We consider the average age of information in G/G/1/1 systems under two service discipline models. In the first model, if a new update arrives when the service is busy, it is blocked; in the second model, a new update preempts the current update in service. For the blocking model, we first derive an exact age expression for G/G/1/1 systems. Then, using the age expression for G/G/1/1 systems, we calculate average age expressions for special cases, i.e., M/G/1/1 and G/M/1/1 systems. We observe that deterministic interarrivals minimize the average age of G/M/1/1 systems for a given mean interarrival time. Next, for the preemption in service model, we first derive an exact average age expression for G/G/1/1 systems. Then, similar to blocking discipline, using the age expression for G/G/1/1 systems, we calculate average age expressions for special cases, i.e., M/G/1/1 and G/M/1/1 systems. Average age for G/M/1/1 can be written as a summation of two terms, the first of which depends only on the first and second moments of interarrival times and the second of which depends only on the service rate. In other words, interarrival and service times are decoupled. We prove that deterministic interarrivals are optimum for G/M/1/1 systems for a given mean interarrival time. On the other hand, we observe for non-exponential service times that the optimal distribution of interarrival times depends on the relative values of the mean interarrival time and the mean service time. Finally, we propose a simple to calculate upper bound to the average age for the preemption in service discipline.

J-8

Scaling Laws for Age of Information in Wireless Networks

We study age of information in a multiple source-multiple destination setting with a focus on its scaling in large wireless networks. There are \(n\) nodes uniformly and independently distributed on a fixed area that are randomly paired with each other to form \(n\) source-destination (S-D) pairs. Each source node wants to keep its destination node as up-to-date as possible. To accommodate successful communication between all \(n\) S-D pairs, we first propose a three-phase transmission scheme which utilizes local cooperation between the nodes along with what we call mega update packets to serve multiple S-D pairs at once. We show that under the proposed scheme average age of an S-D pair scales as \( O(n^{\frac{1}{4}} \log n) \) as the number of users, \(n\), in the network grows. Next, we observe that communications that take place in Phases I and III of the proposed scheme are scaled-down versions of network-level communications. With this along with scale-invariance of the system, we introduce hierarchy to improve this scaling result and show that when hierarchical cooperation between users is utilized, an average age scaling of \( O(n^{\alpha(h)} \log n) \) per-user is achievable, where \(h\) denotes the number of hierarchy levels and \( \alpha(h) = \frac{1}{3·2^h + 1} \). We note that \( \alpha (h) \) tends to \(0\) as \(h\) increases, and asymptotically, the average age scaling of the proposed hierarchical scheme is \( O(\log n) \). To the best of our knowledge, this is the best average age scaling result in a status update system with multiple S-D pairs.

We consider the age of information in a multihop multicast network where there is a single source node sending timesensitive updates to \(n^L\) end nodes, and \(L\) denotes the number of hops. In the first hop, the source node sends updates to n first-hop receiver nodes, and in the second hop each first-hop receiver node relays the update packets that it has received to \(n\) further users that are connected to it. This network architecture continues in further hops such that each receiver node in hop \(\ell\) is connected to \(n\) further receiver nodes in hop \(\ell + 1\). We study the age of information experienced by the end nodes, and in particular, its scaling as a function of \(n\). We show that, using an earliest \(k\) transmission scheme in each hop, the age of information at the end nodes can be made a constant independent of \(n\). In particular, the source node transmits each update packet to the earliest \(k_1\) of the \(n\) first-hop nodes, and each first-hop node that receives the update relays it to the earliest \(k_2\) out of \(n\) second-hop nodes that are connected to it and so on. We determine the optimum \(k_1\) stopping value for each hop \(\ell\) for arbitrary shifted exponential link delays.

J-6

Nonoverlay Heterogeneous Network Planning for Energy Efficiency

Mahmut Demirtas, and Alkan Soysal

Wireless Communications & Mobile Computing, Jan 2017

In this paper, we introduce nonoverlay microcell/macrocell planning that is optimally designed for improving energy efficiency of the overall heterogeneous cellular network. We consider two deployment strategies. The first one is based on a fixed hexagonal grid and the second one is based on a stochastic geometry. In both of our models, microcells are placed in those areas where the received signal power levels of macrocell common pilot channels are below a certain threshold. Thus, interference between microcells and macrocells is minimized. As a result, addition of microcells increases the achieved number of bits per unit energy. Under such deployment assumptions, we investigate the effects of certain parameters on the energy efficiency. These parameters include the user traffic, the Intersite Distance (ISD), the size of microcells and the number of microcells per macrocell for the grid model, and macrocell density and microcell density for the stochastic model. The results of our performance analyses show that utilizing microcells in a sparse user scenario is worse for the energy efficiency whereas it significantly improves both energy and spectral efficiencies in a dense user scenario. Another interesting observation is that it is possible to choose an optimum number of microcells for a given macrocell density.

J-5

Capacity Bounds on MIMO Relay Channel With Covariance Feedback at the Transmitters

Bengi Aygun, and Alkan Soysal

IEEE Transactions on Vehicular Technology, Jun 2013

In this paper, the source and relay transmit covariance matrices are jointly optimized for a fading multiple-antenna relay channel when the transmitters only have partial channel state information (CSI) in the form of covariance feedback. For both full-duplex and half-duplex transmissions, we evaluate lower and upper bounds on the ergodic channel capacity. These bounds require joint optimization over the source and relay transmit covariance matrices. The methods utilized in the previous literature cannot handle this joint optimization over the transmit covariance matrices for the system model considered in this paper. Therefore, we utilize matrix differential calculus and propose iterative algorithms that find the transmit covariance matrices to solve the joint optimization problem. In this method, there is no need to specify first the eigenvectors of the transmit covariance matrices. The algorithm updates both the eigenvectors and the eigenvalues at each iteration. Through simulations, we observe that lower and upper bounds are close to each other. However, the distance between the lower and upper bounds depends on the channel conditions. If the mutual information on the source-to-relay channel and the broadcast channel get closer to each other, the bounds on capacity also get closer.

J-4

Joint Channel Estimation and Resource Allocation for MIMO systems-Part II: Multi-User and Numerical Analysis

This is the second part of a two-part paper on the joint channel estimation and resource allocation problem in MIMO systems with noisy channel estimation at the receiver side and partial CSI, in the form of covariance feedback, available at the transmitter side. We consider transmit-side correlated MIMO channels with block fading, where each block is divided into training and data transmission phases. In this paper, we extend the single-user results of Part I to the multiple access channel. For the data transmission phase, we propose an iterative algorithm to solve for the optimum system resources such as time, power and spatial dimensions. This algorithm updates the parameters of the users in a round-robin fashion. In particular, the algorithm updates the training and data transmission parameters of a user, when those of the rest of the users are fixed, in a way to maximize the achievable sum-rate in a multiple access channel; and iterates over users in a round-robin fashion. Finally, we provide a detailed numerical analysis to support the analytical results of both parts of this two-part paper.

J-3

Joint Channel Estimation and Resource Allocation for MIMO Systems-Part I: Single-User Analysis

Multiple antenna systems are known to provide very large data rates, when the perfect channel state information (CSI) is available at the receiver. However, this requires the receiver to perform a noise-free, multi-dimensional channel estimation, without using communication resources. In practice, any channel estimation is noisy and uses system resources. We shall examine the trade-off between improving channel estimation and increasing the achievable data rate. We consider transmit side correlated multi-input multi-output (MIMO) channels with block fading, where each block is divided into training and data transmission phases. The receiver has a noisy CSI that it obtains through a channel estimation process, while the transmitter has partial CSI in the form of covariance feedback. In Part I of this two-part paper, we consider the single-user case, and optimize the achievable rate jointly over parameters associated with the training phase and data transmission phase. In particular, we first choose the training signal to minimize the channel estimation error, and then, develop an iterative algorithm to solve for the optimum system resources such as time, power and spatial dimensions. Specifically, the algorithm finds the optimum training duration, the optimum allocation of power between training and data transmission phases, the optimum allocation of power over the antennas during the data transmission phase.

J-2

Optimality of Beamforming in Fading MIMO Multiple Access Channels

We consider the sum capacity of a multi-input multi-output (MIMO) multiple access channel (MAC) where the receiver has the perfect channel state information (CSI), while the transmitters have either no or partial CSI. When the transmitters have partial CSI, it is in the form of either the covariance matrix of the channel or the mean matrix of the channel. For the covariance feedback case, we mainly consider physical models that result in single-sided correlation structures. For the mean feedback case, we consider physical models that result in in-phase received signals. Under these assumptions, we analyze the MIMO-MAC from three different viewpoints. First, we consider a finite-sized system. We show that the optimum transmit directions of each user are the eigenvectors of its own channel covariance and mean feedback matrices, in the covariance and mean feedback models, respectively. Also, we find the conditions under which beamforming is optimal for all users. Second, in the covariance feedback case, we prove that the region where beamforming is optimal for all users gets larger with the addition of new users into the system. In the mean feedback case, we show through simulations that this is not necessarily true. Third, we consider the asymptotic case where the number of users is large. We show that in both no and partial CSI cases, beamforming is asymptotically optimal. In particular, in the case of no CSI, we show that a simple form of beamforming, which may be characterized as an arbitrary antenna selection scheme, achieves the sum capacity. In the case of partial CSI, we show that beamforming in the direction of the strongest eigenvector of the channel feedback matrix achieves the sum capacity. Finally, we generalize our covariance feedback results to double-sided correlation structures in the Appendix.

J-1

Optimum Power Allocation for Single-User MIMO and Multi-User MIMO-MAC with Partial CSI

We consider both the single-user and the multi-user power allocation problems in MIMO systems, where the receiver side has the perfect channel state information (CSI), and the transmitter side has partial CSI, which is in the form of covariance feedback. In a single-user MIMO system, we consider an iterative algorithm that solves for the eigenvalues of the optimum transmit covariance matrix that maximizes the rate. The algorithm is based on enforcing the Karush-Kuhn-Tucker (KKT) optimality conditions of the optimization problem at each iteration. We prove that this algorithm converges to the unique global optimum power allocation when initiated at an arbitrary point. We, then, consider the multi-user generalization of the problem, which is to find the eigenvalues of the optimum transmit covariance matrices of all users that maximize the sum rate of the MIMO multiple access channel (MIMO-MAC). For this problem, we propose an algorithm that finds the unique optimum power allocation policies of all users. At a given iteration, the multi-user algorithm updates the power allocation of one user, given the power allocations of the rest of the users, and iterates over all users in a round-robin fashion. Finally, we make several suggestions that significantly improve the convergence rate of the proposed algorithms.