Distributed Rate Allocation for Wireless Networks

# Distributed Rate Allocation for Wireless Networks

## Abstract

This paper develops a distributed algorithm for rate allocation in wireless networks that achieves the same throughput region as optimal centralized algorithms. This cross-layer algorithm jointly performs medium access control (MAC) and physical-layer rate adaptation. The paper establishes that this algorithm is throughput-optimal for general rate regions. In contrast to on-off scheduling, rate allocation enables optimal utilization of physical-layer schemes by scheduling multiple rate levels. The algorithm is based on local queue-length information, and thus the algorithm is of significant practical value.

The algorithm requires that each link can determine the global feasibility of increasing its current data-rate. In many classes of networks, any one link’s data-rate primarily impacts its neighbors and this impact decays with distance. Hence, local exchanges can provide the information needed to determine feasibility. Along these lines, the paper discusses the potential use of existing physical-layer control messages to determine feasibility. This can be considered as a technique analogous to carrier sensing in CSMA (Carrier Sense Multiple Access) networks. An important application of this algorithm is in multiple-band multiple-radio throughput-optimal distributed scheduling for white-space networks.

[theorem]Lemma

## 1Introduction

The throughput of wireless networks is traditionally studied separately at the physical and medium access layers, and thus independently optimized at each of these two layers. As a result, conventionally, data-rate adaptation is performed at the physical layer for each link, and link scheduling is performed at the medium access layer. There are significant throughput gains in studying these two in a cross-layer framework [27]. This cross-layer optimization results in a joint rate allocation for all the links in the network.

Maximum Weighted (Max-Weight) scheduling introduced in the seminal paper [27] performs joint rate allocation and guarantees throughput-optimality1. However, Max-Weight algorithm and its variants have the following disadvantages. (a) It requires periodic solving of a possibly hard optimization problem. (b) The optimization problem is centralized, and thus introduces significant overhead due to queue-length information exchanges. Thus, in order to overcome these disadvantages, we need efficient distributed algorithms for general physical-layer interference models [19].

The goal of this paper is to perform joint rate allocation in a decentralized manner. A related problem is distributed resource allocation in networks, and this problem has received considerable attention in diverse communities over years. In data and/or stochastic processing networks, resource-sharing is typically described in terms of independent set constraints. With such independent set constraints, the resource allocation problem translates to medium access control (or link scheduling) in wireless networks. For such on-off scheduling, recently, efficient algorithms have been proposed for both random access networks [12] and CSMA networks [21]. More recently, with instantaneous carrier sensing, a throughput-optimal algorithm with local exchange of control messages that approximate Max-Weight has been proposed in [25], and a fully decentralized algorithm has been proposed in [15]. The decentralized queue-length based scheduling algorithm in [15] and its variants have been shown to be throughput-optimal in [14]. This body of literature on completely distributed on-off scheduling has been extended to a framework that incorporates collisions in [16]. Further, this decentralized framework has been validated through experiments in [18].

However, independent set constraints can only model orthogonal channel access which, in general, is known to be sub-optimal [5] (Section ). For wireless networks, the interaction among nodes require a much more fine-grained characterization than independent set constraints. This can be fully captured in terms of the network’s rate region, i.e., the set of link-rates that are simultaneously sustainable in the network. As long as the data-rates of links are within the rate region, simultaneous transmission is possible even by neighboring links in the network. Therefore, it is crucial to perform efficient distributed joint rate allocation (and not just distributed link scheduling) in wireless networks. Although distributed rate allocation is a very difficult problem in general, in this work, we show that this problem can be solved by taking advantage of physical-layer information.

In this work, we consider single-hop2 wireless networks. We develop a simple, completely distributed algorithm for rate allocation in wireless networks that is throughput-optimal. In particular, given any rate region for a wireless network, we develop a decentralized (local queue-length based) algorithm that stabilizes all the queues for all arrival rates within the throughput region. Thus, we can utilize the entire physical-layer throughput region of the system with distributed rate allocation. To the best of our knowledge, this is the first paper to obtain such a result. This is a very exciting result as our decentralized algorithm achieves the same throughput region as optimal centralized cross-layer algorithms. The algorithm requires that each link can determine the global feasibility of increasing its data-rate from the current data-rate. In Section 8.1, we provide details on techniques to determine rate feasibility, and explain reasons for using this approach in practice.

The framework developed in this paper generalizes the distributed link scheduling framework. As discussed before, the current distributed link scheduling algorithms primarily deal with binary (on-off) decisions whereas our algorithm performs scheduling over multiple data-rates. Similar to these existing distributed link scheduling algorithms, our algorithm is mathematically modeled by a Markov process on the discrete set of data-rates. However, with multiple data-rates for each link, the appropriate choice of the large number of transition rates is very complicated. Thus, a key challenge is to design a Markov chain with fewer parameters that can be analyzed and appropriately chosen for throughput-optimality. We overcome this challenge by showing that transition rates with the following structure have this property. For link , the transition rate to a data-rate from any other data-rate is , where is a single parameter associated with link that is updated based on its queue-length. The transition takes place only if the new data-rate is feasible. As expected, this reduces to the existing algorithmic framework in the special case of binary (on-off) decisions.

For the general framework mentioned above, at an intuitive level, the techniques required for proving throughput-optimality remain similar to existing techniques. However, there are few additional technical issues that arise while analyzing the general framework. First, we need to account for more general constraints that arise from the set of possible rate allocation vectors. Next, the choice of update rules for with time based on local queue-lengths that guarantee throughput-optimality does not follow directly. The mixing time of the rate allocation Markov chain plays an important role in choosing the update rules. For arbitrary throughput regions, any rate allocation algorithm that approach -close (for arbitrarily small ) to the boundary possibly requires an increasing number of data-rates per link. This leads to a potential increase in the mixing time due to the increase in the size of the state-space. Thus, the analysis performed in this paper is more general and essential to establish throughput-optimality of the algorithms considered.

An important application of this algorithmic framework is for networks of white-space radios [7], where multiple non-adjacent frequency bands are available for operation and multiple radios are available at the wireless nodes. A scheduler needs to allocate different radios to different bands in a distributed manner. This problem introduces multiple data-rates for every link even in the CSMA framework, and hence, existing distributed algorithms cannot be directly applied. We demonstrate that our framework provides a throughput-optimal distributed algorithm in this setting.

Our main contributions are the following:

• We design a class of distributed cross-layer rate allocation algorithms for wireless networks that utilize local queue-length and physical-layer measuring.

• We show that there are algorithms in this class that are (a) throughput-optimal, and (b) completely decentralized.

• We demonstrate that an adaptation of these algorithms are throughput-optimal for multiple-band multiple-radio distributed scheduling.

### 1.1Notation

Vectors are considered to be column vectors and denoted by bold letters. For a vector and matrix , , where is the transpose of . For vectors, , , , and are defined component-wise. denotes all-zeros vector and denotes all-ones vector. Other basic notation used in the paper is given in Table ?. Notation specific to proofs is introduced later as needed.

### 1.2Organization

The next section describes the system model. Section 3 explains the distributed rate allocation algorithm. Section 4 introduces relevant definitions and known results. Section 5 describes the rate allocation Markov chain and the optimization framework. Section 6 establishes the throughput-optimality of the algorithm. The algorithm for multiple-band multiple-radio scheduling is given in Section 7. Further discussions and simulation results are given in Section 8. We conclude with our remarks in Section 9. For readability, the proofs of the technical lemmas in Section 5 and Section 6 are moved to the Appendix.

## 2System Model

Consider a wireless network consisting of nodes, labeled . In this network, we are interested in single-hop flows that correspond to wireless links labeled . Since we have a shared wireless medium, these links interact (or interfere) in a potentially complex way. For single-hop flows, this interaction among links can be captured through a -dimensional rate region for the network, which is formally defined next.

In this paper, we assume that the rate region is fixed3 (i.e., not time-varying). We denote the rate region associated with the network by . By definition, this rate region is compact. We assume that the rate region has the following simple property: if , then for all and . The above property states that rates can be decreased component-wise. Such an assumption is fairly mild, and is satisfied by rate regions resulting from most physical-layer schemes. Next, we define the throughput region of the network.

We use a continuous-time model to describe system dynamics. Time is denoted by Every (transmitter of) link is associated with a queue , which quantifies the information (packets) remaining at time waiting to be transmitted on link . Let the cumulative arrival of information at the -th link during the time interval be with . Rate allocation at time is defined as the rate vector in the rate region at which the system is being operated at time . Let the rate allocation corresponding to the -th link at time be . Then, for every link , the queue dynamics is given by

where . The vector of queues in the system is denoted by . The queues are initially at .

We consider arrival processes at the queues in the network with the following properties.

• We assume every arrival process is such that increments over integral times are independent and identically distributed with

• We assume that all these increments belong to a bounded support , i.e., for all .

Based on these properties, the (mean) arrival rate corresponding to the -th link is . We denote the vector of arrival rates by . Without loss of generality4, we assume . It follows from the strong law of large numbers that, with probability ,

In summary, our system model incorporates general interference constraints through a arbitrary rate region and focuses on single-hop flows. We proceed to describe the rate allocation algorithm and the main results of this paper.

## 3Rate Allocation Algorithm & Main Results

The goal of this paper is to design a completely decentralized algorithm for rate allocation that stabilizes all the queues as long as the arrival rate vector is within the throughput region. By assumption, every link can determine rate feasibility, i.e., every link can determine whether increasing its data-rate from the current rate allocation results in a net feasible rate vector. More formally, every link at time , if required, can obtain the information More details on determining rate feasibility are given in Section 8.

The rate allocation vector at time is denoted by . For decentralized rate allocation, we develop an algorithm that uses only local queue information for choosing over time . Further, we perform rate allocation over a chosen limited (finite) set of rate vectors that are feasible. We choose a finite set of rate levels corresponding to every link, and form vectors that are feasible. The details are as follows:

1. For each link , a set of rate levels are chosen from with , and . Here, is the maximum possible transmission rate for the -th link, i.e., , and is the number of levels other than zero. Since the rate region is compact, without loss of generality5, we assume .

2. The set of rate allocation vectors, denoted by , is given by

The convex hull of the set of rate allocation vectors is denoted by . Define the set of strictly feasible rates. For rate regions that are polytopes, the partitions can be chosen such that . For any compact rate region, it is fairly straightforward to choose partitions with such that if . The trivial partition with as step size in all dimensions satisfy the above property. Thus, for any given , we can obtain a set of rate allocation vectors such that

and if .

Before describing the algorithm, we define two notions of throughput performance of a rate allocation algorithm.

Next, we describe a class of algorithms to determine as a function of time based on a continuous-time Markov chain. Recall that is the set of possible rates/states for allocation associated with the -th link. In these algorithms, the -th link uses independent exponential clocks with rates/parameters6 (or equivalently exponential clocks with mean times ). The clock with (time varying) parameter is associated with the state . Based on these clocks, the -th link obtains as follows:

1. If the clock associated with a state (say ) ticks and further if transitioning to that state is feasible, then is changed to ;

2. Otherwise, remains the same.

The above procedure continues, i.e, all the clocks run continuously. Define . It turns out that the appropriate structure to introduce is as follows:

where We denote the vector consisting of these new set of parameters by .

A distributed algorithm needs to choose the parameters in a decentralized manner. For providing the intuition behind the algorithm, we perform this in two steps. In the first step, we develop the non-adaptive version of the algorithm that has the knowledge of . This algorithm is called non-adaptive as the algorithm requires the explicit knowledge of . The rate allocation at time is set to be . This algorithm uses at all times which is a function of , and is given by

We show in Section 5 that, given , the above optimization problem has a unique solution that is finite, and therefore has a valid . An important result regarding this non-adaptive algorithm is the following theorem.

In the second step, we develop the adaptive algorithm, where is obtained as a function of time denoted by This algorithm is called adaptive as the algorithm does not require the knowledge of . The values of are updated during fixed (not random variables) time instances for . We set and . During interval the algorithm uses . The length of the intervals are . During interval , let the empirical arrival rate be

and the empirical offered service rate be

The update equation corresponding to the algorithm for the -th link is given by

where , i.e., is the projection of to the closest point in , and are the step sizes. Thus, the algorithm parameters are interval lengths , step sizes and .

The following theorem provides -optimal performance guarantee for the adaptive algorithm.

## 4Definitions & Known Results

We provide definitions and known results that are key in establishing the main results of this paper. We begin with definitions on two measures of difference between two probability distributions.

Next, we provide two known results that are used later. Result ? follows directly from [3](Theorem ), and Result ? is in [3](Theorem ).

Lastly, we provide the definition of positive Harris recurrence. For details on properties associated with positive Harris recurrence, see [22].

## 5Rate allocation Markov chain & Rate Stability

Rate allocation Markov chain: The main challenge is to design a Markov chain with fewer parameters that can be analyzed and appropriately chosen for throughput-optimality. First, we identify a class of Markov chains that are relatively easy to analyze. Consider the class of algorithms introduced in Section 3. The core of this class of algorithms is a continuous-time Markov chain with state-space , which is the (finite) set of rate allocation vectors. Define

where , and are the parameters introduced in Section 3. Now, the transition rate from state to state can be expressed as

And, the diagonal elements of the rate matrix are given by for all . This follow directly from the description of the algorithm. This class of algorithms are carefully designed such that it is tractable for analysis. In particular, the following lemma shows that this Markov chain is reversible and the stationary distribution has exponential form.

The offered service rate vector under the stationary distribution is . In general, for , we expect to find values for parameters as a function of and such that . Due the exponential form in ( ?), it turns out that the right structure to introduce is

where , and obtain suitable values for as a function of and such that . To emphasize the dependency on , from now onwards, we denote the stationary distribution by and the offered service rate vector by

Substituting (Equation 8), we can simplify ( ?) to obtain

Optimization framework: We utilize the optimization framework in [15] to show that values for exist such that . In particular, we show that the unique solution to an optimization problem given by has the property . Next, we describe the intuitive steps to arrive at the optimization problem. If , then can be expressed as a convex combination of , i.e., there exists a valid probability distribution such that . For a given distribution , we are interested in choosing such that is close to . We consider the KL divergence of from given by . Minimizing over the parameter is equivalent in terms of the optimal solution(s) to maximizing over the parameter as is a constant. Simplifying leads the optimization problem as follows:

Here, follows from (Equation 10) and follows from the assumption . Now onwards, we denote the objective function by . To summarize, the optimization problem of interest is, given ,

The following lemma regarding the optimization problem in (Equation 11) is a key ingredient to the main results.

The important observations are that the objective function is concave in and the gradient with respect to is . With offered service rate equal to arrival rate, the next step is to show that the queues drain at rate equal to .

### 5.1Proof of Theorem

Rate stability of the non-adaptive algorithm: We establish the rate stability of the non-adaptive algorithm with the result given in Lemma ? as follows.

Consider time instances for with , and interval length . The queue at the -th link can be upper bounded as follows. The offered service during the time interval is is used to serve the arrivals during the time interval alone. Consider a time , and choose such that . Using (Equation 1) and the above upper bounding technique, we obtain

where

For each interval , define the following two random variables:

It follows from the strong law of large numbers that, with probability , . From Lemma ? and ergodic theorem for Markov chains, it follows that, with probability , Since the arrival process is non-decreasing and the increments are bounded by , we have

Rewriting (Equation 12) with above defined random variables and applying (Equation 13) along with and , we obtain

In (Equation 14), the second term on the right hand side (RHS) goes to zero as as . The first term on the RHS of (Equation 14) goes to zero with probability as , and Thus, for any given , with probability ,

which completes the proof.

This result is important due to the following two reasons.

1. The result shows that this algorithm has good performance, and an algorithm that approaches the operating point of this algorithm has the potential to perform “well.” Essentially, this aspect is utilized to obtain the adaptive algorithm.

2. The non-adaptive algorithm does not require the knowledge of the number of nodes or , as required by the adaptive algorithm. This suggests the existence of similar gradient-like algorithms that perform “well” with different algorithm parameters that may not depend on the number of nodes or . We do not address this question in the paper, but the non-adaptive algorithm will serve as the starting point to address such issues.

## 6Throughput Optimality of Algorithm

In this section, we establish the throughput-optimality of the adaptive algorithm for a particular choice of parameters. The algorithm parameters used in this section are dependent on the number of links and . It is evident from the theorem that determines how close the algorithm is to optimal performance. Define

We set all the step sizes (irrespective of interval) to

and used in the projection to

All the interval lengths (irrespective of interval) are set to

for some large enough constant .

We start with the optimization framework developed in the previous section. For the adaptive algorithm, the relevant optimization problem is as follows: given such that ,

The following result is an extension of Lemma .

The update step in (Equation 6), which is central to the adaptive algorithm, can be intuitively thought of as a gradient decent technique to solve the above optimization problem. Technically, it is different as the arrival rate and offered service rate are replaced with their empirical values for decentralized operation. The algorithm parameters can be chosen in order to account for this. This forms the central theme of this section.

### 6.1Within update interval

Consider a time interval . During this interval the algorithm uses parameters . For simplicity, in this subsection, we denote by and the vector by and by . For the rate allocation Markov chain (MC) introduced in Section 5, we obtain an upper bound on the convergence time or the mixing time.

To obtain this bound, we perform uniformization of the CTMC (continuous-time MC) and use results given in Section 4 on the mixing time of DTMC (discrete-time MC). The uniformization constant used is . The resulting DTMC has the same state-space with transition probability matrix . The transition probability from state to state is , and from state to itself is . With our choice of parameters given by (Equation 8), we can simplify (Equation 7) to

For all , clearly . Since at most elements in every row of the transition rate matrix of the CTMC is positive for all . Therefore, is a valid probability transition matrix.

The DTMC has the same stationary distribution as the CTMC. In addition, the CTMC and the DTMC have one-to-one correspondence through an underlying independent Poisson process with rate In this subsection, time denotes the time within the update interval, i.e., denotes global time . Let be the distribution over given by the CTMC at time , and be a Poisson random variable with parameter . Then, we have

where is the identity matrix. Next, we provide the upper bound on the mixing time of the CTMC.

Lemma ? is used to show that the error associated with using empirical values for arrival rate and offered service rate in the update rule (Equation 6) can be made arbitrarily small by choosing large enough . This is formally stated in the next lemma.

Thus, the important result is that due to the mixing of the rate allocation Markov chain, the empirical offered service rate is close to the offered service rate. The next step is to address whether the offered service rates over multiple update intervals is higher than the arrival rates.

### 6.2Over multiple update intervals

We consider multiple update intervals, and establish that the average empirical offered service rate is strictly higher than the arrival rate. This result follows from the observation that, if the error in approximating the true values by empirical values are sufficiently small, then the expected value of the gradient of over sufficiently large number of intervals should be small. In this case, we can expect the average offered service rate to be close to . Since, is strictly higher than arrival rates, we can expect the average offered service rate to be strictly higher than the arrival rate. The result is formally stated next.

Now, we proceed to show that the appropriate ‘drift’ required for stability is obtained.

### 6.3Proof of Theorem

Consider the underlying network Markov chain consisting of all the queues in the network, the update parameters, and the resulting rate allocation vectors at time , i.e., for It follows from the system model and the algorithm description that is a time-homogenous Markov chain on an uncountable state-space The -field on considered is the Borel -field associated with the product topology. For more details on dealing with general state-space Markov chains, we refer readers to [22].

We consider a Lyapunov function of the form, for . In order to establish positive Harris recurrence, for any such that , we use multi-step7 Lyapunov and Foster’s drift criteria to establish positive recurrence of a set of the form , for some From the assumption on the arrival processes, it follows that is a closed petite set (for definition and details see [22]). It is well known that these two results imply positive Harris recurrence [22].

Next, we obtain the required drift criteria. For simplicity, we denote by in the rest of this section. Consider

Here, follows from the fact that over unit time queue difference belong to . Now, we look at two cases. If , clearly during interval as service rate is less than or equal to . For this case, from Lemma ?,

Here, is trivial, but the extra term is added to ensure that the RHS evaluates to a non-negative value for . If , then clearly Since the bounds for each case do not evaluate to negative values for the other case, we have

Since both and are bounded, there exists some fixed such that

Summing up over all , we obtain

This shows that there exists some such that for all with there is strict negative drift. Hence, the set is positive recurrent. Since , clearly . This completes the proof of Theorem ?.

In summary, given any rate region for a wireless network, the (queue-length based) algorithm has -optimal performance.

## 7Applications: White-space Networks

An important application of our algorithmic framework is in the domain of white-space networks [23]. White-space radios are typically required to sense the environment [9]. Therefore, these radios are designed with highly accurate sensing capabilities. Even though these are primarily designed for sensing the presence of primary radios, the same capability can exploited for sensing secondary radios. In this section, we consider a networks of secondary nodes that use the same spectrum, but different from that used by primary nodes. In particular, we assume that the secondary nodes have already found spectrum that are not utilized by primary nodes.

Since such a white-space network of secondary nodes are not centrally controlled, it is desirable to obtain simple distributed algorithms. However, the scheduling problem in these white-space networks is different from the link scheduling problem in traditional wireless networks [7]. First, the available spectrum for the operation of this network is fragmented with different propagation characteristics. Second, these secondary nodes are usually equipped with multiple radios to operate simultaneously in different bands. This is referred to as the multiple-band multiple-radio scheduling problem. Next, we describe the multiple-band multiple-radio scheduling problem in detail.

Consider the network model introduced in Section 2. Define functions that maps links to source nodes, and that maps links to destination nodes. The available spectrum for the operation of this network is fragmented. The spectrum consists of bands, labeled , with bandwidths The transmission from a node to another node gets different spectral efficiencies on different bands. For a link , let be the spectral efficiency that node gets when it transmits on band to node . The link interference graphs are also different on different bands. Let be the link interference graph on band , i.e, the transmission of link interfere with the transmission of link in band if We assume that the link interference is symmetric, i.e., if then . These capture the frequency dependent propagation characteristics and the spatial variation of the quality of spectrum. Further, each node is equipped with radios.

At time , the decision whether link is operated in band is represented by binary decision variables , with representing “true” and representing “false”. The decision variables has to satisfy the constraints that arise from the following. (i) Interference constraints: In every band, the set of allocated links must be non-interfering. (ii) Radio constraints: The total number of radios at each node is limited, and these radios are half-duplex, i.e., a link requires its end nodes to dedicate one radio each for a transmission to happen. More formally, the set of constraints are:

For a feasible schedule, the rate of flow supported on link is

We denote the vector of above rates by . The throughput region is defined as the convex hull of the set of all feasible rate vectors. Note that the queue dynamics is exactly same as described in Section 2.

### 7.1Distributed Algorithm

In this section, we present an adaptation of the developed algorithm that is throughput-optimal for multiple-band multiple-radio scheduling. For simplicity, we assume that perfect and instantaneous carrier sensing is possible on every band. The scheduling vector corresponding to link is . For this link, the possible states are

The link uses an independent exponential clock corresponding to each state with transition rate for state . Based on these clocks, the link obtains as follows:

1. If the clock associated with a state (say ) ticks and transitioning to that state is feasible8, then is changed to ;

2. Otherwise, remains the same.

The above procedure continues. The parameter is updated over time as a function of the queue-length as described in Section 3. This makes the algorithm completely distributed. The vector of is denoted by .

In order to establish that this algorithm is throughput-optimal, we show a correspondence between it and the rate allocation algorithm in Section 3. Consider a fixed . The above algorithm forms a Markov chain on the set of feasible states. Let denote the matrix formed by vectors , and denote the set of feasible matrices satisfying (Equation 21) and ( ?). The transition rate from state to state can be expressed as

where

And, the diagonal elements of the rate matrix are given by for all .

Now, the following lemma is immediate.

The offered service rate vector under the stationary distribution is Thus, we show a one-to-one correspondence to the rate allocation algorithm. As a consequence, we establish the throughput-optimality of the algorithm described in this section based on Theorem ?.

## 8Discussion & Simulation

### 8.1Determining Rate Feasibility

In order to further explain “channel measuring”, we consider an example with a simplified physical-layer model. In this model, a transmitter can potentially communicate with a receiver if the receiver is within distance . This transmitter can communicate at data-rate if there are no other transmitters within distance to it. We consider and . In this setting, for channel measuring, a transmitter needs to simply determine the distance of the nearest active transmitter. Even though we used an over simplified physical-layer model, this shows that channel measuring is a very natural technique for determining rate feasibility. Furthermore, it suggests that slightly more complicated schemes than carrier sensing may be enough to obtain significant throughput gains.

For complex physical-layer interactions, we acknowledge that channel measuring requires a well-designed physical-layer control architecture, which, by itself, is a fairly non-trivial problem. However, radios that perform complex physical-layer signaling are increasingly common and each node has access to current channel interference level, information from beacons, pilot signals and its own location. These will definitely help such radios to perform channel measuring using existing physical-layer control overhead.

### 8.2Algorithm Parameters

In this paper, we show that the algorithm provide throughput-optimal performance for particular choice of algorithm parameters. Although this has significant theoretical value, these parameters may not be directly suitable in practice. In particular, we may have to limit the update interval length and attempt rates as large values of update interval can result in large queue-lengths, and large attempt rates can result in frequent changes in data-rates. There are certain hardware and physical-layer coding limitations on frequently changing data-rates, and frequent attempts lead to increased sensing/measuring overhead. These limitations can be easily dealt with through modified algorithm parameters.

Our approach in the paper motivates a more general class of algorithms that can be throughput-optimal for appropriate choice of parameters. We can consider the general class with update rule

for some function Next, we provide a “good” choice of this function based on simulation results.

### 8.3Simulation

Consider the same Gaussian multiple access channel example with two links as before. This is shown in Figure ?. This is simply an illustrative example to show scheduling over multiple data-rate levels. Similar simulation results apply for any number of users. Let the average power constraint at the transmitters be and noise variance at the receiver be . The information-theoretic capacity region of this channel is the pentagon shown in Figure ? where . The set of rate levels chosen by both transmitters are where and . The only infeasible rate allocation pair is . Consider the following arrival processes at both the transmitters. At integral times, the queues are incremented by an i.i.d. Bernoulli random variable such that the arrival rate is , where represents the load in the system. Clearly, the network will be unstable for

For this system, we perform Monte-Carlo simulations with update interval and update rule This function results in linear update near origin and prevents the rapid growth of with queue-length. We provide a trace of the queue-length process for and in Figure ?. We observe that the algorithm supports load in the system without large increase in queue-lengths. Intuitively, this symmetric operating point is one of the difficult operating points for a distributed algorithm. More importantly, the sum-rate obtained is close to the information-theoretic sum-capacity of this system. Thus, simulations show that our algorithm is of significant practice value.

## 9Conclusion

Decentralized algorithms that use local sensing-based information are highly desirable in practice for wireless networks. In this paper, we develop such an algorithm that guarantees throughput-optimality. Thus, we show that efficient network algorithms can be designed that fully utilizes underlying physical-layer schemes. The algorithm is of practical value due to its decentralized nature, and due to its applications both in the newly introduced channel measurement framework, and already existing carrier sensing framework. Since this paper improves the current state-of-the-art in distributed resource allocation to account for more complex resource-sharing constraints, it has applications in other areas as well, for example, in performing resource allocation in energy networks. The algorithmic framework in this paper can be used to perform utility maximization, i.e., adaptively choose the arrival rates at the links such that a certain utility function is maximized.

The channel measurement framework introduced in this paper motivates further research. First, we need to better understand the feasibility of channel measurement with existing and newly developed radios. This needs development of good physical-layer architectures that minimize the probability of inaccurate measurement and measurement delay. Further, we need to study the impact of imperfect channel measurement on throughput.

## Acknowledgment

The authors would like to thank S. Deb and V. Srinivasan at Bell labs, India for constructive discussions on the multiple-band multiple-radio scheduling problem.

## AProof of Lemmas

### a.1Optimization framework

#### Proof of Lemma

The steps involved are the following. First, we prove that, for any fixed , the objective function is strictly concave in . Next, we show that for any fixed , the optimal value lies inside a compact subset of . These two statements show the existence of a unique solution that is finite. This along with certain necessary condition for optimality completes the proof.

For notational simplicity, we denote by and the normalization constant or partition function by Using calculus, it is straightforward to obtain the gradient (first-order partial derivatives) and the Hessian (second-order partial derivatives) of in the following form:

Here, in (Equation 22) is the offered service rate vector given by (Equation 9), and for any matrix, vector or scalar .

In order to establish that is strictly concave in , we show that the Hessian is negative definite, i.e., for any non-zero , . Since is the negative of a covariance matrix, it is clear that is negative semi-definite, i.e., from (Equation 23),

We next prove that the Hessian is negative definite by contradiction. Consider a fixed . Suppose that there exists such that . Then, from (Equation 24), it follows that the random variable is zero with probability . For any fixed , all feasible states have non-zero probability. In particular, and for all Therefore, the random variable must evaluate to zero at and , i.e.,

which implies . This provides a contradiction and establishes that the Hessian is negative definite.

Next, we prove that the optimal value belongs to a compact set. Let for some . Note that for any there exists such a . Consider a Define , , and . Let

Clearly, , and hence, there exists a distribution on such that