## 1 Introduction  
Machine type communication (MTC), also known as machine to machine (M2M) communication, is envisioned as a main enabler for the Internet of Things (IoT) [1; 2]. Featuring no (or little) human intervention, M2M applications have been experiencing rapid growth in various domains, such as intelligent transport systems, smart cities, e-Health, smart grids, industry automation, monitoring and control systems, etc. [3; 4]. It is expected that there will be more than 50 billion M2M devices at 2020 [5]. Unlike traditional human to human (H2H) communications, M2M traffic is characterized by massive devices, low mobility, small data transmissions and large uplink-to-downlink traffic ratio [6]. Therefore, the long term evolved (LTE) and LTE-Advanced (LTE-A) networks have to be enhanced to accommodate such differentiated M2M services and the coexisting H2H traffic [7; 8; 9]. A simplified model of the random access procedure in LTE/LTE-A networks is depicted in Figure 1a, which shows that four handshake messages are needed before a successful data transmission [10]. If two or more devices select the same preamble in Msg1, they will receive the same random access response (RAR) message from the base station (BS) and then send Msg3 on the same granted uplink channel. However, the BS cannot decode the superposed Msg3 accurately. Hence the colliding devices fail in this random access attempt and would send retransmission requests on the next random access channel (RACH). As a result, the main concern of most works is the RACH congestion problem caused by simultaneous random access requests from massive devices [11; 12; 13; 14; 15; 16]. The existing overload control mechanisms can be classified as push-based schemes (such as access class barring (ACB), dynamic RACH resources allocation, backoff, and slotted access), pull-based scheme (group paging) and a combination of these methods. The packet sizes of many M2M applications are usually small, some of them even on the order of bits or bytes (e.g., health or environment monitors and alarm devices) [17]. As to the random access procedure, shown in Figure 1b, just Msg1 occupies six resource blocks (RBs, a time-frequency resource unit which consists of 180 KHz in frequency domain and a time slot, i.e., 0.5 ms in time domain) in frequency and 1 ms in time (that is 12 RBs in total). Not to mention Msg3 and the retransmissions caused by collisions which can result in much more resource consumption. Thus the ratio of signaling overhead to effective payload may be unacceptable in practice [18]. However, a comprehensive investigation of both RACH congestion and the following data transmission is still lacking attention. A few works have been aware of this problem [13; 14; 19; 20; 21]. The authors in [19] demonstrated that the supportable arrival rate of one-stage protocol, like slotted-ALOHA (S-ALOHA), is higher than that of two-stage protocol, like time division multiple access (TDMA), if the payload size is small. In [20], the LTE/LTE-A random access procedure was tailored exclusively for M2M traffic to reduce the signaling overhead. Along with the surveys of various MAC protocols for M2M communications in [13; 14], we present a novel hybrid S-ALOHA and TDMA MAC protocol (HSTMAC) for M2M and H2H coexisting scenario in [21]. This paper differs from [21] in the following aspects: (i) we assume that H2H traffic follows a Poisson distribution while M2M traffic follows a Beta distribution, which represents the worst case for random access procedure; (ii) we rigorously derive the system equilibrium under any retransmission limit (\(Q\geq 1\)) instead of large value assumption (\(Q\geq 9\)) in [21]; (iii) based on the equilibrium analysis, a dynamic pre-backoff (DPBO) algorithm is designed for load balance of M2M traffic and quality of service (QoS) guarantee of H2H traffic.