In this paper, in work strongly related with that of Coffman et al. [5], Bruss and Robertson [2], and Rhee and Talagrand [15], we focus our interest on an asymptotic distributional comparison between numbers of ‘smallest’ i.i.d. random variables selected by either on-line or off-line policies. Let X
1,X
2,… be a sequence of i.i.d. random variables with distribution function F(x), and let X
1,n
,…,X
n,n
be the sequence of order statistics of X
1,…,X
n
. For a sequence (c
n
)
n≥1 of positive constants, the smallest fit off-line counting random variable is defined by N
e
(c
n
) := max {j ≤ n : X
1,n
+ … + X
j,n
≤ c
n
}. The asymptotic joint distributional comparison is given between the off-line count N
e
(c
n
) and on-line counts N
n
τ for ‘good’ sequential (on-line) policies τ satisfying the sum constraint ∑
j≥1
X
τ
j
I
(τ
j
≤n) ≤ c
n
. Specifically, for such policies τ, under appropriate conditions on the distribution function F(x) and the constants (c
n
)
n≥1, we find sequences of positive constants (B
n
)
n≥1, (Δ
n
)
n≥1 and (Δ'
n
)
n≥1 such that
for some non-degenerate random variables W and W'. The major tools used in the paper are convergence of point processes to Poisson random measure and continuous mapping theorems, strong approximation results of the normalized empirical process by Brownian bridges, and some renewal theory.