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A Stochastic Process Arising in the Study of Muscular Contraction

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Assume two parallel line segments of indefinite length, one of which is fixed and the other of which is movable, moving in a linear direction, parallel to the fixed line. Along the fixed line there are special points equally spaced which we shall denote as positions. Similarly, on the moving line, there exist equally spaced points which we denote as sites. A site may be in either of two states which we refer to as vacant or filled. After a sites initial state, which is determined, it can change states only after interactions with the positions along the fixed line in the following manner. A site which is vacant can become filled at certain positions. Call these load positions. Similarly, a site which is filled can become vacant at the remaining positions. We denote these as release positions. We assume an arbitrary fixed starting position on the fixed line with the positions numbered consecutively beyond the starting position and, further, the release and load positions alternate so that the odd numbered positions are release positions and the even numbered ones are load positions. Then, under certain assumptions stated in the next section, the question posed is the probability a site will be filled or vacant after a transit of n positions. The model will then be extended to the case where a site can be in any one of m 1 states and the analogous question posed is the probability the site will be empty, filled or in any arbitrary state j after a transit of n positions.

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  • Statistics and Probability
  • Medicine and Medical Research

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