A Convergence Result of Dropping Time of a De- Coder with a Fixed Amount of Storage

We consider the stochastic behavior of dropping time of a decoder withM frame buffers. It can be formulated as the following stochastic problem. Suppose X1, X2, . . . , are independent and identically distributed random variables. Suppose W0 and Wi = max{Wi−1 + Xi, 0}, i = 1, 2, . . . . For a fixed positive threshold Γ, the dropping time D is defined as D = min{n,Wn ≥ Γ}. Our main result is to show that the ratio P{D ≥ n + 1}/P{D ≥ n}, which also is the decay rate of the tail probabilities P{D ≥ n}, converges to a constant as n goes to positive infinity. Consequently, it implies that the tail probability sequence P{D ≥ n} is asymptotically a geometric sequence. Our result helps to understand some stochastic phenomena in applications. For example, for a decoder with fixed amount of storage, its frame-dropping time can be formulated as in this model. Our result provides theoretical foundations to develop fast approximation to certain quantities that are of interest. For example, we demonstrate how this result leads to fast approximations to the average dropping time E[D], and the average dropping rate E[1/D]. Simulation results are reported to consolidate our belief.