A Markov Model for Predicting Availability of GPS Satellites in Primary Orbital Slots

Curtis A. Shively

Abstract:	One measure for characterizing the performance of GPS is the probability that a healthy satellite is operating in 21 or more of the 24 primary orbital slots of the constellation. The DOD has assured that this performance measure will be met with probability 0.98. This paper describes a simplified approach using a Markov model for predicting the future performance of this slot availability measure (denoted PGE21Slots). The Markov model is used to determine the availability “states” of each orbital plane, accounting for the redundancy provided by spare satellites already on-orbit or to be launched in the future. Because the number of states in such a model can become very large, the exact time evolution of the plane state probabilities is first determined directly from Markov models for up to 4 slots and 1 spare. Results from this initial work are used to develop closed-form approximations to characterize the full model for 4 slots and 2 spares. The analysis methodology is illustrated for a hypothetical example including both on-orbit satellites and new satellites to be launched in the future. This example includes a launch schedule, assignment of new satellites to provide redundancy for individual slots or entire planes, and several groups of satellites with varying reliability characteristics. In this illustrative example the evolution of PGE21Slots over time is consistent with the schedule and satellite reliability characteristics assumed. Furthermore, the procedure used to assign newly launched satellites produces a balanced distribution of performance over the constellation slots and planes. Based on slot probabilities from the illustrative example, the service availability of LAAS for Category I (CAT I) precision approaches is predicted for an example location. These illustrative results suggest that even when PGE21Slots meets the required level of 0.98, LAAS CAT I service availability might not meet the corresponding requirement of 0.999. A similar observation is made that other performance measures relating to availability of satellites in the constellation may require a higher availability in each primary slot than would be needed to meet the PGE21Slots = 0.98 requirement.
Published in:	Proceedings of the 2008 National Technical Meeting of The Institute of Navigation January 28 - 30, 2008 The Catamaran Resort Hotel San Diego, CA
Pages:	1040 - 1052
Cite this article:	Shively, Curtis A., "A Markov Model for Predicting Availability of GPS Satellites in Primary Orbital Slots," Proceedings of the 2008 National Technical Meeting of The Institute of Navigation, San Diego, CA, January 2008, pp. 1040-1052.
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