Delay minimization based uplink resource allocation for device-to-device communications considering mmWave propagation

System model and problem formulation

Figure 1 represents the system model of D2D communications which reuse the uplink resources of equipments in wireless networks. In these systems, there are two types of equipments: pairs of D2D user equipments (DUEs) and cellular user equipments (CUEs). The first one reuse radio resource allocated to the CUEs and communicate directly without communication load on the base station (BS).

This system is composed of M CUEs, including real and virtual CUEs, N DUEs, K idle resource blocks (RBs) scheduled for transmission during a transmission time interval (TTI) and M − K real CUEs. The RBs correspond to the minimum allocation unit in a wireless communication system, and each idle RB, not used by any real CUEs, is regarded as a virtual CUE with zero transmit power and no signal-to-interference-plus-noise ratio (SINR) requirement. The CUEs are denoted by C₁,C₂,…,C_M and the DUEs by D₁,D₂,…,D_N. The DUE pair tdransmitter is denoted by D_n,Tx and the receiver by D_n,Rx. In order to facilitate the comprehension of the proposed approach and of the considered system model, Table 1 presents the variables and parameters considered in this paper.

Before sharing the RBs to DUEs during transmission, it is pre-allocated a set of uplink RBs for each CUE. The allocated bandwidth denoted by W_m for each CUE m is proportional to the number of allocated RBs. Each CUE can share its RBs with several pairs of DUEs, making it possible a large number of simultaneous connections between DUEs. This characteristic describes the multi-sharing resource allocation.

The CUE and DUE relationship is specified thought a set denoted as Θ_n, which represents the set of CUEs that share their RBs with DUE pair n, and a set denoted as Δ_m, which represents the set of the DUE pairs that reuse the RBs allocated to CUE m. These sets Δ₁,Δ₂,…,Δ_M are subsets of $\left\{1,2,...,N\right\}$.

The set of DUEs only can reuse uplink RBs from CUE m if the interference on m’s transmission obey its SINR requirement. The received SINR for any CUE $m\in \left\{1,2,...,M\right\}$ must be superior to the following threshold:

(1) $${\displaystyle \frac{P_m{G}_{mB}}{{\sigma }_m^2+\sum _{n\in {\mathrm{\Delta }}_m}{P}_n{G}_{nB}}\ge {\gamma }_m,}$$where P_m is the transmit power for CUE m, G_mB is the channel gain between CUE m and the BS, σ²_m is the noise power for CUE m, P_n is the transmit power for DUE pair n, G_nB is the channel gain between DUE pair n transmitter and the BS and γ_m is the SINR threshold required by CUE m.

There are also SINR requirements for the DUE pairs. Certain RBs can be reused by a DUE pair n only if the received SINR is higher than:

(2) $$\frac{P_n{G}_{nn}}{{\sigma }_n^2+P_m{G}_{mn}+{\displaystyle \sum _{n^{\prime }\in {\Delta }_m-\left\{n\right\}}{P}_{n^{\prime }}}{G}_{n^{\prime }n}}\ge {\gamma }_n,\forall m\in {\Theta }_n,$$where P_n is the transmit power for DUE pair n, G_nn is the channel gain between the DUE pair n two ends, σ²_n is the noise power for DUE pair n, P_m is the transmit power for CUE m, G_mn is the channel gain from CUE m to DUE pair n, G_n′n is the channel gain from D_n′,Tx to D_n,Rx and γ_n is the SINR threshold for DUE pair n.

It is assumed in this work that the BS knows the channel properties such as transmit power, channel gain and noise power of the communication link for CUEs m and DUE pairs n, i.e., the BS knows the channel state information (CSI) and the SINR threshold for each user.

The multi-sharing resource allocation problem objective is determine which DUEs reuse RBs from CUEs such that the total system throughput F is maximized. The total system throughput F is defined as the sum of all CUEs’ and DUEs’ Shannon capacities and is formulated as follows:

(3) $$\begin{array}{l}F=\underset{{\Delta }_1,{\Delta }_2,\dots ,{\Delta }_M}{\max }{\displaystyle \sum _{m=1}^M\{}{W}_m\log 2(1+\frac{P_m{G}_{mB}}{{\sigma }_m^2+{\displaystyle \sum _{n\in {\Delta }_m}{P}_n}{G}_{nB}})+\\ {\displaystyle \sum _{n\in {\Delta }_m}{W}_m}\log 2(1+\frac{P_n{G}_{nn}}{{\sigma }_n^2+P_m{G}_{mn}+{\displaystyle \sum _{n^{\prime }\in {\Delta }_m-\left\{n\right\}}{P}_{n^{\prime }}}{G}_{n^{\prime }n}})\},\end{array}$$subject to:

(4) $$\sum _{n\in {\mathrm{\Delta }}_m}{P}_n{G}_{nB}\le {\displaystyle \frac{P_m{G}_{mB}}{{\gamma }_m}-{\sigma }_m^2,\mathrm{\forall }m\in \left\{1,2,...,M-K\right\},}$$

(5) $$P_m{G}_{mn}+\sum _{n^{\mathrm{\prime }}\in {\mathrm{\Delta }}_m-\left\{n\right\}}{P}_{n^{\mathrm{\prime }}}{G}_{n^{\mathrm{\prime }}n}\le \frac{P_n{G}_{nn}}{{\gamma }_n}-{\sigma }_n^2,\mathrm{\forall }n\in \left\{1,2,\dots ,N\right\},\mathrm{\forall }m\in {\mathrm{\Theta }}_n,$$

(6) $$n\in {\mathrm{\Delta }}_m,\mathrm{\forall }n\in \left\{1,2,\dots ,N\right\},\mathrm{\forall }m\in {\mathrm{\Theta }}_n,$$where constraint (4) represents the maximum tolerable interference on m’s transmission and constraint (5) represent the maximum tolerable interference on n’s transmission.

The multi-sharing resource allocation problem described by Eq. (3) under constraints (4), (5) and (6) is non-deterministic polynomial-time hard. That is, it is NP-hard as proved in Sun et al. (2013) and Ciou et al. (2015), which justifies the need to find an efficient and fast algorithm to solve the problem, such as the one presented in this paper.

Related works

In this section, we briefly comment about some works related to D2D communications and multi-sharing resource allocation. The greedy throughput maximization plus algorithm (GTM+) is an iterative algorithm proposed by Ciou et al. (2015) which attempt to find a solution to the multi-sharing resources allocation problem. This algorithm is based on the maximization of a utility function given in terms of the system throughput considering that RBs are reused (Ciou et al., 2015). The authors in Ciou et al. (2015) use the heuristic algorithm proposed by Basagni (2001) to obtain a maximal weight independent set, with time complexity O(n³). The worst-case complexity of GTM+ is O(n⁴), because in each iteration, at least one DUE pair is granted to reuse the RBs. Contrary to our proposal, the GTM+ algorithm does not consider the system delay as an optimization metric.

Zhang, Hu & Qian (2016) proposed to use a distance based power control scheme for D2D communication in order to achieve expected performance gain without causing performance degradation to the primary cellular users due to system interference. The authors applies the Poisson point process (PPP) model, a stochastic geometry model, to get tractable analysis results. Numeric results presented in simulations show that the proposed scheme is benefit for both CUEs and DUEs. The initial results demonstrate the advantages of using the power control scheme, although comparisons are not made with state-of-the-art schemes in terms of throughput and delay for resource allocation in D2D communication.

Mishra, Pandey & Biswash (2016) proposed a resource allocation scheme for D2D communication in the uplink which consist of two phases. In the first phase, if multihop communication (two-hops) is required, a relay is selected from the available relays between cell edge device and BS. The relay selection scheme selects the parameters such as battery power and reliability to minimize packet loss. In the second phase, an efficient resource allocation scheme is proposed that reduces the upload time and optimizes the number of resource blocks. In this work, we also address resource allocation scheme for D2D communication in the uplink but the throughput maximization occurs via a delay minimization based algorithm.

In Pan et al. (2018), the authors studied power control and the problem of resource allocation in D2D communication underlaying a non-orthogonal multiple access (NOMA) cellular network. The objective of the work is to maximize the throughput sum of all D2D pairs by meeting the minimum rate constraints of users and using techniques of successive interference cancelation. The optimal conditions for power control of cellular users in each subchannel are derived first. Then, it is proposed a dual-based iterative algorithm to solve the resource allocation problem. The results show that the proposed scheme outperforms the conventional scheme for the network with high data requirements. Different from Pan et al. (2018), we consider in this work an uplink of CP-OFDM (cyclic prefix - orthogonal frequency-division multiple access) based communication system. However, we also aim to enhance network QoS parameters such as throughput and delay through the application of our resource allocation approach.

Khuntia & Hazra (2019) proposed a Deep Q-learning with an extended Kalman filter (EKF) scheme to solve the channel and power allocation issue for D2D communication devices when the prior traffic information is not known to the BS. They explore an optimal policy for resource and power allocation with the aim of maximizing the total throughput of the system. The resource allocation scheme comprises of four phases, i.e., cell splitting, clustering, queuing model and channel allocation simultaneously with power allocation. It is used EKF together with Deep Q-Network to incorporate weight uncertainty of the Q-function as well as the state uncertainty during transition, helping the learner in achieving an optimal policy. The authors show the advantage of the resource sharing scheme over other existing schemes through numerical simulations.

The author proposed in Sreedevi & Rama Rao (2019) to use reinforcement-learning based latency controlled D2D connectivity (RL-LCDC) algorithm and its Q-Learning approach in an indoor D2D communication network for 5G connectivity with minimum latency. The algorithm discovers the neighbors, decides the D2D link and adaptively controls the communication range for maximum network connectivity. Results presented in Sreedevi & Rama Rao (2019) show that RL-LCDC optimizes the connectivity with lower delay and better energy efficiency when compared with other conventional schemes. Although both approaches (Sreedevi & Rama Rao, 2019; Khuntia & Hazra, 2019) produce interesting results, we alternatively propose an algorithm in this paper that is not based on reinforcement learning that can solve the problem with a lower computational complexity.

Song et al. (2019) proposed a joint uplink and downlink (JUAD) resource allocation scheme which maximizes system capacity and guarantees the SINR constraint for CUEs and DUEs. The authors formulate the optimization problem as a mixed integer nonlinear programing problem (MINLP) and divide it into two sub-problems, the power allocation and channel assignment. At first, the optimal transmission power is obtained through the convex objective function. Then, it is developed the Hungarian algorithm to achieve joint uplink and downlink channel assignment, improving system capacity performance and increasing spectrum efficiency. Authors show through simulations that the performance of the algorithm is better than that of schemes for independent allocation. The JUAD algorithm has among its advantages the fact that it works with allocation in the uplink and downlink.

Li et al. (2019) proposed to use a resource allocation scheme that integrates a probabilistic approach to a quasi-convex optimization algorithm based on channel probability statistical characteristics for D2D communication mode selection and resource optimization. The authors also proposed a sub-optimal allocation algorithm when number of users is too large, in order to save costs and improve efficiency. The presented results show that the algorithm optimizes total throughput of the system and reduces communication interference between users. However, in the same way as JUAD and the other mentioned algorithms, the algorithm proposed in Li et al. (2019) does not consider the user’s system delay in the optimization process, an important parameter in any resource allocation system, especially if we consider that D2D communication takes advantage of the proximity between users to increase spectral efficiency. That is, by decreasing communication delays of the links, spectral efficiency can be increased.

MFBAP envelope process

In general, fractals are described in the literature according to a set of behaviors and characteristics, such as self-similarity, phenomena with patterns that are repeated at different scales and with irregular structures. Multifractals are characterized by a set of fractal dimensions and are used to treat phenomena that occur in multiple scales and dimensions (Feldmann, Gilbert & Willinger, 1998).

Network traffic traces captured at small scales in general tend to be multifractal. This means that they present highly dependent structure between samples with burst incidences at various scales. These characteristics can degrade network performance in relation to traffic flows considered Gaussian and short-dependent (Feldmann, Gilbert & Willinger, 1998). In this work, we propose to use a multifractal envelope process once it was shown that it can better describe real traffic envelope processes at the time scale considered in this paper than monofractal based or short-range based envelope processes (Santos & Vieira, 2015).

The envelope process for incoming packet traffic is an upper bound for the actual accumulated packet traffic process. For a deterministic envelope process, the function $\hat{A}(t)$ corresponds to the maximum value of a flow A(t) in the time slot [s, s + t], and is defined by equation as follows (Le Boudec & Thiran, 2004):

(7) $$\hat{A}(t)=\underset{s\ge 0}{sup}A\left[s,s+t\right],$$where $sup$ is an operator that returns the maximum value of A[s, s + t] without establishing an upper bound value s ≥ 0 in this case.

The multifractal bounded arrival process (MFBAP) is a deterministic alternative to obtain the envelope process that limits the volume of traffic in a given time interval, calculated as follows (Santos & Vieira, 2015):

(8) $${\hat{A}}_{MFBAP}(t)=\overline{a}t+k\sigma t^{H(t)}+B,$$where H(t) is the Hölder exponent (Murali Krishna, Gadre & Desai, 2003), that represents the degree of the singularity of the function, t is the instant of time, ā and σ are respectively the mean and standard deviation of incoming traffic, k is the constant related to the probability of violation (for ε = 10⁻⁶) of the envelope process and B is the buffer size.

Delay bound estimation

The service curve concept has been explored as estimation tool involving various technologies and scenarios, mainly in the area of deterministic and statistical network calculus (Santos & Vieira, 2015). The advantage of the network calculus theory is due to the very intuitive convolution formulas that can be used to determine traffic output processes of a communication system from its arrival envelope process and service curve (Le Boudec & Thiran, 2004).

The upper bound on delay, denoted by $\hat{d}$, is given by Santos & Vieira (2015):

(9) $$\hat{d}=inf\left\{d\ge 0|\mathrm{\forall }t\ge 0:A^{\ast }(t-d)\le S(t)\right\},$$where inf is an operator that returns, in this case, the lowest value of d ≥ 0 obeying A* (t − d) ≤ S(t). A* is the MFBAP envelope process, calculated according to Eq. (8) and S is the generalized service curve of an OFDM (orthogonal frequency-division multiple access) system for any user served for the same time interval T, and can be denoted as Costa (2013):

(10) $$S_{n_S}(t)=cTP+cTmin\left\{{\displaystyle \frac{max\left[t-P{N}_{S}T-(n_S-1)T;0\right]}{T};1}\right\},$$where c is the average service rate on the system server and N_S is the number of time slots T per complete cycle P given by $P=\lfloor {\displaystyle \frac{t}{N_{S}T}}\rfloor$. The operator represents the smallest integer closest to ${\displaystyle \frac{t}{N_{S}T}}$. The use of service curve allows to obtain network performance and behavior parameters in an analytical way, as well as estimates of the delay and backlog bounds.

Delay minimization conflict graph algorithm

In this section, we propose a resource allocation algorithm for the uplink of communication system with D2D users that considers the minimization of delay, called DMCG (delay minimization conflict graph) algorithm. It is proposed in this paper to use Eq. (9) to estimate the delay due to its precision as verified in previously works such as Santos & Vieira (2015) and Gonzaga Ferreira & Teles Vieira (2020), which turns it possible to make early decisions on wireless networks resource scheduling.

The DMCG algorithm is started by randomly allocating idle RBs, i.e., one DUE pair is randomly chosen for each idle RB. Then, the algorithm decides how to reuse RBs of all CUEs. A group of unmarked pair m is formed by adding each unallocated DUE pair n to it in order to optimize utility function (9), forming a set denoted by γ_m. A set of DUE pairs that maximize the total utility is contained into the largest group m′ with unallocated DUE pairs, taking the maximum weight independent set of the conflict graph corresponding to group m′. It is allowed that DUE pairs reuse RBs of CUE m′. Candidates are removed one by one until the SINR requirements are met. Then, finishing an iteration, the set of DUE pairs that will reuse RBs of CUE m′, denoted as Δ_m′, ends up being the remaining candidates and group m′ is marked done. Another iteration is effectuated if there are unmarked groups. Thus, the DMCG algorithm works in an iterative way by forming conflict graphs G_m′ and seeking to meet the SINR requirements at each iteration

The set of vertices of the conflict graph G_m′ corresponds to the DUE pairs in group m′. A weight value is assigned to each vertex according to the utility of the corresponding DUE pair when it joins group m′. In the conflict graph, an edge is added for every two vertices if the mutual interference does not meet the threshold. The same RB could not be reused by two DUE pairs. This explains why the edge connects two vertices.

The conflict graph G_m′ formed by DUE pairs representing the vertices helps to determine the candidates. Thus, the objective is to remove the DUE pairs that cannot coexist due to the large mutual interference and to keep the DUE pairs that maximize the throughput of the system. Candidates are chosen as the maximum weight independent set of the conflict graph G_m′. Finally, the set Δ₁,Δ₂,…,Δ_M of DUE pairs n that reuse RBs allocated to CUE m is returned as the algorithm result.

Algorithms 1 and 2 show the pseudo-code for the proposed multi-sharing resource allocation algorithm, an efficient solution to the maximization problem described by Eq. (3) and subject to constraints (4), (5) and (6).

The operation of Algorithm 1 is similar to that of the GTM+ algorithm, but with the difference of using the estimated delay function described by Eq. (9) instead of a utility function based on throughput. The proposed algorithm seeks to find a solution that minimizes the delay bound estimated to each user and simultaneously meeting the constraints imposed by mutual interference.

The proposed multi-sharing allocation method for uplink transmission described in Algorithm 1 can not be directly applied for a downlink transmission scenario. For this, it would be necessary to reformulate the problem described by Eq. (3) and subject to constraints (4), (5) and (6). This issue will be addressed in future works. Notice also that the DMCG algorithm deals with the reuse of idle resource blocks of CUEs by DUEs, that is, its focus is on determining pairs of DUEs in the network. The joint use of the method proposed in this paper with other techniques available in the literature for resource allocation between devices and base station would solve the allocation problem in a generalized allocation scenario and will be investigated later.

Channel modeling for mmWaves

Recent studies developed by 3GPP regarding high frequency bands between 0.5 and 100 GHz defined two channel models for this scenario: clustered delay line (CDL) and tapped delay line (TDL). These channel models are applicable for system-level simulations subject to the following conditions (European Telecommunications Standards Institute (ETSI), 2017):

• For system-level simulations, the supported scenarios are microcell and macrocell urban, indoor office and macrocell rural;

• Bandwidth is supported up to 10% of the center frequency, but not greater than 2 GHz.

The CDL model is a type of channel modeling where the received signal is composed of several separate clusters in delay, as shown in Fig. 2. Each cluster contains a number of multipath components with the same delay value, but with different arrival and departure angles.

CDL models are defined for the frequency range between 0.5 GHz and 100 GHz with a maximum bandwidth of 2 GHz. CDL models can be implemented through the generation of coefficients or the generation of TDL model using spatial filtering.

Three CDL models, called CDL-A, CDL-B and CDL-C, are defined to represent three different channel profiles for non-line of sight (NLOS) environments, while CDL-D and CDL-E models are built for line of sight (LOS) environments (European Telecommunications Standards Institute (ETSI), 2017).

It is considered in this work the multipath model clustered delay line A (CDL-A) (European Telecommunications Standards Institute (ETSI), 2018a, 2018b) , suitable for the mmWaves scenario, and the Rayleigh fading in order to verify the channel modeling impacts via simulations. The carrier frequency of 26 GHz was chosen based on recent studies by Brazilian National Telecommunications Agency (ANATEL) with this frequency range, which should also be adopted in Europe (Tecnoblog, 2019).

System parameters and results

The simulations were conducted using MATLAB software version R2018a and a microcomputer with the following configuration: Intel Core i7-4785T CPU 2.20 GHz, 8 GB RAM, SSHD SATA III and Windows 10 64 bits. The simulation functions and routines were implemented instead of using available network simulation tools in order to have more control on the simulation scenario configuration and CDL channel modeling parameters than some commercial softwares.

It was compared the simulation results of the proposed DMCG algorithm with those of the greedy throughput maximization plus (GTM+) presented in Ciou et al. (2015), a genetic algorithm (GA) based approach, which aims to find a solution for the maximization problem described by Eq. (3) subject to constraints (4), (5) and (6), and a random reuse scheme. The GA-based algorithm was developed using 30 individuals and 100 iterations.

All CUEs and DUE pairs are randomly distributed in a single cell with the BS at the center and are set to have the same parameters such as SINR threshold, transmission power and noise spectral density, varying only the location in the cell. Each figure reports simulation results averaged over 1,000 transmission time intervals (TTIs) in order to reflect the average system performance. The system simulation parameters are given in Table 2. Most parameter values were set according to references (Sun et al., 2013) and (Ciou et al., 2015).

In this work, it is simulated an uplink based on CP-OFDM (cyclic prefix - orthogonal frequency-division multiple access) whose configuration consists of subcarrier spacing of 15 kHz and normal cyclic prefix (CP), as described in European Telecommunications Standards Institute (ETSI) (2018b). Each radio frame occupies 10 ms, divided in 10 subframes of 1 ms, and each subframe is divided in two time slots of 0.5 ms with 7 symbols for each time slot when using normal CP. In the frequency domain the resources are aggregated in 12 subcarriers with 15 kHz bandwidth, totalizing 180 kHz bandwidth, defined as a resource block (RB), the basic unity of resource allocation.

Five traces of real transmission control protocol/Internet protocol (TCP/IP) network traffic were used to represent users’ data traffic during the simulation of the algorithms, which were aggregated in the time domain at 1 ms intervals and assigned to users randomly. These series represent TCP/IP traffic between the University of Waikato (2020) and external networks and were collected between 20/05/2011 and 29/10/2011.

The total throughput values of the system calculated according to Eq. (3) and the sum of all devices throughput are shown in Figs. 3 and 4. It can be seen that the proposed algorithm has the highest values in the two simulated scenarios, both with Rayleigh and CDL-A modeling. The GTM+ and the DMCG algorithms presented higher throughput values compared to the GA-based allocation, showing that the conflict graph strategy is more efficient than solving the problem represented by Eqs. (3)–(6) by directly applying optimization algorithms. The proposed DMCG outperforms the GTM+ because it is considered the conflict graph strategy in conjunction to delay minimization. In this way, in order to reduce system delay, the proposed DMCG algorithm must increase system throughput similar as done by the GTM+ or find a solution that optimizes the allocation process yielding the system delay minimization. The performance of the proposed algorithm improves as the number of CUEs in the network increases, while the performance of the GA-based algorithm tends to become worse.

It can be observed a similar behavior regarding the D2D devices throughput, as shown in Figs. 5 and 6. The proposed algorithm improves considerably in terms of throughput compared to the GTM+ algorithm.

Fairness is an important measure that determines how fair the distribution of resources among system users is, calculated as described in Jain, Chiu & Hawe (1998). It can be seen in Figs. 7 and 8 that the proposed algorithm presents the highest fairness values in all simulated scenarios, showing that it presents a fair distribution of resources among the users of the system. The algorithm based on GA presents the lowest fairness values, a fact expected once the algorithm considers only the system total throughput and the interference constraints, not taking into account a fair distribution of resources.

Figures 9 and 10 present the average delay values calculated for the compared algorithms. The values presented by the proposed DMCG algorithm are the lowest values, even lower than the values presented by the GTM+ algorithm. This result proves that the DMCG algorithm is efficient in its strategy of using the estimated delay as a utility function in the allocation system, resulting in a considerable drop in the delay value in both simulated scenarios. The delay values of the GA-based algorithm and the random algorithm vary widely in the simulated scenarios. In the tests, the presence of these peaks of values was verified due to the characteristic of these algorithms of not distributing resources in a fair way, i.e., the characteristic of penalizing some users in terms of delay in detriment of total throughput of the system.

The performance improvement of the proposed algorithm in relation to the other studied algorithms is mainly due to its characteristic of reducing the average delay of the system through the verification of the utility function defined by the estimated delay parameter. The reduction of the average delay of the system has a direct consequence in the improvement of the system throughput and in the fairness parameter.

Regarding the processing time, shown in Figs. 11 and 12, it can be noticed that the algorithm based on GA has the highest values. GA-based heuristics is the one that demands the most processing among the compared algorithms, due to the large number of variables involved in the system. The processing time tends to increase as the number of CUEs in the network increases. The proposed DMCG algorithm had a considerable performance improvement in processing time when compared to the GTM+ algorithm, although both have computational complexity O(n⁴). The simulation results confirm that the proposed algorithm based on delay minimization outperforms the GTM+ algorithm for all considered performance parameters.