This paper considers the first two moments of R( β,N)=A( β,N)/N, A( β,N) denoting the random number of unoccupied sites of a one-dimensional array of N compartments saturated by randomly placed β-bell particles ( β≥2). It is shown that, as N → ∞, the mean of R( β,N) approaches a (non-zero) limit L( β) while its variance tends to zero thus yielding the stochastic convergence of R( β,N) to L( β). L( β) is explicitly determined and its behavior for large β is also studied.