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36
Polling Systems in Heavy Traffic: A Bessel Process Limit
, 1996
"... This paper studies the classical polling model under the exhaustiveservice assumption; such models continue to be very useful in performance studies of computer/communication systems. The analysis here extends earlier work of the authors to the general case of nonzero switchover times. It shows tha ..."
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This paper studies the classical polling model under the exhaustiveservice assumption; such models continue to be very useful in performance studies of computer/communication systems. The analysis here extends earlier work of the authors to the general case of nonzero switchover times. It shows that, under the standard heavytraffic scaling, the total unfinished work in the system tends to a Besseltype diffusion in the heavytraffic limit. It verifies in addition that, with this change in the limiting unfinishedwork process, the averaging principle established earlier by the authors carries over to the general model. September 13, 1995 Polling Systems in Heavy Traffic: A Bessel Process Limit E. G. Coffman, Jr., y A. A. Puhalskii, # and M. I. Reiman y y AT&T Bell Laboratories, Murray Hill, NJ 07974 # Institute for Problems in Information Transmission, Moscow, Russia 101447 1 Introduction In classical polling models, M 2 queues are visited by a single server in cyclic or...
Mean value analysis for polling systems in heavy traffic
 In Proceedings of ValueTools, Pisa Article
, 2006
"... In this paper we present a new approach to derive heavytraffic asymptotics for polling models. We consider the classical cyclic polling model with exhaustive service at each queue, and with general servicetime and switchover time distributions, and study its behavior when the load tends to one. F ..."
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In this paper we present a new approach to derive heavytraffic asymptotics for polling models. We consider the classical cyclic polling model with exhaustive service at each queue, and with general servicetime and switchover time distributions, and study its behavior when the load tends to one. For this model, we explore the recently proposed mean value analysis (MVA), which takes a new view on the dynamics of the system, and use this view to provide an alternative way to derive closedfrom expressions for the expected asymptotic delay; the expressions were derived earlier in [32], but in a different way. Moreover, the MVAbased approach enables us to derive closedform expressions for the heavytraffic limits of the covariances between the successive visit periods, which are key performance metrics in many application areas. These results, which have not been obtained before, reveal a number of insensitivity properties of the covariances with respect to the system parameters under heavytraffic assumptions, and moreover, lead to simple approximations for the covariances between the successive visit times for stable systems. Numerical examples demonstrate that the approximations are accurate when the load is close enough to one.
Analysis and Control of Mobile Communications with Time Varying Channels in Heavy Traffic
 IEEE Trans. Autom. Control
, 2000
"... Consider a system with a xed number (K) of remote units and a single base transmitter with time varying (and perhaps correlated) connecting channels. Data to be transmitted to the remote units arrives according to some random process and is queued according to its destination. The uplink is treated, ..."
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Consider a system with a xed number (K) of remote units and a single base transmitter with time varying (and perhaps correlated) connecting channels. Data to be transmitted to the remote units arrives according to some random process and is queued according to its destination. The uplink is treated, for specificity. Power is to be allocated to the K channels in a time varying way. The modeling and control problem can be quite difficult. The channel time variations (fading) are fast and the bandwidth and data arrival rate are high. Owing to the complexity of the physical problem and the high speed of both the fading and arrival and service rates, an asymptotic or averaging method is promising. A heavy traffic analysis is done. By heavy traffic, we mean that on the average there is little server idle time and little spare power. Heavy traffic analysis has been very helpful in simplifying analysis of both controlled and uncontrolled problems in queueing and communications networks. It tends to elimin...
Scheduling Policies for an Antiterrorist Surveillance System
, 2008
"... This paper concerns scheduling policies in a surveillance system aimed at detecting a terrorist attack in time. Terrorist suspects arriving at a public area are subject to continuous monitoring, while a surveillance team takes their biometric signatures and compares them with records stored in a ter ..."
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This paper concerns scheduling policies in a surveillance system aimed at detecting a terrorist attack in time. Terrorist suspects arriving at a public area are subject to continuous monitoring, while a surveillance team takes their biometric signatures and compares them with records stored in a terrorist database. Because the surveillance team can screen only one terrorist suspect at a time, the team faces a dynamic scheduling problem among the suspects. We build a model consisting of an M/G/1 queue with two types of customers—red and white—to study this problem. Both types of customers are impatient, but the reneging time distributions are different. The server only receives a reward by serving a red customer, and can use the time a customer has spent in the queue to deduce its likely type. In a few special cases, a simple service rule—such as firstcomefirstserve—is optimal. We explain why the problem is in general difficult, and develop a heuristic policy motivated by the fact that terrorist attacks tend to be rare events.
Beyond the cµ Rule: Dynamic Scheduling Of A TwoClass Loss Queue
, 1997
"... We consider scheduling a single server in a twoclass M/M/1 queueing system with finite buffers subject to holding costs and rejection costs for rejected jobs. We use dynamic programming to investigate the structural properties of optimal policies. Provided that the delay of serving a job is alwa ..."
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We consider scheduling a single server in a twoclass M/M/1 queueing system with finite buffers subject to holding costs and rejection costs for rejected jobs. We use dynamic programming to investigate the structural properties of optimal policies. Provided that the delay of serving a job is always less costly than rejecting an arrival, we show that the optimal policy has a monotonic threshold type of switching curve; otherwise, numerical analysis indicates that the threshold structure may not be optimal.
A mathematical programming approach to stochastic and dynamic optimization problems
, 1994
"... We survey a new approach that the author and his coworkers have developed to formulate generic stochastic and dynamic optimization problems as mathematical programming problems. The approach has two components: (a) it produces bounds on the performance of an optimal policy, and (b) it develops tech ..."
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We survey a new approach that the author and his coworkers have developed to formulate generic stochastic and dynamic optimization problems as mathematical programming problems. The approach has two components: (a) it produces bounds on the performance of an optimal policy, and (b) it develops techniques to construct optimal or nearoptimal policies. The central idea for developing bounds is to characterize the region of achievable performance (or performance space) in a stochastic and dynamic optimization problem, i.e., find linear or nonlinear constraints on the performance vectors that all admissible policies satisfy. With respect to this goal we review recent progress in characterizing the performance space and its implications for the following problem classes: Indexable systems (the multiarmed bandit problem and its extensions), polling systems, multiclass queueing networks and loss networks. We propose three ideas for constructing optimal or nearoptimal policies: (1) for systems for which we have an exact characterization of the performance space we outline an adaptive greedy algorithm that gives rise to indexing policies (we illustrate this technique in the context of indexable systems); (2) we use integer programming to construct policies from the underlying descriptions of
Flexible Servers in Tandem Lines with Setups
"... We study the dynamic assignment of flexible servers to stations in the presence of setup costs that are incurred when servers move between stations. We focus on tandem lines with two stations and two servers with the goal of maximizing the longrun average profit. We investigate how the optimal serv ..."
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We study the dynamic assignment of flexible servers to stations in the presence of setup costs that are incurred when servers move between stations. We focus on tandem lines with two stations and two servers with the goal of maximizing the longrun average profit. We investigate how the optimal server assignment policy for such systems depends on the magnitude of the setup costs, as well as on the homogeneity of servers and tasks. More specifically, for systems with either homogeneous servers or homogeneous tasks, small buffer sizes, and constant setup cost, we prove the optimality of “multiple threshold ” policies (where servers ’ movement between stations depends on both the number of jobs in the system and the locations of the servers) and determine the values of the thresholds. For systems with heterogeneous servers and tasks, small buffers, and constant setup cost, we provide results that partially characterize the optimal server assignment policy. Finally, for systems with larger buffer sizes and different service rate and setup cost configurations, we present structural results for the optimal policy and provide numerical results that strongly support the optimality of multiple threshold policies.
Polling Systems with Periodic Server Routing in HeavyTraffic: Renewal Arrivals
"... This paper considers heavytraffic limit theorems for polling models with periodic server routing with general renewal arrivals and with mixtures of gated and exhaustive service policies, in a general parameter setting. We provide a strong conjecture for the limiting waitingtime distribution in a g ..."
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This paper considers heavytraffic limit theorems for polling models with periodic server routing with general renewal arrivals and with mixtures of gated and exhaustive service policies, in a general parameter setting. We provide a strong conjecture for the limiting waitingtime distribution in a general parameter setting when the load tends to 1, under proper heavytraffic scalings. The results in this paper, which generalize those in [14] for the special case of Poisson arrivals, are obtained by following the nonPoisson transient systems approach developed in [6]. Numerical results with simulations support the correctness our results, and moreover, show that the expressions for the waitingtime distributions in the limiting case lead to sharp approximations for the waitingtime distributions for stable polling systems with medium and heavy load. Keywords: polling system, polling table, periodic routing, delay, waiting time, heavytraffic, This paper considers heavytraffic limit theorems for polling models with periodic server routing under renewal arrivals. As such, we generalize results in Olsen and Van der Mei [14], which were proven for these systems under Poisson arrivals in steadystate. In such systems the order in which
Control Problems in Telecommunications: The Heavy Traffic Approach
, 2000
"... The goal of this chapter is to demonstrate the usefulness of analytical and numerical methods of stochastic control theory in the design, analysis and control of telecommunication networks. The emphasis will be concentrated on the heavy traffic approach for queueing type systems in which there is li ..."
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The goal of this chapter is to demonstrate the usefulness of analytical and numerical methods of stochastic control theory in the design, analysis and control of telecommunication networks. The emphasis will be concentrated on the heavy traffic approach for queueing type systems in which there is little idle time and the queue length processes can be approximated by reected diffusion processes under suitable scaling. Three principal problems are considered: the multiplexer system, controlled admission in multiserver systems such as ISDN, and the polling or scheduling problem.