The serial correlation coefficients of waiting times in a stationary single server queue

Abstract

Summary: In a stationaryGI/G/1 queueing system in which the waiting time variance is finite, it can be shown that the serial correlation coefficients {ρ_n} of a (stationary) sequence of waiting times are non-negative and decrease monotonically to zero. By means of renewal theory we find a representation for Σ^∞₀ρ_nfrom which necessary and sufficient condition for its finiteness can be found. InM/G/1 rather more can be said: {ρ_n} is convex sequence, the asymptotic form of ρ_ncan be given in a nearly saturated queue, and a simple explicit expression for Σ^∞₀ρ_nexists. For the stationaryM/M/1queue we find the ρ_n's explicitly, illustrate them numerically, and derive a representation which shows that {ρ_n} is completely monotonic.