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17
Logarithmic Asymptotics For SteadyState Tail Probabilities In A SingleServer Queue
, 1993
"... We consider the standard singleserver queue with unlimited waiting space and the firstin firstout service discipline, but without any explicit independence conditions on the interarrival and service times. We find conditions for the steadystate waitingtime distribution to have smalltail asympt ..."
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Cited by 176 (14 self)
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We consider the standard singleserver queue with unlimited waiting space and the firstin firstout service discipline, but without any explicit independence conditions on the interarrival and service times. We find conditions for the steadystate waitingtime distribution to have smalltail asymptotics of the form x  1 logP(W > x)  q * as x for q * > 0. We require only stationarity of the basic sequence of service times minus interarrival times and a Ga .. rtnerEllis condition for the cumulant generating function of the associated partial sums, i.e., n  1 log Ee qS n y(q) as n , plus regularity conditions on the decay rate function y. The asymptotic decay rate q * is the root of the equation y(q) = 0. This result in turn implies a corresponding asymptotic result for the steadystate workload in a queue with general nondecreasing input. This asymptotic result covers the case of multiple independent sources, so that it provides additional theoretical support for a concept of effective bandwidths for admission control in multiclass queues based on asymptotic decay rates.
Dynamic Scheduling of a System with Two Parallel Servers in Heavy Traffic with Resource Pooling: Asymptotic Optimality of a Threshold Policy
 Annals of Applied Probability
, 1999
"... This paper concerns a dynamic scheduling problem for a queueing system that has two streams of arrivals to infinite capacity buffers and two (nonidentical) servers working in parallel. One server can only process jobs from one buffer, whereas the other server can process jobs from either buffer. Th ..."
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Cited by 118 (6 self)
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This paper concerns a dynamic scheduling problem for a queueing system that has two streams of arrivals to infinite capacity buffers and two (nonidentical) servers working in parallel. One server can only process jobs from one buffer, whereas the other server can process jobs from either buffer. The service time distribution may depend on the buffer being served and the server providing the service. The system manager dynamically schedules waiting jobs onto available servers. We consider a parameter regime in which the system satisfies both a heavy traffic condition and a resource pooling condition. Our cost function is a mean cumulative discounted cost of holding jobs in the system, where the (undiscounted) cost per unit time is a linear function of normalized (with heavy traffic scaling) queue length. We first review the analytic solution of the Brownian control problem (formal heavy traffic approximation) for this system. We "interpret" this solution by proposing a threshold contro...
On Dynamic Scheduling of a Parallel Server System with Complete Resource Pooling
 In Analysis of Communication Networks: Call Centres, Traffic and Performance
, 2000
"... scientific noncommercial use only for individuals, with permission from the authors. We consider a parallel server queueing system consisting of a bank of buffers for holding incoming jobs and a bank of flexible servers for processing these jobs. Incoming jobs are classified into one of several dif ..."
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Cited by 71 (5 self)
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scientific noncommercial use only for individuals, with permission from the authors. We consider a parallel server queueing system consisting of a bank of buffers for holding incoming jobs and a bank of flexible servers for processing these jobs. Incoming jobs are classified into one of several different classes (or buffers). Jobs within a class are processed on a firstinfirstout basis, where the processing of a given job may be performed by any server from a given (classdependent) subset of the bank of servers. The random service time of a job may depend on both its class and the server providing the service. Each job departs the system after receiving service from one server. The system manager seeks to minimize holding costs by dynamically scheduling waiting jobs to available servers. We consider a parameter regime in which the system satisfies both a heavy traffic and a complete resource pooling condition. Our cost function is an expected cumulative discounted cost of holding jobs in the system, where the (undiscounted) cost per unit time is a linear function of normalized (with heavy traffic scaling) queue length. In a prior work [40], the second author proposed a continuous review threshold control policy for use in such a parallel server system. This policy was advanced as an “interpretation ” of the analytic solution to an associated Brownian control problem (formal heavy
Scheduling a multiclass queue with many exponential servers: Asymptotic optimality in heavytraffic,” The Annals of Applied Probability
, 2004
"... We consider the problem of scheduling a queueing system in which many statistically identical servers cater to several classes of impatient customers. Service times and impatience clocks are exponential while arrival processes are renewal. Our cost is an expected cumulative discounted function, line ..."
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Cited by 42 (13 self)
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We consider the problem of scheduling a queueing system in which many statistically identical servers cater to several classes of impatient customers. Service times and impatience clocks are exponential while arrival processes are renewal. Our cost is an expected cumulative discounted function, linear or nonlinear, of appropriately normalized performance measures. As a special case, the cost per unit time can be a function of the number of customers waiting to be served in each class, the number actually being served, the abandonment rate, the delay experienced by customers, the number of idling servers, as well as certain combinations thereof. We study the system in an asymptotic heavytraffic regime where the number of servers n and the offered load r are simultaneously scaled up and carefully balanced: n ≈ r + β √ r for some scalar β. This yields an operation that enjoys the benefits of both heavy traffic (high server utilization) and light traffic (high service levels.)
Heavy traffic analysis of open processing networks with complete resource pooling: asymptotic optimality of discrete review policies
 ANN. APPL. PROBAB
, 2005
"... We consider a class of open stochastic processing networks, with feedback routing and overlapping server capabilities, in heavy traffic. The networks ..."
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Cited by 25 (0 self)
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We consider a class of open stochastic processing networks, with feedback routing and overlapping server capabilities, in heavy traffic. The networks
Conditions for the Propagation of Memory Parameter from Durations to Counts and Realized Volatility. Working Paper
, 2006
"... We establish sufficient conditions on durations that are stationary with finite variance and memory parameter d ∈ [0, 1/2) to ensure that the corresponding counting process N(t) satisfies VarN(t) ∼ Ct 2d+1 (C> 0) as t → ∞, with the same memory parameter d ∈ [0, 1/2) that was assumed for the durat ..."
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Cited by 15 (4 self)
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We establish sufficient conditions on durations that are stationary with finite variance and memory parameter d ∈ [0, 1/2) to ensure that the corresponding counting process N(t) satisfies VarN(t) ∼ Ct 2d+1 (C> 0) as t → ∞, with the same memory parameter d ∈ [0, 1/2) that was assumed for the durations. Thus, these conditions ensure that the memory in durations propagates to the same memory parameter in counts and therefore in realized volatility. We then show that any Autoregressive Conditional Duration ACD(1,1) model with a sufficient number of finite moments yields short memory in counts, while any Long Memory Stochastic Duration model with d> 0 and all finite moments yields long memory in counts, with the same d. Finally, we present a result implying that the only way for a series of counts aggregated over a long time period to have nontrivial autocorrelation is for the shortterm counts to have long memory. In other words, aggregation ultimately destroys all autocorrelation in counts, if and only if the counts have short memory.
Heavy Traffic Analysis for EDF Queues with
, 2007
"... This paper presents a heavytraffic analysis of the behavior of a singleserver queue under an EarliestDeadlineFirst (EDF) scheduling policy, in which customers have deadlines and are served only until their deadlines elapse. The performance of the system is measured by the fraction of reneged wor ..."
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Cited by 5 (1 self)
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This paper presents a heavytraffic analysis of the behavior of a singleserver queue under an EarliestDeadlineFirst (EDF) scheduling policy, in which customers have deadlines and are served only until their deadlines elapse. The performance of the system is measured by the fraction of reneged work (the residual work lost due to elapsed deadlines), which is shown to be minimized by the EDF policy. The evolution of the lead time distribution of customers in queue is described by a measurevalued process. The heavy traffic limit of this (properly scaled) process is shown to be a deterministic function of the limit of the scaled workload process, which, in turn, is identified to be a doubly reflected Brownian motion. This paper complements previous work by Doytchinov, Lehoczky and Shreve on the EDF discipline, in which customers are served to completion even after their deadlines elapse. The fraction of reneged work in a heavily loaded system and the fraction of late work in the corresponding system without reneging are compared using explicit formulas based on the heavy
DIFFUSION APPROXIMATION FOR A HEAVILY LOADED MULTIUSER WIRELESS COMMUNICATION SYSTEM WITH COOPERATION
, 2008
"... Abstract. A cellular wireless communication system in which data is transmitted to multiple users over a common channel is considered. When the base stations in this system can cooperate with each other, the link from the base stations to the users can be considered a multiuser multipleinput multi ..."
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Cited by 5 (0 self)
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Abstract. A cellular wireless communication system in which data is transmitted to multiple users over a common channel is considered. When the base stations in this system can cooperate with each other, the link from the base stations to the users can be considered a multiuser multipleinput multipleoutput (MIMO) downlink system. For such a system, it is known from information theory that the total rate of transmission can be enhanced by cooperation. The channel is assumed to be fixed for all transmissions over the period of interest and the ratio of anticipated average arrival rates for the users, also known as the relative traffic rate, is fixed. A packetbased model is considered where data for each user is queued at the transmit end. We consider a simple policy which, under Markovian assumptions, is known to be throughputoptimal for this coupled queueing system. Since an exact expression for the performance of this policy is not available, as a measure of performance, we establish a heavy traffic diffusion approximation. This diffusion process is a semimartingale reflecting Brownian motion (SRBM) living in the positive orthant of Ndimensional space (where N is the number of users). Nominally, this SRBM has one direction of reflection associated with each of the 2 N − 1 boundary faces. We show that, in fact, only those directions associated with the (N − 1)–dimensional boundary faces matter for the heavy traffic limit.
Accuracy of state space collapse for earliestdeadlinefirst queues
 Annals of Applied Probability
, 2006
"... This paper presents a secondorder heavy traffic analysis of a single server queue that processes customers having deadlines using the earliestdeadlinefirst scheduling policy. For such systems, referred to as realtime queueing systems, performance is measured by the fraction of customers who meet ..."
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Cited by 5 (2 self)
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This paper presents a secondorder heavy traffic analysis of a single server queue that processes customers having deadlines using the earliestdeadlinefirst scheduling policy. For such systems, referred to as realtime queueing systems, performance is measured by the fraction of customers who meet their deadline, rather than more traditional performance measures, such as customer delay, queue length or server utilization. To model such systems, one must keep track of customer lead times (the time remaining until a customer deadline elapses) or equivalent information. This paper reviews the earlier heavy traffic analysis of such systems that provided approximations to the system’s behavior. The main result of this paper is the development of a secondorder analysis that gives the accuracy of the approximations and the rate of convergence of the sequence of realtime queueing systems to its heavy traffic limit. 1. Introduction.
On the Performance of a Two User MIMO Downlink System in Heavy Traffic
"... A MIMO downlink system in which data is transmitted to two users over a common wireless channel is considered. The channel is assumed to be fixed for all transmissions over the period of interest and the ratio of anticipated average arrival rates for the two users, also known as the relative traffi ..."
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Cited by 4 (1 self)
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A MIMO downlink system in which data is transmitted to two users over a common wireless channel is considered. The channel is assumed to be fixed for all transmissions over the period of interest and the ratio of anticipated average arrival rates for the two users, also known as the relative traffic rate, is the system design parameter. A packetbased traffic model is considered where data for each user is queued at the transmit end. A queueing analogue for this system leads to a coupled queueing system for which a simple policy is known to be throughputoptimal under Markovian assumptions. Since an exact expression for the performance is not available, as a measure of performance (in heavy traffic), a diffusion approximation is established. This diffusion process is a twodimensional semimartingale reflecting Brownian motion living in the positive quadrant of twodimensional space.