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Order optimal delay for opportunistic scheduling in multiuser wireless uplinks and downlinks
 Proc. of Allerton Conf. on Communication, Control, and Computing (invited paper
, 2006
"... Abstract — We consider a onehop wireless network with independent time varying channels and N users, such as a multiuser uplink or downlink. We first show that general classes of scheduling algorithms that do not consider queue backlog necessarily incur average delay that grows at least linearly wi ..."
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Cited by 40 (6 self)
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Abstract — We consider a onehop wireless network with independent time varying channels and N users, such as a multiuser uplink or downlink. We first show that general classes of scheduling algorithms that do not consider queue backlog necessarily incur average delay that grows at least linearly with N. We then construct a dynamic queuelength aware algorithm that stabilizes the system and achieves an average delay that is independent of N. This is the first analytical demonstration that O(1) delay is achievable in such a multiuser wireless setting. The delay bounds are achieved via a technique of queue grouping together with basic Lyapunov stability and statistical multiplexing concepts.
On maxima and ladder processes for a dense class of Lévy processes
 Journal of Applied Probability
"... Abstract. Consider the problem to explicitly calculate the law of the first passage time T(a) of a general Lévy process Z above a positive level a. In this paper it is shown that the law of T(a) can be approximated arbitrarily closely by the laws of T n (a), the corresponding first passages time for ..."
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Cited by 16 (0 self)
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Abstract. Consider the problem to explicitly calculate the law of the first passage time T(a) of a general Lévy process Z above a positive level a. In this paper it is shown that the law of T(a) can be approximated arbitrarily closely by the laws of T n (a), the corresponding first passages time for X n, where (X n)n is a sequence of Lévy processes whose positive jumps follow a phasetype distribution. Subsequently, explicit expressions are derived for the laws of T n (a) and the upward ladder process of X n. The derivation is based on an embedding of X n into a class of Markov additive processes and on the solution of the fundamental (matrix) WienerHopf factorisation for this class. This WienerHopf factorisation can be computed explicitly by solving iteratively a certain fixed point equation. It is shown that, typically, this iteration converges geometrically fast.
Conditional tail expectations for multivariate phasetype distributions
 Journal of Applied Probability
, 2005
"... The conditional tail expectation in risk analysis describes the expected amount of risk that can be experienced given that a potential risk exceeds a threshold value, and provides an important measure for righttail risk. In this paper, we study the convolution and extreme values of dependent risks ..."
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Cited by 15 (4 self)
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The conditional tail expectation in risk analysis describes the expected amount of risk that can be experienced given that a potential risk exceeds a threshold value, and provides an important measure for righttail risk. In this paper, we study the convolution and extreme values of dependent risks that follow a multivariate phase type distribution, and derive explicit formulas of several conditional tail expectations of the convolution and extreme values for such dependent risks. Utilizing the underlying Markovian property of these distributions, our method not only reveals structural insight, but also yields some new distributional properties for multivariate phase type distributions.
A largedeviations analysis of the GI/GI/1 SRPT queue. Queueing Systems: Theory and Applications
, 2006
"... We consider a GI=GI=1 queue with the shortest remaining processing time discipline (SRPT) and lighttailed service times. Our interest is focused on the tail behavior of the sojourntime distribution. We obtain a general expression for its largedeviations decay rate. The value of this decay rate cr ..."
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Cited by 13 (6 self)
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We consider a GI=GI=1 queue with the shortest remaining processing time discipline (SRPT) and lighttailed service times. Our interest is focused on the tail behavior of the sojourntime distribution. We obtain a general expression for its largedeviations decay rate. The value of this decay rate critically depends on whether there is mass in the endpoint of the servicetime distribution or not. An auxiliary priority queue, for which we obtain some new results, plays an important role in our analysis. We apply our SRPT results to compare SRPT with FIFO from a largedeviations point of view.
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 12 (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.
Fluctuations of spectrally negative Markov additive processes
, 2008
"... For spectrally negative Markov Additive Processes (MAPs) we generalize classical fluctuation identities developed in Zolotarev (1964), Takács (1967), ..."
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Cited by 10 (1 self)
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For spectrally negative Markov Additive Processes (MAPs) we generalize classical fluctuation identities developed in Zolotarev (1964), Takács (1967),
ASYMPTOTIC EXPONENTIALITY OF THE DISTRIBUTION OF FIRST EXIT TIMES FOR A CLASS OF MARKOV PROCESSES WITH APPLICATIONS TO QUICKEST CHANGE DETECTION
, 2007
"... We consider the first exit time of a nonnegative Harrisrecurrent Markov process from the interval [0,A] as A → ∞. We provide an alternative method of proof of asymptotic exponentiality of the first exit time (suitably standardized) that does not rely on embedding in a regeneration process. We show ..."
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Cited by 10 (6 self)
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We consider the first exit time of a nonnegative Harrisrecurrent Markov process from the interval [0,A] as A → ∞. We provide an alternative method of proof of asymptotic exponentiality of the first exit time (suitably standardized) that does not rely on embedding in a regeneration process. We show that under certain conditions the moment generating function of a suitably standardized version of the first exit time converges to that of Exponential(1), and we connect between the standardizing constant and the quasistationary distribution (assuming it exists). The results are applied to the evaluation of a distribution of run length to false alarm in changepoint detection problems.
Modelling default contagion using Multivariate PhaseType distributions
, 2007
"... We model dynamic credit portfolio dependence by using default contagion in an intensitybased framework. Two different portfolios (with 10 obligors), one in the European auto sector, the other in the European financial sector, are calibrated against their market CDS spreads and the corresponding CD ..."
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Cited by 10 (4 self)
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We model dynamic credit portfolio dependence by using default contagion in an intensitybased framework. Two different portfolios (with 10 obligors), one in the European auto sector, the other in the European financial sector, are calibrated against their market CDS spreads and the corresponding CDScorrelations. After the calibration, which are perfect for the banking portfolio, and good for the auto case, we study several quantities of importance in active credit portfolio management. For example, implied multivariate default and survival distributions, multivariate conditional survival distributions, implied default correlations, expected default times and expected ordered defaults times. The default contagion is modelled by letting individual intensities jump when other defaults occur, but be constant between defaults. This model is translated into a Markov jump process, a so called multivariate phasetype distribution, which represents the default status in the credit portfolio. Matrixanalytic methods are then used to derive expressions for the quantities studied in the calibrated portfolios.
A Semidefinite Optimization Approach to the SteadyState Analysis of Queueing Systems
, 2005
"... Computing the steadystate distribution in Markov chains for general distributions and general state space is a computationally challenging problem. In this paper, we consider the steadystate stochastic model W d = g(W,X) where the equality is in distribution. Given partial distributional informa ..."
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Cited by 9 (1 self)
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Computing the steadystate distribution in Markov chains for general distributions and general state space is a computationally challenging problem. In this paper, we consider the steadystate stochastic model W d = g(W,X) where the equality is in distribution. Given partial distributional information on the random variables X, we want to estimate information on the distribution of the steadystate vector W. Such models naturally occur in queueing systems, where the goal is to find bounds on moments of the waiting time under moment information on the service and interarrival times. In this paper, we propose an approach based on semidefinite optimization to find such bounds. We show that the classical Kingman’s and Daley’s bounds for the expected waiting time in a GI/GI/1 queue are special cases of the proposed approach. We also report computational results in the queueing context that indicate the method is promising.