## 2 System Model  
Stochastic geometry is a powerful tool to model and analyze wireless networks assuming that the locations of nodes or the network structure are random in nature because of their unpredictable spatial characteristics [32]. Poisson point process (PPP), where the number of points (nodes) inside any compact set is a Poisson random variable and the points are uniformly distributed in the compact set, is the most widely used spatial model for networks with uniform node density such as ad hoc networks [33; 34; 35; 36; 37] and cellular networks [38]. For this reason, we also assume PPP for the spatial distribution of MTC devices. To be specific, we consider a multi-hop M2M network using mm-wave as shown in Figure 1, where nodes (devices), which are indicated by the black dots, are uniformly scattered according to a two-dimensional of intensity \(\lambda\). Let \(N\) be the number of nodes that exist in an area \(\mathcal{A}\). The probability mass function (PMF) of \(N\) follows the Poisson distribution as  
\[\Pr[N=n]=e^{-(\lambda\,\mathcal{A})}\frac{(\lambda\,\mathcal{A})^{n}}{n!}, \tag{1}\]  
where \(\Pr[\cdot]\) denotes the probability of a certain event. For data communications, we assume that all the nodes have the same maximum transmission range of \(R\), which is a function of transmit power, antenna gain, and path loss. In addition, as in [39; 31], for the analytical tractability, the antenna pattern  
Figure 1: System model. of the devices is approximated by a sectored model, which has the same gain within its beamwidth \(\Phi\) (i.e., \(|\phi|\leq\frac{\Phi}{2}\)) and zero gain outside as indicated by the gray area in Figure 1b. Each device is equipped with an electronically steerable directional antenna. Before the route construction, a node operates with \(\Phi=2\pi\) to receive short control packets for route construction and periodic beacon, by which each node knows the locations of their neighbors. For example, in the Ad hoc On-Demand Distance Vector (AODV) routing protocol [40], a periodic beacon called Hello message is used to detect each other in their neighborhoods. In general, such control packets are designed to be more easily decoded compared to data packets by using different code rates, modulation techniques, transmit power levels, and packet sizes. Thus, even without beamforming, the transmission range in the neighbor discovery phase can have the same or even extended transmission range compared to the data transmission phase [41]. Similar to [13], in the routing schemes that we consider, a node transmits a packet to one of its blockage-free neighbors that lie within a sector \(r\leq R\) and \(|\phi|\leq\frac{\Phi}{2}\), where \(\phi\) is with respect to the transmitting node-destination axis as in Figure 1b, assuming all the nodes know the location of the destination, which may be conveyed in a route request packet from the source (e.g., a route request (RREQ) from the source in AODV [40]). In other words, for route construction, the source steers its beam towards the destination and selects one node as a relay following certain criteria, which will be discussed in Section 4. Then, the selected relay also steers its beam towards the destination and recruits its next-hop node within a sector \(\Phi\) toward the destination in the same manner. Thus, the route built in this way is a zig-zag path, the efficiency of which will be further discussed in Section 5. For the data transmission after the route construction, the nodes in the route steer the beam to their previous-hop and next-hop nodes to receive and forward data packets, respectively. As in AODV [40], we assume that there exists only a single pair of transmitter and receiver in each hop. In other words, the source or a relay can only transmit or forward the data packet to a single node (either the next-hop relay or the destination).