decentralised e ciency: distributed wireless...

Decentralised Efficiency:

Distributed Wireless Networking Protocols

Martin Gould

1

Decentralised Efficiency Distributed Wireless Networking Protocols

1 Introduction

Day by day, the distances between us are getting smaller. As emails reach our friends and relatives thousandsof miles away in a matter of seconds, it is easy to forget that just half a century ago the same messages wouldhave taken days to arrive at their recipients via pen and paper. Colleagues no longer need share cramped officespaces because a simple connection to the Internet will provide them with all of their company’s electronicresources — wherever and whenever they need them.

Crucial to the developments central in this global revolution has been a better understanding of how toharness the power of communications networks. In 1960 J. C. R. Licklider published his visions of accessing“The functions of present-day libraries . . . [from] a network of [computers], connected to one another by wide-band communication lines” in his seminal work “Man-Computer Symbiosis” [1]. In the following decade a paper[2] emerged from the laboratories of Xerox Parc, giving details of “a system for carrying digital packets amonglocally distributed computing stations” — Ethernet was born.

Fast-forward to 1994, and communications networks were making headlines again as emerging giant Microsoftannounced their plans to invest heavily in research into wireless communication. [3] gives an account of theexcitement as it unfolded: suddenly, wireless communication presented itself as the perfect augmentation ofwired networks. Although never meant to completely replace their wired predecessors, the enormous additionalbenefits of wireless networking suddenly became clear. Indeed, such benefits are the primary reasons thatwireless networks continue to be so popular today: mobility, ease and speed of deployment, flexibility and costare just some of the major features explored in [4]. Add to this the technological advances which are continuallyallowing us to hold ever more processing power in the palms of our hands, and it becomes clear why “the wirelessrevolution” was the next natural step in the evolution of communication.

2 The Problem

The host of benefits that wireless networks offered over their wired predecessors was clear from the outset, butso too was the fact that they came at a price. In a word, that price is uncertainty. Computers which areconnected via a wire behave in a very specific, very certain manner. There will be some upper bound on therate that information can be transferred between the terminals (based on many factors, from physical ones suchas distance and the material that the wires are made from to theoretical ones such as the protocols used by thenetwork), but generally speaking once the network is up and running it will behave in a certain manner until itis altered in some way.

In a wireless network, however, data is sent via radio waves. While superior to many other electro-magneticpossibilities (infra-red light was used in early networks, for example, but was deemed inappropriate because itfailed to propagate through walls [5]), using the air around us as a network medium presents an immediatedifficulty. The capacity of the air for carrying such waves varies randomly with time and location, and indeedthis variation is generally asynchronous for different users in the same network. What’s more, because all of theterminals in the network are sharing the air as their network medium, if multiple terminals attempt transmissionsimultaneously then their signals interfere with each other, and none of the transmitted packets are deliveredto their intended recipients. In this essay, I examine how different mathematical models have been developed inorder to build a quantitative framework for assessing the disruption caused by interference in wireless networks.

The other topic which I explore is that of data transfer rate optimisation. There are really only two practicalways of achieving faster performance in a network which is dependent on wave transmission: the first (and mostobvious) of these is to use a larger portion of the frequency spectrum. Wireless devices, however, are constrainedto operate in certain frequency bands - so called “Industrial, Scientific and Medical” (ISM) bands [4]. The useof frequencies outside of these bands is strictly regulated by central government, and using regulated frequenciescomes at a hefty price. This forces research to focus on the only other option: to somehow “better pack” the


information for transmission into the frequency spectrum that is available. To this end, I present in this essaya comparison of the relative strengths and weaknesses of numerous scheduling algorithms.

Throughout the essay, I use a very general network model which can be employed in a variety of situations.I will highlight how certain familiar setups may be considered “special cases” of my model along the way.Consider a network of N = {1, . . . , z} terminals. If terminal i ∈ N is able to send information to terminalj ∈ N , I say that there exists a directed link (i, j).1 Let L be the set of all such directed links. Throughout theessay I assume that (i, j) ∈ L⇒ (j, i) ∈ L (that is, every terminal can receive transmissions from every terminalto which it can send them). It is then very natural to present a network as an undirected graph G = (N,L).

Broadly speaking, “real world” networks fall into one of two categories. The first kind of network is builtaround some central “hub”, and all communication which takes place between terminals does so via the hub.An example of such a network is displayed in Figure 1.

2 3

45

1

Figure 1: Network in which terminals are only able to send information via a central “hub”

Although [6] describes some wireless networks which fall into this category, far more common is it for wirelessnetworks to be implemented in an “ad hoc” manner. This is the other kind of network: there is no central hub,so instead terminals simply communicate with each other directly. An example of such a network is displayedin Figure 2.

1

2

34

5

Figure 2: ad hoc network in which terminals are able to send information to a variety of other terminals

Indeed, an ad hoc setup is a very natural implementation of wireless communication for the very reason that it isparticularly effective at exploiting the mobility benefits that not being constrained to a hard-wired infrastructureoffers. In an ad hoc network, there is no need to stay within transmission range of some centralised piece ofhardware (i.e. the hub). So long as a terminal remains within transmission range of the other terminals withwhich it wishes to communicate, the users are free to move around as they desire.

But what does such a setup mean for scheduling? When all terminals only communicate with the centralhub, decisions about how to schedule transmissions can be made by some centralised algorithm, run at the hub.But in an ad hoc setup, such a “centre” of the network doesn’t exist — instead, a distributed algorithm must beadopted, so that decisions can be made locally at the terminals themselves. The development and analysis ofsuch algorithms has been a prominent focus of recent research. In this essay, I use some theoretical results to

1There are numerous reasons why a given terminal might not be able to send information to some other terminal: a commonexample is that there is too great a physical distance between them.


motivate an examination of some recently proposed distributed algorithms — one of which can even functionunder the current 802.11 hardware.

3 Interference

As described above, a primary difficulty of wireless networking is dealing with interference to transmissionsin the air. Such interference is twofold: first, “background noise” from outside the network means that thereis random fluctuation in the air’s capacity for carrying the required electromagnetic waves. Secondly, inter-fering transmissions from different terminals within the network present a difficult problem, as simultaneoustransmissions from nearby terminals result in no useful information being delivered.

In dealing with the problem of background noise, Shakkottai et al. assert in [7] that although the maximalrate achievable for each user is always fluctuating randomly, in any given short interval of time it is reasonableto assume that it is constant. As such, they propose that time be “slotted” into intervals of the form [t, t+ δ),and consider the maximum possible data transfer rate on some link l ∈ L. Their assumption is that this rate isequal at all times in any given interval [t, t+ δ), for sufficiently small δ > 0.

An alternative way of building this same randomness into a model is presented in [8]. Instead of trying to workwith exact rates of transfer, Jiang and Walrand instead consider the time it would take for one packet of datato be sent from one terminal in the network to another, and allow this to be the random variable. Specifically,the length of time taken to transmit one packet is assumed to be an exponential random variable, and by arescaling of time the mean transmission time is made to be 1. The properties of the exponential distributioncertainly make it viable for such a use,2 and this approach also avoids some potential simplifications induced byShakkottai et al.’s method.3 For this reason, I favour Jiang and Walrand’s approach over Shakkottai et al.’s,and indeed I will adopt it to model the randomness of transmission rates later in the essay.

Conversational Cliques: An Intuitive Model

The other problem that using the air as the network medium presents is the fact that transmitted signalscan interfere with each other. This is a phenomenon familiar to us all, due to the fact that sound is anothertransmission which uses air as its medium. Consider the following example: Upon arriving at a party, you finda large group of people. The people have divided into smaller subsets (call them “cliques”), and within eachclique certain individuals are engaging in hearty conversation. For such conversation to function properly, onlyone person within a clique may speak at a time. If multiple people within the same clique speak simultaneously,the listeners are uncertain as to who they should be focussing on and the sounds get superimposed in the air.Clever multiplexing this is not: everyone in the clique gets confused, and people have to repeat themselves inorder for their message to be understood.

This familiar description is an example of an interference model. In wireless networks, properly establishing arobust interference model is an imperative consideration because radio wave messages may get superimposed inthe air. Choosing an appropriate interference model – that is, defining exactly when simultaneous transmissionscause messages to become confused – is a difficult task, and is highly dependent on the physical setup of thenetwork to be studied. Looking back to the “conversational cliques”, interference only occurred when twoindividuals from the same clique attempted to speak at once, whereas two individuals from different cliquescould speak at the same time without confusion occurring. For the network in Figure 1, it might be reasonableto say that interference will occur if any two terminals attempt simultaneous transmission, as all transmissionsmust involve the central hub (which may only be capable of dealing with one transmission at a time). The

2Although, interestingly, [8] shows that it is not actually necessary to have this “exponential” assumption for the model tofunction correctly.

3In particular (so as to formulate a tractable model) Shakkotai et al. are forced to assume that there are only a finite set ofpossibilities for the maximum transfer rate through the air, when in reality there are infinitely many such possibilities.


network in Figure 2 might have this interference model as well, or it might have something entirely different:it might, for example, be permissible for terminal 1 to transmit to terminal 2 at the same time as terminal3 transmits to terminal 4. A key point to note is that different interference models may result in differentconclusions being drawn about the efficiency of scheduling algorithms.

While many authors adopt a purely algebraic approach, in order to initially present interference models in avery general way I adopt the methodology used in [8]: that of a link contention graph. A link contention graph(LCG) is a graphical description of an interference model:

Definition. Given a network (N,L), a link contention graph H is a graph where each element of L is drawn asa node, and any pair of nodes representing links for which simultaneous transmission will result in interferenceare joined by an arc.

Notice that in a network graph, links are represented by arcs, but in an LCG links are represented by nodes.In order to clarify the definition, I present the process of forming the network graph and LCG in a specific“conversational cliques” type example — but first it is necessary to refine the model slightly, in order to make itbehave more like a wireless network. Previously, when an individual in a clique spoke it was (tacitly) assumedthat they spoke to all other members of the clique. In a wireless network, however, each message will have aunique recipient. To reflect this idea, assume that individuals in a clique direct each message to a single listener.The other members of the same clique aren’t directly involved in the transmission, other than the fact that theystill can’t communicate with each other whilst this is happening.

To be concrete, assume that there are 9 people at the party, who have formed cliques of sizes 4, 3 and 2.Then Figure 3 displays the network graph G for the party.4

1 2

34 7

6

5

9

8

Figure 3: Network Graph G for specific “conversational cliques” example

Taking the links to be the nodes (and recalling that only a single member of each clique may speak at any giventime without interference occurring) yields the LCG shown in Figure 4.5

(5,6)

(8,9)

(5,7)

(6,7)

(1,2) (1,3)

(1,4)

(2,3)(2,4)

(3,4)

Figure 4: LCG H for specific “conversational cliques” example

4Recall that I use undirected arcs because I assume throughout that (i, j) ∈ L⇒ (j, i) ∈ L, ∀ i, j ∈ N .5For clarity, I will maintain the format that I have introduced in Figures 3 and 4: I will use circular nodes in network graphs

and rectangular nodes in LCGs.


Wireless Interference Models

I now present some specific examples of interference models which feature commonly in work on wireless net-works. In what follows, the idea of an active link will play a central role:

Definition. A link l = (i, j) ∈ L is said to be active if either terminal i is transmitting to terminal j, orterminal j is transmitting to terminal i.

The first interference model which I explore is the primary interference model,6 which is described algebraicallyin [9] as follows:

“A transmission on link (i, j) is successful only if no links ofthe form (i, i∗), (i∗, i), (j, i∗) or (i∗, j) are also active, for any i∗ ∈ N .”

To illustrate this definition, I introduce a concrete example and deduce the corresponding LCG. Consider thenetwork whose network graph is shown in Figure 5.

1

2

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Figure 5: Network Graph

Under the primary interference model, if link (1, 2) is active then it is not possible to usefully activate any linkwhich ends at either node 1 or node 2. This means that in the LCG, the node (1, 2) will have arcs connectingit to nodes (1, 4), (1, 5) and (2, 3). Similar deductions can by made for each link, resulting in the LCG shownin Figure 6.

(1,2) (1,4)

(1,5)

(2,3)(3,4)

(4,5)

Figure 6: LCG for Network shown in Figure 5

Similar to the primary interference model is the two hop interference model, in which two links interfereif they share a node or if there is a link that connects any of the end nodes of the two links. Under a twohop interference model for the network shown in Figure 5, it would never be possible for any two links in thenetwork to be simultaneously active — in other words, the LCG would be complete.7

6The primary interference model is also known as the “node exclusive interference model” in some publications.7Recall that a graph is complete if every pair of nodes is connected by an arc [10].


Choosing an Appropriate Model

There are, of course, many other interference models which might apply to certain networks in a particularsituation. Primary interference models are appropriate for networks in which terminals are positioned a longway away from each other — the primary interference constraint provides a good reflection of the physicalfact that wireless transmissions only travel a finite distance, and so are unlikely to reach terminals beyond theirimmediate neighbours when the distances between them are large. They are also appropriate for networks wheresome form of spread spectrum multiplexing is available to enable superimposed transmissions to not becomeconfused, but where each terminal only has the ability to send or receive a single message at a time (generallydue to power constraints). A full discussion of the mechanics of such networks is outside the scope of thisessay, but [11] provides a clear explanation of such multiplexing via “Code Division Multiple Access”, and [12]explores how a primary interference model does indeed provide an accurate description of such a network.

The two hop interference constraint also has some particularly useful applications. Most notable is its abilityto provide a theoretical framework absent of any “hidden terminal” difficulties.8 In doing so, however, it actuallyintroduces something called the “exposed terminal” problem. [13] defines and discusses both of these issues ina wireless context, but as I turn my attention now to the main topic of the essay, I end my discussion of themhere.

4 The Aim of the Algorithms

Since Licklider’s vision in 1960, the primary goal of research into communications networks has been the transferof useful information within them. The problem that I explore now is the fact that often, many differentterminals want to communicate at the same time. As the work in Section 3 has highlighted, inter-terminalinterference provides a great obstacle for wireless networks in this situation. I now formalise the conceptskey to understanding how scheduling algorithms attempt to overcome such difficulties and enable effectivecommunication to take place.

New Ideas

Recall from Section 2 the graphical description of the network G = (N,L) and define a transmission schedule:

Definition. A transmission schedule X ⊆ L is a set of links which are active for data transmission. Atransmission schedule is also commonly represented by a vector x =

(x1, . . . , x|L|

), where:

xl =

{1 if l ∈ X0 if l /∈ X

for l = 1, . . . , |L| .

Notice from the definition that there is no guarantee that a transmission schedule X is free from interference,as X is no more than a set of links. In order to rectify this, it will be helpful to revisit some of the ideas aboutinterference already introduced, but in a more algebraic setting. To this end, define the neighbours of a link:

Definition. For a given interference model, define the neighbours of link l ∈ L to be the set:

N (l) := {l′ ∈ L | simultaneous transmission by l and l′ would result in interference}

Equivalently:l′ ∈ N (l)⇔ l and l′ are joined by an arc in the LCG

8In a two hop interference model, any terminal which may be hidden to a specific terminal cannot cause interference with it,and thus the fact that it is “hidden” is unimportant.


Interference considerations are brought together with the idea of a transmission schedule through the conceptof a feasible schedule:

Definition. A transmission schedule X is feasible if every link in X could be simultaneously active withoutany interference occurring. That is:

X is feasible ⇔ N (l) ∩X = ∅, ∀ l ∈ X

Denote the set of all feasible transmission schedules asM. Intuitively, the function of a scheduling algorithm isto use information about the current state of the network to somehow determine which transmission schedule(feasible or otherwise) is to be used. But what is meant by “the current state of the network”? And whatexactly is a scheduling algorithm trying to achieve?

Queues

Queueing is a phenomenon familiar to us all. Be it in a shop or a theme park, at times when there are morepeople requiring service than there are servers, a queue will form. A similar phenomenon plays a central role inwireless networks, where instead of “people” it is the packets of data which form the queue.

At any given time, numerous links within a network will have packets awaiting transmission along them,but clearly not all links can be active simultaneously so some will be forced to wait. Such packets must beheld in a short-term memory store called a buffer. While some authors (notably Kim and Krunz in [14]) haveexplored how imposing a finite buffer size affects wireless network performance by examining the number ofpackets which are dropped, I instead assume that buffer size is infinite.9 As such, I consider queue lengths tobe the primary quality of service (QoS) indicator in a network. Therefore, for the purposes of this essay, “thecurrent state of the network” is simply the current queueing state of the network:

Definition. The queueing state of the network is the vector Q(t) =(Q1(t), . . . , Q|L|(t)

), where Ql(t) is the

number of packets waiting in the queue for transmission on link l at time t, for l = 1, . . . , |L|.

With this definition in place, it is possible to formally define a scheduling algorithm:

Definition. A scheduling algorithm is a mapping A which takes a queueing state of the network at a giventime into a fixed probability distribution A (Q(t)) on the set of transmission schedules.

As I shall explore later, many scheduling algorithms only use local information about queue lengths in orderto make their decisions, and indeed some use no information at all. For example, the rule “always scheduleno links for transmission” is certainly a scheduling algorithm (albeit a perfectly inefficient one) which has noneed to know the entire queueing state of the network to function. So while the formal definition of schedulingalgorithm remains, it is often more illuminating to consider scheduling algorithms as using “some”, if not “all”,of the information about the queueing state of the network to make their decisions.

Service and Stability

With the terminals in a wireless network each performing complex tasks which vary depending on how userscontrol them, the arrival of packets at links must be viewed as a stochastic process. Formally, fix some smallδ > 0 and for l = 1, . . . , |L| define α(δ)

l (t) to be the number of packets which arrive at link l during the timeinterval [t, t+ δ). Then:

Definition. The arrival rate for the network is the vector λ =(λ1, . . . , λ|L|

), where λl := E

(α

(δ)l

), for

l = 1, . . . , |L|.9Such an assumption is not unreasonable given the tiny size of a packet when compared with the typical memory capacity of a

modern computer.


Because packet arrivals occur in a stochastic manner and the scheduling algorithm chooses how to scheduletransmission based on the queueing state of the network, service in the network is also a stochastic process.10

The definition of service rate for the network can therefore be formulated in the same spirit as the arrival ratewas above: for the same δ > 0 and for l = 1, . . . , |L|, define β(δ)

l (t) to be the number of packets which could betransmitted on link l during the time interval [t, t + δ), assuming there are sufficiently many packets in queuel.11 Then:

Definition. The service rate for the network is the vector µ =(µ1, . . . , µ|L|

), where µl := E

(β

(δ)l

), for

l = 1, . . . , |L|.

To avoid problems with the order that arrivals and departures occur in when queues empty, I assume thatall departures occur before all arrivals in any given interval.12 With arrivals and service clearly defined, it isnow possible to formulate an algebraic expression for the evolution of the queue at a given link:

Ql(t+ δ) = Ql(t)−min{β

(δ)l (t), Ql(t)

}+ α

(δ)l (t), for l = 1, . . . , |L| (1)

Given the exact values of Ql(t), β(δ)l (t) and α(δ)

l (t) for each l ∈ L, equation (1) allows explicit calculation of thequeueing state of the network at time t+δ to be made. The problem is, α(δ)

l (t) and β(δ)l (t) are random variables,

and so equation (1) can’t be used to predict future behaviour in a deterministic setting. However, throughoutthe essay I will assume that the “randomness” of arrivals in the network satisfies the following properties:

• α(δ)l (t) is independent of α′(δ)l (t), for all l, l′ ∈ L

• α(δ)l (t) is independent of α(δ)

l (t+ 1), for all l ∈ L

Then (as time is considered to be “slotted” in intervals of length δ) Q(t) is a discrete time Markov chain on acountable state space:13 clearly the state of the process at the next time step depends only on the present, notthe past — hence the familiar Markov property.14 The use of an infinite capacity buffer at each link means thatthe state space for Q(t) is simply Z|L|≥0.

Viewing Q(t) like this is very useful, as concepts and properties from Markov Chains can now be freelyapplied. In particular, recall the definition of ergodicity of a Markov Chain:

Definition. In a discrete time Markov Chain on state space S, a state s ∈ S is said to be ergodic if it isaperiodic and positive recurrent. If all states in a Markov chain are ergodic, then the chain is said to be ergodic.

Clearly ergodicity is something which is important in an effective wireless network: if some links’ queue lengthsare allowed to increase without limit, so too will the delay in queueing time for packets in that queue. A clearobjective of a scheduling algorithm is to attempt to ensure that Q(t) is ergodic, given the network arrival rateλ. A scheduling algorithm which succeeds in doing so is said to stabilise the network, for arrival rate λ.

But clearly it would be unfair to dismiss a scheduling algorithm as “inefficient” simply because it wasincapable of stabilising the network in some particularly extreme situation. After all, even the best schedulingalgorithm will fail to stabilise a network whose network arrival rate is simply too high. This is where the ideaof the capacity region, as discussed by Tassiulas and Ephremides in [16], comes in:

10A deterministic function of a stochastic process is itself a stochastic process, and certainly a stochastic function of a stochasticprocess is a stochastic process. So regardless of whether a scheduling algorithm be deterministic or stochastic, service in the networkis a stochastic process.

11If there are fewer packets in the queue than this number, it is assumed that the link continues to be occupied by “dummy” or“empty” packets for the remainder of the period. In this way, interfering links still cannot transmit in the period, but the queuelength for the link which is transmitting is not reduced beyond zero.

12This is equivalent to assuming that packets are not “ready” for transmission until the beginning of the interval after theirinitial arrival, which is a common assumption in the literature.

13There are a finite number of links, each with a queue length taking values in Z≥0, and the finite union of countable sets is itselfcountable.

14For a full discussion, see [15].


Definition. The capacity region Λ of a network is the set of arrival rates for which there exists some schedulingalgorithm which can stabilise the network. Arrival rates in the capacity region are known as feasible.

This idea leads to the concept of throughput optimality:

Definition. A scheduling algorithm is throughput optimal if it can stabilise the network for any arrival ratein the capacity region.

The central result of [16] explicitly connects the capacity region with the arrival and service rates as follows:

Theorem 1. Λ ={λ | ∃ some Y =

(Y1, . . . , Y|L|

)∈ conv (M) s.t. λl < Yl, ∀ l = 1, . . . , |L|

}where “conv (M)” denotes the convex hull of the set of feasible transmissions (when considered as vectors x).In other words, a scheduling algorithm only ever has any chance of stabilising the network if the arrival ratecan be expressed as some linear combination of the feasible schedules φ1x1 + φ2x2 + . . . + φ|M|x|M|, withφi ≥ 0 ∀ i = 1, . . . , |M| and

∑i∈M φi = 1.

5 MaxWeight Scheduling

The first scheduling algorithm which I examine in detail is MaxWeight Scheduling. As the name suggests, thestrategy of MaxWeight Scheduling is to choose for service in each time slot the feasible schedule which hasmaximal “weight”, where “weight” is a function of both the queueing state and the service rate of the network(as explored below). Time is assumed to be slotted into slots sufficiently short that the (randomly fluctuating)capacity of the air for carrying information in a given slot can be assumed constant. In particular, in any slotthe air is assumed to be in a “state” m, and associated with each state is a maximum rate of data transfer foreach link (which can be different for different links). MaxWeight Scheduling is formally defined as follows:

Definition. In each time slot, MaxWeight Scheduling chooses a feasible schedule x ∈M for transmission, suchthat:

x ∈ arg maxi∈M

|L|∑l=1

γlwl(i) (2)

where wl(i) := µml (i)δ (Ql)

ξ, and γ1, . . . , γ|L| and ξ are strictly positive constants.15 µml (i) denotes the servicerate which can be given to link l if the air state is m and feasible schedule i ∈ M is chosen. “Ties” are brokenarbitrarily by choosing the tied link with smallest index.

In most published literature about MaxWeight Scheduling, time intervals are chosen as one second in lengthand so µml (i) is simply the (integer) number of packets which are transmitted per second on link l if the air stateis m and feasible schedule i is selected. In my model, however, intervals are of length δ, so µml (i)

δ such packetsare transmitted per second. For a fixed δ, rescaling in this way will not affect the scheduling decisions made bythe algorithm, but I have decided to keep my work in the µml (i)

δ form so that direct parallels may be drawn withthe algorithms which I explore later, where time slots will be rescaled to be the duration of an average packettransmission.16

Theoretical Benefits, Practical Drawbacks

Some immediately appealing theoretical results about MaxWeight Scheduling are:

Theorem 2. MaxWeight Scheduling:

• Is throughput optimal for any arbitrary set of positive constants γ1, . . . , γ|L| and ξ.

15These constants are the parameters of the algorithm which can be set to tweak performance as desired.16So choosing an appropriate δ in the above model yields an equivalent setup.


• Can minimise maxl∈L alQl, for any set of positive constants a1, . . . , a|L|.

A generalisation of the proof of the first part of Theorem 2 is presented in [17], and a proof of the second partis presented in [18]. Both proofs are long and technical in spirit, so I omit them here.

For all of the benefits that results such as Theorem 2 seem to imply, even the briefest examination ofMaxWeight Scheduling reveals that it has limited – if any – practical use. The main problems are threefold:

• Centralised Computation: In order to calculate which feasible x ∈M provides the maximum weight,some comparison between all possible such xs needs to be drawn. Such a comparison is difficult – if notimpossible – in an ad hoc network: the absence of a hub means that there is no natural “leader” in termsof making scheduling decisions.

• Computational Difficulty: Even when a hub is present, finding a feasible schedule with maximal weightis an NP-hard problem.17 Solving such a problem is an exceptionally difficult task in general.

• Required a priori knowledge: In order to minimise maxl∈L alQl, the algorithm’s parameters need tobe set according to the arrival rate for the network. Knowing exactly what to set them to requires a prioriknowledge of such information, which is not always available in real-world networks.

In [7], a successor to MaxWeight Scheduling, called “The EXP-Rule”, is proposed. The EXP-Rule has manybenefits over MaxWeight Scheduling — not least that it continues to be throughput optimal and to minimisemaxl∈L alQl, for any set of positive constants a1, . . . , a|L|, without needing any a priori knowledge of the arrivalrate for the network. The drawback of the EXP-Rule, however, comes from its inability to deal with complexnetworks. All of the results in [7] are only applicable to networks with a central hub, and this hub is onlyable to communicate with one other terminal at any given time. Scheduling decisions can still be made ateach link using local information,18 but the methodology employed in the proofs involved doesn’t currentlyextend to a general ad hoc network. This is not to say that it won’t, however, as Shakkottai’s research groupcontinue to make advances in generalising their work to enable it to deal with “multi-server systems”. Over thecoming years, I think that it will be very interesting to see whether the “deterministic” spirit of the EXP-Rulewill eventually emerge as a successor to the current random access scheme approaches — to which I turn myattention now.

6 Random Access Schemes

Random Access Schemes take an entirely different approach to the problem of making scheduling decisions: thename “random access” describes a large class of scheduling algorithms which use a stochastic decision makingprocess in order to select which transmission schedule should be used at any given time. As well as beingefficient, I have found that the relative simplicity of random access schemes (when compared to MaxWeightScheduling) makes them very attractive for both theoretical study and real-world implementation. RandomAccess Schemes are still a primary focus of current research, with Shah et al ’s recent publication “ALOHAThat Works” [20] citing the “extreme simplicity and distributed nature” of the algorithms as the reason fortheir continued popularity.

The first important paper on the area that I examine is [9], in which Marbach, Eryilmaz and Ozdaglarexplore Random Access Static CSMA Policies. CSMA stands for Carrier Sensing Multiple Access, where acarrier sensing network is one in which each link l ∈ L can sense which other links l′ ∈ N (l) are currentlytransmitting. A carrier sensing mechanism is present in current 802.11 hardware, and sensing capabilities areclearly very helpful for a network in the process of selecting transmission schedules.

17For a discussion of the theory surrounding NP-hard problems, see [19].18i.e. it isn’t necessary to rely on the hub to perform multiple complex scheduling calculations at each step of the algorithm.


For the remainder of this essay, I consider time to be rescaled so that the average transmission durationfor a single packet is equal to one unit of time. In a discrete setting, let time-slots lengths equal this singleunit. Here however, unlike in [7], [17] and [18], Marbach, Eryilmaz and Ozdaglar approach the problem in acontinuous time setting. It is important to note (as is highlighted in [21]) that the practical application of suchan algorithm is therefore limited: under the current design of 802.11 hardware, backoff times are not continuous,but measured in multiples of “mini-slots”. The length of such a mini-slot is determined by the sensing time19

between nodes, and in reality this is not arbitrarily small. I hereafter refer to this sensing time as η > 0. Despitethis physical problem, the paper does still present an important result, which I explore now.

Operating under the primary interference model, the first step for a Static CSMA algorithm is that eachnode i ∈ N senses which links from the set

{l ∈ L | l either starts or ends at node i

}are transmitting. Call

the set of such links at node i that are not transmitting Li. Then, for each l ∈ Li, the node senses whetherany neighbouring link l′ ∈ N (l) is transmitting. If none of them are, link l ∈ Li is called idle. Define the endof an idle period for a link to be the event that the link has been idle for an amount of time exactly equal tosome positive integer multiple of the sensing time, η. With this setup in place, define a Static CSMA Policy asfollows:

Definition. A Static CSMA Policy is given by a vector of probabilities p =(p1, . . . , p|L|

). At any given time

t ≥ 0, each node i ∈ N will “mark” a link l ∈ Li with probability ql, where:

ql =

{pl if link l is at the end of an idle period at time t0 otherwise

and where each node marks links independently of all other nodes in the network. If a node marks a singlelink l, it will activate that link for a single packet transmission — this transmission will be “successful” only ifno other links in N (l) are also active at the same time. If it marks multiple links, a node “breaks the tie” bychoosing the tied link with smallest index.

The main result of [9] is that Random Access Static CSMA Scheduling is asymptotically throughput optimal,for large networks under the primary interference constraint with many small flows and a small sensing time.In this section I explore how, in their proof of this result, Marbach, Eryilmaz and Ozdaglar show that StaticCSMA Scheduling reduces the analysis of complex scheduling down to precisely the problem that was tackledat The University of Hawaii in 1970 by researchers of the ALOHA protocol.

Slotted ALOHA

As is described in [22], the ALOHA protocol was one of the first scheduling algorithms which facilitated effectivecommunication between terminals in a network. The “Slotted ALOHA” update enabled greater throughput tobe achieved in the same network, and operates as follows: assume that there are m terminals in a network, allof which are sharing a single communication channel.20 Terminals have no buffer, so they hold either zero orone packets. Packets arrive as a Poisson process, rate λ, and an arriving packet is randomly assigned to oneof the free terminals (where each free terminal is equiprobable). It is assumed that there are sufficiently manyterminals that a newly arriving packet will always be able to find a terminal with a free space for it, so nopackets are lost.21 Time is slotted, and there is no carrier sensing mechanism. At the beginning of each timeslot:

• Each terminal that received a new packet during the previous time slot will attempt to transmit this packet(independent of other terminals’ behaviour) with probability 1. If this is the only attempted transmission

19As in Carrier Sensing Multiple Access20i.e. the LCG is complete.21This assumption is often referred to as the “infinite set of nodes” or “m = ∞” assumption.


in the whole network, the transmission is successful and the terminal becomes free. If not, the transmissionfails and the terminal is deemed to be “backlogged”.

• Each backlogged terminal will attempt to transmit its packet (independent of other terminals’ behaviour)with probability q. If this is the only attempted transmission in the whole network, the transmissionis successful and the terminal becomes free. If not, the transmission fails and the terminal remains“backlogged”.

A quantity of interest is the number of backlogged stations, n. Clearly n(t) can be viewed as a discrete timeMarkov Chain with stationary transition probabilities, by “sampling” the number of backlogged terminals atthe end of each time slot.

CSMA Slotted ALOHA

Making use of the carrier sensing abilities of a wireless network enables a far more efficient scheduling algorithmcalled “CSMA Slotted ALOHA” to be used. In the standard Slotted ALOHA protocol just described, there isa positive probability that no packets – and thus no useful information – will be transmitted on a link for anentire time-slot. Using the carrier sensing mechanism in the network, however, terminals are able to detect iflinks are idle in just η time slots. Then if η is much less than the length of a time slot (which is certainly true of802.11 hardware capabilities), there is no point in forcing terminals to wait until a whole time-slot passes if thelink is idle. Instead, the next slot may as well begin as soon as it has been detected that this is the case. CSMAslotted ALOHA therefore exploits the carrier sensing mechanism in the network by allowing two different slotlengths to exist: slots which are purely idle last η time units and slots which are filled with a transmissionfollowed by an idle period last a total of 1 + η time units.

But if carrier sensing is to be used, it is clear that only “sampling” n at the end of every unitary time slotis inappropriate. Instead, it is necessary to sample the value of n at the end of every idle period. So now thetime between successive state “samplings” is either 1 + η (in the case of a successful transmission followed byan idle period) or simply η (in the case of no successful transmission, only an idle period).

In CSMA Slotted ALOHA, if a packet arrives at a node while a transmission is in progress, the packet isimmediately regarded as “backlogged”. Then:

Theorem 3. Let E(n) bet the expected length of time between successive samplings, given that there are currentlyn backlogged terminals. Then E(n) = 1 + η − e−λη (1− q)n .

Proof. Let W (k, n) := P (k transmissions attempted in a given slot, given n backlogged terminals). Then:

W (0, n) = P (No arrivals during previous idle slot) P (No retransmissions, given n terminals backlogged)

W (0, n) =e−λη (λη)0

0!·

(n

0

)q0 (1− q)n

= e−λη (1− q)n , so:

E(n) = ηe−λη (1− q)n + (1 + η)(1− e−λη (1− q)n

)= 1 + η − e−λη (1− q)n

There are two immediate and useful corollaries to this result:

Corollary 1. The expected number of arrivals between successive samplings, given that there are n backloggedterminals, is λ

(1 + η − e−λη (1− q)n

).


Proof. Arrivals occur as a Poisson process, so the expected number of arrivals per unit time is equal to thearrival rate λ. But in this setup, the expected length of time between successive samplings, given that there aren backlogged terminals, is 1 + η − e−λη (1− q)n. The result follows.

Corollary 2. The expected fraction of time for which the channel is idle is η1+η−e−λη(1−q)n .

Proof. The channel is idle for η time units between successive samplings, irrespective of whether these samplingsare η or 1+η time units apart. The expected length of time between successive samplings is 1+η−e−λη (1− q)n.Hence the result.

I now wish to examine ergodicity of CSMA Slotted ALOHA. To do so, I will consider a quantity called drift:

Definition. The drift Dn of a CSMA Slotted ALOHA network is the expected change in the number of back-logged terminals between the current sampling and the next sampling, given that there are currently n backloggedterminals.

The Markov Chain n(t) can only be ergodic if there exists some n∗ such that the drift is always less than orequal to zero when n ≥ n∗: otherwise the expected number of backlogged terminals will increase without limit.The following theorem is very useful in the examination of ergodicity:

Theorem 4. Let g(n) be the rate of transmission attempts, given that there are n backlogged terminals. Thenfor small q, the approximation Dn ≈ λ

(1 + η − e−g(n)

)− g(n)e−g(n) holds.

Proof. g(n) = E (# arrivals during previous idle slot)+E (# of retransmission attempts in current slot), so g(n) =λη + nq. Then:

Dn = E (# arriving packets)− E (# successfully transmitted packets, given n backlogged terminals)

= E (# arriving packets)− E(I{Packet successfully transmitted, given n backlogged terminals}

)= E (# arriving packets)− P (Packet successfully transmitted, given n backlogged terminals)

E (# arriving packets) is already known to be λ(1 + η − e−λη (1− q)n

), by Corollary 1. Recall the notation

W (k, n) = P (k transmissions attempted in a given slot, given n backlogged terminals). Then:

W (1, n) = P (Packet successfully transmitted, given n backlogged terminals)

=e−λη (λη)1

1!

(n

0

)(1− q)n q0 +

e−λη (λη)0

0!

(n

1

)(1− q)n−1

q1

= e−λη (1− q)n(λη +

nq

1− q

)

So, combining the above, Dn = λ(1 + η − e−λη (1− q)n

)− e−λη (1− q)n

(λη + nq

1−q

). Then using (for small q)

the approximations (1− q)n−1 ≈ (1− q)n, and (1− q)n ≈ e−qn:

Dn ≈ λ(

1 + η − e−(λη+qn))− e−(λη+qn)λη − e−(λη+qn)nq

≈ λ(

1 + η − e−g(n))− g(n)e−g(n), as required.

Using the definition of g(n) as outlined in Theorem 4, the results from Corollaries 1 and 2 imply that the arrivalrate is approximately 1 + η − e−g(n) and the fraction of time for which the channel is idle is approximately

η1+η−e−g(n) .


A natural target for a scheduling algorithm is to ensure that ergodicity is maintained for as high an arrivalrate as possible. Recall also that for ergodicity, it is necessary for the drift to “eventually” be less than or equalto zero. But:

λ(

1 + η − e−g(n))− g(n)e−g(n) < 0⇔ λ <

g(n)e−g(n)

η + 1− e−g(n)(3)

[9] describes how, by maximising g(n)e−g(n)

η+1−e−g(n) , it is possible to find the greatest arrival rate for which the schedulingalgorithm can maintain ergodicity — and hence the capacity region of the algorithm. The maximisation isachieved by varying the expected number of transmission attempts in a given interval, which in turn is controlledby varying the probability q of a retransmission attempt, because g(n) = λη + qn.

Theorem 5. For small η, using g =√

2η maximises ge−g

η+1−e−g .

Proof. A graphical approach, as I include in Appendix A, highlights the validity of this statement. It is alsopossible to find an analytically tractable approximation to the solution of this optimisation problem if theapproximation eg ≈ 1 + g + g2

2 is used:

ge−g

η + 1− e−g=

g

(η + 1) eg − 1

≈ g

(η + 1)(

1 + g + g2

2

)− 1

Differentiating with respect to g, and setting equal to zero:

⇒(η + 1)

(1 + g + g2

2

)− 1− g (η + 1) (1 + g)(

(η + 1)(

1 + g + g2

2

)− 1)2 = 0

⇒ (η + 1)(

1 + g +g2

2

)− 1 = (η + 1)

(g + g2

)⇒ g =

√2η, as required.

An important comment to make about the approximations in this section is that they are all asymptoticallyaccurate, which is to say that in the limit of the number of terminals in the network, the approximations thatI have made actually become equalities. This is explored by the authors in [9].

The CSMA Fixed Point

I now demonstrate how I have used the work above in my study of Static CSMA Scheduling. Consider a specificlink in a standard ad hoc network under Static CSMA Scheduling. Recall that if a node i is not currentlyengaged in another transmission, it will mark each link of the form (i, j) or (j, i) with probability p(i,j) or p(j,i)

(respectively) at the end of an idle period for that link. Thus, at the end of each idle period one of two thingswill happen to (i, j):

• The first possibility is that no nodes will mark (i, j), in which case (i, j) will be idle for η time slots beforethe nodes around it have the opportunity to mark it again.

• The other possibility is that (i, j) will be marked by one or more nodes. In this case, (i, j) will be occupiedfor 1 time slot for the transmission,22 and then there will be an idle period of η time slots before the nodesaround it have the opportunity to mark it again.

22Whether the transmission is successful or not will depend on whether other terminals attempt to transmit on this link, orindeed other links in N ((i, j)).


The similarities between Static CSMA Scheduling and CSMA Slotted ALOHA are now apparent: firstly, theslot lengths described above have precisely the same structure as those arising from CSMA Slotted ALOHA.Furthermore, if it is assumed that the times at which a given node is idle are independent of the processes at theother nodes, then the resulting behaviour observed at any given link is very close to a CSMA Slotted ALOHAsetup with capture rate g appropriately redefined. Recall that under the CSMA Slotted ALOHA model, g wasthe total rate of transmission attempts from all nodes together. Now it instead makes sense to consider rateslocalised to specific nodes, because if node i has been idle for η time units then there is a positive probabilitythat it will make an attempt to capture some link (i, j) ∈ L. It will only do so if node j is also free,23 andthen if node j is free the capture will only be attempted by node i with probability p(i,j), as set out in thedefinition of the specific Static CSMA Policy. Therefore (by the assumption of independence of behaviour ofnodes), defining ρj to be the fraction of time for which node j is free, the probability that node i makes anattempt to capture link (i, j) ∈ L after i has been idle for η time units is p(i,j)ρj . Also, the probability that nodej makes an attempt to capture link (j, i) ∈ L, given that i has been idle for η time units, is p(j,i)ρj . Combiningthese, the rate at which node i makes or receives capture attempts under Static CSMA policy p, after it hasbeen idle for η time units, can be approximated by Gi(p) as shown in equation (4).

Gi(p) =∑

j s.t.(i,j)∈L

(p(i,j) + p(j,i)

)ρj , i = 1, . . . , |N | (4)

Recalling that the proportion of time for which the communication channel was idle was given by ηη+1−e−g

under CSMA Slotted ALOHA, and using the appropriately redefined rate parameter, an approximation for thefraction of time ρi(p) that node i is idle under static CSMA policy p is given by:

ρi(p) =η

η + 1− e−Gi(p), i = 1, . . . , |N | (5)

Equation (5) is what Marbach, Eryilmaz and Ozdaglar call “The CSMA Fixed Point Equation”.The CSMA Fixed Point is only an approximation because of the assumption that the behaviour of every node

is independent of the transmission status of every other node. While this assumption is clearly not entirely true(for example, it is impossible for an odd number of terminals to ever be transmitting because each transmissionrequires a pair of terminals to be involved), it is clear that as the number of terminals tends to infinity theinter-dependencies between terminals become weaker. For example, in a network with only three terminals theability for any given terminal to transmit is entirely determined by whether the other pair are currently involvedin a transmission. On the other hand, in a huge network with hundreds of terminals operating over a massiveexpanse the status of a specific terminal will have very little effect on the actions of all but the closest terminals.Indeed, this intuitive example proves to be representative of a rigorous limiting behaviour: Marbach, Eryilmazand Ozdaglar prove in [9] that when the number of terminals tends to infinity and η tends to zero,24 the CSMAfixed point is actually asymptotically accurate.

With this key result in place, the following theorem provides a crucial link in working towards an explorationof throughput optimality:

Theorem 6. For every static CSMA policy p, there exists a unique CSMA fixed point ρ(p).

The proof of Theorem 6 is presented in [23]. It makes use of The Brouwer Fixed Point Theorem (see [24])and The Implicit Function Theorem. Due to space considerations, I do not prove the result here.

23Here is an example of the model’s dependence on the interference model used. If a secondary interference model was used, itwould also be necessary to ensure that no other interfering links were currently transmitting, greatly increasing the complexity ofthe problem.

24Recall that η is the sensing time between terminals. If the “location” occupied by the wireless network remains the same size,it is clear that as the number of terminals in the network increases, the average distance between terminals decreases. Using thefact that electromagnetic waves travel at finite speed, the sensing time decreases correspondingly.


Now I explore the important result that for any large network with a small sensing time, for any arrival ratein the capacity region there exists a static CSMA policy which will stabilise the queueing state of the network.The following technical lemma will play an important part in the proof:

Lemma 1. The service rate at a given link, under Static CSMA Policy p, is bounded below. Specifically:

µ(i,j)(p) ≥p(i,j)e

−(Gi(p)+Gj(p))ρj(p)1 + η − e−Gi(p)

Proof. Using the CSMA fixed point approximation leads to:

µ(i,j)(p) ≈p(i,j)ρj(p)e−(GRi (p)+Gj(p))

1 + η − e−Gi(p)

where GRi (p) =∑j s.t. (i,j)∈L p(j,i)ρj(p) is the rate that node i receives capture attempts on links of the form

(j, i), andGj(p) is the total rate of capture attempts on links into and out of node j. Then p(i,j)ρj(p)e−(GRi (p)+Gj(p))

is the probability that node i attempts to capture the link (i, j), given that it has been idle for η time units, andthat this capture attempt does not overlap with any other capture attempt either to or from node j. The in-equality follows from this approximation by noting that Gi(p) ≥ GRi (p), and hence that e−Gi(p) ≤ e−GRi (p).

Before exploring the result about throughput optimality, consider the following definitions:

Definition. Define the arrival rate at node i, ψi, to be∑j s.t. (i,j)∈L

(λ(i,j) + λ(j,i)

).

Definition. Define the service rate of node i under Static CSMA Policy p to be the fraction of time that nodei is involved in successful transmissions under p (i.e. time spent involved in successful transmissions

total time ). Denote this byτi(p).25

Definition. Define the service rate of the network under Static CSMA Policy p to be τ(p) :=∑i∈N τi(p).

It is shown in [9] (and, indeed, suggested by the relationship presented in equation (3) for a CSMA SlottedALOHA network) that the CSMA fixed point can be used to approximate the service rate of the network as:

τ (G(p)) =G(p)e−G(p)

η + 1− e−G(p)(6)

With these definitions and lemmas in place, the key result about Static CSMA Scheduling is:

Theorem 7. Let Γ(η) :={λ|ψi < τ (G+(η)) e−G

+(η), i = 1, . . . , |N |}

, where G+(η) is the G which maximisesthe network service rate. Then for every λ ∈ Γ(η) there exists a static CSMA policy p such that λ(i,j) ≤ τ(i,j)(p)for all (i, j) ∈ L.

Proof. By definition of Γ, the inequality:

ψi < τ(G+(η)

)e−G

+(η) (7)

holds for all i ∈ N . Consider the function f(G∗i ) := e(G∗i−G

+(η))τ (G∗i ) e−G+(η). Clearly f(0) = 0, and notice

that f is continuous. But f (G+(η)) = τ (G+(η)) e−G+(η), so, by (7), f (G+(η)) > ψi for all i ∈ N . Therefore

(by The Intermediate Value Theorem, [25]) for each i ∈ N there exists some Gi ∈ [0, G+(η)) such that:

f(Gi) = e(Gi−G+(η))τ (Gi) e−G

+(η) = ψi (8)

25Notice that under the rescaling of time such that the average duration of a packet transmission is 1, if there is always at leastone packet awaiting transmission at every node then τi(p) is precisely equal to the throughput at node i, for all i = 1, . . . , |N | .


Recall ρi = ηη+1−e−Gi , and define a Static CSMA Policy p by:

p(i,j) =λ(i,j)ηe

2G+(η)

ρiρjfor each (i, j) ∈ L (9)

I wish to investigate the form of Gi(p) under this policy. Using (4):

Gi(p) =∑

j s.t.(i,j)∈L

(p(i,j) + p(j,i)

)ρj

=∑

j s.t.(i,j)∈L

λ(i,j) + λ(j,i)

ρiηe2G

+(η), by (9).

=ηe2G

+(η)

ρiψi. Then, using the Gi chosen in (8):

=ηe2G

+(η)

ρie(Gi−G

+(η))τ(Gi)e−G+(η)

=ηeGiτ(Gi)

ρi

= ηeGiGie−Gi

η + 1− e−Gi· η + 1− e−Gi

η

= Gi

So Gi(p) = Gi for all i ∈ N , and so G(p) = G and ρ(p) = ρ. Therefore the choices G = (G1, . . . , G|N |) andρ = (ρ1, . . . , ρ|N |) define the CSMA fixed point of Static CSMA Policy (9).

Recall the inequality given in Lemma 1:

µ(i,j)(p) ≥p(i,j)e

−(Gi(p)+Gj(p))ρj(p)1 + η − e−Gi(p)

. Using the above:

=p(i,j)e

−(Gi+Gj)ρj

1 + η − e−Gi. Then, using Static CSMA Policy (9):

=λ(i,j)

ρiρjηe2G

+(η) ρje−(Gi+Gj)

1 + η − e−Gi

= λ(i,j)e(G+(η)−Gi)e(G

+(η)−Gj)

But recall that G+(η) ≥ Gn for all n ∈ N . So λ(i,j)e(G+(η)−Gi)e(G

+(η)−Gj) ≥ λ(i,j), and hence µ(i,j)(p) ≥ λ(i,j)

for all (i, j) ∈ L. So the Static CSMA Policy defined in (9) does indeed have the required property.

This result then leads directly to the fact that as η → 0, the set of arrival rates which can be stabilised bysome static CSMA policy is equal to the capacity region of the network. A formal derivation (including all thetechnical details) is presented in the closing part of [9], leading to the conclusion that “static CSMA algorithmsare asymptotically throughput optimal for large networks under the primary interference constraint, with manysmall flows and a small sensing time”. It is, however, possible to consider the general idea (without the finerdetails) as follows: The methods used to prove Theorem 5 all carry over to a Static CSMA Scheduling framework(after the necessary adjustment to the rate, as described above). Therefore G+(η) =

√2η, and τ(

√2η) =

√2ηe−

√2η

η+1−e−√

2η . Then (e.g. by L’Hopital’s Rule) limη→0 τ(√

2η) = 1, so limη→0 Γ(η) = {λ|ψi < 1, i = 1, . . . , |N |}.But recall that time was rescaled so that one time slot was equal in length to the average duration of a singlepacket transmission. Hence the whole capacity region is in limη→0 Γ(η), and there exists a static CSMA policywhich can stabilise the queueing state of the network for any arrival rate in Γ(η). Hence the result.


Adaptive CSMA: An Implementable Approach

Although rooted in a physical setup, Static CSMA Scheduling has a serious limitation as a practical tool: itrequires a priori knowledge of the long term arrival rates at the links in order to make scheduling decisions. Suchknowledge may be very difficult to obtain — particularly if new terminals are being added and removed fromthe network on a daily basis, as is typical of many wireless networks in industry. So while the above examinationof Static CSMA Scheduling certainly provides a key theoretical result (namely “static CSMA algorithms areasymptotically throughput optimal for large networks under the primary interference constraint, with manysmall flows and a small sensing time”), it falls short of providing a solution to the problem of administering apractical implementation.

In [8], Jiang and Walrand attempt to overcome this problem by proposing a new approach. Their “AdaptiveCSMA Scheduling” algorithm – which dynamically adapts its parameters according to observations of thecurrent arrival rate – requires no such a priori knowledge. Recall the vector form of the definition of a feasibletransmission schedule from Section 4. Using the assumption that η = 0 (the implications of which are discussedlater), the algorithm begins with the initial feasible transmission state of x = 0, and proceeds (at each linkl ∈ L) as follows:

1. The link generates a random variable Tl ∼ Exp (Rl) , as discussed below.

2. The link begins counting down from Tl, pausing the countdown whenever any link l′ ∈ N (l) is transmittingand resuming it when all such neighbouring transmissions are complete.

3. When the countdown reaches zero, the link begins the transmission of a packet. This is assumed to take anexponential amount of time to transmit (rather than the guaranteed unitary time slot duration assumedin [9]).

4. When the transmission is complete, the link returns to step 1.

If it is assumed that operation proceeds in continuous time and that η = 0, the above algorithm maintains afeasible transmission state at all times. Under this setup the probability of packet collisions is zero — but thisis not an indication of the algorithm working exceptionally well, but rather an indication that the assumptionslead to an imperfect representation of the reality of the situation. Specifically, the sensing time being zeroviolates fundamental physical laws about the maximum speed of electromagnetic waves and the backoff timesbeing drawn from a continuous distribution ignores the physical limitations of 802.11 hardware.

It is clear that the transmission state evolves as a continuous time Markov process with stationary transitionrates on the set of feasible transmission states. Specifically, let xc be a feasible transmission state in M. Atany given time, if the current transmission state is xc and some xcl = 0, then if no link l′ ∈ N (l) is transmitting,xc transits to state xc + el at rate Rl, where el is a vector consisting of all zeroes, except for a one in the lth

position. Similarly, if xcl = 1, then xc transits to state xc − el at rate 1.The problem then remains to choose the Rls in such a way that the algorithm has useful properties —

and indeed these are the parameters which the algorithm dynamically adjusts during its operation. For conve-nience, define rl = log (Rl), and call this the “transmission aggressiveness” of link l. Define the “transmissionaggressiveness of the network” as r =

(r1, . . . , r|L|

). It is useful to examine the stationary distribution of the

underlying Markov Process:

Theorem 8. The stationary distribution of this Markov Process, under transmission aggressiveness r, is:

π (xc|r) =exp

(∑|L|l=1 x

cl rl

)∑d∈M exp

(∑|L|l=1 x

dl rl

)


Proof. Given a transmission state xc, consider the set A = {l | xcl = 1} (i.e. the set of links which are trans-mitting under the given transmission state). Label the elements of A as l1, . . . , ln. Notice that xc =

∑lc∈A elc .

Then, using the reversibility of the process,26 solve detailed balance (I omit the r after the first line, forbrevity):

π (xc|r) q (xc,xc − el1 |r) = π (xc − el1 |r) q (xc − el1 ,xc|r)

⇒ π (xc) = π (xc − el1)q (xc − el1 ,xc)q (xc,xc − el1)

= π (xc − el1)Rl1

But: π (xc − el1) q (xc − el1 ,xc − (el1 + el2)) = π (xc − (el1 + el2)) q (xc − (el1 + el2) ,xc − el1)

⇒ π (xc − el1) = π (xc − (el1 + el2))q (xc − (el1 + el2) ,xc − el1)q (xc − el1 ,xc − (el1 + el2))

= π (xc − (el1 + el2))Rl2

And so on, up to:

π(xc −

(el1 + . . .+ eln−1

))= π (xc − (el1 + . . .+ eln))Rln

But notice that xc − (el1 + . . .+ eln) = 0. So the expression simplifies to:

π (xc) = π (0)Rl1Rl2 . . . Rln

=|L|∏l=1

Rxcll π (0)

= exp

|L|∑l=1

xcl rl

π (0)

Summing both sides over all feasible transmission schedules:

1 =∑d∈M

exp

|L|∑l=1

xdl rl

π (0)

So π (0) = 1Pd∈M exp

nP|L|l=1 x

dl rl

o , and the result follows.

The service rate at a given link can now be stated explicitly as follows:

Corollary 3. For a given link l ∈ L, the service rate, under transmission aggressiveness r, is given by:

µl(r) =

∑c∈M xcl exp

(∑|L|l=1 x

cl rl

)∑d∈M exp

(∑|L|l=1 x

dl rl

)Proof. Clearly µl(r) =

∑c∈M xclπ (xc|r).27 Then use Theorem 8.

Recall the definition of the capacity region Λ of the network from Section 4. Notice that under Adaptive CSMAScheduling, an arrival rate λ is feasible (i.e. in the capacity region) if there exists some vector π =

(π1, . . . , π|M|

),

where πd ≥ 0 for all d = 1, . . . , |M| and∑d∈M πd = 1, such that λ =

∑d∈M πdx

d. Define λ to be strictlyfeasible if in addition πd > 0 for all d = 1, . . . , |M|. The key result from [9] is:

26As outlined in [8].27I assume, as before, that if a particular link’s queue becomes empty then that queue simply transmits “dummy” packets until

new real packets arrive.


Theorem 9. If λ is strictly feasible, then µl(r∗) ≥ λl ∀ l ∈ L, and hence the queueing state of the network isergodic.

Proof. Consider the function F (r) :=∑c∈M πc log (π (xc|r)). As well as being a log likelihood function for

estimating r given a large number of observations of xcs,28 F (r) has many useful properties for proving thetheorem. The first step in the proof is to use a result from Jiang and Walrand which states that whenever λis strictly feasible, there exists a finite r∗ ≥ 0 such that the equality supr≥0 F (r) = F (r∗) holds. This is atechnical result and a proof is presented in the Appendix of [8], so for brevity I will omit it here.

Knowing that there is a finite r∗ at which the supremum is attained, Lagrangian methods can be used tolocate it. First, expand F :

F (r) =∑c∈M

πc log (π (xc|r))

=∑c∈M

πc log

exp(∑|L|

l=1 xcl rl

)∑d∈M exp

(∑|L|l=1 x

dl rl

)

=∑c∈M

πc

|L|∑l=1

xcl rl − log

∑d∈M

exp

|L|∑l=1

xdl rl

=∑c∈M

πc

|L|∑l=1

xcl rl −∑c∈M

πc log

∑d∈M

exp

|L|∑l=1

xdl rl

But because λ is (strictly) feasible, it holds that for all l ∈ L, λl =

∑c∈M πcx

cl . So:

F (r) =|L|∑l=1

λlrl −∑c∈M

πc log

∑d∈M

exp

|L|∑l=1

xdl rl

=|L|∑l=1

λlrl − log

∑d∈M

exp

|L|∑l=1

xdl rl

because

∑c∈M πc = 1.

F (r) is concave in r, so I proceed using Lagrange Multipliers: Let L (r;h) := F (r) + hTr, where h is avector of non-negative Lagrange multipliers. Then:

L (r;h) =|L|∑l=1

λlrl − log

∑d∈M

exp

|L|∑l=1

xdl rl

+|L|∑l=1

hlrl

∂L∂rl

∣∣∣∣r=r∗,h=h∗

= λl −

∑d∈M xdl exp

(∑|L|l=1 x

dl r∗l

)∑d∈M exp

(∑|L|l=1 x

dl r∗l

) + h∗l

= λl − µl(r∗) + h∗l , by Corollary 3.

At the supremum, λl − µl(r∗) + h∗l = 0. Then, by the definition of h as having non-negative entries, it musthold that for all l ∈ L, λl − µl(r∗) ≤ 0, and hence that λl ≤ µl(r∗) as required. So the queueing state of thenetwork is ergodic.

These theoretical results show that if the transmission aggressiveness of the network can be dynamically updatedin order to accurately “track” r∗, then the network will be able to maintain stability of all queue lengths without

28Such an observation is not crucial in the development of the theory here, but I include a derivation of the idea in Appendix B.


prior knowledge of the arrival rates, because observations of the (randomly fluctuating) quantities of interestare sufficient.

The aim of Adaptive CSMA Scheduling is to regularly “sample” the information required in order to deter-mine how r should change, and to update r accordingly. There is not even any need to record long runninginformation about how the process has behaved in the past — simply define λ′l(t) and µ′l(t) to be the averagearrival and departure rates (respectively) in a given interval (between times t and t+ ω, say). Then λ′l(t) andµ′l(t) can be used, instead of the true arrival rates λl and µl(r), to update rl as follows:

For each l ∈ L, let: rl(t+ ω) = [rl(t) + a · (λ′l(t)− µ′l(t))]+ (10)

where a is a positive constant “step size” for the algorithm.To understand why this works, consider a typical relationship between r and F (r), such as the one shown in

Figure 7. Then, using again the fact that ∂F (r)∂rl

= λl − µl(r∗), notice that if, at time t, F (r(t)) < F (r∗), thenλl(t) − µl(r∗) > 0, so the algorithm sets rl(t + ω) > rl(t). Hence F (r(t+ ω)) > F (r(t)), and so movement isindeed towards the supremum. Similarly, if F (r(t)) > F (r∗), then λl(t)− µl(r∗) < 0, so rl(t+ ω) < rl(t) andso F (r(t+ ω)) > F (r(t)), and so movement is again towards the supremum.

F ( )

∗

r

rr

Figure 7: Example Relationship between r and F (r∗)

Adaptive CSMA Scheduling is a distributed algorithm because all of the information needed by a given linkin order to manage its transmissions is available locally. The algorithm governing the progression of the linkfrom idle, to waiting, to transmitting and back to idle again only uses information from the link itself and fromits neighbours N (l).29 The algorithm for adapting the transmission aggressiveness for a specific link only usesinformation available at the link itself.

Recall from Section 4 that equation (1) can be used to describe the evolution of the queueing state of thenetwork. Notice that it is now possible to rewrite equation (1) using the quantities just defined, and in a formvery similar to that of equation (10):

For each l ∈ L, Ql(t+ ω) = [Ql(t) + ω · (λ′l(t)− µ′l(t))]+ (11)

Then, after choosing an initial condition (i.e. an initial r), equations (10) and (11) can be combined to give thesingle equality:

rl(t) =a

ωQl(t) (12)

Equation (12) highlights an immediate and important property: if r is stable, then Q is stable. Oneremaining difficulty arises in the inevitable random fluctuation of λ′l(t) and µ′l(t): “outlier” observations ofthese parameters in the short run can cause the algorithm to temporarily poorly track the actual long run

29Such information about the status of neighbouring links is available via the carrier sensing mechanism.


arrival and departure rates. One way around this is to allow a to be very small, so that updates which thealgorithm makes to the rl are only very small, and therefore short term irregular behaviour only slightly affectsthe parameters before the arrivals and departures settle back down. A different approach is to let ω be verylarge, so that the actual periods of time over which observations are averaged (so as to estimate the rate) arevery long. By The Central Limit Theorem [26], increasing ω would result in individual estimates of arrivalsand departures being closer to the true long run rates, because more information would be included into eachestimate.

Jiang and Walrand note in [9] that despite such potential difficulties, their simulations suggest that thealgorithm performs well even with small values of ω and moderate values of a. They do indicate, however,that more research into this area would provide a deeper understanding of how important it is for the randomprocesses to be “close to” their stationary distributions in order for useful inferences to be drawn.

7 Collisions

Under some assumptions, the Adaptive CSMA algorithm described in Section 6 has been shown to be capableof stabilising the queueing state of the network for any strictly feasible arrival rate. While undoubtedly animportant result, by operating in continuous time and assuming that the sensing delay in the network is zero,the algorithm fails to address the possibility of packet collisions. The assumptions built into the model do notreflect the reality that under current 802.11 hardware, backoff times must be multiples of mini-slot lengths,where these mini-slots have length equal to the sensing time η > 0. Therefore the distribution of backoff timesis discrete, and so collisions can occur with positive probability. I now explore two different approaches toovercoming this problem: the first is a minor alteration to the work proposed above, while the second uses a“two-phase” approach to scheduling in order to minimise the disruption that collisions cause.

Adapting the Algorithm

One way to address the practical shortcomings identified above is by modifying the Adaptive CSMA algorithmpresented in Section 6. The problem with the algorithm as it stands is that an exponential random variable isbeing generated — the backoff time is therefore a continuous random variable, when what is really required isan integer multiple of η.

In the closing part of [8], Jiang and Walrand propose the use of the discrete uniform distribution to solvethis problem. In particular, they say that a contention window Wl should be chosen such that if the backofftime is sampled from the discrete uniform distribution on [0,Wl − 1], then the expected backoff time is equalto the expected backoff when chosen from the (continuous) exponential distribution (up to integer rounding).That is:

E (Backoff time under exponential setup) = E (Backoff time under discrete uniform setup)

1Rl

= ηWl − 1

2

Solve for Wl:Wl =

2ηRl

+ 1 (13)

Due to the different structures on which they are built, it is difficult to draw parallels between the “continuousbackoff” and the “discrete backoff” versions of the model at a theoretical level. For this reason, I have exploredthe relative performance of the different versions of Adaptive CSMA Scheduling via computer simulations.Specifically, I used R to run simulations of Adaptive CSMA Scheduling over a time horizon of 10000 average


packet transmission durations,30 and compared the total number of successful transmissions:

1. Under “idealised” (continuous time) Adaptive CSMA Scheduling, as presented in Section 6.

2. Under the discrete time modification of Adaptive CSMA Scheduling, but still counting colliding transmis-sions as being successful.

3. Under the discrete time modification of Adaptive CSMA Scheduling, but not counting colliding transmis-sions as being successful.

The “average packet transmission duration” to “mini-slot length” ratio is taken to be 100:1, as an approximationto the physical operation of 802.11a hardware, as detailed in [27]. I performed each simulation on two differentnetworks, whose network graphs are shown in Figure 8.

1 2 3 4

5

6

78910

11

12

1

2

34

5

Network 1 Network 2

Figure 8: Network Graphs used in Simulations

For each network I used the primary interference model, as described in Section 3. This results in the LCGsshown in Figure 9.

(1 2),

(1 5),

(2 3),

(2 4),(3 4),

(3 5),

(4 5),

(1,2) (2,3) (3,4) (4,5)

(5,6)

(6,7)

(7,8)(8,9)(9,10)(10,11)

(11,12)

(12,1)

Network 1 Network 2

Figure 9: LCGs used in Simulations

The source codes for all of the simulations in this essay are available on http://www.martingould.webs.com.The results of these simulations are displayed (as tests a, b and c respectively) in Figure 11 and Table 1 inAppendix C.

The discrete time adaptation of the algorithm did not perform as well as the original continuous time versionin either network, even when colliding transmissions were deemed to still be successful. Then, as expected,banning colliding transmissions further reduced the number of packets which were successfully transmitted.

I do not consider the reduction in throughput exhibited in changing from the continuous to the clash-freediscrete time version of the model to be surprising — although the move to discrete uniform backoff times allowsthe idea to be implemented under current 802.11 hardware, such a “fix” is forcing the algorithm to operate underconditions different to those for which it was designed. I find the reduction in throughput when disallowingclashing transmissions to be similarly unsurprising, but it also motivates a new problem for me: whether usingan algorithm which actively seeks to minimise collisions might lead to an increase in throughput.

30The counters in the codes show different stopping times because the time frame used is dependent on whether that code iscounting in average packet transmission durations, in mini-slot lengths (i.e. η) or in some combination of the two.


Collision Avoidance

As is highlighted above, colliding transmissions erode the efficiency of a network. Links become active so thatterminals can transmit packets, but the colliding transmissions result in no useful information being received bythe destination terminals. Because such colliding transmissions waste time, Collision Avoidance mechanismscan be built into scheduling algorithms in an attempt to minimise this disruption.

The first collision avoidance mechanism that I explore is proposed at the end of [8] as an extension to thediscrete time version of Adaptive CSMA Scheduling. Here, the collision avoidance tactic exploits the fact thatlarger contention windows result in lower collision probabilities. This is because if a link is to choose its backofftime and a neighbouring link is already part-way through a countdown, the probability that the links eventuallycollide their transmissions is equal to the probability that the link just starting its backoff chooses a backofftime exactly equal to the remainder of the backoff time of the other link: this is 1

Wl.

Equation (13) reveals that under the model setup described, the only way to make Wl large is to make Rlsmall — but the values of r, and hence Rl, are updated dynamically by the algorithm so setting them directlyis not possible. However, it is possible to prevent them from becoming too large by bounding them above(componentwise) by some rmax. In particular, bounding r ≤ rmax guarantees:

Wl ≥2

ηermaxl+ 1

To examine the effectiveness of this collision avoidance mechanism, I again ran simulations in R (under the samesetup as is described above). The results are displayed as test d in Figure 11 and Table 1 in Appendix C.

Interestingly, imposing an rmax upper bound does not serve to increase the number of packets successfullytransmitted in either case. Examination of the clashed counters in my code shows that although the number oftransmissions which collide does fall (in both networks) when the upper bound is applied, the reduction that isachieved is very small. This reduction comes at the expense of links choosing backoff times with higher means,and so I conclude from the results that (for the networks studied and under the primary interference model)the additional waiting times imposed at the links are more harmful to throughput than the wasted time fromthe (now avoided) collisions.

Collision Free Scheduling

Having noted one particular collision avoidance mechanism to be ineffective for the networks in my simulations,I now turn my attention to a distributed scheduling algorithm which ensures that collisions are avoided entirely.The algorithm, proposed by Ni and Srikant in [21], is called CSMA Collision Avoidance Scheduling. The keyfeature of the algorithm is that it ensures only feasible schedules are chosen for the transmission of packets. Itdoes so by dividing time not only into slots, but into two distinct phases within each time slot: the controlphase and the data phase. During the control phase, links use the algorithm to make scheduling decisions (tobe used in the data phase) based only on information available locally.31 This is done in a manner very similarto Adaptive CSMA Scheduling, and the result is the selection of a feasible schedule which will be used in thedata phase for the transmission of packets. Thus x(t) evolves as a Markov Chain, but by allowing collisions inthe control phase, CSMA Collision Avoidance Scheduling ensures that there are no collisions whatsoever in thedata phase. Herein lies the unique feature of the algorithm: rather than having collisions occur during packettransmission (when they are costly in terms of time wasted), it only allows them to occur in the control phase,as part of the process of selecting a feasible schedule for the data phase.

CSMA Collisions Avoidance Scheduling can achieve throughput optimality by dynamic adaptation of thealgorithm’s parameters. The ratio of the length of the control phase to the length of the data phase is thena precise, quantitative measure for the loss in network performance necessary to maintain feasibility of the

31Hence the “distributed” nature of the algorithm.


transmission schedule: the longer the control phase (relative to the data phase), the more time the algorithmspends doing things other than actually transmitting packets. It was shown in [21] that a control phase oflength 2η is all that is required per data phase (in which a single packet is sent) in order to achieve throughputoptimality. This figure has since been reduced to just η following a modification to the algorithm by Libin Jiang,although this modification was only made very recently and is yet to be thoroughly researched, so I restrict myattention to the 2η version (aside from some simulations later).

The remarkably small ratio of control phase to data phase length is independent of the size of the network —a property which is very useful when dealing with ad hoc networks to which terminals are added and removed ona regular basis. In addition, the results proven about CSMA Collision Avoidance Scheduling hold for a generalinterference model, and so are very attractive from both theoretical and practical standpoints.

I now describe how Ni and Srikant’s algorithm proceeds during the control phase, for each link l:

1. Link l independently selects a backoff time Tl = klη by sampling kl from the discrete uniform distributionon [0,Wl − 1] (where Wl is explored below).

2. Link l starts counting down from Tl, in such a way that all links start their countdowns simultaneously.

3. If, during this countdown, l hears an “INTENT” message (using its carrier sensing mechanism) from anylink l′ ∈ N (l), it stops counting and sets xl(t) = xl(t − 1) (so it will behave in the coming data phaseexactly how it behaved in the previous one).

4. If l’s countdown reaches 0, it broadcasts an “INTENT” message (which can be heard by all links l′ ∈ N (l)).

5. If l hears an “INTENT” message from any link l′ ∈ N (l) within η of broadcasting its own “INTENT”message, l knows that this “INTENT” was actually broadcast at exactly the same time as l broadcast itsown (because each link chose kl as an integer), so l sets xl(t) = xl(t− 1).

6. If l does not hear an “INTENT” message from another link in N (l) within η of broadcasting its own, l isconsidered to be in the changing schedule, M .

7. After a total time of Wlη has passed since the countdowns began, if l is in M it determines xl(t) as follows:

• If there exists some link l′ ∈ N (l) which transmitted during the previous data slot, then xl(t) = 0.

• If no links l′ ∈ N (l) transmitted during the previous data phase, then xl(t) = 1 with probabilitypl,32 else xl(t) = 0.

8. Now xl(t) is determined for all l ∈ L, so proceed to the data phase.

It should be clear from this description that:

x(t− 1) feasible ⇒ x(t) feasible33

Define M0 to be the set of all possible changing schedules M , and define α(M) to be the probability thatin a given control phase, the chosen changing schedule is M ∈M0. Then:

Lemma 2. A feasible transmission schedule X can make a transition to another feasible transmission scheduleX ′ iff X ∪X ′ is feasible and there exists some changing schedule M such that (X \X ′) ∪ (X ′ \X) ⊆M .

32Where pl is a dynamically updated parameter, as explored later.33A formal proof is laid out in [21], but intuitively it is enough to notice that the only links which change their transmission

status from 0 to 1 (and hence the only links which could cause a problem for feasibility) have no transmitting neighbours, andhence are links for whom a change from 0 to 1 won’t cause a problem for feasibility.


This is a “book-keeping” lemma, and the proof is simply a case of keeping track of the algorithm’s behaviour,so I omit it here. The interested reader may find the complete proof in [21].

Lemma 2 provides a concrete way to identify which states the Markov Chain x(t) may make transitionsbetween. Then:

Theorem 10. If X can make a transition to X ′, then the probability of such a transition being made is givenby:

∑M s.t. (X\X′)∪(X′\X)⊆M

α(M)

∏l∈X\X′

(1− pl)

∏k∈X′\X

pk

( ∏i∈M∩X∩X′

pi

) ∏j∈M\(X∪X′)\N (X∪X′)

(1− pj)

Proof. For a link l ∈ M , if there is some link l′ ∈ N (l) which transmitted during the previous data slot (i.e.xl′(t − 1) = 1), then certainly xl(t) = 0. But all other links can change their transmission status with somepositive probability. The possible changes are 1 → 0, 0 → 1, 1 → 1 and 0 → 0, and links only change theirstatus if they are in M . Consider these cases in turn:

• A 1→ 0 change happens when l ∈ X but l /∈ X ′. The conditional probability of such a change, given thatk ∈M , is pk.

• A 0 → 1 change happens when k /∈ X but k ∈ X ′. The conditional probability of such a change, giventhat l ∈M , is (1− pl).

• A 1 → 1 change happens when i ∈ X ∩X ′. But a 1 → 1 transmission status would also be observed ifi /∈ m and xi(t − 1) = 1, so the “correct” type of change only happens when i ∈ M . The conditionalprobability of such a change, given that i ∈M , is pi.

• A 0 → 0 change happens when j ∈ M \ (X ∪ X ′) AND j does not have its transmission state decidedby the fact that a neighbouring link transmitted during the previous data phase, or is about to transmitin the coming data phase (which would be the case if j ∈ N (X ∪X ′)). So the conditional probability ofsuch a change, given that j ∈M , j /∈ (X ∪X ′) and j /∈ N (X ∪X ′) is (1− pj).

Then, as each link makes its decision independently of all other links, the overall conditional probability ofa transition X → X ′, given the relevant memberships of M , is the product of the individual link transitionconditional probabilities. The Law of Total Probability [28] then produces the final result.

As an intermediate step in proving throughput optimality, Ni and Srikant present the following lemma:

Lemma 3. The Markov Chain x(t) is irreducible and aperiodic iff L = ∪M∈M0M , and in this case the resultingstationary distribution is given by:

π(X) =

∏i∈X

pi1−pi∑

X∈M∏i∈X

pi1−pi

(14)

Proof. The proof of “irreducible and aperiodic” is simple book-keeping (the details can be found in [21]). Thederivation of the stationary distribution uses precisely the same “detailed balance” technique as I used in theproof of Theorem 8.

The central result of the paper is then:

Theorem 11. The Markov Chain x(t) achieves the stationary distribution given in equation (14) wheneverWl ≥ 2 ∀ l ∈ L.

Proof. Define a maximal feasible transmission schedule to be a feasible transmission schedule such that noadditional link could be added to the transmission schedule without violating the interference constraint. Denotethe set of all maximal feasible transmission schedules as Mmax.


In any maximal feasible transmission schedule, if a link l is inactive then at least one other link in N(l)must be active, otherwise X would not be maximal. Therefore X would certainly be chosen as the changingschedule if Tl = 0 ∀ l ∈ X and Tl′ = 1 ∀ l′ /∈ X, and so α(X) ≥ P (Tl = 0 ∀ l ∈ X, Tl′ = 1 ∀ l′ /∈ X). ButWl ≥ 2 ∀ l ∈ L⇒ P (Tl = 0 ∀ l ∈ X, Tl′ = 1 ∀ l′ /∈ X) > 0. In other words, if Wl ≥ 2 ∀ l ∈ L there is a strictlypositive probability of X occurring as a changing schedule, in which case it holds that X ∈Mmax ⇒ X ∈M0.So Mmax ⊆M0.

Notice that ∪X∈MmaxX = L, because given any l ∈ L it is possible to construct some maximal feasibletransmission schedule containing l.34 Then:

Wl ≥ 2 ∀ l ∈ L⇒Mmax ⊆M0 ⇒ ∪X∈M0X = L

Lemma 3 then provides the result of the theorem.

Throughput Optimality

The results in this section have clearly shown that CSMA Collision Avoidance Scheduling is particularly at-tractive, both in the sense that it is easy to implement and also that it leads to straightforward “product-form”stationary distributions. Ni and Srikant demonstrate in [21] that when pl := ewl(t)

ewl(t)+1, where wl(t) is a weight

function similar to the ones described in Section 5, CSMA Collision Avoidance Scheduling is throughput optimal.In light of this result, I firmly believe that the algorithm is a very attractive candidate for practical imple-

mentation. Then – to examine how CSMA Collision Avoidance Scheduling fared in comparison to AdaptiveCSMA Scheduling – I performed simulations in R under the same setup as is described above. Figure 11 andTable 1 in Appendix C display the results for the two simulations that I ran: test e is using the “2η length”control slot (as outlined above), and test f is using Libin Jiang’s modification which reduces the control slotlength to η.

I find the results of these simulations to be particularly interesting. In Network 1, the use of CSMA CollisionAvoidance Scheduling results in significantly fewer packets being successfully transmitted than under AdaptiveCSMA Scheduling in discrete time. In Network 2, however, the result is precisely the opposite, with CSMACollision Avoidance Scheduling achieving a success rate almost matching the theoretical upper bound exhibitedby continuous time Adaptive CSMA Scheduling. In Network 1, the “overly cautious” approach of CSMACollision Avoidance Scheduling means that most links remain inactive for long periods when they could beusefully transmitting. On the other hand, in Network 2 the extra effort taken to maintain feasibility provesto be worthwhile in the long run. As such, I have found clear evidence that the use of collision avoidancemechanisms can help to significantly increase the throughput achieved in some networks — but also that a“blind” application of these ideas is unwise, as such an improvement is far from guaranteed. Once again, theintricate interactions between network topology, interference model and scheduling algorithm have been broughtinto the limelight.

8 Conclusions

Throughout this essay I have examined the mechanics of a variety of different scheduling algorithms proposed inpublished literature, including what their assumptions are and how realistic these assumptions are in reflectingthe reality of an actual wireless network. I have found that although many of the results from recent researchpapers are built upon assumptions which are not accurate reflections of current hardware capabilities, thefindings still hold theoretical importance as an indication of what could be achievable under perfect conditions.Furthermore, such examinations have been instructive as to where I should direct my research next — for

34An algorithm for doing this is to set l ∈ X, then to keep adding links to X until it is not possible to add more links withoutviolating the interference constraint.


example, Adaptive CSMA Scheduling’s reliance on an assumption of continuous time operation motivated myexploration for what would happen if such a requirement were relaxed, and subsequently directed my worktowards an examination of collision avoidance techniques.

The theoretical result that “static CSMA algorithms are asymptotically throughput optimal for large net-works under the primary interference constraint, with many small flows and a small sensing time” provideda clear indication that random access algorithms could potentially guarantee to satisfy Quality of Service de-mands such as throughput optimality, but I ultimately found that the required a priori knowledge of arrivalrates meant that the algorithm had little use as a practical tool. Adaptive CSMA, on the other hand, is a“true” distributed algorithm with no such prior knowledge requirement, and indeed my simulations have shownthat Adaptive CSMA Scheduling is both attractive and implementable. Finally, employing techniques to avoidcollisions entirely was shown to be a good strategy in some (but not all) networks by the superior performanceof Network 2 (and inferior performance of Network 1) in tests e and f .

My findings motivate further research in a number of areas. Firstly, all of the work that I’ve undertakenfocusses on how, in isolation, scheduling affects network performance. In [8], however, Jiang and Walrand presentan interesting exploration of how network performance can be further enhanced by employing an algorithm whichalso controls other elements of the data transmission process, such as routing and flow control. I think that itwould be very interesting to conduct further study of how network efficiency could be improved by employingalgorithms with more control over the many interacting components of the data transmission process.

Secondly, I have focussed much of my theoretical assessment of scheduling algorithms around the concept ofthroughput optimality. While throughput optimality is clearly a desirable property for a scheduling algorithmto possess, my simulations have shown that the performance of different throughput optimal algorithms can stillvary greatly over a moderate finite time horizon. It would, therefore, be interesting to work towards formallyproving whether such algorithms have (or indeed do not have) other useful properties (such as MaxWeightScheduling’s proven ability to minimise maxl∈L alQl, for any set of positive constants a1, . . . , a|L|).

A difficulty which has presented itself a number of times in my work is the assumption that some evolvingMarkov Processes are always (and immediately) in their stationary state. For example, throughput optimalityof Adaptive CSMA Scheduling relies on the assumption that r∗ perfectly tracks r, which in reality only happensif the Markov Process is allowed to settle into equilibrium between each successive sampling. The authors of[29] assert that an approximation argument applied to the stochastic sub-gradient algorithm presented in [8]implies that such an assumption is reasonable, but I think that it would be an interesting experiment to contrastthe performance of Adaptive CSMA Scheduling against a version of the algorithm with prior knowledge of thecorrect arrival rates in order to examine the loss of efficiency which results from employing the “adaptive”element of the algorithm.

Indeed, while [8], [9], [17] and [21] all demonstrate throughput optimality in some sense, they each alsoinclude some assumptions which are not entirely faithful representations of current 802.11 hardware, or indeedof standard configurations of networks themselves. [17] is only able to prove throughput optimality for networksin which there is a single communication channel shared by all terminals, and where there is a hub to scheduletransmissions – this clearly doesn’t provide a solution to the problem of finding a throughput optimal distributedscheduling algorithm. The algorithm proposed in [9] requires a priori knowledge of the arrival rates in thenetwork, which is clearly an unreasonable assumption to have in an ad hoc setting. [8] and [21] both rely onthe “immediate convergence of Markov Processes” idea just discussed – in conclusion, I feel that although allof the above papers have provided important theoretical results, the problem of designing a throughput optimaldistributed scheduling algorithm which operates only under assumptions which reflect the reality of currenttechnological capabilities is still an open one.

A common theme in my findings is that there is a strong relationship between the chosen interference modeland the performance of a given scheduling algorithm. As such, on a theoretical level it is an important openproblem to extend many of the results presented here to a more general interference framework. On a practical


level, the importance of understanding how real-world wireless infrastructures result in particular interferencemodels has become particularly evident. I firmly believe that a deeper understanding of these principles couldresult in far more effective wireless networking solutions being deployed in industry.

Designing throughput optimal scheduling algorithms is a difficult task in general, and the added difficultiesof producing such algorithms which operate in a distributed manner only serve to make the problem harder still.As such, it is only as our mathematical models of the complex, interacting procedures involved improve that wecan begin to truly harness the real power of the technology. Whether it be via Random Access Schemes or evenvia deterministic approaches such as the EXP-Rule, I believe that the key advances from future research willcome from – and indeed foster – a greater understanding of the interactions between networks, their interferencemodels and their scheduling algorithms.


9 Appendices

A MATLAB Output

Using MATLAB, I found the maximum of ge−g

η+1−e−g , using η = 0.01, to be 0.136 to 3 significant figures. Theactual value of

√2η is 0.141 to 3 significant figures in this setup, so the approximation is indeed close to being

correct. A graph of ge−g

η+1−e−g is displayed in Figure 10.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0 136 ≈√

2. ηg

g exp(−g)/(0.01+1−exp(−g))

Figure 10: Graphical Output from MATLAB

B Derivation that F (r) is a log likelihood Function

As stated in the essay, as well as being useful in the proof of ergodicity of the queueing state of the network, F (r)has a specific interpretation in its own right: it can be thought of as a log likelihood function for estimatingr when the πis are known, given a large number of observations of the xis chosen by the Adaptive CSMAalgorithm.

The likelihood function for r (after observing just a single xi chosen by the scheduling algorithm) is simplygiven by L(r|xi) = π(xi|r). Consider repeating the same observation n times independently, and labelling thenumber of times that xi was observed as ni, for each i = 1, . . . , |M|. Then the overall likelihood function wouldbe given by

∏i∈M π(xi|r)ni , and hence the log likelihood function would be given by

∑i∈M ni log π(xi|r).


Finally, as the overall number of observations increases, ni will tend to nπ(xi|r). So in the limit as n→∞, thelog likelihood function becomes

∑i∈M nπi log π(xi|r). Therefore (up to a factor of n, which is always known

and will not affect likelihood maximisation) F (r) can be viewed as a log likelihood function for estimating rgiven a large number of observations of xis, as required.

C Simulation Results

13200

13400

13600

13800

14000

Network 1

12600

12800

13000

34000

34500

35000

35500

36000

Network 2

32500

33000

33500

a b d ec f a b d ec f

#Su

cces

sful

Tra

nsm

issi

ons

#Su

cces

sful

Tra

nsm

issi

ons

Figure 11: Results from Simulations

Letter Description Network 1 Result Network 2 Resulta Adaptive CSMA, Continuous Time 13906 35922b Adaptive CSMA, Discrete Time, Collision Effects Ignored 13536 34106c Adaptive CSMA, Discrete Time, Collision Effects Acknowledged 13455 33813d As in c, but with an rmax bound 13279 33135e CSMA Collision Avoidance, with control phase length 2η 13169 35031f CSMA Collision Avoidance, with control phase length η 12755 35238

Table 1: Table of R Simulation Results


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