Coordinated Multi Point Transmission and Reception for Mixed-Delay Traffic

This paper analyzes the multiplexing gains (MG) for simultaneous transmission of delay-sensitive and delay-tolerant data over interference networks. In the considered model, only delay-tolerant data can profit from coordinated multipoint (CoMP) transmission or reception techniques, because delay-sensitive data has to be transmitted without further delay. Transmission of delay-tolerant data is also subject to a delay constraint, which is however less stringent than the one on delay-sensitive data. Different coding schemes are proposed, and the corresponding MG pairs for delay-sensitive and delay-tolerant data are characterized for Wyner's linear symmetric network and for Wyner's two-dimensional hexagonal network with and without sectorization. For Wyner's linear symmetric also an information-theoretic converse is established and shown to be exact whenever the cooperation rates are sufficiently large or the delay-sensitive MG is small or moderate. These results show that on Wyner's symmetric linear network and for sufficiently large cooperation rates, the largest MG for delay-sensitive data can be achieved without penalizing the maximum sum-MG of both delay-sensitive and delay-tolerant data. A similar conclusion holds for Wyner's hexagonal network only for the model with sectorization. In the model without sectorization, a penalty in sum-MG is incurred whenever one insists on a positive delay-sensitive MG.


I. INTRODUCTION
O NE of the main challenges for future wireless communication systems is to accommodate heterogeneous data streams with different delay constraints. This is also the focus of various recent works, notably [1]- [7]. In particular, [1], [2] study a cloud radio access network (C-RAN) under mixed-delay-constraints traffic. Specifically, users close Manuscript  to base stations (BS) transmit delay-sensitive data, which is directly decoded at the BSs, and users that are further away send delay-tolerant data, which is decoded at the central processor. In this paper we refer to delay-tolerant data as "slow" messages, and to delay-sensitive data as "fast" messages. In [4], we extended above C-RAN models to allow each user to send both "fast" and "slow" messages, and to time-varying fading channels. The results in [4] show that at any "fast" rate, the stringent delay constraint on "fast" messages penalizes the overall performance (sum-rate) of the system. The work in [5] proposes a superposition approach over a fading channel to communicate "fast" messages within single coherence blocks and "slow" messages over multiple blocks. In [6] a scheduling algorithm is proposed for a K-user broadcast network that gives preference to the communication of "fast" messages over "slow" messages. A related work was performed in [7], where "fast" messages can be stored in a buffer during a single scheduling period.
The focus of the current work is on the benefits of cooperation for mixed-delay traffics, assuming that only the transmissions of "slow" messages can profit from cooperation between terminals, but not "fast" messages. Networks with transmitter-(Tx) and/or receiver-(Rx) cooperation have been considered in many recent works including [8]- [18], [40] but mostly only with a single type of messages, namely the messages that we call "slow" messages. Huleihel and Steinberg [8] considered two types of messages: one type that has to be decoded whether or not the Rx-cooperation link is present, and the other that only has to be decoded when the cooperation link is present. Inspired by this model, we studied Wyner's soft-handoff model [22], [23] with mixed-delay traffics in [19], where the Tx-cooperation messages can only depend on the "slow" messages in the system and not on the "fast" messages, and "fast" messages have to be decoded prior to the Rxcooperation phase, whereas "slow" messages can be decoded thereafter. Moreover, in [19] the total number of Tx-and Rxcooperation rounds is constrained also for the "slow" messages as proposed [18]. The problem setup that we consider in this paper is different from the setups in [9]- [18] as each Tx wishes to send both "fast" and "slow" messages, and is different from the setup in [19] as it covers networks with arbitrary interference graphs whereas in [19], we focus on Wyner's soft-handoff model where the signal sent by each transmitter interferes only the observed signal by the receiver to its right. The results in [19] show that, in the high signal to This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ noise ratio (SNR) regime, when both the Txs and the Rxs can cooperate, and for sufficiently large cooperation rates, it is possible to accommodate the largest possible rate for "fast" messages without penalizing the maximum sum-rate of both "fast" and "slow" messages. When only Txs or only Rxs can cooperate, transmitting also "fast" messages causes no penalty on the sum-rate at low "fast" rates, but the sum-rate decreases linearly at high "fast" rates. Notice that the standard approach to combine the transmissions of "slow" and "fast" messages is to time-share (schedule) the transmission of "slow" messages with the transmission of "fast" messages. In this approach, the sum-rate decreases linearly with the rate of the "fast" messages and attains the maximum sum-rate only when no "fast" messages are transmitted.
The focus of this paper is on the pairs of Multiplexing Gains (MG), also called degrees of freedom or capacity prelogs, that are simultaneously achievable for "fast" and "slow" messages. We propose a general coding scheme for any interference network with Tx-and Rx-cooperation that simultaneously accommodates the transmissions of "slow" and "fast" messages, and characterize their achievable MG pairs for two specific cellular network models: Wyner's linear symmetric model [22], [23] and Wyner's two-dimensional hexagonal model [22] with and without sectorization. In Wyner's linear symmetric model, cells are aligned in a line and interference is short range, so that transmissions in a cell are interfered only by transmissions from the two neighbouring cells. Such a model is adequate for systems deployed along highways, railroads or long corridors, see also [38]. In Wyner's two-dimensional models, cells are of hexagonal shapes and transmissions in a cell are interfered by the transmissions in the six neighbouring cells. Sectorization occurs in this model if the BS employs directional antennas pointed to three different directions, so that transmissions in these sectors do not interfere. The hexagonal models are adequate for systems where BSs are spread in two dimensions. For all three considered models, in this paper we assume that the various users of the same cell are scheduled in different frequency bands. Interference thus occurs only from the mobile users in neighbouring cells that are scheduled on the same frequency band.
We establish information-theoretic converses for all three models. For Wyner's symmetric network the converse bound matches the proposed set of achievable MG pairs when the cooperation links are of sufficiently high prelogs or when the MG of "fast" messages is small. These results show that when the prelog of the cooperation links is sufficiently large, for Wyner's linear symmetric model, as for Wyner's linear soft-handoff model [19], it is possible to accommodate the largest possible MG for "fast" messages without penalizing the maximum sum MG of both "fast" and "slow" messages. Our achievable schemes suggest that the same also holds for the sectorized hexagonal model considered in this paper where each cell is divided into three non-interfering sectors by employing directional antennas at the BSs [30]. In contrast, for the considered non-sectorized hexagonal model, there seems to be a penalty in maximum sum MG whenever the "fast" MG is larger than 0.
To achieve the described performances, we propose a novel coding scheme where we assign "fast" and "slow" messages to different sets of transmitters in a way that "fast" transmissions are interfered only by "slow" transmissions. Then we use precoding and successive interference cancellation techniques to transmit each of the "fast" messages at full MG without disturbing the transmission of "slow" messages. The transmission of "slow" messages can benefit from cooperation by applying CoMP reception/ transmission in small subnets to jointly decode/encode the "slow" messages at different Rxs/Txs. More specifically, in our coding scheme, we identify a set of the Txs whose signals do not interfere. The chosen Txs send "fast" messages and the others send "slow" messages or nothing. Communication of "fast" messages is thus only interfered by transmissions of "slow" messages and this interference can be described during the Tx-conferencing phase and precanceled at the "fast" Txs. Also, "fast" Rxs decode their messages immediately and can describe their decoded messages during the Rx-conferencing phase to their adjacent "slow" Rxs allowing them to subtract the interference from "fast" messages before decoding their own "slow" messages. As a result, "fast" messages can be decoded based on interference-free outputs and moreover, they do not disturb the transmission of "slow" messages. CoMP transmission or reception [20], [21] for limited clusters is then employed to convey the "slow" messages.

A. Organization
The rest of this paper is organized as follows. We end this section with some remarks on notation. The following Sections II and III consider general interference networks and describe the problem setup and the proposed coding scheme and its multiplexing gain region for such a general network. Sections IV-VI specialize the results to the symmetric linear Wyner model and to the two-dimensional hexagonal Wyner model. Section VII concludes the paper.

B. Notation
We use the shorthand notations Rxfor "Receiver" and Tx for "Transmitter". The set of all integers is denoted by Z, the set of positive integers by Z + and the set of real numbers by R. For other sets we use calligraphic letters, e.g., X . Random variables are denoted by uppercase letters, e.g., X, and their realizations by lowercase letters, e.g., x. For vectors we use boldface notation, i.e., upper case boldface letters such as X for random vectors and lower case boldface letters such as x for deterministic vectors.) Matrices are depicted with sans serif font, e.g., H. We use [K] to denote the set {1, . . . , K}. We also write X n for the tuple of random variables (X 1 , . . . , X n ) and X n for the tuple of random vectors (X 1 , . . . , X n ).

II. PROBLEM DESCRIPTION
Consider a cellular interference network with K cells each consisting of one Tx/Rx pair. Txs and Rxs are equipped with L antennas and we assume a regular interference pattern except at the network borders. As an example, Fig. 1 shows Wyner's the rates of transmissions of the "fast" and "slow" messages.
We consider a cooperation scenario where neighbouring Txs cooperate during D Tx > 0 rounds and neighbouring Rxs during D Rx > 0 rounds. The total cooperation delay is constrained: where D ≥ 0 is a given parameter of the system and the values of D Tx and D Rx are design parameters and can be chosen arbitrary such that (1) is satisfied.
To describe the encoding at the Txs, denote by N Tx (k) the set of all Txs that have a direct cooperation link with a given Tx k ∈ [K]. We refer to N Tx (k) as the Tx-neighbouring set of Tx k. Neighbouring Txs can communicate to each other during D Tx > 0 rounds, where this communication can only depend on "slow" messages but not on "fast" messages. In each conferencing round j ∈ {1, . . . , D Tx }, Tx k sends a cooperation message T The cooperation communication is assumed noise-free but rate-limited: for a given Tx-conferencing prelog μ Tx > 0 and where H(·) denotes the entropy function and P > 0 is the average blockpower constraint. Tx k computes its channel inputs X n k = (X k,1 , . . . , X k,n ) ∈ R L×n as a function of its "fast" and "slow" messages and of the D Tx |N Tx (k)| obtained cooperation messages: →k , . . . , T →k } ∈NTx(k) . (3) The channel inputs have to satisfy the average block-power constraint almost surely: To describe the decoding, denote the Rx-neighbouring set of a given Rx k ∈ [K], i.e., the set of all receivers that can directly exchange cooperation messages with Rx k, by N Rx (k). Also, define the interference set I k as the set of all Txs whose signals interfere at Rx k.
Decoding takes place in two phases. During the fastdecoding phase, each Rx k decodes its "fast" message M and Z n k,k is i.i.d. standard Gaussian noise, and the fixed L-by-L full-rank matrix Hk ,k models the channel from Txk to the receiving antennas at Rx k. So, Rx k produces: using some decoding function g (n) k on appropriate domains. In the subsequent slow-decoding phase, each Rx k ∈ [K] sends a conferencing message Q The cooperative communication is noise-free, but rate-limited: for given Rx-conferencing prelog μ Rx > 0. Each Rx k decodes its desired "slow" message aŝ using some decoding function b (n) k on appropriate domains. Throughout this article we assume short range interference and thus: Given power P > 0, maximum delay D ≥ 0, and cooperation prelogs μ Rx , μ Tx ≥ 0, average rates (R  (10) and encoding, cooperation, and decoding functions for these rates satisfying constraints (1), (2), (4), and (7) and so that the probability of error (11) tends to 0 as n goes to ∞.
An MG pair (S (F ) , S (S) ) is called achievable, if for every positive integer K and power P > 0 there exist achievable average rates {R The closure of the set of all achievable MG pairs (S (F ) , S (S) ) is called optimal MG region and denoted S (μ Tx , μ Rx , D).
Remark 1: In (12a), we let K → ∞ to remove the boundary effects. The coding scheme proposed in the following Section III can be implemented for arbitrary values of K.

III. CODING SCHEMES AND ACHIEVABLE MULTIPLEXING GAINS
We describe various coding schemes that either transmit both "fast" and "slow" messages (Subsections III-A and III-B) or only "slow" messages (Subsection III-C), and a scheme that does not use any kind of cooperation (Subsection III-D).
An important building block in our coding schemes is CoMP transmission or CoMP reception. Depending on which of the two is used, the scheme requires more Tx-or Rxcooperation rates. So, depending on the application, any of the two can be advantageous. In some applications, cooperation rates might however be too low to employ either of the two. In this case, the proposed schemes can be time-shared with alternative schemes that require less or no cooperation rates at all. Alternatively, the proposed schemes can be employed with a smaller number of cooperation rounds D < D, which also reduces the required cooperation prelog in all our schemes.

A. Coding Scheme to Transmit Both "Fast" and "Slow" Messages With CoMP Reception
Split the total number of conferencing rounds between Tx-and Rx-conferencing as:

1) Creation of Subnets and Message Assignment:
Each network is decomposed into three subsets of Tx/Rx pairs, T silent , T fast and T slow , where • Txs in T silent are silenced and Rxs in T silent do not take any action. • Txs in T fast send only "fast" messages. The corresponding Txs/Rxs are called "fast".
• Txs in T slow send only "slow" messages. The corresponding Txs/Rxs are called "slow". We choose the sets T silent , T fast , and T slow in a way that: • the signals sent by the "fast" Txs do not interfere; and • silencing the Txs in T silent decomposes the network into non-interfering subnets such that in each subnet there is a dedicated Rx, called master Rx, that can send a cooperation message to any other "slow" Rx in the same subnet in at most DRx−1 2 cooperation rounds. For example, consider Wyner's symmetric model (described in detail in Section IV) where Txs and Rxs are aligned on a grid and cooperation is possible only between neighbouring Txs or Rxs. Interference at a given Rx is only from adjacent Txs. The network is illustrated in Figure 2. This figure also shows a possible decomposition of the Tx/Rx pairs into the sets T silent (in white), T fast (in yellow) and T slow (in blue) when D = 6. The proposed decomposition creates subnets with 7 active Tx/Rx pairs where the Rx in the center of any subnet (e.g. Rx 4 in the first subnet) can serve as a master Rx as it reaches any slow (blue) Rx in the same subnet in at most D Rx − 1/2 = (D − 2)/2 = 2 cooperation rounds. As required, transmissions from "fast" (yellow) Txs are only interfered by transmissions from "slow" (blue) Txs.
2) Precanceling of "Slow" Interference at "Fast" Txs: Any "slow" Tx k quantizes its pre-computed input signal X n k (how this signal is generated will be described under item 5)) and describes the quantised signalX n k during the last Tx-cooperation round to all its neighbouring "fast" Txs, which then precancel this interference on their transmit signals. (Here, there is only a single Tx-cooperation round, but this item will be reused in later subsections where D Tx > 1.) Fig. 2 illustrates the sharing of the described quantization information with neighbouring "fast" Txs for Wyner's symmetric model.
To describe this formally, for each k ∈ [K], we define the "slow" interfering set Also, we denote by U n k (M (F ) k ) the non-precoded input signal precomputed at a given "fast" Tx k. (The following item 3) explains how to obtain U n k (M over the channel. Since each "fast" Rx k is not interfered by the signal sent at any other "fast" Tx, the precoding in (15) makes that a "fast" Rx k observes the almost interference-free signal where the variance of above disturbance is around noise level and does not grow with P. 3) Transmission of "Fast" Messages: Each "fast" Tx k encodes its desired message M from a Gaussian point-to-point code of power P. The corresponding Rx k applies a standard point-to-point decoding rule to directly decode this "fast" codeword without Rx-cooperation from its "almost" interference-free outputs Y k , see (16). 4) Canceling "Fast" Interference at "Slow" Rxs: According to the previous item 3), all "fast" messages are decoded directly from the outputs without any Rx-cooperation. During the first Rx-cooperation round, all "fast" Rxs can thus share their decoded messages with all their neighbouring "slow" Rxs, which can cancel the corresponding interference from their receive signals. More formally, we define the "fast" interference set as the set of "fast" Txs whose signals interfere at Rx k. Each "slow" Rx k forms the new signal and decodes its desired "slow" message based on this new signal following the steps described in the following item 5). Fig. 2 illustrates with yellow arrows the sharing of decoded "fast" messages with neighbouring "slow" Rxs in Wyner's symmetric model.  (18). CoMP reception is employed to decode all "slow" messages in a given subnet. That means, each "slow" Rx k applies a rate-L 2 log(1 + P) quantizer to the new output signalŶ n k , and sends the quantization information over the cooperation links to the master Rx in its subnet. Each master Rx reconstructs all the quantized signals and jointly decodes the "slow" messages, before sending them back to their intended Rxs. By item 4) the influence of "fast" transmissions has been canceled on the "slow" receive signals.

5) Transmission and Reception of "Slow" Messages Using
6) MG Analysis: In the described scheme, all transmitted "fast" and "slow" messages can be sent reliably at MG L because all interference is cancelled (up to noise level) either at the Tx or the Rx side, and because Txs and Rxs are equipped with L antennas each.
The presented coding scheme thus achieves the MG pair where The scheme we described so far requires different cooperation rates on the various Tx-or Rx-cooperation links.
To evenly balance the load on the Tx-cooperation links and on the Rx-cooperation links, different versions of the scheme with different choices of the sets T silent , T fast , and T slow and different cooperation routes can be time-shared. The main quantity of interest is then the average cooperation load, which for the scheme above is characterized as follows. During the single Tx-cooperation round, each "fast" Tx k receives a quantised version of the transmit signal of each of its "slow" interfererŝ k ∈ I (S) k . Since each quantisation message is of prelog L, the average required Tx-cooperation prelog equals where Q K,Tx denotes the total number of Tx-cooperation links in the network.
There are three types of Rx-cooperation messages. In the first Rx-cooperation round, each "slow" Rx k obtains a decoded message from each of its "fast" interferersk ∈ I The total number of messages sent in this first round is thus k∈Tslow |I (F ) k | and each is of prelog L. In Rx-cooperation rounds 2, . . . , DRx−1 2 +1, "slow" Rxs send quantized versions of their output signals to the master Rx in the same network. Each of these messages is of prelog L and the total number of such messages equals k∈Tslow γ Rx,k , where γ Rx,k denotes the number of cooperation rounds required for "slow" Rx k to reach the master Rx in its subnet. In rounds DRx−1 2 + 2, . . . , D Rx , the master Rx sends the decoded messages to all the "slow" Rxs in the subnet. Each of these messages is again of prelog L and the total number of such messages is again k∈Tslow γ Rx,k . To summarize, each of the transmitted messages is of prelog L and thus the average cooperation prelog required per Rx-cooperation link is: where Q K,Rx denotes the total number of Rx-cooperation links in the network. Remark 2: If the master Rx of a subnet is a "fast" Rx, it does not have to send its decoded message to its "slow" neighbours, because it decodes all "slow" messages jointly. In this case, less Rx-cooperation prelog is required.

B. Coding Scheme to Transmit Both "Fast" and "Slow" Messages With CoMP Transmission
This second scheme splits the total number of cooperation rounds D as: Similarly to the previous Subsection III-A, the scheme is described by 5 items: 1) Creation of Subnets and Message Assignment: This item is similar to item 1) of Subsection III-A, but the sets T silent , T fast and T slow are chosen in a way that: • as before, the signals sent by the "fast" Txs do not interfere; and • silencing the Txs in T silent decomposes the network into non-interfering subnets so that in each subnet there is a dedicated master Tx that can send a cooperation message to any other "slow" Tx in the same subnet in at most DTx−1 2 cooperation rounds.
Items 2)-4) remain as described in Subsection III-A. Item 5) is replaced by the following item.
5) Transmission and reception of "slow" messages using CoMP transmission: "Slow" messages are transmitted using standard CoMP transmission techniques that can ignore interference from "fast" Txs (due to the post-processing in item 4)) but account for the modified interference graph and the modified channel matrix between slow messages caused by the precanceling performed under item 2). The receivers decode based on the new outputsŶ n k in (18). We describe CoMP transmission in this context more formally. During the first DTx−1 2 Tx-cooperation rounds, each "slow" Tx of a subnet, sends its message to the master Tx of the subnet. This latter encodes all received "slow" messages using individual Gaussian codebooks and precodes them so as to cancel all the interference from other "slow" messages at the corresponding Rxs. I.e., it produces signals so that when they are transmitted over the active antennas in the cell, the signal observed at each "slow" Rx only depends on the "slow" message sent by the corresponding Tx but not on the other "slow" messages. The master Tx applies a Gaussian vector quantizer on these precoded signals and sends the quantization information over the cooperation links to the corresponding Txs during the Tx-cooperation rounds DTx−1 2 + 1 to D Tx − 1. This is possible by the way we defined the master Txs. All "slow" Txs reconstruct the quantized signalsX n k intended for them and send them over the network: X n k X n k . Each "slow" Rx k decodes its desired message from the modified output sequenceŶ n k defined in (18) using a standard point-to-point decoder.
Analysis: Similarly to Subsection III-A, each transmitted message can be sent reliably at MG L, and thus the scheme achieves the MG pair in (19).
The load on the different cooperation links is again unevenly distributed across links, and thus, by time-sharing and symmetry arguments, the average Rx-and Tx-cooperation rates are the limiting quantities. The required average Rx-cooperation rate is easily characterized as: because Rx-cooperation takes place in a single round, during which each "slow" Rx k learns all decoded "fast" messages that interfere their receive signals and these messages are of MG L. To calculate the required average Tx-cooperation rate, define for each k ∈ T slow the positive parameter γ Tx,k to be the number of cooperation hops required from Tx k to reach the master Tx in its subnet. During the first DTx−1 , now from the master Tx to the "slow" Txs in the subnet. During the last round, "slow" Txs convey their messages to the adjacent "fast" Txs that are interfered by their signals. Some of these signals, however have already been shared during Tx-cooperation rounds DTx−1 2 + 1, . . . , 2 DTx−1 2 , and thus do not have to be sent again. The total number of cooperation messages during the last Tx-cooperation rounds is thus only equal to k∈Tfast |I where q denotes the number of the messages that have already been sent in previous rounds. We will characterize the value of q when we analyze specific networks. To summarize, the average required Tx-cooperation rate of our scheme is:

C. Coding Scheme to Transmit Only "Slow" Messages With CoMP Reception and Transmission
In principle, since any "fast" message satisfies the constraints on "slow" messages, we can use the schemes provided in Subsections III-A and III-B to send only "slow" messages. Sometimes, the following scheme however performs better because it requires less Tx-or Rx-cooperation rates. Choose a set T silent ⊆ [K] and silence the Txs in this set, which decomposes the network in non-interfering subnets. The remaining Txs in T slow := [K]\T silent send only "slow" messages using CoMP transmission or reception. The set T silent thus has to be chosen such that in each subnet there is a dedicated master Rx (or master Tx), which can be reached by any other Rx (Tx) in the subnet in at most D 2 cooperation rounds. Both versions achieve the MG pair where The CoMP-reception scheme requires no Tx-cooperation but average Rx-cooperation prelog and the CoMP-transmission scheme requires no Rx-cooperation but average Tx-cooperation prelog where recall that γ Rx,k , γ Tx,k ∈ {1, . . . , D 2 } denote the number of cooperation hops required from a Rx k or a Tx k to reach the master Rx or the master Tx in its subnet.

D. Coding Scheme Without Cooperation
Choose a set of Txs T silent ⊆ [K] so that the remaining Txs T active := [K]\T silent do not interfere, and send "slow" or "fast" over the resulting interference-free links. The scheme requires no cooperation and achieves for any β ∈ [0, 1] the MG pair where IV. WYNER'S SYMMETRIC LINEAR MODEL Consider Wyner's symmetric linear cellular model where cells are aligned in a single dimension and signals of users that lie in a given cell interfere only with signals sent in the two adjacent cells. See Figure 1 where the interference pattern is illustrated by black dashed lines. We assume that the various mobile users in a cell are scheduled on different frequency bands, and focus on a single mobile user per cell (i.e., on a single frequency band). We shall further assume that the number of cells K and the maximum delay D are even.
The input-output relation of the network is where X 0,t = 0 for all t, and the interference set at a given user k is where indices out of the range [K] should be ignored. In this model, Rxs and Txs can cooperate with the two Rxs and Txs in the adjacent cells, so Fig. 1 illustrates the interference pattern of the network and the available cooperation links. As can be seen from this figure, Txs 1 and K and Rxs 1 and K have a single outgoing cooperation link and all other Txs and Rxs in this network have two outgoing cooperation links. Thus, the total numbers of Tx-and of Rx-cooperation links both are A. Choice of Tx/Rx Sets for the Schemes in Section III

1) "Fast" and "Slow" Messages With CoMP Reception:
For the mixed-delay scheme, choose the Tx/Rx set association in Fig. 2, where "fast" Tx/Rx pairs are in yellow, "slow" in blue, and silenced in white. I.e., set For this choice, transmissions of "fast" messages are interfered only by transmissions of "slow" messages and for any , the Tx/Rx pairs in form a subnet for which Rx (D + 2) + D/2 + 1 can act as the master Rx because it can be reached by any "slow" Rx (i.e., even Rx) in its subnet in at most (D Rx − 1)/2 cooperation hops. By (20) and (36), the scheme achieves the MG pair (S (F ) = S .
When D/2 + 1 is odd, the sum in (42) evaluates to D 2 8 . Moreover, in this case, the master Rx is a "fast" Rx. It does not have to send its decoded message to any neighbour, as it locally decodes all "slow" messages of the subnet. So, (see also Remark 2), the nominator in (22) can be reduced by 2. Putting all these together, we obtain μ (r) 2) "Fast" and "Slow" Messages With CoMP Transmission: Choose the same cell association as for the CoMP reception scheme described in (36) and depicted in Fig. 2. Under this cell association, Tx D/2 + 1 can act as a master Tx because it can be reached by any "slow" (even) Tx in its subnet in at most (D Tx − 1)/2 cooperation rounds. Since the same cell partitioning is used, namely (36), this scheme achieves the same MG pair as with CoMP reception, see (38). Moreover, by (24) and (41) in the limit as K → ∞, the required average Rx-cooperation prelog is Similarly, consider (25) and (39) and notice that for D/2+1 even, whereas for D/2 + 1 odd, this sum evaluates to D 2 8 . We consider the q-term in (25), which characterizes the number of quantization messages describing the "slow" signals that are counted twice: once for the CoMP transmission and once for the interference mitigation at "fast" transmitters. In each subnet, D/2 − 1 such messages are double-counted, when D/2 + 1 is even, and D/2 messages are double-counted when D/2 + 1 is odd. Therefore, and according to (25), (39), (46), when K → ∞, the average Tx-cooperation prelog required by the scheme is irrespective of whether D/2 + 1 is even or odd.

3) Transmitting Only "Slow" Messages With CoMP Reception and Transmission: Consider the scheme in
Subsection III-C that transmits only "slow" messages, either using CoMP transmission or CoMP reception. For both schemes we regularly silence every D + 2nd Tx, i.e., as in the two previous subsections, With CoMP reception, this scheme does not use any Tx-cooperation. To calculate the Rx-cooperation prelog, we use the fact that Rx k = (D + 2) + i, for positive integers and i ≤ D + 1, reaches the master Rx in its subnet in γ Rx,k = |D/2 + 1 − i| hops. Since: by (28), in the limit as K → ∞, the average Rx-cooperation prelog tends to Similar conclusions show that when CoMP transmission is used instead of CoMP reception, the scheme requires zero Rx-cooperation prelog and a Tx-cooperation prelog of μ and According to the arguments in the previous subsection, the following regions of MG pairs are achievable depending on the available cooperation prelogs μ Tx and μ Rx .
By Corollary 1, for large cooperation prelogs μ Tx and μ Rx , imposing a stringent delay constraint on the "fast" messages never penalizes the maximum achievable sum-MG of the system: the same sum-MG can be achieved as if only "slow" messages were sent.
The next corollary characterizes the optimal MG region S (μ Tx , μ Rx , D) when one of the two cooperation prelogs (μ Tx or μ Rx ) is small and the other large, and when S (F ) lies below a certain threshold. The corollary shows that also in this regime the same maximum sum-MG can be achieved as if only "slow" messages were sent. When S (F ) exceeds this threshold, our  achievable MG region in (59) shows a penalty in sum-MG which increases linearly with the "fast" MG. In this regime we do not have a matching converse result.
Corollary 2: Assume that Rx,S and μ Tx < μ Tx,S and μ Rx < μ For any S (F ) ∈ [0, α· L 2 ], where α is defined in (52), the pair (S (F ) , S (S) ) lies in the optimal MG region S (μ Tx , μ Rx , D) if, and only if, it is in the trapezoid described on the LHS of (59).
Proof: Follows directly by Theorem 1, see (59), and by Proposition 1, and because the sum S D+2 coincides with the maximum sum MG. Figures 3 and 4 illustrate the inner and outer bounds (Theorem 1 and Proposition 1) on the MG region with D = 6 and D = 10, and different values of μ Rx and μ Tx . As can be seen in Figure 3 and as also explained in Corollary 1, when μ Rx ≥ 2.625 and μ Tx ≥ 1.125, or when μ Rx ≥ 1.125 and μ Tx ≥ 2.25 the inner bound in (58) and the outer bound match. In the former case, the inner bound is achievable using the scheme in Subsection IV-A.1 based on CoMP reception, and in the latter case it is achievable using the scheme in Subsection IV-A.2 based on CoMP transmission. As explained in Corollary 2, when only one of the two cooperation prelogs is large and the other small (e.g., μ Rx ≥ 4.5 and μ Tx = 0.5; or μ Tx ≥ 4.5 and μ Rx = 0.5) the inner bound in (59) matches the outer bound of Proposition 1 only for S (F ) < α · L 2 , where α is defined in (52). For larger values of S (F ) , the maximum "slow" MG S (S) achieved by our schemes decreases linearly with S (F ) . For example, for D = 6 and (μ Rx ≥ 4.5, μ Tx = 0.5) or (μ Tx ≥ 4.5, μ Rx = 0.5) when S (F ) ≥ α · L 2 increases by Δ then S (S) decreases by approximately 1.75Δ and the sum-MG by 0.75Δ. The behaviour changes again when both μ Rx and μ Tx are moderate or small, e.g., μ Rx = 0.5 and μ Tx = 1 or μ Rx = 1 and μ Tx = 0.5. In this case, the sum-MG achieved by our inner bound is constant over all regimes of S (F ) . We finally notice that in these small cooperation-prelog regimes our inner bounds remain unchanged for D = 6, 8, 10. The reason is that in this regime, even when D > 6, it is more advantageous to reduce the number of cooperation rounds to 6 in order to satisfy the cooperation prelogs than to time-share different schemes with D > 6 cooperation rounds. The brown dotted line is the result of time-sharing the scheme in Subsection IV-A.3 that transmit only "slow" messages with the scheme in Subsection IV-A.4 when only "fast" messages are transmitted. The scheme in Subsection IV-A.3 is based on CoMP reception and to achieve the maximum "slow" MG requires μ Rx ≥ 4.5 for D = 6 and μ Rx ≥ 7.5 for D = 10. The sum-MG in this scheme decreases linearly with the "fast" MG.
V. HEXAGONAL NETWORK Consider a network with K hexagonal cells, where each cell consists of one single mobile user (MU) and one BS. The signals of users that lie in a given cell interfere with the signals sent in the 6 adjacent cells. The interference pattern of our network is depicted by the black dashed lines in Fig. 5, i.e., the interference set I k contains the indices of the 6 neighbouring cells whose signals interfere with cell k. The input-output relation of the network is as in (97).
Each Rx k (BS of a cell) can cooperate with the six Rxs in the adjacent cells, i.e., |N Rx (k)| = 6. Thus, the number of Rx-cooperation links Q K,Rx in this network is approximately equal to 6K (up to edge effects). Similarly, each Tx (MU of a cell) can cooperate with the six Txs in the adjacent cells and thus |N Tx (k)| = 6 and Q K,Tx ≈ 6K.
To describe the setup and our schemes in detail, we parameterize the locations of the Tx/Rx pair in the k-th cell by a number o k in the complex plane C. Introducing the coordinate vectors as in Figure 5, the position o k of Tx/Rx pair k can be associated with integers (a k , b k ) satisfying The interference set I k and the neighbouring sets can then be expressed as For simplicity we assume an even-valued D satisfying Other cases can be treated in a similar way. We specify the Tx/Rx set associations for the schemes in Section III. See [39] for a detailed analysis. For the no-cooperation scheme in Subsection III-D choose and T silent = [K]\T active . The corresponding cell association is shown in Figure 6a where active cells are in yellow and silenced in white. By (31), the sum-MG achieved by this scheme is We next explain the Tx/Rx set association for sending only "slow" messages as in Subsection III-C, see also [34]. Set τ = D 2 + 1 and choose Tx k (Rx k) as a master Tx (Rx), if it belongs to To describe the silenced set T silent , we define for any integers x and τ ≥ 0: where mod denotes the standard modulo operator. In fact, the operator x [−τ,2τ ] resembles the standard mod 3τ operator, but it shifts every number into the interval [−τ, 2τ ) and not into [0, 3τ ). We then set and T slow = [K]\T silent . Figure 6b shows the proposed cell association for D = 6: blue or yellow are the active "slow" cells and white the silenced cells. Master Txs (Rxs) are in green pattern. We observe that the choice in (76) silences all Tx/Rx pairs which lie D 2 + 1 hops away from a master Tx/Rx pair. As we detail out in [39], by (26) and (27), this choice establishes an achievable MG pair of (S (F ) = 0, Moreover, by (28) and (29), with CoMP reception or CoMP transmission the scheme requires average Rx-or Tx-cooperation prelogs equal to .
Finally, we turn to the scheme that sends both "fast" and "slow" messages in Subsections III-A and III-B. Here, we set τ = D 2 and choose the set of master Txs (Rxs) as in (74), but for this new value of τ . Similarly, we choose the silenced set T silent as in (76) but again for the new value τ = D 2 . The "fast" transmit set T fast is chosen in the same way as T active in (72), and T slow = [K]\{T silent ∪ T fast }. The cell association is depicted in Figure 6b for D = 8, where "fast" cells are in yellow, "slow" cells in blue, and master cells are in green pattern. As detailed out in [39], by (20), the proposed cell association achieves the MG pair (S (F ) = S and by (21) and (22) and with CoMP transmission they are

A. Achievable MG Region
Recall the definitions of S no-coop , S Also, define Theorem 2 (Achievable MG Region: Hexagonal Model): Assume D ≥ 2, even, and D 2 − 1 mod 3 = 0. When μ Rx ≥ max{μ Proof: Follows by extending the converse in [34,Theorem 2] to the hexagonal model without sectors and with both Tx-and Rx-cooperation. See [39, Appendix C] for details. Figure 7 illustrates the inner and outer bounds (Theorem 2 and Proposition 2 ) on the MG region for D = 8, and different values of μ Rx and μ Tx . We observe that, unlike Wyner's symmetric model, the sum-MG of this network always decreases as S (F ) increases, irrespective of the cooperation prelogs μ Tx , μ Rx . Moreover, maximum S (F ) = L 3 in our bound is only achieved for S (S) = 0. We remark here that for certain channel matrices (in fact for many but not for all) "fast" MG S (F ) is achievable using interference alignment [35]- [37]. For these channel matrices of course our inner bound can be improved accordingly.
In Figure 7, we can distinguish 4 behaviours for the achieved MG region: 1) If both μ Rx and μ Tx are above given thresholds, for D = 8 and either (μ Tx ≥ 0.6, μ Rx ≥ 2. is thus not reduced as long as S (F ) remains small; for larger values of S (F ) the maximum achievable S (S) however suffers significantly. The reason is that our schemes that send both "fast" and "slow" messages inherently require both Txand Rx-cooperation of sufficiently high cooperation prelogs. As a consequence, the maximum S (S) that our schemes achieve for large S (F ) highly depends on the smaller of the two cooperation prelogs μ Tx and μ Rx . 3) When both μ Tx , μ Rx are moderate, we can still achieve the MG pair (S  achieved under our proposed scheme show that the penalty on the sum-MG caused by the transmission of "fast" messages is large in this scheme.

VI. SECTORIZED HEXAGONAL MODEL
Reconsider the cellular network with K hexagonal cells and cell coordinate system spanned by the vectors e x and e y introduced in the previous section. Here, each cell consists of three sectors denoted by "S", "W", and "E", see Figure 8a, and we also number the sectors from 1 to 3K. A single 3L-antenna Rx (BS) is associated to each cell and a single L-antenna Tx to each sector. Each Rx decodes the 3 "slow" and the 3 "fast" messages of the Txs in the 3 sectors corresponding to its cell. Rxs are equipped with directional antennas, where each set of L antennas at a given Rx (BS) points to one of the three sectors of its cell. Therefore, communications from different sectors in the same cell do not interfere, see Fig. 8a where interference is depicted by dashed lines. Interference is shortrange, and transmission in the grey-shaded sector of Fig. 8a is, e.g., interfered by the transmissions in the four adjacent pink-shaded sectors. The interference set I Tx,k of sector k is thus the set of indices of the 4 adjacent sectors that lie in a different cell.
For the purpose of this section, we thus modify the setup in Section II in that we have 3K Txs and K Rxs and each Rx k observes the output signals Y n k := (Y n k1 , Y n k2 , Y n k3 ), where k 1 , k 2 , k 3 denote the three sectors in cell k, and Hk ,ki X n k + Z n ki , i ∈ {1, 2, 3}.
We consider per-sector MGs, and accordingly the average rates in (10) are normalized with respect to 3K and not K. All other definitions of Section II remain unchanged.
Each Rx k (BS of a cell) can cooperate with the Rxs in the six adjacent cells, i.e., |N Rx (k)| = 6 and Q K,Rx ≈ 6K. Each Tx (MU of a cell) can cooperate with the four Txs in the adjacent sectors of different cells, i.e. |N Tx (k)| = 4 and since there are 3K Txs, Q K,Tx ≈ 12K. Assume D even.
The coding schemes and results in Section III apply also to this modified setup, if T silent , T active , T fast , T slow ⊆ [3K] and the MG results (20), (27), and (31) are normalized with respect to 3K and not K. We only consider CoMP reception, and thus T master ⊆ [K].

A. Tx/Rx Set Associations and MG Region
We specify the Tx/Rx set associations for our schemes of Section III. For the no cooperation scheme, define the active set T active as the set of either the "W" sectors, the "E" sectors, or the "S" sectors of all cells. This achieves the sum-MG For the cooperative schemes, we pick the set of master cells T master as in (74) for τ = D 2 . Unlike in the hexagonal model in Section V, it suffices to silence certain sectors of layer D/2 around each master cell (but not necessarily entire cells). Consider the subnet that has its master cell k master at the origin a kmaster = b kmaster = 0. For this subnet, we keep active all 3 sectors of the corner cells in layer D/2 that have coordinates (a k = D/2, b k = 0), (a k = 0, b k = D/2), and (a k = −D/2, b k = −D/2), and we silence all 3 sectors of the remaining 3 corner cells of this layers, which have coordinates (a k = D/2, b k = D/2), (a k = −D/2, b k = 0), and (a k = 0, b k = −D/2). We further silence in this layer D/2: the "S" sector of all non-corner cells with coordinates |b k | = D/2 and sign(a k ) = sign(b k ); the "E" sector of all non-corner cells with coordinates |a k | = D/2 and sign(a k ) = sign(b k ); and the "W" sector of all non-corner cells with coordinates sign(a k ) = sign(b k ). As for the hexagonal model, all Txs that lie less than D/2 cell hops from a master cell are kept active. The proposed sector association splits the entire network into equal non-interfering subnets (up to edge effects that vanish as K → ∞), each consisting of a master cell, all sectors of the cells in the D/2 − 1 surrounding layers, and none or one sector in each cell of layer D/2. The proposed cell and sector association is shown in Figure 8b for D = 8, where yellow and blue sectors are active and white are silenced. The borders of the subnets are shown by red lines.
As shown in [39], when sending only "slow" messages the proposed sector association achieves (S (F ) = 0, and it requires an average Rx-cooperation prelog of μ (r) In the scheme sending both "fast" and "slow" messages, the Txs in the "yellow" sectors of Figure 8b send "fast" messages and the Txs in the "blue" sectors send "slow" messages. We describe the cell association more formally for a subnet whose master cell is at the origin. All other subnets are equal. All active sectors in layer-D/2 of this subnet send "fast" messages, but all sectors in the cells satisfying one of the three following conditions only send "slow" messages: (a k ≥ 0 and b k = 0) or (a k = 0 and b k ≥ 0) or (a k = b k ≤ 0). All other cells have exactly one "fast" sector and two "slow" sectors. Specifically, cells with a k , b k > 0 send a "fast" message in their "W" sector; cells with a k < 0 and b k > a k send a "fast" message in their "S" sector; and cells with b k < 0 and a k > b k send a "fast" message in their "E" sector.
We prove in [39] that the proposed sector association achieves the MG pair The following theorem is proved in [39]. .
Proof: Follows by an extension of the converse in [34, Theorem 2] to both Tx-and Rx-cooperation. See [39, Appendix D] for details. Figure 9 illustrates the inner and outer bounds (Theorem 3 and Proposition 3) on the MG region for D = 4, and different values of μ Rx and μ Tx . As can be seen from this figure, when μ Rx ≥ 2.25 and μ Tx ≥ 0.75, there is no penalty in sum MG even at maximum "fast" MG. It also can be seen from this figure that transmitting "fast" messages using the traditional time-sharing scheme (brown dotted line) at any "fast" MG has a penalty on sum-MG.

VII. CONCLUSION
We proposed a coding scheme for general interference networks that accommodates the transmission of both delay-sensitive and delay-tolerant messages. We characterized the MG region of Wyner's symmetric network for certain parameters and derived inner bounds on the achievable MG region for general parameters, as well as for the sectorized and non-sectorized hexagonal model. The results for Wyner's symmetric model showed that it is possible to accommodate the largest possible MG for delay-sensitive messages, without penalizing the maximum sum MG of both delay-sensitive and delay-tolerant messages. Our proposed scheme suggests a similar behaviour for the sectorized hexagonal model, when one restricts to one-shot interference alignment. For the non-sectorized hexagonal model this does not seem to be the case, and our results always show a penalty in sum MG whenever the delay-sensitive MG is not zero. These results indicate that each network needs to be carefully analyzed to determine whether a sum MG penalty exists under mixeddelay traffics. Nevertheless, in this paper we proposed a joint coding scheme that accommodates mixed-delay traffics for general networks while significantly improving the sum MG compared to a classical scheduling approach.
Our proposed coding schemes suggest that in the regime of high delay-sensitive MGs, it is important to have sufficiently high cooperation prelogs both at the Tx-and the Rx-side to attain the same sum MG as when only delay-tolerant messages are sent. Moreover, in this regime, Tx-cooperation seems to be slightly more beneficial under mixed-delay traffics than Rxcooperation. An interesting line of future research is to analyze the effect of delay-sensitive messages on generalized Wyner models with fading coefficients and finite precision channel state information. Here also the notion of generalized degrees of freedom (GDoF) is of interest, see also [33].