Medium Access Control ProtocolsWith Memory

# Medium Access Control Protocols With Memory

Jaeok Park and Mihaela van der Schaar The authors are with Electrical Engineering Department, University of California, Los Angeles (UCLA), 420 Westwood Plaza, Los Angeles, CA 90095-1594, USA. e-mail: {jaeok, mihaela}@ee.ucla.edu.
###### Abstract

Many existing medium access control (MAC) protocols utilize past information (e.g., the results of transmission attempts) to adjust the transmission parameters of users. This paper provides a general framework to express and evaluate distributed MAC protocols utilizing a finite length of memory for a given form of feedback information. We define protocols with memory in the context of a slotted random access network with saturated arrivals. We introduce two performance metrics, throughput and average delay, and formulate the problem of finding an optimal protocol. We first show that a TDMA outcome, which is the best outcome in the considered scenario, can be obtained after a transient period by a protocol with -slot memory, where is the total number of users. Next, we analyze the performance of protocols with 1-slot memory using a Markov chain and numerical methods. Protocols with 1-slot memory can achieve throughput arbitrarily close to 1 (i.e., 100% channel utilization) at the expense of large average delay, by correlating successful users in two consecutive slots. Finally, we apply our framework to wireless local area networks.

Access control, access protocols, communication systems, distributed decision-making, multiaccess communication.

## I Introduction

In multiaccess communication systems, multiple users share a communication channel and contend for access. Medium access control (MAC) protocols are used to coordinate access and resolve contention among users. We can categorize MAC protocols111Since we deal with MAC protocols exclusively in this paper, we use the term “protocol” to represent “MAC protocol” hereafter. into two classes, centralized and distributed protocols, depending on the existence of a central entity that coordinates the transmissions of users. Time division multiple access (TDMA) is an example of centralized protocols, where a scheduler assigns time slots to users. Centralized control can achieve a high level of channel utilization by avoiding collisions, but it requires large overhead for the communication of control messages. Slotted Aloha and IEEE 802.11 distributed coordination function (DCF) are examples of distributed protocols. In slotted Aloha, users transmit new packets in the next time slot while retransmitting backlogged packets with a fixed probability. In DCF, carrier sense multiple access with collision avoidance (CSMA/CA) and binary slotted exponential backoff (EB) are used for users to determine their transmission times. These distributed protocols can be implemented without explicit control messages, but coordination is limited in that collisions may occur or the channel may be unused when some users have packets to send.

In this paper, we aim to improve the degree of coordination attainable with distributed protocols by introducing memory into the MAC layer. Under a protocol with memory, a node dynamically adjusts its transmission parameters based on the history of its local information. The idea of utilizing histories at the MAC layer can be found in various existing protocols. For example, the slotted Aloha protocol [1] and its generalized version [2] adjust the transmission probabilities of nodes depending on whether the current packet is new or backlogged. The pseudo-Bayesian algorithm of [3] utilizes channel feedback to update the estimated number of backlogged packets in the system, based on which the transmission probability is determined. The EB protocols in [4] use the results of transmission attempts to adjust the contention window and current window sizes of nodes or their transmission probabilities.

The above protocols, however, utilize available past information in a limited way. We can consider the above protocols as the current state of a node determining its transmission parameters and the state transition occurring based on its local observations. Although this structure makes implementation simple in that nodes can simply keep track of their states in order to make transmission decisions, there may be many possible paths that lead to the same state, and important information may be lost by aggregating different histories into a single state. For example, in slotted Aloha, a node with a backlogged packet uses the same transmission probability following a slot in which it waited and following a slot in which it transmitted and collided. However, these two outcomes are observable by the node, and a significant performance improvement may be achieved by using different transmission probabilities following the two outcomes. Another limitation of the above protocols is that they are designed assuming a particular form of feedback information. In case that more informative feedback is available, utilizing the additional information may result in performance gains. For example, the EB protocols in [4] prescribe that a node should not update its parameters following a waiting slot. If a node can sense the channel while waiting, utilizing the information obtained from sensing may improve the performance of the EB protocols.

In order to overcome the limitations of the existing protocols utilizing memory, we provide a systematic framework to express and evaluate protocols with memory in the context of a slotted multiaccess system with saturated arrivals where nodes make transmission decisions based on their transmission probabilities. Our framework allows us to formally express a protocol utilizing memory of any finite length and operating under any form of feedback information. Also, we introduce two performance metrics, throughput and average delay, based on which we can evaluate protocols with memory. The two main results of this paper can be summarized as follows.

1. In the considered scenario with saturated arrivals, TDMA is the best protocol in a sense that there is no other protocol that achieves a higher throughput or a smaller average delay. A TDMA outcome can be obtained after a transient period by a protocol with -slot or -slot memory, where is the total number of nodes in the system.

2. A protocol with 1-slot memory can achieve throughput arbitrarily close to 1 (i.e., 100% channel utilization) at the expense of large average delay, by correlating successful nodes in two consecutive slots (i.e., a node that has a successful transmission in the current slot has a high probability of success in the next slot).

The proposed protocols with memory can be related to splitting algorithms [5] and reservation Aloha [6]. In splitting algorithms such as tree algorithms [7], backlogged nodes are divided into groups, one of which transmits in the next slot. Protocols with memory use histories to split nodes into groups. As nodes randomly access the channel based on transmission probabilities, histories will evolve differently across nodes as time passes. The probability of a successful transmission can be made high by choosing transmission probabilities in a way that the expected size of the transmitting group is approximately one most of the time. In reservation Aloha, nodes maintain frames with a certain number of slots, and a successful transmission serves as a reservation for the same slot in the next frame. Reservation Aloha can thus be expressed as a protocol with memory whose length is equal to the number of slots in a frame, provided that all nodes can learn successful transmissions in the system. Protocols with memory are more flexible than reservation Aloha in that protocols with memory can specify different transmission probabilities in non-reserved slots, can make reservations in a probabilistic way, and can be implemented with an arbitrary form of feedback information.

The rest of this paper is organized as follows. In Section II, we describe the considered slotted multiaccess model and define protocols with memory. In Section III, we introduce performance metrics and formulate the problem of finding an optimal protocol. In Section IV, we show that a protocol with -slot or -slot memory can achieve the performance of TDMA. In Section V, we analyze the properties of protocols with 1-slot memory using numerical methods. In Section VI, we show that protocols with memory can be applied to the wireless local area network (WLAN) environment and can achieve a performance improvement over DCF. We conclude the paper in Section VII.

## Ii System Model

### Ii-a Setup

We consider a slotted multiaccess system as in [2]. The system has total contending users, or transmitter nodes, and the set of users is denoted by . We assume that the number of users is fixed over time and known to users. Users share a communication channel through which they transmit packets. Time is slotted, and users are synchronized in their slot transmission times. We label slots by . A user always has a packet to transmit and can attempt to transmit one packet in each slot. The set of actions available to a user in a slot is denoted by , where stands for “transmit” and for “wait.” We denote the action of user by and an action profile, or a transmission outcome, by . The set of (transmission) outcomes is denoted by . A transmission is successful if it is the only transmission in the slot, and two or more transmissions in the same slot result in a collision.

### Ii-B Feedback Information

After a user transmits a packet, it learns whether the packet is successfully transmitted or not using an acknowledgement (ACK) response. If the user receives an ACK from the receiver node, it learns that its transmission was successful. Otherwise, it concludes that its packet has collided. We assume that there is no error in the transmission and the reception of ACK responses so that a user always learns the correct results of its transmission attempts. Formally, we represent ACK feedback to user by , which takes the value if , where is the outcome in which only user transmits, and otherwise.

At the end of each slot, users obtain channel feedback about the number of transmissions in that slot. Let be the number of transmissions in outcome . The set of possible numbers of transmissions in a slot is . Let be an information partition of for user [8]. Then channel feedback to user in a slot with outcome is given by the element of that contains , which we denote by . The definition of information partitions requires that for all . That is, channel feedback never leads users to regard the actual number of transmissions as impossible. However, when channel feedback has errors, it is possible for users to obtain incorrect channel feedback such that . In the presence of the hidden terminal problem (also called “erasures” in [9]), users may interpret a success or a collision slot as an idle slot. Under noise errors [9], users may interpret an idle or a success slot as a collision slot. It is also possible that users obtain channel feedback in a probabilistic way, in which case is determined according to a probability distribution on , .

All the aforementioned channel feedback settings can be incorporated in the formulation of the protocol designer’s problem developed in Section III. Although modeling general settings can be of importance in practice, we impose the following simplifying assumptions on channel feedback for analytic convenience. First, we assume that channel feedback is generated in a deterministic way. Then channel feedback to user can be represented by a mapping from to , instead of . Second, we assume that for all so that channel feedback always contains the actual number of transmissions. Finally, we assume that every user obtains the same channel feedback, i.e., , which requires . Under these three assumptions, a channel feedback model is completely described by an information partition of .

Consider a partition of , , where . Each element of the partition represents a type of channel feedback, and users can potentially learn whether there has been zero packet (idle), one packet (success), or more than one packet (collision) transmitted in the slot. With an abuse of notation, we will use 0 and 1 to represent channel feedback that corresponds to and , respectively [5]. We say that the channel possesses ternary feedback, or feedback, if all three types of feedback are available. Binary feedback is also possible, and given the three possible types of channel feedback, we can consider three kinds of binary feedback: success/failure (S/F) feedback, which informs users whether there was a successful transmission (1) or not (); collision/no collision (C/NC) feedback, which informs users whether there was a collision () or not (); and empty/not empty (E/NE) feedback, which informs users whether the current slot was empty (0) or not () [9]. We can also consider no channel feedback, which does not give any channel information to users and corresponds to information partition , and -ary feedback, which informs users of the exact number of transmissions and corresponds to information partition , the finest partition of . (See [10] for a similar list of channel feedback models in a multiple reception scenario.)

The feedback information of user consists of ACK feedback and channel feedback, i.e., . We define a feedback technology as a rule that generates feedback information for each user depending on transmission outcomes. The above three assumptions on channel feedback allow us to represent a feedback technology by a mapping , where , and write the feedback information of user more compactly as . Since if , we can set . Since user can distinguish the outcomes in 1 from those in using ACK feedback whenever it transmits, we can set if and if . Let , which is independent of by the symmetry assumption. represents the set of feedback information that a user can obtain with feedback technology . The feedback information of user is denoted by . We use to denote the set of all feedback technologies, which is equivalent to the set of all partitions of .

### Ii-C Protocols With Memory

A user decides whether to transmit or not in each slot using a transmission probability, which lies in . A protocol is a rule based on which users determine their transmission probabilities. We assume that control or coordination messages cannot be used in the system. Then the transmission action and the feedback information of user are all the information that it obtains in a slot. The -slot history for user in slot is given by

 Lti=(at−Mi,zt−Mi;…;at−1i,zt−1i),

for and .222In addition to actions and feedback, a user knows the transmission probabilities it has used. We do not include past transmission probabilities in histories because we focus on protocols that do not depend on past transmission probabilities directly. We set for as initialization. Let be the set of possible action-feedback pairs under feedback technology , i.e., . Then the set of -slot histories is given by . A stationary decision rule based on -slot histories is defined by a mapping

 f:LMρ→[0,1],

where represents the transmission probability for a user whose -slot history is .

Since the set of -slot histories is affected by feedback technology , the set of stationary decision rules based on -slot histories depends on the feedback technology of the system. In particular, as the feedback technology is more informative in a sense that the corresponding information partition is finer, users can distinguish more outcomes, and thus more decision rules can be deployed. We use to denote the set of all stationary decision rules based on -slot histories under feedback technology . We define a protocol with -slot memory as a profile of stationary decision rules based on -slot histories given a feedback technology , i.e., . We say that a protocol is symmetric if it prescribes the same decision rule to every user, i.e., . We will sometimes use to represent a symmetric protocol with a common decision rule when there is no confusion.

### Ii-D Automaton Representations of Protocols With Memory

A protocol with memory can be described by a finite automaton, which consists of a finite set of states, an initial state, an action rule, and a state transition rule [11]. We can represent a symmetric protocol as a finite automaton by defining states as possible -slot histories, the initial state as the -slot history obtained from idle slots, the action rule as the decision rule , and the state transition rule as to specify the new state as the -slot history updated based on the transmission action and the feedback information in the current slot.

Fig. 1LABEL:sub@fig:auto-a shows the automaton representation of a protocol with memory in the simplest case of 1-slot memory and no channel feedback (i.e., ACK feedback only). With ACK feedback only, there are three possible action-feedback pairs, or 1-slot histories: , , and , where corresponds to channel feedback . Hence, there are three states in the automaton, corresponding to the three possible action-feedback pairs, with the initial state specified as . The transmission probability in each state is given by . The state in the next slot is determined by the action-feedback pair in the current slot.

[2] proposes generalized slotted Aloha protocols, which we call two-state protocols. Under a two-state protocol, users choose their transmission probabilities depending on whether their current packets are new (a free state) or have collided before (a backlogged state). Thus, having ACK feedback suffices to implement a two-state protocol. The automaton representation of a two-state protocol is shown in Fig. 1LABEL:sub@fig:auto-b. In general, a two-state protocol cannot be expressed as a protocol with finite memory. Since the state remains the same following a waiting slot, the transmission probability for the current slot is determined by the result of the most recent transmission attempt, which may have occurred arbitrarily many slots ago. However, [2] shows that under saturated arrivals two-state protocols achieve the maximum throughput when and , where and are transmission probabilities in the free and backlogged states, respectively. When a two-state protocol specifies , which is also the case in the original slotted Aloha protocol, the action-feedback pair cannot occur in the free state and thus we can find an equivalent protocol with 1-slot memory such that and . In effect, a two-state protocol with aggregates the two different action-feedback pairs, and , into one state, the backlogged state, and assigns the same transmission probability following these two action-feedback pairs. On the contrary, a protocol with 1-slot memory can fully utilize all the available information from the previous slot in that three different transmission probabilities can be assigned following the three action-feedback pairs.

As will be shown later in Section V.D., the limited utilization of past information by two-state protocols results in performance limitations. In particular, throughput, or the fraction of success slots, is bounded from above by under two-state protocols (Theorem 2 of [2]) while protocols with 1-slot memory can achieve throughput arbitrarily close to 1. When memory longer than 1 slot or channel feedback is available, there are more distinguishable histories for each user. Two-state protocols aggregate different histories into just two states whereas protocols with memory can assign as many transmission probabilities as the number of possible histories given the length of memory and the feedback technology. Thus, the performance gap between the two kinds of protocols will be larger when longer memory or more informative feedback is available.

## Iii Problem Formulation

### Iii-a Performance Metrics

#### Iii-A1 Throughput

We define the throughput of a user as the fraction of slots in which it has a successful transmission and total throughput as the fraction of slots in which there is a successful transmission in the system, which is equal to the sum of the individual throughput of users. When users follow a protocol with memory , throughput can be computed using a Markov chain. Let be the length of memory used by protocol and be the associated feedback technology. Then we can consider a Markov chain whose state space is given by . We write an element of as . Let be the -slot history for user when the outcomes in the recent slots are , i.e.,

 Li(¯a)=(ai,1,ρ(ai,1,k(a1));…;ai,M,ρ(ai,M,k(aM))),

where is the th element of . Given as its -slot history, user transmits with probability . The transmission probabilities of users yield a probability distribution on the outcome in the current slot. The probability that the current outcome is under when the outcomes in the recent slots are is given by

 P(a′|¯a;f)=N∏i=1[1{a:ai=T}(a′)fi(Li(¯a))+1{a:ai=W}(a′)(1−fi(Li(¯a)))], (1)

where is an indicator function such that if and 0 otherwise. The transition probability from to under protocol is given by

 Q(¯a′|¯a;f)={P(a′M|¯a;f)if a′m=am+1, for all m=1,…,M−1,0otherwise.

Let be the probability distribution on the state space in slot induced by protocol . By the initialization in the definition of protocols with memory, the initial distribution has element 1 for and 0 elsewhere, where denotes the idle outcome, . Let be the transition matrix of the Markov chain under protocol . Then the probability distribution on in slot can be computed by

 vt(f)=v0(f)Q(f)t,

for . Let be the subset of with user ’s success as the most recent outcome, i.e., . Then the probability of user ’s success in slot is given by . The fraction of slots with user ’s success, or the throughput of user , is given by

 τi(f)≜limJ→∞1JJ∑t=1τi,t(f),

assuming that the limit exists. If is chosen so that the induced Markov chain has only one closed communicating class, then there exists a unique stationary distribution , independent of the initial distribution , which satisfies

 v(f)=v(f)Q(f) and v(f)e=1, (2)

where is the column vector of length whose elements are all 1 [12]. Then the expression for the throughput of user is reduced to . Finally, the total throughput of the system under protocol is given by

 τ(f)=N∑i=1τi(f).

#### Iii-A2 Average Delay

The average delay of a user is defined as the average waiting time, measured in the unit of slots, until the beginning of its next successful transmission starting from an arbitrarily chosen time. Average delay under a protocol with memory can be computed using a Markov chain. Consider a slot to which an arbitrarily chosen time belongs, and let be the outcomes in the recent slots. Given , yields a probability distribution on the outcome in the next slot, , given by (1). Using , we can compute the probability that user succeeds for the first time after slots when the outcomes in the recent slots are given by and users follow ,

 μi(r;¯a,f)≜Pr{at+r=ai and at+r′≠ai, for all r′=1,…,r−1|¯at=¯a;f}, (3)

for . For example,

 μi(1;¯a,f) =P(at+1=ai|¯at=¯a;f) and μi(2;¯a,f) =∑a∈A∖{ai}P(at+2=ai|¯at+1=(¯at,a);f)P(at+1=a|¯at=¯a;f),

where is obtained by deleting the first outcome in and adding as the most recent outcome. Using (3), we can compute the average number of slots until user ’s next success starting from a slot with the recent outcomes :

 di(¯a;f)≜∞∑r=1μi(r;¯a,f)r.

Since the distribution on the recent outcomes in slot is given by , the average delay of user under protocol can be computed as

 Di(f)≜limJ→∞1JJ∑t=1∑¯a∈AMvt(¯a;f)di(¯a;f)−0.5,

where is subtracted to take into account that the average time staying in the initial slot is starting from an arbitrarily chosen time. When the Markov chain has a unique stationary distribution , the expression for the average delay of user is reduced to

 Di(f)=∑¯a∈AMv(¯a;f)di(¯a;f)−0.5.

#### Iii-A3 Discussion on Throughput and Average Delay

The throughput of the system can be considered as an efficiency measure of a protocol as it gives the channel utilization over time. However, as its definition suggests, throughput reflects the performance averaged over a long period of time and does not contain much information about the short-term and medium-term performance. As an illustration, consider two sequences of outcomes generated by repeating and . In both sequences, each user succeeds in one out of slots over a long period of time, as measured by throughput . However, users may prefer the first sequence to the second one as the first exhibits the steadier performance over a short period of time. For example, suppose that a user counts its successes in consecutive slots from an arbitrary slot. The first sequence guarantees one success in consecutive slots regardless of the initial slot. On the contrary, in the second sequence, consecutive slots contain no success with a high probability and many successes with a low probability. The variation in the performance over a short period of time is captured by average delay.

A widely-used measure of delay in queueing theory is the inter-packet interval. In our model, we can compute the average inter-packet time of user , i.e., the average number of slots between two successes of user , using

 ~Di(f)=∑¯a∈Siv(¯a;f)∑¯a∈Siv(¯a;f)di(¯a;f). (4)

We can interpret as the average time to enter one of the states in starting from the state under protocol . Then can be interpreted as the average time to enter starting from an arbitrary state, where the starting state is chosen following the stationary distribution. Similarly, can be interpreted as the average time to return starting from a state in .

Note that satisfies

 di(¯a;f)=1+∑¯a′∈¯¯¯¯¯SiQ(¯a′|¯a,f)di(¯a′;f), (5)

for all , where is the complement of , i.e., . Using (2) and (5), we can show that for any protocol with memory , and thus (4) can be rewritten as

 ~Di(f)=1τi(f), (6)

which can be regarded as a version of Little’s Theorem [5]. Therefore, the average inter-packet time is completely determined by throughput, and thus it provides no additional information beyond that provided by throughput.

The waiting time paradox [13] suggests that not only the mean of the inter-packet time but also the variance of the inter-packet time matters for the average waiting time measured from an arbitrarily chosen time. Let be the random variable that represents the inter-packet time of user with support . By definition, we have . The Pollaczek-Khinchine (P-K) formula [14] gives the average residual time until the next success, or average delay, by

 Di=E(X2i)2E(Xi)=12(1+κ2i)~Di, (7)

where represents the coefficient of variation of , i.e., . The P-K formula shows that average delay is increasing in the variance of the inter-packet time for a given level of throughput (or equivalently, for a given mean of the inter-packet time).

Consider the following three examples with a fixed level of the average inter-packet time and different levels of the coefficient of variation . (i) Suppose that user has periodic successful transmissions. Then it transmits a packet successfully once in slots, and its average delay is since . Note that is the smallest average delay achievable with protocols that yield the average inter-packet time . (ii) Suppose that successes for user are completely random in a sense that follows a geometric distribution. Such a random variable can be generated by a memoryless protocol that prescribes a fixed transmission probability for each user. In this case, , and thus . (iii) Suppose that the successes of user are highly correlated over time. Then , and thus . In this case, user has frequent successes for a short period of time but sometimes has to wait for a long period of time until its next success.

It is reasonable to assume that users prefer to have a steady stream of transmissions as well as a high transmission rate. As the above examples illustrate, the coefficient of variation of the inter-packet time measures the volatility in successful transmissions over time. Since average delay reflects the coefficient of variation of the inter-packet time, we use it as a second performance metric to complement the long-term performance metric, throughput.

### Iii-B Problem of the Protocol Designer

We formulate the problem faced by the protocol designer as a two-stage procedure. First, the protocol designer chooses the length of memory, slots, and the feedback technology, . Next, the protocol designer chooses a protocol from the class of available protocols given the stage-one choice . For simplicity, we assume that the protocol designer considers only symmetric protocols. For a symmetric protocol , we have and . Total throughput is given by for any , and we use to denote the average delay of each user.

The problem of the protocol designer can be written formally as

 maxM∈N,ρ∈R{(maxf∈FM,ρU(τ(f),D(f)))−C(M,ρ)}. (8)

represents the utility function of the protocol designer, defined on total throughput and average delay. We assume that is increasing in total throughput and decreasing in average delay so that the protocol designer prefers a protocol that yields high total throughput and small average delay. represents the cost function of the protocol designer, defined on the length of memory and the feedback technology. We assume that is increasing as memory is longer and as the feedback technology is more informative. Note that from a practical point of view the cost of expanding memory is vanishingly small compared to the cost associated with the feedback technology, which implies that the system is more likely to be constrained by the available feedback technology rather than by the size of memory. Hence, it is more natural to interpret the cost associated with memory as the cost of implementing protocols with a certain length of memory. For instance, the protocol designer may prefer protocols with short memory to protocols with long memory because the former is easier to program and validate than the latter. We say that a protocol is optimal given if it attains . We say that is an optimal protocol for the protocol designer if solves (8).

In order to analyze the performance of protocols with memory, we approach the protocol designer’s problem from two different directions. In Section IV, we first find the feasible throughput-delay pair most preferred by the protocol designer. Then we show that the most preferred throughput-delay pair can be achieved by a protocol with memory. In Section V, we focus on the simplest class of protocols with memory, namely protocols with 1-slot memory, and investigate their properties using numerical methods.

## Iv Most Preferred Protocols

Suppose that the protocol designer can choose any protocol at no cost. Then the protocol designer’s problem becomes

 maxf∈FU(τ(f),D(f)), (9)

where denotes the set of all symmetric protocols including those that cannot be expressed as protocols with memory. We say that is a most preferred protocol if solves (9). Consider a fixed level of total throughput , which implies individual throughput by the symmetry assumption on protocols. Note that the P-K formula (7) was obtained without imposing any structure on protocols, and it shows that for a given level of throughput, average delay is minimized when there is no variation in the inter-packet time of a user, i.e., when . Hence, combining (6) and (7), we can express the minimum average delay given total throughput as

 Dmin(τ)=~Di2=N2τ.

Since is decreasing in , setting total throughput at the maximum level yields the minimum feasible average delay . In other words, there is no protocol that attains . Hence, if there exists a protocol that achieves the maximum total throughput 1 and the minimum average delay at the same time, then is a most preferred protocol.

In order to obtain the most preferred throughput-delay pair , a protocol needs to provide each user with a successful transmission in every slots. TDMA is a protocol that achieves such a sequence of outcomes. Since the labels of users are arbitrary, we can describe the TDMA protocol as having user transmit in slot if and wait in all other slots. Then each user has one successful transmission in every slots, and thus the TDMA protocol is a most preferred protocol. However, the TDMA protocol requires coordination messages to be sent to users in order to assign time slots, and thus it does not belong to the class of protocols with memory defined in Section II.C. The following theorem establishes the existence of a most preferred protocol in the class of protocols with memory.

###### Theorem 1

Assume -slot memory and success/ failure (S/F) binary feedback . Denote an -slot history by , and let be the number of successes in , i.e., . Define a protocol with memory by

 f(L)={0if L contains (T,1)% ,1/(N−n(L))otherwise.

Then and .

###### Proof:

Since if contains , a user waits for slots following its success. Also, users that have no success in recent slots compete among themselves with the transmission probability equal to the reciprocal of the number of such users. Suppose that there is a success in each of the recent slots for the first time in the system. Since a user waits for slots following its success, the successes must be by different users. Then the only user that has not had a success in the recent slots transmits with probability 1 while other users wait in the current slot. From that point on, a cycle of length containing one success of each user is repeated. Since the probability of having consecutive successes before slot approaches 1 as goes to infinity, achieves a TDMA outcome after a transient period with probability 1, and thus it is a most preferred protocol. \qed

Theorem 1 shows that with -slot memory and S/F feedback, users can determine their transmission slots in a distributed way without need of explicit coordination messages, thereby achieving the same outcome as TDMA after a transient period. By trial and error, users find their transmission slots in a self-organizing manner and stabilize to transmit once in every slots. Note that is the minimum length of memory required to emulate TDMA using a protocol with memory. A cycle of successes by users cannot be generated with memory shorter than slots. Also, S/F feedback is necessary to guarantee a success after consecutive successes.

The expected duration of the transient period can be shortened by using -slot memory. Consider a protocol with -slot memory, , defined by

 ~f(L)=⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩1if (a1,z1)=(T,1),0if (a1,z1)=(W,1),0if z1≠1 and (am,zm)=(T,1) for some m=2,…,N,1/(N−n(L))if z1≠1 and (am,zm)≠(T,1) for all m=2,…,N, (10)

where is now the number of successes in recent slots. The first and the third lines of (10) state that once a user succeeds, it waits for slots before the next transmission. The second and the fourth lines state that a user with no success in recent slots waits if the current slot is already “reserved” by some user that succeeded slots ago and contends with other such users if no user succeeded slots ago. Hence, under , once a user succeeds in slot , it is guaranteed to succeed in slots , , and so on, whereas under in Theorem 1 a user that succeeded slots ago has to compete with other users with no success in recent slots during the transient period. Again, S/F feedback is necessary to let users know whether the current slot is reserved or not. can be considered as a generalization of reservation Aloha with a frame of slots, where users can adjust their transmission probabilities depending on the number of contending users for non-reserved slots.

In practice, the number of users may vary over time as users join and leave the system. When ternary feedback is available, users can find the exact number of users in the system using the protocol as long as the number of users does not change too frequently. The initial estimate on the number of users is set to be the maximum number of users that the system can allow. When users follow based on their initial estimate, there will be empty slots in a cycle obtained after a transient period. Once a cycle is repeated, users can construct a new cycle by deleting the empty slots, adjusting the number of slots in a frame equal to the actual number of users. If some users leave the network, empty slots will appear in a frame, and the remaining users can reduce the length of a frame by deleting the empty slots. If some users join the network, we require them to transmit immediately. Then a collision will occur, and it serves as a signal to notify the existing users that there arrived a new user in the system. Once a collision occurs after a transient period, users reset their estimates to the maximum number of users and repeat the procedure from the beginning in order to increase the length of a frame by the number of new users.

## V Protocols With 1-Slot Memory

### V-a Structure of Delay-Efficient Protocols

We now examine protocols with 1-slot memory, which are the simplest among protocols with memory. Under protocols with 1-slot memory, users determine their transmission probabilities using their action-feedback pairs in the previous slot. In Appendix A, we explain in detail how to compute throughput and average delay under symmetric protocols with 1-slot memory using a Markov chain. Since the protocol designer prefers a protocol with small average delay for a given level of throughput, we consider the following reduced problem of the protocol designer:

 D∗(τ)≜minf∈F1,ρD(f)subject to τ(f)=τ, (11)

for and some feedback technology . We say that a protocol is delay-efficient if it solves (11) for , i.e., if . Also, we call the set of points the delay-efficiency boundary of protocols with 1-slot memory. Since computing and for a given protocol involves solving matrix equations, it is in general difficult to solve (11) analytically, and thus we rely on numerical methods.

To obtain numerical results, we consider ternary feedback, denoted by , and five users, i.e., . In order to guarantee the existence of a stationary distribution for transition matrix , we restrict the range of (i.e., the set of possible transmission probabilities) to be , instead of . We use the function fmincon of MATLAB to solve the restricted version of (11).333We vary from 0.99 to 0.01 with a step size of . We choose the initial protocol for fmincon as for and use the solution for as the initial protocol for from to . Fig. 2LABEL:sub@fig:deleff-a depicts the delay-efficient boundary of protocols with 1-slot memory, shown as a U-shaped curve. Since the optimization problem to find delay-efficient protocols is not necessarily convex, it is possible that the numerical results locate local minima instead of global minima. To validate the delay-efficiency of the protocols obtained by the fmincon function, we generate 5,000 symmetric protocols with 1-slot memory where transmission probabilities are randomly chosen on and plot total throughput and average delay under those protocols as dotted points in Fig. 2LABEL:sub@fig:deleff-a (in the figure, only 3,631 points with average delay less than 100 are shown). The results in Fig. 2LABEL:sub@fig:deleff-a suggest that the numerically computed protocols are indeed (at least approximately) delay-efficient.444Another interesting point to notice from Fig. 2LABEL:sub@fig:deleff-a is that most of the randomly chosen protocols yield throughput-delay pairs close to those achieved by memoryless protocols. It suggests that protocols with 1-slot memory need to be designed carefully in order to attain total throughput above the maximum level achievable with memoryless protocols. Fig. 2LABEL:sub@fig:deleff-b plots the transmission probabilities under the delay-efficient protocols, denoted by , as varies. It shows that the structure of changes around , which is also the turning point of the delay-efficiency boundary. There is numerical instability for between 0.41 and 0.48 in that the solution depends highly on the initial protocol. In fact, for in that region we can find protocols that yield average delay slightly smaller than those in Fig. 2LABEL:sub@fig:deleff-a by specifying different initial protocols.

In the following, we provide an explanation for the shape of the delay-efficiency boundary by investigating the structure of the delay-efficient protocols. In the low throughput region, (), the structure of is given by

 f∗(W,0)∈[0,0.2], f∗(W,1)=1, f∗(W,e)=0, f∗(T,1)=0, f∗(T,e)∈[0.65,1].

Since and , a success lasts for only one slot and is followed by a collision. Since is high, a collision is likely to be followed by another collision. Similarly, an idle slot is likely to be followed by another idle slot since is low. This means that total throughput is kept low by inducing many idle or collision slots between two successes. Hence, as total throughput increases in the low throughput region, the expected number of idle or collision slots between two successes is reduced, and as a result users transmit their packets more frequently. According to the P-K formula (7), average delay is determined by the mean and the coefficient of variation of the inter-packet time. On the left-hand side of the delay-efficiency boundary (), the average inter-packet time is reduced while the coefficient of variation of the inter-packet time remains about the same as increases, resulting in the inverse relationship between throughput and average delay.

In the high total throughput region (), the structure of is given by

 f∗(W,0)≈1N, f∗(W,1)≈0, f∗(W,e)≈1N−2, f∗(T,1)=1, f∗(T,e)=0. (12)

In a slot following an idle slot, users transmit with probability close to . In a slot following a success, the successful user transmits with probability 1 while other users wait with high probability. The transmission probability of other users approaches 0 as becomes close to 1. In a slot following a collision, users that transmitted in the collision wait while other users transmit with probability close to . Since a collision involving two transmissions is most likely among all kinds of collisions, setting maximizes the probability of success given that colliding users wait. and is the key feature of protocols with 1-slot memory that achieve high total throughput. By correlating successful users in two consecutive slots, protocols with 1-slot memory can yield total throughput arbitrarily close to 1. However, higher throughput is achieved by allowing a successful user to use the channel for a longer period, which makes other users wait longer until they transmit next time. On the right-hand side of the delay-efficiency boundary (), the coefficient of variation of the inter-packet time increases without bound as approaches 1, resulting in the positive relationship between throughput and average delay.

Since the protocol designer prefers a protocol that yields high total throughput for a given level of average delay, optimal protocols must lie on the right-hand side of the delay-efficiency boundary. As an illustrative example, suppose that the utility function of the protocol designer is given by

 U(τ,D)=−max{200(1−τ),D}.

Then an increase in total throughput by 0.1 has the same utility consequence as a decrease in average delay by 20 slots. In Fig. 3, dashed curves depict the indifference curves of the utility function, each of which represents the throughput-delay pairs that yield the same level of utility, while the arrow shows the increasing direction of the utility function. The throughput-delay pair that maximizes the utility of the protocol designer is , marked with an asterisk in Fig. 3. The protocol designer can determine the optimal protocol given by finding a protocol that yields , which is

 fo(W,0)=0.20, fo(W,1)=0.03, fo(W,e)=0.34, fo(T,1)=0.99, fo(T,e)=0. (13)

### V-B Robustness Properties of Delay-Efficient Protocols

#### V-B1 Unknown Number of Users

We relax the assumption that users know the exact number of users in the system. Instead, we assume that each user has an estimate on the number of users and executes the prescribed protocol based on the estimate.555In other words, the protocol designer specifies a protocol as a function of the numbers of users, not a fixed protocol. We consider a scenario where there are ten users and ternary feedback is available. For simplicity, we assume that the objective of the protocol designer is to achieve total throughput 0.9 and that all users have the same estimate on the number of users, which will be the case when users use the same estimation method based on past channel feedback. Fig. 4 shows a portion of the delay-efficiency boundary with and plots the throughput-delay pairs when the ten users follow delay-efficient protocols that are computed to achieve total throughput 0.9 based on the estimated number of users, denoted by , between 7 and 13. The results from Fig. 4 suggest the robustness of delay-efficient protocols with respect to variations in the number of users in a sense that as the estimated number of users is close to the actual one, the protocol designer obtains a performance close to the desired one.

Based on the observations from Fig. 4, we can consider the following procedure to dynamically adjust the estimates of users. Users update their estimates periodically by comparing the actual total throughput since the last update with the desired total throughput.666Note that users can compute the actual total throughput using ternary feedback. Users increase (resp. decrease) their estimates by one if the actual total throughput is lower (resp. higher) than the desired total throughput by a certain threshold level. When designed carefully, this estimation procedure will make the estimated number of users converge to the actual number of users because the actual total throughput is lower than (resp. equal to, higher than) the desired total throughput if the estimated number of users is smaller than (resp. equal to, larger than) the actual number. This procedure can be regarded as an extension of the pseudo-Bayesian algorithm of [3] in which users adjust their estimates in every slot based on the channel feedback of the previous slot. In the proposed estimation procedure, memory is utilized not only to coordinate transmissions but also to estimate the number of users.

#### V-B2 Errors in Feedback Information

So far we have restricted our attention to deterministic feedback technologies. We relax this restriction and consider stochastic feedback technologies. As discussed in Section II.B., stochastic channel feedback for user can be represented by a mapping from to . If the structure of channel feedback, , is known to the protocol designer, it can be modeled in the transition matrix in Section III by extending the state space from to . Then the protocol designer can find an optimal protocol taking into account randomness in feedback information.

Here we introduce random errors in channel feedback, which are not modeled by the protocol designer, and examine the performance of an optimal protocol in the presence of random errors. We assume that ternary feedback is available but subject to random errors. In particular, a user obtains the correct feedback signal with probability and each of the two incorrect signals with probability , for a small . For example, if there is no transmission in the system, then a user receives feedback with probability and each of feedback and with probability . We assume that the feedback signals of users are independent. We continue to assume that ACK feedback is perfect, and thus transmitting users always learn the correct results of their transmission attempts, regardless of the realization of channel feedback.

Table I shows the performance of the optimal protocol in (13) at the various levels of when . To obtain the simulation results, we generate transmission decisions and feedback information for slots, for each level of . Table I suggests that delay-efficient protocols have a robustness (or continuity) property with respect to random errors in feedback information, since the obtained performance is close to the desired one when the error level is small. Note that an error occurring to a waiting user following a success induces the user to transmit with a higher probability (i.e., to transmit with probability or instead of ), making consecutive successes last shorter. Thus, as the error level increases, both total throughput and average delay decrease. The obtained throughput-delay pairs remain close to the delay-efficiency boundary, suggesting that errors following an idle or a collision slot cause little performance degradation.

### V-C Comparison of Channel Feedback

We now analyze the impact of the different forms of channel feedback on the performance of protocols with 1-slot memory. As mentioned in Section II.C., the set of available protocols expands as the feedback technology becomes more informative (in other words, a protocol can always be replicated by another protocol if is more informative than ). This relationship implies that a more informative feedback technology yields a lower delay-efficiency boundary for a given length of memory. We consider six feedback technologies with ACK feedback and different channel feedback models: no channel feedback, S/F binary feedback, C/NC binary feedback, E/NE binary feedback, ternary feedback, and -ary feedback, as introduced in Section II.B. Fig. 5 depicts the delay-efficiency boundaries of protocols with 1-slot memory under the six feedback technologies. As expected, for a given level of throughput, the minimum average delay becomes smaller as we move from no feedback to binary feedback, to ternary feedback, and to -ary feedback.777In Fig. 5, average delay is smallest under C/NC binary feedback for and , which is a result of numerical instability around the turning point of the delay-efficiency boundaries, as pointed out in Section V.A.

Fig. 5 shows that in the operating region of the protocol designer (i.e., the right-hand side of the delay-efficiency boundaries), protocols with 1-slot memory perform worse under no channel feedback and C/NC binary feedback than under other considered channel feedback. In order to obtain high throughput with 1-slot memory, we need a high correlation between successful users in two consecutive slots, which requires and . However, under no channel feedback or C/NC binary feedback, a waiting user cannot distinguish between idle slots and success slots, leading to . This makes idle slots last long once one occurs, resulting in large average delay. Under S/F binary feedback, a user is constrained to use . As can be seen in Fig. 2LABEL:sub@fig:deleff-b, the values of and are not much different in delay-efficient protocols under ternary feedback. Thus, using a single probability instead of two different probabilities, and , causes only minor performance degradation. Under E/NE binary feedback, a delay-efficient protocol with 1-slot memory that achieves high throughput has the following structure:

 f(W,0)≈1N, f(W,1∪e)=0, f(T,1)≈1, f(T,e)≈12. (14)

Following a collision, only colliding users transmit under (14) whereas only noncolliding users transmit under (12). Fig. 5 shows that the restriction that E/NE binary feedback imposes compared to ternary feedback has a small impact on performance. Fig. 5 also shows that the improvement in performance from having -ary feedback over ternary feedback is only marginal. The ability of users to distinguish the exact numbers of transmissions in collisions does not help much because collisions involving three or more transmissions rarely occur under ternary feedback in the high throughput region.

### V-D Comparison of Protocols

We consider and compare the performance of four different kinds of protocols: protocols with 1-slot memory, two-state protocols in [2], memoryless protocols, and TDMA. In Fig. 6, the delay-efficiency boundary of protocols with 1-slot memory is shown assuming no channel feedback (i.e., ACK feedback only) because two-state protocols can be implemented without using channel feedback. [2] proposes two-state protocols of the form and , where is called a short-term fairness parameter, which measures the average duration of consecutive successes. We vary from 0.01 to 1 with a step size of 0.01 to generate the throughput-delay pairs under two-state protocols plotted in Fig. 6. The result confirms that total throughput achievable with a two-state protocol is bounded from above by , whereas protocols with 1-slot memory can attain any level of total throughput between 0 and 1. Moreover, in the range of total throughput achievable with a two-state protocol, delay-efficient protocols with 1-slot memory yield smaller average delay than two-state protocols. Since two-state protocols with can be considered as imposing a restriction of , these results suggest that there is a significant performance degradation by assigning the same transmission probability following the two action-feedback pairs, and . Moreover, by combining Fig. 6 with Fig. 5, we can see that the performance degradation from using a two-state protocol instead of a protocol with 1-slot memory is severer when channel feedback is available.

A symmetric memoryless protocol can be represented by a single transmission probability, which is used regardless of past histories. The throughput and the average delay of user under memoryless protocol are given by and , respectively. The optimal memoryless protocol is thus the transmission probability that maximizes and minimizes at the same time, which is given by [15]. The throughput-delay pairs under memoryless protocols in Fig. 6 are obtained by varying from 0 to 1. The lower-right point of the throughput-delay curve corresponds to the throughput-delay pair under the optimal memoryless protocol, . Protocols with 1-slot memory also outperform memoryless protocols in that protocols with 1-slot memory support a wider range of achievable total throughput than memoryless protocols do. In Section IV, we have seen that TDMA achieves the most preferred throughput-delay pair, . Protocols with 1-slot memory cannot achieve the most preferred throughput-delay pair because memory of length at least slots is necessary to obtain it.

## Vi Application to Wireless Local Area Networks

In the idealized slotted multiaccess system considered so far, all packets are of equal size, the transmission of a packet takes the duration of one slot, and users receive immediate feedback information. We relax these assumptions to apply protocols with memory to WLANs. In particular, packet sizes may differ across packets, and we take into consideration propagation and detection delay as well as overhead such as a packet header and an ACK signal. We consider a WLAN model where users follow a random access scheme using transmission probabilities based on CSMA/CA. In the WLAN model, the duration of slots depends on the channel state (idle, success, or collision). Let , , and be the duration of a slot when the channel state is idle, success, and collision, respectively. The expressions for and can be found in (14) and (17) of [16], depending on whether the RTS/CTS mechanism is disabled or not. Total throughput is expressed as

 τ=P1E[P]P0σ0+P1σ1+P2σ2, (15)

where is the average packet transmission duration and , , and are the fractions of idle, success, and collision slots, respectively. In the idealized slotted model, we assume that the size of each packet is equal to the slot duration and ignore overhead so that , and thus the expression for total throughput in (15) is reduced to , the fraction of success slots. [16] shows that the IEEE 802.11 DCF protocol can be approximated by a protocol that prescribes a single transmission probability, i.e., a memoryless protocol. The transmission probability corresponding to DCF is determined as a function of the minimum and maximum contention window sizes by solving (7) and (9) of [16] simultaneously. In Appendix B, we derive the expressions for throughput and average delay under symmetric protocols with 1-slot memory and memoryless protocols in the WLAN model.

Fig. 7 depicts the delay-efficiency boundaries of protocols with 1-slot memory under ternary feedback and E/NE binary feedback. It also plots the throughput-delay pairs under memoryless protocols as well as the throughput-delay pair achieved by the memoryless protocol corresponding to DCF. To obtain numerical results, we consider and use parameters specified by IEEE 802.11a PHY mode-8 [17], which are tabulated in Table II. The values of , , , and are 341.33, 9, 419.56 and 400.48, respectively, in s. To find the transmission probability corresponding to DCF, we use the minimum and maximum contention window sizes and . An upper bound on total throughput can be obtained by setting , which yields . We compute delay-efficient protocols with 1-slot memory for between 0.01 and 0.81 with a step size of 0.01 using the same numerical method as in Section V.

Comparing Fig. 7 with Fig. 6, we can see that by utilizing a carrier sensing scheme, memoryless protocols resolve contention more efficiently in the WLAN model than in the slotted multiaccess model. Since idle slots are much shorter than collision slots in WLAN, idle slots have much smaller effects on total throughput and average delay than collision slots have. Thus, in WLAN, transmission probabilities can be set low to achieve a success without experiencing many collisions. Carrier sensing also induces the turning point of the delay-efficiency boundaries in WLAN to occur around , yielding the narrower and steeper right-hand side of the delay-efficiency boundaries. The main findings in Section V.D. that protocols with 1-slot memory can achieve smaller average delay for a given level of total throughput and a wider range of total throughput compared to memoryless protocols remain valid in the WLAN model. On the right-hand side of the delay-efficiency boundaries, i.e., for , the delay-efficient protocols with 1-slot memory under ternary feedback have the structure of , and thus the performance is not affected by having E/NE binary feedback instead of ternary feedback.

## Vii Conclusion

In this paper, we have investigated how memory can be utilized in MAC protocols to achieve coordination without relying on explicit control messages. With -slot memory, users can share the channel as in TDMA. With 1-slot memory, high throughput can be obtained by correlating successful users in two consecutive slots, which results in large average delay. Generalizing these results, with -slot memory, where , we can have the first successful users use the channel alternatingly while a collision created by a non-successful user with a small probability leads to a potential change of hands for the collision slot.

Our framework can be extended in several directions. First, we can consider asymmetric protocols to provide quality of service differentiation across users. Second, as the literature on repeated games suggests, memory can also be used to sustain cooperation among selfish users. By utilizing memory, users can monitor the behavior of other users and punish misbehavior. Lastly, the basic idea of this paper can be carried over to a general multi-agent scenario where it is desirable to have one agent behave in a different way than others.

## Appendix A Derivation of Throughput and Average Delay Under Symmetric Protocols With 1-Slot Memory in the Slotted Multiaccess System

Exploiting the symmetry of protocols, we can consider a Markov chain for a representative user whose state is defined as a pair of its transmission action and the number of transmissions, instead of a Markov chain with the state space as in Section III. Let user be the representative user. The state of user in the current slot when the outcome of the previous slot was is given by . We use to denote the set of all states, as defined in Section II.B. There are total states, which we list as

 (T,1),…,(T,N),(W,0),…,(W,N−1). (16)

Suppose that users follow a symmetric protocol . Then we can express the transition probabilities across states in terms of . If user is in state , then users including user transmit with probability , and users with probability . Hence, a transition from to occurs with probability

 Q((T,k′)|(T,k);f)=f(T,ρ(T,k))∑{(x,y)∈N20:x+y=k′−1,x≤k−1,y≤N−k} (17) [(k−1x)f(T,ρ(T,k))x(1−f(T,ρ(T,k)))k−1−x(N−ky)f(W,ρ(W,k))y(1−f(W,ρ(W,k)))N−k−y]

for , and

 Q((W,k′)|(T,k);f)=(1−f(T,ρ(T,k)))∑{(x,y)∈N20:x+y=k′,x≤k−1,y≤N−k} (18) [(k−1x)f(T,ρ(T,k))x(1−f(T,ρ(T,k)))k−1−x(N−ky)f(W,ρ(W,k))y(1−f(W,ρ(W,k)))N−k−y]

for , where is the set of nonnegative integers. Similarly, if user is in state , then users transmit with probability , and users including user with probability . A transition from to occurs with probability

 Q((T,k′)|(W,k);f)=f(W,ρ(W,k))∑{(x,y)∈N2+:x+y=k′−1,x≤k,y≤N−k−1} (19)