The Diversity-Multiplexing-Delay Tradeoff in MIMO Multihop Networks with ARQ

# The Diversity-Multiplexing-Delay Tradeoff in MIMO Multihop Networks with ARQ

Yao Xie,  Yao Xie (Email: yaoxie@stanford.edu) is with the Department of Electrical Engineering at Stanford University.    Deniz Gündüz, Deniz Gündüz (Email: dgunduz@princeton.edu) was with the Department of Electrical Engineering, Stanford University, and with the Department of Electrical Engineering, Princeton University. He is now with Centre Tecnològic de Telecomunicacions de Catalunya (CTTC), Spain.     Andrea Goldsmith Andrea Goldsmith (Email: andrea@wsl.stanford.edu) is with the Department of Electrical Engineering at Stanford University.This work is supported by the Interconnected Focus Center, ONR grant N000140910072P00006, DARPA’s ITMANET program, and the Stanford General Yao-Wu Wang Graduate Fellowship.
July 5, 2019
###### Abstract

We study the tradeoff between reliability, data rate, and delay for half-duplex MIMO multihop networks that utilize the automatic-retransmission-request (ARQ) protocol both in the asymptotic high signal-to-noise ratio (SNR) regime and in the finite SNR regime. We propose novel ARQ protocol designs that optimize these tradeoffs. In particular, we first derive the diversity-multiplexing-delay tradeoff (DMDT) in the high SNR regime, where the delay is caused only by retransmissions. This asymptotic DMDT shows that the performance of an node network is limited by the weakest three-node sub-network, and the performance of a three-node sub-network is determined by its weakest link, and, hence, the optimal ARQ protocol needs to equalize the performance on each link by allocating ARQ window sizes optimally. This equalization is captured through a novel Variable Block-Length (VBL) ARQ protocol that we propose, which achieves the optimal DMDT.

We then consider the DMDT in the finite SNR regime, where the delay is caused by both the ARQ retransmissions and queueing. We characterize the finite SNR DMDT of the fixed ARQ protocol, when an end-to-end delay constraint is imposed, by deriving the probability of message error using an approach that couples the information outage analysis with the queueing network analysis. The exponent of the probability of deadline violation demonstrates that the system performance is again limited by the weakest three-node sub-network. The queueing delay changes the consideration for optimal ARQ design: more retransmissions reduce decoding error by lowering the information outage probability, but may also increase message drop rate due to delay deadline violations. Hence, the optimal ARQ should balance link performance while avoiding significant delay. We find the optimal fixed ARQ protocol by solving an optimization problem that minimizes the message error subject to a delay constraint.

Submitted to IEEE Trans. Info. Theory, April, 2011.

## I Introduction

Multihop relays are widely used for coverage extension in wireless networks when the direct link between the source and destination is weak. The coverage of relay networks can be further enhanced by equipping the source, relays and destination with multiple antennas and using multiple-input-multiple-out (MIMO) techniques for beamforming. Indeed, MIMO can be used either for beamforming, which improves the reliability, or for spatial multiplexing, which increases the data rate [1]. These dual uses of MIMO gives rise to a diversity-multiplexing tradeoff in point-to-point and multihop MIMO systems, as discussed in more detail below.

Recovery of packets received in error in multihop networks is usually achieved by automatic retransmission (ARQ) protocols. With an ARQ protocol, on each hop, the receiver feeds back to the transmitter a one-bit indicator signifying whether the message can be decoded or not. In case of failure the transmitter retransmits the same message (or incremental information, e.g., using a Raptor code [2][3]) until successful packet reception. The ARQ protocol can be viewed as either a one-bit feedback scheme from the receiver to the transmitter, or as a time diversity scheme employed by the transmitter. The ARQ protocol improves system reliability at a cost of increased delay. In order to design an effective ARQ protocol for multihop relay networks with MIMO nodes, first the fundamental tradeoffs between reliability, data rate, and delay of such systems must be determined, and then the protocol performance can be compared to this theoretical performance limit.

A fundamental tradeoff in designing point-to-point MIMO systems is the tradeoff between reliability and data rate, characterized by the diversity-multiplexing tradeoff (DMT). The asymptotic DMT was introduced in [4] focusing on the asymptotically high SNR regime. The finite SNR DMT was presented in [5]. The DMT has also been used to characterize the performance of classical three-node relay networks, with a direct link between the source and the destination, when the nodes have single-antenna (SISO) or multiple antennas for various relaying strategies [6], [7], [8]. The DMTs for the amplify-and-forward (AF) and decode-and-forward (DF) relaying strategies are discussed in [6]. Several extensions of the amplify-and-forward strategy have been proposed recently, including the rotate-and-forward relaying [9] and flip-and-forward relaying [10] strategies, which employ a sequence of forwarding matrices to create an artificial time-varying channel within a single slow fading transmission block in order to achieve a higher diversity gain. A dynamic decode-and-forward (DDF) protocol, in which the relay listens to the source transmission until it can decode the message and then transmits jointly with the source, is proposed in [11] and its DMT performance is shown to dominate the fixed AF and DF schemes. The DDF protocol is shown to achieve the optimal DMT performance in MIMO multihop relay networks in [8]. In this paper, we restrict our attention to multihop networks using the DF relaying strategy, since it enables us design an optimal ARQ protocol for MIMO multihop relay networks, as we will show later.

Here we consider the diversity-multiplexing-delay tradeoff (DMDT), which was introduced in [12] as an extension of the DMT to include the delay dimension. Here the notion of delay is the time from the arrival of a message at the transmitter until the message is successfully decoded at the receiver, also known as the “sojourn time” in queueing systems. Delays are incurred for two reasons: (1) ARQ retransmissions: messages are retransmitted over each hop until correctly decoded at the corresponding receiver, and (2) queueing delay: ARQ results in a queue of messages to be retransmitted at the transmitter. Most works on DMDT assume infinite SNR for the asymptotic analysis, and the queueing delay has been largely neglected. This is because in the high SNR regime, retransmission is a rare event [13]. With this asymptotic infinite SNR assumption, [12] presents the DMDT for a point-to-point MIMO system with ARQ, [14] studies the DMDT for cooperative relay networks with ARQ and single-antenna nodes, and [15] proves the DMDT-optimality of ARQ-DDF for the multiple access relay and the cooperative vector multiple access channels with single antenna nodes. However, the asymptotically high SNR regime does not capture the operating conditions of typical wireless systems in practice, where errors during transmission attempts are not rare events [13]. Hence, to fully characterize the DMDT performance in the finite SNR regime, we must bring the queuing delay into the problem formulation. For a point-to-point MIMO system with a delay constraint and no feedback link, the tradeoff between the error caused by outage due to insufficient code length, and the error caused by delay exceeding a given deadline, has been studied in [16] using large deviation analysis.

One of the goals of our paper is to study the effects of dynamic ARQ on the DMDT in relay networks. Hence, we consider a line network in which a node’s transmission is only received by adjacent nodes in the line. This is a reasonable approximation for environments where received power falls off sharply with distance (i.e., the path loss exponent is large). For this multi-hop channel model we show that the optimal ARQ protocol requires dynamic allocation of the ARQ transmission rounds based on the instantaneous channel state, and we obtain its exact DMDT characterization. The more general case where non-adjacent nodes receive a given node’s transmission is significantly more complicated, and the optimal DMT is unknown for this case even with a single relay [14].

The contribution of this paper is two-fold: (1) we characterize the DMDT of multihop MIMO relay networks in both the asymptotically high SNR regime and in the finite SNR regime where, in the latter, queuing delay is incorporated into the analysis; (2) we design the optimal ARQ protocol in both regimes. Our work extends the DMDT analysis of a point-to-point MIMO system presented in [13] to MIMO multihop relay networks. In the first part of the paper, we derive the DMDT in the asymptotic high SNR regime, where the delay is caused by retransmissions only. For a certain multiplexing gain, the diversity gain is found by studying the information outage probability. An information outage occurs when the receiver fails to decode the message within the maximum number of retransmission rounds allowed. Based on this formulation, for some multihop relay networks a closed-form expression for the DMDT can be identified, whereas for general multihop networks, determining the DMDT can be cast as an optimization problem that can be solved numerically. The DMDT of a general multi-hop network can be studied by decomposing the network into three-node sub-networks. Each three-node sub-network consists of any three neighboring nodes in the network and the corresponding links between them. The asymptotic DMDT result shows that the performance of the multihop MIMO network, i.e., its DMDT, is determined by the three-node sub-network with the minimum DMDT. The DMDT of the three-node sub-network is again determined by its weakest link. Hence, the optimal ARQ protocol should balance the link DMDT performances on each hop by allocating ARQ window sizes among the hops. From this insight, we present an adaptive variable block-length (VBL) ARQ protocol and prove its DMDT optimality.

Next, we study the DMDT in the finite SNR regime, in which the delay is caused by both retransmissions and queueing. We introduce an end-to-end delay constraint such that a message is dropped once its delay exceeds this constraint. We characterize the finite SNR DMDT by studying the probability of message error, which is dominated by two causes: the information outage event and the missing deadline event, when the block length is sufficiently long [4]. Our approach couples the information-theoretical outage probability [5] with queueing network analysis. In contrast to the analysis under asymptotically high SNR, this does not yield closed-form DMDT expressions; however, it leads to a practically more relevant ARQ protocol design. The end-to-end delay that takes the queueing delay into consideration introduces one more factor into the DMDT tradeoff and the associated optimal ARQ protocol design. Specifically, allocating more transmission rounds to a link may improve its diversity gain and, hence, lower the information outage probability; however, it also increases the queueing delay and, hence, may also increase the overall error probability as more messages are dropped due to the violation of the deadline. Thus, an optimal ARQ protocol in the finite SNR regime should balance these conflicting goals: our results will show that this leads to equalizing the DMDT performance of the links.

We formulate the optimal ARQ protocol design as an optimization problem that minimizes the probability of message error under a given delay constraint. The end-to-end delay constraint requires us to take into account the message burstiness and queueing delays, which are known to be the main obstacles in merging the information-theoretical physical layer results with the network layer analysis [17]. We bridge this gap by modeling the MIMO multihop relay network as a queueing network. However, unlike in traditional queueing network theory, e.g., [18, 19], the multihop network with half-duplex relay nodes is not a standard queue tandem, because node along the multihop queue tandem must wait to complete reception of the previous message by the node before it can transmit to node in the tandem. Another difference between our analysis and traditional queueing theory is that we study the amount of time a message waits in the queue (similar to [13]) rather than just the number of messages awaiting transmission. This poses a challenge because the distribution associated with these random delays is hard to obtain [20], unlike the distribution of the number of messages for which a product form solution is available [18]. In [21] delay is studied by using a closed queue model and diffusion approximation. We derive the exponent of the deadline missing probability in our half-duplex multihop MIMO network by adapting the large deviation argument used in [22]. The expression of the exponent again demonstrates that the system performance (in terms of the exponent) of multi-hop network with half-duplex relays is determined by the three-node sub-network with the minimum exponent.

The remainder of this paper is organized as follows. Section II introduces the system model and the ARQ protocol. Section III presents the asymptotic DMDT analysis for various ARQ protocols while proving the DMDT optimality of the VBL ARQ. Section IV presents the finite SNR DMDT with queueing delays, including some illustrative examples. Finally, Section V concludes the paper and discusses some future directions.

## Ii System Model and ARQ Protocols

### Ii-a Channel Model

Consider an -node multihop MIMO relay network. Node 1 is the source, node is the destination, while nodes 2 through serve as relays. Node has antennas for The system model is illustrated in Fig. 1. We denote this MIMO relay network as . At the source, the message is encoded by a space-time encoder and mapped into a sequence of matrices, , where is the block length, i.e., the number of channel uses of each block, and is the maximum number of end-to-end total ARQ rounds that can be used to transmit each message from the source to the destination. The rate of the space-time code is .

We define one ARQ round as the transmitter sending a whole block code of the message to the receiver. We assume that the relays use the DF protocol: node , , decodes the message, and reencodes it with a space-time encoder into a sequence of matrices . The channel between node and node () is given by:

 Yi,l=√SNRMiHi,lXi,l+Wi,l,1≤l≤L, (1)

where , , is the received signal at node in the th ARQ round. Channels are assumed to be frequency non-selective, block Rayleigh fading and independent of each other, i.e., the entries of the channel matrices are independent and identically distributed (i.i.d.) complex Gaussian with zero mean and unit variance. The additive noise terms are also i.i.d. circularly symmetric complex Gaussian with zero mean and unit variance. The forward communication links and ARQ feedback links only exist between neighboring nodes.

Other assumptions we have made for the channel model are as follows:

• We consider half-duplex relays, that is, the relays cannot transmit and receive at the same time.

• We assume a short-term power constraint at each node for each block code, given by , . Here denotes expectation, and denotes the Hermitian transpose. A long-term power constraint would allow us to adapt the transmit power and achieve power control gain, as we briefly discuss later in the paper. In the following results we assume a short-term power constraint in order to focus on the diversity gain achieved by the ARQ protocol.

• We consider both the long-term static channel model, in which for all , i.e. the channel state remains constant during all the ARQ rounds for a hop and is independent from hop to hop; and the short-term static channel model, where are i.i.d. but not identical for the same . The long-term static channel assumption is the worst-case in terms of the achievable diversity with a maximum of ARQ rounds [12], because there is no time diversity gain. The long-term static channel model may be suitable for modeling indoor applications such as Wi-Fi, while the short-term static channel model suits applications with higher mobility, such as outdoor cellular systems.

### Ii-B Multihop ARQ Protocols

Consider a family of multihop ARQ protocols, in which the following standard ARQ protocol is used over each hop. The receiver in each hop tries to decode the message after or during one round, depending on whether the synchronization is per-block based or per-channel-use based. Once it is able to decode the message, a one bit acknowledgement (ACK) is fed back to the transmitter that triggers the transmission of the next message. After one ARQ round, if the receiver cannot decode the message, a negative acknowledgement (NACK) is fed back to the transmitter. Then the transmitter sends the next block of the code that carries additional information for the same message. The retransmission over the th hop continues for a maximum number of rounds, called the ARQ window size. Once the ARQ window size is reached without successful decoding of the message, the message is discarded, causing an information outage. Then the next message is transmitted. The sum of the ARQ window sizes is upper bounded by , where

 N−1∑i=1Li≤L. (2)

We consider several ARQ protocols with different ways to allocate the available ARQ windows among different hops:

• A fixed ARQ protocol, which allocates a fixed ARQ window size of for the transmitter of node , such that .

• An adaptive ARQ protocol, in which the allocation of the ARQ window size per hop is not fixed but adapted to the channel state. The transmitter of a node can keep retransmitting as long as the total ARQ window size of has not been reached. We further consider two types of adaptive ARQs based on different synchronization levels:

• Fixed-Block-Length (FBL) ARQ protocol: The synchronization is per-block based. The transmission of a message over each hop spans an integer number of ARQ rounds.

• Variable-Block-Length (VBL) ARQ protocol: The synchronization is per-channel-use based. The receiver can send an ACK as soon as it can decode the message, and the transmitter starts transmitting a new message without waiting until the beginning of the next channel block. VBL has a finer time resolution than FBL and is more efficient in using the available channel block, at a cost of higher synchronization complexity.

We assume that the ARQ feedback links has zero-delay and no error.

## Iii Asymptotic DMDT

We characterize the tradeoff among the data rate (measured by the multiplexing gain ), the reliability (measured by the diversity gain ), and the delay by the asymptotic DMDT of a system with ARQ. Following the framework of [4] and [12], we assume that the rate of transmission depends on the operating SNR , and consider a family of space time codes with block rate scaling with the logarithm of SNR as

 R(ρ)=rlogρ. (3)

### Iii-a Diversity Gain

In the high SNR regime, the diversity gain is defined as the SNR exponent of the message error probability [4]. It is shown in [4] that the message error probability is dominated by the information outage probability whenÊ the block-length is sufficiently large. In the following we make this assumption. The information outage event is the event that the accumulated mutual information at the receiver within the allowed ARQ window size does not meet the data rate of the message and, therefore the receiver cannot decode the message. Hence, the diversity gain for a family of codes is defined as:

 d(r)≜−limρ→∞logPe(ρ)logρ. (4)

The DMT of an MIMO system is denoted by and defined as the supremum of the diversity gain over all families of codes. DMT of a point-to-point MIMO system is characterized in [4] by the following theorem:

###### Theorem 1.

For a sufficiently long block-length, theÊ DMT is given by the piece-wise linear function connecting the points for .

### Iii-B Asymptotic DMDT

To characterize the asymptotic DMDT for a multihop network in the high SNR regime, we need the following quantity. Assume that the channel inputs at both the source and the relays are Gaussian with identity covariance matrices. Define , for . For the long-term static channel, let be the nonzero eigenvalues of , for . Suppose , for , . At high SNR, we can approximate the channel capacities as 111Here the exponential equality is defined as , if . The exponential inequalities and are defined similarly. , where

 Si(αi)≜M∗i∑j=1(1−αi,j)+, (5)

, and the vector This plays an important role in the asymptotic DMDT analysis. The closer the SNR exponents ’s are to unity, the closer the channel matrix is to being singular. Similarly, we can define in the short-term static channel model and the corresponding matrix as .

Proofs for the asymptotic DMDT analysis rely on the notion of decoding time, which is the time at which the accumulated information reaches . In the case of the short-term static channel, for the FBL ARQ and other block-based ARQ protocols, the decoding time for the th node is given by

 ti≜inf{t∈Z+:t∑l=1Ci(Hi,l)≥rlogρ}≐inf{t∈Z+:t∑l=1Si(αli)≥r}, (6)

where denotes the set of positive integers. For the VBL ARQ and other non-block-based ARQ protocols, the decoding time is given by

 ti≜inf⎧⎨⎩t∈R:⌊t⌋∑l=1Si(αli)+(t−⌊t⌋)Si(α⌊t⌋+1i)≥r⎫⎬⎭, (7)

where denotes the largest integer smaller than . Similarly we can define the decoding time for the long-term static channel model. We can view the accumulated mutual information as a random walk with random increments and stopping boundary .

In the following, we first state our results for the three-node network , and then extend them to the general -node network.

#### Iii-B1 Long-Term Static Channel

The DMDT of the fixed ARQ protocol in the case of the long-term static channel is given by the following theorem:

###### Theorem 2.

With the long-term static channel assumption, the DMDT of the fixed ARQ protocol for a three-node MIMO multihop network with window sizes and , , , is given by:

 d(M1,M2,M3)F(r,L1,L2|L1+L2≤L)=mini=1,2{d(Mi,Mi+1)(rLi)}. (8)

Proof: See Appendix A.

Consistent with our intuition, (8) shows that the performance of a three-node network is limited by the weakest link. This implies that if there were no constraint for the ’s to be integers, the optimal choice should equalize the diversity-multiplexing tradeoff of all the links, i.e.,

 d(M1,M2)(rL1)=d(M2,M3)(rL2). (9)

With the integer constraint we choose the integer ’s such that the minimum of for is maximized.

The DMDT of the FBL ARQ protocol is a piece-wise linear function characterized by the following theorem:

###### Theorem 3.

With the long-term static channel assumption, the DMDT of the FBL ARQ protocol for a three-node MIMO multihop network is given by

 d(M1,M2,M3)FBL(r,L)=minli∈Z+:l1+l2=L−1{d(M1,M2)(rl1)+d(M2,M3)(rl2)}. (10)

Proof: See Appendix B.

The DMDT of the VBL ARQ protocol cannot always be expressed in closed-form, but can be written as the solution of an optimization problem, as stated in the following theorem.

###### Theorem 4.

With the long-term static channel assumption, the DMDT of the VBL ARQ protocol for a three-node MIMO multihop network is given by

 d(M1,M2,M3)VBL(r,L)=inf(α1,α2)∈Oh({αi,j}), (11)

where . The set is defined as

 (12)

and this is the optimal DMDT for a three-node network in the long-term static channel.

Proof: See Appendix C.

Note that the DMDT of the VBL ARQ protocol in the three-node network, under the long-term static channel assumption, is similar to the DMT of DDF without ARQ given in [8], with proper scaling of the multiplexing gain. The optimization problem in (11) can be shown to be convex using techniques similar to Theorem 3 in [8].

We have closed-form solutions for some specific cases where the optimization problem in (11) has a simple form and can be solved analytically. For example, for a network (11) becomes

 d(M1,1,M3)VBL(r,L)=infα1,1,α2,1 M1α1,1+M3α2,1 subject to (1−α1,1)+(1−α2,1)+(1−α1,1)++(1−α2,1)+

The DMDT for this case (and two other special cases) is given by the following corollary:

###### Corollary.

With the long-term static channel assumption the DMDT of the VBL ARQ protocol

• for a MIMO multihop network is given by

 d(M1,1,M3)VBL(r,L)=⎧⎨⎩min{M1,M3}1−2r/L1−r/L,0≤r≤L/2;0,otherwise. (14)
• for a MIMO multihop network is given by

 d(1,M,1)VBL(r,L)=⎧⎨⎩M1−2r/L1−r/L,0≤r≤L/2;0,otherwise. (15)
• for a (2, 2, 2) MIMO multihop network is given by

 d(2,2,2)VBL(r,L)=⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩2(4−5r/L)2−r/L,0≤r≤L/2;3−4r/L1−r/L,L/2≤r≤2L/3;4(1−r/L)2−r/L,2L/3≤r≤L;0,otherwise.

#### Iii-B2 Short-Term Static Channel

The DMDT of the fixed and the FBL ARQ under the short-term static channel assumption are similar to those under the long-term static channel assumption, with additional scaling factors for DMDTs of each hop due to the time diversity gain.

###### Theorem 5.

With the short-term static channel assumption, the DMDT of the FBL ARQ protocol for a three-node MIMO multihop network is given by

 d(M1,M2,M3)FBL(r,L)=minli∈Z+:l1+l2=L−1{2∑i=1lid(Mi,Mi+1)(rli)}.

Proof: See Appendix D.

###### Theorem 6.

With the short-term static channel assumption, the DMDT of the VBL ARQ protocol for a three-node MIMO multihop network is given by

 d(M1,M2,M3)VBL(r,L)=inf(α1,α2)∈G~h({αli,j}), (16)

where , and the set is defined as

 G ≜ {(α1,α2)∈RM∗1×L×RM∗2×L: (17) αli,1≥⋯≥αli,M∗i≥0,∀i,l,t1+t2>L}.

The ’s, defined in (7), depend on the ’s. This is the optimal DMDT for a three-node MIMO multihop network in the short-term static channel.

Proof: See Appendix E.

### Iii-C DMDT of an N-Node Network and Optimality of VBL

Next, we extend our DMDT results to general -node MIMO multihop networks. Note that in our model, since each transmitted signal is received only by the next node in the network, the transmission over the th hop does not interfere with other transmissions. We will show the DMDT of this more general network is a minimization of the DMDTs of all its three-node sub-networks, due to half-duplexing and multihop diversity. The multihop diversity [8] captures the fact that we allow simultaneous transmissions of multiple node pairs in half-duplex relay networks. For example, while node is transmitting to node , node can also transmit to node . This effect allows us to split a message into pieces, which are transmitted simultaneously in the network to increase the multiplexing gain. Using this rate-splitting scheme, we can prove the DMDT optimality of the VBL ARQ protocol. Due to their fixed block length, we are only able to provide upper and lower bounds for DMDTs of fixed ARQ and FBL ARQ in an -node network.

###### Theorem 7.

With the long-term or short-term static channel assumption, the DMDT of the VBL ARQ for an -node MIMO multihop network is given by

 d(M1,⋯,MN)VBL(r,L)=mini=1,⋯,N−2d(Mi,Mi+1,Mi+2)VBL(r,L), (18)

and this is the optimal DMDT for an -node network.

###### Proof.

See Appendix F. ∎

Theorem 7 says that the DMDT of an -node system is determined by the smallest DMDT of its three-node sub-networks. The minimization in Theorem 7 is over all possible three-node sub-networks instead of pairs of nodes, due to the half-duplex constraint: each low-rate piece of message has to wait for the previous piece to go through two hops before it can be transmitted. Theorem 7 also says that the VBL ARQ is the optimal ARQ protocol in the general multi-hop network.

###### Theorem 8.

With the long-term or short-term static channel assumption, the DMDT of fixed ARQ for an -node network is lower bounded and upper bounded, respectively, by

 (19)

and

 d(M1,⋯,MN)F(r,L1,⋯,LN−1)≤mini=1,⋯,N−2d(Mi,Mi+1,Mi+2)VBL(r,L), (20)

where .

###### Proof.

See Appendix G. ∎

###### Theorem 9.

With the long-term or short-term static channel assumption, the DMDT of the FBL ARQ for an -node network is lower bounded and upper bounded, respectively, by

 d(M1,⋯,MN)FBL(r,L)≥mini=1,⋯,N−2d(Mi,Mi+1,Mi+2)VBL(r,L−N), (21)

and

 d(M1,⋯,MN)FBL(r,L)≤mini=1,⋯,N−2d(Mi,Mi+1,Mi+2)VBL(r,L). (22)
###### Proof.

See Appendix H. ∎

An intuitive explanation for the DMDT optimality of the VBL ARQ is as follows. Recall that is the number of channel blocks, including retransmissions, needed to decode the message over the th hop. For a three-node network, we can illustrate the information outage region in the region of values as in Fig. 2. The outage region of the VBL ARQ is smaller than those of the fixed and the FBL ARQ. Due to its per-block based synchronization, the outage region boundary of the FBL ARQ is a piecewise approximation to that of the VBL ARQ. In the high SNR regime, we formalize the above intuition in the following corollary to Theorem 9.

###### Corollary.

With the long-term or short-term static channel assumption, for an -node MIMO multihop network, the DMDT of the FBL ARQ converges to that of the VBL ARQ when .

###### Proof.

Using (21) and (22), when ,

 mini=1,⋯,N−2d(Mi,Mi+1,Mi+2)VBL(r,L)≥d(M1,⋯,MN)FBL(r,L)≥mini=1,⋯,N−2d(Mi,Mi+1,Mi+2)VBL(r,L[1−N/L])L→∞−−−→mini=1,⋯,N−2d(Mi,Mi+1,Mi+2)VBL(r,L). (23)

### Iii-D Power Control Gain with Long Term Power Constraint

With the long-term power constraint and channel state information at the transmitter (CSIT), we can employ a power control strategy to further improve diversity. Let the SNR in the th round be , where is the average SNR, and is the function defining the power control strategy. In the high SNR regime, similar to (5) we can approximate channel capacities as , where . Hence, with power control, all the asymptotic DMDT results in the previous sections hold with replaced by .

### Iii-E Examples for Asymptotic DMDT

In this section we show some illustrative examples for the asymptotic DMDT. We first consider the long-term static channel model. For a three-node multihop network with , Fig. 3 shows the DMDT of the fixed ARQ with , of the per-hop-performance-equalizing and satisfying (9), as well as the DMDTs of the FBL and the VBL ARQs. Note that the DMDT of the VBL ARQ in Fig. 3 is the optimal DMDT for the (4, 1, 3) network. We also consider a (2, 2, 2) network, whose DMDTs are shown in Fig. 4.

Fig. 5 presents the three-dimensional DMDT surface of the VBL and the FBL ARQs, respectively, for the multihop network. Note that as increases, the diversity gain at a given increases for both the FBL and the VBL ARQ protocols. Also note that the DMDT surface of the FBL ARQ is piecewise and that of the VBL ARQ is smooth due to their different synchronization levels. Fig. 6 illustrates the cross sections of the surfaces in Fig. 5 at and , which demonstrates the convergence of the DMDTs proved in Theorem III-C.

Next we consider the short-term static channel model. The DMDT of the (4,1,3) multihop network using the FBL ARQ is shown in Fig. 7. Note that the asymptotic DMDT in the short-term static channel model is not necessarily a multiple of the corresponding DMDT in the long-term static channel model, which differs from the point-to-point MIMO channel [12], where the asymptotic DMDT in the short-term static channel model is a multiple of the corresponding DMDT in the long-term static channel model.

## Iv Finite SNR DMDT With Delay Constraint

Ê In analyzing the finite SNR DMDT, we add a practical end-to-end delay constraint: each message has to reach the destination before the deadline; otherwise it is discarded. We characterize the finite SNR DMDT by studying the probability of message error. With the delay constraint brought into the picture, the probability of message error has two components: the information outage probability and the deadline missing probability, which we analyze using the finite SNR DMT introduced in [5] and the queueing network analysis, respectively. In the finite SNR regime, the multiplexing gain is defined as:

 r≜Rlog2(1+Mrρ), (24)

where is the number of antennas at the receiver. In the following we only consider the long-term static channel model and the fixed ARQ protocol. We first introduce the queueing network model.

### Iv-a Queueing Network Model

The messages enter the network at the source node and exit from the destination node, forming an open queue. Messages arrive at the source node as a Poisson process with a mean message inter-arrival time of blocks. As in the previous sections, the unit of time is one block of the channel consisting of channel uses. The end-to-end delay constraint is blocks. Each node can be viewed as a service station transmitting (possibly with several retransmissions) a message to the next node. The time node spends to successfully transmit a message to node is called the service time of the th node, which depends on the channel state and is upper bounded by the ARQ window size . The allocated ARQ window sizes satisfy .

As an approximation, we assume that the random service time at node for each message is i.i.d. with an exponential distribution of mean (the actual service time has value distributed in the interval of ). Here , which we derive later, is the actual average service time of the ARQ process when the ARQ window size is . With these assumptions we can treat each node as an queue. This approximation makes the problem tractable and characterizes the qualitative behavior of MIMO multihop networks. The messages enter the buffer and are processed based on the first-come-first-served (FCFS) rule. We assume , , so the queues are stable, i.e., the waiting time at a node does not grow unbounded.

Burke’s theorem (see [18]) says that in an queue with Poisson arrivals, the messages leave the server as a Poisson process. Hence messages arrive at each relay (and the destination node) as a Poisson process with rate , where is the probability that a message is dropped. When the SNR is reasonably high, we can assume the message drop probability is small and hence .

### Iv-B Probability of Message Error

Denote the total queueing delay experienced by the th message transmitted from the source to the destination as , and the number of transmissions needed by node to transmit the th message to node as , if the transmitter can use any number of rounds. For the th message, if it is not discarded due to information outage, the total “service time” is and the random end-to-end delay is Recall for the fixed ARQ, the message is dropped once the number of retransmissions exceeds the ARQ window size of any hop, or the end-to-end delay exceeds the deadline. Hence, the message error probability of the th message can be written as

 Pe = P{∪N−1i=1{tin>Li}}+P{∩N−1i=1{tin≤Li}∩{Dn>k}}. (25)

The first term in (25) is the message outage probability:

 Pout({Li}|ρ)≜P{∪N−1i=1{tin>Li}}, (26)

which is identical for any message since channels are i.i.d. The second term in (25) is related to the deadline missing probability, and can be rewritten as

 P{∩N−1i=1{tin≤Li}∩{Dn>k}}=[1−Pout({Li}|ρ)]P{Dn>k∣∣∩N−1i=1{tin≤Li}}. (27)

Define the stationary deadline missing probability:

In the following we derive (26) and (28).

#### Iv-B1 Information Outage Probability

Since channels in different hops are independent, (26) becomes

 Pout({Li}|ρ)=N−1∑i=1P{tin>Li}=N−1∑i=1P{LiCi(Hi)

which is a sum of the per-hop outage probabilities . Using results in [5] for point-to-point MIMO, we have

 Pout,i(Li|ρ)=sup(b1,⋯bM∗i)∈BiM∗i∏l=1γ(Mi+1−Mi+2l−1,Miρ[(1+Mi+1ρ)bl−(1+Mi+1ρ)bl−1])Γ(Mi+1−Mi+2l−1), (30)

where the set is given by

 Bi=⎧⎪⎨⎪⎩(b1,⋯,bM∗i)∣∣ ∣∣bl−1

is the incomplete gamma function, and for a positive integer . For orthogonal space-time block coding (OSTBC), we can derive a closed-form using techniques similar to [5]:

 Pout,OSTBC({Li}|ρ)=N−1∑i=1P{rsLilog2(1+ρMi∥Hi∥2F)≤rlog2(1+Mi+1ρ)}, (32)

where denotes the Frobenius norm of a matrix , and the spatial code rate is equal to the average number of independent constellation symbols transmitted per channel use. For example, for the Alamouti space-time code [23]. When is Rayleigh distributed, its Frobenius norm has the Gamma(1, ) distribution. Hence, (32) becomes:

 Pout,OSTBC({Li}|ρ)=N−1∑i=11(Mi⋅Mi+1−1)!γ(Miρ[(1+Mi+1ρ)rrsLi−1],Mi⋅Mi+1). (33)

For a three-node network with half-duplex relay, there is only one queue at the source that incurs the queueing delay. For given and , we can derive the stationary deadline missing probability using a martingale argument:

###### Theorem 10.

For a half-duplex three-node MIMO multihop network with Poisson arrival of rate and ARQ rounds and , the probability that the end-to-end delay exceeds the deadline is given by

Proof: See Appendix I.

For general multihop networks with any number of half-duplex relays, the analysis for (28) is more involved. Due to half-duplexing the neighboring links cannot operate simultaneously, and this effect is not captured in the standard queueing network analysis. Here we adapt the proof in [22], which uses large deviation techniques, to derive the following theorem for the exponent of the deadline missing probability in half-duplex relay networks:

###### Theorem 11.

For a half-duplex -node MIMO multihop network, with Poisson arrival of rate and ARQ rounds ’s, the probability that the end-to-end delay exceeds the deadline is given by

 (35)

where , and , .

Proof: See Appendix J.

This theorem again demonstrates that the performance of the -node multihop network with a half-duplex relay (here the performance metric is in terms of the deadline-missing probability exponent) is determined by the smallest exponent of each three-node sub-network. By Theorem 11, for finite we can approximate (28) as

Also note that in the special case with nodes, for finite

which is identical with (34) up to a multiplicative constant . This constant is typically not identifiable by large deviation techniques such as the one used in Theorem 11.

#### Iv-B3 Mean Service Time Calculation

The above analysis requires , which we will derive in this section. For a given and message , the cumulative distribution function (CDF) of is given by Ê

 P{tin≤t}=P{tCi(Hi)≥rlog2(1+Mi+1ρ)}=1−Pout,i(t|ρ), (38)

where is given in (30) (or a term in the summation in (33) for OSTBC). Differentiating (38) gives the desired probability density function (PDF):

 P{tin=t}=Mirρrst2(Mi⋅Mi+1−1)!fMi⋅Mi+1−1ie−fi(1+Mi+1ρ)r/trslog2(1+Mi+1ρ), (39)

where . Using this we have

 μ(Li)=∫Lit=1tP{tin=t}dt+Li∫∞t=LiP{tin=t}dt. (40)

### Iv-C Optimal Fixed ARQ Design at Finite SNR

Based on the above analysis, we formulate the optimal fixed ARQ design in the finite SNR regime as an optimization problem that allocates the total ARQ window size among hops to minimize the probability of message error subject to the queue stability and the end-to-end delay constraint :

The terms in (41) are given by (29), (33), (34), and (36). This optimization problem can be solved numerically. In particular, for a three-hop network with OSTBC, (41) becomes

 min{Li} 2∑i=11(MiMi+1−1)!γ(Miρ[(1+ρ)rrsLi−1],MiMi+1)+μ(L1)+μ(L2)λe−k(1μ(L1)+μ(L2)−1λ) subject to 1≤μ(Li)≤λ,N−1∑i=1Li≤k. (42)

As we demonstrate in the following examples, the information outage probability is decreasing in , and the deadline missing probability is increasing in . Hence the optimal ARQ window size allocation at each node should trade off these two terms. Moreover, the optimal ARQ window size allocation should equalize the performance of each link.

### Iv-D Numerical Example

We first consider a point-to-point (2, 2) MIMO system at the source and the destination. Assume dB and blocks. An OSTBC with is used, for which the information outage probability is given in (33). The deadline constraint is blocks. For , the information outage probability (26) and the deadline missing probability (28) are shown in Fig. 8. Note that (26) decreases, while (28) increases with the ARQ window size.

Next we consider the (4, 1, 3) MIMO multihop network. Assume dB and blocks. An OSTBC with is used. The optimal fixed ARQ protocol is obtained by solving (42). For and a deadline constraint of blocks, the optimal fixed ARQ has and , and the optimal probability of message error is 0.1057. For and blocks, the optimal fixed ARQ has and , and the optimal probability of message error is 0.0355. For all and , the probability of message error is plotted as a surface in Fig. 9. This surface is the DMDT for the three-node network in the finite SNR regime, which has an interesting similarity to the high SNR asymptotic DMDT surface in Fig. 6, since indeed the high SNR DMDT represents the SNR exponent of the finite SNR DMDT.

## V Conclusions

We have analyzed the asymptotic diversity-multiplexing-delay tradeoff (DMDT) for the -node MIMO multihop relay network with ARQ, under both long-term and the short-term static channel assumptions. We show that the asymptotic DMDT can be cast into an optimization problem that can be solved numerically in general, and closed-form asymptotic DMDT expressions are obtained in some special cases. We also proposed the VBL ARQ protocol which adapts the ARQ window size among hops dynamically and proved that it achieves the optimal DMDT under both channel assumptions. We also show that the DMDT for general multihop networks with multiple half-duplex relays can be found by decomposing the network into three-node sub-networks such that each sub-network consists of three neighboring nodes and its corresponding two hops. The DMDT of the relay network is determined by the minimum of the DMDTs of the three-node sub-networks. We have also shown that the DMDT of the three-node subnetwork is determined by its weakest link. Hence, the optimal ARQ should equalize the link performances by properly allocating the per-hop ARQ window sizes dynamically.

We then studied the DMDT in the finite SNR regime for fixed ARQ protocols. We introduced an end-to-end delay constraint such that a message is dropped once its delay exceeds the delay constraint. Since in the finite SNR regime retransmission is not a rare event, we incorporated the queueing delay into the system model, and modeled the system as a queueing network. The finite SNR DMDT is characterized by the probability of message error, which consists of the information outage probability and the deadline missing probability. While the information outage probability can be found through finite SNR DMDT analysis, we have also found the exponent for the deadline missing probability. Our result demonstrates that the performance of a multihop network with half-duplex relays in the finite SNR regime is also determined by the performance of the weakest three-node sub-network. It has been shown that, based on these analyses, the optimal fixed ARQ window size allocation can be solved numerically as an optimization problem, which should balance the per-hop diversity performance and avoid a long per-hop delay.

The difficulty in merging the network layer analysis with the physical layer information theoretic results stems from the bursty nature of the source and the end-to-end delays. By modeling the multihop relay network with ARQ as a queuing network, we have tried to answer a question posed at the end of [13]: how to couple the fundamental performance limits of general multihop networks with queueing delay. Our work provides a step towards bridging the gap between network theory and information theory. Future work includes developing an optimal dynamic ARQ protocol that can adapt to the channel state and the message arrival rate. The problem can be formulated as a dynamic programming problem or analyzed using a heavy traffic approximation.

## Appendix A Proof of Theorem 2

With fixed-ARQ protocol and half-duplex relays, the system is in outage if any hop is in outage. The probability of message error , using the decoding time definition in (6), can be written as:

 Pe(ρ) ≐P{{t1>L1}∪{t2>L2}} (47) =P{t1>L1}+P{t2>L2} ≐P{L1S1(α1)

where (47) is due to the independence of each link, and (47) follows from the method used in [4], and the fact that

 P{ti>Li}=P{Li∑l=1Si(αli)

since and for the long-term static channel. The last equality follows since when SNR is high, the dominating term is the one with the smaller SNR exponent. Using (47) and the definition of diversity in (4) we obtain the DMDT stated in Theorem 2.

## Appendix B Proof of Theorem 3

For the FBL ARQ protocol with two hops, the probability of message error is given by

 Pe(ρ)≐P{t1+t2>L}=L−1∑k=1P{t1=k}P{t2>L−k}+P{t1≥L}. (49)

In the long-term static channel model we have

 (50)

which follows from the fact that is monotone decreasing, i.e., . If we plug (50) into (49) we get

 Pe(ρ) ≐P{C1(H1)≥rlogρ}P{C2(H2)≤rlogρL−1} (52) +L−1∑k=2P{(L−k)C2(H2)

where we have used the fact that . From the definition of diversity in (4) the DMDT in Theorem 3 follows.