On the Discrete-Timeg/Gi/∞ Queue

The discrete-timeG/GI/∞ queue model is explored. Jobs arrive to an infinite-server queuing system following anarbitraryinput processX; job sizes are general independent and identically distributed random variables. The system's output processY(of job departures) and queue processN(tracking the number of jobs present in the system) are analyzed. Various statistics of the stochastic mapsX↦YandX↦Nare explicitly obtained, including means, variances, autocovariances, cross-covariances, and multidimensional probability generating functions. In the case of stationary inputs, we further compute the spectral densities of the stochastic maps, characterize the fixed points (in theL2sense) of the input–output map, precisely determine when the output and queue processes display either short-ranged or long-ranged temporal dependencies, and prove a decomposition result regarding the intrinsicL2structure of general stationaryG/GI/∞ systems.