Proportional Fair Multiuser Scheduling in LTE

IEEE SIGNAL PROCESSING LETTERS, VOL. 16, NO. 6, JUNE 2009

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Proportional Fair Multiuser Scheduling in LTE Raymond Raymond Kwan, Kwan, Cyril Leung, Leung, and and Jie Zhang Zhang

 Abstract—The challenge of scheduling user transmissions on the downlink downlink of a Long Term Evolution Evolution (LTE) (LTE) cellular cellular communi communicatio cation n system is addressed. A maximum rate algorithm which does not consider fairness among users was proposed in [1]. Here, a multiuser scheduler with proportional fairness (PF) is proposed. Numerical results show that the proposed PF scheduler provides a superior fairness performance with a modest loss in throughput, as long as the user average SINRs are fairly uniform. A suboptimal PF scheduler is also proposed, which has a much lower complexity at the cost of some throughput degradation.  Index Terms—Digital Digital communicat communication, ion, fading fading channels, channels, radio communication, scheduling.

block (SB). Each RB consists of 12 adjacent subcarriers with an inter subcarrier spacing of 15 kHz. AnSB has has a dura durati tionof onof 1 msandconsi msandconsist stss of (typ (typic ical ally12 ly12 or 14) 14) OFDM symbols. symbols. Let Let be the the total total number number of subcarriers subcarriers and and be the the numbe umberr of data data-c -car arry ryin ing g subca ubcarr rrie iers rs for for symbol , where . Also, let be the code rate rate associ associate ated d with with Modula Modulatio tion n and Coding Coding Scheme Scheme (MCS) (MCS) be the the conste constella llatio tion n size size of the the MCS MCS and be the the OFDM OFDM symbol symbol durat duration ion.. Then, Then, the the bit rate, rate, , that that corresponds to a single SB is given by (1)

I. INTRODUCTION SCHEDULIN SCHEDULING G algorithm algorithm for maximizing maximizing the throughput throughput on the downlink of a multiuser Long Term Evolution (LTE) cellular communication system is studied in [1]. However, the issue of fairness among users was not addressed. In this paper, we propose and compare two scheduling algorithms: 1) a maximum-rat maximum-ratee scheduler scheduler which has similar similar throughput throughput performance to the algorithm in [1] but is simpler to implement 2) a proportional-rate scheduler intended to improve fairness among users. This paper is not intended to provide comprehensive system-level simulation results. Rather, it examines some scheduling schemes, focussing on how the physical resource blocks are assigned. The results show that the PF scheduler is effective in reducing variations in user bit rates with little average bit rate degradation as long as user average SINRs are fairly uniform.

A

II. II. SYSTEM M ODEL The LTE system model adopted is the one described in [1]. The main aspects are summarized as follows. The modulation scheme scheme is orthogonal orthogonal frequency frequency division division multiplexin multiplexing g (OFDM). (OFDM). In order to reduce signalling overhead, subcarriers are grouped into resource blocks (RBs) [2], [3]. The scheduler allocates resources to users in quanta of two consecutive RBs; for convenience, we will refer to the two consecutive consecutive RBs as a scheduling scheduling

Let Let be the the numb number er of simu simult ltan aneo eous us user users, s, and and be the the total number of SBs that are available during each Transmission Time Interv Interval al (TTI (TTI). ). In In addit addition ion,, let let be a subset subset of the the SBs whose whose channe channell qualit quality y index index (CQI) (CQI) value valuess are to be report reported ed back back by user user ; the the size size of is deno denote ted d by and and dete determ rmin ines es the the feed feedba back ck over overhe head ad.. It is assu assume med d that that the the high highes estt SB CQI CQI values are fed back. Let be a real scalar or vector sent back back by user user to indicat indicatee the collecti collective ve channe channell qualitie qualitiess of  all the subcarr subcarriers iers within within the th reported reported SB. Furtherm Furthermore, ore, let be the index of the highest-rate MCS MCS that that can can be be supp suppor orte ted d by use userr for for the the -th -th SB at at CQI CQI value , i.e., . For convenience, we assume that the MCS rate increa increases ses monot monotoni onical cally ly with with and that that the rate rate of MCS 1 is zero. SBs whose CQI values are not reported back are assigned to MCS 1. An  important  constraint   constraint in LTE downlink (non-MIMO configuration) scheduling is that all SBs allocated to a given user in any given scheduling period have to use the same MCS. If MCS is to be used for user user , then only certain certain SBs can can be assigned assigned to the the user user.. For For examp xample le,, supp suppos osee , and and

(2) Manuscript received October 12, 2008; revised January 26, 2009. Current version published April 24, 2009. This work was supported in part by the Natural Sciences and Engineering Research Council (NSERC) of Canada under Grant OGP0001731, by the UBC PMC-Sierra Professorship in Networking and Communi Communicatio cations, ns, and by a Marie Curie Curie Internatio International nal Incoming Incoming Fellowshi Fellowship. p. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Markku Renfors. R. Kwan and J. Zhang are with the University of Bedfordshire, Luton LU1 3JU, U.K. (e-mail: [email protected]; [email protected]; [email protected]). C. Leung is with the Department of Electrical and Computer Engineering, Univers University ity of British British Columbia Columbia,, Vancouve ancouver, r, BC V6T 1Z4 Canada Canada (e-mail: (e-mail: [email protected]). Digital Object Identifier 10.1109/LSP.2009.2016449 10.1109/LSP.2009.2016449

Then, if MCS is used, only SBs 3 and 5 can can be allo alloca cate ted d to user user sinc sincee only only thes thesee SBs SBs havegood havegood enou enough gh channel qualities to support an MCS index of or high higher er.. Sele Select ctin ing g SBs SBs 1 or 2 with with MCS MCS would result in unacceptably high error rates for these SBs. On the other hand, if , all 4 SBs can be selected, at the expense of a lower bit rate for SBs 1, 3, and 5. Thus, there is an optimal optimal value value of which maximizes maximizes the the total total bit rate rate for user user .

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IEEE SIGNAL PROCESSING LETTERS, VOL. 16, NO. 6, JUNE 2009

III. SCHEDULING A LGORITHMS In this section, a number of approaches to designing schedulers for the LTE downlink are discussed.

rate indices, one for each SB for user at time . The users are then ranked according to their priority index values,

 A. Single User Optimization

In single user optimization, the aim is to determine the MCS (rate) index, and the set of SBs to be allocated to user so as to maximize the assigned bit rate, , given the set of channel qualities . Let , and let be the MCS vector for user , where

The optimal , which maximizes the total bit rate for user , is obtained by solving Problem . (3) subject to (4) (5) The formulation in (3) allows the selected bit rate for SB to be less than what can potentially support, as may be the case if user is assigned more than one SB during a TTI. Constraint (4) ensures that the MCS for user can only take on a single value between 1 and . The above optimization problem can be easily solved as follows. Let be an matrix with -th element . Denote the sum of the elements in the -th column of by (6)

(9) where is the average bit rate up to time , and is the bit rate assigned to user at time . The first line in the RHS of (9) corresponds to proportional fair (PR) scheduling ([2, p. 113]), [4],[5], whereas the second line corresponds to maximum rate scheduling ([2, p. 111]). The term is a function which returns the highest bit rate that user can support based on , as discussed in Section III-A, i.e., . For notational convenience, let be the ranked version of , and be a function which maps the ordered user index back to the original user index . In the second stage, the allocation of resources is done in a sequential fashion, one user at a time, according to the following user order: . Thus, starting with user , and the initial set of SBs, , where corresponds to the complete set of available SBs, the MCS index and the set of SBs, , are determined as described in Section III-A, and assigned to user . The remaining SBs, , are then made available to user . The resource allocation process continues until all SBs have been assigned. C. Joint Optimization

To assess the effectiveness of the sequential scheduling algorithm in Section III.B, we now consider the joint optimization of allocation of BSs and MCSs among all users. The joint optimization problem can be formulated as (10) subject to (4) and

Then the optimal MCS for user is

(11) (7)

and the corresponding maximum bit rate is SBs allocated to user is given by

. The set,

(12)

, of  The term

is given by (13)

(8)

 B. Multiuser Sequential Suboptimal Scheduling

The proposed suboptimal multiuser scheduler consists of  two stages. In the first stage, the scheduler determines the set, , of maximum

It is known that the PF scheduler is asymptotically optimal [4], [6]. Other multi-user schedulers have been proposed in [7], [8]. In problem , and is a binary decision variable, with value 1 if SB is assigned to user and 0 otherwise. Problem is

KWAN et al. : PROPORTIONAL FAIR MULTIUSER SCHEDULING IN LTE

Fig. 1. Average total bit rate as a function of  SINRs of 14 dB, 15 dB, and 16 dB.

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for three users with average

Fig. 2. Fairness index as a function of  for three users with average SINRs of  14 dB, 15 dB, and 16 dB.

nonlinear due to the product in (10). Although solutions can be obtained using optimization techniques such as Branch-and-Bound [9], global optimality cannot be guaranteed. To overcome this difficulty, Problem can be transformed into an   equivalent  linear problem by introducing an auxiliary variable , i.e., (14) subject to (4), (11), (12) and (15) (16) (17) where is a large positive real value. Problem can then be solved using standard integer linear programming techniques [9]. IV. NUMERICAL R ESULTS For illustration purposes, we assume SBs per TTI, subcarriers per SB, and that the normal cyclic prefix configuration is used [2]. The fading amplitude for each subcarrier of any user follows the Nakagami-m model [10], with a fading figure equal to 1. It is assumed that the signal-to-interference plus noise ratios (SINRs) for all subcarriers of any user are correlated, but identically distributed (c.i.d.), and that the resource blocks follow the localized configuration [2]. The correlation coefficient between a pair of subcarriers is given by , where and are the subcarrier indices. For simplicity, the SINR of a given subcarrier is assumed to be independent at every scheduling period, and constant within a scheduling period. This independent assumption is reasonable for the purpose of comparing the long-term fairness for the Max-Rate and PF schedulers. The set of MCSs consists of  QPSK 1/2 and 3/4, 16-QAM 1/2 and 3/4, as well as 64-QAM 3/4 [11], and the L1/L2 control channels are mapped to the first

Fig. 3. Average total bit rate as a function of  SINRs of 10 dB, 15 dB, and 20 dB.

for three users with average

OFDM symbol within each subframe. Furthermore, each subframe consists of eight reference symbols [2]. The feedback  method is based on the Exponential Effective SINR Mapping (EESM) [12], with parameter values obtained from [13]. Let be the total bit rate at time , and be the corresponding value averaged over channel realizations. Similarly, let be the average bit rate for user , and be the Jain’s fairness index [14] for the average user bit rates. The value of lies in the range ; an value of 1 corresponds to all users having the same average (over scheduling periods) bit rates. Figs. 1 and 2 show the average total bit rate, , and fairness index, , as a function of for three users, with average user SINRs of 14 dB, 15 dB, and 16 dB. It can be seen from Fig. 1 that the bit rates for all schedulers increase with . This can be explained as follows. The motivation behind EESM is to map a set of subcarrier SINRs, , to a single effective SINR, , in such a way that the block error probability (BLEP) due to can be well approximated by that at in additive white Gaussian noise (AWGN) [11], [12]. The value of tends to be skewed towards the weaker subcarriers in order to maintain an acceptable BLEP. At a low value of , subcarriers with large

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Fig. 4. Fairness index as a function of  for three users with average SINRs of  10 dB, 15 dB, and 20 dB.

IEEE SIGNAL PROCESSING LETTERS, VOL. 16, NO. 6, JUNE 2009

Fig. 6. Fairness index as a function of the number of users with average SINRs of 7 dB for all users.

and

a slightly lower throughput than the sequential Max-rate scheduler but a higher fairness index. V. CONCLUSION The bit rate and fairness characteristics of a Max-Rate and a PF scheduler were studied. A jointly optimal as well as a simpler, suboptimal problem formulations were considered. It was found that the PF scheduler is effective in reducing variations in user bit rates with little average bit rate degradation relative to the Max-Rate scheduler as long as user average SINRs are fairly uniform. REFERENCES

Fig. 5. Average total bit rate as a function of the number of users with and average SINRs of 7 dB for all users.

SINRs are not effectively utilized, leading to a relatively poor performance. It can also be seen that the bit rate for the jointly optimal PF scheduler is almost as good as that for the jointly optimal Max-Rate scheduler. In comparison, the bit rates for the sequential Max-Rate and PF schedulers are about 5% and 10% lower. Fig. 2 shows that the fairness index, , is significantly higher for the two PF schedulers than for their Max-Rate counterparts, indicating that the PF schedulers are quite effective in promoting fairness among users. Similar plots are shown in Figs. 3 and 4, for user average SINRs of 10 dB, 15 dB, and 20 dB respectively. Here, the variation among user average SINRs is larger than in Figs. 1 and 2. Fig. 3 shows that there is now a larger gap between the bit rates for the jointly optimal PF and Max-Rate schedulers. This is due to the increased effort needed to maintain fairness. It can be seen from Fig. 4 that the two PF schedulers provide significantly better user fairness than the Max-Rate schedulers. The average total bit rate and fairness index are plotted as a function of the number of users in Figs. 5 and 6, respectively, with an average SINR of 7 dB for all users. In this case, the results show that the jointly optimizedMax-rate andPF schedulers provide similar performances. The sequential PF scheduler has

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