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to continuous introduction of mobile devices and services , future cellular systems are facing a significantly increased number of mobile devices requesting large data volume . to accommodate such a large growth of mobile devices , there are active researches on the 5th generation ( 5 g ) cellular system . new targets for the 5 g cellular system are to support latency - sensitive applications such as tactile internet @xcite and low energy consumption for machine - type communication ( mtc ) @xcite or the internet of things ( iot ) @xcite . unfortunately , a cellular system can not achieve the two targets simultaneously , but a non - trivial tradeoff can exist . although this tradeoff is very important to 5 g cellular system designers , related researches are rare . this is because it is often hard to deal with the latency and the energy consumption analytically so that intensive simulation - based network plannings are widely spread @xcite , @xcite . however , this approach becomes impractical when the network components , such as the number of users and bs antennas are scaled up . more viable approach is to analyze the network . this paper mainly concentrates on the analysis about the tradeoff between the latency and the energy consumption in a promising 5 g cellular system .    in 5 g cellular systems , there has been great interest to a large - scale antenna system ( lsas ) , a.k.a . massive multiple - input multiple - output ( mimo ) , in which very large number of antennas are equipped at a base station ( bs ) to serve many users simultaneously @xcite . its inherent merits come from massive spatial dimensions , which include i ) generating sharp beams for intended users to improve spectral efficiency by suppressing unintended interference @xcite , @xcite , ii ) reducing transmit energy while guaranteeing quality of service ( qos ) @xcite , and iii ) allowing a complexity - efficient transceiver algorithm @xcite . in order to achieve such advantages , an appropriate channel state information ( csi ) acquisition process is essential . to acquire csi , a widely - accepted approach is the training - based transmission in which a frame is divided into two phases : one is the training phase , in which users transmit known training signals and the bs estimates the csi , and the other is the data transmission phase , in which the users transmit information - bearing signals and the bs extracts the information by utilizing the estimated csi . even if the training - based transmission is not optimal in information - theoretic point of view , it gives an efficient way to acquire the csi as well as to provide the optimal degrees of freedom in the high signal - to - noise ratio ( snr ) regime @xcite .    in order to analyze the latency in the training - based lsas , it is necessary to optimize the user scheduling policy as well as the resource allocation under reasonable and practical constraints . if this optimization is not performed , it often gives an inappropriate cellular system design . the optimization of the training - based transmission is firstly investigated by hassibi and hochwald @xcite . they consider the mimo point - to - point channel with a capacity - approaching transmitter / receiver pair and successfully derive the optimal system parameters as a function of snr and other parameters . later , this results are extended to the mimo broadcast channel @xcite , multiple access channel @xcite , relay channel @xcite , and interference channel @xcite . however , these works optimize the energy and time dedicated to the training phase only under a given user set so that it can not be directly applied to the latency - energy tradeoff in the lsas . in order to evaluate the latency of the lsas , it is necessary to further optimize those variables under the optimal scheduling policy . the scheduling policies to minimize the latency ( or delay ) under a minimum rate constraint or to maximize spectral efficiency under a maximum latency constraint have been widely investigated in literature under various system models . in @xcite , the system average delay is optimized by using combined energy / rate control under average symbol - energy constraints . in @xcite , delay - optimal energy and subcarrier allocation is proposed for orthogonal frequency division multiple access ( ofdma ) . in @xcite , the energy minimizing scheduler , by adapting energy and rate based on the queue and channel state is proposed . however , most of them assume perfect csi at transmitter and receiver so that it often overestimates the network - wise performance . also , their scheduling policies are too complicated to be analyzed for providing an intuitive insight on the network - wise performance . thus , a practically optimal scheduling policy for the training - based lsas is needed and an intuitive analysis is desired to provide an insight on the latency - energy tradeoff in the lsas . decreasing the latency in the lsas is closely related to increasing the spectral efficiency , because higher spectral efficiency results in a smaller transmission completion time if the number of users and their rate constraints are given . in addition , the spectral efficiency of a multiple - access channel with @xmath1 bs antennas and @xmath10 scheduled users is asymptotically expressed as @xmath11 as @xmath12 , which implies that the spectral efficiency can be enhanced by scheduling users as many as possible in the lsas . however , most literature assumes that _ orthogonal _ pilots are allocated to users so that the maximum number of scheduled users is limited by the number of available pilots in practice . actually , there is no reason that orthogonal pilots are optimal for the latency - energy tradeoff so that it is natural to consider _ non - orthogonal _ pilots in general . there are a few results related to the case using non - orthogonal pilots . in @xcite , optimal non - orthogonal pilots for minimizing channel estimation error are derived and it turns out that finding the optimal non - orthogonal pilots is equivalent to solving the grassmannian subspace packing problem . in @xcite , an iterative algorithm is proposed to find optimal non - orthogonal pilots for maximizing the number of users with a minimum rate constraint in a downlink lsas . however , they still do not address the effect of the non - orthogonal pilots on the latency and it would be very interesting to find whether the use of non - orthogonal pilots can reduce the latency and when and how much reduction can be obtained over the case of using orthogonal pilots .    in this paper , we are interested in an uplink training - based lsas serving many users with an average energy constraint , in which each user has a limited average energy for transmitting a frame . in addition , we assume a block rayleigh fading model and a practical receiver such as the maximum ratio combining ( mrc ) or the zero - forcing ( zf ) receiver and focus on the resource allocation and multiple access strategy ( to be specific , pilot allocation , user grouping and scheduling , and energy allocation ) . the main target of this paper is to address the following question : * _ how much time is needed for guaranteeing the minimum throughput to all users in the uplink training - based lsas ? _ * it is , in general , hard to address this question analytically so that we look into the two asymptotic regimes , high and low energy regimes and successfully derive the effect on the network latency according to the energy consumption .    the main contributions of this paper are summarized as follows :    * we optimize the uplink scheduling policy for minimizing the latency with guaranteeing the minimum rate constraint . the optimizing variables are the scheduling groups in which users are simultaneously scheduled , the scheduling portion indicating how often each scheduling group actually transmits , and the energy allocation indicating how much portion of energy is dedicated to the training phase . this problem is transformed into an equivalent problem of maximizing the spectral efficiency with the rate constraint . the optimal scheduling policy is obtained by solving the binary integer programming ( bip ) and it is proved that the optimal solution of the original bip can be obtained by a linear programming relaxation with a polynomial - time complexity . * we investigate the asymptotic performance of the proposed optimal uplink scheduling policy for a large number of users . we derive a simple close - form expression for the asymptotic network latency and find the optimal parameters for the proposed optimal uplink scheduling policy . then , we identify four operating regimes of the training - based lsas according to the growth or decay rate of the average received signal quality , @xmath0 , and the number of bs antennas , @xmath1 . it turns out that orthogonal pilot sequences are optimal only when the average received signal quality is sufficiently good and the number of bs antennas is not - so - large . in other regimes , it turns out that non - orthogonal pilot sequences become optimal . in fact , the use of non - orthogonal pilots can reduce the network latency by a factor of @xmath6 when the received signal quality is quite poor ( @xmath4 ) or by a factor of @xmath7 when the received signal quality is sufficiently good ( @xmath8 ) and the number of bs antennas is sufficiently large ( @xmath13 ) . the remainder of this paper is organized as follows . in section ii , a detailed model description is presented including channel , energy , and signal models for a training - based lsas . in section iii , the uplink scheduling policy problem is formulated and its optimal solution is provided . numerical experiments to verify the superiority of the proposed uplink scheduling policy are shown in section vi . in section v , an asymptotic analysis provides the closed - form network latency and optimal parameters for the proposed uplink scheduling policy . finally , conclusion is drawn in section vi . matrices and vectors are respectively denoted by boldface uppercase and lowercase characters . also , @xmath14 , @xmath15 , and @xmath16 stand for the transpose , conjugate transpose , and cardinality of a set , respectively , and @xmath17 and @xmath18 are the natural logarithm and the logarithm with base 2 , respectively . also , @xmath19 denotes the function rounding @xmath20 towards the nearest integer , @xmath21 , and @xmath22 denotes the indicator function . @xmath23 denotes the distribution of a circularly symmetric complex gaussian random vector with mean vector @xmath24 and covariance matrix @xmath25 and @xmath26 $ ] and @xmath27 denotes the statistical expectation and the statistical variance , respectively . finally , standard order notations in @xcite are used . for better readability , frequently used symbols are summarized in table i.            ? symbols & descriptions + @xmath1 & # of bs antennas + @xmath28 & # of total users and their index set , @xmath29 + @xmath30 & # of frames in a uplink scheduling period and its minimum with guaranteeing @xmath31 bits to all of users . + @xmath32 & network latency , optimized network latency and optimized asymptotic network latency normalized by @xmath33 with guaranteeing @xmath31 bits to all of users . + @xmath34 & # of sub - frames in a frame + @xmath35 & # of symbols in a sub - frame + @xmath36 & # of symbols in the training phase or the data transmission phase + @xmath37 & bandwidth and the symbol duration + @xmath38 & bandwidth inefficiency with @xmath39 + @xmath40 & the average energy constraint of user @xmath41 . + @xmath42 , { r}_k[f;t ] $ ] & the achievable rate of users @xmath41 in sub - frame @xmath43 of frame @xmath44 and its approximation ( bits / hz ) + @xmath31 & throughput threshold ( bits ) + @xmath45 & the scheduling group and its scheduling portion + @xmath46 & the energy dedicated to the training symbols or data symbols ( joule / symbol ) + we consider an uplink lsas consisting of a bs with @xmath1 antennas , and @xmath33 single - antenna users as illustrated in fig . it is assumed that the users are randomly distributed on the cell coverage region and they want the quality of service ( qos ) on their own data traffic ( rate , latency , and reliability ) so that the bs serves these users persistently and try to guarantee their qos .    a two - phase frame structure with training and data transmission phases , illustrated as in fig . 2 , is used . for every uplink scheduling period , the bs broadcasts the scheduling information and then the users transmit @xmath47 frames in uplink direction step by step . the frame of time length @xmath48 seconds and bandwidth @xmath49 hz is divided into @xmath34 equal - bandwidth sub - frames by partitioning frequency domain by using the orthogonal division multiplex access ( ofdma ) , single - carrier frequency domain multiple access ( sc - fdma ) or any good one of the newly considered waveforms @xcite . the sub - frame consists of @xmath35 symbols of time period @xmath50 seconds . in the training phase of time period @xmath51 seconds , the scheduled users send @xmath52 training symbols , and the bs estimates the uplink channels . then , in the data transmission phase of the remaining time period @xmath53 seconds , all of the scheduled users transmit @xmath54 data symbols to the bs simultaneously in a space division multiple access ( sdma ) manner . sub - frames and @xmath35 symbols in the sub - frame can be arbitrarily configured both in time and frequency domains . for example , in the long - term evolution ( lte ) , one sub - frame includes @xmath55 symbols ( 14 symbols in time domain and 12 sub - carriers in frequency domain ) and one frame includes @xmath56 sub - frames in time domain . ] so , the transmit signal vector of user @xmath41 , who is allocated to the @xmath43th sub - frame in frame @xmath44 , is written as @xmath57 = { \left [ { { { \left ( { { \bf{x}}_j^{{\rm{tr}}}[f;t ] } \right)}^h},{{\left ( { { \bf{x}}_j^{{\rm{dt}}}[f;t ] } \right)}^h } } \right]^h},\ ] ] where @xmath58 $ ] is the @xmath59 training symbol vector for the training phase and @xmath60 $ ] is the @xmath61 data symbol vector for the data transmission phase .    in every sub - frame , at most @xmath1 users are scheduled and the set of users scheduled in sub - frame @xmath43 of frame @xmath44 is denoted as @xmath62 $ ] . assume that @xmath63|=k$ ] . within one block of @xmath35 symbols , the @xmath64 received signal matrix , denoted as @xmath65=\left[\mathbf{y}_1[f;t],\mathbf{y}_2[f;t],\dots,\mathbf{y}_n[f;t]\right]$ ] , is given as @xmath66 & = \mathbf{g}_{\mathcal{s}[f;t]}[f;t]\mathbf{x}^h[f;t ] + \mathbf{v}[f;t ] , \end{split}\ ] ] where @xmath67}[f;t ] = \left[\mathbf{g}_{j}[f;t]\right]_{j\in\mathcal{s}[f;t]}$ ] is the @xmath68 channel matrix , @xmath69=[\mathbf{x}_1[f;t],\dots,\mathbf{x}_k[f;t ] ] $ ] is the @xmath70 transmitted signal matrix and @xmath71=\left[\mathbf{v}_{1}[f;t],\mathbf{v}_{2}[f;t],\dots,\mathbf{v}_{n}[f;t]\right]$ ] is the @xmath64 noise matrix , whose elements are independent and identically distributed ( i.i.d . ) random variables with @xmath72 . the @xmath73 flat - fading channel vector between the bs and user @xmath41 at the sub - frame @xmath43 of frame @xmath44 , @xmath74 $ ] , can be written as @xmath75 = \sqrt { { \beta _ { j } } } { { \mathbf{h}}_{j}}[f;t],\ ] ] where @xmath76 \in { \mathbb{c}^{m \times 1}}$ ] is the short - term csi whose elements are i.i.d . random variables with @xmath72 and @xmath77 is the long - term csi depending on the path - loss and shadowing . the long - term csi between the bs and user @xmath41 is modeled as @xmath78 , where @xmath79 is the wireless channel path - loss exponent and @xmath80 is the distance between the bs and user @xmath41 . we assume that rayleigh block fading model , where the short - term csi of each user remains constant within a given frame but is independent across different frames , while the long - term csi does not vary during a much longer interval . further , it is assumed that the long - term csi of all users is perfectly known at the bs .    since each of users has a different limited battery capacity , recharge process , or radio frequency ( rf ) transmitter , they are assumed to be limited to spend energy for transmitting each sub - frame . let @xmath81 be the average allowed energy level ( in joule ) of user @xmath41 per sub - frame of length @xmath48 . the energy is consumed during both the training phase of length @xmath82 and the data transmission phase of length @xmath83 . so , the consumed energy transmitting each sub - frame needs to meet @xmath84\|^2 = \mathbb{e}\|\mathbf{x}_j^{\rm{tr}}[f;t]\|^2+\mathbb{e}\|\mathbf{x}_j^{\rm{dt}}[f;t]\|^2\le e_j,~\forall j.\ ] ] letting @xmath85\|^2/l$ ] and @xmath86\|^2/(n - l)$ ] be the transmit energy of each training symbol and data symbol , respectively , the constraint is represented as @xmath87 in the sequel , we drop the sub - frame index @xmath43 and the frame index @xmath44 if there is no ambiguity .      to estimate the csi of @xmath10 scheduled users , the bs allocates @xmath10 pilot sequences with length of @xmath52 . let @xmath88 $ ] be the @xmath89 pilot matrix with normalized columns , i.e. , @xmath90 for all @xmath41 . the training symbol vector of scheduled user @xmath41 during the training phase is @xmath91 . for equalizing the difference of all users channel estimation quality ( maximizing the worst ) , the _ channel - inversely power - controlled pilots _ are assumed similarly as in @xcite , in which the average received signal energy of the users in @xmath62 $ ] is set to the common target received energy , @xmath92 . so , the transmit energy at the training phase is set by @xmath93 then , the @xmath94 received signal matrix in frame @xmath44 at the bs during the training phase , denoted as @xmath95 $ ] , can be written as @xmath96}\boldsymbol\psi _ { l , k}^h + { { \mathbf{v}}}^{\rm{tr}},\end{aligned}\ ] ] where @xmath97 } = \left[\mathbf{h}_{j}\right]_{j\in\mathcal{s}[f;t]}$ ] , and @xmath98 $ ] is the noise vector during the training phase . using the minimum mean - square error ( mmse ) channel estimator @xcite , the estimated short - term csi of user @xmath41 , denoted as @xmath99 , can be written as @xmath100 denote the channel estimation error by @xmath101 . the following lemma informs the property of the mmse channel estimation .    with the channel - inversely power - controlled pilots , @xmath102 and @xmath103 are mutually independent and the channel estimation error variance @xmath104 is given by @xmath105 where @xmath106 are the eigenvalues of @xmath107 . see the proof of theorem 1 in @xcite .      during the data transmission phase , the @xmath108th received signal vector at sub - frame @xmath43 of frame @xmath44 , @xmath109 , is given by @xmath110 = \sum\limits_{j\in\mathcal{s}[f;t ] } \sqrt{p_j^{\rm{dt}}\beta_j}\mathbf{h}_j s_{jl}[f;t ] + \mathbf{v}_l,\ ] ] where @xmath111\sim\mathcal{cn}(0,1)$ ] is the @xmath108th information - bearing data symbol of user @xmath41 at sub - frame @xmath43 of frame @xmath44 .      with the channel - inversely power - controlled non - orthogonal pilots , the achievable rate of user @xmath122 of the mrc or zf receiver is approximated by @xmath127 \approx \left\ { { \begin{array}{*{20}{l } } { \log_2\left ( { 1 + \gamma _ k[f;t ] } \right),}&{{\text{if } } k \in { \cal s}[f;t],}\\ { 0,}&{{\text{if } } k \notin { \cal s}[f;t ] , } \end{array } } \right.\ ] ] where @xmath128 $ ] is given as @xmath129=   \left\ { { \begin{array}{*{20}{l } } { \frac{{\left ( { 1 - \sigma _ { { \rm{tr}}}^2 } \right)\left ( { m - k } \right)p_k^{{\rm{dt}}}{\beta _ k}}}{{1 + \sigma _ { { \rm{tr}}}^2\sum\limits_{j \in { \cal s}[f;t ] } { p_j^{{\rm{dt}}}{\beta _ j } } } } } , & { { \text{if zf}}},\\ { \frac{{\left ( { 1 - \sigma _ { { \rm{tr}}}^2 } \right)\left ( { m - 1 } \right)p_k^{{\rm{dt}}}{\beta _ k}}}{{1 + { \sum\limits _ { j \in \mathcal{s}[f;t]}}p_j^{{\rm{dt}}}{\beta _ j } - \left ( { 1 - \sigma _ { { \rm{tr}}}^2 } \right)p_k^{{\rm{dt}}}{\beta _ k}}}},&{{\text{if mrc}}}. \end{array } } \right . \end{split}\ ] ]    the proof is similar as in @xcite , so omitted for brevity . note that ( [ eq14 ] ) is valid only when @xmath130 for the zf receiver ( @xmath131 for the mrc receiver ) . in fact , when such an approximation is used as the utility function of a scheduler , it can not select @xmath132 users , which under - utilizes the resource , especially when @xmath1 is small . however , this paper deals with the lsas so that this problem becomes less significant . additionally , to overcome this problem for a small @xmath1 , @xmath133 can be replaced by @xmath134 , which results in a small approximation error . ( similar approximation is shown in theorem 5 in @xcite . ) furthermore , this approximated version of the achievable rate becomes more accurate as the number of bs antennas increases . [ rate achievability ] since the low latency communications are of interest , codewords can not be spread sufficiently in time domain . also , the transmitter does not acquire its instantaneous csi , the transmission strategy can not be adapted according to the channel realization . in this paper , we assume that user @xmath41 transmits information - bearing data symbols with a fixed rate of @xmath135 regardless of the channel realization . the bs can successfully receive this data symbol if and only if @xmath136 . from the strong law of large number , we can show that @xmath136 almost surely as @xmath137 so that the rate @xmath135 is achievable in the lsas with high probability . this is called _ channel hardening effect _ of the lsas , which implies that the instantaneous rate is approaching a deterministic quantity , i.e. , its mean , as @xmath138 . see @xcite for more rigorous discussions .    to maximize the average achievable rate , we need to design pilot sequences . optimal pilot sequences @xmath139 for maximizing @xmath135 satisfies @xmath140    since the case @xmath141 is trivial so we omit . let @xmath142 . the channel estimation error variance ( [ eq7 ] ) is simplified into @xmath143 obviously , @xmath144 is a decreasing function with @xmath104 within @xmath145 . so , the maximization of @xmath146 is equivalent to the maximization @xmath147 under @xmath148 , which is obtained when the eigenvalues are the same , i.e. , @xmath149 . note that the above condition is identical to the condition minimizing the channel estimation error @xcite . in fact , this condition is known as the welch bound equality ( wbe ) @xcite . so , from lemma 3 , the channel estimation error variance is minimized at @xmath150 and the sinr is maximized at @xmath151 = \left\ { { \begin{array}{*{20}{l } } { \frac{{l\left ( { m - k } \right)p_k^{{\rm{dt}}}{\beta _ k}\overline { { p^{{\rm{tr } } } } } } } { { \left ( { 1 + { { \left ( { k - l } \right)}^ + } \overline { { p^{{\rm{tr } } } } } } \right)\left ( { 1 + \sum\limits_{j \in \mathcal{s}[f;t ] } { p_j^{{\rm{dt}}}{\beta _ j } } } \right ) + l\overline { { p^{{\rm{tr } } } } } } } , } & { { \text{if zf}},}\\ { \frac{{l\left ( { m - 1 } \right)p_k^{{\rm{dt}}}{\beta _ k}\overline { { p^{{\rm{tr } } } } } } } { { \left ( { 1 + \max \left\ { { l , k } \right\}\overline { { p^{{\rm{tr } } } } } } \right)\left ( { 1 + \sum\limits_{j \in \mathcal{s}[f;t ] } { p_j^{{\rm{dt}}}{\beta _ j } } } \right)-l\overline{p^{\rm{tr}}}p_k^{\rm{dt}}\beta_k}},}&{{\text{if mrc}}. } \end{array } } \right . \end{split}\ ] ] if the pilot sequences are under - utilized ( orthogonal pilots are used ) , i.e. , @xmath152 , @xmath104 is simplified into @xmath153 , which means that the channel estimation error depends only on the energy dedicated to the training phase @xmath154 . however , if the pilot sequences are over - utilized ( non - orthogonal pilots are used ) , i.e. , @xmath155 , additional interference @xmath156 occurs . [ optimal non - orthogonal pilots ] obviously , if @xmath152 , the optimal pilot sequences can be obtained from arbitrarily chosen @xmath10 columns of an @xmath157 unitary matrix . in the case of @xmath155 , one option is to over - sample an @xmath157 unitary matrix . although there are infinitely many sequences holding the wbe , one simple example is the discrete fourier transform ( dft ) based sequences , obtained as @xmath158,\ ] ] where @xmath159 and @xmath160 are arbitrarily chosen integers . note that the dft - based sequences are widely used in the design of the unitary - space time modulation @xcite or the feedback codebook @xcite . the accuracy of the approximated rate in ( [ eq13 ] ) and ( [ eq17 ] ) is presented in fig . 3 , by setting @xmath161 , @xmath162 , @xmath163 , @xmath164db , @xmath165 , @xmath166 , and @xmath167 , @xmath168 . in fig . 3 , we plot the exact rate in ( [ eq12 ] ) and the approximated rate in ( [ eq13 ] ) according to the number of bs antennas , @xmath1 . the dashed line represents the average of the exact rate and each error bar represents its standard deviation . we can easily notice that the approximated rate is quite well fitted even at a small number of antennas in a wide range of average allowed energy level @xmath169 .    ) and ( [ eq17 ] ) . the line marked by a circle represents the results using the zf receiver and the line marked by a square represents the results using the mrc receiver . ] according to the level of accessible information , uplink scheduling policies can be classified into two types . a _ dynamic _ user scheduling , which is based on the _ instantaneous _ csi , can provide a substantial rate gain primarily because it allows the bs to select a subset of users whose channels are nearly orthogonal and the achievable uplink performance increases with the number of users . however , obtaining the instantaneous csi at a bs will incur a large amount of uplink training resource cost . in fact , due to the nature of the limited channel coherence time , it is hard for every user to participate in the scheduling pool without incurring non - negligible overhead . on the contrary , uplink scheduling exploits the long - term information only , such as the csi statistics and/or the average allowed energy levels , which can be easily acquired via infrequent feedback with negligible overhead .    a static scheduling policy for @xmath47 frames is defined as @xmath170 , where @xmath171 denotes the set of scheduling groups , @xmath172 denotes the set of scheduling portions , @xmath173 denotes the set of energy allocations for all users , and @xmath52 determines the sub - frame configuration . due to the nature of the static uplink scheduling , the output of the static scheduling policy should satisfy the following two constraints : 1 ) @xmath174 if @xmath175 , and 2 ) @xmath176 , where @xmath177 . also , the scheduling portion is defined as @xmath178 = \mathcal{o}_q\}},\ ] ] so that it must satisfy @xmath179 .    note that @xmath180 is the portion of the sub - frames allocated to scheduling group @xmath181 , @xmath182 , during one scheduling period consisting of @xmath183 sub - frames . in fact , since @xmath184 is not an integer , @xmath185 sub - frames may be allocated . we assume that @xmath186 is sufficiently large so that the error @xmath187 is negligibly small .      in general , a user tries to send some of information to the bs via a wireless medium within various constraints . however in cellular systems , the limited wireless medium is shared so that severe delay often occurs for waiting transmission turns and also for transmitting the target data volumes . thus , to guarantee a low latency is one of the main hurdles to be addressed in future cellular systems . the network latency ( delay ) is defined as @xmath188 ( sec ) , where @xmath189 is the minimum number of frames required to deliver the target throughput of @xmath31 bits for each user . our definition is different to @xcite , @xcite , in which the delay is defined as the scheduling delay ( the waiting time for transmission turns ) only and the delay for transmitting the target throughput volume is ignored , and also to @xcite , @xcite , in which the delay is defined as the transmission completion time for the target throughput only and ignores the effect of the scheduling delay . our definition includes both scheduling delay and transmission completion time .    to minimize the latency under a given throughput constraint @xmath31 , it is sufficient to minimize the number of frames , @xmath47 . thus , the optimization problem can be constructed as follows :    [ eq19 ]    align ( ) &  s ,   & + & ( n - l)_t=1^s _ f=1^f r_j[f;t]_th , j , & + & p_j^tr + ( 1-)p_j^dt ,  p_j^tr , p_j^dt0 , j , & + & _ p _ q = ,  _ p=1^q_p= , & + & _ q=1^q d_q = 1 ,  d_q 0 , q , & + & l\{1,2,  ,n-1}. &    the constraint ( [ eq19]b ) is to guarantee the required throughput @xmath31 for each of users , ( [ eq19]c ) is to meet the average energy constraint , and ( [ eq19]d)-([eq19]e ) are to meet the static scheduler condition . the variables in the optimization problem ( p ) are as follows :    * which of users are simultaneously scheduled ? ( @xmath190 ) * how many sub - frames are allocated to each group ? ( @xmath172 ) * how the energy is allocated to the training and data transmission phases ? ( @xmath191 for @xmath168 ) * how much the symbols are dedicated to the training phase ? ( @xmath52 )    the problem ( p ) is obviously non - convex and is very complicated , but can be transformed into an equivalent problem . since we deal with a static uplink scheduling policy , the achievable rate of user @xmath41 is independent to the sub - frame index @xmath43 and frame index @xmath44 if user @xmath41 is scheduled . thus , we use the notation @xmath192 instead of @xmath193 $ ] if @xmath194 $ ] . let @xmath195 and define the spectral efficiency of user @xmath41 ( in bps / hz ) as the average rate of user @xmath41 served by the bs normalized by time and bandwidth , given as @xmath196 } } } = \frac{1-{\frac{l}{n}}}{\eta } d_q r_j , \label{eq20 }   \end{aligned}\ ] ] where @xmath197 denotes the bandwidth inefficiency ( such as the cyclic prefix overhead ) , and the second equality comes from ( [ eq18 ] ) . since every user has the same throughput constraint @xmath31 , it is required that @xmath198 for @xmath199 , where @xmath200 is the common rate for scheduled group @xmath181 . inserting @xmath198 into ( [ eq20 ] ) , we have @xmath201 in order to guarantee the same minimum rate for all users , we further set @xmath202 from ( [ eq21 ] ) and ( [ eq22 ] ) , every user is guaranteed to exceed the common spectral efficiency @xmath203    our approach is first to maximize the spectral efficiency while providing the common spectral efficiency @xmath204 to every user in a cell at each possible value of @xmath52 . to meet the target throughput @xmath31 for every user , the bs needs @xmath205 sub - frames so that the latency is given as @xmath206 where that last approximation is valid when @xmath207 is large . then , an equivalent optimization problem can be formulated as follow :    [ eq25 ]    align ( )  & & & , + & & & r_j_q ,  j _ q ,  q , + & & & - . note that a similar transformation is shown in @xcite . in the sequel , we devise the optimal uplink scheduling policy by solving ( p - eq ) . assume that @xmath52 is fixed and users @xmath209 are scheduled in the @xmath181th scheduling group and they are arranged in the descending order of @xmath210 , i.e. , @xmath211 , without loss of generality . note that each scheduled user should have non - zero transmit energy for both the training and data transmission phases . in order to find the optimal transmit energy allocation in ( p - eq ) , the following sub - problem is considered . [ eq26 ] @xmath212    to obtain the optimal solution , the following observations are helpful .    * since @xmath192 is an increasing function of the transmit energy , the optimal transmit energy are obtained when @xmath213 for @xmath214 ( no waste ) , i.e. , @xmath215 * since @xmath216 for any @xmath217 , if @xmath218 is feasible , then @xmath219 is also feasible . so , when @xmath10 users are scheduled , the optimal energy of the @xmath10th user should satisfy @xmath220    under the assumption that @xmath10 users are scheduled and the above two observations , the objective function of ( p - a ) in ( [ eq26]a ) can be written as @xmath221 from ( [ eq17 ] ) , where @xmath222 with @xmath223 and @xmath224 are defined as in table ii . since the objective function of ( p - a ) is a single - variable function , it can be easily solved and the following theorem states the optimal transmit energy allocation .    for given @xmath52 and scheduling user set @xmath182 , the optimal transmit energy of user @xmath225 during the training and data transmission phases is given as @xmath226 where @xmath227 is given in the bottom of the this page . @xmath228    see appendix a.    for later use , we define @xmath229 as the optimal common rate for given @xmath182 and @xmath52 . note that by inserting ( [ eq31 ] ) into ( [ eq29 ] ) and using variables in table ii , it can be seen that @xmath230 is a non - decreasing function of @xmath231 and is independent to @xmath210 , @xmath232 . even though the optimal transmit energy allocation strategy for given users are derived in ( [ eq30 ] ) , the size of the search space is too large to be exhaustively searched . to reduce the search space , we need to find implicit properties for the optimal scheduling groups .    from ( [ eq23 ] ) , the objective function of ( p - eq ) is given by @xmath234 where @xmath235 is the sets of the optimal scheduling groups . suppose that the cardinality of each of the optimal scheduling groups is given , i.e. , @xmath236 . then , the problem ( p - eq ) is reduced to the following cardinality - constrained problem for each possible value of @xmath52 :    [ eq33 ] @xmath237    to obtain the optimal scheduling groups , the following theorem is quite helpful . denote @xmath238 , @xmath239 , as the optimal solution of ( p - b ) at a given value of @xmath52 . then the optimal solution has the following properties :    * for @xmath181 with @xmath240 , if @xmath241 , then there is no @xmath242 such that @xmath243 . * for @xmath181 with @xmath244 , if @xmath245 and there exists @xmath242 such that @xmath243 , then @xmath246 .    see appendix b.    from theorem 2 , it is shown that the optimal uplink scheduling policy is to select users having similar product values of the average allowed energy level @xmath247 and the path - loss @xmath248 and it significantly reduces the search space . more detailed discussions on the search space will be given in sec . iii - d . although theorem 2 indicates some useful implicit properties for the optimal scheduling groups , it does not provide the exact solution explicitly , and we still need to find the optimal scheduling groups among the reduced search space . fortunately , it can be transformed into a binary integer programming ( bip ) with the following generic form :    [ eq34 ] @xmath249    where @xmath250 $ ] is the @xmath251 state matrix , @xmath252 @xmath253 is the @xmath254 cost vector given by @xmath255^t},\end{aligned}\ ] ] @xmath256 is the @xmath254 all - one vector , @xmath257 denotes the number of candidate scheduling groups , and @xmath258 denotes the optimal common rate at given @xmath259 and @xmath52 , defined in ( [ eq32 ] ) . the optimizing variable @xmath260 informs which candidate scheduling groups are selected , i.e. , if @xmath261 , the corresponding candidate scheduling group @xmath182 is selected as one of the optimal scheduling groups . such a bip has been widely researched in literature and a variety of efficient algorithms are summarized in @xcite . unfortunately , finding the optimal solution in a bip is known as np - hard in general . however , due to the special structure of our bip , it will be shown that a linear programming ( lp ) relaxation using @xmath262^{c\times 1}$ ] does not affect the optimality . to show this , we introduce the following definition and two lemmas and then conclude the optimality of the proposed algorithm . a matrix @xmath263 is _ totally unimodular _ if every square sub - matrix of @xmath263 has a determinant of @xmath264 , @xmath265 , or @xmath266 . [ @xcite , example 7 ] if every column of a binary matrix @xmath263 has consecutive ones only without being interrupted by @xmath264s , then @xmath263 is totally unimodular . [ @xcite , theorem 19.1 ] if @xmath267 is totally unimodular and @xmath268 is an integer vector , then the polytope described by @xmath269 , @xmath262^{c\times 1}$ ] , has integer vertices only . now , we are ready to state the optimality of the proposed algorithm using the lp relaxation in ( [ eq34 ] ) . the optimal solution of the bip in ( [ eq34 ] ) is identical to the solution obtained by using the lp relaxation on ( [ eq34 ] ) .    from the properties of theorem 2 , every column of the matrix @xmath270 has consecutive ones only , which implies that @xmath270 is totally unimodular from lemma 4 . using lemma 5 , the feasible region of ( [ eq34 ] ) is a polytope with integer vertices only , which guarantees that the solution obtained by using the lp relaxation on ( [ eq34 ] ) does not affect the optimality . the proposed algorithm for obtaining the optimal static uplink scheduling policy is outlined in algorithm 1 . the proposed optimal algorithm obtains @xmath271 and corresponding @xmath272 for each @xmath52 , and then find the optimal @xmath273 maximizing @xmath272 . the algorithm for obtaining @xmath271 is composed of the two parts . the first part finds the candidate scheduling groups , denoted as @xmath274 , and their corresponding common rate by using ( [ eq32 ] ) obtained by using the optimal energy allocations in theorem 1 . then , the second part finds the optimal combination of the selected scheduling groups that maximizes the spectral efficiency by applying the lp relaxation by virtue of theorems 2 and 3 . _ example : _ here , we explain a toy example . suppose that the network has @xmath275 users , @xmath276 , and @xmath277 and @xmath278kbits , @xmath279khz , @xmath280 , @xmath281 , @xmath282 , and @xmath283 . the first part returns the candidate scheduling groups as @xmath284 and suppose that the corresponding common rate is determined as @xmath285 , respectively . then , the cost vector and the state matrix are respectively set as @xmath286^t}\ ] ] and @xmath287.\ ] ] to satisfy the constraints ( [ eq34]b ) , there exist five feasible solutions : @xmath288^t$ ] , @xmath289^t$ ] , @xmath290^t$ ] , @xmath291,{\bf{x}}_5 = \left [ { 0,0,0,0,1,0,1 } \right]$ ] . since @xmath292 , @xmath293 and @xmath294 are selected as the optimal uplink scheduling groups . and @xmath295 and @xmath296 are the optimal scheduling portions and @xmath297 bps / hz . then , the latency is @xmath298 frames . note that among @xmath299 sub - frames @xmath300 sub - frames are allocated to @xmath301 and the remained sub - frames are allocated to @xmath302 . define @xmath303 as the whole search space for finding the optimal scheduling groups in ( [ eq19 ] ) without theorem 2 , where @xmath304 is the collection of @xmath305-ary partitions of @xmath306 with at most @xmath1 elements , i.e. , each scheduling group size is no greater than the number of antennas , given by @xmath307 note that @xmath308 is well - defined only when @xmath309 and @xmath310 , where @xmath311 denotes the stirling number of the second kind . for a fixed @xmath305 , @xmath312 increases exponentially with @xmath33 . thus , the whole search space is given as @xmath313 for a large @xmath33 . now , define @xmath314 as the reduced search space for finding the optimal scheduling groups in ( [ eq19 ] ) with the aid of theorem 2 . then , we can show that the cardinality of @xmath314 can be represented as the following recursive formula : @xmath315 which is known as the generalized fibonacci number @xcite . with help of the binet s formula @xcite , we arrive at @xmath316 where @xmath317 is the unique positive root of @xmath318 . after some algebraic manipulations , @xmath319 and @xmath320 as @xmath138 so that @xmath321 , which implies that the reduced search space still increases exponentially with the number of total users , @xmath33 . however , combined with theorem 3 , the following dramatical complexity reduction can be obtained .    * the reduction gain of theorem 2 itself also increases exponentially with @xmath33 . in fact , the reduction gain is at least @xmath322 for a large @xmath33 . * without theorem 2 , the number of candidate scheduling groups in ( [ eq34 ] ) is @xmath323 , ( for @xmath324 , @xmath325 ) , which increases exponentially with @xmath33 . however , due to theorem 2 , it reduces into @xmath326 , ( for @xmath324 , @xmath327 ) , which increases only squarely with @xmath33 . * as shown in theorem 3 , applying the lp relaxation on ( [ eq34 ] ) still provides the optimal solution of the original bip ( [ eq34 ] ) , which can dramatically reduce the exponential - time complexity into a polynomial - time complexity . now , we are ready to quantify the computational complexity of algorithm 1 . the computational complexity of algorithm 1 consists of the following three parts , namely 1 ) the sorting operation ( line 1 ) , 2 ) the optimal energy allocation ( lines 3 - 9 ) and 3 ) solving the relaxed lp ( lines 10 - 15 ) . the worst - case computational complexity for sorting @xmath33 samples is @xmath328 . since the optimal energy allocation requires @xmath329 iterations , the worst - case computational complexity of the second part is @xmath330 . finally , the worst - case computational complexity of the lp is @xmath331 by using the karmarkar s algorithm @xcite . thus , the total worst - case computational complexity for the proposed algorithm is @xmath332 . in this section , we present some numerical results to verify the superiority of the proposed uplink scheduling policy . one frame is set to occupy @xmath56mhz and @xmath266ms in the frequency and time domains and consists of @xmath333 sub - frames with @xmath334khz and @xmath266ms . the number of symbols in each sub - frame is set to @xmath335 by assuming @xmath336 ( 25% cp overhead ) . there are @xmath337 users each requesting @xmath338kbits date volume . we use the pathloss model @xmath339 , where @xmath340 , @xmath341 and @xmath80 is given by @xmath342 with @xmath343 . this pathloss model reflects the bs located at the origin and the users are located uniformly along the line @xmath344 $ ] . all of users have the same transmit energy constraint , @xmath345 for @xmath168 . according to the simulation setting , the received signal energy of the worst - case user at the bs is @xmath264db when @xmath346db energy is equally spread over the symbols in a sub-frame.dbm/hz for the noise spectral density , @xmath347db means only @xmath348dbm ( @xmath349mw ) per sub - frame in this simulation setting . ]     when @xmath350 and @xmath351 . the solid line represents the results using the zf receiver and the dashed line represents the results using the mrc receiver . ] when @xmath352db and @xmath351 . the solid line represents the results using the zf receiver and the dashed line represents the results using the mrc receiver . ] when @xmath350 and @xmath351 . ] when @xmath352db and @xmath351 . ] the following three schemes are compared and simulation results using the zf receiver are summarized in tables iii - v :    * ( random - equal ( re ) ) @xmath10 users are randomly selected and the transmit energy is equally used during the training and data transmission phases . the training length , @xmath52 , and @xmath10 are exhaustively searched . * ( random - optimal ( ro ) ) @xmath10 users are randomly selected and the transmit energy is optimized by using theorem 1 . the training length , @xmath52 , and @xmath10 are exhaustively searched . * ( proposed ) the optimal uplink scheduling policy in algorithm 1 is used . 4 depicts the latency and spectral efficiency of the three schemes as a function of @xmath169 when @xmath350 and @xmath351 . the solid line represents the results using the zf receiver and the dashed line represents the results using the mrc receiver . when the zf receiver is employed , it is observed that at @xmath353db ( @xmath354dbm per sub - frame ) , the proposed uplink scheduling policy provides about @xmath355 or @xmath356 times smaller latency over the random - equal or the random - optimal scheme . the major gain comes from the optimal energy allocation . when the mrc receiver is employed , it is observed that at @xmath353db the proposed uplink scheduling policy provides about @xmath357 or @xmath358 times smaller latency over the random - equal or the random - optimal scheme . similarly as in the zf case , the major gain comes from the optimal energy allocation . the difference is that the common spectral efficiency of the zf receiver increases logarithmically with @xmath169 , while that using the mrc receiver is saturated at high @xmath169 due to the uncanceled interference . so , when high @xmath169 is available , the zf receiver clearly outperforms the mrc receiver . the gain of the zf receiver over the mrc receiver is @xmath359 and it becomes @xmath360 at @xmath353db . 5 depicts the latency and spectral efficiency of the three schemes as a function of @xmath1 when @xmath352db and @xmath351 . when the zf receiver is employed , it is observed that at @xmath361 , the proposed uplink scheduling policy provides about @xmath362 or @xmath363 times smaller latency over the random - equal or the random - optimal scheme . the gain of the proposed one over the random - optimal scheme diminishes when the number of bs antennas becomes high because all users can be scheduled with sharing the same resource . similar trends can be observed when the mrc receiver is employed and the latency and common spectral efficiency of the two receivers become identical . now , discussions on the behavior of the proposed uplink scheduling policy are provided . 6 depicts the optimal scheduling groups , @xmath364 $ ] , and the optimal training length , @xmath273 , as a function of @xmath169 when @xmath350 , @xmath351 , and the zf receiver is employed . 6 shows that small - size scheduling groups are preferred at low @xmath169 , while large - size ones are preferred at high @xmath169 , because high array gain is required at low @xmath169 . in spite of the pathloss difference , the size of each optimal scheduling group is nearly identical . although a longer training period is required in high @xmath169 , it is less than or equal to @xmath365 , which is the half of each sub - frame . over a wide range of @xmath169 , @xmath273 is larger than or equal to @xmath366 for @xmath239 , which implies that orthogonal pilots can be used for not - so - large number of @xmath1 . also , small - size scheduling groups are preferred at low @xmath1 , while large - size ones are preferred at high @xmath1 because higher array gain is available . 7 illustrates the optimal scheduling groups , @xmath364 $ ] , and the optimal training length , @xmath273 , as a function of @xmath1 when @xmath352db , @xmath351 , and the zf receiver is employed . similar trends are observed as in fig . 6 for the size of each optimal scheduling group . on the other hand , the optimal training length first increases with @xmath1 but it becomes decreasing if @xmath1 increases further . interestingly , @xmath273 becomes smaller than @xmath366 for large @xmath1 , which implies that non - orthogonal pilots become beneficial . in such a case , very high array gain is available and the optimal uplink scheduling policy provides an efficient non - orthogonal multiple access among users so that a low - latency ultra - reliable communication can be provided . suppose that users are partitioned into @xmath367 for @xmath368 and figs . 8 ( a)-(c ) visualize the effect of the training length @xmath52 and the scheduling group size @xmath10 on the common spectral efficiency . note that the support of @xmath369 is partitioned into two regions separated by the line @xmath370 , because non - orthogonal pilots are used in @xmath155 , while only orthogonal pilots are used in @xmath152 . in figs . 8 ( a ) and ( b ) , it is seen that the maximum points are belongs to the orthogonal region and the choice of the training length does not affect much for low @xmath169 and not high @xmath1 . however , as can be seen in fig . 8 ( c ) , the optimal choice of @xmath52 does matter and it belongs to the non - orthogonal region at high @xmath1 . so far , the superiority of the proposed uplink scheduling policy is verified by using some numerical examples , which indicates that the behavior of the optimal uplink scheduling policy changes significantly according to the system parameters @xmath169 and @xmath1 . in order to provided better insight on this , an asymptotic analysis would be fruitful . it is interesting to consider the case that the total number of users , @xmath33 , and the number of bs antennas , @xmath1 are simultaneously large , but @xmath33 is far larger than @xmath1 . the product of each user s location ( or pathloss @xmath371 ) and the transmit energy constraint @xmath372 is considered as independent random process with a common distribution @xmath373 and we denote @xmath374 $ ] as the averages of the transmit and the receive energies , respectively .    the following theorem states the asymptotic behavior of the static uplink scheduling policy and its network latency . let @xmath375 denote a random variable with cdf @xmath373 . suppose that @xmath376 $ ] exists for @xmath377 given as in the bottom of this page . then , as @xmath378 , the followings hold :    * the asymptotically optimal pilot length , @xmath273 , and the optimal scheduling group size , @xmath379 , are respectively given by @xmath380 * and the network latency normalized by @xmath33 converges to @xmath381 where @xmath382    see appendix c.    theorem 4 implies that a set of equi - sized scheduling groups become asymptotically optimal as @xmath378 . so , it gives a hint to construct an easy way to implement an asymptotically optimal scheduling policy , which is outlined in algorithm 2 . the computational complexity of the proposed asymptotically optimal scheduling policy is just @xmath383 , mainly comes form the sorting operation .    since theorem 4 is involved with a complicated function @xmath377 , it is not easy to gain a good insight . for a further insight , we restrict the random variable @xmath210 with @xmath384 , which implies that almost all realizations of @xmath210 are scaled as @xmath0 . in addition , the number of bs antennas is assumed to be large , but is much smaller than @xmath33 . then , the following four scaling regimes according to the number of bs antennas , @xmath1 , and the average received energy constraint level @xmath0 , can be classified :    * @xmath2 and @xmath385 : sufficiently high @xmath0 and not - so - large @xmath1 , * @xmath2 and @xmath386 : sufficiently high @xmath0 and large @xmath1 , * @xmath4 and @xmath387 : sufficiently low @xmath0 and not - so - large @xmath1 , * @xmath4 and @xmath388 : sufficiently low @xmath0 and large @xmath1 .    then , the following theorem states the asymptotic behavior of the proposed optimal static uplink scheduling policy . suppose that @xmath389 . as @xmath378 and @xmath137 with @xmath390 , the followings hold . * the asymptotically optimal training length and the scheduling group size are respectively given by @xmath391 where @xmath108 is an arbitrary integer among @xmath392 , @xmath393 and @xmath394 denotes the lambert w function , known as the inverse function of @xmath395 @xcite . * the asymptotically optimal network latency is given as @xmath396    see appendix d.    from theorem 5 , some implications can be discussed as follows :    * _ * regime i ) * _ : here , the transmit energy constraint is sufficiently high and the number of bs antennas is not sufficiently high . so , this regime can be interpreted as the scenario that each bs equipped with a not - so - large number of antennas serves users with sufficient energy in a small - sized cell . in this case , the asymptotically optimal policy is to configure the half of each sub - frame as the training phase and to serve @xmath397 users simultaneously , which implies that orthogonal pilots are optimal . note that this result is consistent to those in previous literature , such as theorem 2 in @xcite and sec . v in @xcite . also , equal - energy allocation over all symbols , i.e. , @xmath398 is nearly optimal , but the network latency can not be reduced as the number of bs antennas increases because the growth rate of the bs antennas is too slow . in case @xmath399 , the target throughput can be increased as @xmath400 while keeping the latency requirement , which is also consistent to the classical point of view . * _ * regime ii ) * _ : the transmit energy constraint and the number of bs antennas are both sufficiently high . so , this regime can be interpreted as the scenario that each bs equipped with a very large number of antennas serves users with sufficient energy in a small - sized cell . in this case , it is asymptotically optimal to configure one - third of a sub - frame as the training phase and to serve @xmath401 users simultaneously for each sub - frame , which implies that non - orthogonal pilots become optimal . also , the optimal energy allocation is given as @xmath402 which implies that although @xmath403db higher energy per symbol is dedicated to each training symbol , but @xmath403db higher energy is allocated to the data transmission phase . in this regime , the network latency can be arbitrarily reduced by increasing the number of bs antennas . however , allowing more energy is not beneficial . in case @xmath399 , the target throughput can be increased as @xmath404 while keeping the latency requirement or the latency is reduced as @xmath405 while keeping the target throughput . * _ * regime iii ) * _ : here , the transmit energy constraint is quite tight and the number of bs antennas is not so high . this regime can be interpreted as the scenario that each bs equipped with a not - so - large number of antennas serves users with limited energy in a large - sized cell . in this case , it turns out that optimal pilots are non - orthogonal and the optimal energy allocation is @xmath406 which implies that the energy allocation is identical to that in regime i ) , i.e. , equal - energy allocation over all symbols becomes optimal if @xmath407 is selected as in region i ) . in case @xmath399 , in order to meet the latency requirement , the target throughput needs to be scaled as @xmath408 . * _ * regime iv ) * _ : the transmit energy constraint is quite tight , but the number of bs antennas is sufficiently high . this regime can be interpreted as the scenario that each bs equipped with a very large number of antennas serves users with limited transmit energy in a large - sized cell . in this case , optimal pilots are non - orthogonal with the same asymptotically optimal scheduling policy to that for regime iii ) , but with different optimal energy allocation is given as @xmath409 however , if @xmath410 is selected as in regime i ) , the equal - energy allocation becomes optimal . in case @xmath399 , in order to meet the latency requirement , the target throughput needs to be scaled as @xmath411 . it is worth noting that our analysis may be regarded as the results of the capacity - approaching receiver , even if we deal with simple linear receivers only because as @xmath412 , the lower - bound of the achievable rate of the zf or mrc receiver converges to the exact achievable rate , i.e. , @xmath413 and the achievable rate also converges to the capacity . the only required condition is @xmath414 , which is valid in regimes i ) and ii ) . however , this condition is not satisfied in regimes iii ) and iv ) , which implies that linear receivers become strictly sub - optimal if the network is operated in a limited energy regime .    from the above , it is shown that orthogonal pilots become optimal only in regime i ) , i.e. , only in a classical cellular system scenario , but non - orthogonal pilots become optimal in regimes ii)-iv ) , i.e. , in new scenarios for future cellular systems . to quantify the advantages of using non - orthogonal pilots , the asymptotically optimal network latency using orthogonal pilots only is given as follows . suppose that @xmath415 but the network does not allow non - orthogonal pilots ( i.e. , @xmath416 ) . as @xmath378 and @xmath137 with @xmath390 , the asymptotically optimal network latency is given as @xmath417    since non - orthogonal pilots are not allowed , the asymptotically optimal network latency is obtained by using @xmath418 and @xmath419 . as the number of bs antennas increases , the gain obtained by allowing non - orthogonal pilots becomes quite dramatic in regimes ii)-iv ) . in regimes iii ) and iv ) , @xmath1 times lower network latency can be achieved . in regime ii ) , the use of non - orthogonal pilots makes the network latency decrease sub - linearly rather than logarithmically with @xmath1 . thus , an important design guideline can be derived : _ for a latency - sensitive application , it would be better to serve more users by employing non - orthogonal pilots , which is quite dramatic in the case of being operated in a high energy regime with a high target throughput , and is still quite meaningful even in the case of being operated in a low energy regime_.    in order to prevent the misunderstanding on the above results , we would like to emphasize that each of users is assumed to have an independent transmit energy source for transmission in this paper . so , as the number of scheduled users increases , more energy is consumed at a given sub - frame . thus , in a low energy regime , i.e. , regime iii ) or iv ) , the gain coming from scheduling more users is larger than the channel estimation quality degradation , which is clearly different compared to the case in @xcite , where the total energy dedicated for a transmission is fixed regardless of the number of scheduled users . in this paper , the latency - optimal static uplink scheduling policy is provided and its network latency is analyzed in an uplink training - based lsas employing simple zf or mrc receiver . the optimal uplink scheduling problem considered in this paper is to minimize the network latency when each user is constrained with a target throughput and energy limit and the corresponding optimal solution provides the optimal scheduling groups with their own scheduling portions , the optimal energy allocation between the training and data phases for each user , and the optimal frame configuration for the training based lsas . the optimal energy allocation is derived in a simple close - form for a given scheduling group and the optimal scheduling groups are found to be comprised of users with similar received signal quality . then , a low - complexity uplink scheduling algorithm providing the exact optimal solution is proposed with polynomial - time complexity of @xmath420 . via numerical examples , it is shown that the proposed uplink scheduling algorithm can provide an optimal policy which can provide several times lower network latency at given throughput and energy constraints over the conventional non - optimized scheduling algorithms in realistic environments , which shows that the proposed work can be a key enabler for the oncoming 5 g communication networks . in addition , the proposed uplink scheduling policy and the corresponding optimal network latency are analyzed asymptotically to provide better insights on the system behavior . as the number of users increases , it is shown that the network latency , normalized by the number of total users , converges to a deterministic quantity and asymptotically optimal frame configuration and scheduling policy can be obtained , which gives a way to construct much simpler asymptotically optimal uplink scheduling policy with complexity of @xmath328 . further , four operating regimes are classified according to the growth or decay rate of the average received signal quality and the number of bs antennas . it turns out that orthogonal pilot sequences , widely used in current systems , are optimal only when the average received signal quality is sufficiently good and the number of bs antennas is not - so - large , i.e. , only in a conventional scenario . in other regimes representing new service and system scenarios , it turns out that non - orthogonal pilot sequences become optimal . by using non - orthogonal pilots , the network latency can be reduced by a factor of @xmath6 when the received signal quality is quite poor ( @xmath4 ) or by a factor of @xmath7 when the received signal quality is sufficiently good ( @xmath8 ) and the number of bs antennas is sufficiently large ( @xmath13 ) . thus , this work proves that , in order to minimize the network latency , it is better to schedule users more than the amount of available training resource by employing non - orthogonal pilots for a training - based lsas , which would be a critical guideline for designing 5 g cellular communication systems supporting massive mtc ( or iot ) with low energy or latency - sensitive ultra - reliable tactile internet services . for brevity , we consider the zf receiver only and drop the indices @xmath10 and @xmath52 . by taking the derivative of ( [ eq29 ] ) , we obtain @xmath421 since ( [ eq29 ] ) is non - negative and continuous on @xmath422 $ ] and @xmath423 , @xmath424 is obtained by finding a real root of the quadratic function @xmath425 as long as ( [ eq30 ] ) is feasible , i.e. , @xmath426 . first , show that @xmath424 is real . to do this , it is sufficient that @xmath427 after inserting values in table ii , ( [ eq48 ] ) becomes equivalent to @xmath428 which is always true since @xmath429 . now , show that @xmath430 . if @xmath431 , it is trivial so we omit . suppose that @xmath432 . then , the condition @xmath430 can be written as @xmath433 which is also always true since @xmath432 . similarly @xmath430 when @xmath434 . finally , show that @xmath435 . the case @xmath436 is again trivial so that we omit it . assume that @xmath437 . then , we have @xmath438 which holds since @xmath439 for any @xmath440 and inserting it in ( [ eq31 ] ) shows @xmath435 . similarly for @xmath441 , we have @xmath442 which completes the proof . without loss of generality , we assume @xmath443 and @xmath444 and rewrite the objective function as @xmath445 note that @xmath446 is a monotonically decreasing function of @xmath447 for @xmath448 and is independent to @xmath449 . since @xmath450 , in order to minimize @xmath451 , @xmath452 should be @xmath453 similarly , @xmath454 is successively determined once @xmath455 , @xmath456 , are determined , which concludes that the two properties in theorem 2 hold for all @xmath457 . before proving theorem 4 , some preliminary results about a quantile function are introduced . suppose that @xmath458 are i.i.d . real - valued random variables with cdf @xmath34 and the order statistics of @xmath458 are denoted by @xmath459 . for @xmath460 , the @xmath461th quantile of @xmath34 is defined as @xmath462 . correspondingly , the sample quantile is defined as the @xmath461th quantile of the empirical cdf @xmath463 with @xmath464 samples , @xmath465 , which can also be expressed as @xmath466 . let @xmath467 be i.i.d . random variables from a cdf @xmath34 satisfying @xmath468 for any @xmath469 . then , for every @xmath470 and @xmath471 , @xmath472 where @xmath473 and @xmath474 is a positive constant .    the proof is directly obtained by applying the dvoretzky - kiefer - wolfowitz inequality @xcite .    now , show the almost - sure convergence of @xmath475 as @xmath476 . let @xmath467 be i.i.d . random variables from a cdf @xmath34 . then , @xmath477 .    for any given @xmath478 and sufficiently large @xmath35 , we have @xmath479 which can be made arbitrarily small by increasing @xmath35 because @xmath480 is convergent . thus , @xmath481 as @xmath476 , which implies @xmath477 .    for a riemann - integrable function @xmath482 , we have @xmath483    from lemma 7 , as @xmath476 , @xmath484 where the last convergence comes from the definition of the riemann integral .    let @xmath485 be the @xmath305 samples of @xmath33 i.i.d random variables @xmath486 with @xmath487 . define @xmath488 and assume that @xmath489 as @xmath378 . then , @xmath490    since each @xmath491 and @xmath492 , @xmath493 as @xmath378 . from lemma 8 , we have @xmath494 since @xmath495 is continuous and each @xmath491 , small variation in the length of each interval ( @xmath496 vs. @xmath497 ) does not affect the convergence of the riemann integral so that @xmath498 which concludes the proof .      in this proof , we omit the index @xmath52 by assuming @xmath273 is used in the symbols @xmath499 , @xmath500 , @xmath501 , @xmath502 , and @xmath224 for simplicity . assume that @xmath503 and let @xmath504 be the sequence of positive finite integers such that @xmath505 and @xmath506 , @xmath239 , where @xmath507 for @xmath508 . we first prove that the optimal scheduling groups can be selected among equi - sized ones as @xmath378 . from ( [ eq23 ] ) and ( [ eq24 ] ) , we have @xmath509 where @xmath305 is the number of scheduling groups ( @xmath493 as @xmath378 ) and @xmath510 note that in ( [ eq29 ] ) , @xmath511 is given as @xmath512 since @xmath513 depends only on @xmath514 , denote @xmath515 by allowing some notational abuse . from lemma 9 as @xmath378 , @xmath516 which implies that the equi - sized scheduling group with size of @xmath379 can achieve the optimal network latency asymptotically . also from lemma 8 , the asymptotically optimal network latency can be expressed as @xmath517 first consider the case of @xmath518 and @xmath519 so that @xmath520 = 0 . then , @xmath521 is given as @xmath522 inserting ( [ eq56 ] ) into ( [ eq54 ] ) , we obtain @xmath523 by using ( [ eq55 ] ) , we obtain @xmath524 which concludes the proof for the case . the other cases can be shown in similar ways . for @xmath525 , we obtain @xmath529 using the equality @xmath530 simplifies the case @xmath152 , @xmath531 , and inserting @xmath532 into ( [ eq61 ] ) results in @xmath533 by which we arrive at ( [ eq59 ] ) . on the other hand , by using taylor expansion , @xmath534 for @xmath527 . then , for @xmath535 , we obtain @xmath536 to simplify the case @xmath537 , we use the identity @xmath538 by which we arrive at ( [ eq60 ] ) .      by using the chebyshev s inequality , we obtain , as @xmath476 @xmath542 } \right| > c\mathbb{e}[{x_n } ] } \right ) \le \frac{{{\rm{var}}\left ( { { x_n } } \right)}}{{{c^2}{{\left ( { \mathbb{e}[{x_n } ] } \right)}^2 } } } \to 0,\ ] ] where @xmath543 is a finite and positive constant , which implies that the realizations of @xmath544 is included in the set @xmath545\le x_n\le(1+c)\mathbb{e}[x_n]\}$ ] almost - surely . thus , the minimum and maximum in @xmath546 is scaled as @xmath547 $ ] almost - surely so that its transformation , i.e. , @xmath548 is also scaled as @xmath549)$ ] almost - surely by continuous mapping theorem @xcite , which completes the proof . _ regime i ) _ : consider the case @xmath152 and let @xmath550 . we obtain @xmath551 & = \theta \left ( { \frac{1}{{\log \left ( { 1 + a{\rho } } \right ) } } } \right)\\ & = \theta \left ( { \frac{1}{{\log { \rho } } } } \right),\label{eq63}\end{aligned}\ ] ] where the first equality comes from lemma 11 and the last equality comes from @xmath552 . so , @xmath553 $ ] becomes independent to @xmath10 and @xmath52 asymptotically . by using theorem 5 and ( [ eq63 ] ) , we obtain @xmath554}{{k\left ( { 1 - \frac{l}{n } } \right ) } }   \\&=\mathop { \arg \max } \limits_{k \le l } k\left ( { 1 - \frac{l}{n } } \right ) \nonumber \\&= \left(\frac{n}{2},\frac{n}{2}\right),\label{eq64}\end{aligned}\ ] ] and by inserting ( [ eq64 ] ) into ( [ eq43 ] ) , we obtain @xmath555\\ & = \theta \left ( \frac{\mathcal{t}_{\rm{th}}}{w{\log \rho } } \right),\label{eq65}\end{aligned}\ ] ] where the last equality comes from lemma 11 and @xmath556 . now , consider the case @xmath557 . by using lemma 10 , we can rewrite ( [ eq41 ] ) as @xmath558 where @xmath559 if @xmath1 is finite , @xmath10 is also finite so that @xmath560 is maximized at @xmath561 and @xmath562 . inserting it into ( [ eq43 ] ) , we obtain @xmath563 comparing ( [ eq67 ] ) with ( [ eq65 ] ) indicates that the asymptotically lower network latency can be achieved when @xmath141 . now , consider the case when @xmath564 . for a fixed @xmath52 and @xmath564 , @xmath565 can be approximated as @xmath566 which is maximized at @xmath567 for some positive @xmath568 . ignoring the non - dominant term @xmath569 , inserting @xmath570 , and replacing @xmath52 with @xmath571 $ ] in ( [ eq68 ] ) yields @xmath572 by which we obtain optimal @xmath573 instead of using ( [ eq68 ] ) . since ( [ eq69 ] ) is unimodal , it is sufficient to find the point @xmath574 satisfying @xmath575 and @xmath576 , which are respectively given as @xmath577 subtracting ( [ eq70 ] ) from ( [ eq71 ] ) results in @xmath578 and @xmath579 should satisfy @xmath580 for @xmath581 . after some mathematical manipulations , we arrive at @xmath582 from which we have @xmath583 and thus @xmath584 by introducing the lambert w function . by inserting @xmath585 in ( [ eq43 ] ) , we obtain @xmath586 combining ( [ eq65 ] ) and ( [ eq72 ] ) , we obtain @xmath587 . _ regime ii ) _ : consider the case @xmath588 . then , we obtain both ( [ eq65 ] ) and ( [ eq72 ] ) . now , consider the case @xmath589 . for @xmath155 , we still obtain ( [ eq72 ] ) . however , for @xmath152 , @xmath590 so that @xmath591 = \theta(\frac{1}{\log m})$ ] . although ( [ eq63 ] ) still holds , ( [ eq65 ] ) becomes @xmath592 , which , together with ( [ eq72 ] ) , results in @xmath593 . _ regime iii ) and iv ) _ : from lemma 10 , ( [ eq41 ] ) can be rewritten as @xmath594}}{{k\left ( { 1 - \frac{l}{n } } \right)}}\\   & = \mathop{\arg\min}\limits_{{\scriptstyle 1\le l < n,\atop \scriptstyle 1\le k\le m } } \frac{{4n{{\log } _ 2}e}}{{k\left ( { m - k } \right)}}\mathbb{e}\left [ { { x^ { - 2 } } } \right],\label{eq73}\end{aligned}\ ] ] where the second equality comes from that @xmath595 for small @xmath20 . finally , @xmath596 and @xmath273 can be selected arbitrarily among integers between @xmath266 and @xmath597 . inserting @xmath598 , we obtain @xmath599 , which competes the proof .    1 g. p. fettweis , `` the tactile internet : applications and challenges , '' _ ieee veh . technol . mag . _ , vol . 9 , no . 1 , pp . 6470 , mar . t. taleb and a. kunz , `` machine type communications in 3gpp networks : potential , challenges , solutions , '' _ ieee commun . mag . 3 , pp . 178184 , mar . l. atzori , a. iera , and g. morabito , `` the internet of things : a survey , '' _ comput . 15 , pp . 27872805 , oct . j. c. ikuno , m. wrulick , and m. rupp , `` system - level simulation of lte networks , '' _ in proc . ieee vtc  spring _ , taipei , taiwan , may 2010 , pp . 15 . g. piro , a. grieco , g. boggia , f. capozzi , and p. camarda , `` simulating lte cellular systems : an open source framework , '' _ ieee trans . veh . 498513 , feb . 2011 . t. l. marzetta , `` noncooperative cellular wireless with unlimited numbers of base station antennas , '' _ ieee trans . wireless commun . _ , vol . 9 , no . 11 , pp . 35903600 , nov . f. rusek , d. persson , b. k. lau , e. g. larsson , t. l. marzetta , o. edfors , and f. tufvesson , `` scaling up mimo : opportunities and challenges with very large arrays ,  _ ieee signal process . mag . _ , vol . 30 , no . 40 - 46 , jan . 2013 . n. q. ngo , e. g. larsson , and t. l. marzetta , ' ' energy and spectral efficiency of very large multiuser mimo systems , `` _ ieee trans . 4 , pp . 14361449 , apr . 2012 . a. muller , a. kammoun , e. bjornson , and m. debbah , ' ' linear precoding based on polynomial expansion : large - scale multi - cell mimo systems , `` _ ieee j. sel . topics signal process . _ , vol . 8 , no . 5 , pp . 861875 , oct . 2014 . l. zheng and d. tse , ' ' communication on the grassmann manifold : a geometric approach to the noncoherent multiple - antenna channel " , _ ieee trans . inf . theory _ , 2 , pp . 359383 , feb . 2002 . b. hassibi and b. m. hochwald , `` how much training is needed in multiple - antenna wireless links ? '' _ ieee trans . inf . theory _ , vol . 4 , pp . 951963 , apr . 2003 . m. kobayashi , n. jindal , and g. caire , `` training and feedback optimization for multiuser mimo downlink , '' _ ieee trans . 59 , no . 8 , pp . 22282240 , aug . s. murugesan , e. u .- biyikoglu , and p. schniter , `` optimization of training and scheduling in the non - coherent simo multiple access channel , '' _ ieee j. sel . areas commun . 25 , no . 7 , pp . 1441456 , sep . 2007 . s. sun , and y. jing , `` channel training design in amplify - and - forward mimo relay networks , '' _ ieee trans . wireless commun . 10 , pp . 33803391 , oct . o. ayach , a. lozano , r. w. heath , `` on the overhead of interference alignment : training , feedback , and cooperation , '' _ ieee trans . wireless commun . _ , vol . 11 , no . 11 , pp . 41924203 , nov . i. bettesh and s. shamai , `` optimal energy and rate control for minimal average delay : the single - user case , '' _ ieee trans . inf . theory _ 9 , pp . 41154141 , sep . 2006 . v. k. n. lau and c. ying , `` delay - optimal energy and subcarrier allocation for ofdma systems via stochastic approximation , '' _ ieee trans . wireless commun . _ , vol . 9 , no . 1 , pp . 227233 , jan . d. rajan , a. sabharwal , and b. aazhang , `` delay - bounded packet scheduling of bursty traffic over wireless channels , '' _ ieee trans . inf . theory _ , 1 , pp . 125144 , jan . h. wang , w. zhang , y. liu , q. xu , and p. pan , `` on design of non - orthogonal pilot signals for a multi - cell massive mimo system , '' _ ieee wireless commun . 2 , pp . 129132 , apr . shen , j. zhang , and k. b. letaief , `` downlink user capacity of massive mimo unser pilot contamination , '' _ ieee trans . wireless commun . 14 , no . 6 , pp . 31833193 , jun . 2015 . p. banelli , s. buzzi , g. colavolpe , a. modenini , f. rusek , and a. ugolini , `` modulation formats and waveforms for 5 g networks : who will be the heir of ofdm ? : an overview of alternative modulation schemes for improved spectral efficiency , '' _ ieee signal process . mag . 6 , pp . 8093 , nov 2014 . b. m. hochwald , t. l. marzetta , and v. tarokh , `` multiple - antenna channel hargening and its implications for rate feedback and scheduling , '' _ ieee trans . theory , _ vol . 9 , pp . 18931909 , sep . 2004 . b. m. hochwald , t. l. marzetta , t. j. richardson , w. sweldens , and r. urbanke , `` systematic design of unitary space - time constellation , '' _ ieee trans . inf . theory _ , 19621973 , sep . p. xia , s. zhou , and g. b. giannakis , `` achieving the welch bound with difference sets , '' _ ieee trans . inf . theory _ , 5 , pp . 19001907 , may 2005 . d. calabuig , r. h. gohary , and h. yanikomeroglu , `` optimum transmission through the multiple - antenna gaussian multiple access channel , '' _ ieee trans . inf . theory _ , 1 , pp . 230243 , jan .
in this paper , an uplink scheduling policy problem to minimize the network latency , defined as the air - time to serve all of users with a quality - of - service ( qos ) , under an energy constraint is considered in a training - based large - scale antenna systems ( lsas ) employing a simple linear receiver . 
 an optimal algorithm providing the exact latency - optimal uplink scheduling policy is proposed with a polynomial - time complexity . via numerical simulations 
 , it is shown that the proposed scheduling policy can provide several times lower network latency over the conventional ones in realistic environments . 
 in addition , the proposed scheduling policy and its network latency are analyzed asymptotically to provide better insights on the system behavior . 
 four operating regimes are classified according to the average received signal quality , @xmath0 , and the number of bs antennas , @xmath1 . 
 it turns out that orthogonal pilots are optimal only in the regime @xmath2 and @xmath3 . in other regimes ( @xmath4 or @xmath5 ) , it turns out that non - orthogonal pilots become optimal . 
 more rigorously , the use of non - orthogonal pilots can reduce the network latency by a factor of @xmath6 when @xmath4 or by a factor of @xmath7 when @xmath8 and @xmath9 , which would be a critical guideline for designing 5 g future cellular systems .    large - scale antenna system , training - based transmission , network latency minimization , uplink scheduling policy , non - orthogonal pilots