A family of queues is studied for which the service time B/sub n/ of customer n and the interarrival time I/sub n/ between customers n-1 and n exhibit some sort of proportionality. The focus is on dependencies that arise naturally in the context of communication systems, where the finite speed of the communication links constrains the amount of data that can be received in a given time interval. The simple case of a deterministic proportionality relation between the service time of a customer and its preceding interarrival time is considered. This is extended to allow the addition of an independent, generally distributed overhead to the service time of each customer. Several models that capture the ON-OFF behavior of communication links in packet networks are considered. In all cases, expressions for the delay experienced by a packet in the system are provided. Numerical examples illustrate the impact of dependencies through comparison with less accurate models. The results should be of relevance to environments other than communication as well. >

Analysis of a correlated queue in a communication system

In [1] the authors dealt with a particular queueing system in which arrivals occurred in a Poisson stream and the probability differential of a service completion was pao when the queue contained n customers. Much of the theory could not be carried out further analytically for a general r,, which is a purely n-dependent quantity. To carry the analysis further to the extent of finding the "effective" service time and the waiting time distribution when a,, is a linear function of n, (which is considered to be rather general and sufficient for practical purposes), constitutes the subject matter of this paper.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/884BFBA0F4F70882818C8072CAB36D54/S0021900200026644a.pdf/div-class-title-on-the-service-time-distribution-and-the-waiting-time-process-of-a-potentially-infinite-capacity-queueing-system-div.pdf

On the service time distribution and the waiting time process of a potentially infinite capacity queueing system

A family of queues for which the service time B/sub n/ of customer n depends on the interarrival time I/sub n/ between customers n-1 and n and the random variables I/sub n/ and B/sub n/ exhibit a proportionality relation is studied. In particular, the focus is on dependencies that arise naturally in communication systems, where the finite speed of the communication links constrains the amount of data that can be received in a given time interval. The simple case of a deterministic proportionality relation between the service time of a customer and its preceding interarrival time is considered and extended to allow the addition of an independent, generally distributed overhead to the service time. Several models that capture the on-off behavior of communication links in packet networks are then addressed. In all cases, expressions for the delay experienced by a packet in the system and illustrative numerical examples are provided. >

We analyze a family of queueing systems where the interarrival time In+1 between customers n and n + 1 depends on the service time Bn of customer n. Specifically, we consider cases where the dependency between In+1 and Bn is a proportionality relation and Bn is an exponentially distributed random variable. Such dependencies arise in the context of packet-switched networks that use rate policing functions to 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 are 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. In all cases, we provide expressions for the Laplace-Stieltjes Transform of the waiting time of a customer in the system. Numerical results are provided and compared to those of an equivalent system without dependencies.

On Queues with Interarrival Times Proportional to Service Times

A family of queuing systems in which the interarrival time I/sub n+1/ between customers n and n+1 depends on the service time B/sub n/ of customer n is considered. Specifically, cases where the dependency between I/sub n+1/ and B/sub n/ is a proportionally relation and B/sub 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. >

B. W. Conolly

Papers

A Comparison of the Operational Features of Conventional Queues with a Self-Regulating System