Maximal Scheduling in Wireless Networks with Priorities
We consider a general class of low complexity distributed scheduling algorithms in wireless networks, maximal scheduling with priorities, where a maximal set of transmitting links in each time slot are selected according to certain pre-specified static priorities. The proposed scheduling scheme is simple, which is easily amendable for distributed implementation in practice, such as using inter-frame space (IFS) parameters under the ubiquitous 802.11 protocols. To obtain throughput guarantees, we first analyze the case of maximal scheduling with a fixed priority vector, and formulate a lower bound on its stability region and scheduling efficiency. We further propose a low complexity priority assignment algorithm, which can stabilize any arrival rate that is in the union of the lower bound regions of all priorities. The stability result is proved using fluid limits, and can be applied to very general stochastic arrival processes. Finally, the performance of the proposed prioritized maximal scheduling scheme is verified by simulation results.
design of efficient scheduling in wireless networks has attracted much attention over the past few years (e.g. ). As the core subproblem in the cross-layer optimization for wireless networks , the MAC-layer scheduling plays a key role in achieving efficient and fair utilization of the wireless network resources. On the other hand, the MAC-layer scheduling is very challenging, due to the complicated conflicting relationships between transmitting links, due to the fundamental broadcast nature of wireless communications. That is, the transmission of any link will be received by any unintended receiver, which can be arbitrarily located in the network, as interference, and thereby impairing its own communication quality.
Despite the numerous efforts made in the past, optimal MAC-layer scheduling in wireless networks is still hard to achieve. For example, the popular optimal max-weight scheduling by Tassiulas and Ephremides  is hard to implement even in a centralized manner. This is because the algorithm requires solving a max-weight independent set (MWIS) problem in each time slot, which is well-known to be NP-hard for wireless networks under general interference constraints. To resolve the complexity issue, several attempts were made to achieve optimal distributed scheduling using constant computation per time slot. For example, Tassiulas  proposed an optimal random ‘pick-and-compare’ scheduling with linear complexity per time slot. Recently, Jiang et al.  and Ni et al.  proposed CSMA based optimal scheduling schemes, which only requires constant computation per time slot. However, all such algorithms suffer from the large delay (exponential in the size of the network) in the worst case, which is inevitable , since, intuitively, it takes an exponential number of time slots for such amortized ‘constant computation’ based schedules to converge to an optimal schedule, due to the NP-hardness of the scheduling problem .
As such, distributed suboptimal scheduling, even if it achieves only a fraction of the maximum throughput region, is still very attractive, due to the low complexity and ease of implementation. Recently, Chaporkar et al.  have shown that a class of simple scheduling policies, maximal scheduling, can achieve a guaranteed fraction of the optimal stability region for general wireless networks, which can even be constant for large-scale wireless networks under the ubiquitous 802.11 protocols . The scheduling scheme is very simple. Under the popular interference graph model, a maximal scheduler simply chooses a maximal set of backlogged links that form an independent set of the interference graph. A set of links is maximal if it can not be further augmented. The scheduling is otherwise arbitrary. For example, consider the wireless network in Fig. ? (a) and its interference graph in Fig. ? (b). The set is a maximal independent set. The set is a also maximal, but in addtion, is a maximum independent set, since it is the independent set with the largest cardinality. Compared with the optimal scheduling schemes , maximal scheduling is very attractive, as it can achieve quite good throughput performance with distributed implementation, with low complexity , or even constant overhead .
On the other hand, maximal scheduling suffer from small throughput guarantees under certain topologies, due to the ad hoc choice of maximal schedules. One example is the general star-shaped interference graph such as in Fig. ? (b) with peripheral links, where it has been shown  that the worst case maximal scheduler can only achieve of the optimal stability region. In the extreme case, as , the worst maximal scheduling can not achieve any positive fraction of the optimal stability region for certain packet arrival processes . Thus, in order to improve the throughput performance of maximal scheduling, it is essential to carefully design the scheduling scheme by choosing the schedules carefully according to the network parameters, such as topology and packet arrivals.
In this paper, we improve the performance of maximal scheduling by using static priorities, which are chosen based on the network topology and packet arrival rates. During scheduling, the scheduler simply considers the links in a sequence specified by a priority vector , and adds back-logged links to the schedule whenever there is no conflict. It can be easily shown that the resulting independent set is maximal. The scheduling is simple, and is easily amendable for distributed implementation. For example, the ubiquitous 802.11 protocols has already defined a set of Inter Frame Space (IFS) parameters to provide prioritized wireless channel access. A related algorithm is the Longest Queue First (LQF) scheduling (also referred to as the greedy maximal scheduling) , where the priorities are chosen according to queue lengths in each time slot, so that a link with higher queue length will receive higher priority. Compared to the LQF scheduling, where the priorities can change globally every time slot, the static priority approach requires much smaller overhead, and is therefore easier for distributed implementation.
Simple as the prioritized maximal scheduling scheme is, the assignment of the static priorities is highly nontrivial, one has to search over a set of possible priorities for a network with links. For networks with moderate size, a naive search over all possible priorities simply becomes impossible. As a main contribution of this paper, we show that, somewhat surprisingly, the optimal priority assignment can be achieved online, and with only linear complexity. The optimality is in the following sense. For maximal scheduling with a priority vector , we associate a tight lower bound stability region , and prove that, using fluid limits, the network is stable under whenever the general stochastic packet arrivals have average rate . Now, suppose a packet arrival rate vector is given. We claim that, as long as , where is the set of priority vectors, our proposed priority assignment algorithm can produce a stabilizing priority vector , such that is stable under . In other words, by obtaining the maximal scheduler based on a carefully chosen priority vector, we can improve the guaranteed throughput region of maximal scheduling from to . Such throughput increase can significantly improve the cross-layer optimization results  by providing upper layers with larger guaranteed throughput regions.
Based on the obtained stability region , we next analyze its scheduling efficiency, which is defined as the largest fraction of the optimal throughput region. We show that the scheduling efficiency can be bounded by . Here is the prioritized interference degree of the network, which is similar to the ‘interference degree’ metric defined in . However, the lower bound can be much larger than that in . For example, it can be shown that the prioritized interference degree of any acyclic interference graph is 1, in which case the prioritized maximal scheduling scheme is globally optimal. On the other hand, the interference degree of the same network can be arbitrarily large (e.g., for the star shaped network in Fig. ?), in which case the scheduling efficiency of the maximal scheduler is close to zero. Finally, the proposed bound is the same as the one used for LQF scheduling , so that the static priority based scheduling can achieve similar throughput guarantee as the LQF scheduling, but with much simpler design and lower scheduling overhead.
The organization of the rest of this paper is as follows: In Section 2 we formulate the system model for prioritized maximal scheduling, and in Section 3 we analyze the performance of maximal scheduling assuming a fixed priority vector. Section 4 proposes an online priority assignment algorithm and proves its throughput guarantees, Section 5 demonstrates simulation results, and finally Section 6 concludes this paper.
In this section, we describe the system model and formulate the prioritized maximal scheduling problem. We begin with the network model.
We consider a single-hop wireless network, whose topology can be modeled as a directed graph , where is the set of user nodes, and is the set of communication links. Fig. ?(a) shows a wireless network consisting of 9 links. Note that it is not hard to generalize the analysis in this paper to the multi-hop scenarios using standard techniques . In this paper, the focus on single-hop network is mainly for the simplicity of exposition. The interference constraint is modeled by an undirected interference graph , where represents the transmitting links (edges) in , and is the set of pairwise conflicts. Thus, two links if and only if they are not allowed to transmit together, due to the strong interference one may cause upon the other. In any time slot, the set of scheduled links must form an independent set of . Fig. ?(b) illustrated the interference graph corresponding to the wireless network in Fig. ?(a). The interference graph model is widely adopted for the scheduling problem in many types of wireless networks, such as the Blue-tooth or FH-CDMA networks, where the interference graph is built based on the node exclusive interference model , which specifies that a node can either transmit or receive in each time slot, but not both. It can also be applied to the ubiquitous 802.11 networks, where a two-hop interference model  is used, i.e., links within two hops are not allowed to transmit together, due to the exchange of RTS/CTS control messages.
We assume that time is slotted. Each link is associated with an exogenous stochastic packet source, which is specified by upper layer protocols. The packet arrivals happen only at the end of each time slot. We have the following weak assumptions on the packet arrival processes. First, the number of arrived packets in a time slot is uniformly bounded with probability 1 (w.p.1):
where is the total number of packets that have arrived at link during the first time slots, and is a large positive constant. Further, we assume that Strong Law of Large Numbers (SLLN) can be applied to the arrival processes:
Note that these assumptions on the arrival processes are quite mild, since the packet arrivals are allowed to be arbitrarily correlated across links as well as over time slots. In each time slot, a scheduler chooses an independent set of back-logged links for transmission. With an abuse of notation, we also denote as an vector of indicator functions for the independent set. That is, if link is in the independent set, otherwise . Thus, we can write the total departures until the end of slot in a vector form as . Finally, the queueing dynamics of the network is as follows:
where is the queue length vector at time slot .
2.2Maximal Scheduling with Static Priorities
As described in Section 1, we propose to use a priority vector to assist maximal scheduling. Here, is a permutation of , where is the priority of link . We say that link has higher priority than link if . Thus, the link with has the highest priority, while the link with has the lowest priority. Given , the prioritized maximal scheduler computes the schedule by considering the links sequentially, from the highest priority ’’ to the lowest priority ’’, and adds each back-logged link to the schedule if none of its higher priority neighbors have already been scheduled when it is considered. Denote as the set of higher-priority neighbors of link according to . The following is a key property for proving the throughput guarantee of the scheduling scheme:
The throughput performance of a scheduler can be represented by its stability region, which is the set of stable arrival rates under the scheduler. In this paper, we are interested in rate stability . A link is said to be rate stable if , so that a throughput of can be achieved at link . The network is rate stable if all links in the network are rate stable. It has been shown  that the largest achievable stability region is the convex hull of all independent sets, i.e., , where denotes convex hull, and is the set of independent sets of the interference graph. For the suboptimal schedulers that we consider here, the exact stability regions are hard to obtain, as one needs to prove stability for all possible arrival processes with the same average rates. Instead, lower bounds are often provided. For maximal scheduling, it has been shown  that the following stability region can be achieved by any maximal scheduler:
where is the set of neighbors of link . As a comparison, we will prove later in this paper that a maximal scheduler with priority can achieve the following stability region:
where is the indicator function, i.e., , . One can easily see that for any priority vector , so that . In the following sections, we will propose a low complexity priority assignment scheme which can achieve .
Another performance metric for a suboptimal scheduler is its scheduling efficiency, which is defined as , where is the stability region associated with the scheduler . Thus, denotes the largest achievable fraction of by the scheduler. In particular, if the scheduling is optimal. It has been shown  that the worst case maximal scheduler has , where is defined as the ‘interference degree’ of the network, and is the cardinality of the largest independent set in the subgraph induced by the links . For example, in the star shaped network in Fig. ?, the interference degree of link 1 is , since there are at most 8 independent links in the subgraph induced by . Similarly, . It is easy to see that , and therefore the worst case maximal scheduling can guarantee a scheduling efficiency of .
For maximal scheduling with priority vector , we will show that its scheduling efficiency is , where is the prioritized interference degree associated with priority . That is, is the cardinality of the largest independent set in the subgraph induced by links . Note that for any priority and any link , and therefore, the scheduling efficiency of the prioritized scheme is always better than the worst case. In Section 4, we will show that a simple priority assignment scheme with the maximal scheduling can achieve a scheduling efficiency of , where .
3Maximal Scheduling with Fixed Priorities
In this section, we analyze the performance of maximal scheduling assuming a fixed priority is always used. The optimal priority assignment will be presented in the next section. We begin by obtaining the stability region associated with a fixed priority vector .
3.1Lower Bound Stability Region
We first show that the stability region in (Equation 3) is a lower bound for maximal scheduling with .
We first discuss the intuition behind the proof. According to Lemma ?, for any back-logged link , maximal scheduling with priority will schedule at least one packet departure from . On the other hand, the definition of in (Equation 3) implies that the total arrival rate in is no more than 1. Thus, assuming link always has packets, the average total departure from the set is no smaller than the average total arrival, from which the stability result may follow. However, rigorous proof is needed to show that the stability always holds without assuming that link is back-logged, under general stochastic arrivals.
Having proved that is a lower bound stability region, we next show its tightness result.
Thus, the lower bound stability region is tight. In Section 4, we will show that the union of such regions can be achieved by a priority assignment algorithm. Before we do that, we analyze the scheduling efficiency represented by the lower bound region .
We next show that the scheduling efficiency associated with can be lower bounded by .
We continue to show that is a lower bound on the scheduling efficiency of .
Compared to the interference degree , the prioritized interference degree can be much smaller, thereby achieving much larger scheduling efficiency. For example, in Fig. ? (b), the interference degree is , since link 1 has 8 independent neighbors. On the other hand, the prioritized interference degree implies that (globally optimal), which is achieved by assigning link 1 the highest priority. In fact, we have the following result:
We next briefly discuss the distributed implementation of proposed scheduling scheme. As a specific type of maximal scheduling, the prioritized maximal scheduling allows easy and distributed implementation in wireless networks. A direct implementation of the priority mechanism is as follows. We partition each time slot into a contention period and a data transmission period. Given a wireless network with priorities, we can divide the contention period of each time slot into mini-slots, such that a back-logged link with priority will back-off until the end of -th mini-slot, and schedule itself for transmission if it does not hear any transmission intent (such as RTS) within its neighborhood. Otherwise link will wait until the next time slot. Note that it is possible to reduce the number of contention mini-time slot by introducing randomized back-off, such as , in which case the priority mechanism is implemented in a ‘soft’ manner, to further reduce the scheduling overhead. It is also possible implement the prioritized maximal scheduling asynchronously by assigning back-off parameters such as the IFS in the 802.11 protocols. Such implementation topics are beyond the scope of this paper, and will be addressed in future research.
Summarizing the results in this section, we conclude that we can achieve a lower bound region of and scheduling efficiency bound of using maximal scheduling with static priorities, which is easily implemented in a distributed fashion. On the other hand, such performance guarantees hold only if the optimal priority is used. In order to maximize the throughput performance, the priority vector has to be carefully chosen. At the first glance, such priority assignment seems to be quite hard, since it involves an optimization over a set of priority vectors. In the next section, we show that, somewhat surprisingly, the structure of allows one to obtain the optimal priority very efficiently.
In this section, we propose an online priority assignment algorithm to achieve . For the simplicity of exposition, we first consider a simple off-line case, where the estimated packet arrival rate is fixed, and show that the priority assignment algorithm can output a stabilizing priority as long as . Later we will prove stability in the online case, where arrival rates must be estimated from general stochastic packet sources.
4.1An Offline Assignment
The priority assignment algorithm is shown in Algorithm ?. At each step, the algorithm chooses a link with the smallest ‘total neighborhood arrival rate’ in the reduced interference graph , and assigns it the lowest priority that is available locally, i.e., link only needs to have higher priority than the neighboring links which are already removed. The algorithm then removes from and repeats. We next show that Algorithm ? implicitly solves a min-max optimization problem:
As an application of Theorem ?, we next prove that Algorithm ? can achieve .
Thus, we conclude that if is fixed, Algorithm ? can output a stabilizing priority as long as . Note that one important feature of Algorithm ? is that the priority can be reused, as it is locally assigned. This can achieve a significant reduction in the total number of priorities and scheduling overhead. For example, the star network in Fig. ? (b) only requires 2 distinct priorities. Further, for general wireless networks, one can often get a small number of priority levels by trading off a certain fraction of stability region. In fact, one can easily show that Algorithm ? only needs at most levels of priorities to achieve the worst case maximal scheduling performance bounds and , where is the maximum degree of the interference graph. The detailed analysis of such trade-off, on the other hand, is beyond the scope of this paper, and will be addressed in future research.
4.2Online Priority Assignment
Having demonstrated the optimality of the offline priority assignment, we next implement it online with estimated arrival rates from stochastic packet arrival processes, and prove that the same optimality result still holds. To start with, we partition time into frames, each of which has length . Throughout each frame , a fixed priority is used, which is computed as follows. For the first frame, we assume is arbitrary. At the beginning of each subsequent frame, if the estimated arrival rate satisfies , where
Otherwise we set , where is returned by Algorithm ? with . We next show network stability in the following theorem:
In this section, we evaluate the performance of the proposed priority scheduling scheme by MATLAB simulation. All the simulation results in this section are obtained from 30 independent simulations over a period of time slots. The packet arrival processes are i.i.d and independent across different links.
We first consider a wireless network with 11 links as shown in Fig. ?, where the center link 1 is at the intersection of two cliques. Thus, link 1 interfere with both local clusters, and is the bottleneck of the network. We assume that every link other than link 1 has an arrival rate of , so that each clique has a total arrival rate of . We simulate three types of scheduling algorithms: 1) a maximal scheduler with a suboptimal priority vector, as an upper bound on the worst-case throughput performance of maximal scheduling, 2) maximal scheduling with the optimal priority vector obtained by the online priority assignment algorithm, and 3) the LQF scheduling. For the prioritized maximal scheduling, we choose . The priority converges after a few time frames as the arrival processes are i.i.d. Figure 1 shows the maximum queue lengths under different values of over time slots simulation, along with confidence intervals.
The network is unstable under the worst-case maximal scheduling, which can be clearly observed by the significant larger queue lengths than the other two schedulers (note that time slots were simulated). On the other hand, the network is always stable under maximal scheduler with the optimal priority. In fact, for this topology, the optimal priority scheduling scheme is globally optimal, since one can easily verify that . Thus, we can obtain significant throughput improvement by properly optimizing the priorities.
The network is also stable under LQF scheduling. In fact, it can be shown that LQF scheduling is throughput optimal for such topology, due to the ‘local pooling’ condition . It has also been widely observed that the LQF scheduling can achieve good throughput performance in general wireless networks, at the expense of frequent update of priorities, which may incur large scheduling overhead. Compared to LQF scheduling, the static priority based maximal scheduling can achieve similar throughput performance, with smaller scheduling overhead.
We next consider a random wireless network with 8 links, whose communication graph is shown in Figure 2. In the figure, the square nodes are the transmitters, and the round nodes are the receivers. The interference graph is constructed by placing a guard zone  around the receiver of each link, so that two links form an edge if one’s transmitter is inside the guard zone associated with the other. We consider all the scheduling algorithms in the previous case, and the optimal max-weight scheduling  as a benchmark. The simulation results are shown in Figure 3, where the maximum queue lengths are shown after time slots, with confidence intervals. Similar convergence results are observed for the online priorities, due to the fast convergence of the arrival processes.
The network is unstable under the worst-case maximal scheduling with arrival rate above . From the queue lengths of the max weight scheduling, it can be observed that the boundary of the stability region is around . Thus, for this random network, the worst case maximal scheduling can achieve a scheduling efficiency of no more than around .
Prioritized Maximal Scheduling
Maximal scheduling with optimal priority achieves the same maximum uniform throughput as the max-weight scheduling, although with larger queue lengths. Thus, compared to the worst-case maximal scheduling, we conclude that we can achieve significant throughput improvement by choosing the priority vectors carefully. Further, compared to the max-weight scheduling, the optimal priority based maximal scheduling can achieve the same throughput with low complexity.
The LQF scheduling also achieves the network stability for all arrival rates, with smaller queue lengths than the optimal prioritized maximal scheduling. However, note that this is achieved at the expense of more control overhead associated with frequent priority updates, due to the changes in the queue lengths, which induces global changes to the priorities in each time slot. On the other hand, the priority updates of our scheduler with the estimated arrival rates happen very infrequently, and therefore requires much smaller scheduling overhead.
In this paper, we proposed a static priority based scheme to improve the performance of maximal scheduling in wireless networks. The scheduling has low complexity, and is easily amendable for distributed implementation. We first formulated a tight lower bound stability region for maximal scheduling with a fixed priority, and then discussed its scheduling efficiency. We next introduced an online priority assignment scheme, which can compute the optimal priority based on estimated packet arrival rates. Future research will focus on the trade-off between the throughput performance and the scheduling overhead, such as the number of distinct priorities.
7Proof of Lemma
In this section we briefly introduce fluid limits, which is a general framework to analyze stochastic queueing systems. For details, please refer to  and the references therein.
8.1Construction of Fluid Limits
Given the network dynamics , we first extend the support from to using linear interpolation. For a fixed sample path , define the following fluid scaling
where the function can be or . It can be verified that these functions are uniformly Lipschitz-continuous, i.e., for any and , we have
where the positive constant is for functions and and 1 for functions . Thus, these functions are equi-continuous. According to the Arzéla-Ascoli Theorem , any sequence of equi-continuous functions contains a subsequence , such that
with probability 1, where is a uniformly continuous function (and therefore differentiable almost everywhere ). Define any such limit as a fluid limit.
8.2Properties of Fluid Limits
The following properties holds for any fluid limit
where (Equation 4) is because of (Equation 2) and the functional SLLN, and ( ?) is because any regular point with achieves the minimum value (since ), and therefore has zero derivative. We further have the following lemma, which provides a sufficient condition about rate stability :
9Proof of Theorem
10Proof of Theorem
11Proof of Theorem
12Proof of Corollary
13Proof of Corollary
14Proof of Theorem
15Proof of Theorem
16Proof of Theorem
(S’07) received the B.Engg. degree from the Department of Electronics Information Engineering, Tsinghua University, Beijing China, in 2006. He received the M.S. degree from the Department of Electrical and Computer Engineering, Carnegie Mellon University, Pittsburgh, PA USA, in 2008.
He is currently a Ph.D. Candidate in the Department of Electrical and Computer Engineering, Carnegie Mellon University. His research interests include distributed algorithms, smart grid technologies, and wireless networking.
(S’98-M’00) received the B.Tech. degree in electrical engineering from the Indian Institute of Technology, Bombay, in 1995. He received the M.S. and Ph.D. degrees from Stanford University, CA, in 1996 and 2000, respectively, both in electrical engineering.
Since 2000, he has been with the Electrical and Computer Engineering Department, Carnegie Mellon University, Pittsburgh, PA, where he is a Professor. His research interests include signal processing, coding for communications systems, information theory, networking, cross-layer optimization, and sensor networks. Dr. Negi received the President of India Gold Medal in 1995.
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