Opportunistic Temporal Fair Mode Selection and User Scheduling for Full-duplex Systems
Shahram Shahsavari, Farhad Shirani, Mohammad A Khojastepour, Elza Erkip
11 Opportunistic Temporal Fair Mode Selectionand User Scheduling for Full-duplex Systems
Shahram Shahsavari † , Farhad Shirani † , Mohammad A. (Amir) Khojastepour (cid:5) ,Elza Erkip †† NYU Tandon School of Engineering, (cid:5)
NEC Laboratories America, Inc.Emails: † {shahram.shahsavari,fsc265,elza}@nyu.edu, (cid:5) [email protected] Abstract
In-band full-duplex (FD) communications — enabled by recent advances in antenna and RF circuitdesign — has emerged as one of the promising techniques to improve data rates in wireless systems.One of the major roadblocks in enabling high data rates in FD systems is the inter-user interference(IUI) due to activating pairs of uplink and downlink users at the same time-frequency resource block.Opportunistic user scheduling has been proposed as a means to manage IUI and fully exploit themultiplexing gains in FD systems. In this paper, scheduling under long-term and short-term temporalfairness for single-cell FD wireless networks is considered. Temporal fair scheduling is of interestin delay-sensitive applications, and leads to predictable latency and power consumption. The feasibleregion of user temporal demand vectors is derived, and a scheduling strategy maximizing the systemutility while satisfying long-term temporal fairness is proposed. Furthermore, a short-term temporal fairscheduling strategy is devised which satisfies user temporal demands over a finite window-length. Itis shown that the strategy achieves optimal average system utility as the window-length is increasedasymptotically. Subsequently, practical construction algorithms for long-term and short-term temporalfair scheduling are introduced. Simulations are provided to verify the derivations and investigate themultiplexing gains. It is observed that using successive interference cancellation at downlink usersimproves FD gains significantly in the presence of strong IUI.
I. I ntroduction
The application of full-duplex (FD) radios enables simultaneous uplink (UL) and downlink(DL) communication over a common frequency band which can potentially lead to significant
This work is supported by NYU WIRELESS Industrial A ffi liates and National Science Foundation grants 1547332 and1527750. a r X i v : . [ c s . I T ] M a y multiplexing gains [1]. Self-interference at the FD base station (BS), as well as inter-userinterference (IUI) between the UL and DL users are among major roadblocks in achieving thesemultiplexing gains. Recent advances in antenna and radio frequency circuit design have led tosignificant progress in e ffi cient self-interference mitigation (SIM) [2]. Furthermore, opportunisticscheduling along with successive interference cancellation (SIC) decoding methods at the DLreceiver can be used to further reduce IUI [1].In FD systems, the DL user either treats the UL interference (i.e. IUI) as noise or implementsSIC methods to mitigate the IUI [3]. Consequently, as shown in Figure 1, the FD system inwhich the BS is FD and users are half-duplex (HD) may operate in four modes of operation ateach resource block: a) HD-UL mode, where only UL transmission occurs, b) HD-DL, whereonly a DL user is activated, c) FD-IN, where IUI is treated as noise, and d) FD-SIC, where SICis used to decode and cancel IUI in DL, An opportunistic scheduler selects the active users ateach resource block as well as their mode of operation so as to maximize the resulting systemutility subject to fairness criteria. In the FD-SIC and FD-IN modes, the scheduler activates pairsof UL and DL users such that the impact of IUI is minimized.There has been a large body of research on quantifying fairness in user scheduling. Variouscriteria on the users’ quality of service (QoS) have been proposed to model fairness of thescheduling strategies. For HD systems, scheduling under utilitarian [4], [5], proportional [6],[7], and temporal [8], [9] fairness criteria have been studied. Temporal fairness is of interest indelay sensitive applications, where a system with predictable latency may be more desirable thana system with highly variable latency, but potentially higher throughput [10]. Short-term temporalfair schedulers guarantee that each user is activated in at least a predefined fraction of time-slots ateach finite scheduling window, whereas under long-term temporal fairness, the temporal demandsare met over infinitely large window-lengths. Temporal fair scheduling have been investigatedin HD wireless local area networks (WLAN) [11], [12] and HD cellular systems [13], [14].Furthermore, scheduling under long-term temporal fairness in FD systems is considered in [15],where a heuristic scheduler is provided by modifying an optimal temporal fair HD scheduler.However, optimal long-term and short-term temporal fair scheduling in FD systems, which is thetopic of this paper, have not been investigated before. In our recent works, we have investigatedoptimal user scheduling under long-term [16] and short-term [17] fairness in non-orthogonalmultiple access (NOMA) systems. In this paper, we extend our results to FD systems. Severalof the results in [16] and [17] depend on the underlying NOMA system constraints which are 𝑮 𝒊 (a) HD-UL, 𝐯 $% 𝑮 𝒋 (b) HD-DL, 𝐯 %’ 𝑮 𝒊 (c) FD-NI, 𝐯 $’ 𝑮 𝒋 𝚿 𝒃 𝑯 𝒊𝒋 𝑮 𝒊 (d) FD-SIC, 𝐯 $’ 𝑮 𝒋 𝚿 𝒃 cancelled Fig. 1: Figures (a) and (b) show the HD UL and HD DL modes, (c) is the FD mode when IUIis treated as noise, (d) shows the FD mode with SIC in DL, and (e) is the FD mode when thesame user is activated in UL and DL.essentially di ff erent in the FD system model considered in the paper as articulated in SectionsIII and IV. The main contributions of this paper are as follows: • We prove that a class of scheduling strategies called threshold based strategies (TBS) achieveoptimal system utility under long-term temporal fairness in FD systems. • We provide an iterative algorithm for construction of optimal TBSs in practice. • We devise a scheduling strategy under short-term temporal fairness constraints whose aver-age utility is shown to converge to optimal utility as the scheduling window-length growsasymptotically large. • We present simulation results which verify the derivations under practical FD system modelsand parameters, and demonstrate the e ff ectiveness of SIC in recovering the FD gains in thepresence of strong IUI. Notation:
We represent random variables by capital letters such as X , U . Sets are denotedby calligraphic letters such as X , U . The set of numbers { , , · · · , n } , n ∈ N is represented by[ n ]. The vector ( x , x , · · · , x n ) is represented by x n . Particularly, vectors of length two such as( x , x ) are represented by x . The random variable A is the indicator function of the event A .Sets of vectors are shown using sans-serif letters such as V .II. S ystem M odel We consider a single-cell scenario consisting of n HD users and one FD BS. The user setis denoted by U = { u , u , · · · , u n } . While this paper considers HD users, the results can begeneralized when an arbitrary fraction of the users have FD capability. Subsets of distinct userswhich can be activated simultaneously in UL and DL are called virtual users . The set of virtual users is denoted by V ⊆ { v i , j | i , j ∈ [ n ] ∪ { }} where v i , j signifies the instance when the i th user isactivated in UL and the j th user is activated in downlink. If i ( j ) is equal to zero, then the BSoperates in the HD UL (DL) mode. We note that since the users are HD, v i , i (cid:60) V , ∀ i . We denotethe propagation channel coe ffi cient between user u i and the BS which captures small-scale andlarge-scale fading e ff ects at time-slot t by G i , t . Similarly, we denote the channel between users u i and u j at time-slot t by H i , j , t , i (cid:44) j . It is assumed that the channel coe ffi cients G i , t , i ∈ [ n ]and H i , j , t , i , j ∈ [ n ] are independent over time. Additionally, we assume that the channels arereciprocal. A. Communication Modes
As mentioned in Section I, the system may operate in several communication modes.
1) HD-UL and HD-DL modes:
In these modes a single user is scheduled either in UL orDL. Virtual user v i , ( v , j ) represents the case when user i ( j ) is activated in UL (DL) in HDmode; see Figure 1(a) (Figure 1(b)). If v i , is activated at time t , the SINR for user u i isSINR HD,UL i , , t = P UL i , t | G i , t | N UL , (1)where P UL i , t is the UL transmit power and N UL is the UL noise power. Similarly, if v , j is activatedat time t , the SINR for user u j is as follows,SINR HD,DL0 , j , t = P DL j , t | G j , t | N DL , (2)where P DL j , t is the BS transmit power and N DL is the DL noise power.
2) FD-IN mode:
In this mode, two di ff erent users are activated, one in each direction, andthe DL user treats IUI as noise. If virtual user v i , j , i (cid:44) j is activated in FD-IN mode as depictedin Figure 1(c), the SINRs of users u i and u j areSINR FD-IN,UL i , j , t = P UL i , t | G i , t | P DL j , t Ψ b + N UL , (3)SINR FD-IN,DL i , j , t = P DL j , t | G j , t | P UL i , t H i , j , t + N DL , (4)respectively, where Ψ b is the e ff ective channel between transmit and receive terminals at the BSwhich is inversely proportional to the level of SIM at the BS.
3) FD-SIC mode:
We assume that an arbitrary fraction of the users are capable of SIC enablingFD-SIC mode. In this mode, two di ff erent users are activated, one in each direction, and the DLuser performs SIC to cancel IUI. If virtual user v i , j , i (cid:44) j is activated in FD-SIC mode (Figure1(d)), the SINR of users u i and u j areSINR FD-SIC,UL i , j , t = min P UL i , t | G i , t | P DL j , t Ψ b + N UL , P UL i , t | H i , j , t | P DL j , t | G j , t | + N DL , (5)SINR FD-SIC,DL i , j , t = P DL j , t | G j , t | N DL , (6)respectively. FD-SIC imposes two restrictions on the UL transmission rate. First, SIC requiresthe DL user to be able to decode the UL signal. Second, the BS needs to be able to decodethe UL signal. Consequently, the UL rate must be chosen such that the UL signal is decodableat both DL user and the BS. As a result, the SINR of the UL user is defined as the minimumbetween the SINRs over UL and inter-user channels as in Equation (5). As implied in Equation(6), the DL transmission is interference-free since IUI is cancelled. B. System Utility
At any given time-slot, the channel realization, choice of the active virtual user, and thecommunication mode determine the resulting system utility at that time-slot. In this paper, wetake the resulting sum-rate as a measure of the system utility. The following describes theresulting utility from activating each of the virtual users.The performance value corresponding to the HD virtual users v i , and v , j at time-slot t aredefined as R i , , t = C (cid:16) SINR HD , ULi , , t (cid:17) , (7) R , j , t = C (cid:16) SINR HD , DL , j , t (cid:17) , (8)respectively, where C ( x ) = max { log (1 + x ) , γ max } is the truncated Shannon rate and γ max modelsthe maximum feasible spectral e ffi ciency in the system. Virtual user v i , j , i (cid:44) j , can be activatedin either of FD-IN and FD-SIC modes if user u j is capable of SIC. The BS activates the virtualuser in the mode which leads to the highest utility. Hence, the utility due to activating v i , j , i (cid:44) j is given as R i , j , t = max X ∈X (cid:20) C (cid:16) SINR X , ULi , j , t (cid:17) + C (cid:16) SINR X , DLi , j , t (cid:17) (cid:21) , (9) where, X = { FD-IN,FD-SIC } if u j is capable of SIC, and X = { FD-IN } otherwise.In a given time-slot, the system utilities due to activating di ff erent virtual users may dependon each other since each user is included in multiple virtual users. However, the performancevalues in di ff erent time-slots are independent of each other due to the independence of channelcoe ffi cients over time. The matrix of system utilities due to activating each of the virtual usersis called the performance matrix . The performance matrix is random and its value depends onthe realization of the underlying time-varying channel. Definition 1 ( Performance Matrix).
The matrix of jointly continuous variables ( R i , j , t ) i , j ∈ [ n ] ∪{ } , t ∈ N is the performance matrix of the virtual users at time t. The sequence ( R i , j , t ) i , j ∈ [ n ] ∪{ } , t ∈ N isa sequence of independent matrices distributed identically according to the joint density f R n × n . Remark 1.
In practical scenarios, the performance matrix is a matrix of discrete variables dueto discrete modulation and coding schemes. The results of this paper can be extended to thecase of discrete performance matrices as in [16, Sec. VI].C. Opportunistic Temporal Fair Scheduling
Under temporal fairness, it is required that the fraction of time-slots in which each user isactivated in UL or in DL is bounded from below (above). The vector of UL lower (upper)bounds w n UL ( w n UL ) is called the UL lower (upper) temporal demand vector. Similarly, the vector w n DL ( w n DL ) is called the DL lower (upper) temporal demand vector. The objective is to design ascheduling strategy satisfying the temporal fairness constraints in a given window-length whilemaximizing the resulting system utility. Accordingly, a scheduling strategy is defined as follows. Definition 2 ( s -scheduler). Consider the scheduling setup parametrized by ( n , V , w nUL , w nUL , w nDL , w nDL , f R n × n ) .A scheduling strategy Q = ( Q t ) t ∈ [ s ] with window-length s ∈ N (s-scheduler) is a family of (possiblystochastic) functions Q t : R n × n × t → V , t ∈ [ s ] , where: • The input to Q t , t ∈ [ s ] is the sequence of performance matrices R n × n × t which consists of tindependently and identically distributed matrices with distribution f R n × n . • The temporal demand constraints are satisfied:P (cid:16) w UL , i ≤ A QUL , i , s ≤ w UL , i , i ∈ [ n ] (cid:17) = , (10) P (cid:16) w DL , i ≤ A QDL , i , s ≤ w DL , i , i ∈ [ n ] (cid:17) = , (11) where, the UL and DL temporal shares of user u i , i ∈ [ n ] up to time t ∈ [ s ] is defined asA QUL , i , t = t t (cid:88) k = n (cid:88) j = (cid:8) v i , j = Q k ( R n × n × k ) (cid:9) , ∀ i ∈ [ n ] , t ∈ [ s ] , (12) A QDL , j , t = t t (cid:88) k = n (cid:88) i = (cid:8) v i , j = Q k ( R n × n × k ) (cid:9) , ∀ j ∈ [ n ] , t ∈ [ s ] . (13)In the context of Definition 2, an ∞ - scheduler is a scheduler which satisfies long-term temporalfairness constraints. A scheduling setup where the user temporal shares are required to takea specific value, i.e. A Qi , s = w i , i ∈ [ n ], is called a setup with equality temporal constraints and is parametrized by ( n , V , w n UL , w n UL , w n DL , w n DL , f R n × n ). The following defines the set of feasiblewindow-lengths and temporal demand vectors given a virtual user set. Definition 3. (Feasible Temporal Demands)
For a virtual user set V , the window-length s, andtemporal demand vector ( w nUL , w nDL ) are called feasible if a scheduling strategy satisfying theequality temporal demand constraints exists. The set of all feasible window-lengths and demandvectors ( s , w nUL , w nDL ) is denoted by S ( V ) . Particularly, the set of feasible demand vectors forasymptotically large window lengths is defined as follows: S ∞ ( V ) = lim t →∞ (cid:110)(cid:98) w n |∃ s : t ≤ s & ( s , (cid:98) w n ) ∈ S ( V ) (cid:111) , where (cid:98) w n = ( w nUL , w nDL ) . Furthermore, under inequality temporal constraints, the schedulingsetup with upper and lower temporal demand vectors ( w nUL , w nUL , w nDL , w nDL ) and window-length sis said to be feasible if: ∃ w nUL , w nUL : ( s , w nUL , w nUL ) ∈ S ( V ) , w nUL ≤ w nUL ≤ w nUL , w nDL ≤ w nDL ≤ w nDL . The average system utility of an s -scheduler is: Definition 4 ( System Utility).
For an s-scheduler Q: • The average system utility up to time t, is defined asU Qt = t t (cid:88) k = (cid:88) i ∈ [ n ] (cid:88) j ∈ [ n ] R i , j , k (cid:8) v i , j = Q k ( R n × n × k ) (cid:9) . (14) • The variable U Qs is called the average system utility for the s-scheduler. An s-scheduler Q ∗ s is optimal if and only if Q ∗ s ∈ argmax Q ∈Q s U Qs , where Q s is the set of all s-schedulers for thescheduling setup. The optimal utility is denoted by U ∗ s . The objective is to study properties of U ∗ s and design scheduling strategies achieving themaximum average system utility under temporal fairness constraints.III. F easibility of T emporal D emands In this section, we study the set of feasible temporal demand vectors under long-term and short-term fairness constraints. We first consider the feasible demand region under long-term fairnessconstraints (i.e. S ∞ ( V )). Then, we investigate feasibility under short-term fairness constraints. A. Feasibility under Long-term Fairness
The following theorem characterizes the feasible temporal demand region under long-termtemporal fairness, i.e. S ∞ ( V ). Theorem 1.
For the scheduling setup with virtual user set V = (cid:8) v i , j | i (cid:44) j , i , j ∈ { } ∪ [ n ] (cid:9) , thefollowing holds: ( w nUL , w nUL ) ∈ S ∞ ( V ) ⇐⇒ (cid:80) i ∈ [ n ] w UL , i ≤ , (cid:80) i ∈ [ n ] w DL , i ≤ , (cid:80) i ∈ [ n ] w UL , i + w DL , i ≥ , w UL , i + w DL , i ≤ . The proof is provided in Appendix A.
B. Feasibility under Short-term Fairness
In the next step, we consider feasibility under short-term temporal fairness constraints, wherethe fairness constraints must be satisfied in window-length s . The following theorem providesthe feasible region. Theorem 2.
For the scheduling setup with virtual user set V = (cid:8) v i , j | i (cid:44) j , i , j ∈ { } ∪ [ n ] (cid:9) andfairness window-length s, temporal demand vectors ( w nUL , w nUL ) are feasible, i.e. ( s , w nUL , w nUL ) ∈ S ( V ) , if and only if: ∃ a i , j , a , j , a , i ∈ [ s ] : sw UL , i = (cid:80) j ∈ [ n ] a i , j + a i , , i ∈ [ n ] , sw DL , j = (cid:80) i ∈ [ n ] a i , j + a , j , j ∈ [ n ] , (cid:80) i , j ∈ [ n ] a i , j + (cid:80) k ∈ [ n ] ( a k , + a , k ) = sa i , i = , i ∈ [ n ] a i , j ≥ , i , j ∈ [ n ] . (15)Note that in Theorem 2, we must have w UL , i = k UL , i s , i ∈ [ n ], w DL , i = k DL , i s , i ∈ [ n ], where k UL , i , k DL , i ∈ [ s ] , i ∈ [ n ]. The variable a i , j can be viewed as the temporal share of the user v i , j .The forward proof follows by noting that the first two bounds ensure that the temporal fairnessconstraints are satisfied, the third bound must holds since a virtual user must be activated ateach time-slot. The fourth set of constraints follows from the fact that the users do not havethe FD capability. The converse proof follows by constructing a round robin scheduler whichactivates v i , j for a i , j of the time. The complete proof is provided in Appendix B. We note thatthe proof depends on the structure of virtual users. Hence, the arguments provided in [17] forNOMA systems are not directly applicable.IV. T hreshold B ased S trategies In this section, we show that a class of schedulers called threshold based schedulers achieveoptimal utility under long-term temporal fairness constraints. Furthermore, we introduce a sched-uler which satisfies short-term fairness constraints whose performance converges to optimalperformance as the scheduling window-length is increased asymptotically.
A. Scheduling under Long-term Fairness Constraints
The following defines threshold based schedulers.
Definition 5 ( TBS).
For the scheduling setup ( n , V , w nUL , w nUL , w nDL , w nDL , f R n × n ) a threshold basedstrategy (TBS) is characterized by the pair ( λ nUL , λ nDL ) ∈ R n . The strategy Q T BS ( λ nUL , λ nDL ) = ( Q T BS , t ) t ∈ N is defined as: Q T BS , t (cid:0) R n × n × t (cid:1) = argmax v i , j ∈ V M (cid:0) v i , j R i , j , t (cid:1) , t ∈ N , (16) where M (cid:0) v i , j R i , j , t (cid:1) = R i , j , t + λ UL , i + λ DL , j is the ‘scheduling measure’ corresponding to the virtualuser v i , j , and λ UL , = λ DL , = . The resulting temporal shares are represented as w UL , i = A Q TBS UL , i , i ∈ [ n ] and w DL , i = A Q TBS DL , i , i ∈ [ n ] . The utility of the TBS is written as U w n ( λ nUL , λ nDL ) . Theset of threshold based strategies is denoted by Q T BS . The following states that the optimal utility under long-term temporal fairness is achievedusing TBSs.
Theorem 3.
For the scheduling setup ( n , V , w nUL , w nUL , w nDL , w nDL , f R n × n ) , assume that ( w nUL , w nUL , w nDL , w nDL ) is feasible. Then, there exists an optimal threshold based strategy Q T BS . The proof is provided in Appendix C. Note that the arguments provided in [16] to prove asimilar statement for NOMA systems are not directly applicable since the proof depends on theset of virtual users which is di ff erent in FD systems. The following Corollary provides su ffi cientoptimality conditions which will be used for devising a low complexity algorithm to estimatethe optimal thresholds in the next sections. Corollary 1.
For the scheduling setup ( n , V , w nUL , w nUL , w nDL , w nDL , f R n × n ) , assume that there existpositive thresholds ( λ nUL , λ DL n ) satisfying the complimentary slackness conditions:i ∈ [ n ] : λ UL , i (cid:16) A Q TBS UL , i − w UL , i (cid:17) = , w UL , i ≤ A Q TBS UL , i ≤ w UL , i , j ∈ [ n ] : λ DL , j (cid:16) A Q TBS DL , j − w DL , j (cid:17) = , w DL , j ≤ A Q TBS DL , j ≤ w DL , j , where Q T BS is the TBS corresponding to the threshold vector ( λ nUL , λ nDL ) . Then, Q T BS is an optimalscheduling strategy.
Note that the complementary slackness conditions in Corollary 1 are written only in terms ofthe lower temporal demands. Similar su ffi cient conditions can be derived in terms of the uppertemporal share demands. B. Practical Construction Algorithms
The optimal thresholds in TBS depend on the statistics of the performance matrix R n × n , whichis typically unavailable in practice. In this section, we propose an online algorithm to find theoptimal thresholds in an online fashion only by observing the realization of the performancematrix at each time-slot which can be obtained after channel estimation. Algorithm 1 constructsan optimal TBS using the complementary slackness conditions provided in Corollary 1. Thealgorithm starts with a vector of initial thresholds (e.g. all-zero thresholds). At time-slot t , itchooses virtual user v i ∗ , j ∗ , t to be activated based on the threshold vector ( λ n UL , t , λ n DL , t ). It updates the temporal shares and thresholds based on the scheduling decision at the end of the time-slot(line 2-7). The update rule for the thresholds given in lines 6 and 7 are based on a variationof the Robbins-Monro update described in [16]. The parameter c is the step-size. Lines (8-23)verify that the temporal demand constraints and dual feasibility conditions are satisfied. Thecomputational complexity of the algorithm is proportional to the number of virtual users whichis O ( n ). Algorithm 1
Heuristic Threshold Optimization in TBS
Initialization : λ UL , i , = λ DL , j , = i , j ∈ [ n ] for t ∈ N do v i ∗ , j ∗ , t = Q t ( λ n UL , t , λ n DL , t ) A Q UL , i , t + = A Q UL , i , t + + t + (cid:16) { i = i ∗ } − A Q UL , i , t (cid:17) , i ∈ [ n ] A Q DL , j , t + = A Q DL , j , t + + t + (cid:16) { j = j ∗ } − A Q DL , j , t (cid:17) , j ∈ [ n ] λ min = min i ∈ [ n ] { λ UL , i , t , λ DL , i , t } λ UL , i , t + = λ UL , i , t − c (cid:16) λ UL , i , t − λ min (cid:17)(cid:16) { i = i ∗ } − w UL , i (cid:17) , i ∈ [ n ] λ DL , j , t + = λ DL , j , t − c (cid:16) λ DL , j , t − λ min (cid:17)(cid:16) { j = j ∗ } − w DL , j (cid:17) , j ∈ [ n ] for i = n do if λ UL , i , t = λ min and A Q UL , i , t + < w UL , i then λ UL , i , t + = λ UL , i , t + c (cid:16) w UL , i − A Q UL , i , t + (cid:17) end if if λ UL , i , t = λ min and λ min < then λ UL , i , t + = λ UL , i , t + + c end if end for for j = n do if λ DL , j , t = λ min and A Q DL , j , t + < w DL , j then λ DL , j , t + = λ DL , j , t + c (cid:16) w DL , j − A Q DL , j , t + (cid:17) end if if λ DL , j , t = λ min and λ min < then λ DL , j , t + = λ DL , j , t + + c end if end for end for C. Scheduling under Short-term Fairness Constraints
In this section, we provide a class of scheduling strategies called augmented threshold basedstrategies (ATBS) for FD systems which satisfy hard short-term temporal fairness constraints.More precisely, the strategy satisfies the temporal fairness constraints in a given window-length s with probability one. It is shown that the average utility due to the proposed scheduler convergesto the optimal utility as the length of the scheduling window is taken to be asymptotically large.The scheduling strategy has two phases of operation, i) TBS phase, and ii) compensation phase.In the TBS phase, the strategy operates similar to the optimal TBS designed for long-termfairness constraints. In the compensation phase, the strategy activates virtual users in a way soas to ensure that the temporal fairness criteria are satisfied regardless of the resulting utility. TheATBSs are formally defined below. Definition 6 ( ATBS).
For the scheduling setup ( n , V , w nUL , w nUL , w nDL , w nDL , f R n × n ) with window-length s ∈ S , an ATBS is characterized by the pair of vectors ( λ nUL , λ nDL ) ∈ R n . The strategyQ AT BS ( s , λ nUL , λ nDL ) = ( Q AT BS , t ) t ∈ N is defined as:Q AT BS , t (cid:0) R n × n × t (cid:1) = argmax v i , j ∈ V t M (cid:0) v i , j R i , j , t (cid:1) , t ∈ N , (17) where M (cid:0) v i , j R i , j , t (cid:1) is the scheduling measure in Definition 5, and V t is the ‘feasible virtual userset’ at time t and consists of all virtual users v i , j satisfying the following conditions:s − t ≥ (cid:88) i ∈ [ n ] (cid:100) sw UL , i (cid:101) − ( t − A QUL , i , t − − (cid:88) j ∈ [ n ] ∪{ } A i , j + , (18) s − t ≥ (cid:88) i ∈ [ n ] (cid:100) sw DL , i (cid:101) − ( t − A QDL , i , t − − (cid:88) j ∈ [ n ] ∪{ } A i , j + , (19) s − t ≤ (cid:88) i ∈ [ n ] (cid:98) sw UL , i (cid:99) − ( t − A QUL , i , t − − (cid:88) j ∈ [ n ] ∪{ } A i , j + (cid:88) i ∈ [ n ] (cid:98) sw DL , i (cid:99) − ( t − A QDL , i , t − − (cid:88) j ∈ [ n ] ∪{ } A i , j , (20) s − t ≥ max i ∈ [ n ] (cid:32)(cid:16) (cid:100) sw UL , i (cid:101) − ( t − A QUL , i , t − − (cid:88) j ∈ [ n ] ∪{ } A i , j (cid:17) + (cid:16) (cid:100) sw DL , i (cid:101) − ( t − A QDL , i , t − − (cid:88) j ∈ [ n ] ∪{ } A i , j (cid:17)(cid:33) , (21) where x + = x × x ≥ , A i , j = { v i , j = Q T BS , t ( R n × n × t ) } , Q T BS is the TBS with threshold vector ( λ nUL , λ nDL ) . Algorithm 2
Augmented Threshold Based Strategy V = V for t = s with step-size 1 do V t = φ for i , j : v i , j ∈ V t − do if Equations (18), (19), (20) and (21) are satisfied then V t = V t ∪ {V j } end if end for Q AT BS , t (cid:0) R n × n × t (cid:1) = argmax v i , j ∈ V t M (cid:0) v i , j , R t , i , j (cid:1) end for For a given pair of threshold vectors ( λ n UL , λ n DL ) the steps in the corresponding ATBS strategyis described in Algorithm 2. In this algorithm, at each time-slot Equations (18) and (19) ensurethat the uplink and downlink lower temporal demands can be satisfied if the TBS with thresholdvector ( λ n UL , λ n DL ) is used in the next time-slot. Equation (20) ensures the satisfaction of the uppertemporal demands. Equation (21) ensures that the users are not required to be scheduled in ULand DL simultaneously in the remaining time-slots in order to satisfy the temporal demands. Thefollowing Theorem shows that for asymptotically large scheduling window-lengths, the utilityof the ATBSs converge to the optimal utility which is achieved by TBSs. Theorem 4.
For the scheduling setup ( n , V , w nUL , w nUL , w nDL , w nDL , f R n × n ) let ( λ ∗ nUL , λ ∗ DLn ) be thethreshold vectors of the TBS which achieves optimal average utility under long-term fairnessconstraints (i.e. s → ∞ ) and ( w ∗ ULn , w ∗ DLn ) the corresponding temporal share vectors. Then, lim s →∞ U ∗ s = U ∗ . The proof follows by similar arguments as in the proof of Theorem 4 in [17] and is omitteddue to space limitations. V. S imulation R esults We consider a single 50 m ×
50 m square cell with a FD BS in the center and four HD usersdistributed around the BS with an exclusion of central disk with radius r min = ffi ce building. We adopt channel model ofindoor RRH / Hotzone scenario from [18]. We assume that there are no upper temporal demand constraints. To investigate the impact of user distribution, we consider two models: i ) uniformmodel where the users are distributed uniformly inside the cell and ii ) hotspot model where thereare n h randomly located hotspots within the cell and n / n h users are distributed uniformly withina circle of radius 10 m around each hotspot [15]. Table I lists the simulation parameters. Theuser SINRs are modeled as described in Section II and the network utility is assumed to be thesum-rate. At each time-slot prior to the scheduling, a max-min power optimization is performedfor each virtual user including two users and for both FD-IN and FD-SIC modes [19]. For a givenvirtual user and a mode of operation, we find UL and DL transmit powers which maximizesthe minimum individual user rates in that virtual user. Assuming a non-line of sight (NLOS)channel, maximum UL and DL transmit powers are chosen such that the average SNR of 0 dBis achievable when a single user is active on the boundary of the cell, i.e. at d = √ Parameter Value
Bandwidth 10 MHzNoise spectral density −
174 dBm / HzNoise figure BS: 8 dB, user: 9 dBNumber of hotspots 1 , d in km) LOS: 89 . + . ( d )NLOS: 147 . + . ( d )Small-scale fading model Rayleigh block fadingMaximum spectral e ffi ciency 6 bps / Hz A. Long-term Fairness
In this section, we consider long-term temporal fairness where s → ∞ and apply Algorithm 1to find the optimal thresholds in TBS. The step-size c is taken to be 0 . w UL , i = / , w DL , i = / , i ∈ [4]. One can easily check the feasibility of the fairnessconstraints using Theorem 1. Figure 2 illustrates the temporal shares of the users in UL andDL directions after 4 × time-slots. It can be seen that the temporal demand constraints aresatisfied. Fig. 2: Long-term temporal share of the users versus their lower temporal demands.Next, we study the impact of availability of FD BS on the system throughput when the userdistribution is uniform. We assume that the users are not able to perform SIC, hence the availablemodes are HD-UL, HD-DL, and FD-IN. We use a HD system (HD BS and users) as a base-lineto evaluate the FD gains in the system throughput. Furthermore, as a benchmark to FD-IN, weuse the heuristic temporal fair scheduler proposed in [15]. We do not provide the details due tothe lack of space and refer the reader to [15]. According to [15], this heuristic scheduler requires (cid:80) i w UL , i + (cid:80) j w DL , j ≤
1, since it uses an underlying HD scheduler. Consequently, there is a setof feasible temporal demands in S ∞ ( V ) characterized in Section III-A, which are not achievableby this heuristic method whereas according to Theorem 3, the optimal TBS provided in SectionIV can achieve any choice of temporal demands belonging to S ∞ ( V ). Furthermore, the heuristicalgorithm cannot guarantee upper temporal demands unlike our proposed TBS.Figure 3 illustrates the average percentage gain in the system throughput when comparingvarious schedulers with the base-line HD scheduler for di ff erent levels of SIM at the BS. Weassume that w UL , i = w DL , i = / , i ∈ [4]. Note that these temporal demands are feasible forthe base-line HD as well as the heuristic scheduler. We observe that both the optimal and theheuristic scheduler lead to significant improvements for large enough values of SIM at the BS.Additionally, this improvement increases with SIM level at the BS. The reason is that higherSIM leads to less interference for UL reception which in turn improves UL performance valuein FD-IN mode.Next, we investigate the impact of user distribution on the multiplexing gains provided by FDoperations. We assume that SIM is 80 dB at the BS. Moreover, we consider two scenarios. In A v e r a g e F D T h r oughpu t G a i n ov e r HD ( % ) Heuristic Scheduler Optimal Scheduler
Fig. 3: The average gain in the system throughput when comparing various schedulers with thebase-line HD scheduler for di ff erent levels of SIM at the BS.the first scenario (Scenario 1), we assume that the users cannot perform SIC. Hence the availablemodes are HD-UL, HD-DL, and FD-IN. In the second scenario (Scenario 2), we assume thatall modes namely HD-UL, HD-DL, FD-IN, and FD-SIC are available. We consider the optimalTBS in both scenarios, where Algorithm 1 is used to find the corresponding thresholds. Figure4 illustrates the average system throughput gain when comparing optimal TBS and the base-lineHD scheduler for di ff erent user distributions. We observe that when users are located around asingle hotspot, the throughput gain achieved in Scenario 1 is limited since IUI is strong and thescheduler tends to use HD-UL and HD-DL modes more frequently. In contrary, the throughputgain is close to 100% in Scenario 2 since the DL user can cancel IUI in the FD-SIC modewhich incentivizes the scheduler to select the pairs in this mode more frequently. When thereare two hotspots, the throughput gain is higher in Scenario 1 as UL and DL users can be chosenfrom di ff erent hotspots. However, Scenario 2 still leads to higher improvements. An interestingobservation from Figure 4 is that SIC is beneficial even if the users are distributed uniformly.The reason is that when SIM level is not very large (e.g. less than 80 dB), performing SIC canimprove DL rate at no cost for the UL rate since the UL channel will be the bottleneck for theUL transmission rate rather than the inter-user channel. In other words, with a high probabilitythe first term in (5) will be the minimum of the two terms. B. Short-term Fairness
Next, we consider a limited fairness window-length s and use Algorithm 2 to ensure theshort-term fairness. We assume that w UL , i = w DL , i = / , i ∈ [4]. Furthermore, we consider s ∈ { , , , } . It is straightforward to show that these window-lengths are feasible. Figure One hotspot Two hotspots Uniform A v e r a g e F D T h r oughpu t G a i n ov e r HD ( % ) Scenario 1: HD, FD-INScenario 2: HD, FD-IN, FD-SIC
Fig. 4: The average system throughput gain when comparing optimal TBS and the base-line HDscheduler for di ff erent user distribution models. S y s t e m U tilit y ( bp s / H z ) Fairness window-length s (time-slot) Short-term (ATBS) Optimal Long-term
Fig. 5: System utility as a function of window-length s for the proposed ATBS using the samethresholds as the optimal TBS.5 depicts the average system utility (throughput) as a function of fairness window-length s forATBS, described in Algorithm 2, using the same thresholds as the optimal TBS. As a benchmark,Figure 5 also illustrates the optimal long-term system utility U ∗∞ corresponding to the optimallong-term fair scheduling (i.e. optimal TBS). Note that optimal long-term utility is not a functionof window-length s . It is not di ffi cult to show that U ∗∞ provides an upper-bound for the utility ofATBS. We observe that the utility of ATBS approaches the optimal long-term utility as window-length increases, confirming Theorem 4. Furthermore, we can see that the gap between the utilityof ATBS and optimal long-term utility is small even for relatively small window-lengths suchas s =
80. VI. C onclusion
In this paper, we have studied opportunistic mode selection and user scheduling in single-cellFD systems under short-term and long-term temporal fairness constraints. We have proved that a class of scheduling strategies called threshold based strategies achieve optimal system utilityunder long-term temporal fairness. Furthermore, we have provided a low-complexity online algo-rithm for construction of the optimal schedulers under long-term temporal fairness. Additionally,we have provided a scheduling strategy under short-term temporal fairness constraints whoseaverage utility is shown to converge to optimal utility as the scheduling window-length growsasymptotically large. Simulation results have demonstrated the e ff ectiveness of the proposedscheduling algorithms. A natural extension to this work is multi-cell scheduling in FD systems.The methods proposed here may be extended and applied to centralized and distributed FDsystems. Particularly, scheduling for multi-cell FD systems with limited base station cooperationis an interesting avenue for future work. Another avenue is to consider FD capability for theusers which can potentially improve the performance in the scenarios where the user distributionis concentrated around a few hotspots and SIC is not available.A ppendix AP roof of T heorem w n UL , w n UL ) must satisfy the bounds provided in the theorem statement.). The first bound (cid:80) i ∈ [ n ] w UL , i ≤ (cid:80) i ∈ [ n ] w DL , i ≤ (cid:80) i ∈ [ n ] w UL , i + (cid:80) i ∈ [ n ] w DL , i ≥ w UL , i + w DL , i ≤ w n UL , w n UL ) satisfying the bounds provided in the theoremstatement is feasible.). As a first step, we prove the theorem for k =
0, when all users are HD.We provide a detailed description for the two-user case. For more than two users, we provide anoutline of the proof. Assume that n =
2, it is straightforward to see that a round-robin schedulersatisfying the temporal demands exists. To elaborate, let α = (cid:80) i ∈ [ n ] w UL , i + (cid:80) i ∈ [ n ] w DL , i −
1. Thevariable α indicates the fraction of resource blocks the BS operates in FD mode. The roundrobin scheduler activates v , for the first a = min( w UL , , w DL , , α ) fraction of the resourceblocks. Next, it activates v , for a = min( w UL , , w DL , , α − a ) resource blocks. Note that a + a = α since w UL , i + w DL , i ≤ , i ∈ { , } . The scheduler activates v , , v , , v , ,and v , for a = w DL , − a , a = w UL − a , a = w DL , − a and a = w UL , − a fraction of the resource blocks. It can be verified that this is a valid allocation since a i ≥ (cid:80) i ∈ [6] a i =
1, andall of the temporal demand constraints are satisfied. For n >
2, a round robin scheduler canbe constructed using the same idea as in the previous case. First, the round robin scheduleroperates in the FD mode for α = (cid:80) i ∈ [ n ] w UL , i + (cid:80) i ∈ [ n ] w DL , i − v , for a , = min ( α, w UL , , w DL , ) fraction of the time. Then, it activates v , for a , = min ( α − a , , w UL , − a , , w DL , ) fraction of the time. The scheduler proceeds in thismanner until α fraction of the resource blocks are scheduled. The fact that the process can becontinued until α fraction of the resource blocks is guaranteed from the bounds in the theoremstatement. The scheduler proceeds by activating the users in the HD mode until the temporalconstraints are satisfied. The proof for k > ppendix BP roof of T heorem w n DL , w n UL ) satisfies Equation (15), then it is feasible.Let ( a i , j , a , j , a , i ) , i , j ∈ [ n ] be the corresponding coe ffi cients satisfying Equation (15). Theround robin strategy characterized by Q t ( R n × n × t ) = V i t , j t , t ∈ [ s ], where ( i t , j t ) is the uniqueindex for which the inequality (cid:80) i ≤ i t , j < j t sa i , j + ≤ t ≤ (cid:80) i ≤ i t , j ≤ j t sa i , j achieves temporal fairnessin the scheduling window-length s . Conversely, assume that ( w n UL , w n UL ) is feasible, then it isstraightforward to see that a round robin strategy satisfying the temporal fairness constraintsexists. Let ( a i , j , a , j , a , i ) , i , j ∈ [ n ] be the shares of the corresponding virtual users for thisscheduling strategy. Then, we argue that the coe ffi cients ( a i , j , a , j , a , i ) , i , j ∈ [ n ] along withtemporal demand vectors ( w n DL , w n UL ) satisfy Equation (15). The first two eqaulities hold sincethe UL and DL temporal demand are satisfied and and the next three equations hold by thedefinition of the virtual user temporal shares.A ppendix CP roof of T heorem λ UL , i , λ DL , i ∈ [ − M , M ] , ∀ i ∈ [ n ] , then it is optimal, where M is a limitedupper-bound on the performance value of any virtual user. Fix (cid:15) >
0. Let (cid:15) (cid:48) = nM (cid:15) . Let (cid:98) Q ∈ Q T BS be a TBS characterized by the threshold vectors ( λ n UL , λ n DL ) ∈ [ − M , M ] n and let Q be an arbitrary scheduling strategy. From (11) we know that | A Qi − w i | ≤ (cid:15), ∀ i ∈ [ n ]. Also, byassumption, λ i ≤ M , ∀ i ∈ [ n ]. As a result, λ i ( A Qi − w i ) + (cid:15) (cid:48) n ≥ , ∀ i ∈ [ n ]. We have, U Q ≤ U Q + n (cid:88) i = (cid:0) λ UL , i ( A Q UL , i − w UL , i ) (cid:1) + n (cid:88) i = (cid:0) λ DL , i ( A Q DL , i − w DL , i ) (cid:1) + (cid:15) (cid:48) ≤ lim inf t →∞ (cid:34) t t (cid:88) k = (cid:88) i , j ∈ [ n ] ∪{ } (cid:16) R i , j , k (cid:8) Q k ( R n × n × k ) = v i , j (cid:9)(cid:17)(cid:35) + n (cid:88) i = λ UL , i · lim inf t →∞ t (cid:34) t (cid:88) k = (cid:88) j ∈ [ n ] ∪{ } (cid:16) (cid:8) v i , j = Q k ( R n × n × k ) (cid:9)(cid:17)(cid:35) + n (cid:88) i = λ DL , i · lim inf t →∞ t (cid:34) t (cid:88) k = (cid:88) j ∈ [ n ] ∪{ } (cid:16) (cid:8) v j , i = Q k ( R n × n × k ) (cid:9)(cid:17)(cid:35) − (cid:88) i ∈ [ n ] ∪{ } ( λ UL , i w UL , i + λ DL , i w DL , i ) + (cid:15) (cid:48) ( a ) ≤ lim inf t →∞ (cid:34) t t (cid:88) k = (cid:88) i , j ∈ [ n ] ∪{ } (cid:16) R i , j , k (cid:8) Q k ( R n × n × k ) = v i , j (cid:9)(cid:17) + (cid:88) i , j ∈ [ n ] ∪{ } (cid:16) ( λ UL , i + λ DL , j ) (cid:8) v i , j ∈ Q k ( R n × n × k ) (cid:9)(cid:17)(cid:35) − (cid:88) i ∈ [ n ] ∪{ } ( λ UL , i w UL , i + λ DL , i w DL , i ) + (cid:15) (cid:48) = lim inf t →∞ t (cid:34) t (cid:88) k = (cid:88) i , j ∈ [ n ] ∪{ } (cid:16) R i , j , k + λ UL , i + λ DL , j (cid:17) (cid:8) v i , j ∈ Q k ( R n × n × k ) (cid:9)(cid:35) − (cid:88) i ∈ [ n ] ∪{ } ( λ UL , i w UL , i + λ DL , i w DL , i ) + (cid:15) (cid:48) ( b ) ≤ lim inf t →∞ t (cid:34) t (cid:88) k = (cid:88) i , j ∈ [ n ] ∪{ } (cid:16) R i , j , k + λ UL , i + λ DL , j (cid:17) (cid:8) v i , j ∈ (cid:98) Q k ( R n × n × k ) (cid:9)(cid:35) − (cid:88) i ∈ [ n ] ∪{ } ( λ UL , i w UL , i + λ DL , i w DL , i ) + (cid:15) (cid:48) ( c ) = lim inf t →∞ (cid:34) t t (cid:88) k = (cid:88) i , j ∈ [ n ] ∪{ } (cid:16) R i , j , k (cid:8) (cid:98) Q k ( R n × n × k ) = v i , j (cid:9)(cid:17)(cid:35) + n (cid:88) i = λ UL , i · lim inf t →∞ t (cid:34) t (cid:88) k = (cid:88) j ∈ [ n ] ∪{ } (cid:16) (cid:8) v i , j = (cid:98) Q k ( R n × n × k ) (cid:9)(cid:17)(cid:35) + n (cid:88) i = λ DL , i · lim inf t →∞ t (cid:34) t (cid:88) k = (cid:88) j ∈ [ n ] ∪{ } (cid:16) (cid:8) v j , i = Q k ( R n × n × k ) (cid:9)(cid:17)(cid:35) − (cid:88) i ∈ [ n ] ∪{ } ( λ UL , i w UL , i + λ DL , i w DL , i ) + (cid:15) (cid:48) ≤ U (cid:98) Q + n (cid:88) i = (cid:0) λ UL , i ( A (cid:98) Q UL , i − w UL , i ) + λ DL , i ( A (cid:98) Q DL , i − w DL , i ) (cid:1)(cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) ≤ (cid:15) (cid:48) + (cid:15) (cid:48) ≤ U (cid:98) Q + (cid:15) (cid:48) , where (a) holds since limit inferior satisfies supper-additivity, (b) holds due to the rearrangementinequality, and finally, (c) follows from the existence of the limit inferior. As a result: U Q ≤ U (cid:98) Q + (cid:15) (cid:48) , ∀ (cid:15) > , ⇒ U Q ≤ U (cid:98) Q . The proof of the existence of the threshold vector follows by similar arguments as in the proofof Theorem 1 in [16]. We provide an outline of the proof for the case when (cid:80) i ∈ [ n ] w UL , i > (cid:80) i ∈ [ n ] w DL , i >
1. We use the following Lemma: Lemma 1 ( Avoiding Cones Conditions [20] ) . Let n ∈ N . Consider the set of continuousfunctions f i : R n → R , i ∈ [ n ] . Assume that for each function f i , i ∈ [ n ] , there are positive realsM + i and M − i such that i) for any point x n such that x i = M + i , either the function f i is positive or ∃ j (cid:44) i : f j ( x n ) (cid:44) , and ii) For any point x n such that x i = − M − i , either the function f i is negativeor ∃ j (cid:44) i : f j ( x n ) (cid:44) . Then, the function f n = ( f , f , · · · , f n ) has a root in the n-dimensionalcube (cid:81) ni = [ − M − i , M + i ] . Alternatively: ∃ x ∗ , . . . , x ∗ n ∈ n (cid:89) i = [ − M − i , M + i ] : f i ( x ∗ , . . . , x ∗ n ) = , ∀ i ∈ [ n ] . Take f UL , i ( λ nUL , λ nDL ) (cid:44) A Q TBS UL , j − w UL , j , ∀ i ∈ [ n ] and f DL , i ( λ nUL , λ nDL ) (cid:44) A Q TBS DL , j − w DL , j , ∀ i ∈ [ n ].First we construct M − UL , i and M − DL , i , i ∈ [ n ] satisfying the conditions in Lemma 1. Note that byassumption e UL (cid:44) (cid:80) i ∈ [ n ] w UL , i − n > e DL (cid:44) (cid:80) i ∈ [ n ] w DL , i − n >
0. Furthermore, it is straightforwardto show that there exists α nUL , α nDL > (cid:80) i ∈ [ n ] α UL , i = (cid:80) i ∈ [ n ] α DL , i = w UL , i − α UL , i e UL , w DL , i − α DL , i e DL > , i ∈ [ n ]. Define w (cid:48) UL , i = w UL , i − α UL , i e UL , w (cid:48) DL , i = w DL , i − α DL , i e DL , i ∈ [ n ]. Then, by construction, (cid:80) i ∈ [ n ] w (cid:48) UL , i = (cid:80) i ∈ [ n ] w (cid:48) DL , i =
1. So, the temporal demands ( w (cid:48) nUL , w (cid:48) nDL )can be satisfied by activating a single user at each slot. Note that if λ UL , i , λ DL , i ≤ − M , ∀ i ∈ [ n ],then only the individual users will be chosen by the threshold strategy. The reason is that inthis case the scheduling measures for the HD virtual users are larger than that of joint userswith probability one. Let λ − UL , i , λ − DL , i ∈ [ − M , − M ] , i ∈ [ n ] be the thresholds for an HD scheduler(with virtual users V , i , V i , , i ∈ [ n ]) satisfying the temporal constraints ( w (cid:48) nUL , w (cid:48) nDL ). Then, M − UL , i = λ UL , i , M − DL , i = λ DL , i , i ∈ [ n ] satisfy the condition that f UL , i ( λ nUL , λ nDL ) < , λ UL , i = M − UL , i ,and f DL , i ( λ nUL , λ nDL ) < , λ DL , i = M − i ∀ i ∈ [ n ]. Furthermore, M + UL , i = M + DL , i = M , i ∈ [ n ] satisfythe conditions in Lemma 1. The reason is that for instance if M + UL , i = M , then the scheduleronly activates FD virtual users or HD users in UL. Then, at each time-slot, either the i th useris chosen in UL or there exists user u j for which f UL , j ( λ nUL , λ nDL ) > (cid:80) i ∈ [ n ] w UL , i <
2. So,we have shown that the conditions for Lemma 1 are satisfied which proves the existence ofthreshold values for which f UL , i ( λ nUL , λ nDL ) = A Q TBS UL , j − w UL , j = , ∀ i ∈ [ n ] and f DL , i ( λ nUL , λ nDL ) = A Q TBS DL , j − w DL , j = , ∀ i ∈ [ n ]. R eferences [1] S. Goyal, P. Liu, S. S. Panwar, R. A. Difazio, R. Yang, and E. Bala, “Full duplex cellular systems: will doubling interferenceprevent doubling capacity?” IEEE Communications Magazine , vol. 53, no. 5, pp. 121–127, 2015. [2] J. Zhou, T. Chuang, T. Dinc, and H. Krishnaswamy, “Integrated wideband self-interference cancellation in the RF domainfor FDD and full-duplex wireless,” IEEE Journal of Solid-State Circuits , vol. 50, no. 12, pp. 3015–3031, Dec 2015.[3] A. E. Gamal and Y. H. Kim,
Network Information Theory . Cambridge University Press, 2011.[4] X. Liu, E. K. Chong, and N. B. Shro ff , “A framework for opportunistic scheduling in wireless networks,” Elsevier ComputerNetworks , vol. 41, no. 4, pp. 451–474, 2003.[5] Z. Zhang, Y. He, and E. K. Chong, “Opportunistic scheduling for OFDM systems with fairness constraints,”
EURASIPJournal on Wireless Communications and Networking , vol. 2008, p. 25, 2008.[6] F. P. Kelly, A. K. Maulloo, and D. K. Tan, “Rate control for communication networks: shadow prices, proportional fairnessand stability,”
Journal of the Operational Research society , vol. 49, no. 3, pp. 237–252, 1998.[7] P. Viswanath, D. N. C. Tse, and R. Laroia, “Opportunistic beamforming using dumb antennas,”
IEEE Transactions onInformation Theory , vol. 48, no. 6, pp. 1277–1294, 2002.[8] X. Liu, E. K. P. Chong, and N. B. Shro ff , “Opportunistic transmission scheduling with resource-sharing constraints inwireless networks,” IEEE Journal on Selected Areas in Communications , vol. 19, no. 10, pp. 2053–2064, 2001.[9] S. S. Kulkarni and C. Rosenberg, “Opportunistic scheduling for wireless systems with multiple interfaces and multipleconstraints,” in
Proc. ACM Intl. Workshop on Modeling Analysis and Simulation of Wireless and Mobile Systems , 2003.[10] S. Shahsavari, F. Shirani, and E. Erkip, “Opportunistic temporal fair scheduling for non-orthogonal multiple access,” in , Oct 2018, pp. 391–398.[11] T. Joshi, A. Mukherjee, Y. Yoo, and D. P. Agrawal, “Airtime fairness for IEEE 802.11 multirate networks,”
IEEETransactions on Mobile Computing , vol. 7, no. 4, pp. 513–527, 2008.[12] T. Issariyakul and E. Hossain, “Throughput and temporal fairness optimization in a multi-rate TDMA wireless network,”in , vol. 7. IEEE, 2004, pp. 4118–4122.[13] S. Shahsavari and N. Akar, “A two-level temporal fair scheduler for multi-cell wireless networks,”
IEEE Wireless Commun.Letters , vol. 4, no. 3, pp. 269–272, 2015.[14] S. Shahsavari, N. Akar, and B. H. Khalaj, “Joint cell muting and user scheduling in multicell networks with temporalfairness,”
Wireless Communications and Mobile Computing , vol. 2018, 2018.[15] S. Shahsavari, D. Ramirez, and E. Erkip, “Joint user scheduling and power optimization in full-duplex cells with successiveinterference cancellation,” in , Oct 2017, pp. 1099–1104.[16] S. Shahsavari, F. Shirani, and E. Erkip, “A general framework for temporal fair user scheduling in NOMA systems,”
IEEEJournal of Selected Topics in Signal Processing , pp. 1–1, 2019.[17] S. Shahsavari, F. Shirani, and E. Erkip, “On the fundamental limits of multi-user scheduling under short-term fairnessconstraints,” arXiv preprint arXiv:1901.07719 , 2019.[18] 3GPP, “Technical specification group radio access network; evolved universal terrestrial radio access (E-UTRA); furtheradvancements for E-UTRA physical layer aspects (TR 36.814 V9.0.0),” 3GPP, Tech. Rep., 2010.[19] X. Zhang, T. Chang, Y. Liu, C. Shen, and G. Zhu, “Max-min fairness user scheduling and power allocation in full-duplexofdma systems,”
IEEE Transactions on Wireless Communications , pp. 1–1, 2019.[20] A. Fonda and P. Gidoni, “Generalizing the Poincaré–Miranda theorem: the avoiding cones condition,”