Triggered batch movement in queueing networks

In the queueing network literature there have recently appeared a number of examples of networks in which events are triggered by the occurrence of other events. For example Henderson and Taylor [16] discuss queueing networks with external "interruptions", which arrive from outside the network, end the service of a customer at a queue, and route it through or out of the network even though it has not received a complete service. The interruptions are events which trigger the movement of customers. Gelenbe [8] describes a network of single server queues with triggers in which a customer, upon the completion of its service, can trigger, or force, the movement of a second customer from one queue to another in the network. The networks of Gelenbe [7], incorporating negative customers, are a special case of those in [8] in which the released customer is always deposited outside the network. Gelenbe [9] and Henderson, Northcote and Taylor [12] allow the forced release of batches of customers from a single node and Henderson [10] and Henderson, Northcote and Taylor [13] incorporate a degree of state dependence into the service and arrival rates of customers in the networks described in [7,8,9,12]. The surprising aspect of all of the above networks is that they have product form equilibrium distributions and yet do not fit into the general framework described by, for example, Jackson [20] and Kelly [21]. Recent work by Chao and Pinedo [3], however, has established that the queues in these networks are quasireversible and that an appropriate coupling of these queues explains the product form equilibrium distribution for the networks. Each network described thus far restricts positive (normal) customers to be routed through the network singly. This requirement has been relaxed by a number