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A mathematical model for the clock phase and frequency deviation based on the theory of stochastic differential equations (SDEs) is discussed. In particular, we consider a model that includes what are called the 'white and random walk frequency noises' in time metrology, which give rise in a mathematical context to a Wiener and an integrated Wiener process on the clock phase. Due to the particular simple expression of the functions involved an exact solution exists, and we determine it by considering the model in a wider theoretical context, which is suitable for generalizations to more complex instances. Moreover, we determine an iterative form for the solution, useful for simulation and further processing of clock data, such as filtering and prediction. The Euler method, generally applied in literature to approximate the solution of SDEs, is examined and compared to the exact solution, and the magnitude of the approximation is evaluated. The possible extension of the model to other noise sources, such as the flicker and white phase noises, is also sketched.

The statistical generation of an accurate, stable and reliable time scale from a clock ensemble calls for an efficient algorithm. Here we present a comparative study which points out conceptual differences and similarities between two different types of time scale algorithm used worldwide:1. ALGOS (BIPM), which produces, bimonthly, the international reference TAI (temps atomique international) at the Bureau International des Poids et Mesures (Sèvres, France), and2. AT1 (NIST), from which is obtained, in real-time, the AT1 time scale at the National Institute of Standards and Technology (Boulder, Colorado, USA).These two algorithms rely basically upon weighted averages of clock readings, but the weight determination and the frequency prediction are of different types because they are adapted to different needs and to the peculiarities of the available time measurements.