On the Existence of Moments for the Busy Period in a Single-Server Queue

We obtain sufficient conditions for the finiteness of the moments of the single-server queue busy period in discrete and continuous time. In the cases of power and exponential moments, the conditions found coincide with the known necessary ones. 1. Introduction and notation. Consider a sequence t n n of random vectors with nonnegative coordinates which are i.i.d. in n, but any dependence between the components is allowed. Put a = Et 1 − E 1. Assume that a > 0. Let n = n − t n , S n = n i=1 i , and = minn ≥ 1 S n ≤ 0 < a.s. be the first passage time to the nonpositive half-line. Here E 1 = −a < 0. Let B = i=1 i. In what follows, we investigate conditions for the finiteness of the moments E expgB and E expg, where the function g is taken from a rather wide class which includes the logarithmic (ggx = log x, ≥ 1), the linear functions (ggx = cx, c > 0), and functions between the logarithmic and linear (e.g., ggx = x