The relation between the probability of collision-free broadcast transmission in a wireless network and the stirling number of the second kind

Abstract

The broadcast performance of the 802.11 wireless protocol depends on several factors. One of the important factor is the number of nodes simultaneously contending for the shared channel. The Medium Access Control (MAC) technique of 802.11 is called the Distributed Coordination Function (DCF). DCF is a Carrier Sense Multiple Access with Collision Avoidance (CSMA/CA) scheme with binary slotted exponential backoff. A collision is the result of two or more stations transmitting simultaneously. Given the simplicity of the DCF scheme, it was adapted for Dedicated Short Range Communication (DSRC) based vehicular communication. A broadcast mechanism is used to disseminate emergency and safety related messages in a vehicular network. Emergency and safety related messages have a strict end-to-end latency of 100 ms and a Packet Delivery Ratio (PDR) of 90% and above. The PDR can be evaluated through the packet loss probability. The packet loss probability PL is given by, PL = 1-(1 - Pe)(1 - PC), where Pe is the probability of channel error and PC is the probability of collision. Pe depends on several environmental and operating factors and thus cannot be improved. The only way to reduce PL is by reducing PC. Currently, expensive radio hardware are used to measure PL. Several adaptive algorithms are available to reduce PC. In this paper, we establish a closed relation between PC and the Stirling number of the second kind. Simulation results are presented and compared with the analytical model for accuracy. Our simulation results show an accuracy of 99.9% compared with the analytical model. Even on a smaller sample size, our simulation results show an accuracy of 95% and above. Based on our analytical model, vehicles can precisely estimate these real-time requirements with the least expensive hardware available. Also, once the distribution of PC and PL are known, one can precisely determine the distribution of Pe.