I Introduction
Channel state information at the transmitter is important for achieving high throughput in wireless systems. Preferably, CSIT is known before a transmission takes place since, in that case, the transmitter is able to optimize the transmission parameters such as rate and power. The transmitter may acquire an estimate of the CSI in advance in various ways; for example, by using channel reciprocity or via explicit feedback from the receiver. This is referred to as
prior CSIT. A wireless channel is, however, dynamic and in many cases the channel changes from the time the CSI has been acquired to the time at which the channel is actually used for transmission [2, pp. 211–213]. In addition, even if the channel is static, during the transmission there may be an unpredictable amount of interference at the receiver. In such cases, prior CSI is different from the actual conditions at the receiver when the data transmission takes place and thus of limited use for adapting the transmission parameters. On the other hand, it is viable to assume that the transmitter gets feedback about the CSI after the data transmission has been made. We refer to this as delayed CSIT as it carries information to the transmitter about the conditions at the receiver in the past. The simplest form of delayed CSIT is the bit feedback used in ARQ protocols: (ACK) the transmission was successful, i.e., the channel could support the chosen data rate and (NACK) the channel could not support the data rate. In the most elementary form of ARQ, a failed packet is retransmitted in the subsequent time slots until it is successfully decoded or until a strict decoding time constraint is violated. In order to increase throughput compared to ARQ, one can use chase combining (CC) or send incremental redundancy (IR) instead of retransmissions that consist of pure packet repetition. Such extensions are referred to as HARQCC and HARQINR, respectively [3]. In this paper, we focus on IRbased protocols.The ergodic capacity represents an upper bound on the throughput for any communication protocol and can be approached by fixedlength coding across many time slots. HARQtype protocols attempt to get as close as possible to this upper bound while keeping the average or maximum decoding time as small as possible. Specifically, as the rate , which is used in the first transmission opportunity, tends to infinity, the average decoding time of HARQINR also tends to infinity and the throughput of HARQINR approaches the ergodic capacity of the underlying channel provided that there is no strict constraint on the decoding time. If a strict or average decoding time constraint is present, the achievable throughput is strictly lower than the ergodic capacity.
The purpose of this paper is to put forth and investigate a type of retransmission protocol which is fundamentally different from conventional HARQ protocols and uses rate adaptation based on delayed CSIT to achieve high throughput subject to an average decoding time constraint. As with most prior work in the area of HARQINR, we assume the channel is modeled by a Gaussian blockfading channel, with each time slot consisting of channel uses. The channel gain is kept constant during a single time slot but varies independently from time slot to time slot. Feedback, such as delayed CSIT or acknowledgements (ACKs), can only be received by the transmitter at the end of each time slot. The main problem with an HARQINR protocol for a blockfading channel is that resources are wasted when the receiver sends NACK, while it only needs a small amount of additional information to be able to decode. This results in underutilization of the last time slot and may significantly reduce the throughput when the average decoding time is small. Our key idea is to append new information bits in each time slot such that the last time slot is rarely underutilized and the throughput degradation is reduced. We achieve this by using delayed CSIT which allows the transmitter to estimate the amount of unresolved information at the beginning of each time slot.
Ia Prior work
Caire and Tuninetti [3] were among the first who analyzed HARQ from an informationtheoretic perspective. Here, the throughput measure was defined through the renewalreward theorem (see also [4] and [5]) and achievability and converse results were proved for the HARQINR protocol. Several lines of works has since improved the throughput of HARQINR by using available side information in combination with either power adaptation or rate adaptation.
One line of work uses power or rate adaptation to enhance the throughput of HARQINR with either prior or no CSIT. For example, [6]
investigates HARQINR protocols that maximize the throughput over a blockfading channel with independent channel gains under both a strict decoding time constraint and a longterm power constraint. The longterm power constraint allows the use of slotbased power allocation. It is found that HARQINR in combination with slotbased power allocation increases the throughput. The key idea is that the probability of having to retransmit
times is decreasing in . This implies that the throughput is increased by using more power in the first slots. In addition, it is shown that if the single feedback bit is used to convey a onebit quantization of the prior CSI rather than an ACK/NACK message, then this can result in significant throughput gains. The results from [6] are further extended to any number of feedback bits per slot in [7]. Under the same channel conditions, [8] considers rate adaptation for an HARQINR protocol without prior nor delayed CSIT. Dynamic programming is used to maximize the throughput under an outage constraint and it is found that rate adaptation provides significantly lower outage probabilities. The assumption of independent channel gains is relaxed in [9], where optimal rate adaptation policies are found for the cases in which the channel gains are correlated.Although prior CSIT improves the throughput of HARQINR remarkably, CSIT is often delayed when it is obtained by the transmitter. This has led to another line of work which studies the benefits of delayed CSIT in context of HARQINR protocols. Specifically, [10] and [11] considers a pointtopoint channel with independent blockfading in a setting identical to ours. Apart from the statistics of the channel gain, the transmitter has no knowledge about the current CSI, but the transmitter is informed about the CSI of the previous slot. In their protocol, the channel uses of each slot are divided among a large number of parallel HARQINR instances transmitting separate messages in a time division multiplexing (TDM) fashion. In particular, for a specific HARQINR instance, the number of channel uses used for the th retransmission is some percentage of the number of channel uses spend in the first transmission. This implies that new HARQINR instances, with new data, can be initiated in each slot. The objective is to maximize the throughput under a constraint on the outage probability. It is found that delayed CSIT significantly decreases the outage probabilities. A similar setting was considered in [12], where power adaptation was investigated. Here, the authors used a conventional HARQINR instance, but adapted the power in each slot according to the delayed CSIT. In contrast to [10], in which the authors design composite protocols based on a large number of HARQINR instances, the protocol proposed in [12] only uses a single HARQINR instance with power adaptation which is optimized using dynamic programming. Rate adaptation can also be achieved using superposition coding. A multilayer broadcast approach to fading channels without prior CSIT is proposed in [13]. Specifically, a transmission is initiated in large number of superposition coded layers and the number of decoded layers at the receiver depends on the actual CSI, which is assumed not to be known in advance. This approach provides an alternative to HARQ protocols in the sense that it provides variablerate transmission with a fixed transmission length of one slot. The approach, however, has the disadvantage that the throughput in practical implementations suffer as the number of layers increases. A more practical approach is taken in [14] which combines the approach in [13] for few layers with HARQINR. Specifically, the proposed protocols initiate an HARQINR instance in each layer. In a certain slot, the receiver feeds back the number of decoded layers and, in the subsequent slot, the transmitter only conveys IR for the layers not decoded. For the layers that are decoded, the transmitter initiates new HARQINR instances with new data. Finally, although not directly related to our work, it was shown in [15] that delayed CSIT, which is possibly completely independent of the current channel state, increases the multiplexing gains in a multipleinput multipleoutput (MIMO) broadcast channel with transmit antennas and receivers each with one receive antenna.
In contrast to previous works, this paper is motivated by the backtrack retransmission request (BRQ) protocol proposed in [1]. BRQ is suited for systems in which the transmission opportunities come in slots of a predefined number of channel uses. This prevents conventional HARQINR to optimize the throughput, as the number of channel uses cannot be adapted to the required amount of IR. BRQ overcomes this problem by appending additional new information bits before the information bits sent in previous slot have been decoded. The number of new information bits is adapted according to the reported delayed CSIT. Our approach in this paper combines the idea of appending new data during a transmission for HARQ in [1], [10], and [14] with streaming codes proposed in [16] and [17]. The streaming codes in [16] and [17] are a family of codes that allow the transmitter to append new information bits during a transmission in such a way that all information bits can be jointly decoded as one code. In [16], each message has the same absolute deadline at which all messages need to be decoded. In [17], each message is required to be decoded within a certain number slots after arrival. Both [16] and [17] use a transmission scheme that enlarges the message space in each slot. In coding theory, streaming codes, as those investigated in [16] and [17], are also known as crosspacket codes. Crosspacket codes based on Turbo codes and LDPC codes have previously been considered in the context of HARQ in [18] and [19], respectively. The EMS protocols proposed in this paper extend streaming codes to an HARQINR setting in which the amount of new information bits that are appended within a retransmission is adaptive, as it depends on the delayed CSIT in manner similar to BRQ.
EMS protocols are thus variablerate protocols in a sense similar to [10] and [14]. However, to the best of our knowledge, all previously proposed protocols that allow for rate adaptation are composite protocols based on a conventional HARQINR protocol as building block, where rate adaptation is achieved by using a large number of parallel HARQINR instances in a TDM fashion or in superposition coded layers. These approaches incur rate penalties in practical implementations because each HARQINR instance only uses a small fraction of the available resources (channel uses/power) in each slot. In contrast, EMS protocols differ fundamentally from HARQINR in the way new information bits are appended in each slot. This implies that, in principle, one can use our scheme instead of HARQINR as a building block and devise protocols similar to [10] and [14]. Consequently, we consider HARQINR and HARQINR with power adaptation based on delayed CSIT as relevant baseline protocols for comparison.
IB The backtrack retransmission protocol
Since our work is motivated by BRQ, we shall provide a brief description of the protocol below. Suppose the transmitter sends to the receiver in slots, where each slot is a fixed communication resource that consists of channel uses. The channel is modeled as a Gaussian blockfading channel with channel gains of the slots being independent and identically distributed. Assume also that the transmitter uses unit transmission power such that is the SNR in the th slot. The channel gain is fed back to the transmitter by the end of the th slot. The BRQ protocol uses a single channel code with blocklength and a fixed rate in each slot such that the receiver can decode if , where we have defined
(1) 
In the first slot, the transmitter sends bits of new information using the fixed channel code. If the realized channel gain satisfies , the receiver decodes the packet, extracts the information bits, and the protocol terminates with a decoding time of one slot. On the other hand, if , the receiver cannot decode the packet, it feeds back the CSI of the first slot, and the protocol continues in slot . Considering the th slot, with and assuming that for all , the transmitter forms the packet of bits for the th slot as follows:

The first bits are IR that allow the decoding of the packet in slot .

The remaining bits are new information bits.
Note that is fed back to the transmitter by the end of slot and thereby known at the transmitter in slot . If , the receiver feeds back the CSI of the slot and the protocol continues in slot . If , the receiver can decode the packet in slot and it can recover the information bits. It also recovers the bits of IR for the packet in slot . At this time, the receiver can decode the packet conveyed in the th slot using the side information from the IR bits in slot . Next, the decoder sequentially decodes the packets in a similar fashion, thereby recovering all the bits. Over the same slots, one could have transmitted information bits if the channel gains had been available a priori (and assuming that power adaptation was not used). The loss in throughput by BRQ is thus only due to the difference . The throughput of BRQ, reported in [1], is restated in Theorem 3.
We note that the IR bits and the new information bits are only separable in the digital domain, but not at the physical layer. Hence, the receiver needs to decode the whole packet, which is transmitted using the fixed channel code with rate , in order to extract the IR bits and the new information bits.
We observe that BRQ relies on appending information bits to the parity bits. The transmission rate used in BRQ is predefined to be in each slot. The number of appended information bits is computed based on delayed CSIT but chosen such that the a priori probability of decoding a certain slot is kept constant. Hence, the BRQ protocol ends a transmission as soon as the CSI is above a level that is sufficient for decoding the predefined rate .
IC Contribution
In this paper, we generalize the BRQ protocol from [1]
. First, we propose a family of EMS protocols that allow the transmitter to expand the message space in manner similar to BRQ. In contrast to BRQ, however, the EMS protocols are based on streaming codes and all information bits are decoded jointly. The notion of an EMS protocol introduced here is sufficiently general to include protocols like ARQ, HARQINR, and BRQ. Next, we prove a converse and an achievability result for the EMS protocols, and it is shown that the throughput of the optimal zero outage EMS protocol given a constraint on the average decoding time and full delayed CSIT is identical to the throughput of BRQ. Then, we address the same problem with only a finite number of feedback messages in each slot. In this case, we put forth heuristic EMS protocols which have a structure similar to BRQ, but are designed to work with a finite number of feedback messages. Finally, the throughput of BRQ and the proposed finite feedback EMS protocols are evaluated and compared to relevant baseline protocols. Specifically, we compute the throughput in terms of SNR and in terms of average decoding time. Our numerical results confirm that the throughput of BRQ converges to the ergodic capacity faster than the throughput of HARQINR. Moreover, the proposed finite feedback EMS protocol using only three feedback messages per slot achieves throughput which is only slightly worse than that of BRQ. We remark that EMS protocols have previously been introduced in
[20], where we used finite blocklength analysis to investigate a protocol similar to BRQ in a simplified setup. In a similar setting, optimal rate adaptation policies were optimized using error exponents in [21].Notation
Vectors are denoted by boldface (e.g., ), while their entries are denoted by roman letters (e.g., ). The transpose of a vector is denoted by , the length of a vector by , and the tuple , for , is denoted by
. Similarly, we denote a tuple of random variables
, , by . We adopt the convention that and likewise we let be the empty tuple. Let be the natural numbers, be the reals, and be the nonnegative reals. Moreover, the range of integers , , is denoted by . We also use the standard asymptotic notation and which means that and that , respectively. Finally, we let .Ii System Model
We consider a singleuser blockfading channel with Gaussian noise. The transmitter sends to the receiver in slots of channel uses, where is sufficiently large to offer reliable communication that is optimal in an informationtheoretic sense. The received signal vector in slot is given by
(2) 
where is an
dimensional noise vector distributed according to the Gaussian distribution with zero mean and identity covariance matrix,
is the transmitted vector satisfying(3) 
and denotes the instantaneous channel gain, drawn independently from a smooth probability density with support on
. The cumulative distribution function of
is given by . The instantaneous channel gain is unknown at the transmitter prior to the transmission of but is known at the receiver after observing . Moreover, the receiver is able to provide feedback based on the CSI. Specifically, we assume that feedback is given by a sequence of feedback functions that maps to a feedback alphabet such that is observed at the transmitter before transmission in the th slot. The feedback cost is defined as the cardinality of the feedback alphabet and may be finite, countably infinite, or uncountably infinite. The transmitter is said to have full delayed CSIT if can be recovered from .If a transmission is to be done over slot alone, the maximum supported rate is given by , whereas the maximum achievable rate if a transmission is done over many slots approaches the ergodic capacity [22]
(4) 
as the number of slots tends to infinity. Here, denotes a random variable distributed according to . If, however, a transmission is to be done over few slots, high throughput cannot be achieved without either layered transmissions as in [14] or a HARQ technique. The latter approach is commonly applied in practical systems due to its relative simplicity compared to the layered transmissions.
A comment on the blockfading assumption is in order. The blockfading channel model is an abstraction of a practical system model. In particular, if slots are transmitted consecutively in time as this model suggests, the channel gains cannot be assumed to be independent. In practical systems, however, the delay of ACK/NACK feedback can often spread over multiple slots in time. Therefore, in wireless systems such as LTE, multiple HARQ instances are interleaved in time [23, Ch. 12]; while the transmitter waits for feedback from one HARQ instance, it transmits to other users. In the uplink in LTE, a synchronous version of HARQ is employed [23, Ch. 12]. This ensures that the time between each retransmission is fixed and known by both the transmitter and the receiver. The fact that each transmission opportunity is spaced apart by a fixed number of slots implies that channel gains can be assumed to be independent for many scenarios. In addition to these considerations, one cannot expect that each transmission opportunity occurs in the same frequency slot; this further justifies the use of a blockfading model.
An EMS protocol is now defined by

A sequence of feedback functions that maps to the feedback alphabet such that
(5) 
A sequence of rate selection functions that satisfy , for all , and for some positive constant . We also define the cumulative rates .

A sequence of encoding functions such that
(6) Here, denotes all binary vectors (of arbitrary length), i.e., , where denotes the vector of length ; are independent Bernoulli variables with parameter ; and the tuple is denoted by .

A sequence of decoding functions .

A sequence of nonnegative integervalued random variables , which are stopping times with respect to the filtration (see e.g. [24, p. 488]) and satisfy and .
The error event of an EMS protocol is given by
(7) 
We also define the limiting rate selection functions and stopping time of an EMS protocol:
(8)  
(9) 
The limit of exists because is nondecreasing in and bounded above by . On the other hand, we define as the supremum over since the existence of the limit of cannot be guaranteed because only is bounded above for increasing .
The random variables correspond to the binary sequence of information bits, which size in bits is unbounded. We assume that all the information bits are available prior to the transmission in the first slot. This implies that the stopping time is also the decoding time and the transmission time in slots. In the remainder of this paper, we shall refer to as a decoding time. We note that our definition of decoding time deviates from some other works. For example, in [8] and [10], the decoding time is measured as the time from the information bits are appended to the time at which they are decoded.
As an implication of the definition of an EMS protocol, becomes a function of the information bits . This enables the encoder to combine IR and new information bits, i.e., in each slot the encoder fetches information bits and encodes them jointly with the previously encoded information bits. This message structure is different from other works on HARQINR protocols. In light of [25], HARQINR can be seen as fixedtovariable coding because the number of transmitted information bits are prespecified while the number of channel observations at the receiver depends on channel realization. On the other hand, for an EMS protocol, both the number of information bits and the number of channel observations depend on the channel realization. This concept has previously been used in [10] and [14]; however, none of these works alter the conventional HARQINR protocol. They rather use it as a building block and initiate a large number of HARQINR instances which run in parallel consecutively in time or in multiple superposition coded layers.
Following other HARQ works [3, 5, 14], we define the throughput of an EMS protocol in terms of a renewalreward process. A renewal event occurs at time and the reward is the sum of all rates appended since time . Likewise, the interrenewal time corresponds to the decoding time . Hence, we define the throughput of an EMS protocol as . This leads us to the definition of a zero outage EMS protocol.
Definition 1
An EMS protocol is called an zero outage EMS protocol if there exists a nondecreasing integervalued sequence such that , , ,
(10) 
and
(11) 
Our focus is on the characterization of optimal zero outage EMS protocols:
(12) 
The condition in (10) ensures that the outage probability of the EMS protocol is zero, while the condition in (11) ensures that the conditional probability of error given a decoding time vanishes uniformly for all decoding times except for . We note that our converse result does not hinge on the condition in (11); it is only introduced to strengthen the achievability result.
We note that most other HARQ works consider strict latency constraints which naturally arise in wireless communication systems having either a strict deadline or a limited buffer size. We consider average decoding time constraints and zero outage protocols for two reasons:

A strict latency constraint does not naturally arise in systems without a strict deadline or limited buffer size, and hence, in such applications, there is no reason to choose a specific deadline in the strict decoding time constraint . For example, consider an application that requires high reliability. In this case, imposing a strict latency constraint for the HARQ protocol only implies that the receiver will request a retransmission of the data in outage. This is the case for LTE, which uses HARQ in the medium access control (MAC) layer, while it implements an ARQ protocol on a higher layer – in the radio link control (RLC) layer – that requests retransmissions for data in outage [23, Ch. 12]. In that sense, LTE attempts to achieve an outage probability close to zero, and an average decoding time constraint is therefore a natural constraint which attempts to keep the decoding time low on average but does not give any strict guarantees. As previously mentioned, LTE employs synchronous HARQ in the uplink which implies that the decoding time is indeed proportional to real decoding time in a system. We also point out that the customary metric for latency in queuing theory is the average waiting time.
Iii Achievability and Converse
In this section, we state converse and achievability results that we shall apply in the subsequent sections. The achievability and converse results state conditions for when the probability of error tends to zero or one, respectively. In order to state the results, it is convenient to introduce some notation. In particular, given rate selection functions and feedback functions, let
(13) 
for and let for . Intuitively, is the remaining amount of information needed to decode the information bits appended up to time given the channel gains . We also define
(14)  
(15) 
We prove the following results in Appendix A and Appendix B.
Lemma 1 (converse)
Given an EMS protocol, we have
(16) 
for every satisfying and given that .
Remark 1
Remark 2
The smallest limiting decoding time of a zero outage EMS protocol which is not ruled out by Lemma 1 is given by
(17) 
To show that an EMS protocol with is not ruled out by Lemma 1, note that by the definition of , we must have
(18) 
and
(19) 
Thus, using the fact that for every , we find that the conditions in Lemma 1 cannot be simultaneously satisfied.
Lemma 2 (achievability)
Let decoding times , rate selection functions , and feedback functions be given. Suppose that there exist positive sequences , , and such that is a nondecreasing sequence satistying and such that
(20) 
as . Moreover, define the event
(21) 
and assume for all sufficiently large that
(22) 
Then, there exists an EMS protocol satisfying
(23) 
Iv Full Delayed CSIT
In this section, we consider the case in which the feedback alphabet is the positive reals, , and the feedback functions are given by
(24) 
This provides the transmitter with full delayed CSIT. In the following, we characterize the tradeoff between throughput and the average decoding time for optimal zero outage EMS protocols. First, we specify an EMS protocol and we shall later show that it is an optimal zero outage EMS protocol. The EMS protocol is specified as follows
(25) 
for a positive constants and . The decoding times are given by
(26) 
where
(27)  
(28) 
for an arbitrary constant . The particular choice of the rate selection functions has a simple operational interpretation when neglecting the vanishing term . Consider a transmitter using a fixedrate codebook with rate in each slot such that the minimal channel gain required to decode a slot is . Based on the delayed CSI, in slot , the transmitter sends the exact amount of IR that is required to decode the previous packet, i.e., bits, along with bits of new information bits. This protocol resembles the BRQ protocol previously described in Section IB but formulated as an EMS protocol.
The operation of BRQ is illustrated and compared to HARQINR in Fig. 1. Initially, HARQINR transmits at a rate . The receiver accumulates information until the amount of unresolved information reaches zero. BRQ starts the transmission at a rate , but in contrast to HARQINR, it uses the delayed CSI to append new information bits in each slot to ensure that the amount of unresolved information, before the receiver observes and , remains . Note that, in order to attain the same average decoding time for BRQ and HARQINR, needs to be chosen smaller than since no additional information bits are appended during transmission in the HARQINR protocol. This is why we have chosen in the figure. For the particular realization of channel gains depicted in Fig. 1, it is seen that HARQINR does not fully utilize the supported rate since the unresolved information, before and are observed, is significantly smaller than the supported rate in that slot. This phenomenon reduces the throughput at low average decoding times. The problem is partially circumvented in BRQ by ensuring that the amount of unresolved information, before and are observed, is kept constant.
In contrast to BRQ, the EMS protocol specified by (24) and (25) uses joint decoding over all slots. Since the EMS protocol specified by (24) and (25) and BRQ are closely related, we shall refer to the proposed EMS protocol as “BRQEMS” to emphasize its relation to BRQ.
The following result analyzes the tradeoff between throughput and average decoding time of BRQEMS. Specifically, we find that the throughput is identical to that of BRQ. Furthermore, we apply the converse result in Lemma 1 and we demonstrate using dynamic programming that BRQEMS is optimal within the class of zero outage EMS protocols.
Theorem 3
For , we have
(29) 
where
(30) 
Moreover, we have that if
(31) 
for every . Here, denotes the derivative of .
Remark 4
Remark 5
Remark 6
Proof:
We shall first use Lemma 2 to show that there exists an zero outage EMS protocol with rate selection and feedback functions given by (24) and (25), respectively. Then, we apply the converse result in Lemma 1 to show that under the condition in (31).
Fix positive constants and , and as in (30). We first show that an EMS protocol specified by (24)–(26) has throughput and average decoding time upperbounded by . Since is a nondecreasing sequence of random variables and since , Lebesgue’s monotone convergence theorem [24, Th. 16.2] implies that
(32)  
(33)  
(34)  
(35) 
Similarly, we also have that
(36)  
(37)  
(38)  
(39)  
(40)  
(41) 
Here, (37) follows from Lebesgue’s monotone convergence theorem [24, Th. 16.2] because and are nondecreasing in . Moreover, (39) follows from Tonelli’s theorem [24, Th. 18.3] and (41) follows because
(42) 
and because . As a result of (35) and (41), we obtain the throughput
(43) 
To show the existence of an zero outage EMS protocol, we need to demonstrate that BRQEMS satisfies (10) and (11). Both of these conditions follow from (23) if the conditions of Lemma 2 can be verified. Let . Then, we shall first show that implies that , which in turn implies that
(44) 
for , where is defined in (21). Because for every and because when , this follows from
(45) 
and from the following chain of inequalities^{1}^{1}1We use the convention that for all and for all integers .
(49)  
(50) 
Here, (49) follows from (13) and (25), (49) follows because (28) implies that when , (49) follows because for , and (50) follows because . Next, we show that satisfies (22):
(51)  
(52)  
(53)  
(54)  
(55) 
Here, (52) follows because implies that the event occurs. It also follows that (20) is satisfied:
(56) 
as . As a consequence of (55) and (56), Lemma 2 implies that there exists an EMS protocol satisfying (23). In addition, the EMS protocol is also an zero outage EMS protocol, which follows because the condition in (11) is implied by (23) and (44):
(57)  
(58)  
(59) 
as . Here, denotes the complement of the event and (58) follows (44). The condition in (10) now follows from (59) and because as .
V Finite Number of Feedback Messages
Full delayed CSIT feedback is not always an viable assumption. This section addresses the case where the feedback cost is finite. While HARQINR does not allow for rate adaptations, EMS protocols with three or more feedback messages can be used to signal ACK/NACK, but also to instruct the transmitter to append additional information bits in the subsequent slot. The key difference from the case with full delayed CSIT is that the optimal amount of new information to be appended cannot be specified through the feedback. We provide a heuristic choice of the rate selection functions, feedback functions, and decoding times and demonstrate the existence of a zero outage EMS protocol. In Section VI, it is numerically shown that the throughput of the finite feedback cost EMS protocol is comparable with that of the BRQ protocol.
We shall construct an EMS code with feedback cost , where . Specifically, we define the rate selection and feedback functions as
(60) 
and
(61)  
Here, is a predefined constant, , and for an arbitrary positive constant . The decoding time is given by
(62) 
where
(63)  
(64) 
Here, is an arbitrary positive constant and the feedback designates an ACK message. Since can take at most values, the corresponding EMS protocol has feedback cost . We define the composite ratefeedback function as
(65) 
With this definition, we can write
(66) 
for all and such that .
The tradeoff between throughput and average decoding time achievable by an EMS protocol is characterized by the following theorem which provides a way to compute the throughput and average decoding time by solving two integral equations. Varying the parameter determines the tradeoff between throughput and average decoding time.
Theorem 4
Define and through the integral equations
(67) 
and
(68) 
Here,
denotes the probability density function of
. Then, there exists an zero outage EMS protocol with(69) 
and
(70) 
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