ABSTRACT. The basic queueing system considered in this paper is the MIG/1 processor sharing (PS) queue with or without impatience and with finite or infinite capacity. Under some mild assumptions, a criterion for the validity of the RSR (Reduced Service Rate) approximation is established when service times are heavy tailed. This result is applied to various models based on MIG/1 processor sharing queues.

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We study in this paper the integration of elastic and streaming traffic on a same link in an IP network. We are specifically interested in the computation of the mean bit rate obtained by a data transfer. For this purpose, we consider that the bit rate offered by streaming traffic is low, of the order of magnitude of a small parameter ε ≪ 1 and related to an auxiliary stationary Markovian process (X(t)). Under the assumption that data transfers are exponentially distributed, arrive according to a Poisson process, and share the available bandwidth according to the ideal processor sharing discipline, we derive the mean bit rate of a data transfer as a power series expansion in ε. Since the system can be described by means of an M/M/1 queue with a time-varying server rate, which depends upon the parameter ε and process (X(t)), the key issue is to compute an expansion of the area swept under the occupation process of this queue in a busy period. We obtain closed formulas for the power series expansion in ε of the mean bit rate, which allow us to verify the validity of the so-called reduced service rate at the first order. The second order term yields more insight into the negative impact of the variability of streaming flows.

We investigate an overloaded processor sharing queue with renewal arrivals and generally distributed service times. Impatient customers may abandon the queue, or renege, before completing service. The random time representing a customer's patience has a general distribution and may be dependent on his initial service time requirement. We propose a scaling procedure that gives rise to a fluid model, with nontrivial yet tractable steady state behavior. This fluid model captures many essential features of the underlying stochastic model, and we use it to analyze the impact of impatience in processor sharing queues. We show that this impact can be substantial compared with FCFS, and we propose a simple admission control policy to overcome these negative impacts.

File-sharing networks are distributed systems used to disseminate files among nodes of a communication network. The general simple principle of these systems is that once a node has retrieved a file, it may become a server for this file. In this paper, the capacity of these networks is analyzed with a stochastic model when there is a constant flow of incoming requests for a given file. It is shown that the problem can be solved by analyzing the asymptotic behavior of a class of interacting branching processes. Several results of independent interest concerning these branching processes are derived and then used to study the file-sharing systems.