On queues with inter-arrival times proportional to service times

A family of queuing systems in which the interarrival time I _n+1 between customers n and n+1 depends on the service time B_n of customer n is considered. Specifically, cases where the dependency between I_n+1 and B_n is a proportionally relation and B_n is an exponentially distributed random variable is considered. Such dependencies arise in the context of packet-switched networks from employing rate policing functions which regulate the amount of data that can arrive at a link within any given time interval. The models developed and the associated solutions are, however, of independent interest and potentially applicable to other environments. Several scenarios that consist of adding an independent random variable to the interarrival time, allowing the proportionality to be random, and the combination of the two are considered. Numerical results are compared to those for an equivalent system without dependencies