Elsevier

Computer Communications

Volume 179, 1 November 2021, Pages 22-34
Computer Communications

A randomized algorithm for joint power and channel allocation in 5G D2D communication

https://doi.org/10.1016/j.comcom.2021.07.018Get rights and content

Abstract

We formulate the joint power and channel allocation problem (JPCAP) for device to device (D2D) communication as a cost minimization problem, where cost is defined as a linear combination of the number of channels used and total power requirement. We first show that JPCAP is a NP-hard problem and providing n1/ε approximation for JPCAP ε>0 is also NP-hard. Then we propose a mixed integer linear programming (MILP) to solve this problem. As solving MILP is a NP-hard problem we propose a greedy channel and power allocation (GCPA) algorithm to assign channels and powers to the links. We design GCPA in such a fashion that there exists an order of the links for which it produces optimum solution. We show that an order is equivalent to many orders and hence design an incremental algorithm (IA) to efficiently search good orders. Finally using IA we develop a randomized joint channel and power allocation (RJCPA) algorithm. We show that if a certain condition holds we can find the optimum in expected polynomial time else a slowly growing exponential time with very high probability. We then theoretically calculate the expected cost and energy efficiency (EE) produced by RJCPA. Through simulation, we show that RJCPA outperforms two existing approaches with respect to both cost and EE significantly. Finally we validate our theoretical findings through simulation.

Introduction

In 5G device to device (D2D) communication, two users residing within the transmission range of each other can communicate directly among themselves over a common channel without involving the base station (BS) [1], [2], [3], [4], [5], [6], [7], [8], [9], [10]. Also, users not residing within the transmission range of each other can use other devices to relay signal to them [11]. A cellular user communicates with the BS by forming a cellular link (CL) and a pair of D2D users communicates among themselves by forming a D2D link (DL). In D2D underlaid cellular network, DLs reuse the same uplink channel resources of CLs [12], [13], [14]. Each link (CL or DL) includes a transmitter and a receiver. The transmitter of each link has to be allocated sufficient power such that it can communicate with its receiver in the presence of noise and interference from other links operating on the same channel. More specifically, the allocated power of a transmitter must satisfy the required signal to interference plus noise ratio (SINR) at the receiver of that link. Each link requires certain level of SINR depending on its data rate requirement. Moreover, the allocated power at a transmitter must not exceed the residual power available at it.

Note that under a BS only one CL can use a particular channel. However, the channel of a CL may be shared by multiple DLs provided the required SINR is satisfied for each link sharing the channel. Hence we have to find a channel vector C=(c(li)) for the n links l1,l2,,ln, where c(li) represents the channel allocated to link li. Due to the scarcity of channels, we always have to minimize total number of distinct channels Y used in the communication.

It is evident that maximum power that can be allocated to the transmitter of a link is a limited quantity. If link li is activated with power x(li) then x(li)δ(li), where δ(li) is the residual power available at the transmitter of link li. Hence we have to find a power vector X=(x(li)) for the n links l1,l2,,ln such that the total power requirement P=i=1nx(li) is minimized.

The links activated with the same channel will interfere to each other. If more links are activated with the same channel to keep the channel requirement at low, the power requirement of the corresponding links would be high. On the other hand, if each link is activated with a different channel, the power requirement will be minimum but the channel requirement will be maximum. It is thus evident that Y and P have a natural trade-off. Owing to this natural trade-off, we define our minimization objective as a cost function f=Y+αP. Here α is a constant reflecting the relative weights of Y and P, where the weight of Y is normalized to 1. The joint power and channel allocation problem (JPCAP) deals with the problem of finding the channel vector C and the power vector X such that the required SINR criteria for each link is satisfied and the cost f=Y+αP is minimized.

Several authors [15], [16], [17], [18], [19] have studied the problem of sizing the orthogonal channels in D2D communication. A nice survey of various resource allocation schemes in D2D communication can be found in [7]. In [20] authors discussed an analytical model of resource allocation. In [16], [17], [21], the energy efficient mode selection techniques were discussed. In [22], [23], [24], [25], [26] authors adopted graph coloring approach to solve channel assignment problem in D2D communication. In [27], a power minimization solution with joint sub-carrier allocation, adaptive modulation, and mode selection was proposed. In [28], [29] different strategies to minimize power were discussed. In [30] authors adopted a energy saving coding design. Spectral efficiency (SE) [31] and energy efficiency (EE) [32], [33] are two well adopted maximization objective in power and channel allocation in D2D communication. In SE data rate per spectrum is maximized whereas, in EE, data rate per spectrum per energy unit is maximized. In [34] authors formulate the resource allocation problem as a non-convex optimization problem to minimize power. In [35], [36], [37] authors propose channel assignment algorithms where both cellular and D2D users share channels. In this underlaid scenario, cellular and D2D users may interfere with each other. Here, one channel could be used by one cellular user only whereas, a single channel might be used by multiple D2D users. In [18] authors discussed the method to maximize the minimum weighted energy efficiency of D2D links while ensuring maximum data rate in cellular links. The main features of the existing approaches which are similar to our proposed approach are summarized in Table 1. The difference between these approaches and our proposed approach are also summarized in Table 1.

In contrast to the above mentioned existing approaches, we propose a randomized joint channel and power allocation (RJCPA) algorithm to solve the power and channel allocation problem in a combined setup where multiple DLs can be paired with one CL. More specifically our contributions are summarized below.

We formulate JPCAP as a cost minimization problem where the cost f=Y+αP is designed such a manner that by properly tuning α we can set the goal of JPCAP to minimize Y only or P only or a joint objective of Y and P. Then we reduce JPCAP to the classical graph coloring problem and thereby show that it is NP-hard and also providing n1/ε approximation to JPCAP ε>0 is NP-hard. Next we propose a mixed integer linear programming (MILP) formulation for JPCAP and subsequently develop a greedy channel and power allocation (GCPA) algorithm for it. GCPA works by taking an order of the links as input. We show that there exists an order of the links on which if GCPA is applied it will provide an optimal solution. Then we develop a method to search orders efficiently. We show that an order is equivalent to many orders. We develop an incremental algorithm (IA) which searches orders from different equivalent sets and thereby evaluating less number of orders, it essentially explores large number of orders. Finally, using IA, we design a randomized joint channel and power allocation (RJCPA) algorithm to find the near optimum solution. We also theoretically calculate the expected cost produced by RJCPA. Moreover, we identify some special cases where RJCPA can produce optimal result in expected polynomial time. We also compute the expected energy efficiency (EE) produced by RJCPA. We perform extensive simulations to show that RJCPA outperforms both the two-step approach [14] and RSBI algorithm [9] with respect to both cost and EE. Finally we validate our theoretical findings through simulation.

Suppose there are n links within the coverage region of a BS where each link is either a CL or a DL. Let SCL be the set of CLs and SDL be the set of DLs where n=|SCLSDL|. Let channels are represented by positive integers and Sc={1,2,,n} be the set of available channels. Each link li requires a channel c(li)Sc and power x(li)[0,δ(li)] for its activation, where δ(li) is the residual power available at the transmitter of link li. It is evident that x(li) consists of power consumption of transmitter of link li and power loss at circuitry blocks of both transmitter and receiver of link li [9]. When communication is not taking place x(li) is considered to be 0 by neglecting the minute leakage current [44]. We assume that each link undergoes distance dependent pathloss and small-scale fading. If x(lj) power is allocated at the transmitter of link lj then the power received at the receiver of link li can be expressed as x(lj)Gli,lj, where Gli,lj=hslow(li,lj)hfast(li,lj)dliljβ [34] is the gain at the receiver of link li from the transmitter of link lj, hslow(li,lj) is a log-normally distributed random variable representing slow fading, hfast(li,lj) is a exponentially distributed random variable representing fast fading, dlilj is the Euclidean distance between the transmitter of link lj and the receiver of link li and β>1 is a path loss exponent. Note that our solution technique is independent of how Gli,lj is computed. For simulation purpose we have computed Gli,lj as stated in [34].

Note that each link li will receive interference from every other link lj for which c(li)=c(lj) where ij. Let γ(li) be minimum SINR required at the receiver of link li to satisfy its data rate requirement. Link li can be activated with c(li) and x(li) if SINR(li), the SINR received at the receiver of link li, is greater than or equals to γ(li). That is, SINR(li)=x(li)Gli,liσ2+j:ji&c(li)=c(lj)x(lj)Gli,ljγ(li),where σ2 is the constant noise over each link.

We denote y(c)=1 if channel cSc is allocated to at least one link, else 0. Note that each CL requires a different channel to communicate [14] and hence c(li)c(lj) for all li,ljSCL where ij. But a DL may share channel with other CL and/or DL. Clearly total number of distinct channels used is given by Y=c=1ny(c) and total power used in the communication is given by P=i=1nx(li).

Given a constant α, our objective is to find a channel vector C=(c(li)) and a power vector X=(x(li)) such that (1) cost f=Y+αP is minimized, (2) each activated CL gets different channel and (3) SINR(li), the SINR received at the receiver of each activated link li, satisfies Constraint (1).

Rest of the paper is organized as follows. In Section 2 we prove the hardness of JPCAP and propose a mixed integer linear programming formulation of JPCAP. In Section 3 we propose GCPA and present a detailed analysis of the algorithm. In Section 4 we present the randomized joint channel and power allocation (RJCPA) algorithm. In Section 5 we find the expected cost and energy efficiency generated by RJCPA. In Section 6 we simulate RJCPA and compare with two existing approaches of [14] and [9] respectively. Finally in Section 7 we conclude the paper. All notations used in this paper are summarized in Table 2.

Section snippets

Hardness and MILP formulation

In this section we first formally show that JPCAP is NP-hard and then provide a MILP formulation of this problem.

Greedy channel and power allocation algorithm

In this section, we propose a greedy channel and power allocation (GCPA) algorithm to allocate channels and powers to the links. Let S=(l1,l2,,ln) be an arbitrary order of the links. In GCPA, we visit the links one by one following S and allocate channels and powers to them. Thus while allocating link li, all the links l1,l2,,li1 have already been allocated. In other words, c(lj) and x(lj), 1ji1, are already known before the allocation of c(li) and x(li). Let kSc be a channel. We now

Randomized algorithm

In this section we first propose an incremental algorithm (IA) to minimize the cost and then propose a randomized joint channel and power allocation algorithm (RJCPA) which uses IA with parameter ρ to further minimize the cost. Next we present the analysis of RJCPA. Since the optimum hitting probability of RJCPA is a function of ρ2, we then compute the expected value of ρ2 to find the optimum.

Expected cost and energy efficiency

We now analyze the expected cost and energy efficiency (EE) produced by RJCPA. For the analysis purpose we will first state some assumptions and notations. We assume that n DLs are situated inside a R radius cell with the base station placed at its center and κ=|SDL|n. Let g=E[Gli,li], where liSDL, and G=E[Gli,lj], where ij and li,ljSDL. Also Γ=E[γ(li)]. Let E[Y]=Y¯, E[P]=P¯, E[x(li)]=x¯=P¯n, E[f]=f¯ and E[δ(li)]=Δ. Let EE be the energy efficiency and E[EE]=EE¯. We also assume that expected

Simulation

This section comprises of three subsection. In the first subsection we elaborate the simulation environment. In the second subsection we compare RJCPA with two existing algorithms. In the third subsection we validate our analytical findings through simulation.

Conclusion

In this paper, we have formulated the joint power and channel allocation problem (JPCAP) in D2D underlaid cellular network as a cost minimization problem, where cost is defined as a linear combination of Y and P. We first showed that JPCAP is NP-hard and even providing a n1/ε approximation for it is also NP-hard. Then we proposed a MILP formulation of this problem. As solving MILP is also NP-hard we proposed a GCPA algorithm which runs on an order of links. We proved that there exists an order

CRediT authorship contribution statement

Subhankar Ghosal: Conceptualization, Methodology, Software, Data curation, Writing - original draft, Visualization, Investigation. Sasthi C. Ghosh: Supervision, Editing, Validation, Visualization, Investigation.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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    A preliminary version of this paper appeared in Proceedings of the 18th IEEE International Symposium on Network Computing and Applications, IEEE NCA, 2019: 1-5 (Ghosal and Ghosh, 2019).

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