Abstract-In this paper we analyze the M/G/1 processor sharing queue with heavy tailed services and with impatient customers. It is assumed that impatience depends on the value of the service required. We prove that a reduced service rate (RSR) approximation holds for estimating the sojourn time of a customer in the system, when the queue capacity is finite or infinite. This allows us to evaluate the reneging probability of customers with very large service times. We then use these results to investigate the impact of admission control on a link of a packet network. Admission control simply consists of limiting the number of simultaneous connections. It turns out that there is a real benefit for the efficiency of the system to perform admission control: It globally increases the fraction of customers, who complete their service (i.e. without being impatient). Finally, we investigate the fairness of the system and propose a criterion to assess the capacity of the system so as to allow the completion of very large service times.

We give in this paper an algorithm to compute the sojourn time distribution in the processor sharing, single server queue with Poisson arrivals and phase type distributed service times. In a first step, we establish the differential system governing the conditional sojourn times probability distributions in this queue, given the number of customers in the different phases of the PH distribution at the arrival instant of a customer. This differential system is then solved by using a uniformization procedure and an exponential of matrix. The proposed algorithm precisely consists of computing this exponential with a controlled accuracy. This algorithm is then used in practical cases to investigate the impact of the variability of service times on sojourn times and the validity of the so-called reduced service rate (RSR) approximation, when service times in the different phases are highly dissymmetrical. For two-stage PH distributions, we give conjectures on the limiting behavior in terms of an M/M/1 PS queue and provide numerical illustrative examples.

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We analyse in this paper the fluid weighted fair queueing system with two classes of customers, who arrive according to Poisson processes and require arbitrarily distributed service times. In a first step, we express the Laplace transform of the joint distribution of the workloads in the two virtual queues of the system by means of unknown Laplace transforms. Such an unknown Laplace transform is related to the distribution of the workload in one queue provided that the other queue is empty. We explicitly compute the unknown Laplace transforms by means of a Wiener—Hopf technique. The determination of the unknown Laplace transforms can be used to compute some performance measures characterizing the system (e.g. the mean waiting time for each class) which we compute in the exponential service case.

Networks based upon the asynchronous transfer mode (ATM) provide for high flexibility to cope with a wide range of applications, some of them producing highly sporadic traffic. Therefore, the problem of burstiness has become in the last few years a key issue for such networks. A basic question is how to dimension network buffers in the presence of bursty traffic? In this paper we investigate the concept of burstiness and its impact on resource management. In burstiness characterization encountered in the literature, special attention has been given to the squared coefficient of variation of inter‐arrival time Cv2 in a cell arrival process. In order to observe the impact of bursty traffic on a queue, we develop in the present paper a ‘bursty’ traffic model, namely the two‐stage hyper‐Bernoulli cell arrival process, HBP2, for short. We numerically solve the HBP2/D/1/K queue. We especially derive the rejection probability Ploss. Numerical results are then thoroughly studied and we discuss the relevance for burstiness characterization of Cv2 and peak to mean rate ratio B. We draw attention to the concept of local overload, i.e. when the arrival rate is greater than the server rate. This seems to be the most relevant phenomenon in the impact of a bursty traffic on a queue. These results are finally applied to the problem of resource management in ATM networks.