This paper considers the scheduling of soft real-time sporadic task systems under global EDF on an iden-tical multiprocessor. Though Pfair scheduling is theoretically optimal for hard real-time task systems on multiprocessors, it can incur significant run-time overhead. Hence, other scheduling algorithms that are not optimal, including EDF, have continued to receive considerable attention. However, prior research on such algorithms has focussed mostly on hard real-time systems, where, to ensure that all deadlines are met, ap-proximately 50 % of the available processing capacity will have to be sacrificed in the worst case. This may be overkill for soft real-time systems that can tolerate deadline misses by bounded amounts (i.e., bounded tardiness). In this paper, we derive tardiness bounds under preemptive and non-preemptive global EDF on multiprocessors when the total utilization of a task system is not restricted and may equal the number of pro-cessors. Our tardiness bounds depend on per-task utilizations and execution costs — the lower these values, the lower the tardiness bounds. As a final remark, we note that global EDF may be superior to partitioned EDF for multiprocessor-based soft real-time systems in that the latter does not offer any scope to improve system utilization even if bounded tardiness can be tolerated.

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Consider a single server queue with renewal arrivals and i.i.d. service times in which the server operates under a processor sharing service discipline. To describe the evolution of this system, we use a measure valued process that keeps track of the residual service times of all jobs in the system at any given time. From this measure valued process, one can recover the traditional performance processes, including queue length and workload. We show that under mild assumptions, including standard heavy traffic assumptions, the (suitably rescaled) measure valued processes corresponding to a sequence of processor sharing queues converge in distribution to a measure valued diffusion process. The limiting process is characterized as the image under an appropriate lifting map, of a one-dimensional reflected Brownian motion. As an immediate consequence, one obtains a diffusion approximation for the queue length process of a processor sharing queue. 1. Introduction. Consider

Prior work has shown that the global earliest-deadline-first (GEDF) scheduling algorithm ensures bounded deadline tardiness on multiprocessors with no utilization loss; therefore, GEDF may be a good candidate scheduling algorithm for soft real-time workloads. However, such workloads are often implemented assuming an average-case provisioning, and in prior tardiness-bound derivations for GEDF, worst-case execution costs are assumed. As worst-case costs can be orders of magnitude higher than average-case costs, using a worstcase provisioning may result in significant wasted processing capacity. In this paper, prior tardiness-bound derivations for GEDF are generalized so that execution times are probabilistic, and a bound on expected (mean) tardiness is derived. It is shown that, as long as the total expected utilization is strictly less than the number of available processors, the expected tardiness of every task is bounded under GEDF. The result also implies that any quantile of the tardiness distribution is also bounded. 1

This paper presents a heavy traffic analysis of the behavior of multi-class acyclic queueing networks in which the customers have deadlines. We assume the queueing system consists of J stations, and there are K different customer classes. Customers from each class arrive to the network according to independent renewal processes. The customers from each class are assigned a random deadline drawn from a deadline distribution associated with that class and they move from station to station according to a fixed acyclic route. The customers at a given node are processed according to the earliest-deadline-first (EDF) queue discipline. At any time, the customers of each type at each node have a lead time, the time until their deadline lapses. We model these lead times as a random counting measure on the real line. Under heavy traffic conditions and suitable scaling, it is proved that the measure-valued lead-time process converges to a deterministic function of the workload process. A two-station example is worked out in details, and simulation results are presented to illustrate the predictive value of the theory. This work is a generalization of Doytchinov, Lehoczky and Shreve [5], which developed these results for the single queue case.

Motivated by interest in making delay announcements to arriving customers who must wait in call centers and related service systems, we study the performance of alternative real-time delay estimators based on recent customer delay experience. The main estimators considered are: (i) the delay of the last customer to enter service (LES), (ii) the delay experienced so far by the customer at the head of the line (HOL), and (iii) the delay experienced by the customer to have arrived most recently among those who have already completed service (RCS). We compare these delay-history estimators to the estimator based on the queue length (QL), which requires knowledge of the mean interval between successive service completions in addition to the queue length. We characterize performance by the mean squared error (MSE). We do analysis and conduct simulations for the standard GI/M/s multi-server queueing model, emphasizing the case of large s. We obtain analytical results for the conditional distribution of the delay given the observed HOL delay. An approximation to its mean value serves as a refined estimator. For all three candidate delay estimators, the MSE relative to the square of the mean is asymptotically negligible in the many-server and classical heavy-traffic limiting regimes.

Abstract. We investigate a processor sharing queue with renewal arrivals and generally distributed service times. Impatient jobs may abandon the queue, or renege, before completing service. The random time representing a job’s patience has a general distribution and may be dependent on its initial service time requirement. A scaling procedure that gives rise to a fluid model with nontrivial yet tractable steady state behavior is presented. This fluid model model captures many essential features of the underlying stochastic model, and it is used to analyze the impact of impatience in processor sharing queues.

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In this paper, we present a functional fluid limit theorem and a functional central limit theorem for a queue with an infinity of servers M/GI/∞. The system is represented by a point-measure valued process keeping track of the remaining processing times of the customers in service. The convergence in law of a sequence of such processes after rescaling is proved by compactness-uniqueness methods, and the deterministic fluid limit is the solution of an integrated equation in the space S ′ of tempered distributions. We then establish the corresponding central limit theorem, that is, the approximation of the normalized error process by a S ′-valued diffusion. We apply these results to provide fluid limits and diffusion approximations for some performance processes. 1. Introduction. The

This paper considers a GI/GI/1 processor sharing queue in which jobs have soft deadlines. At each point in time, the collection of residual service times and deadlines is modeled using a random counting measure on the right half-plane. The limit of this measure valued process is obtained under diffusion scaling and heavy traffic conditions and is characterized as a deterministic function of the limiting queue length process. As special cases, one obtains diffusion approximations for the lead time profile and the profile of times in queue. One also obtains a snapshot principle for sojourn times. 1. Introduction. Congestion