The problem of Carrier Sense Multiple Access (CSMA) with multi-packet reception (MPR) is studied. Most prior work has focused on the homogeneous case, where all the mobile users are assumed to have identical packet arrival rates and transmission probabilities. The inhomogeneous case remains largely open in the literature. In this work, we make a first step towards this open problem by deriving throughput and delay expressions for inhomogeneous CSMA, with a particular focus on a family of MPR models called the "all-or-nothing" symmetric MPR. This family of MPR models allows us to overcome several technical challenges associated with conventional analysis and to derive accurate throughput and delay expressions in the large-systems regime. Interestingly, this family of MPR models is still general enough to include a number of useful MPR techniques-such as successive interference cancellation (SIC), compute-and-forward (C&F), and successive computeand-forward (SCF)-as special cases. Based on these throughput and delay expressions, we provide theoretical guidelines for meeting quality-of-service requirements and for achieving global stability; we also evaluate the performances of various MPR techniques, highlighting the clear advantages offered by SCF.
In this paper, we analyze mathematical models of digital loops used to track the phase and timing of communications and navigations signals. The limits on the accuracy of phase and timing estimates play a critical role in the accuracy achievable in telemetry-based ranging applications. We describe in detail a practical algorithm to compute the loop parameters for discrete update (DU) and the continuous
update (CU) loop formulations, consistent with the development of [3], and we show that a simple power-series approximation to the DU model is valid over a large range of time-bandwidth product ( ) . Several numerical examples compare the estimation error variance of the DU and CU models to each other and to Cramér-Rao lower bounds.Finally, the results are applied to the problem of ranging, by evaluating the performance of a phase-locked loop designed to track a typical ambiguity-resolving PN code received and demodulated at the spacecraft, on the uplink part of the twoway ranging link.
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