We consider a collaborative opportunistic scheduling problem in a decentralized network with heterogeneous users. While most related researches focus on solutions for optimizing decentralized systems’ total performance, we proceed in another direction. Two problems are specifically investigated. (1) With heterogenous users having personal demands, is it possible to have it met by designing distributed opportunistic policies? (2) With a decentralized mechanism, how can we prevent selfish behaviors and enforce collaboration? In our research, we first introduce a multiuser network model along with a scheduling problem constrained by individual throughput requirement at each user’s side. An iterative algorithm is then proposed to characterize a solution for the scheduling problem, based on which collaborative opportunistic scheduling scheme is enabled. Properties of the algorithm, including convergence, will be discussed. Furthermore in order to keep the users staying with the collaboration state, an additional punishment strategy is designed. Therefore selfish deviation can be detected and disciplined so that collaboration is enforced. We demonstrate our main findings with both analysis and simulations.
Opportunistic scheduling of resource access within a multiuser network has been investigated during the last several years. This is very well motivated by examples and problems in wireless networks with shared medium and decentralized cognitive users, where how to effectively allocate the shared resource (e.g., frequency, time, energy, etc.) to users is essential for optimizing network’s utilization. To answer this question, many research efforts have been made towards increasing system’s total throughput (which in addition may increase service provider’s revenue). Moreover, along with the development of centralized solutions for such classical scheduling problems [
An interesting observation from the above discussions is that individual requirement of each user has not been well studied in existing researches. Consider a system with multiple users: while each individual unit of the system tries to help achieve optimal performance (in fact this may not even be a goal), each of them may have an individual requirement over their own share of total network performance. This is however critical; otherwise users would have no incentive at all to help system reach the global optimum. This framework has a large area of applications. For example, consider a video streaming network. Different classes of subscribed users may demand different transmission qualities or bandwidth. Although providing maximum overall throughput can make the largest profit for the network service provider, an unresolved problem is whether users would choose this provider’s service or not, which is coupled with the satisfaction of their individual requirement. Our research is particularly motivated from resolving issues of scenarios where each user is associated with its own demand over service quality. In our work we consider a model with performance requirement captured by users’ acquired throughput. Another interesting evaluation criteria and natural extension is considering each user’s delay performance. This could be solved in the same way as presented in our paper; nevertheless it remains an interesting topic for future research.
According to the above discussions, we address the following issues in our research: (1) designing an opportunistic scheduling scheme so that channel diversities can be exploited as much as possible along with individual requirement for users being satisfied and (2) targeting a mechanism that can distinguish those unfeasible demands (i.e., requirements beyond the channel capacity). Moreover, considering the decentralized structure we considered, two more challenges have to be resolved: (1) the solution has to be a distributed one and (2) while selfish behaviors exist in a distributed network, coordination strategies have to be designed to prevent malicious competition and enforce collaboration. It is in this sense that we call it a distributed approach with collaborative opportunistic scheduling. More specifically in this paper, we first rigorously model and analyze multiuser network with individual throughput requirement. Then we try to tackle the distributed scheduling problem with goals satisfying each individual user’s demand, which we will show later could be solved by stopping theory based threshold policies. We show by our designed algorithm optimal solutions can be effectively derived in a distributed way.
Since our scheduling policy requires collaborations and asks all users to follow a prespecified threshold strategy profile, problems arise for such a system with no coordination in the following sense: due to the selfish nature of strategic users, each of them could have incentive to deviate from the collaboration state (prespecified strategy), for example, when the assigned access strategy is not a Nash equilibrium (N.E.) which is in general the case. Therefore we proceed to the second step of our system design: try to enforce collaboration in a decentralized way. Since we are more interested in decentralized network, we will not assume any centralized coordinator which counters the essence of mechanism design for a distributed system. Instead in our work we show, by designing a punishment based mechanism enabled by group efforts, that users would be deterred from any selfish deviation. Theoretical analysis and simulation results are provided in our work to verify our design and claims.
The rest of the paper is organized as follows. Backgrounds and system model are provided in Section
The idea of opportunistic scheduling originates from exploiting multiuser diversity which is firstly discussed by Tse in [
Recent studies concerning overall capacity optimization can be mainly categorized into two groups: (1) centralized mechanisms are used to solve scheduling problems in cellular communications systems and highspeed dedicated systems [
More specifically the research topics for opportunistic scheduling can be categorized into four branches based on different system model and performance evaluation criteria. Details can be found in Table
A summary of former opportunistic scheduling researches.
Evaluation  Model  

Centralized  Decentralized  
Original OS [ 
DOS [  
Global optimization  HSDPA [ 
ADOS [ 
CAWS [ 



IS approach [ 

Individual satisfaction  Dual approach [ 

Virtual queue [ 
In wireless networks, channel conditions vary from time to time. While users are not aware of channels’ instantaneous condition, it is hard to decide whether and how they may accomplish each transmission, let alone to optimize channel utilization by designing and following specific channel access policies. Thus prepositive channel probing becomes a necessary step in opportunistic scheduling. Probing packets are used to detect current channel state, as well as for claiming medium access to avoid interferences (e.g., IEEE 802.11 CSMA/CA). To be more specific, users will send out a carrier sensing packet to reserve the transmission right over a certain channel before transmission. If the carrier sensing packet is correctly received, decoded, and acknowledged, the pair of users (transmitter/receiver pair, and we refer to it as link in the following part) successfully obtain the transmission right.
To simplify the above model and make our analysis tractable, we make the following assumptions. First of all, we consider networks consisting of multiple links (pairs of source/destination users, instead of users in an ad hoc network). Secondly, the system works in a discrete time fashion; that is, links try to access the channel and make decision at
An example of channelaware scheduling in wireless network.
The singlehop wireless network with
Following the system model, packets arrive at MAC layer of each link in a stochastic manner. In this case, random access can be a natural distributed solution. However, considering the individual throughput requirement, it is important to find out whether random access can fulfill these specific requirements or not. In this section, we first present a method for modeling each links’ throughput so that analytical results can be derived to evaluate the performance of the widely adopted random access mechanism. Then by characterizing a threshold enabled policy, we show that, besides optimizing overall throughput of the system, links’ individual throughput can also be optimized with threshold enabled policies.
With channel probing probability
In this section, we design a distributed multithreshold opportunistic policy to achieve our goal: improving individual throughput as compared with random access. The distributed scheduling rule is a modified edition of random access and is explained as follows. After one link wining the channel competition, instead of an immediate transmission as in random access, a decision has to be made between the following two options.
When the link’s current transmission rate is
When the link’s current transmission rate is
Consider a threshold set denoted by
Therefore an average waiting time before a transmission is given by
For any link
The proof can be found in Appendix
With Proposition
We first derive the thresholds to achieve global satisfaction. For link
The uniqueness of the solution is another interesting point worth addressing in this problem. Unfortunately, this is not necessarily true as a result of the complex distribution of different links’ conditions. Even though it is possible to provide some sufficient conditions, with which unique solution can be established, it is out of interest of this paper since the major task is to achieve certain satisfaction level of individual throughput requirement.
In the following part, we focus on an iterative algorithm for deriving one of the feasible threshold solution sets under individual requirement. Two rounds of iterations are adopted in the algorithm. The first round is at the whole threshold set level, of which at step
Initialize
As presented in the proof of Proposition
Based on the comparison between current
if
if
if
If the algorithm does not terminate, continue with the same procedure to calculate for the link
For a given and feasible set of requirements
The proof can be found in Appendix
(2) The number of iterations to achieve the convergence is also important for the algorithm. Unfortunately, it is not always finite due to the nonmonotonicity of the output at each step. However, the transmission rate is always digitalized in wireless network, and a certain level of accuracy is practicably enough. To show the performance of this part, experiments will be shown later.
(3) It can be observed that computing the threshold set requires the knowledge of all links in the network. Exchanging information with channel probing can be an easy solution for solving this problem; however it is also uncertain since fraud remains a potential problem. To this end, a more realistic solution is to apply an online learning algorithm. By observing previous channel contention and transmission, approximate channel state information can be practically learned. This mechanism is a wellinvestigated topic; an example can be found in Section IV.E of [
(4) Following the above discussions, the threshold set can be calculated in a distributed manner. By updating the local estimation of the overall network condition, approximate synchronization can be achieved while everyone in the network collaborates for the purpose of global satisfaction.
(5) By designing a multithreshold policy we have a collaborative opportunistic scheduling policy in the sense that everyone needs to follow a prespecified strategy instead of behaving in their own way.
We provide insights and discussions on the algorithm’s complexity as well as its convergence speed. We start with discussing its complexity. The major computation complexity comes from solving the fixed point equation in Step (1). Consider the following fixed point equation (via setting
We now turn to the convergence rate of the algorithm. The overall algorithm could be viewed as solving a system of fixed point equations. Therefore the convergence speed is determined by eigenvalues of the derivative matrix. The diagonal terms are determined as above. For the rest of the terms (
It has been shown above that, with our mechanism, threshold policy can be used to achieve better individual throughput for links in wireless network. However, there is certainly a limitation in the sense that extreme cases can never be satisfied with limited channel resource. Therefore as mentioned before, another problem arises as how to distinguish achievable requirements from infeasible ones.
Interestingly, our iterative algorithm presented in the above section can be used directly for this problem.
For a given set of requirements
The proof is similar to the proof of Proposition
According to the above discussions, it can be concluded that individual thresholds can be calculated distributively by each link; and by performing the threshold enabled scheduling policy all links can achieve their throughput requirements. However, this satisfaction requires all links to be unselfish. If one individual link deviates from the prescribed policy, no throughput guarantee can be made to anyone else. Moreover since we already characterized the average throughput per link mathematically with (
In this section we focus on designing decentralized mechanism that deter selfish behaviors (which we will detail later) of users. We assume links’ objective is to maximize their discounted sum reward over infinite time horizon; that is,
Denote the N.E. of the access problem by
Deviate from attempt probability
Deviate from transmission threshold
We start with describing our mechanism by defining the following states.
Consider the following mechanism
Links start at state
At any time
For detecting deviations, links essentially use past observed access statistics in a certain short period of time to estimate other links’ current attempt rate. Though this would be imprecise in practice, for now we ignore this deficiency while focusing more on presenting the basic idea with a clean model.
At state
At state
Denote this mechanism by (M) and now we want to see whether there will be link deviating under (M); that is, we are interested in whether (M) is enforceable or not. Before proceeding to the proof we put some requirements on choosing over
(1)
(2) A second restriction over selecting
(3) A third restriction on selecting
Now we show, with appropriately chosen parameter, that (M) enforces the strategy profile
The proof can be found in Appendix
In this section, we provide simulations to validate our results. First of all, AWGN channel models are adopted to capture the stochastic nature of diversified links. Based on such a model, normalized SNR
Furthermore, it can be derived from (
Obviously, the number of iterations needed to achieve the fixed point is important in an iterative algorithm. We present an example in Table
Convergence behavior of the iterative algorithm.
Link 








Link 1  20  2.666  2.526  2.505  2.500  2.500  2.500 
Link 2  20  4.795  4.743  4.734  4.733  4.733  4.733 
Convergence speed of the iterative algorithm.
Rounds  2  3  4–6  7–9  10–15  Others 


Percentage  11.1%  16.7%  25%  27.8%  16.7%  2.7% 
With the similar scenario, we present Figure
The effective area.
With the same scenario used in the numerical results, we simulate the scheduling scheme in our simulator developed in MATLAB according to [
Throughput performance comparison.
Throughput of link 1 under different individual requirements
Throughput of link 2 using our approach
To show the advantage of our methodology, we provide Figures
The throughput of link 2 in random access and DOS.
Throughput in random access
Throughput in DOS
Moreover, our framework is capable of achieving better throughput under more complicated network scenarios. In Figure
Throughput comparison among different approaches.
More general simulation cases are also done for completeness of our study. In particular networks with multiple links are generated (normalized SNR is randomly selected), and our iterative algorithm is applied to achieve better network efficiency in these networks. Part of the results is shown in Figure
Threshold calculation in multiple scenarios.
To verify our collaboration and punishment strategy, we consider a twolink network. Without loss of generality, we assume that
Through simulation we observe that by deviating (link 1) both links’ average throughputs become
In this paper, we have investigated the distributed scheduling problem under individual throughput requirement in order to balance individuals’ resources within the networks. The idea of opportunistic scheduling has been applied so that it is possible to satisfy links’ individual requirements as much as possible. Meanwhile since there is no centralized regulator, selfish behavior is designed to be punished according to our strategy; and thus collaboration can be enforced in a decentralized network. We show with analysis and simulations that our solutions can achieve the above goals.
Derived from (
To examine the influence of threshold set
First, we prove that the iterative algorithm converges to a certain point. As we have shown in Appendix
Second, we prove that the iteration can achieve a fixed point according to the algorithm, and this fixed point must satisfy the following two terms for all
Next we prove that, for a threshold set satisfying (
If we have
Then we prove that if
Then assuming that issue (
Furthermore, we show that the convergence point of the algorithm can satisfy (
For item (
For item (
To prove the enforceability of
Next we analyze the punishment phases. Consider
First consider link
The last step is to check whether link
The authors declare that they have no conflict of interests regarding the publication of this paper.
This work was supported in part by the International Researcher Exchange Project of the National Science Foundation of China and the French Centre National de la Recherche Scientifique (NSFCCNRS) under the Grant no. 61211130104 and the National Science Foundation of China under Grants nos. 60932003 and 61271220.