# Adaptive Mechanism for Distributed Opportunistic Scheduling

Andres Garcia-Saavedra, Albert Banchs, Pablo Serrano and Joerg Widmer A. Garcia-Saavedra is with Hamilton Institute, Ireland. P. Serrano is with University Carlos III of Madrid. A. Banchs is with University Carlos III of Madrid and Institute IMDEA Networks. J. Widmer is with Institute IMDEA Networks.This paper is an extended version of our paper [1], which was presented at IEEE INFOCOM 2012.
###### Abstract

Distributed Opportunistic Scheduling (DOS) techniques have been recently proposed to improve the throughput performance of wireless networks. With DOS, each station contends for the channel with a certain access probability. If a contention is successful, the station measures the channel conditions and transmits in case the channel quality is above a certain threshold. Otherwise, the station does not use the transmission opportunity, allowing all stations to recontend. A key challenge with DOS is to design a distributed algorithm that optimally adjusts the access probability and the threshold of each station. To address this challenge, in this paper we first compute the configuration of these two parameters that jointly optimizes throughput performance in terms of proportional fairness. Then, we propose an adaptive algorithm based on control theory that converges to the desired point of operation. Finally, we conduct a control theoretic analysis of the algorithm to find a setting for its parameters that provides a good tradeoff between stability and speed of convergence. Simulation results validate the design of the proposed mechanism and confirm its advantages over previous proposals.

## I Introduction

DOS lets stations contend for channel access and, upon successful contention, a station uses its local information about channel conditions to decide whether to transmit data or give up the transmission opportunity. This decision is taken based on a pure threshold policy, i.e., a station gives up its transmission opportunity if the bit rate allowed by the channel falls below a certain threshold. By giving up a transmission opportunity and allowing recontention, it is likely that the channel is taken by a station with better channel conditions, resulting in a higher aggregate throughput. Furthermore, since no coordination between stations is required, DOS protocols are simpler to implement and have a lower control overhead compared to centralized approaches.

The seminal work of [5] provides valuable insights and a deeper understanding of DOS techniques and their performance. Several works [6, 7, 8, 9] extend the basic mechanism of [5] to analyze the case of imperfect channel information [7], improve channel estimation through two-level channel probing [6], and incorporate delay constraints [8]. In turn, [9] proposes the idea of effective observation points to avoid the assumption of independent observations during the probing phase used in [5]. A fundamental drawback of these works is that they only aim to maximize total throughput, an objective that may cause the starvation of those stations with poor link conditions. Heterogeneous links are considered in [10] and [11]. The authors of [10] study the asymptotic sum-rate capacity of MIMO systems that exploit opportunism with a threshold policy, including non-homogeneous users, which requires some global information (like the number of links contending in the network) and assume a Gaussian channel model; in contrast, our approach relies on local information only and does not take any assumption on the distribution of the channel. The authors of [11] consider two types of links which may have different QoS constraints but only optimize the thresholds and do not consider non-saturated stations, whereas we jointly optimize access probabilities and thresholds and support different traffic loads.

The contributions of this paper are the following:

• While previous works only optimize the transmission rate thresholds, we perform a joint optimization of both the thresholds and the access probabilities. Our optimization provides a proportionally fair allocation [12] that achieves a good tradeoff between total throughput and fairness in heterogeneous topologies. Although the derivation of the optimal configuration follows similar ideas as [13], here we use a different approximation which helps us to remove dependencies on global information without compromising performance.

• The second contribution is the design of ADOS, a light adaptive scheme based on control theory, that drives the system to the optimal point of operation with the following advantages:

• ADOS performs well in networks with non-saturated stations.111A saturated station always has data ready for transmission while a non-saturated station may at times have nothing to send. The analysis and design of previous approaches require the assumption that all stations are always saturated, resulting in overly conservative behavior under non-saturation conditions. In contrast, our approach adapts to the actual network load instead of the number of stations, and hence increases the network capacity when there are non-saturated stations.

• ADOS adapts the configuration of the system to the dynamics of the environment, such as mobility or stations joining and leaving the network. In contrast, all previous works (including [1]) assume static radio conditions and therefore can only be applied in scenarios with little or no mobility.

• ADOS only relies on information that can be observed locally, in contrast to previous approaches which need global information and hence require substantial signaling.

• The third contribution of the paper is the control theoretic analysis of the proposed mechanisms. This analysis guarantees the convergence and stability of the mechanism, and provides a configuration of its parameters that achieves a good tradeoff between stability and speed of convergence. Prior approaches [5, 6, 7, 8, 9, 10, 11] do not provide these guarantees.

This paper extends very substantially the work we recently presented in [1]. First, we design a new light algorithm to adapt to changing radio conditions. Previous approaches, including [1], require to re-compute the threshold with some periodicity which can be computationally very costly (e.g., the iterative algorithm proposed in [5], and used in [1], requires solving definite integrals), which precludes a quick adaptation to changes in the channel conditions. The proposed adaptive algorithm is based on control theory, like the algorithm designed in [1] to adjust the access probability. However, both the design of the algorithm and its analysis are entirely novel, as the conditions that determine the optimal point of operation (and hence the algorithm design to drive the system to this point) as well as the system dynamics (and thus the control theoretic analysis to guarantee an appropriate reaction to changing conditions) are different from [1]. Second, we discuss the implementability of ADOS using off-the-shelf devices in §VI. Third, we significantly extend the performance evaluation of the mechanism:

• In addition to comparing ADOS to the team-game approach (TDOS) proposed in [5], we also compare it against the non-cooperative approach (NDOS) of [5] and CSMA/CA, and show that it not only outperforms TDOS, but it performs far better than NDOS and CSMA/CA. This result is very relevant because ADOS, NDOS and CSMA/CA use only local information whereas TDOS requires global information (and thus involves substantial signaling).

• In addition to analyzing and validating the configuration of the algorithm to adapt the thresholds to changing radio conditions, we also compare its performance with the algorithm we presented in [1] for a mobile scenario with different speeds and number of stations.

• We evaluate the proposed algorithm under different load conditions and show that the gains obtained with the proposed approach are even higher than those given in [1] when the load of non-saturated stations is small.

• We assess the performance of all the mechanisms in the presence of channel estimation errors and show that ADOS outperforms all other approaches in this case too.

The rest of the paper is organized as follows. §II presents the analysis of our DOS system and optimizes its configuration in terms of proportional fairness. §III proposes a novel adaptive mechanism, Adaptive Distributed Opportunistic Scheduling (ADOS), that drives the system to the configuration obtained previously. ADOS is analyzed in §IV from a control theoretic standpoint to derive a configuration of the mechanism that provides a good tradeoff between stability and reaction to changes. Its performance is validated via simulations in §V. §VI explains how to implement ADOS with commodity devices. Finally, §VII concludes the paper.

## Ii DOS Optimal Configuration

In the following, we compute the optimal configuration of the access probabilities and transmission rate thresholds of a DOS system for a proportionally fair throughput allocation, which is a well known allocation criterion to provide a good tradeoff between maximizing total throughput (which may be unfairly distributed among stations) and a purely fair allocation (that may waste capacity) [12]. While the analysis conducted in this section assumes saturation conditions, the mechanism that we devise in the next section also takes into account the non-saturated case.

### Ii-a System Model

Similarly to [5, 6, 7, 8, 13], we model our system as a single-hop contention-based wireless network with stations where time is divided into mini slots of fixed duration . At the beginning of each slot, station contends for channel access with a given channel access probability, . A slot can be empty if none of the stations attempt to access the channel. If stations access the channel in the same slot, a collision occurs and the channel is freed for the next slot. There is a successful contention if only one station accesses the medium, which then probes the channel. After this channel probing (which we assume takes one slot), the station has perfect knowledge of the instantaneous link conditions which can be mapped into a reliable transmission bit rate at time . If the available rate is below a given threshold , station gives up its transmission opportunity and frees up the channel for re-contention. Otherwise, the station transmits data for a fixed duration of time . We illustrate the operation of DOS in Fig. 1.

Our model, like that of [5, 6, 7, 8, 13], assumes that remains constant for the duration of a data transmission and that different observations of are independent.222The assumption that remains constant during a transmission is a standard assumption for the block-fading channel in wireless communications [14], while the assumption of independent observations is justified in [5] through numerical calculations. From [5], we have that the optimal transmission policy is a threshold policy: given a threshold , station only transmits after a successful contention if .

With the above model, stations’ throughputs are a function of the access probabilities, , and the transmission rate thresholds, . Given that a proportionally-fair allocation maximizes [12], where is the throughput of station , we define our problem as the following unconstrained optimization problem:

 max¯R,p∑ilogri (1)

### Ii-B Optimal pi configuration

We start by computing the optimal configuration of the parameters. The analysis to compute these parameters follows that of [13], but it relies on different approximations, which are needed for the adaptive mechanism design that we present in §III. To compute the optimal configuration, we start by expressing the throughput as a function of . Let be the average number of bits that station transmits upon a successful contention and be the average time it holds the channel. Then, the throughput of station is

 ri=ps,ili∑jps,jTj+(1−ps)τ

where is the probability that a mini slot contains a successful contention of station and is the probability that it contains any successful contention, .

Both and depend on . Upon a successful contention, a station holds the channel for a time in case it transmits data and in case it gives up the transmission opportunity. In case the station uses the transmission opportunity, it transmits a number of bits given by . Thus, and can be computed as and where is the pdf of . Similarly as in [13], let us define as

 wi=ps,ips,1 (2)

where we take station 1 as reference. From the above equation, we have that ; substituting this into (II-B) yields

 ri=wipsli∑jwjpsTj+∑jwj(1−ps)τ

In a slotted wireless system such as the one of this paper, the optimal access probabilities satisfy (see [15]), which results in the following optimal success probability :

 ps=∑ipi∏j≠i1−pj≈∑ipie−∑jpj=e−1 (3)

With the above, the problem of finding the configuration that maximizes the proportionally fair rate allocation is thus equivalent to finding the values that maximize , given that . To obtain these values, we impose which yields . Combining this expression for and , we obtain

 wiwj=psTj+(1−ps)τpsTi+(1−ps)τ

Under the assumption of small ’s (the case of interest to exploit multiuser diversity with an opportunistic scheduler), , and thus , which leads to . Moreover, given that , the above can be rewritten as

 pipj=Tj+(e−1)τTi+(e−1)τ (4)

Furthermore, the probability that a given mini slot is empty can be computed as follows,

 pe=∏i1−pi≈e−∑ipi=e−1 (5)

We use a different approximation than [13]’s in order to remove any dependency on the number of stations, a result that we will exploit to design an algorithm that works well under non-saturation conditions too. Our simulation results show a very small performance impact for using this approximation instead, practically negligible for scenarios with stations.

With the above, we solve the optimization problem by finding the values that solve the system of equations formed by (4) and (5). The uniqueness of the solution of this system of equations can be proved as follows. Without loss of generality, let us take the access probability of station 1, , as reference. From (4) we have that for can be expressed as a continuous and monotone increasing function of . Applying this to (5), we have that the term () is a continuous and monotone decreasing function of that starts at 1 and decreases to 0, while the right hand side is the constant value . From this, it follows that there is a unique value of that satisfies this equation. Taking the resulting and computing from (4), we have a solution to the system. Uniqueness of the solution is given by the fact that all relationships are bijective and any solution must satisfy (5), which (as we have shown) has only one solution.

Hereafter, we denote the unique solution to the system of equations by . Note that determining requires computing , which depend on the optimal configuration of the thresholds . In the following section we address the computation of the optimal , which we denote by .

### Ii-C Optimal ¯Ri configuration

In order to obtain the optimal configuration of , we need to find the transmission rate threshold of each station that, given the computed above, optimizes the overall performance in terms of proportional fairness.

To this aim, we rely on Theorem 1 in [13] to find that the optimal configuration of the transmission rate thresholds is given by , where is the transmission rate threshold that optimizes the throughput of station when it is alone in the channel and contends with (under the assumption that different channel observations are independent). This is done in [5], which uses optimal stopping theory and finds that the optimal threshold can be obtained by solving the following fixed point equation:

 (6)

Note that the above allows computing the threshold of a station based on local information only, as (6) does not depend on the other stations in the network and their radio conditions. In particular, the optimal threshold configuration is independent of the access probabilities , which is crucial as it allows us to independently design the mechanisms to adjust the configuration of and , respectively, as we explain in the sequel.

In this section, we present the ADOS mechanism, which consists of two independent adaptive algorithms. The first algorithm determines the access probability used by a station, , adjusting the value when the number of active stations in the network or their sending behavior change. The second algorithm determines the transmission rate threshold of a station, , adapting its value to the changing radio conditions of the station. Both algorithms together aim to drive the system to the optimal point of operation. One of the key features of these algorithms is that they do not require to know the number of stations in the network, and they do not need to keep track of the behavior of the other stations or their channel conditions.

### Iii-a Non-saturation conditions

The optimal configuration obtained in the previous section corresponds to the case where all stations are saturated. We next discuss how to consider the case when some of the stations are not saturated. As we explained above, when all the stations are saturated, the optimal channel empty probability takes a constant value equal to , independent of the number of stations. The first key approximation is to assume that this also holds when some of the stations are not saturated. The rationale behind this assumption is that the impact of the aggregated load of several non-saturated stations is similar to the impact of a smaller number of saturated stations. Given that, as we show in §II, the optimal does not depend on the number of stations in saturated conditions, we can assume that when there are non-saturated stations too.

We have also seen in the previous section that, under saturation, the optimal transmission rate thresholds are constant values that only depend on the local radio conditions. The second key approximation is to assume that the optimal transmission rate thresholds take the same constant values under non-saturation. The rationale is as follows. Proposition 3.1 in [5] shows that, additionally to the local radio conditions, the optimal threshold also depends on the number of slots prior to a successful channel access. As the mechanism we describe below drives the system to a point of operation where even if there are non-saturated stations, we can assume that the optimal threshold in this case is the one given by (6) for saturated stations.

We next present the design of the algorithms to adjust and that consider both saturation and non-saturation conditions following the two approximations exposed above.

### Iii-B Adaptive algorithm for pi

Following the first approximation above, with ADOS each station implements an adaptive algorithm to configure the access probability , with the goal of driving the channel empty probability to , as given by (3).

Driving the channel empty probability toward a constant optimum value fits well with the framework of classic control theory. With these techniques, we measure the output signal of the system and, by judiciously adjusting the control signal, we aim at driving it to the reference signal. A key advantage of using such techniques is that they provide the means for achieving a good tradeoff between the speed of reaction and stability while guaranteeing convergence, which is a major challenge when designing adaptive algorithms.

Fig. 2 depicts our algorithm to adjust , where each station computes the error signal by subtracting the output signal from the reference signal (the functions in the figure are given in the domain). The output signal is combined with a noise component of zero mean, modeling the randomness of the channel access algorithm. In order to eliminate this noise, we follow the design guidelines from [16] and introduce a low-pass filter . The filtered error signal is then fed into the controller of each station, which provides the control signal , defined as the average time between two transmission of station . Station then computes its access probability as . With the of each station, the wireless network provides the output signal , which closes the loop.

In the above system, we need to design the reference and output signals and , as well as the transfer functions of the low-pass filter and the controller, and . We address next their design with the goal of ensuring that the empty probability is driven to .

In our system, time is divided into intervals such that the end of an interval corresponds to a transmission (either a success or a collision). Given that the target empty probability is equal to , the target average number of empty mini slots between two transmissions (i.e., our reference signal) is equal to . In this way, after the -th transmission, each station computes the output signal at interval , denoted by , as the number of empty mini slots between the -th and the -th transmission. The error signal for the next interval is computed as

 Ep(n+1)=Rp−Op(n). (7)

With the above, if is too large then will be larger than in average, yielding a negative error signal that will decrease for the next interval, which will increase the transmission probability and therefore reduce (and vice-versa). This ensures that will be driven to the optimal value.

For the low-pass filter , we use a simple exponential smoothing algorithm of parameter [17], given by the following expression in the time domain, , which corresponds to the following transfer function in the domain: . For the transfer function of the controllers , we use a simple controller from classical control theory, namely the Proportional Controller [18], which has already been used in a number of networking problems (e.g. [19, 20]), i.e., , where is a per-station constant.

In addition to driving the empty probability to , we also impose that the access probabilities satisfy (4). Since we feed the same error into all stations, and the proportional controller simply multiplies this error by a constant to compute , the following equation holds for all :

 pipj=Kp,jKp,i

Therefore, by simply setting as , we ensure that (4) is satisfied.

### Iii-C Adaptive algorithm for ¯Ri

Following the second approximation of §III-A, the adaptive algorithm of ADOS to adjust the threshold aims to drive the threshold of all (saturated and non-saturated) stations to the optimal value given by (6). Note that (6) is equivalent to the following equation:

 E[(Ri(t)−¯R∗i)+−¯R∗iτT/e]=0 (8)

In the following, we design an adaptive algorithm that drives to the value given by the above equation. The algorithm is depicted in Fig. 3. Similarly to the adaptive algorithm for , we base the algorithm design on control theory. The key difference between the two algorithms is that, since the optimal value of threshold of a station depends on local information only and hence does not depend on the threshold value of the other stations, we can consider each station separately (in contrast to Fig. 2).

In order to ensure that the configuration of satisfies (8), we design the output signal of the algorithm, , equal to the term , and the reference signal, , equal to the term . Thus, by driving the difference with these two terms (i.e., the error signal) to zero, we ensure that (8) is satisfied.

Following the above, upon its successful contention, a station measures the channel transmission rate and computes the output signal as

 OR(n)={Ri(n)−¯Ri(n),if Ri(n)>=¯Ri(n)0,otherwise

From the above output signal, it then computes the error signal as

 ER(n+1)=OR(n)−¯Ri(n)τT/e

Due to the randomness of the radio signal, the output signal carries some noise . In order to filter out this noise, we apply (like in the previous case) a low pass-filter on the error signal, which yields . Also like in the previous case, the error signal is introduced into a proportional controller, , where is the constant of the controller.

The controller gives the threshold configuration as output. As mentioned above, by driving the error signal to 0, the controller ensures the threshold value satisfies (8) and thus achieves the objective of adjusting the treshold to the optimal value obtained in §II.

## Iv Control Theoretic Analysis

With the above, we have all the components of the ADOS mechanism fully designed. The remaining challenge is the setting of its parameters, namely the parameters of the adaptive algorithm for ( and ) and the adaptive algorithm for ( and ). In this section, we conduct a control theoretic analysis of the algorithms to find a suitable parameter setting.

As discussed in §II, the setting of the optimal threshold does not depend on the configuration of . Based on this, we analyze the closed-loop behavior of the two adaptive algorithms independently. For the adaptive algorithm to adjust , the behavior is independent of the configuration. For the algorithm to adjust , we consider that the values of are fixed, as their configuration depends only on the radio conditions, and analyze the convergence of to the optimal configuration corresponding to these values.

In the following, we first analyze the adaptive algorithm to adjust and then we analyze the one to adjust ; these analyses provide good values for the parameters of the respective algorithms.

### Iv-a Analysis of the algorithm for pi

We next conduct a control theoretic analysis of the closed-loop system of the algorithm for to find good values for the parameters and . Fig. 4 depicts the closed-loop system for this algorithm. Note that the term in the figure shows that the error signal at a given interval is computed with the output signal of the previous interval.

In order to analyze this system from a control theoretic standpoint, we need to characterize the transfer function , which takes as input and gives as output. The following equation gives a nonlinear relationship between and :

 Op=11−pe−1

where .

To express the above relationship as a transfer function, we linearize it when the system suffers small perturbations around its stable point of operation. Then, we study the linearized model and force that it is stable. Note that the stability of the linearized model guarantees that our system is locally stable.333We assess stability from a control theory standpoint (a similar approach was used in [21] to analyze RED), in contrast to other analyses of schedulers such as [22] which look at the stability of the system queues from a queuing theory perspective.

We express the perturbations around the stable point of operation as follows:

 ti=t∗i+Δti

where is the stable point of operation of , and are the perturbations around this point of operation.

With the above, the perturbations suffered by can be approximated by where

 ∂Op∂tj=∂Op∂pj∂pj∂tj=pep2j(1−pj)(1−pe)2.

Given that , the above can be rewritten as

 ΔOp=(∑j(Tj+(e−1)τ)pep2j(Ti+(e−1)τ)(1−pj)(1−pe)2)Δti

With the above, we have characterized :

 Hp,i=∑j(Tj+(e−1)τ)pep2j(Ti+(e−1)τ)(1−pj)(1−pe)2

The closed-loop transfer function for station is then given by

 Tp,i(z)=−z−1Cp,i(z)Fp(z)Hp,i(z)1+z−1Cp,i(z)Fp(z)Hp,i(z)

Substituting the expressions for , and yields

 Tp,i(z)=−αpHp,iKp,iz−(1−αp−αpKp,iHp,i) (9)

To guarantee stability, we need to ensure that the zero of the denominator of falls inside the unit circle [23], which implies

 Kp<2−αpαp1∑j(Tj+(e−1)τ)pep2j(1−pj)(1−pe)2

The problem with the above upper bound is that it depends on the number of stations and their channel conditions. In order to assure stability, we need to obtain an upper bound that guarantees stability independent of these parameters. To do this, we observe that the right hand side of the above inequality takes a minimum value when and . Therefore, by setting as follows, we guarantee that the above inequality will be met independent of the number of stations and their channel conditions:

 Kp

In order to set to a value that provides a good tradeoff between the speed of reaction to changes and stability, we follow the Ziegler-Nichols rules [18], which are widely used to configure proportional controllers. According to these rules, this parameter cannot be larger than one half of the maximum value that guarantees stability, which we denote by :

 Kp≤Kstabilityp=Kmaxp2 (10)

In addition to the above, also needs to be set to eliminate the noise from the system. Noise is generated by the randomness of the output signal, which is given by the number of empty mini slots between two transmissions and hence follows a geometric random variable of factor . Hence, the noise at the input of the low-pass filter has a zero mean and a variance given by:

 E[W2p]=pe(1−pe)2=1/e(1−1/e)2

The noise at the output of the controller can be obtained from the noise at the input of the low-pass filter with the following transfer function:

 TWp(z)=−z−1Cp,i(z)Fp(z)1+z−1Cp,i(z)Fp(z)Hp,i(z)

Substituting , and into the above yields

 TWp(z)=−z−1αpKp,i1−z−1(1−αp(1+Kp,iHp,i))

With the above transfer function, we can compute the variance of the noise at the output of the controller, denoted by , as follows:

 E[W2p,c]=α2pK2p,i1−(1−αp(1+Kp,iHp,i))2E[W2p]

From the above equation, and taking into account from (9) and (10) that we can obtain the following upper bound for :

 E[W2p,c]≤αpKp,i(1−αp/2)Hp,iE[W2p]

To limit the impact of the noise, we impose a gain factor of at least of the signal level at the output of the controller, , over the noise level at the same point, :

 E[S2p]E[W2p,c]≥Gp

The signal at the output of the controller is equal to , which yields . Combining this with the inequality of (IV-A), we have that the following condition is sufficient to provide the desired gain:

 t2i(1−αp/2)Hp,iαpKp,iE[W2p]≥Gp

Isolating from the above yields

 Kp≤t2i(1−αp/2)GpαpE[W2p]∑j(Tj+(e−1)τ)pep2j(Ti+(e−1)τ)2(1−pj)(1−pe)2

which is satisfied as long as the following condition holds,

 Kp≤1−αp/2Gpαp∑jTj+(e−1)τ(Ti+(e−1)τ)2

To find an upper bound that is independent of the number of stations and their conditions, we observe that the right hand side of the above inequality takes a minimum for and , which leads to the following upper bound, which we denote by ,

 Kp≤Knoisep=1−αp/2Gpαp(T+eτ)

The analysis conducted in this section has given two upper bounds, and , which guarantee that on the one hand the system is stable and on the other hand the noise level is not excessive. As these bounds depend on and , we also need to find a setting for these parameters. In order to provide a good level of protection against noise, needs to be sufficiently large. Additionally, in order to allow sufficiently large values, which is needed to avoid a large steady state error at the input of the controllers, needs to be sufficiently small. Following these considerations, we set and . With we aim to mitigate the effect of the noise sufficiently, without compromising the speed of reaction to changes (i.e., in the order of magnitude of 1000 samples). With we set an upper bound to the noise power, i.e., we enforce a gain of the output signal of the controllers which is 100 times larger than the noise. With these and values, we then configure as follows:

 Kp=min(Knoisep,Kstabilityp)

which ensures that the two objectives concerning stability and noise are met.

### Iv-B Analysis of the algorithm for ¯Ri

We next conduct a control theoretic analysis of the closed-loop system of the algorithm for , depicted in Fig. 5. This analysis follows the same steps as the one above.

The perturbations around the point of equilibrium can be expressed as and the perturbations suffered by can be approximated by where

 HR= ∂ER∂¯Ri=∂∂¯Ri((Ri−¯Ri)+−¯RiτT/e)=∂(Ri−¯Ri)+∂¯Ri−τT/e

To compute , we note that expresses an average value, as the variations around this average value are captured by another component, namely the noise . For the calculation of the average, we take all possible values weighted by ’s pdf, , which yields

 ∂(Ri−¯Ri)+∂¯Ri =∂∂¯Ri∫∞¯Ri(r−¯Ri)fRi(r)dr=−∫∞¯RifRi(r)dr

With the above, can be expressed as , where and .

The closed-loop transfer function of the system is given by

 TR(z)=CR(z)FR(z)HR(z)1−z−1CR(z)FR(z)HR(z)

where

 FR(z)=αR1−(1−αR)z−1,   CR(z)=KR.

Substituting the expressions for , and yields

 TR(z)=−αRKR(HR,1+HR,2)1−z−1(1−αR−KRαR(HR,1+HR,2))

To guarantee stability, we need to ensure that the zero of the denominator of falls inside the unit circle , which implies

 KR<2−αRαR(HR,1+HR,2)

In order to find a sufficient condition that holds for all cases, we consider the worst case , which leads to

 KR<2−αRαR(1+eτ/T)

According to Ziegler-Nichols rules, to guarantee stability we take a value equal to half of the above value,

 KstabilityR=2−αR2αR(1+eτ/T)

The noise introduced into the system, , is given by the randomness in the transmission rate values . If we assume that the available transmission rate for a given SNR is given by the Shannon channel capacity, then , where is a constant parameter, is the SNR and is the normalized random gain of the channel (). Note that the values of below are eliminated from the system by the module that performs the operation , which reduces the noise in the system. In what follows, we do not consider this effect in order to obtain an upper bound on the noise, which provides a worst case analysis.

If we represent the SNR as the sum of its average value () plus some noise of zero mean (which we denote by ), then we can express the transmission rates as which we can approximate at the stable point of operation () by

 Ri≈Clog(1+ρ)+Wh∂Ri∂Wh∣∣∣Wh=0

Since the noise introduced into the system is given by the variations of around its average value, from the above we have that we can approximate by

 WR≈Wh∂Ri∂Wh∣∣∣Wh=0=C1+ρWh

With the above approximation, we can compute the variance of as follows,

 E[W2R]=C2(1+ρ)2E[W2h]

If we assume that the channel follows a Rayleigh fading model, then corresponds to an exponential random variable of rate . With this, we have that , which yields .

If we denote the noise at the output of the controller by , we have

 WR,c(z)=FR(z)CR(z)1−z−1FR(z)CR(z)HR(z)WR(z)

from which

 WRc(z)=αRKR1−z−1(1−αR−KRαR(HR,1+HR,2))WR(z)

From the above, the variance of the noise at the output of the controller can be computed as

 E[W2R,c]=(αRKR)21−(1−αR(1+KR(HR,1+HR,2)))2E[W2R]

Given that , we can obtain the following upper bound on :

 E[W2R,c]≤αRKR(HR,1+HR,2)(1−αR/2)E[W2R] (11)

In order to guarantee a gain of of the signal over the noise at the output of the controller, we impose

 E[S2R]E[W2R,c]≥GR (12)

where the signal is the threshold , which we approximate by the average transmission rate, . With this and the upper bound of (11) for , we can obtain the following sufficient condition to guarantee (12):

 (log(1+ρ)(1+ρ)ρ)2(HR,1+HR,2)(1−αR/2)αRKR≥GR

Isolating from the above yields

 KR≤(log(1+ρ)(1+ρ)ρ)2(HR,1+HR,2)(1−αR/2)αRGR

In order to find a value of that ensures the desired gain for all scenarios, we chose the value that minimizes the right hand side of the above equation and take the worst case value for , which leads to the following upper bound on , which we denote by ,

 KR≤KnoiseR=eτ(1−αR/2)TαRGR

Following the rationale of §IV-A, we set and and choose , which ensures that the two goals in terms of noise and stability are met.

## V Performance Evaluation

In this section, we present a performance evaluation of ADOS by means of simulations. Unless otherwise stated, we assume that different observations of the channel conditions are independent and that the available transmission rate for a given SNR is given by the Shannon channel capacity: , where is the channel bandwidth in Hz, is the normalized average SNR and is the random gain of Rayleigh fading. Unless otherwise stated, we set and , i.e., the same values used in [5], for comparison purposes. We also set and run enough simulations to obtain 95% confidence intervals below 1%.

### V-a Homogeneous scenario

#### V-A1 Saturated stations

We start by considering a homogeneous scenario where all stations are saturated and have the same normalized average SNR (). We compare the performance of ADOS to the following approaches:

• The static optimal configuration obtained from performing an exhaustive search over the space and choosing the best configuration (‘static configuration’).

• An approach that, although it probes the channel too (to avoid long collisions), it never skips a transmission opportunity regardless of the estimated link quality (‘non-opportunistic’).

• A CSMA/CA protocol which does not skip any transmission opportunity but it does not probe the channel so collisions last for the duration of a frame.

• The team game approach proposed in [5] (TDOS). This approach requires that each station knows the channel state of all the stations in the network, and hence incurs substantial signaling overhead. In the simulations we assume that this overhead is non-existent.

• The non-cooperative approach proposed in [5] (NDOS). This approach, like ours, only requires information that can be observed locally, and hence does not involve any signaling.444Since [5] only optimizes the transmission rate thresholds but not the access probabilities, we take the ’s used in the simulations of [5] for TDOS and NDOS. For ‘non-opportunistic’, we choose the access probabilities that maximize performance.

Fig. 6 shows the total throughput as a function of the number of stations in the network. The figure confirms that ADOS is effective in driving the system to the optimal point of operation, providing the same throughput as the benchmark given by the ‘static configuration’. The TDOS and NDOS approaches provide lower throughput as they only optimize the transmission rate thresholds; among them, NDOS performs substantially worse as it has less information. Finally, the ‘non-opportunistic’ approach provides the lowest throughput due to the lack of opportunistic scheduling. In conclusion, the proposed ADOS mechanism provides optimal throughput performance, outperforming the other approaches.

#### V-A2 Non-saturated stations

We now assess the performance in the presence of non-saturation stations (that do not always have data ready for transmission). We first consider a scenario with homogeneous radio conditions () with one saturated station and non-saturated stations. Figs. (a)a and (b)b illustrate the total throughput of the network as a function of the number of stations, when the non-saturated stations transmit at one half and one tenth of their saturation throughput (i.e., the throughput the would obtain if they were saturated). We observe that ADOS significantly outperforms all other approaches and that this effect becomes more accentuated as the throughput of the non-saturated stations decreases. The reason is that the other approaches assume that all stations are always saturated, and thus the access probabilities they use become overly conservative for the non-saturated case.

### V-B Heterogeneous scenario

In the case of heterogeneous channel conditions, performance does not only depend on the total throughput but also on the way this throughput is shared among the stations. To analyze performance in this scenario, we consider saturated stations divided into four groups according to their channel conditions. The normalized SNR of the stations from group is given by , with . Fig. (a)a shows , the figure of merit for proportional fairness, as a function of . We observe that ADOS performs at the same level as the benchmark given by the ‘static configuration’, while the other approaches provide a substantially lower performance. TDOS exhibits an increasing degree of unfairness as grows that harms its performance in terms of proportional fairness. NDOS, in contrast to TDOS, does not show this behavior: with NDOS, each station sets its threshold based on its local radio conditions and therefore the fact that other stations have better radio conditions does not impact fairness. The price that NDOS pays for this non-cooperative behavior, however, is that the overall throughput performance is substantially degraded for all values. The ‘non-opportunistic’ approach and CSMA/CA also provide poor performance, similar to NDOS.

In order to gain additional insight into the throughput distribution with heterogeneous radio conditions, Fig. (b)b depicts the throughput obtained by a station of each group with the different approaches, along with the Jain’s fairness index (JFI) [24] of each distribution. The results confirm that TDOS suffers from high unfairness with heterogeneous radio conditions, since with this approach the stations with worst radio conditions () are almost starved while the stations with best radio conditions () obtain a very large throughput. In contrast, the TDOS, ‘non-opportunistic’ and CSMA/CA approaches do not suffer from unfairness but provide significantly smaller throughputs than ADOS. We conclude that ADOS substantially outperforms all other approaches with heterogeneous radio conditions.

### V-C Performance under realistic models

#### V-C1 Impact of channel coherence time

Our channel model is based on the assumption that different observations of the channel conditions are independent. In order to understand the impact of this assumption, we repeated the experiment of Fig. (a)a using Jakes’ channel model [25] to obtain channel conditions that are correlated over time. The results, for a Doppler frequency of (which roughly corresponds to 100 Km/h at 2.4 GHz), are given in Fig. (a)a where ADOS outperforms all the others. We also observe that the performance is slightly lower than that of Fig. (a)a. This is due to the fact that when the channel is bad, a station does not transmit after a successful contention, and therefore it takes a shorter time until it successfully contends again. Thus, a station accesses the channel more often when the channel is bad than when it is good, which introduces a bias that reduces throughput.

#### V-C2 Discrete set of transmission rates

While all previous experiments assumed continuous rates, the design of ADOS do not rely on any assumption on the mapping of SNR to transmission rates, and therefore any mapping function (continuous or discrete) can be used. We consider the case of a wireless system in which the only transmission rates available are Mbps. For a given SNR, we choose the largest available transmission rate that is smaller than the one given by Shannon channel capacity model. Fig. (b)b shows the result of repeating the experiment of Fig. (a)a with this discrete set of transmission rates. The results confirm that ADOS outperforms the other approaches with different mapping functions.

#### V-C3 Imperfect channel estimation

Our design assumes that the channel state is perfectly known to the transmitter. However, real estimators often have to deal with noisy observations and produce inaccurate results, which may worsen performance or even cause outage in the communication. Yet, according to [7], the optimal threshold still has a threshold structure under these conditions. To assess the performance in the presence of estimation errors, we model the measured SNR as , where is the random estimation error with average , and, following the scheme proposed in [7], we select a linear function to back off from the estimated bit rate which is equal to . We evaluate the same heterogeneous scenario as before for , and plot in Fig. (c)c the performance as a function of for all the schemes under evaluation, revealing that ADOS also outperforms all the others in this case.

### V-D Validation of the configuration proposed for ADOS

The analysis in §IV derives the guidelines to configure the parameters of ADOS ( and ) in order to guarantee a good behavior over time (stability and convergence speed). We next validate such guidelines in contrast to other settings that deviate from them.

#### V-D1 Static conditions

To verify stable behavior in a static environment, we first observe the evolution over time of the access probability of a station for the proposed setting and for a configuration of these parameters 10 times larger, in a homogeneous scenario with saturated stations and . Fig. (a)a shows the evolution of for both cases, sampled over intervals. We observe from the figure that with the proposed setting (labeled “”), shows minor deviations around its average value, while for a larger setting (labeled “”), it shows unstable behavior with drastic oscillations.

Similarly, we also observe the evolution over time of the threshold of a station for the proposed setting and for a configuration of these parameters 10 times larger in the same scenario. The results, depicted in Fig. (b)b confirm that the proposed setting for these parameters is stable while a larger setting is highly unstable. We conclude from these results that the analysis conducted in §IV is effective in guaranteeing stability.

#### V-D2 Changing number of stations

We next investigate the speed with which the system reacts to changes in the number of stations of the network, which triggers the adjustment of the access probabilities . To this aim, we consider a network with initially 5 stations, where 5 additional stations join the network after a time . Fig. (a)a shows the evolution of the access probability of one of the initial stations sampled over intervals. We observe from the figure that with our setting (labeled “”), the system quickly adapts the of the station to the new value. In contrast, for a setting of these parameters 10 times smaller (labeled “”), the reaction is very slow and the system only converges after .