## 1 Introduction  
With the advent of the Internet of Things (IoT), which shifts the paradigm of the Internet from human interconnection into a network of devices, it is predicted that almost 50 billion devices will be connected by 2020 [1]. As a key technology to realize the IoT ecosystem, machine-to-machine (M2M) communication enables wireless devices to constantly interact with each other as well as with their environments without direct human intervention [2; 3; 4; 5; 6; 7; 8]. Furthermore, to enhance network capacity, a fifth generation (5G) cellular system is envisioned to have significantly greater spectrum allocations at millimeter-wave (mmWave) frequency bands. For this reason, in this paper, we investigate multi-hop wireless networks for M2M communications using mmWave. M2M communication is desirable to interact with a large number of remote devices acting as the interface with end customers, utilities, etc. [5]. For example, various machine-type communication (MTC) devices such as smart meters, signboards, cameras, remote sensors, laptops, and appliances can be interconnected to support wide range of applications. Considering the limited capability of energy harvesting technologies as noted in [9], the subsequent wireless communication techniques must be short-range with low data rates, which makes multi-hop ad hoc networking absolutely necessary in realistic deployment of a large volume of MTC devices [8]. In this paper, we consider M2M communications with low-cost devices such as sensors and low-power mobile machines for applications such as healthcare, energy management, and entertainment [10]. As a related topic, device-to-device (D2D) communication is an emerging technology in cellular networks, where the devices in proximity can directly communicate with each other. From an architectural viewpoint, D2D communication may look similar to multi-hop M2M communication such as mobile ad hoc networks [11]. However, D2D communication is mainly for single hop communications with the involvement of the cellular network in the control plane, and it generally does not inherit multi-hop routing issue in multi-hop wireless networks (MWNs) [12]. We note that such types of D2D communication (single or small number of hops) are out of the scope of this paper. Instead, in this work, we consider multi-hop M2M communications using mmWave. A fundamental question in MWNs is whether it is advantageous to route over many short hops (short-hop routing) or over a smaller number of longer hops (long-hop routing), as highlighted in [13; 14]. In general, the long-hop routing is preferred for delay-sensitive applications, while the short-hop routing is desirable to reduce transmit power consumption [15; 16]. For this reason, the hop distance statistics of different routing strategies are extensively studied [13; 14; 15; 16; 17; 18] in microwave systems. For example, in [13], the hop distance statistics are used to derive the transmit energy consumption assuming Rayleigh fading channel. As in the microwave systems, the routing strategy associated with the hop distance characteristics is crucial in the future generation ad hoc networks using mmWave because the M2M networks with battery-powered devices inherently entail the limited transmission range, which requires multi-hop transmissions [19; 20]. To cope with dramatic increase in mobile traffic and extreme device density, the future fifth generation (5G) cellular networks are expected to have mmWave carrier frequency with massive bandwidths and unprecedented number of antennas [21]. Enabled by the availability of a wide spectrum and recent advances in radio frequency integrated circuit (RFIC) technologies, mmWave communication is a key technology to support ever-increasing capacity demand [22; 23; 19; 24]. At mmWave frequency, highly directional transmission using antenna arrays is an effective technique to overcome its heavier path-loss compared to the microwave systems [25; 26; 27; 28].