The ratio of the extreme to the sum in a random sequence

Abstract

For X1 , X2 , ..., Xn a sequence of non-negative independent random variables with common distribution function F(t), X(n) denotes the maximum and Sn denotes the sum. The ratio variate Rn = X(n) / Sn is a quantity arising in the analysis of process speedup and the performance of scheduling. O’Brien (J. Appl. Prob. 17:539–545, 1980) showed that as n → ∞, Rn →0 almost surely iff \({\sf E} X_1\) is finite. Here we show that, provided either (1) \({\sf E} X_1^2 \) is finite, or (2) 1 − F (t) is a regularly varying function with index ρ < − 1, then \({\sf E} R_n \sim { {\sf E} X_{(n)} }/{{\sf E} S_n } ,( n \rightarrow \infty )\). An integral representation for the expected ratio is derived, and lower and upper asymptotic bounds are developed to obtain the result. Since \({\sf E} X_{(n)}\) is often known or estimated asymptotically, this result quantifies the rate of convergence of the ratio’s expected value. The result is applied to the performance of multiprocessor scheduling.