Efficient and Fair Scheduling of Uplink and Downlink in IEEE OFDMA Networks

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1 Efficient and Fair Scheduling of Uplink and Downlink in IEEE OFDMA Networks Vandana Singh Vinod Sharma Department of Electrical Communication Engineering Indian Institute of Science Bangalore, India {vandana, Abstract We develop new scheduling algorithms for the IEEE d OFDMA/TDD based broadband wireless access system, in which radio resources of both time and frequency slots are dynamically shared by all users. Our objective is to provide a fair and efficient allocation to all the users to satisfy their quality of service. Keywords : OFDMA, fair scheduling, IEEE d, resource allocation, QoS. I. INTRODUCTION BROADBAND WIRELESS ACCESS (BWA) is a very flexible and easily deployable high-speed communication system. As against it, wireline broadband access technologies such as copper line, coaxial cable, xdsl and cable modem are very costly and time consuming to deploy [4], [5]. IEEE group aims to unify the BWA solutions [14]. A technical overview of this system is provided in [1]. IEEE group has developed standards in GHZ band for LOS (line of sight) and in 2-11 GHZ for non- LOS communication. In NLOS communication there can be significant multipath fading and path loss which is mitigated via OFDMA. By a good allocation of slots and carriers, exploiting channel diversity, one can significantly improve the system performance. In the standard the scheduling has been left unspecified. This paper addresses this problem. In IEEE , the BS (Base Station) centrally allocates the channels in different slots to different SSs (Subscriber Stations) for uplink and downlink which in turn allocates these resources to the various connections they are supporting at that time. The BS is aware of the channel state of all subchannels for all the SSs. Thus the BS can exploit channeluser diversity by allocating different subchannels to different SSs. The overall system throughput can be maximized by allocating a subchannel to the SS with the best channel state. However, this may not satisfy the QoS (Quality of Service) requirements of different SSs. At a time an SS may be supporting various real and nonreal time applications.the QoS of these users can be satisfied by ensuring certain minimal overall rate to each SS in every frame. If there are insufficient resources to satisfy the QoS of all the users at each SS, then there is the issue of satisfying their requirements in a fair way. We develop scheduling algorithms that provide QoS to different SSs efficiently in a fair way. 984 The problem of slot/carrier allocation for OFDM/OFDMA systems has also been considered in [2], [3], [7], [11]. [7] and [11] consider only the problem of scheduling for non realtime users with minimum rate requirements. [3] considers all the service classes in but considers the GPC (Grant Per Connection) as against GPSS (Grant Per Subscriber Station). GPSS is scalable to a large number of SSs and connections and is the only mode considered recently. The algorithm developed in [3] can not be directly used in GPSS. [2] develops algorithms that minimize total transmit power while providing the required rates to SSs. Power control, though important, is not being considered seriously in the standard at present. The transmit power per frequency band is kept constant. Given the channel states for different stations, the modulation and the coding scheme (and hence rate) are decided by the BS. Routing and scheduling algorithms in mesh mode of IEEE are provided in [16] and [17]. Because of mesh mode (multihop) and because OFDM and not OFDMA is used in the mesh mode, the algorithms and problems in [16] and [17] are quite different from those in the present paper. The remainder of this paper is organised as follows. In Section II, we present our system model and formulate the problem. Section III presents efficient algorithms which provide optimal overall throughput. These algorithms may not be fair to different users. In Section IV we develop fair versions of these algorithms. Section V provides algorithms which an SS can use for a fair allocation of resources to its different users, taking into account their QoS. Concluding remarks and directions for future work are given in section VI. II. SYSTEM OVERVIEW AND PROBLEM FORMULATION A. System Overview /06/$20.00 (c)2006 IEEE We study the IEEE PMP (Point to Multipoint) system where the downlink operates on an PMP basis. The wireless link operates with one central BS and many SSs. In the downlink direction, the BS is the only transmitter and thus need not coordinate with other stations, except for the overall TDD (time division duplex) that divides a frame into uplink and downlink transmission periods. An SS can be an AP (Access Point) of a WLAN in a building. There can

2 be many end users of the network at an SS. We consider only stationary SSs as specified in d (mobility is being supported in e) supports four service classes, described in the next subsection. For each service class, it has different bandwidth request and grant mechanisms. A connection is associated with any one of the service classes depending upon its QoS requirements. An SS requests the BS for bandwidth on a per flow basis for uplink. The BS grants a total bandwidth for all the connections, belonging to that SS. Then the SS redistributes this sum-total grant among its users according to the service class of the user s connection and its QoS requirements. This allocation scheme is known as GPSS (Grant per SS). B. Physical Layer The standard supports three different physical layers: 1) Single Carrier, 2) OFDM/TDMA and 3) OFDMA. We limit ourselves to the third one. This OFDMA/TDD system has a total bandwidth of 6 MHz. For an FFT size of 2048, 1536 data carriers (called subcarriers) are available in the downstream. These are divided into 48 groups for downstream transmission. For upstream transmission there are 53 groups. In Adjuscent subcarrier permutation (ASP) adjuscent groups of subcarriers (number of subcarriers can vary) are taken together to form a subchannel. Thus a subchannel is formed of subcarriers in adjuscent frequency bands. As against this, in Distributed subcarrier Permutation (DSP), the subcarriers in a subchannel are widely apart in the frequency spectrum. A total of 32 subchannels can be formed. The time axis is divided into frames which is further divided into slots. The BS can allocate a channel-slot pair to any SS which in turn allocates to its users (in uplink). The allocation is done for a few frames at a time and takes into account the channel states and demands of different SSs and users. In ASP, the channel states of different subcarriers can be substantially different even for a single SS while in DSP they will be quite similar although will vary from frame to frame. The channel states of the same subchannel can be generally different for different users in the same frame. The BS is aware of different SSs at a time and also of their throughput requirements via the bandwidth request mechanism. In the following we develop algorithms for channel-slot assignments that the BS can use for fair and efficient allocation. We will consider ASP channel formation. This is being emphasised in the standard. The resource allocation problem for DSP is simpler and a special case of the ASP. The algorithms for this case can be developed by simple modification of the following algorithms. C. QoS Service flows in A distinctive feature of is its QoS support. It has 4 service classes to support real time and non-real time communication: UGS (Unsolicited grant service) Flows: UGS is designed to support real-time service flows that generate fixed size data packets on a periodic basis, such as Voice over IP. 985 Real-Time Polling Service (rtps) Flows: rtps is designed to support real-time service flows that generate variable size data packets on a periodic basis, such as MPEG video. Non-real-time Polling Service (nrtps): nrtps is designed to support delay-tolerant data streams consisting of variablesized data packets for which a minimum data rate is required, such as FTP or HTTP (web browsing). Best Effort (BE): BE service is designed to support data streams for which no minimum service level is required and therefore may be handled on a space-available basis. D. Scheduling Policies For each frame depending upon the requirements of different SSs and their channel conditions, the BS schedules the channels and slots to different SSs (although the BS may do scheduling for several frames at a time, for simplicity we will call it one frame). Although, the BS is aware of the class-wise requirement of each SS, it gives SSs the freedom to use those channels/slots in their own way. In the following we propose algorithms which a BS can use to allocate the channels/slots in the uplink and the downlink and subsequently an SS can use for allocation to different users it has, for uplink. Because the channel states and SS requirements for rates will change with time, the BS will need to run these algorithms after ever (say M) frames. For overall system optimization, channel allocation for upstream and downstream should be done jointly. However because of TDD between upstream and downstream traffic, we need to consider them separately. (Of course the boundary between the uplink and the downlink allocation will be flexible and our algorithms will decide this). We will consider the following general approach in our scheduling. Since the real time applications are delay sensitive, we will assume that the requirements of real-time applications (UGS and rtps) need to be met in the same frame while the transmission of packets of non real-time data can be defered. Thus we propose that, given the requirements of SSs for all the classes, the BS will first try to satisfy the needs of the UGS applications in the upstream and downstream. The algorithms for this will be provided below. Next the requirements of the rtps in both directions will be satisfied. After that the minimum requirements of the nrtps will be satisfied and finally the requirements of nrtps without minimum rate requirement and Best Effort traffic will be taken care of. Once the BS does this calculation, it will divide the frame (since there is TDD) into upstream and downstream. Then the overall channel/slot allotment for each SS for upstream is broadcast to them in the beginning of the frame. When an SS knows its upstream channel/slot allocation in a frame, it uses the algorithms developed below to allocate the channels/slots to different users in its local area network (say within its building). In the rest of the section we formulate an optimization problem for allocation of resources to different SSs. Algorithms for its solution are provided in sections III and IV. In section V we will finally develop the complete algorithm (using sections III and IV), which the BS and an SS can use

3 (based on the general approach mentioned above) so as to provide QoS to different end users in a fair manner. E. Problem Formulation Suppose there are n SSs and m subchannels with a BS. Let subscriber station SS i have a total demand of λ i bytes for its UGS connection(s) for upstream in a given frame. Let for SS i, the rate on channel j be R ij bytes/slot in the frame. If the total number of slots alloted to SS i in this frame on channel j is N ij, the unsatisfied demand after this frame for SS i will be + C i = λ i R ij N ij (1) where (x) + = max(0,x). Our objective is to minimize n C i subject to N ij N j (2) for j =1...m, where N ij s are non-negative integers and N j is the total number of slots of subchannel j available for data transmission. The objective function (1) is nonlinear. Thus we reformulate this problem as minimization of C = C i = λ i R ij N ij (3) subject to (2) and R ij N ij λ i (4) for i = 1...n. Now the objective function (3) is linear although we have included extra linear constraints (4). In the next section we develop algorithms to solve this optimization problem. These algorithms will be used by the BS to allocate channels/slots to different SSs in every (say M) frame. Thus the computational complexity will also be an issue. We will compare the different algorithms for their computational complexity. The complexity of these algorithms is much less than directly solving the nonlinear problem 1-2. The standard specifies that the demand by the SSs and the channel states of different channels is available at the BS and hence the solutions of above optimization problem provided below can be used by the BS (and SSs) for channels/slots allocation. However, we can also consider the scenario when the demand of each SS and the channel rates are not known to the BS at the beginning of the frame. Then the BS can use the statistics of these quantities, if that is available. Suppose the demand of SS i is a random number x i and the channel rates are r ij s. Then our objective is to minimize 986 the cost function E x i N ij r ij (5) after each frame, subject to N ij E [r ij ] E [x ij ], and N ij N j N ij 0 i =1...n, j =1...m. The first constraint here is a relaxation of the actual constraint and hence in general, the solution can provide more or fewer channel/slot to a user than needed. The algorithms needed for this problem will be same as for (2)-(4). One advantage of this formulation is that one need not run it in every frame; only when the channel state and/or the traffic statistics change. Of course the performance of this algorithm will be in general worse than that for (2)-(4). III. ALGORITHMS FOR THROUGHPUT OPTIMIZATION In this section we provide algorithms for the optimization problem (2)-(4). A. Linear Programming The cost function and the constraints of the problem (2)- (4) are linear. However the solution N ij needs to have integer values. Thus to get an optimal solution we need to use Integer programming (IP). But the complexity of IP is very high and hence we need to develop a less complex algorithm which can be used by the BS in real time. Ignoring that N ij s should be integer, Linear Programming (LP) can be used and the result rounded off to obtain a feasible solution. The rounded solution may provide a reasonable solution though it sometimes allocates more resources than required. We will observe this in the example below. If we make all N j equal to N then the computational complexity for LP is O(n 3 m3 N) while IP is NP-complete. For a large number of SSs and higher FFT sizes, even solving this LP in real time (for each frame) will be computationally demanding. Thus we consider a simpler solution also. B. Heuristic Algorithm In this algorithm we allocate the different subchannels to SSs one slot at a time. After allocation has been made for t slots, for slot t +1, a subchannel is given to an SS that can transmit maximum amount of data (out of the remaining data it has) on that suchannel. This is a very intuitive algorithm but we will see that its performance is close to optimal. Below we provide the details. STEPS 1) SET N ij =0forall i =1...n,j =1...m, slot = 0, total demand = i λ i and Ω x = {1...n}. 2) SET Ω y = {1...m}. 3) Select a j Ω y and set Ω y =Ω y {j}. 4) Check if i N ij < N j ; otherwise GOTO 3.

4 5) Find i Ω x such that i = arg max i R ij for λ i 0. 6) If (R ij λ i ) then set N ij = N ij +1, λ i = λ i R ij and total demand = total demand R ij. 7) If (R ij >λ i ) then set R ij = λ i and GOTO step 5. 8) If λ i =0then set Ω x =Ω x {i} and R ij =0for all j. 9) If total demand =0then GOTO step ) If Ω y = 0 then set slot = slot +1 and GOTO 2; otherwise GOTO step 3. 11) END. In the above algorithm Ω x is the set of the users having a nonzero demand and Ω y is the set of subchannels available for data transmission in the current slot. If the data of all the SSs has finished then the allocation process stops; otherwise for that slot all the subchannels are exhausted and the algorithm moves to the next slot in step 10. The subchannel selection in step 3 could be (say) sequential (by its index). At the cost of higher computational complexity, one could select more judiciously. The computational complexity of the Heuristic Algorithm with the sequential subchannel selection method is O (n m N) which is much less than the LP algorithm and thus is a good candidate for real time allocations. We consider three examples to compare the three algorithms: 1) Linear Programming, 2) Rounded LP and 3) Heuristic algorithm. The parameter for comparison is the total fraction of data lost (fraction of unsatisfied demand) which is Data loss / total demand. The three examples are (the details of the channel state and the demands of the SSs are not provided due to lack of space) 1) 10 SS, 10 Channels, Frame Length = 24 slots, Total Demand= Mb. 2) 10 SS, 10 Channels, Frame Length = 50 slots, Total Demand= Mb. 3) 50 SS, 32 Channels, Frame Length = 64 slots, Total Demand= Mb. TABLE I COMPARISON OF THE THREE ALGORITHMS FOR THROUGHPUT PERFORMANCE (THE FRACTION OF UNSATISFIED DEMAND) Example LP Algo Rounded LP Heuristic Algo The fraction of unsatisfied demand for the three algorithms are shown in Table I. The solution corresponding to the LP provides the least total unsatisfied demand. Since its solution may be noninteger, it infact provides a solution (nonfeasible) which is better than even the optimal algorithm obtained via Integer Programming (due to rounding off, the second algorithm sometimes provide more bandwidth than needed and hence we observe negative entries for this algorithm at some places). We observe that for all the examples, the heuristic algorithm provides performance quite close to the 987 LP solution and infact in the third example better than the truncated LP. Since the complexity of the heuristic is much less, this is a good candidate that can be implemented in real time. Table II provides the fraction of data lost for all the SSs for the three algorithms for example 2. It can be seen clearly that some of the SSs are able to transmit all their data while others are lossing upto 70 percent. Thus all the three algorithms are not fair to the different SSs. In order to optimize system performance, they favour the SSs with better channels. This motivates us to develop algorithms which provide good overall performance but are also fair to all the SSs. TABLE II COMPARISON OF THE THREE ALGORITHMS FOR FAIRNESS FOR EXAMPLE 2 OF TABLE I: SS No Fraction of data lost SS No Fraction of LP algo data lost Inte.LP algo Heuristic algo IV. FAIR ALGORITHMS As seen above, algorithms which optimize the overall system performance may not be fair to different SSs and users. In this section we modify the algorithms developed above to obtain fairer solutions without sacrificing the overall system performance. Fairness can be defined in various ways. Two of the most popular criteria are Min-Max fairness and Proportional fairness [6]. In min-max fairness we allocate such that the maximum unsatisfied demand (among the different SSs) is minimized. In proportional fairness, we try to provide allocation such that the percentage of satisfied demands of different SSs are same (or as close to each other as possible). We consider proportional fairness to be more appropriate for us. This is in conformance with the QoS requirements of real time traffic where one wants to have the fraction of packets lost to be less than (say) 2%. First we cosider proportional fairness as studied in Kelly([6]). It is the solution of the optimization problem max i log(x i) subject to Ax C over x 0 where A and C are appropriate matrices/vectors. In our case x i = j R ijn ij and the A and C are obtained from (2)-(4) and N ij 0. Unlike (3) and (5), Kelly s faireness requires optimizing a nonlinear cost function with linear constraints. We have found that for log function an optimization algorithm takes considerably longer than the LP. Thus it seems to be completely inappropriate for real time implementation. However

5 we consider this algorithm in this section for comparison purposes. We also consider some other proportional fairness concepts which we consider more appropriate in our context. We will consider solutions where an SS gets a BW proportional to its requirement. This concept has recently been considered in [10] and [9] also. One advantage of this concept is, as we will see below, we can have a control over the tradeoff between the global efficiency and fairness. The following algorithms are modifications of the algorithms developed in section III. Algorithm 1: Consider the optimization problem (2)-(4) along with the additional constraints R ij N ij βλ i (6) for i =1...n for a fixed β with 0 β 1. It is possible that for a given β this problem may not have a feasible solution. Thus we can start with a β close to 0 and then increase till we reach a β with no feasible solution. This will provide a good solution which is also proportionally fair. This is because it will ensure that each SS gets at least a β fraction of its demand. If β is small, the solution obtained will be more globally optimal but less fair. Thus β provides a trade off between global efficiency and fairness. One way to obtain the maximum fairness (largest β) in the above problem in one step is to first solve the LP slot with a larger β. This will provide a more efficient but less fair solution and with lower computational complexity. Keep repeating till we get the largest feasible β. Algorithm 4: We modify our heuristic algorithm such that in each slot, for carrier j and SS i, we compare w i R ij instead of R ij and give carrier j in the slot to the SS with the largest weighted transmission. If w i = λ α i, α>0, a user having bad channels but large demand can have significant priority and thus may get a chance to transmit some of its data. This algorithm will be fairer than the original algorithm. The constant α provides the tradeoff between efficiency and fairness. FLmax FLmin FAIRNESS Vs DATA LOSS Algorithm 3 Algorithm 4 Algorithm 1 Rounded Algorithm 1 Algorithm 2 Kelly s PF Algorithm Maxβ (7) subject to (2), (4) and (6). Then use this β in problem (2)-(4), (6). Algorithm 2: In this scheme we change our cost function to provide fair allocation. Now we minimize C = w i λ i R ij N ij (8) where w i are appropriate constants. For example, to obtain proportional fairness, we take w i = λ α i where α > 0. All the constraints are as in Section III. If λ i is large, minimizing C will provide a solution that will give more ( weightage to SS i, i.e., the solution obtained will reduce λ i ) m R ijn ij more than others. Now, again the cost function and the constraints are linear. Thus we can use IP (Integer Programming), LP(Linear Programming) and LP with rounding-off. Algorithm 3: Consider the heuristic algorithm. Start with a small positive constant β (0 β 1). Run the algorithm with the requirement of SS i as βλ i instead of λ i. If the requirements of all the SSs are met and we still have more channels/slots, then we can increase β and complete the allocation of the remaining channels/slots. A smaller initial β will provide more fairness. Alternatively, start the algorithm from the first FRACTION OF TOTAL DATA LOST Fig. 1. Comparison of Fairness of Kelly s algorithm and Algorithms 1-4 Figure 1 shows the fairness performance of all the schemes proposed with respect to the fraction of data lost. Here we consider a system with 32 channels (example 3 in Table I). The scheduler allocates for two frames at a time. Thus the frame length is 64 slots. The number of subscriber stations is 50. FLmax and FLmin are the maximum and the minimum values of fractional data loss among all the SSs for a given algorithm. A scheme provides good fairness if the difference of FLmax and FLmin is small. Here the value of α (for algorithms 2 and 4) is varying between 0 to 2 for different points in the graph. The starting value of β for Algorithm 1 is varying between 0 to 0.12 and for Algorithm 3 between 1 to As can be seen clearly, for all the schemes, as the fairness improves the data loss increases. It shows a trade-off between the system throughput and fairness. Algorithm 1 is performing very well. Algorithm 3 also performs reasonably at a lower computational cost. Algorithms 2 and 4 perform very badly. Algorithms 1, 3 perform better than Kelly s algorithm especially at somewhat heigher total loss. We have compared these algorithms for other examples given in Section III also and obtained similar results.

6 V. ALGORITHMS FOR SCHEDULING Now we get back to the setup in section II-D and show how we will use the algorithms developed in sections III and IV for uplink and downlink scheduling by the BS. Next we will consider uplink scheduling by the SSs to the different end users. A. Algorithms for BS First we consider the uplink and downlink scheduling by the BS. Joint scheduling of uplink and downlink is more efficient. But we are considering OFDMA/TDD setup and thus we can not assign some carriers to uplink and some to downlink in the same slot. Thus we will consider the uplink and downlink scheduling separately. However they will use the same algorithms and hence we will mention only uplink in the following. As mentioned earlier, the BS has the information about the requirements of each SS for each of the four classes and also of the channel state. The BS will first make allocation for UGS class of each SS (in the uplink and the downlink: separately but before the next class of service is considered. we will follow the same approach for each class.). The reason it is given higher priority than even rtps is that it is the most well behaved traffic (CBR sources) and its requirements have been announced at connection admission time without any ambiguity. Thus this traffic should not suffer just because some rtps connection has a burst of arrivals. Let λ i (in bytes) be the UGS requirements of SS i in the frame under consideration. If there are sufficient computational resources at the BS, it can run LP; otherwise the heuristic in section III. If the channels are sufficiently good, the UGS requirements of all SSs can be met. Otherwise, we can use Algorithm 1 or Algorithm 3 of Section IV to obtain efficient fair algorithms. Since we are giving highest priority to UGS sources and they actually require low BW (these are generally voice sources), in all likelihood, we do not need to use fair versions of the algorithms for UGS (otherwise it is a highly under-engineered system). Next we consider the rtps service class of all the SSs. Again we can use the algorithms of Section III and IV. Now of course we have to use them on the leftover channels and slots. Because of the variable rates of these sources and high BW requirements, we may directly apply Algorithms 1 or 3 of Section IV in this case. If all the SSs can be satisfied, these will provide the solutions obtained by the algorithms of Section III. Next we consider the nrtps with minimum BW requirements and finally the remaining demands. The algorithms of Sections III and IV can be used for these in the same way as explained above. As mentioned before, the allocations for downstream can be done in the same way. For upstream case, these allocations are then announced in the beginning of the frame for the information of the SSs. Once an SS knows the subchannels and slots assigned to it, then it has to allocate these to its various users. In the following we explain the algorithms used by the SS. 989 B. Algorithms for SS Let a total of λ bytes be assigned by the BS to an SS in a given frame. The SS needs to assign this BW to the different users. It will follow the sequence followed by the BS: First satisfy the demands of UGS users, then of rtps, next the minimum requirements of nrtps and finally the requirements of the rest. The trade-off between overall system performance and fairness among different users emerges again. In the following we first discuss the issues involved and then the actual algorithms. Let us first consider the case of UGS users. They will have only one packet at a time in a frame to send (the frame duration is msec). The size of each packet of user k will be fixed, say γ k bytes. γ k s can be different for different users. The standard allows fragmentation of these packets. But because it is real time service, fragmented packets may reach too late at the destination. Thus we will not fragment; either a complete packet of a user will be sent or it will be completely discarded. Thus the relevant problem in this case will be: given an overall allotment of λ bytes in a frame, the SS should allocate the resources so as to satisfy the maximum number of UGS users. We will address this problem in the following. Next let us consider rtps (real time VBR connections) service class. In this class, the standard specifies that the resources make demands at periodic intervals. But the packet size of the connections can change from time to time. We will also allow the possibility that a connection may have multiple number of packets which can be of different sizes. In the allocation of λ a bytes (the leftover from the UGS users) among such users, the SS will need to consider the issues of overall system performance (e.g. maximum number of packets transmitted), fairness to different users, possibilities of packet fragmentation etc. Finally we consider the problem of allocation of λ b bytes assigned to an SS for its nrtps (e.g. web traffic) users. Each user has a requirement of γ k bytes (which may be the sum of the packet lengths of all the packets a user has at that time). Now we will allow fragmentation of packets, even though in the current Internet wisdom it may not be the recommended practice. Then again all the issues mentioned above for the rtps service will be important in the allocation of the channel resources. In the following we develop algorithms which consider the above scenarios. If we allow packet fragmentation, we will call it byte model; otherwise packet model. Algorithms for packet model: First we consider the problem relevant for the UGS service class. Let us assume that the SS has been allocated carriers and slots such that it can send λ bytes in the frame. Let there be M UGS users, each with one packet to transmit. The packet length of k th user is γ k. No fragmentation of packets is allowed. We would like to allocate λ bytes such that maximum number of packets can be transmitted. This is the well known Knapsack Problem[13]. It is known to be NP-Complete. Hence we seek a good heuristic algorithm

7 which is computationally not very demanding. The following is a well known suboptimal algorithm: - Arrange the packets in increasing packet sizes. - Start allocating from the first packet (i.e., the smallest packet will be sent first). Its complexity is O(M) and hence can be used in real time by the SS. Next we consider the situation (relevant for rtps) where a user may have more than one packet to send and the length of different packets of a user can be different. Now the issues of overall performance and fairness are relevant. Let us first develop a scheduling mechanism which maximizes the total number of packets transmitted by the SS. This is again the Knapsack problem which has been addressed above. Thus the suboptimal algorithm given above can be used. However now the issue of fairness arises. A user with smaller packets will be favoured by the above algorithm. Following is a simple variation of the above algorithm that can provide fairness at the cost of system performance: - Order all the packets in increasing size. - Now instead of scheduling these packets sequentially, we schedule in cycles. In the first cycle we transmit the smallest packet of each user (among these packets the smaller packets are sent first). - If there is still BW available, we start the next cycle and schedule in the same way the remaining packets. This modified algorithm is fairer than the original algorithm but may not be proportionally fair. To obtain proportional fairness, we modify a cycle of the previous paragraph as follows: - Instead of scheduling one packet from each user in a cycle, we schedule n k packets from user k (in increasing packet size) where n k is proportional to the number of packets of user k to be transmitted in the current frame. If the number of cycles is large, or we end up having no fractional cycle (due to exhaustion of slots), this will provide proportional fairness in terms of number of packets. Algorithms for Byte Model: We consider the following scenario. The BS has allocated to the SS λ bytes. There are M nrtps users. User k requires a minimum of γ k bytes (sum of lengths of all its packets) to be transmitted in the frame. The packets can be arbitrarily fragmented. If we want the allocation that maximizes the total number of bytes that can be transmitted, then since all packets can be arbitrarily fragmented, we do the following: - Order all packets in increasing packet size. - Schedule their transmission in increasing order and if all packets can not be accomodated, truncate the last one so that we can send a total of λ bytes. This algorithm sends maximum number of packets but is not necessarily fair. We consider proportional fairness. For user k with demand γ k, we will allocate βγ i where 0 β 1 and β should be as high as possible. If k γ k λ then all packets can be served and β =1. Otherwise, let β = λ k γ. k Now, once βγ k has been fixed, to minimize fragmentation, 990 user k can allocate this BW to its packets in the way done earlier (this is Knapsack problem). This algorithm can also be used for nrtps without minimum rate guarantees and the BE traffic. VI. CONCLUSIONS We have developed efficient scheduling algorithms for channel/slot allocation in the IEEE OFDMA/TDD system. These can be used in real time. However these may not be fair. Next we developed fair versions of these algorithms. Finally we explain the overall setup for allocation in upstream and downstream by the BS to different SSs and then by an SS to its different users belonging to the four service classes so that the QoS of the users can be satisfied in a fair and efficient way. REFERENCES [1] C. Eklude, R. B. Marks, K. L. Stanwood and S. Wang. 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