Spatial No-Waiting Stations with Moving Customers

We analyze spatial MAP/G/∞-, spatial MAP/G/c/0¹ and spatial Cox/G/∞-stations with group arrivals over some Polish space X (with Borel σ-algebra X), including the aspect of customer motion in space. For models with MAP-input, characteristic differential equations are set up that describe the dynamics of phase dependent random functions Q _r;ij (u,t;S′), where Q _r;ij (u,t;S′) is the probability to observe, at time u≤t, the number r of those customers in some source set S′∈X, who will be in a destination set S∈X at time t. For Cox/G/∞-stations, i.e., infinite server stations with doubly stochastic input, the arrival intensities as well as service times may depend on some general stochastic process (J′ _t ) _t≥0 with countable state space. For that case we obtain explicit expressions for space–time distributions as well as stationary and non-stationary characteristics.