On queues with inter-arrival times proportional to service times

We analyze a family of queueing systems where the interarrival time I_n+1 between customers n and n + 1 depends on the service time B_n of customer n. Specifically, we consider cases where the dependency between I_n+1 and B_n is a proportionality relation, and B_n is an exponentially distributed random variable. Such dependencies arise in the context of packet-switched networks from employing rate policing functions which regulate the amount of data that can arrive to a link within any given time interval. These controls result in significant dependencies between the amount of work brought in by customers/packets and the time between successive customers. The models developed in the paper and the associated solutions are, however, of independent interest and potentially applicable to other environments. Several scenarios which consist of adding an independent random variable to the inter-arrival time, allowing the proportionality to be random and the combination of the two are considered. In all cases we provide expressions for the Laplace-Stieltjes Transform (LST) of the waiting time of a customer in the system. Numerical results are provided and compared to those of an equivalent system without dependencies.