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3 to satisfy the following system constraints: 1. Each channel can be used for transmission to only one MS at a given time slot; 2. The data transmission rate has to be selected from one of the M discrete transmission rates available (but less than the time and user dependent capacity); 3. On each channel, changing the data rate, or the intended MS, makes the next time slot useless in terms of data transmission; 4. Each MS n can use at most α(n) channels in a time-slot. IV. PREDICTIVE MULTI-USER MULTI-CHANNEL SCHEDULING ALGORITHM Predictive Multi-User Multi-Channel (PMUMC) scheduling algorithm is an online scheduling algorithm that attempts to optimize the throughput by making use of the knowledge of the CS for all users and channels in the current time slot (c n,f (i)), as well as the predicted CS for the following time slot (denoted as c n,f (i + 1) from now on). We have restricted the heuristics in this paper to at most one lookahead, since, as we shall elaborate in subsequent sections, a single look-ahead yields most of the potential benefits of future CS prediction in our experiments, even when the estimates are imperfect. PMUMC PSEUDO-CODE for time slot i Given: (i) a(f, i 1), and x n,f (i 1) n, f: the user assigned to each channel, and the transmission rates for each channel and user in the previous time slot, (ii) c n,f (i), c n,f (i + 1) n, f: the capacity in the current time slot, and the estimated capacity in the next time slot Find: a(f, i) and x n,f (i), n, f: the user assigned to each channel, and the transmission rates for each channel and user in the current time slot. / / Initially, no channel is assigned to any user 1 Set a(f, i)=null f 2 Set A n (i)=0 n / / Assign users and rates to each channel sequentially based on the optimal tentative two-slot schedule. 3 for (each channel f){ 4 Compute { n : x n,f (i), x n,f (i + 1)} using MUWDH 5 Set a(f, i) = arg max n:an(i)<α(n){x n,f (i) + x n,f (i + 1)} 6 Set A a(f,i) (i) = A a(f,i) (i) Set x n,f (i) = 0 n A a(f,i) (i) 8 } Fig. 1. Predictive MUMC scheduling algorithm. PMUMC, whose pseudo-code is given in Figure 1, is an online algorithm which uses three consecutive time-slots as a building block in a sliding-window fashion: the transmission decisions for slot i are determined based on the transmission decisions for slot i 1, and capacity parameter inputs for slots i and i+1. Define the assignee a(f, i) as the user (if there is one) who is granted transmission on channel f in slot i. Let x n,f (i) be the transmission rate used by MS n on channel f in slot i, and A n (i) be the total number of channels assigned to user n in time slot i. By definition, x n,f (i) > 0 only for n = a(f, i). At the beginning of time slot i, PMUMC selects x n,f (i), and hence a(f, i) ( n, f) as follows. Given the assignees and transmission rates assigned to the channels in the previous slot {a(f, i 1),x n,f (i 1) : n, f}, and the channel states in the current and next time slots, {c n,f (i),c n,f (i + 1) : n, f}, it assigns users and transmission rates for slot i to the channels sequentially. For each channel, the assignee is selected as the user that would maximize the transmission over the two time-slot period (i, i + 1) while meeting the constraint on the number of channels that can be assigned to it. For a given channel f, PMUMC utilizes multi-user wait-dominate-hold (MUWDH), which computes for each user n a tentative twoslot schedule x n,f (i), x n,f (i + 1) that maximizes the tentative throughput x n,f (i) + x n,f (i + 1), assuming that user n is granted transmission in both slots on channel f. PMUMC then selects the MS with the highest tentative throughput while not exceeding the channel count limit as a(f, i), the actual user for channel f in slot i, with the corresponding transmission rate x n,f (i). We emphasize that x n,f (i + 1) values computed in slot i are only projections, and they may not materialize later. This is because similar computations will be performed in slot i + 1 with one slot shift in time. We next describe MUWDH, a generalized version of the single-channel single-user algorithm in [6]. For a given channel f, let m n,f (i) denote the maximum usage achievable in the current slot by user n, given a(f, i 1), x n,f (i 1) and c n,f (i). Then, we have three different cases. (I) m n,f (i) = 0 if n = a(f, i 1) and c n,f (i) < x n,f (i 1), (II) m n,f (i) = c n,f (i) if a(f, i 1) =NULL, and (III) m n,f (i) = x n,f (i 1) if n = a(f, i 1) and c n,f (i) x n,f (i 1). Then, the tentative schedule for each user and channel is constructed by choosing one the following three actions. (i) Wait: If the (estimated) next capacity is more than twice the maximum possible current usage, set the current usage to 0 (so that the next capacity can be used fully). (ii) Dominate: If the maximum possible current usage is more than twice the next capacity, set the current usage to maximum possible current usage. Usage for the next time slot will be 0. (iii) Hold: If neither of the above is the case, use the minimum of the two capacities if the channel was not used by any user in the previous slot (so that both the current and next time slot may be potentially used). If the user was already using the channel in the previous time slot, then use the maximum possible current usage. The precise set of rules for choosing x n,f (i), and x n,f (i+1) are as follows. (i) if c n,f (i + 1) > 2m n,f(i), set x n,f (i) = 0, x n,f (i + 1) = c n,f (i + 1) (ii) if 2c n,f (i + 1) < m n,f (i), set x n,f (i) = m n,f (i), x n,f (i + 1) = 0 (iii) if 1/2c n,f (i + 1) m n,f(i) 2c n,f (i + 1), and x n,f (i 1) > 0 set x n,f (i) = m n,f (i). Set x n,f (i + 1) = 0 if m n,f (i) > c n,f (i + 1), and x n,f (i + 1) = m n,f (i) otherwise

6 Fig. 4. Estimation Accuracy (p) vs. throughput for 5 user, 5 channel system Fig. 6. Estimation Accuracy (p) vs. throughput for 40 user, 5 channel system Fig. 5. Estimation Accuracy (p) vs. throughput for 20 user, 5 channel system Fig. 7. Estimation Accuracy (p) vs. throughput for 5 user, 5 channel system with α = 4. resource consumption for obtaining accurate estimates can be assessed based on these observations. We also observed that the presence multiple users favors a greedier approach that gives less weight to the future transmission projections. A natural extension of this work would be an algorithmic complexity analysis of the problem [7], and developing QoSsensitive scheduling algorithms. Complementary techniques to reduce channel monitoring requirements, such as sub-carrier grouping where we have preliminary results [8], also constitute interesting research directions. REFERENCES [1] S. Catreux, V. Erceg, D. Gesbert, and R.W. Heath, Adaptive modulation and MIMO coding for broadband wireless data networks, IEEE Communications Magazine, vol. 40, pp , June [2] V. Tsibonis, L. Georgiadis, and L. Tassiulas, Exploiting wireless channel state information for throughput maximization, in Proceedings of IEEE Infocom 03, April 2003, pp [3] S. Kulkarni and C. Rosenberg, Opportunistic scheduling policies for wireless systems with short term fairness constraints, in Proceedings of Globecom 2003, December 1992, pp [4] X. Liu, E.K.P. Chong, and N. Shroff, Opportunistic transmission scheduling with resource-sharing constraints in wireless networks, IEEE JSAC, vol. 19, no. 10, pp , Oct [5] T. Keller and L. Hanzo, Adaptive multicarrier modulation: a convenient framework for time-frequency processing in wireless communications, Proceedings of the IEEE, vol. 88, pp , August [6] A. Arora and H.A. Choi, Wait-dominate-hold, personal communication. [7] A. Arora, F. Jin, G. Sahin, and H.A. Choi, Throughput maximization in multi-carrier wireless networks with predictive link adaptation, July 2004, in preparation. [8] F. Jin, G. Sahin, A. Arora, and H.A. Choi, The effects of the sub-carrier grouping on multi-carrier channel-aware scheduling, in Proceedings of Broadnets 04, October 2004, to appear.

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