Discrete time tandem networks of queues: Effects of different regulation schemes for simultaneous events
We consider tandem networks of discrete time Bernoulli servers with state dependent service rates and a state dependent Bernoulli arrival stream at the first node of the tandem. We investigate the effects of different regulation schemes for simultaneous events (e.g. joint arrivals and departures at some node, or joint departures at different nodes) on the performance behaviour of the network. The most serious effects occur with respect to arrival theorems which describe the distribution of the other customers present in the network and seen by an arriving customer, or observed by a customer during a jump inside the network. We prove necessary and sufficient conditions on the regulation scheme for a customer to observe always the time stationary behaviour of the network during his jumps. It turns out that we have to distinguish between local and global control for the regulation of simultaneous jumps. For different arrival schemes we compute the joint sojourn time distribution for a customer traversing the tandem. As a consequence we identify from this a regulation scheme which is known in the literature, where Littles formula cannot be applied directly.
simultaneous events state dependent service arrival theorem sojourn time distribution littles formula early arrivals late arrivals departure first arrival first state dependent arrivals product form distribution