A Fully Distributed Algorithm for Throughput Performance in Wireless Networks

A Fully Distributed Algorithm for Throughput Performance in Wireless Networks

Eyjólfur I. Ásgeirsson School of Science and Engineering
Reykjavik University
101 Reykjavik, Iceland
eyjo@ru.is
Magnús M. Halldórsson School of Computer Science
Reykjavik University
101 Reykjavik, Iceland
mmh@ru.is
 and  Pradipta Mitra School of Computer Science
Reykjavik University
Reykjavik 101, Iceland
ppmitra@gmail.com
Abstract.

We study link scheduling in wireless networks under stochastic arrival processes of packets, and give an algorithm that achieves stability in the physical (SINR) interference model. The efficiency of such an algorithm is the fraction of the maximum feasible traffic that the algorithm can handle without queues growing indefinitely. Our algorithm achieves two important goals: (i) efficiency is independent of the size of the network, and (ii) the algorithm is fully distributed, i.e., individual nodes need no information about the overall network topology, not even local information.

1. Introduction

Designing high-performance scheduling algorithms for wireless networks has become an increasingly important topic in recent years. Scheduling in a wireless environment is a non-trivial problem, since simultaneous transmissions interfere with each other in complex ways. A two-fold challenge of appropriately modelling the interference, and then developing algorithms for that model presents itself. Furthermore, in many realistic settings, a centralized controller cannot be assumed, and algorithms that work in a distributed fashion have to be developed.

In this work, we are interested in stability and the associated throughput performance of scheduling algorithms for wireless networks under realistic interference models. We assume that packets arrive at potential senders according to a stochastic process, and the goal of an algorithm is to schedule these transmissions so that the queues of unscheduled packets at each sender remain bounded (in which case, the system is called stable). A rich body of research has been devoted to dealing with this issue in a variety of settings. The seminal work of Tassiulas and Ephremides [23] established that an optimal scheduling policy exists, one that stabilizes the system under all arrival rates for which stability is potentially possible. In most settings, however, such a “perfect” solution is computationally intractable, and additionally a distributed implementation is unlikely.

Hence, the search for efficient and/or distributed algorithms which, if not as good as the optimal algorithm, are nevertheless useful. Since these algorithms may not stabilize all feasible arrival processes, the concept of efficiency of an algorithm has been introduced, being the fraction of that the algorithm can stabilize, where is the space of arrival processes that the optimum algorithm can stabilize. There have been many approaches to developing such algorithms. A natural step in the search for efficient algorithms is to seek maximal solutions. In the context of wireless networks this is known as Greedy Maximal Scheduling (GMS) algorithm [10] or Longest Queue First (LQF) algorithm [4]. The stability and efficiency of LQF has been investigated extensively [10, 16, 4]. Many other approaches have been proposed as well [18, 11, 22].

Most analytic work on wireless networks has been done in graph-based interference models (e.g. [18, 10, 16, 4, 22]). In these models, wireless links (a link is a sender-receiver pair) that are neighbors in a specified link-graph cannot transmit simultaneously. Though interesting in their own right, these models are known to over-simplify interference coupling [17, 20]. As a result, many research communities working on wireless networks have increased their focus on the so-called physical model or the SINR model. In this model, a transmission is considered successful if the signal received at the intended receiver is suitably larger than the cumulative interference due to all other transmissions in the network, plus the ambient noise. In the SINR model, solving the characterization in [23] is equivalent to solving the maximum weighted capacity problem, which is known to be NP-hard [6] and additionally has no known constant factor approximation algorithm (quite apart from the issue of distributed implementation).

In this paper, we develop an algorithm that is completely distributed, with nodes requiring no topological information about the network (not even information about “neighbors”), and achieves an efficiency ratio that is independent of the network size (i.e., the number of links). Thus, the algorithm is scalable in relation to network size. It can also operate in an asynchronous setting, with nodes appearing arbitrarily (as long as the stochastic process meets the required condition).

We give simulation results which lend credence to the theoretical bounds. Our algorithm is extremely robust under fairly high amount of load (achieving efficiency ratios bordering on ).

The only other work (that we are aware of) on stability of algorithms in the SINR model is the recent work by Le et. al. [14], who analyze the stability properties of the LQF algorithm. The authors show that the basic LQF is not efficient, but a variation of it that localizes interference is shown to work, with efficiency similar to ours. In a related work, [15] also consider the SINR model, but the model there is different (links are always feasible, but have different data rates based on the SINR achieved. This sort of problem is rather different from the “combinatorial” situation at hand).

A distinguishing feature of our distributed implementations is that they require almost no additional “infrastructure”. Often distributed algorithms for wireless networks have to assume another underlying information infrastructure that can be used to run a localized and/or distributed algorithm, and that infrastructure, moreover, is not subject to the interference constraints of the original network. This is the case with [14], as well as many other works on the topic ([18, 15] for example). This is a rather strong assumption, especially in light of the fact that in a wireless network, one is usually trying to establish such an infrastructure in the first place. It is interesting that we can do without them while obtaining high throughput performance.

From a technical perspective, we adopt the vocabulary and techniques developed in the context of worst-case algorithmic research on the SINR model ([19, 9, 12, 8]). The concept of “affectance” (defined later) developed in some of these works turns out to be quite effective in this context. This approach may have further applications in the study of stability of wireless networks.

The paper is organized as follows. In Section 2, we describe our algorithm and state the main result. In Section 3 we present the system model, and discuss related work further in Sec. 4. The proof of the stability result is given in Section 5. Finally, in Section 6 we present simulation results.

2. Algorithm and Result

The wireless network is modeled as a set of links, where each link represents a potential transmission from a sender to a receiver , each a point in a metric space.

We assume that packets arrive at the sender of each link according to a stochastic process with average arrival rate .

The extremely simple and fully distributed algorithm is as follows.

1:   (queue of outstanding packets)
2:  for  do
3:     Let be the set of packets that arrive at the beginning of time slot
4:     Add to the end of
5:     if  is non-empty then
6:        Transmit a packet from with probability
7:     end if
8:  end for
Algorithm 1 Reflect (Run by each link in the system)

Our main result is:

Theorem 1.

For all given networks with links on metric spaces, and all sublinear, length-monotone power assignments, Reflect achieves an efficiency ratio independent of .

3. Some Preliminaries

The distance between two points and is denoted . The distance from ’s sender to ’s receiver is denoted . The length of link is denoted simply by . The link set is associated with a power assignment , which is an assignment of a transmission power to be used by the sender of each link . The signal received at point from a sender at point with power is where the constant is the path-loss exponent.

We can now describe the physical or SINR-model of interference. In this model, a receiver successfully receives a message from the sender if and only if the following condition holds:

(1)

where is the environmental noise, the constant denotes the minimum SINR (signal-to-interference-noise-ratio) required for a message to be successfully received, and is the set of concurrently scheduled links in the same slot (we assume that time is slotted). We say that is SINR-feasible (or simply feasible) if (1) is satisfied for each link in .

A power assignment is length-monotone if whenever and sublinear if whenever . This class includes the most interesting and practical power assignments, such as uniform power (all links use the same power), linear power (, known to be energy efficient in the presence of noise), and mean power (, the assignment that produces maximum capacity in this class [8]). Let where and are, respectively, the maximum and minimum lengths of links in .

Definition 2.

The affectance of link caused by another link , with a given power assignment , is the interference of on relative to the signal received, or

where .

We need the following assumption.

Assumption 3.

for any link .

This is is fairly reasonable assumption. It simply says that in the absence of other links, the transmission succeeds comfortably. The constant is not fundamental; any value greater than 1 would suffice.

Since by definition, this implies that

(2)

The definition of affectance was introduced in [5, 9] and achieved the form we use in [13]. When clear from the context we drop the superscript . Also, let . Using affectance, Eqn. 1 can be rewritten as

(3)

for all .

Signal-strength and robustness. A -signal set of links is a set of links where the affectance on any link is at most . A set is SINR-feasible iff it is a 1-signal set. We know:

Lemma 4 ([7]).

Let be links in a -signal set. Then, .

Now we switch to the aspects of this paper related to queueing theory and stochastic processes. We first define stability.

Definition 5.

An algorithm stabilizes a network for a particular arrival process if, under that arrival process, the average queue size is bounded (at any given time).

The throughput region is then the set of all possible arrival rate vectors such that there exists some scheduling policy that can stabilize the network.

As proved in [23], the throughput region is characterized by

where is the set of all maximal feasible schedules (meaning arrival processes that put weight on a single maximal feasible set) and is the convex hull of . Note that and are -dimensional vectors and means each element of is upper bounded by the corresponding one in .

Since fast and/or distributed algorithms might not stabilize all of , one hopes to achieve a large efficiency ratio.

Definition 6.

The efficiency ratio of a scheduling algorithm is stabilized for all , where .

We assume that the arrival process on a link is i.i.d. across time, and different links are independent of each other.

We will use to denote both maximal feasible sets, and characteristic vectors of said sets (the usage being clear from context). For a given efficiency ratio , it must hold for all permissible arrival rate vectors that , where and are weights such that

(4)

It can be easily seen that for any link ,

(5)

4. Related work

As stated in Thm. 1, the efficiency of the algorithm is independent of (the number of links in the system). It is, however, dependent on another network parameter , the ratio between the longest and the shortest link in the system (the proof in the next section contains the exact expression). The only comparable work on this model [14] has the same dependence on (this is not explicitly stated in the paper, but can be seen to be necessary). The main discriminating feature of our work is that it is distributed in a much stronger sense. The algorithm in [14] can be characterized as “localized”, where each link needs to be aware of and have communicated with other links in its neighborhood. We have no need for such infrastructure.

In terms of efficiency ratio, a range of results have been derived in a variety of models. Naturally one seeks efficiency of whenever possible [18], but results for efficiency ratio of under certain conditions [4], or [3], or [10] can be found in the literature. Ratios in terms of certain network characteristics are known as well – such as in terms of the degree of the interference graph [2] or the local pooling factor [10, 14]. In [21], the abstract SINR model (received signal is a general function instead of being length-based) in the context of MIMO networks is studied. An efficiency ratio based on a system-specific value (“effective interference number”) is derived, with no direct comparison with distance-based SINR models.

For the SINR model, an efficiency ratio that is an “unconditional” constant (independent of both and ) is not known.

5. Proof of Stability

We now present a proof of Thm. 1.

Note that the probability used for link in the algorithm is well-defined, since we claim stability with constant efficiency ratio bounded from above by .

We first need the following observation.

Observation 7.

For any two links and using a length monotone, sub-linear power assignment,

Proof.

If , it holds by sub-linearity that . Otherwise, if , then by monotonicity and by definition of , . ∎

The following key lemma shows that no link is affected too much by any single feasible set.

Lemma 8.

Consider a feasible set and a link (not necessarily a member of ). Then,

(6)

for some constant .

Proof.

We use the signal strengthening technique of [9]. For this, we decompose the set to sets, each a -signal set. We prove the claim for one such set; since there are only constantly many such sets, the overall claim holds (with the appropriate increase in the constant factor). Let us reuse the notation to be such a -signal set.

Consider the link such that is minimum. Also consider the link such that is minimum. Let . We claim that for all links with , it holds that

(7)

To prove this, assume, for contradiction, that . Then, , by definition of . Now, again by the definition of , and . Thus and similarly . On the other hand . Now, , contradicting Lemma 4.

Now that we have proven Eqn 7, by the triangle inequality, . Applying Obs. 7, we see that

where follows from Eqn. 2. Finally, summing over all links in ,

where we use by the definition of affectance, and since is feasible and .

This completes the proof setting . ∎

We turn to the proof of Thm. 1.

Proof.

We claim an efficiency ratio of where is the constant from Lemma 8. Thus, it is enough to prove stability for all stochastic processes for which the following holds:

(8)

Consider the affectance on any link during the execution of the algorithm in a single slot, and denote it by . This can be computed as , where is a Bernoulli random variable which is iff link has a non-empty queue and chooses to transmit during the same slot. Now,

where explanations of the numbered (in)equalities are:

  1. by the description of the algorithm.

  2. By Eqn. 5 and rearrangement.

  3. By Lemma 8.

  4. By Eqn. 8.

Thus, with probability at least , (by Markov’s inequality). Hence, if has a non-empty queue, with probability at least , the queue size decreases. Note that the probability can potentially be higher than , but never smaller. Therefore, this system is at least as efficient as the system where in each slot the queue size reduces by with i.i.d. probability exactly .

The queue dynamics on a single link become equivalent to the following single server system with slotted time and an infinite queue. In this system, at the beginning of each time slot, packets arrive, where is a random variable on the non-negative integers, with . At the end of each slot, the server processes packets (or empties the queue), where is a Bernoulli random variable with . Since the departure process is faster than the arrival process, the stability of the queue is guaranteed by basic results in queueing theory [1]. ∎

We note that for linear power assignment, the dependence on in Obs. 7 completely disappears, and thus also in Lemma 8 and Thm. 1.

Corollary 9.

For all given networks with links on metric spaces, using linear power assignment, Reflect achieves an efficiency ratio that is an absolute constant (independent of and ).

Remark: In Reflect we have assumed that each link knows . In practice, this can be easily approximated at time by (where is the number of packet arrivals up to time ), which converges to the right value almost surely.

5.1. Link partitioning

If link lengths are known beforehand and some pre-processing is allowed, the efficiency can be made to have a better dependence on , specifically, we can achieve an efficiency ratio of .

We can partition the link set into a collection of nearly equi-length link sets, i.e., sets where the lengths in the set vary by at most a factor of . It is easy to show that a link set can be be partitioned into at most sets of nearly equi-length links for where contains links of lengths in .

We partition the time slots accordingly, a time slot is used to schedule links from class where .

With the partition, the arrival process on a sequence of slots devoted to a single length class is equivalent to the setting where all links are nearly equi-length and assuming that .111There are certain technicalities here, since a) may not be a maximal feasible set in and b) Two sets and may have the same “projection” in , i.e., it is possible that for . These can be handled in a straightforward way. This partitioning combined with Lemma 8 proves the claimed efficiency.

6. Simulations

To see how the distributed algorithm Reflect performs, we ran simulations on instances based on random topology. The problem instances were created by generating random links in a rectangle with side length . The length of the links were uniform random variables between and , which we set as and respectively. We generated random transmission requests for 100,000 time slots while running the algorithms. Our focus was on the behavior of the maximum queue length, i.e., the largest number of waiting transmission requests over all the links, which was measured every 10,000 rounds.

Figure 1. The maximum queue lengths for the distributed algorithms Distr-SingleLink and Reflect. The problem instances are based on random topology with , , , and .

Figure 1 shows the results of the distributed algorithm Reflect for random instances with links after 10,000, 50,000 and 100,000 time slots. The probability scaling factor, , defines the load on the system. Thus when , the system will, in expectation, receive a maximal feasible set in every round, while if the scaling factor is zero, no requests are generated. The efficiency ratio of an algorithm is then equal to the largest such that the algorithm is stable. We used a granularity of for values of between and and took the average over runs for each value of .

The Reflect algorithm used in Figure 1 approximates the arrival rate of requests for each link, as mentioned in an earlier remark, instead of assuming that the links know the arrival rate of requests.

It is interesting to note, in Figure 1, that there is a very sharp threshold where the algorithm is no longer stable. As soon as the algorithm becomes unstable, the queue lengths increase very rapidly. As expected, the centralized Longest Queue First (LQF) algorithm was more stable than the distributed algorithm, managing to keep the maximum queue length below for all values of and not becoming unstable until . We note here again that though our algorithm has a lower (though still high) throughput, its main feature is the lack of a requirement for a centralized or even localized control.

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