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Statistical-Time A ccess F airness I ndex of One-Bit Fe edback F air S cheduler. Fumio Ishizaki Dept. of Systems Design and Engineering Nanzan University, Japan Email: [email protected] Contents. Introduction System Model Analysis Numerical Results Conclusion. Introduction.

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Statistical-Time Access Fairness Index of One-Bit Feedback Fair Scheduler Fumio IshizakiDept. of Systems Design and Engineering Nanzan University, Japan Email: [email protected] Contents Introduction System Model Analysis Numerical Results Conclusion Introduction Background Since the utilization of multiuser diversity (MD)in wireless networks can increase the information theoretic capacity (ITC), much attention has been paid to scheduling algorithms exploiting MD. MD is a diversity existing between the wireless channel states of different users. MD comes from the fact that the wireless channel state processes of different users are usually independent for the same shared medium. Schedulers exploiting MD For example, MD can be exploited in such a way that the scheduler at the BS (Base Station) selects the MS (Mobile Station) whose received SNR is the best, and transmits packets to the selected MS. This scheduling algorithm maximizes ITC of the overall system, but it is highly unfair. Proportional Fair (PF) scheduler To solve this unfair problem, proportional fair (PF)scheduling was proposed. PF considers the normalized SNRs of MSs (defined by the received SNR /the average received SNR), and selects the MS whose normalized SNR is the largest. Quantized PF (QPF) scheduler In practice, probably the normalized SNR values are quantized, the quantized normalized SNR values are reported to the BS by MSs, and the PF scheduling is performed based on the quantized normalized SNR values. This PF scheduling is called QPF (Quantized PF) scheduling. One-bit feedback fair (1FF) scheduler From a view of reducing the amount of feedback information, a small number of quantization levels is desirable. One-Bit Feedback fair (1FF) scheduling is QPF scheduling with twoquantization levels (i.e., with only one threshold for quantization). It has been reported that the 1FF scheduler can achieve a relatively good ITC, if the quantization threshold is appropriately determined. Long term fairness and short term fairness There exists a tradeoff between ITC and fairness achieved by schedulers exploiting MD. The fairness is classified into short term fairness andlong term fairness. Short term (ST) fairness: the ability of the scheduler on how equally it can distribute network resources (e.g., service times) over multiple MSs in a finite observation period. Long term (LT) fairness: the ability of the scheduler on how equally it can distribute network resources over multiple MSs in an infinite observation period. Significance of short term fairness ST fairness greatly affects packet level performances such as delay and loss probability of individual MSs. LT fairness governs the (long run) average throughput of individual MSs. 1FF scheduler provides an ideal LT fairness property. However, its ST fairness properties have not been sufficiently explored yet. Statistical-time access fairness index (STAFI) As an index of short term fairness, Liu et al. propose a statistical time-access fairness index (STAFI) defined as : the amount of the service in time (not in bits) for flow i in : assigned weight for flow i : some function Purpose of this study Study the short term fairness properties of 1FF schedulers Consider STAFI as a measure of short term fairness In this paper, call the probability on the left hand side of the inequality the STAFI In particular, consider STAFI where the assigned weights are equal to one What we do in this study Develop two numerical methods to examine the transient properties of the STAFI of 1FF scheduler. The first method calculates the exact value of the STAFI by using the inverse discrete FFT method. It enables us to precisely observe how the STAFI changes as the progress of time. The second method estimates the asymptotic decay rate of the STAFI by using the theory of large deviations. It enables us to predict how fast the STAFI approaches to ideal fairness as the progress of time. System Model Consider a wireless network consisting of a BS and K MSs as shown in Fig. 1. BS employs 1FF scheduler for downlink transmission. Focus on downlink transmission and analyze the STAFI of the 1FF scheduler. Channel model We assume that the downlink channel of MS is described by a flat Rayleigh fading channel model. The received SNR process of MS i (i=1,…,K) is a stationary process for any t is according to the following exponential distribution: where denotes the average received SNR of MS i. We assume that the received SNR processes of the K MSs are independent with each other. 1FF scheduler Under 1FF scheduling, the normalized SNR processes of MSs are considered. Each MS quantizes the entire normalized SNR range into 2 grades with (quantization) threshold . MSs with the normalized SNR values greater than or equal to transmit one-bit feedback information to the BS. For downlink transmission, 1FF scheduler at the BS randomly selects one of MSs which feed back. If there are no MS which feed back, 1FF scheduler randomly selects one of KMSs. 2-state DTMCs selects one of MSs which feed back. Let denote the wireless channel state process of MS i. We assume that the wireless channel state processes are well described by stationary discrete-time 2-state Markov chains(MCs). We determine the transition probability matrix of the MCs according to the method shown in [12]. Analysis selects one of MSs which feed back. STAFI of 1FF scheduler selects one of MSs which feed back. Without loss of generality, analyze the STAFI between MS 1 and MS 2. Let denote the STAFI between MS 1 and MS 2 during n slots. The STAFI is then given by for any , where denotes the amount of service for MS i in . Inverse discrete FFT method selects one of MSs which feed back. The STAFI can be expressed in terms of 2n+1 unknown constants (depend on n) as We can determine the unknown constants by using the inverse discrete FFT method [17]. By using this numerical method, we can calculate the exact value of the STAFI. Large deviations selects one of MSs which feed back. Although the numerical method based on the inverse discrete FFT method provides the exact value of the STAFI , it is very time-consuming when is large. By using the theory of large deviations, we can estimate how fast the STAFI decreases as . The following Proposition shows that selects one of MSs which feed back. the STAFI exponentially decreases as . We call the term the asymptotic decay rate (ADR)of the STAFI . Numerical Results selects one of MSs which feed back. Provide numerical results to get insight about the properties of STAFI of 1FF scheduler Fix the parameters mobility-induced Doppler spread of MSs = 10Hz the length of one slot=1msec Effect of threshold value on STAFI properties of STAFI of 1FF scheduler Observe the effect of the threshold on STAFI. Fig.2 shows the STAFI of 1FF as a function of . “1FF(ydB)”: 1FF scheduler whose threshold is equal to dB. For comparison, also shows the STAFI of the random scheduler (RS) which randomly selects a MS among K MSs irrespective of their received SNRs. “RS”: the random scheduler Observation in Fig.2 properties of STAFI of 1FF scheduler For almost whole range of , the STAFIs of 1FF schedulers are greater than that of RS, i.e., short term fairness provided by 1FF scheduler is worse than that provided by random scheduler. This is due to the positive correlation of the normalized SNR process in time. For properties of STAFI of 1FF schedulersmall of , 1FF scheduler with larger threshold provides better fairness than that with smaller threshold. However, the situation is converse for large . 1FF scheduler with large threshold can keep the probability of moderate unfairness lower, but it can cause serious unfairness with higher probability, compared to 1FF scheduler with small threshold. Non-monotonicity in threshold value properties of STAFI of 1FF scheduler If , 1FF scheduler behaves as the random scheduler. If , 1FF scheduler behaves as the random scheduler. Reason of serious unfairness with large threshold properties of STAFI of 1FF scheduler First, suppose that threshold is large. In many sample paths, all MSs including MS 1 and MS 2 start in state 0 and likely to stay in state 0 during a certain period. The realization probability of such sample paths is large. If such sample paths are realized, serious unfairness is not caused, because one MS among all the MSs is randomly selected for service slot-by-slot. On the other hand, there exist sample paths where properties of STAFI of 1FF schedulerMS 1 and all the other MSs including MS 2 start in state 1 and in state 0, respectively. In these sample paths, MS 1 is surely selected for service in the first slot and MS 1 is likely to be continuously selected during a certain period due to the positive correlation of the normalized SNR processes in time. Although the realization probability of such sample paths is small, if such sample paths are realized, serious unfairness is caused. Conversely, suppose properties of STAFI of 1FF schedulerthat threshold is small. In many sample paths, all MSs including MS 1 and MS 2 start in state 1 and likely to stay in state 1 during a certain period. The realization probability of such sample paths is large. If such sample paths are realized, serious unfairness is not caused, because one MS among all the MSs is randomly selected for service slot-by-slot. On the other hand, there exist sample paths where properties of STAFI of 1FF schedulerMS 1 and all the other MSs including MS 2 start in state 0 and in state 1, respectively. In these sample paths, one MS of all the other MSs except for MS 1 is randomly selected for service slot-by-slot during a certain period. Thus, even in such sample paths, the probability of causing serious unfairness is quite low, compared to the case where threshold is large, because MS 2 is selected for service with probability 1/(K-1). Change of STAFI as progress of time properties of STAFI of 1FF scheduler Examine how the STAFI of 1FF scheduler changes as the progress of time. Figs. 3 and 4 exhibit the STAFI as a function of for . Number of MSs=30, threshold =3.78dB in Fig3 and =2.00dB in Fig.4. Observation in Figs. 3 and 4 properties of STAFI of 1FF scheduler Under 1FF scheduling, the STAFI rapidly decreases with the increase of for every . In other words, the STAFI rapidly approaches to the ideal fairness. How unfairness is resolved properties of STAFI of 1FF scheduler Investigate how unfairness is resolved as the progress of time. For this purpose, consider the STAFI as a function of , where is a parameter and denotes the number of MSs. Note here that the term denotes the expected access-time which each user receives when the ideal fairness is achieved. Thus, the STAFI is considered as a measure indicating a deviation from the ideal fairness, where is a parameter. Fig properties of STAFI of 1FF scheduler. 5 displays the STAFI as a function of the observation period for denoted by “h=2.0” and “h=4.0”, respectively. In addition, to confirm that the estimated ADR is identical to the actual ADR, shows the exponential decay lines for denoted by “ADR(h=2.0)” and “ADR(h=4.0)”, respectively, where is the estimated ADR of the STAFI from Proposition 1. Set the number of MSs to 16 and the threshold to 2.74dB. Observation in Fig.6 properties of STAFI of 1FF scheduler In an asymptotic sense, i.e., , the STAFI decreases exponentially as stated in Proposition 1. The actual ADR of the STAFI seems to be identical to the estimated ADR . The ADR of the STAFI is large when the parameter is large. Effect of number of MSs on ADR properties of STAFI of 1FF scheduler Observe the effect of the number of MSs on the ADR of the STAFI . Fig. 6 shows the ADR of the STAFI as a function of the number of MSsfor 1FF schedulers with . Observation in Fig.6 properties of STAFI of 1FF scheduler For the scheduler with dB, the ADR of the STAFI does not significantly change with the increase in the number of MSs. However, for the schedulers with dB, the ADR decreases with the increase in the number of MSs in the range from K=10 to K=40. For the scheduler with large threshold, the ADR of the STAFI greatly decreases with the increase in the number of MSs. For the scheduler with large threshold , the speed approaching to the ideal fairness as the progress of time becomes slow with the increase in the number of MSs. Conclusion properties of STAFI of 1FF scheduler Conclusion properties of STAFI of 1FF scheduler We focus on 1FF scheduler and numerically study the STAFI of 1FF scheduler to understand its short term fairness properties. For this purpose, we develop the two numerical methods. Numerical results show that the threshold greatly affects the short term fairness properties of 1FF scheduler. If rigorous fairness is required even in a relatively short time period, we should consider the short term fairness of the scheduler as well as ITC, when we determine the threshold value.

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