2876 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 63, NO. 6, JULY 2014 Modeling Query-Based Wireless CSMA Networks Through Stochastic Geometry Flavia Martelli, Member, IEEE, Chiara Buratti, Member, IEEE, and Roberto Verdone, Member, IEEE Abstract—A mathematical model for characterizing the per- formance of wireless access networks based on a carrier-sense multiple-access (CSMA) mechanism is derived. The model is de- veloped through a statistical description of the interference power coming from stochastic geometry and by taking into account the hidden node problem. As an example of a specific application of the model, the IEEE 802.15.4 standard is considered. Nodes are assumed to be uniformly distributed according to a Poisson point process (PPP) in a 2-D infinite plane. They have to communicate with the network coordinator through direct links (star topology), following a query-based application. The total interference power is modeled considering the actual density of nodes that are simul- taneously transmitting. This takes into account the operation of the CSMA with collision avoidance (CSMA/CA) algorithm used by nodes to access the channel, according to the IEEE 802.15.4 standard. Results show the impact of node density, packet length, and sensing range on the system performance, which is evaluated in terms of success probability in transmitting packets from nodes to the coordinator. Index Terms—Access protocols, carrier sense multiple access, IEEE 802.15.4, stochastic geometry, wireless sensor networks. I. I NTRODUCTION T HE reference scenario of this paper is a 2-D area where nodes are uniformly distributed with spatial density ρ [m −2 ]. They have to communicate with a central coordinator (CC), which is conventionally located at the origin of the coordinate system. We consider a query-based application and a star topology: the CC periodically triggers all nodes, and those that correctly receive the query will try to access the channel and send their data to the CC through direct links. We assume that the query interval is longer than the maximum delay experienced by any packet reaching the CC after the query; therefore, subsequent query intervals are independent. Manuscript received January 18, 2013; revised June 26, 2013 and September 20, 2013; accepted November 6, 2013. Date of publication December 5, 2013; date of current version July 10, 2014. The review of this paper was coordinated by Dr. P. Lin. F. Martelli was with the Department of Electrical, Electronic and Information Engineering, University of Bologna, Bologna 40126, Italy. She is now with Teko Telecom Srl, Bologna 40024, Italy (e-mail: flavia.martelli@unibo.it). C. Buratti and R. Verdone are with the Department of Electrical, Electronic and Information Engineering, University of Bologna, Bologna 40126, Italy (e-mail: c.buratti@unibo.it; roberto.verdone@unibo.it). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TVT.2013.2294188 We mathematically derive the success probability for a generic node located at a distance r 0 from the CC (reference transmitter TX 0 ) to correctly transmit its data when a carrier- sense multiple-access (CSMA) mechanism is used. In our model, we consider both connectivity and interference issues. • Link-level connectivity is taken into account since only the nodes that receive the query from the CC can attempt their data transmission. • Interference at the CC can happen because of the following causes: 1) hidden node problems (i.e., two nodes con- nected to the CC cannot “hear” each other); and 2) accord- ing to the query-based application considered, all nodes start the procedure to access the channel simultaneously, and this implies that two or more nodes may sense the channel at the same time, finding it free and transmitting simultaneously. The mathematical model presented is based on stochastic geometry, which is a branch of applied mathematics that allows the study of random phenomena in different kinds of applica- tions and subjects [1] and that, in the last decade, has been extensively used also for the analysis and design of wireless networks (see, e.g., [2]–[4]). Focusing in particular on ad hoc networks and wireless sensor networks (WSNs), two main research directions can be identified in this framework: link- level connectivity issues and interference characterization. The former is related to the possibility for nodes of a network to communicate with a central node. Therefore, link-level connec- tivity is intended in this paper as the property of two nodes to hear each other. Several works can be found in the literature, both for single and multisink scenarios, considering infinite or finite regions [5]–[8]. Interference can be one of the main factors affecting perfor- mance in wireless multiuser networks. A classic approach to model interference, which is suitable for situations with a large number of nodes and no dominant contribution, is to consider the sum of the interfering power as a Gaussian random process. In the recent literature, different frameworks were proposed for the cases in which the Gaussian approximation does not hold (see, e.g., [9] and [10]). In [10], in particular, a general framework is developed to characterize the aggregate network interference for nodes distributed according to a homogeneous Poisson point process (PPP), which, in the case of an infinite domain, is shown to follow an alpha-stable distribution. Homogeneous PPP is a useful model for the location of nodes in a network when their position is not known a priori, and it has been widely used in several of the studies based on the stochastic geometry stated previously. For a homogeneous 0018-9545 © 2013 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.