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212
CDMA Uplink Power Control as a Noncooperative Game
, 2002
"... We present a gametheoretic treatment of distributed power control in CDMA wireless systems. ..."
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Cited by 168 (22 self)
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We present a gametheoretic treatment of distributed power control in CDMA wireless systems.
Threshold autoregression with a unit root
 Econometrica
, 2001
"... This paper develops an asymptotic theory of inference for an unrestricted tworegime threshold autoregressive Ž TAR. model with an autoregressive unit root. We find that the asymptotic null distribution of Wald tests for a threshold are nonstandard and different from the stationary case, and suggest ..."
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Cited by 103 (1 self)
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This paper develops an asymptotic theory of inference for an unrestricted tworegime threshold autoregressive Ž TAR. model with an autoregressive unit root. We find that the asymptotic null distribution of Wald tests for a threshold are nonstandard and different from the stationary case, and suggest basing inference on a bootstrap approximation. We also study the asymptotic null distributions of tests for an autoregressive unit root, and find that they are nonstandard and dependent on the presence of a threshold effect. We propose both asymptotic and bootstrapbased tests. These tests and distribution theory allow for the joint consideration of nonlinearity Ž thresholds. and nonstationary Žunit roots.. Our limit theory is based on a new set of tools that combine unit root asymptotics with empirical process methods. We work with a particular twoparameter empirical process that converges weakly to a twoparameter Brownian motion. Our limit distributions involve stochastic integrals with respect to this twoparameter process. This theory is entirely new and may find applications in other contexts. We illustrate the methods with an application to the U.S. monthly unemployment rate. We find strong evidence of a threshold effect. The point estimates suggest that the threshold effect is in the shortrun dynamics, rather than in the dominate root. While the conventional ADF test for a unit root is insignificant, our TAR unit root tests are arguably significant. The evidence is quite strong that the unemployment rate is not a unit root process, and there is considerable evidence that the series is a stationary TAR process.
Estimation in Conditionally Heteroscedastic Time Series Models
 Lecture Notes in Statist. 181
, 2005
"... This paper studies the quasimaximumlikelihood estimator (QMLE) in a general conditionally heteroscedastic time series model of multiplicative form Xt = σt Zt, where the unobservable volatility σt is a parametric function of (Xt−1,...,Xt−p,σt−1,...,σt−q) for some p,q ≥ 0, and (Zt) is standardized i ..."
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Cited by 81 (2 self)
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This paper studies the quasimaximumlikelihood estimator (QMLE) in a general conditionally heteroscedastic time series model of multiplicative form Xt = σt Zt, where the unobservable volatility σt is a parametric function of (Xt−1,...,Xt−p,σt−1,...,σt−q) for some p,q ≥ 0, and (Zt) is standardized i.i.d. noise. We assume that these models are solutions to stochastic recurrence equations which satisfy a contraction (random Lipschitz coefficient) property. These assumptions are satisfied for the popular GARCH, asymmetric GARCH and exponential GARCH processes. Exploiting the contraction property, we give conditions for the existence and uniqueness of a strictly stationary solution (Xt) to the stochastic recurrence equation and establish consistency and asymptotic normality of the QMLE. We also discuss the problem of invertibility of such time series models.
Efficiencydriven heavytraffic approximations for manyserver queues with abandonments
 Management Science
, 2004
"... Motivated by the desire to understand the performance of serviceoriented call centers, which often provide lowtomoderate quality of service, this paper investigates the efficiencydriven (ED) limiting regime for manyserver queues with abandonments. The starting point is the realization that, in ..."
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Cited by 74 (35 self)
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Motivated by the desire to understand the performance of serviceoriented call centers, which often provide lowtomoderate quality of service, this paper investigates the efficiencydriven (ED) limiting regime for manyserver queues with abandonments. The starting point is the realization that, in the presence of substantial customer abandonment, callcenter servicelevel agreements (SLA’s) can be met in the ED regime, where the arrival rate exceeds the maximum possible service rate. Mathematically, the ED regime is defined by letting the arrival rate and the number of servers increase together so that the probability of abandonment approaches a positive limit. To obtain the ED regime, it suffices to let the arrival rate and the number of servers increase with the traffic intensity ρ held fixed with ρ> 1 (so that the arrival rate exceeds the maximum possible service rate). Even though the probability of delay necessarily approaches 1 in the ED regime, the ED regime can be realistic because, due to the abandonments, the delays need not be excessively large. This paper establishes ED manyserver heavytraffic limits and develops associated approximations for performance measures in the M/M/s/r + M model, having a Poisson arrival process, exponential service times, s servers, r extra waiting spaces and exponential abandon times (the final +M). In the ED regime, essentially the same limiting behavior occurs when the abandonment rate α approaches 0 as when the number of servers s approaches ∞; indeed, it suffices to assume that s/α → ∞. The ED approximations are shown to be useful by comparing them to exact numerical results for the M/M/s/r + M model obtained using an algorithm developed in Whitt (2003), which exploits numerical transform inversion.
Source Time Scale and Optimal Buffer/Bandwidth Tradeoff for Heterogeneous Regulated Traffic in a Network Node
, 1996
"... In this paper, we study the problem of resource allocation and control for an ATM node with regulated traffic. Both guaranteed lossless service and statistical service with small loss probability are considered. We investigate the relationship between source characteristics and the buffer/bandwidt ..."
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Cited by 65 (3 self)
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In this paper, we study the problem of resource allocation and control for an ATM node with regulated traffic. Both guaranteed lossless service and statistical service with small loss probability are considered. We investigate the relationship between source characteristics and the buffer/bandwidth tradeoff under both services. Our contributions are the following. For guaranteed lossless service, we find that the optimal resource allocation scheme suggests a time scale separation of sources sharing an ATM node with finite bandwidth and buffer space, with the optimal buffer/bandwidth tradeoff determined by the sources' time scale. For statistical service with a small loss probability, we present a new approach for estimating the loss probability in a shared buffer multiplexor with the so called "extremal" onoff, periodic sources. Under this approach, the optimal resource allocation for statistical service is achieved by maximizing both the benefits of buffering sharing and ba...
Stationary maxstable fields associated to negative definite functions, Ann
 Prob
, 2009
"... Let Wi,i ∈ N, be independent copies of a zeromean Gaussian process {W(t),t ∈ R d} with stationary increments and variance σ 2 (t). Independently of Wi, let ∑∞ i=1 δUi be a Poisson point process on the real line with intensity e −y dy. We show that the law of the random family of functions {Vi(·),i ..."
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Cited by 61 (12 self)
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Let Wi,i ∈ N, be independent copies of a zeromean Gaussian process {W(t),t ∈ R d} with stationary increments and variance σ 2 (t). Independently of Wi, let ∑∞ i=1 δUi be a Poisson point process on the real line with intensity e −y dy. We show that the law of the random family of functions {Vi(·),i ∈ N}, where Vi(t) = Ui +Wi(t) − σ 2 (t)/2, is translation invariant. In particular, the process η(t) = ∨ ∞ i=1 Vi(t) is a stationary maxstable process with standard Gumbel margins. The process η arises as a limit of a suitably normalized and rescaled pointwise maximum of n i.i.d. stationary Gaussian processes as n → ∞ if and only if W is a (nonisotropic) fractional Brownian motion on R d. Under suitable conditions on W, the process η has a mixed moving maxima representation.
Forecasting Multifractal Volatility
 Journal of Econometrics
"... This paper develops analytical methods to forecast the distribution of future returns for a new continuoustime process, the Poisson multifractal. The process captures the thick tails, volatility persistence, and moment scaling exhibited by many nancial time series. It can be interpreted as a stocha ..."
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Cited by 56 (10 self)
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This paper develops analytical methods to forecast the distribution of future returns for a new continuoustime process, the Poisson multifractal. The process captures the thick tails, volatility persistence, and moment scaling exhibited by many nancial time series. It can be interpreted as a stochastic volatility model with multiple frequencies and a Markov latent state. We assume for simplicity that the forecaster knows the true generating process with certainty but only observes past returns. The challenge in this environment is long memory and the corresponding innite dimension of the state space. We introduce a discretized version of the model that has a nite state space and an analytical solution to the conditioning problem. As the grid step size goes to zero, the discretized model weakly converges to the continuoustime process, implying the consistency of the density forecasts. JEL Classication: C22; C53; F31 Keywords: Forecasting; Long memory; Multiple frequencies; Stoch...
HeavyTraffic Limits for the G/H ∗ 2 /n/m Queue
, 2005
"... We establish heavytraffic stochasticprocess limits for queuelength, waitingtime and overflow stochastic processes in a class of G/GI/n/m queueing models with n servers and m extra waiting spaces. We let the arrival process be general, only requiring that it satisfy a functional central limit the ..."
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Cited by 49 (12 self)
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We establish heavytraffic stochasticprocess limits for queuelength, waitingtime and overflow stochastic processes in a class of G/GI/n/m queueing models with n servers and m extra waiting spaces. We let the arrival process be general, only requiring that it satisfy a functional central limit theorem. To capture the impact of the servicetime distribution beyond its mean within a Markovian framework, we consider a special class of servicetime distributions, denoted by H ∗ 2, which are mixtures of an exponential distribution with probability p and a unit point mass at 0 with probability 1 − p. These servicetime distributions exhibit relatively high variability, having squared coefficients of variation greater than or equal to one. As in Halfin and Whitt (1981, Heavytraffic limits for queues with many exponential servers, Oper. Res. 29 567–588), Puhalskii and Reiman (2000, The multiclass GI/PH/N queue in the HalfinWhitt regime. Adv. Appl. Probab. 32 564–595), and Garnett, Mandelbaum, and Reiman (2002. Designing a call center with impatient customers. Manufacturing Service Oper. Management, 4 208–227), we consider a sequence of queueing models indexed by the number of servers, n, and let n tend to infinity along with the traffic intensities �n so that √ n�1 − �n � → � for − � <�<�. To treat finite waiting rooms, we let mn / √ n → � for 0 <�≤�. With the special H ∗ 2 servicetime distribution, the limit processes are
State space collapse and diffusion approximation for a network operating under a fair bandwidthsharing policy, in preparation
, 2004
"... We consider a connectionlevel model of Internet congestion control, introduced by Massoulié and Roberts [36], that represents the randomly varying number of flows present in a network. Here bandwidth is shared fairly amongst elastic document transfers according to a weighted αfair bandwidth sharin ..."
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Cited by 46 (8 self)
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We consider a connectionlevel model of Internet congestion control, introduced by Massoulié and Roberts [36], that represents the randomly varying number of flows present in a network. Here bandwidth is shared fairly amongst elastic document transfers according to a weighted αfair bandwidth sharing policy introduced by Mo and Walrand [37] (α ∈ (0,∞)). Assuming Poisson arrivals and exponentially distributed document sizes, we focus on the heavy traffic regime in which the average load placed on each resource is approximately equal to its capacity. A fluid model (or functional law of large numbers approximation) for this stochastic model was derived and analyzed in a prior work [29] by two of the authors. Here we use the long time behavior of the solutions of this fluid model established in [29] to derive a property called multiplicative state space collapse, which loosely speaking shows that in diffusion scale the flow count process for the stochastic model can be approximately recovered as a continuous lifting of the workload process. Under weighted proportional fair sharing of bandwidth (α = 1) and a mild
Scheduling control for queueing systems with many servers: Asymptotic optimality in heavy traffic
, 2005
"... A multiclass queueing system is considered, with heterogeneous service stations, each consisting of many servers with identical capabilities. An optimal control problem is formulated, where the control corresponds to scheduling and routing, and the cost is a cumulative discounted functional of the s ..."
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Cited by 42 (6 self)
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A multiclass queueing system is considered, with heterogeneous service stations, each consisting of many servers with identical capabilities. An optimal control problem is formulated, where the control corresponds to scheduling and routing, and the cost is a cumulative discounted functional of the system’s state. We examine two versions of the problem: “nonpreemptive,” where service is uninterruptible, and “preemptive, ” where service to a customer can be interrupted and then resumed, possibly at a different station. We study the problem in the asymptotic heavy traffic regime proposed by Halfin and Whitt, in which the arrival rates and the number of servers at each station grow without bound. The two versions of the problem are not, in general, asymptotically equivalent in this regime, with the preemptive version showing an asymptotic behavior that is, in a sense, much simpler. Under appropriate assumptions on the structure of the system we show: (i) The value function for the preemptive problem converges to V, the value of a related diffusion control problem. (ii) The two versions of the problem are asymptotically equivalent, and in particular nonpreemptive policies can be constructed that asymptotically achieve the value V. The construction of these policies is based on a Hamilton–Jacobi–Bellman equation associated with V.