An M/G/1 queue with customer collection

This paper analyses a variant of the M/G/1 queue in which the service times of arriving customers depend on the length of the interval between their arrival and the previous arrival. The dependence structure under consideration arises when individual customers arrive according to a Poisson process, while customer collectors are sent out according to a Poisson process to collect the customers and to bring them to the service facility. In this case collected numbers of customers, and hence total collected service requests, are positively correlated with the corresponding collect intervals. Viewing a batch of collected customers as one (super) customer gives rise to an M/G/1 queue with a positive correlation between service times of such customers and their interarrival times. Both for individual customers and for supercustomers we derive the transforms of the sojourn time, waiting time and queue length distributions. We also compare over results with those for the ordinary M/ G/:1 queue with dependence