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Tardiness bounds under global edf scheduling on a multiprocessor
 In RTSS ’05: Proceedings of the 26th IEEE International RealTime Systems Symposium
, 2005
"... This paper considers the scheduling of soft realtime sporadic task systems under global EDF on an identical multiprocessor. Though Pfair scheduling is theoretically optimal for hard realtime task systems on multiprocessors, it can incur signicant runtime overhead. Hence, other scheduling algorit ..."
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Cited by 62 (38 self)
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This paper considers the scheduling of soft realtime sporadic task systems under global EDF on an identical multiprocessor. Though Pfair scheduling is theoretically optimal for hard realtime task systems on multiprocessors, it can incur signicant runtime 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 realtime systems, where, to ensure that all deadlines are met, approximately 50 % of the available processing capacity will have to be sacriced in the worst case. This may be overkill for soft realtime systems that can tolerate deadline misses by bounded amounts (i.e., bounded tardiness). In this paper, we derive tardiness bounds under preemptive and nonpreemptive global EDF on multiprocessors when the total utilization of a task system is not restricted and may equal the number of processors. Our tardiness bounds depend on pertask utilizations and execution costs the lower these values, the lower the tardiness bounds. As a nal remark, we note that global EDF may be superior to partitioned EDF for multiprocessorbased soft realtime systems in that the latter does not offer any scope to improve system utilization even if bounded tardiness can be tolerated. ¤Work supported by NSF grants CCR 0204312, CCR 0309825, and CCR 0408996. The rst author was also supported by an IBM Ph.D. fellowship.
Diffusion approximation for a processor sharing queue in heavy traffic
, 2004
"... 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 ..."
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Cited by 27 (2 self)
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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 onedimensional 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
A Stochastic Framework for Multiprocessor Soft RealTime Scheduling ∗
"... Prior work has shown that the global earliestdeadlinefirst (GEDF) scheduling algorithm ensures bounded deadline tardiness on multiprocessors with no utilization loss; therefore, GEDF may be a good candidate scheduling algorithm for soft realtime workloads. However, such workloads are often implem ..."
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Cited by 19 (5 self)
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Prior work has shown that the global earliestdeadlinefirst (GEDF) scheduling algorithm ensures bounded deadline tardiness on multiprocessors with no utilization loss; therefore, GEDF may be a good candidate scheduling algorithm for soft realtime workloads. However, such workloads are often implemented assuming an averagecase provisioning, and in prior tardinessbound derivations for GEDF, worstcase execution costs are assumed. As worstcase costs can be orders of magnitude higher than averagecase costs, using a worstcase provisioning may result in significant wasted processing capacity. In this paper, prior tardinessbound 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
Control of EndtoEnd Delay Tails in a Multiclass Network: LWDF Discipline Optimality
 Annals of Applied Probability
"... We consider a multiclass queueing network with N customer classes, each having an arbitrary fixed route through the network. (Thus, the network is not necessarily feedforward.) We show that the largest weighted delay first (LWDF) discipline is an optimal scheduling discipline in the network in the f ..."
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Cited by 18 (2 self)
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We consider a multiclass queueing network with N customer classes, each having an arbitrary fixed route through the network. (Thus, the network is not necessarily feedforward.) We show that the largest weighted delay first (LWDF) discipline is an optimal scheduling discipline in the network in the following sense. Let wi be the (random) instantaneous largest endtoend delay of a class i customer in the network in stationary regime. For any set of positive constants α1,...,αN, the LWDF discipline associated with this set maximizes (among all disciplines) the quantity (1) min αi lim i=1,...,N n→∞ −1 n log P(wi>n) = lim n→∞ −1 log P(r>n), n where r. = maxi wi/αi is the maximal weighted delay in the network. [This result is a generalization of the singleserver result proved by A. L. Stolyar and K. Ramanan in Ann. Appl. Probab. 11 (2001) 1–48.] As the key element of the proof, we establish the following critical node property: In a LWDF network, there exists a most likely path to build large r, which is a most likely path to do so in one of the network nodes in isolation. Such a most likely path has a very simple structure: its parameters [and the optimal value of (1)] can be computed by solving a finitedimensional optimization problem for each network node.
Fluid Limits for Processor Sharing Queues with Impatience
 in "Mathematics of Operation Research", To Appear, 2007. Publications in Conferences and Workshops
"... 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 s ..."
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Cited by 17 (6 self)
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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.
Approximating the GI/GI/1+GI queue with a nonlinear drift diffusion: Hazard rate scaling in heavy traffic
 Mathematics of Operations Research
, 2008
"... We study a single server queue, operating under the FIFO service discipline, in which each customer independently abandons the queue if his service has not begun within a generally distributed amount of time. Under some mild conditions on the abandonment distribution, we identify a limiting heavy tr ..."
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Cited by 16 (3 self)
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We study a single server queue, operating under the FIFO service discipline, in which each customer independently abandons the queue if his service has not begun within a generally distributed amount of time. Under some mild conditions on the abandonment distribution, we identify a limiting heavy traffic regime in which the resulting diffusion approximation for both the offered waiting time process (the process that tracks the amount of time an infinitely patient arriving customer would wait for service) and the queue length process contains the entire abandonment distribution. In order to use a continuous mapping approach to establish our weak convergence results, we additionally develop existence, uniqueness, and continuity results for nonlinear generalized regulator mappings that are of independent interest. We further perform a simulation study to evaluate the quality of the proposed approximations for the steadystate mean queuelength and the steadystate probability of abandonment suggested by the limiting diffusion process.
EarliestDeadlineFirst Service in HeavyTraffic Acyclic Networks
 Annals of Applied Probability
, 2002
"... This paper presents a heavy traffic analysis of the behavior of multiclass 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 ..."
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Cited by 16 (4 self)
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This paper presents a heavy traffic analysis of the behavior of multiclass 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 earliestdeadlinefirst (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 measurevalued leadtime process converges to a deterministic function of the workload process. A twostation 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.
Realtime delay estimation based on delay history
, 2007
"... 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 realtime delay estimators based on recent customer delay experience. The main estimators considered are: (i) the delay of the ..."
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Cited by 14 (4 self)
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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 realtime 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 delayhistory 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 multiserver 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 manyserver and classical heavytraffic limiting regimes.
A multiclass queue in heavy traffic with throughput time constraints: Asymptotically optimal dynamic controls. Queueing Syst
 Theory Appl
, 2001
"... Abstract. Consider a singleserver queueing system with K job classes, each having its own renewal input process and its own general service time distribution. Further suppose the queue is in heavy traffic, meaning that its traffic intensity parameter is near the critical value of one. A system mana ..."
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Cited by 12 (4 self)
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Abstract. Consider a singleserver queueing system with K job classes, each having its own renewal input process and its own general service time distribution. Further suppose the queue is in heavy traffic, meaning that its traffic intensity parameter is near the critical value of one. A system manager must decide whether or not to accept new jobs as they arrive, and also the order in which to serve jobs that are accepted. The goal is to minimize penalties associated with rejected jobs, subject to upper bound constraints on the throughput times for accepted jobs; both the penalty for rejecting a job and the bound on the throughput time may depend on job class. This problem formulation does not make sense in a conventional queueing model, because throughput times are random variables, but we show that the formulation is meaningful in an asymptotic sense, as one approaches the heavy traffic limit under diffusion scaling. Moreover, using a method developed recently by Bramson and Williams, we prove that a relatively simple dynamic control policy is asymptotically optimal in this framework. Our proposed policy rejects jobs from one particular class when the server’s nominal workload is above a threshold value, accepting all other arrivals; and the sequencing rule for accepted jobs is one that maintains near equality of the relative backlogs for different classes, defined in a natural sense. 1. Introduction and