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Scheduling for endtoend deadlineconstrained traffic with reliability requirements in multihop networks technical report.” [Online]. Available: http://somewhere APPENDIX A PROOF OF PROPOSITION 4 To prove Proposition 4, we consider our onehop system as
 Lemma 1: limm!1 cm = c for all A ∈ (p). Proof: For all A ∈ (p), we have ∑f (1 − pf )af < T . We
"... Abstract—We attack the challenging problem of designing a scheduling policy for endtoend deadlineconstrained traffic with reliability requirements in a multihop network. It is wellknown that the endtoend delay performance for a multihop flow has a complex dependence on the highorder statist ..."
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Abstract—We attack the challenging problem of designing a scheduling policy for endtoend deadlineconstrained traffic with reliability requirements in a multihop network. It is wellknown that the endtoend delay performance for a multihop flow has a complex dependence on the highorder statistics of the arrival process and the algorithm itself. Thus, neither the earlier optimization based approaches that aim to meet the longterm throughput demands, nor the solutions that focus on a similar problem for singlehop flows directly apply. Moreover, a dynamic programmingbased approach becomes intractable for such multitime scale QualityofService(QoS)constrained traffic in a multihop environment. This motivates us in this work to develop an alternative model that enables us to exploit the degree of freedom in choosing appropriate service discipline. Based on the new model, we propose two alternative solutions, first based on a Lyapunovdrift minimization approach, and second based on a novel relaxed optimizationformulation. We provide extensive numerical results to compare the performance of both of these solutions to throughputoptimal backpressuretype schedulers and to longest waiting time based schedulers that have provably optimal asymptotic performance characteristics. Our results reveal that the dynamic choice of service discipline of our proposed solutions yields substantial performance improvements compared to both of these types of traditional solutions under nonasymptotic conditions. I.
Large deviations of queues under QoS scheduling algorithms,” Allerton 2006
, 2006
"... Abstract — We consider a model where multiple queues are served by a server whose capacity varies randomly and asynchronously with respect to different queues. The problem is to optimally control large deviations of the queues in the following sense: find a scheduling rule maximizing ..."
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Abstract — We consider a model where multiple queues are served by a server whose capacity varies randomly and asynchronously with respect to different queues. The problem is to optimally control large deviations of the queues in the following sense: find a scheduling rule maximizing
Large deviations of maxweight scheduling policies on convex rate regions
, 2007
"... Abstract—We consider a single server discretetime system with K users where the server picks operating points from a compact, convex and coordinate convex set in ℜ K +. For this system we analyse the performance of a stablising policy that at any given time picks operating points from the allowed ..."
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Abstract—We consider a single server discretetime system with K users where the server picks operating points from a compact, convex and coordinate convex set in ℜ K +. For this system we analyse the performance of a stablising policy that at any given time picks operating points from the allowed rate region that maximise a weighted sum of rate, where the weights depend upon the workloads of the users. Assuming a Large Deviations Principle (LDP) for the arrival processes in the Skorohod space of functions that are rightcontinuous with lefthand limits we establish an LDP for the workload process using a generalised version of the contraction principle to derive the corresponding rate function. With the LDP result available we then analyse the tail probabilities of the workloads under different buffering scenarios. I.
Duedate scheduling: asymptotic optimality of generalized longest queue and generalized largest delay rules
 Operations Research
, 2003
"... Consider the following duedate scheduling problem in a multiclass, acyclic, singlestation service system: Any class k job arriving at time t must be served by its due date t +Dk. Equivalently, its delay �k must not exceed a given delay or leadtime Dk. In a stochastic system, the constraint �k � D ..."
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Consider the following duedate scheduling problem in a multiclass, acyclic, singlestation service system: Any class k job arriving at time t must be served by its due date t +Dk. Equivalently, its delay �k must not exceed a given delay or leadtime Dk. In a stochastic system, the constraint �k � Dk must be interpreted in a probabilistic sense. Regardless of the precise probabilistic formulation, however, the associated optimal control problem is intractable with exact analysis. This article proposes a new formulation which incorporates the constraint through a sequence of convexincreasing delay cost functions. This formulation reduces the intractable optimal scheduling problem into one for which the Generalized c � (Gc�) scheduling rule is known to be asymptotically optimal. The Gc � rule simplifies here to a generalized longest queue (GLQ) or generalized largest delay (GLD) rule, which are defined as follows. Let Nk be the number of class k jobs in system, �k their arrival rate, and ak the age of their oldest job in the system. GLQ and GLD are dynamic priority rules, parameterized by �: GLQ(�) serves FIFO within class and prioritizes the class with highest index �kNk, whereas GLD(�) uses index �k�kak. The argument is presented first intuitively, but is followed by a limit analysis that expresses the cost objective in terms of the maximal duedate violation probability. This proves that GLQ(�∗) and GLD(�∗), where �∗�k = 1/�kDk, asymptotically minimize the probability �k�ns � of maximal duedate violation in heavy traffic. Specifically, they minimize lim inf n→ � Pr�maxk sups∈�0�t � n1/2 � x � for all positive t and Dk x, where �k�s � is the delay of the most recent class k job that arrived before time s. GLQ with appropriate parameter � � also reduces “total variability ” because it asymptotically minimizes a weighted sum of �th delay moments. Properties of GLQ and GLD, including an expression for their asymptotic delay distributions, are presented.
2005) Large deviations methods and the jointheshortestqueue model. Report. Available (JSQreport) from http://www.ee.technion.ac.il/~adam/PAPERS
"... Please send questions and/or remarks of nonscientific nature to driessen@tinbergen.nl. Most TI discussion papers can be downloaded at ..."
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Please send questions and/or remarks of nonscientific nature to driessen@tinbergen.nl. Most TI discussion papers can be downloaded at
Explicit solution for a network control problem in the large deviation regime
 Queueing Systems
, 2004
"... Abstract. We consider optimal control of a stochastic network, where service is controlled to prevent buffer overflow. We use a risksensitive escape time criterion, which in comparison to the ordinary escape time criteria heavily penalizes exits which occur on short time intervals. A limit as the b ..."
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Abstract. We consider optimal control of a stochastic network, where service is controlled to prevent buffer overflow. We use a risksensitive escape time criterion, which in comparison to the ordinary escape time criteria heavily penalizes exits which occur on short time intervals. A limit as the buffer sizes tend to infinity is considered. In [2] we showed that, for a large class of networks, the limit of the normalized cost agrees with the value function of a differential game. In this game, one player controls the service discipline (who to serve and whether to serve), and the other player chooses arrival and service rates in the network. The game’s value is characterized in [2] as the unique solution to a Hamilton–Jacobi–Bellman Partial Differential Equation (PDE). In the current paper we apply this general theory to the important case of a network of queues in tandem. Our main results are: (i) the construction of an explicit solution to the corresponding PDE, and (ii) drawing out the implications for optimal risksensitive and robust regulation of the network. In particular, the following general principle can be extracted. To avoid buffer overflow there is a natural competition between two tendencies. One may choose to serve a particular queue, since that will help prevent its own buffer from overflowing, or one may prefer to stop service, with the goal of preventing overflow of buffers further down the line. The solution to the PDE indicates the optimal choice between these two, specifying the parts of the state space where each queue must be served (so as not to
Large deviations without principle: Join the shortest queue
, 2005
"... We develop a methodology for studying “large deviations type” questions. Our approach does not require that the large deviations principle holds, and is thus applicable to a larg class of systems. We study a system of queues with exponential servers, which share an arrival stream. Arrivals are route ..."
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We develop a methodology for studying “large deviations type” questions. Our approach does not require that the large deviations principle holds, and is thus applicable to a larg class of systems. We study a system of queues with exponential servers, which share an arrival stream. Arrivals are routed to the (weighted) shortest queue. It is not known whether the large deviations principle holds for this system. Using the tools developed here we derive large deviations type estimates for the most likely behavior, the most likely path to overflow and the probability of overflow. The analysis applies to any finite number of queues. We show via a counterexample that this sytem may exhibit unexpected behavior.
Modeling, Scheduling and Simulation of Switched Processing Systems
"... Switched Processing Systems (SPS) serve as canonical models in a wide area of applications such as high performance computing, wireless networking, call centers, and flexible manufacturing. In this paper, we model the SPS by considering both slotted and continuous time and analyze it under fairly mi ..."
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Switched Processing Systems (SPS) serve as canonical models in a wide area of applications such as high performance computing, wireless networking, call centers, and flexible manufacturing. In this paper, we model the SPS by considering both slotted and continuous time and analyze it under fairly mild stochastic assumptions. Two classes of scheduling policies are introduced and shown to maximize the throughput and maintain strong stability of the system. In addition, their performance with respect to the average job sojourn time is examined by simulating small SPS subject to different types of input traffic. By utilizing the simulation result of the proposed policies, a hybrid control policy is constructed to reduce the average job sojourn time when the system has unknown and changing input loads.
Stochastics and Statistics Queueing
, 2001
"... systems with leadtime constraints: A fluidmodel approach for admission and sequencing control ..."
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systems with leadtime constraints: A fluidmodel approach for admission and sequencing control