Batch Bayesian optimisation (BO) is a successful technique for the\noptimisation of expensive black-box functions. Asynchronous BO can reduce\nwallclock time by starting a new evaluation as soon as another finishes, thus\nmaximising resource utilisation. To maximise resource allocation, we develop a\nnovel asynchronous BO method, AEGiS (Asynchronous $\\epsilon$-Greedy Global\nSearch) that combines greedy search, exploiting the surrogate's mean\nprediction, with Thompson sampling and random selection from the approximate\nPareto set describing the trade-off between exploitation (surrogate mean\nprediction) and exploration (surrogate posterior variance). We demonstrate\nempirically the efficacy of AEGiS on synthetic benchmark problems,\nmeta-surrogate hyperparameter tuning problems and real-world problems, showing\nthat AEGiS generally outperforms existing methods for asynchronous BO. When a\nsingle worker is available performance is no worse than BO using expected\nimprovement.\n