## 1 Introduction  
Among the possible medium access control (MAC) techniques for wireless communication systems, the simplicity of random access schemes (_i.e.,_ ALOHA, carrier sense multiple access (CSMA)) make them suited to be implemented in several standards for short range applications [1,2]. Even though mitigation methods can be introduced in random access MAC (_i.e.,_ carrier sense multiple access with collision avoidance (CSMA-CA)), collisions are not completely avoidable. Nevertheless, some receivers have the ability to correctly receive a signal despite a significant level of co-channel interference, and collisions do not always lead to a total loss of the collided packets. This co-channel interference tolerance is called capture effect [3]. In the presence of concurrent transmissions at the same carrier frequency (collisions), packet capture may happen even for low values of the signal-to-interference ratio (SIR). The first papers in the literature about the capture effect mostly consider Frequency Modulation (FM) demodulators [3, 4, 5]. Later, the capture effect has been also studied in a variety of transceivers and MAC schemes, including ALOHA networks [6, 7, 8], IEEE 802.11 devices [9, 10, 11], Bluetooth radios [12] and cellular systems [13]. Focusing on IEEE802.15.4, several papers describing experimental results can be found in the literature [14, 15, 16, 17, 18, 19, 20, 21, 22]. To give some examples, in [14], the packet capture probability of a Chipcom CC1000 transceiver [15] is measured with the aim of exploiting the capture effect for collision detection and recovery. Another study [16], carried out again with CC1000 transceivers, which work in the sub-1 GHz band, obtained a threshold for the capture probability for the case of one interferer, but unstable results for the multiple interferers. This early work seems to suggest that the number of interferers might have an important effect on the capture probability with CC1000 devices. However, in contrast to the previous CC1000 measurements, successive studies, carried out with Chipcom CC2420 transceivers [17], which operate in the 2.4 GHz industrial, scientific and medical (ISM) band, show that the capture effect depends on the total interfering power, but it is independent of the number of interferers [18, 21]; therefore, in our mathematical analysis, we assume that the performance is independent of the number of interferers for the 2.4 GHz physical layer (PHY) of IEEE 802.15.4, and as a result, we use only one interfering signal for the mathematical evaluation. A behavior on the packet capture similar to that reported in [16, 18] is also observed in [20] with Freescale MC1224 transceivers [22], which, again, operate in 2.4 GHz. The experiments conducted in interferer power-dominant (with respect to the noise) environments in [16, 18, 19, 20] show a couple of common behaviors: (i) the receiver starts capturing the useful packet when signal-to-interference-plus-noise ratio (SINR) goes beyond 0 dB; (ii) packet reception rate (PRR) reaches one for values of SINR larger than 4 dB (please see in [16], Figure 5 and Figure 16, for CC1000 and CC2420, respectively; in [18], Figure 4 for CC2420; in [19], Figure 7c for CC2420; and in [20], Figure 3 for MC13192). Although the experimental results in [16, 18, 19, 20] agree with each other, there is no theoretical model, purely based on mathematical analysis, which can be applied to the 2.4 GHz PHY of IEEE 802.15.4 to explain such characteristics. Motivated by this consideration, we propose an analytical framework to investigate the behavior of the IEEE 802.15.4 2.4 GHz PHY layer. A review of low-rate wireless personal area network (LR-WPAN) solutions, including IEEE 802.15.4, can be found in [23]. The impact of interference in wireless sensor networks plays a very important role and can severely degrade the overall performance of the network and the efficiency of the upper layers. In our opinion, this aspect has not been sufficiently addressed in past years.