (and hence ), we adapt the algorithm originally developed for solving the job scheduling problem in [29], which is called YDS algorithm named after the authors, to our spectrum minimization problem. The time complexity of the YDS algorithm is related to the total number of possible intervals. Clearly the optimal interval can only begin from the generation time of a demand and end at the deadline of a demand. So the total number of intervals needed to be checked is . Thus the time complexity of our adaptive YDS algorithm is [29]. But the complexity of general LP algorithms is where is a parameter determined by the coefficients of the LP [30]. Thus, our combinatorial algorithm has much lower complexity than general LP algorithms.

Vi Optimal Solution With D2D

In this section, we formulate the optimization problem to compute the minimum sum spectrum when D2D communication is enabled. In this case, since the traffic can be directed to other BSs via inter-cell D2D links, all BSs are coupled with each other and need to be considered as a whole. We will first define the traffic scheduling policy with D2D and then formulate the problem as an LP.

Vi-a Traffic Scheduling Policy

Given traffic demand set , we need to find a routing policy to forward each packet to BSs before the deadline, which is the traffic scheduling problem. Since we should consider the traffic flow in each slot, we will use the time-expanded graph to model the traffic flow over time [31]. Specifically, denote as the allocated spectrum (unit: Hz) for link at slot for demand . Then the delivered traffic volume from node to node at slot for demand is . For ease of formulation, we set the self-link rate to be . Then the self-link traffic i.e., , is the traffic volume stored in node at slot for demand . But the allocated (virtual) spectrum for self-link traffic, i.e., , will not contribute to the spectrum requirements of BSs (see (6c) later). All traffic flows over time are precisely captured by the time-expanded graph and . Then we define the traffic scheduling policy as follows.

Definition 1

A traffic scheduling policy is the set such that

(5a)
(5b)
(5c)
(5d)
(5e)

where and are the incoming neighbors and outgoing neighbors of node in the time-expanded graph.

Constraint (5a) shows the flow balance in the source node while (5b) shows the flow balance in the destination nodes such that all traffic can reach BSs before their deadlines. Equality (5c) is the flow conservation constraint for each intermediate node in the time-expanded graph. Here we assume that all BSs and all users have enough radios such that they can simultaneously transmit data to and receive data from multiple BSs (or users). This is a strong assumption for mobile users because current mobile devices are not equipped with enough radios. However, multi-radio mobile devices could be a trend and there are substantial research work in multi-radio wireless systems (see a survey in [32] and the references therein). We made this assumption here because wireless scheduling problem for single-radio users is generally intractable and we want to avoid detracting our attention and focus on how to characterize the benefit of D2D load balancing and get a first-order understanding. We remark that this assumption is also made in recent work [21] on spectrum reallocation in small-cell cellular networks.

Vi-B Problem Formulation

Then we formulate the problem of computing the minimum total spectrum to serve all demands in all cells with D2D, named as Min-Spectrum-D2D,

(6a)
s.t.
(6b)
(6c)
(6d)

where the auxiliary variable is the total used spectrum from users to BS at slot , the auxiliary variable is the total used spectrum dedicated to all users in BS at slot , and is the allocated (peak) spectrum for BS . Note that in our case with D2D load balancing, a user can adopt the D2D mode to transmit to another user via a D2D link (e.g., is the allocated spectrum to the D2D link from user to user in slot ) and/or the cellular mode to transmit to its BS via a user-to-BS link (e.g., is the allocated spectrum to the user-to-BS link from user to BS in slot ). In addition, note that we assume a receiver-takeover scheme in the sense that any traffic will consume spectrum resources of the receiver’s BS. Equalities (6b) and (6c) show that BS is responsible for all traffic dedicated to itself and to its users except self-link (virtual) spectrum (see Sec. VI-A). We also remark that although spectrum sharing is one of the major benefits of D2D communication, in this work we do not model the spectrum sharing among D2D links and user-to-BS links to simplify the analysis. Later in Sec. X, we show that our D2D load balancing scheme can significantly reduce the spectrum requirement even without doing spectrum sharing among D2D links and user-to-BS links. If we further do spectrum sharing, the D2D load balancing has more gains.

Given an optimal solution to Min-Spectrum-D2D, we denote as the allocated spectrum for each BS , and thus the total spectrum is The total D2D traffic and total user-to-BS traffic are

(7)
(8)

which are used to calculate the overhead ratio in (2). We further remark that since all traffic demands must reach any BSs, it is easy to see that the user-to-BS traffic is exactly the total volume of all traffic demands, i.e.,

Given the optimal (minimum) total spectrum, i.e., , we next minimize the overhead, named Min-Overhead, by solving the following LP333In other words, minimizing the total spectrum is our first-priority objective and minimizing the corresponding D2D traffic overhead (without exceeding the minimum total spectrum) is our second-priority objective.,

(9a)
(9b)
(9c)
(9d)
(9e)

As compared to Min-Spectrum-D2D in (6), Min-Overhead in (9) adds a constraint (9e) for the given total spectrum and changes the objective to be the total D2D traffic defined in (7). Note that even though we write (9e) as an inequality, it must hold as an equality. This is because is the optimal value of Min-Spectrum-D2D in (6) and any solution in Min-Overhead in (9) is also feasible to Min-Spectrum-D2D in (6).

The number of variables in Min-Spectrum-D2D is and the number of constraints in Min-Spectrum-D2D is . The problem Min-Overhead has the same complexity as Min-Spectrum-D2D. Solving the problem, even though it is an LP, incurs high complexity. We further discuss how to reduce the complexity without loss of optimality in Appendix B. Even with our optimized LP approach, later in our simulation in Sec. X, we show that we cannot solve Min-Spectrum-D2D

for practical Smartone network with off-the-shell servers. Thus, we further propose a heuristic algorithm to solve

Min-Spectrum-D2D with much lower complexity in Sec. VIII. We also provide performance guarantee for our heuristic algorithm. Before that, we show our theoretical results on the spectrum reduction ratio and the overhead ratio in next section.

Vii Theoretical Results

From the two preceding sections, we can compute with the (adaptive) YDS algorithm (Theorem 1) and by solving the large-scale LP problem Min-Spectrum-D2D (Sec. VI-B). Hence, numerically we can get the spectrum reduction and the overhead ratio. In this section, however, we seek to derive theoretical upper bounds on both spectrum reduction and overhead ratio. Such theoretical upper bounds provide insights for the key factors to achieve large spectrum reduction and thus provide guidance to determine whether it is worthwhile to implement D2D load balancing scheme in real-world cellular systems.

Vii-a A Simple Upper Bound for Spectrum Reduction

We can get a simple upper bound for by assuming no cost for D2D communication in the sense that any D2D communication will not consume bandwidth and will not incur delays. Then we can construct a virtual grand BS and all users are in this BS. Then the system becomes similar to the case without D2D. We can apply the YDS algorithm to compute the minimum peak traffic, which is a lower bound for , i.e., where

(10)

Here in (10), is the set of all active traffic demands whose lifetime is within the interval and is the best user-to-BS link. Then we have the following theorem.

Theorem 2

.

Proof:

Please see Appendix D.

Note that both and can be computed by the YDS algorithm, much easier than solving the large-scale LP Min-Spectrum-D2D. Therefore, numerically we can get a quick understanding of the maximum benefit that can be achieved by D2D load balancing.

Vii-B A General Upper Bound for Spectrum Reduction

We next describe another general upper bound for any arbitrary topology and any arbitrary traffic demand set. We will begin with some preliminary notations.

We first define some preliminary notations. Let be the number of BSs and we define a directed D2D communication graph where the vertex set is the BS set and if there exists at least one inter-cell D2D link from user in BS to user in BS . Denote as the in-degree of BS in the graph and define the maximum in-degree of the graph as . In addition, we define some notations in Tab. I to capture the discrepancy of D2D links and non-D2D links for users and BSs. Note that these definitions will be used thoroughly in Appendix E to prove Theorem 3.

Now we have the following theorem.

Theorem 3

For an arbitrary network topology associated with a D2D communication graph and an arbitrary traffic demand set, the spectrum reduction is upper bounded by

(11)
Proof:

Please see Appendix E.

Based on this upper bound, we observe that the benefit of D2D load balancing comes from two parts: intra-cell D2D and inter-cell D2D. More interestingly, we can obtain the individual benefit of intra-cell D2D and inter-cell D2D separately, as shown in the following Corollaries 1 and 2. One can go through the proof for Theorem 3 by disabling inter-cell or intra-cell D2D communication and get the proof of these two corollaries.

TABLE I: Discrepancy Notations ,
Fig. 2: The benefit of intra-cell D2D communications.
Fig. 3: Tradeoff between and .
Corollary 1

If only intra-cell D2D communication is enabled, the spectrum reduction is upper bounded by

(12)

This upper bound is quite intuitive. When , then for any user , there does not exist any intra-cell D2D link with better link quality than its direct link to BS . Therefore, using the user-to-BS link is always the optimal choice. Thus the spectrum reduction is 0. When , larger means more advantages for intra-cell D2D links over the user-to-BS links. Therefore, D2D can exploit more benefit.

Moreover, this upper bound can be achieved by the simple example in Fig. 3. Suppose that user generates one traffic demand with volume and delay at slot 1. Suppose link rates . Then without intra-cell D2D, the (peak) spectrum requirement is . With intra-cell D2D, user transmits traffic to user b from slot 1 to slot and then user transmits all traffic to BS at slot . The (peak) spectrum requirement is . Then the spectrum reduction is

(13)

The benefit of intra-cell D2D communication is widely studied (see [7] [8]). However, in this paper, we mainly focus on the benefit of inter-cell D2D load balancing. Indeed, in our simulation settings in Sec. X, the intra-cell D2D brings negligible benefit.

Corollary 2

If only inter-cell D2D communication is enabled, the spectrum reduction is upper bounded by

The intuition behind the parameter is similar to the effect of parameter in the intra-cell D2D case. In what follows, we will only discuss the effect of parameter , which actually reveals the insight of our advocated D2D load balancing scheme. Now suppose that all the links have the same quality and w.l.o.g. let . Then , meaning that no intra-cell D2D benefit exists. And the benefit of inter-cell D2D is reduced to the following upper bound

(14)

The rationale to understand this upper bound is as follows. On a high level of understanding, the main idea for load balancing is traffic aggregation. If each BS can aggregate more traffic from other BSs, it can exploit more statistical multiplexing gains to serve more traffic with the same amount of spectrum. Since the in-degree for each BS indeed measures its capacity of traffic aggregation, it is not surprising that the upper bound for is related to maximum in-degree .

To evaluate how good the upper bound in (14) is, two natural questions can be asked. The first is: Is this upper bound tight? Another observation is that if we want to achieve unbounded benefit, i.e., , it is necessary to let , which means that . Then the second question is: Can indeed approach 100% as ?

In the rest of this subsection, we will answer these two questions by constructing a specified network and traffic demand set. Specifically, we consider BSs each serving one user only. To facilitate analysis, let be the -th BS and be the user in BS , for all . We consider a singleton-decoupled traffic demand set as follows. Each user has one and only one traffic demand with the same volume and the same delay . Let and the traffic generation time of user be slot . Therefore, the lifetime of user ’s traffic demand is , during which there are no other demands.

Under such settings, we will vary the user-connection pattern such that the D2D communication graph is different. Specifically, we will prove that this upper bound is asymptotically tight in the ring topology for in Fact 1, and 100% in the complete topology as the number of BSs in Fact 2. Moreover, we will also discuss the overhead ratio for these two special topologies.

Fact 1

If and the D2D communication graph forms a bidirectional ring graph, then there exists a traffic scheduling policy such that the spectrum reduction is

(15)

Besides, the overhead ratio in this case is

(16)
Proof:

Please see Appendix F.

Fact 2

If the D2D communication graph forms a bidirectional complete graph, then there exists a traffic scheduling policy such that the spectrum reduction is

(17)

Besides, the overhead ratio in this case is

(18)
Proof:

Please see Appendix G.

Remark: (i) Fact 1 shows the tightness of the upper bound in (14) for the ring-graph topology when . (ii) Fact 2 shows that can indeed approach , implying that in the best case, goes to . This gives us strong motivation to investigate D2D load balancing scheme both theoretically and practically. (iii) For the complete-graph topology, the upper bound is not tight. Indeed, since in the complete-graph topology, we have

(19)

(iv) Let us revisit the toy example in Fig. 1 which forms a complete-graph topology with . It verifies the spectrum reduction and overhead ratio in Fact 2, i.e., and . (v) We also highlight the tradeoff between the benefit and the cost , as illustrated in Fig. 3. Furthermore, Fig. 3 shows that the complete-graph topology outperforms the ring-graph topology asymptotically because and for the ring-graph topology but (larger benefit) and (smaller cost) for the complete-graph topology.

Vii-C An Upper Bound for Overhead Ratio

Previously we study upper bounds for the spectrum reduction. Now we instead propose an upper bound for overhead ratio. Recall that is the maximum demand delay. We then have the following result.

Theorem 4

.

Proof:

Please see Appendix H.

The upper bound in Theorem 4 increases when the maximum demand delay increases. This is reasonable because a traffic demand can travel more D2D links (and thus incurs more D2D traffic overhead) if its delay is large. For our toy example in Fig. 1, we have and thus the upper bound for the overhead ratio is , which is in line with our actual overhead ratio 25%.

Viii A Low-Complexity Heuristic Algorithm for Min-Spectrum-D2D

Our proposed LP formulation for Min-Spectrum-D2D has high complexity due to the size of input traffic demand and cellular network. To reduce the complexity, in this section, we propose a heuristic algorithm which can significantly reduce the number of traffic demands that is needed to be considered. Moreover, our algorithm has a parameter (which is defined shortly) such that we can balance the complexity and the performance.

Our proposed algorithm has three steps.

 

Step I. We solve Min-Spectrum-ND for each BS , and get the optimal solution .

Step II. For each BS with the spectrum profile , we consider the following set,

(20)

where parameter controls the split level. Now we divide all cell- traffic demands into two demand sets

(21)

and

(22)

For all traffic demand in , we schedule them according to without D2D, which results in at most spectrum requirement for BS at slot . Note that no demand in is served in slot set . We thud denote as the already allocated spectrum spectrum for demand set for BS at slot , which satisfies when and when .

Step III. We solve the D2D load balancing problem with traffic demands , according to the following LP, which adaptes Min-Spectrum-D2D in (6) by considering the already allocated spectrum ,

(23a)
s.t.
(23b)
(23c)
(23d)

Similar to the overhead minimization problem Min-Overhead in (9), given the optimal spectrum requirement of (23), denoted as, , we next minimize the overhead by solving the following LP,

(24a)
s.t.
(24b)
(24c)
(24d)
(24e)

 

Note that in (23)/(24), all variables, , have the same meanings of those in (6)/(9). There are two differences between (23)/(24) and (6)/(9). First, the traffic demand set in (23)/(24) is while that in (6)/(9) is . Likewise, the traffic scheduling policy characterized by (5a), (5b), (5c), (5d), (5e) in (23)/(24) is for the traffic demand set while that in (6)/(9) is for the traffic demand set . Second, constraint (23d)/(24d) is different from constraint (6d)/(9d) in that (23d)/(24d) considers the already allocated spectrum . Namely, the spectrum requirement for BS at slot includes the already allocated spectrum to serve the traffic demand and the new allocated spectrum to serve the traffic demand .

Obviously, if the number of traffic demand in is much less than the total number of traffic demands in , which is indeed the case according to our empirical study in Sec. X, we can significantly reduce the number of variables and constraints in (23)/(24) in Step III as compared to the LP problem Min-Spectrum-D2D/Min-Overhead in (6)/(9). After these three steps, the total spectrum is given by the objective value of (23) and the corresponding overhead is given by the objective value of (24). An example of our heuristic algorithm is shown in Appendix I.

We denote the spectrum reduction of our heuristic algorithm as

(25)

Similarly, we denote as the overhead ratio of our heuristic algorithm. We next show that the performance guarantee of our heuristic algorithm.

First, for the spectrum we reduction, we have,

Theorem 5

Proof:

Please see Appendix J.

Theorem 5 shows that when , we have . This is because when , we have , i.e., all demands participate in D2D load balancing in our heuristic algorithm when and thus the objective value of (23) when is exactly . When , since , all traffic demands are served locally without D2D and therefore the objective value of (23) when is exactly . Thus, the lower bound is tight. Further, the lower bound , decreases as increases, but the computational complexity decreases as increases. Thus, this lower bound illustrates the tradeoff between the performance and the complexity of our heuristic algorithm.

Second, we give an upper bound for the overhead ratio444Recall that is the maximum demand delay..

Theorem 6

.

Proof:

Please see Appendix K.

We can see that the upper bound of the overhead ratio is 0 when because , i.e., all traffic demands are served locally without D2D. Moreover, when increases, the upper bound decreases because less traffic demands participate in D2D load balancing.

Overall, our heuristic algorithm reduce the complexity of our global LP approach and has performance guarantee. Moreover, our proposed heuristic algorithm has a controllable parameter to balance the benefit in terms of spectrum reduction, the cost in terms of overhead ratio, and the computational complexity for our D2D load balancing scheme.

Ix Towards Spectrum Reduction with Frequency Reuse

In this paper, we use the sum spectrum to describe how many resources are needed to serve all users’ traffic demands in cellular networks. This may not directly reflect the total required spectrum for cellular operators, because the same spectrum can be spatially reused by multiple BSs who are sufficiently far away from each other. The benefit of spectrum spatial reuse is characterized by the frequency reuse factor , which represents the proportion of the total spectrum that one cell can utilize. For instance, means that any cell can use all spectrum, and means that one cell can only utilize of the total spectrum, to avoid excessive interference among adjacent cells. A back-of-the-envelope calculation suggests that, if the total number of required channels for all BSs is , then distinct radio channels are needed to serve the entire cellular network.

In the case without D2D, the sum spectrum of all BSs is , which corresponds to the total number of channels for all cells. Thus, with frequency reuse factor , distinct channels are needed without D2D.

In the case with D2D, D2D communication can degrade the original frequency reuse pattern if they are sharing the same spectrum with cellular users (which is called underlay D2D [7]). Given the new frequency reuse factor . A back-of-the-envelope analysis suggests that

distinct radio channels are needed with D2D load balancing. Consequently, the spectrum reduction can be estimated as