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Statistical Time-Access Fairness Index of One-Bit
Feedback Fair Scheduler
Department of Systems Design and Engineering diversity comes from the fact that the wireless channel state Since the utilization of multiuser diversity in wireless net- processes of different users are usually independent for the works can increase the information theoretic capacity, much same shared medium. Since the utilization of multiuser di- attention has been paid to schedulers exploiting multiuser versity in wireless networks can increase the information the-oretic capacity, much attention has been paid to schedulers diversity. It is known that there exists a tradeoff between exploiting multiuser diversity (see, e.g., [5, 8, 10, 14, 19] and the capacity and fairness achieved by schedulers exploiting multiuser diversity. Due to its good balance between the To achieve the efficient use of the bandwidth, multiuser capacity and fairness, the one-bit feedback fair scheduler is diversity can be exploited, for instance, in such a way that considered as an attractive choice. The fairness is classi- the scheduler in the BS (Base Station) selects the MS (Mo- fied into short term fairness and long term fairness. It isknown that the one-bit feedback fair scheduler has an ideal bile Station) whose received SNR (Signal-to-Noise Ratio) is long term fairness property. However, the short term fair- the largest, and transmits packets to the selected MS at ness properties of the scheduler have not been sufficiently each time slot. This scheduling scheme maximizes the in- explored yet. Since packet level performances of individ- formation theoretic capacity of the overall system, but it ual MSs (Mobile Stations) are strongly affected by short becomes highly unfair when there are MSs having very dis- term fairness, it is also important to examine the short term parate channel conditions [22]. To solve this unfair problem, fairness of the schedulers. In this paper, we focus on the the proportional fair (PF) scheduling was proposed wherethe scheduler considers the normalized SNR values of MSs, short term fairness of the one-bit feedback fair scheduler.
defined by the received SNR values divided by their corre- As a short term fairness index, we consider the statistical sponding average received SNR values, and selects the MS time-access fairness index (STAFI). We then develop two whose normalized SNR value is the largest. The PF schedul- numerical methods to understand the transient properties ing provides strict fairness among MSs because the normal- of the STAFI of the one-bit feedback fair scheduler. The ized SNR values are i.i.d. (independent and identically dis- first method calculates the exact value of the STAFI by us-ing the inverse discrete FFT method. The second method tributed) among MSs [22]. Here, strict fairness means that estimates the asymptotic decay rate of the STAFI by using the access probabilities of MSs to the wireless channel are In practice, the normalized SNR values are quantized and the quantized normalized SNR values are reported to the BS by MSs. The PF scheduling is then performed at the multiuser diversity, one-bit feedback fair scheduler, short BS based on the quantized normalized SNR values. This PF scheduling is called the QPF (Quantized PF) schedul-ing [4, 9].
In general, the information theoretic capacity achieved by the QPF scheduler increases and approachesto that achieved by the PF scheduler as the number of its In wireless networks, scheduling for efficient bandwidth quantization levels increases. On the other hand, from the utilization is a key component to the success of quality- viewpoint of reducing the feedback overheads (for report- of-service (QoS) guarantees. One way to achieve efficient ing the quantized normalized SNRs) from MSs to the BS, a bandwidth utilization of time-varying wireless channel is to small number of quantization levels is desirable. For the re- exploit diversity. Multiuser diversity [11] is a diversity ex- duction of the feedback overheads, the one-bit feedback fair isting between the channel states of different users. This scheduler, which is the QPF scheduler with two quantizationlevels, is considered as one of good candidates to implementin practice [3, 6, 7, 15, 16, 18, 20, 23]. It is reported that Permission to make digital or hard copies of all or part of this work for the one-bit feedback fair scheduler can achieve a relatively personal or classroom use is granted without fee provided that copies are good capacity if the quantization levels are appropriately not made or distributed for profit or commercial advantage and that copies determined [6]. It is known that the one-bit feedback fair bear this notice and the full citation on the first page. To copy otherwise, to scheduler also provides strict fairness among MSs.
republish, to post on servers or to redistribute to lists, requires prior specific In this paper, we focus on the fairness of the one-bit feed- permission and/or a fee.
QTNA 2011, Aug. 23-26, 2011, Seoul, Korea back fair scheduler. The fairness is classified into short term Copyright 2011 ACM 978-1-4503-0758-1/11/08 .$10.00.
fairness and long term fairness [7, 18]. Short term fairness indicates the ability of the scheduler on how equally it can distribute network resources (e.g., service times) over mul- tiple MSs in a finite observation period. On the other hand,long term fairness indicates the ability of the scheduler on how equally it can distribute network resources over multi- ple MSs in an infinite observation period. Thus, long termfairness governs the (long run) average throughput of indi-vidual MSs, while short term fairness greatly affects packetlevel performances such as delay and loss probability of in- dividual MSs. Although the strict fairness mentioned aboveis ideal fairness for long term fairness, it does not guaranteegood fairness for short term fairness. Hence it is important to examine the short term fairness of schedulers as well asthe long term fairness.
In wireline networks, the proportional fairness index is usually used as a measure of short term fairness, and it char- Figure 1: System model
acterizes the service discrepancy between two flows i and jover any time interval [t1, t2) during which the two flows arecontinuously backlogged. A fair scheduler for wireline net- the STAFI where the assigned weights φi in (2) are all equal works guarantees the proportional fairness index to have a to one. As mentioned above, it is known that the one-bit feedback fair scheduler provides strict fairness, which means an ideal long term fairness property. However, there are only a few studies on the short term fairness properties of the one- bit feedback fair scheduler [7] and they have not been suffi- ciently explored yet, although the packet level performances i(t1, t2) denotes the service in bits that flow i re- of individual MSs are strongly affected by the short term 1, t2), φi denotes the assigned weight for flow fairness. In this paper, we develop two numerical methods i,j is a constant which may depend on i and j. How- ever, the proportional fairness index for wireline networks to understand the transient properties of the STAFI of the is not suitable for wireless networks. First, the hard de- one-bit feedback fair scheduler. The first method calculates terministic guarantee where the proportional fairness index the exact value of the STAFI by using the inverse discrete always satisfies (1) does not take randomness inherent in the FFT method. It enables us to precisely observe how the wireless channel conditions into account. As a result, if we STAFI changes as the progress of time. The second method require the hard deterministic guarantee, schedulers do not estimates the asymptotic decay rate of the STAFI by using have the flexibility to exploit short term channel variations the theory of large deviations. It enables us to predict how and to select users with better channel conditions (i.e., the fast the STAFI approaches to ideal fairness as the progress flexibility to exploit multiuser diversity). Second, the pro- portional fairness index considers fairness of users’ through- The remainder of this paper is organized as follows. In puts rather than channel access times. In wireless networks, Section 2, we describe a system model considered in this pa- users can transmit at different rates depending on their cur- per. We assume that the wireless channel process for each rent channel quality. Thus, to normalize throughputs would user is modeled by a discrete-time two-state Markov chain require allowing the user with the worst channel quality to and the channel processes are stationary.
have a disproportionately large share of channel access time.
sumptions, we analyze the STAFI of the one-bit feedback This degrades the throughput performance of wireless net- fair scheduler in Section 3. We then develop the two nu- works. Hence, instead of the hard deterministic proportional merical methods based on the analysis. Section 4 provides fairness index, we need a more suitable short term fairness numerical results to examine the transient statistical fairness measure for wireless networks. To resolve those problems properties of the one-bit feedback fair scheduler. Conclusion which the proportional fairness index has, Liu et al. [13] consider modifications to the proportional fairness index. Tomake the distinction between temporal and throughput fair- ness, they define α(t1, t2) as the service in time (instead of In this paper, we consider a wireless network consisting of W (t1, t2) as the service in bits) that flow i receives during a BS and K MSs as shown in Fig. 1. We suppose that the [t1, t2). Then, by relaxing the hard deterministic fairness BS employs the one-bit feedback fair scheduler for downlink guarantee to be a statistical one, they propose a statistical transmission from the BS to the MSs. We focus on downlink time-access fairness index (STAFI) defined as transmission and analyze the STAFI of the one-bit feedback We assume that the downlink channel of MS i (i = 1, . . . , K) is described by a Rayleigh fading channel model. Time axis where f (i, j, x) is a probability distribution which may de- is divided into physical (PHY) frames of equal size Tf (sec) and time index is given by t = 0, 1, 2, · · · . The PHY frame In this paper, we focus on the one-bit feedback fair sched- duration Tf is considered to be the unit time in our model.
uler and study the STAFI of the scheduler to explore its Then, the received SNR process {z(i)(t)} (t = 0, 1, . . .) of short term fairness properties. In particular, we consider MS i (i = 1, . . . , K) is a stationary stochastic process and z(i)(t) at time t is according to the following exponential where fd denotes the mobility-induced Doppler spread of MSs and we assume that for all the MSs, the mobility-induced Doppler spreads are identical.
P{z(i)(t) ≤ x} = 1 exp(−x/¯ We next consider the stationary probability vector s =
z(i) denotes the average received SNR of MS i and (s0, s1) of the 2-state discrete-time Markov chain {s(i)(t)}.
z(i) = E[z(i)(t)]. We assume that the received Note here that for all the MSs, the channel state processes SNR processes of the K MSs are independent with each have the same stationary probability vector due to the nor- malization of the received SNRs. From (3), the stationaryprobability vector is given by s0 = 1 − e−γ1 , Under the one-bit feedback fair scheduling, the normal- ized SNR processes of MSs are considered, where the nor- The state transition probabilities are then determined by malized SNR process is defined by the process {z(i)(t)/¯ (i = 1, . . . , K). To reduce the feedback overheads for re- porting the normalized SNR from MSs to the BS, each MSquantizes or partitions the entire normalized SNR range into p0,0 = 1 − p0,1, p1,1 = 1 − p1,0, two grades with threshold denoted by γ1. We assume that i (i = 0, 1) and χ(γ1) are given by (5) and (4), re- 1 is a priori determined. If z(i)(t)/ ¯ spectively. (6) and (7) determine the transition probabil- we say that the wireless channel state of MS i is in state ity matrix P of the 2-state Markov chain, whose stationary
0 at time t. If z(i)(t)/¯ z(i) ≥ γ1, we say that the wireless channel state of MS i is in state 1 at time t. We assumethat perfect channel estimation is possible at each MS andeach MS knows its average SNR ¯ z(i) (i = 1, . . . , K). Then MS i can determine the grade of its channel to the BS with Without loss of generality, we analyze the STAFI between the knowledge of its normalized SNR.
MS 1 and and MS 2. To express the STAFI between MS The one-bit feedback fair scheduler then operates as fol- 1 and MS 2, we introduce an auxiliary i.i.d. (independent and identically distributed) stochastic sequence {v(t)} (t =0, 1, . . .) according to the uniform distribution on [0, 1). We At every time t, MS i estimates its received normalized also define a random variable ν(t) (t = 0, 1, . . .) by SNR z(i)(t)/¯ z(i) and examines if z(i)(t)/¯ I(s(k)(t) = 1), If z(i)(t)/¯z(i) is greater than or equal to the threshold γ1 (i.e., the wireless channel state of MS i is in state where I(·) denotes the indicator function. Note here that the 1), MS i transmits on-bit feedback information to the random variable ν(t) represents the number of MSs in state BS. Otherwise (i.e., if the wireless channel state of MS 1 at time t. Let c(i)(t) (i = 1, . . . , K; t = 0, 1, . . .) denote a i is in state 0), MS i does not feedback any information random variable representing the amount of service of MS to the BS. We assume that the one-bit feedback infor- i at time t, i.e., c(i)(t) = 1 when the one-bit feedback fair mation is transmitted through an error-free feedback scheduler selects MS i for downlink transmission at time t, channel from MSs to the BS without delay.
and c(i)(t) = 0 otherwise. c(1)(t) and c(2)(t) can be expressed For downlink transmission, the one-bit feedback sched- uler at the BS randomly selects one of MSs which feed < 1 (s(1)(t) = 1, v(t) [0, 1(t))), back at time t. If there are no MSs who feed back, then (ν(t) = 0, v(t) [0, 1/K)), the scheduler randomly selects one of the K MSs.
The scheduling is performed PHY frame-by-frame.
> 1 (s(2)(t) = 1, v(t) [s(1)(t)(t), (s(1)(t) + 1)(t)]), c(2)(t) = > 1 (ν(t) = 0,v(t) [1/K,2/K)), In this subsection, we consider a wireless channel state process of MS i (i = 1, . . . , K). Let {s(i)(t)} (t = 0, 1, . . . ; i =1, . . . , K) denote the wireless channel state process of MS i, The amount service α(i)(t0, t0 + n) for MS i in [t0, t0 + n) is where s(i)(t) = 1 if z(i)(t)/¯ 1 and s(i)(t) = 0 oth- erwise. We assume that the channel state process {s(i)(t)} (t = 0, 1, . . . ; i = 1, . . . , K) of MS i is well described by a α(i)(t0, t0 + n) = stationary discrete-time 2-state Markov chain [7, 12]. Let P = (pi,j) (i, j = 0, 1) denote the transition probability
We are now ready to provide an expression of the STAFI of matrix of the 2-state Markov chain. The transition prob- the one-bit feedback fair scheduler. Let G ability matrix P is determined as follows (for the detailed
n(x) (n = 1, 2, . . .) derivation of the transition probabilities, see [12]). We first consider the level crossing rate χ(γ) of the received normal- Gn(x) = P((1)(t0, t0 + n) − α(2)(t0, t0 + n)| ≥ x) for any t0. Note here that since we have assumed that all the channel state processes are stationary, Gn(x) is independent 0. We further define the probability mass function gn(x) series of z as ηn(z) = l=−n lzl, where cl (l = −n, . . . , n) (n = 1, 2, . . .) by is a (unknown) real constant satisfying 0 ≤ cl ≤ 1 and Then the probability mass function gn(x) gn(x) = P((1)(t0, t0 + n) − α(2)(t0, t0 + n)| = x). is expressed as gn(x) = cx + c−x. Thus, if we determine the In what follows, we analyze the STAFI G l=−n, we obtain the probability purpose, we define some matrices and vectors. We first de- mass function gn(x). The STAFI Gn(x) is then given by fine a (K − 1) × (K − 1) matrix R by
n(l) = 1 Px−1 There are several possible methods to determine the un- known real constants {cl}nl=−n. In this paper, we use the [R]i,j
inverse discrete FFT method [17] to determine them. Since n(x) has a finite support, i.e., gn(x) = 0 for x > n, we can calculate the exact value of gn(x) by using the inverse · K − 2 − i pj−kpK−2−i−j+k, discrete FFT method. By using this method, we can pre- cisely observe how the STAFI Gn(x) or the probability massfunction g where [R]
n(x) changes as the progress of time (i.e., with the i,j (i, j = 0, . . . , K − 2) denotes the (i, j)th element of R. Note that R is a transition probability matrix of the
For comparison, we consider a random scheduler which Markov chain {r(t)} (t = 0, 1, . . .), where r(t) is defined randomly selects a MS among K MSs irrespective of their I(s(k)(t) = 1). Thus, [R]
received SNR. For the STAFI of the random scheduler, we conditional probability that j MSs among the (K − 2) MSs (excluding MS 1 and MS 2) are in state 1 at time t given n(z) (n = 1, 2, . . .) by that i MSs among the (K − 2) MSs was in state 1 at time t − 1. Let r denote the stationary probability vector of R.
The stationary probability vector r is given by
which corresponds to ηn(z) of the one-bit feedback fair sched- [r]
uler. Similar to the case of the one-bit feedback fair sched- ηn(z), we can calculate the exact value the STAFI where [r]
n(x) and its probability mass function ˜ j (j = 0, . . . , K − 2) denotes the jth element of r,
and s0 and s1 are given by (5). We then define a 4(K − 1) × Although the numerical method based on the inverse dis- 4(K − 1) matrix Q by
crete FFT method provides the exact value of the STAFI Q = P P R,
Gn(x), it is time-consuming when n is large. It is thus dif-ficult to get the exact value of the STAFI Gn(x) for large where denotes the Kronecker product, P is determined
value of n. However, by using the theory of large deviations, by (6) and (7), and R is defined by (8).
we can estimate how fast the STAFI Gn(nx) decreases as We next define a 4(K − 1) × 4(K − 1) diagonal matrix n → ∞. In other words, we can predict how fast the STAFI D(z) by
approaches to ideal fairness as the progress of time. The D(z) = diag(d
following Proposition shows that the STAFI G 0,0(z), d0,1(z), d1,0(z), d1,1(z)),
nentially decreases as n → ∞ (for the derivation, see, e.g., where di,j (z) (i, j = 0, 1) is a 1 × (K − 1) vector given by
< z + z−1 + K − 2 Proposition 1.
[d
log Gn(nx) = min[Λ(x), Λ(−x)], where Λ(a) is defined by Λ(a) = sup [θa − log δ [d0,1(z)]k =
δC (θ) denotes the Perron-Frobenius eigenvalue of the ma-
trix C
().
We hereafter call the term min[Λ(x), Λ(−x)] in Propo- [d1,0(z)]k =
[d1,1(z)]k =
sition 1 the asymptotic decay rate of the STAFI Gn(nx).
for k = 0, . . . , K − 2.
Remark 1.
From the property of Λ(x), it can be shown We then define 4(K 1) × 4(K − 1) matrix C(z) by
that Λ(x) 0, Λ(x) is convex and it attains its minimumat x = 0 [2]. Also, it can be shown that Λ(x) = Λ(−x).
C (z) = QD(z),
where Q and D(z) are defined by (10) and (11), respectively.
n(z) (n = 1, 2, . . .) by n(nx) = Λ(x). ηn(z) = (s s r)C(z)ne,
We also see that the asymptotic decay rate of the STAFI where e denotes a 4(K − 1) × 1 vector whose elements are all
Gn(nx) increases with the increase in x, when the parame- equal to one, s and r are given by (5) and (9), respectively,
ters K, fd, γ1 and Tf are fixed.
and C (z) is defined by (12).
We are now ready to present the analysis of the STAFI n(x). Note that ηn(z) can also be expressed in the power Table 1: Estimations of STAFI
K = 30, γ1 = 3.78dB K = 30, γ1 = 3.78dB K = 20, γ1 = 3.78dB K = 30, γ1 = 3.78dB K = 30, γ1 = 2.00dB K = 30, γ1 = 2.00dB Figure 3: STAFI G
n(hn) as a function of h (γ1 =
3.78dB)
K = 30, γ1 = 6.00dB Figure 4: STAFI G
n(hn) as a function of h (γ1 =
2.00dB)
Figure 2: STAFI G256(x) as a function of x
bit feedback fair scheduler whose threshold γ1 is equal to In this section, we provide numerical results to get insight x, and “RS” means the random scheduler. In this figure, about the properties of the STAFI of the one-bit feedback we observe the following. For almost whole range of x, the fair scheduler. Throughout this subsection, we fix the pa- STAFIs G256(x) of the one-bit feedback fair schedulers are rameters as fd = 10Hz and Tf = 1msec.
greater than the STAFI G256(x) of the random scheduler. In First, we compare numerical results obtained by the in- other words, the short term fairness provided by the one-bit verse discrete FFT method with simulation results to con- feedback fair schedulers is worse than that provided by the firm the correctness of the numerical results. For various random scheduler. This is due to the positive correlation of n, x, the number K of MSs and the threshold γ1, Table 1 the normalized SNR process {z(i)(t)/¯ z(i)} in time. We also shows the STAFI Gn(x) obtained by the inverse discrete see that for small x of G256(x), the one-bit feedback fair FFT method. For comparison, Table 1 also shows the STAFI scheduler with larger threshold γ1 provides better fairness Gn(x) estimated by Monte Carlo simulation. The estima- than that with smaller threshold. However, the situation is tions by Monte Carlo simulation are the averages of 100 converse for large x of G256(x). Thus, the one-bit feedback estimates, and each estimate is the average of 106 samples.
fair scheduler with larger threshold γ1 can keep the proba- Table 1 shows the variances of the estimates, too. In Table 1, bility of moderate unfairness lower, but it can causes serious we confirm that the inverse discrete FFT method yields cor- unfairness with higher probability, compared to the one-bit feedback fair scheduler with smaller threshold.
Next, we observe the effect of the threshold γ1 on the Next we examine how the STAFI of the one-bit feedback STAFI Gn(x). Fig. 2 shows the STAFI G256(x) of the one- fair scheduler changes as the progress of time. Fig. 3 and bit feedback fair scheduler as a function of x.
4 exhibit the STAFI Gn(hn) as a function of h for n = parison, Fig. 2 also shows the STAFI G256(x) of the ran- 64, 128, 256, 512. We set the number K of MSs to 30, the dom scheduler. In the figure, “1FF(xdB)” means the one- threshold γ1 to 3.78dB in Fig. 3 and to 2.00dB in Fig. 4.
Figure 5: STAFI Gn(hn/K) as a function of h (γ1 =
Figure 6: Asymptotic decay rate of STAFI Gn(hn/K)
2.74dB)
as a function of K
for the scheduler with large threshold value, the asymptotic In Fig. 3 and 4, we observe that under the one-bit feedback decay rate of the STAFI Gn(hn/K) greatly decreases with the increase in the number of MSs. This means that for the n(hn) rapidly decreases with the increase of n for every h. In other words, the STAFI of the one- scheduler with large threshold value, the speed approaching bit feedback fair scheduler rapidly approaches to the strict to the strict fairness as the progress of time becomes slow with the increase in the number of MSs.
We next investigate how unfairness between two users is resolved under the one-bit feedback fair scheduling. For this purpose, we consider the STAFI Gn(hn/K) as a function In this paper, we focus on the one-bit feedback fair sched- of n where h is a parameter. Note here that the term n/K uler and numerically study the STAFI of the scheduler to denotes the expected time-access which a user receives when explore its short term fairness properties. We develop two ideal fairness is achieved. Thus the STAFI Gn(hn/K) is numerical methods to understand the transient properties of considered as a measure indicating a deviation from the ideal the STAFI of the one-bit feedback fair scheduler. The first fairness, where a large value of h means a sever deviation method calculates the exact value of the STAFI by using the from the ideal fairness. Fig. 5 displays the STAFI Gn(hn/K) inverse discrete FFT method. It enables us to precisely ob- as a function of the observation period n for h = 2.00, 4.00 serve how the STAFI changes as the progress of time. The denoted by “h=2.0” and “h=4.0”, respectively. In addition, second method estimates the asymptotic decay rate of the Fig. 5 shows the lines exp(−nΛ(h/K)) for h = 2.00, 4.00 STAFI by using the theory of large deviations. It enables us denoted by “ADR(h=2.0)” and “ADR(h=4.0)”, respectively, to predict how fast the STAFI approaches to ideal fairness where Λ(h/K) is the asymptotic decay rate of the STAFI In the numerical results, we observe that the STAFI of rate Λ(h/K) from Proposition 1.
the one-bit feedback fair scheduler rapidly approaches to the the number K of MSs to 16 and the threshold γ1 to 2.74dB.
strict fairness. We also observe that the threshold γ1 affects We observe in the figure that in an asymptotic sense, i.e., as the short term fairness property of the one-bit feedback fair n → ∞, the STAFI Gn(hn/K) decreases exponentially as scheduler. The one-bit feedback fair scheduler with larger stated in Proposition 1. The asymptotic decay rates of the threshold γ1 can keep the probability of moderate unfair- STAFI Gn(hn/K) seem to be equal to Λ(h/K). We also ness lower, but it can causes serious unfairness with higher see that as mentioned Proposition 1, the asymptotic decay probability, compared to the one-bit feedback fair scheduler rate of the STAFI Gn(hn/K) is large when the parameter with smaller threshold. We finally observe that for the one- h is large and K is fixed.
bit feedback fair scheduler with large threshold value, the Finally we observe the effect of the number K of MSs on asymptotic decay rate of the STAFI Gn(hn/K) becomes the asymptotic decay rate of the STAFI Gn(hn/K). Using small with the increase in the number of MSs. We conclude Proposition 1, we estimate the asymptotic decay rate of the that if rigorous fairness is required even in a relatively short STAFI Gn(hn/K). Fig. 6 shows the estimated asymptotic time period, we should consider the short term fairness of decay rates as a function of the number K of MSs for the the scheduler as well as its information theoretic capacity schedulers with the threshold values γ1 = 2.00, 3.78, 6.00dB.
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