Space filling and depletion

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Abstract
For a givenk≥ 1, subintervals of a given interval [0,X] arrive at random and are accepted (allocated) so long as they overlap fewer thanksubintervals already accepted. Subintervals not accepted are cleared, while accepted subintervals remain allocated for random retention times before they are released and made available to subsequent arrivals. Thus, the system operates as a generalized many-server queue under a loss protocol. We study a discretized version of this model that appears in reference theories for a number of applications, including communication networks, surface adsorption-desorption processes, and reservation systems. Our primary interest is in steady-state estimates of the vacant space, i.e. the total length of available subintervalskX- ∑ℓi, where the ℓiare the lengths of the subintervals currently allocated. We obtain explicit results fork= 1 and for generalkwith all subinterval lengths equal to 2, the classicaldimercase of chemical applications. Our focus is on the asymptotic regime of large retention times.