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Alexander L. Stolyar - One of the best experts on this subject based on the ideXlab platform.
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multiclass multiserver queueing system in the halfin whitt heavy traffic regime asymptotics of the stationary distribution
Queueing Systems, 2012Co-Authors: David Gamarnik, Alexander L. StolyarAbstract:We consider a heterogeneous queueing system consisting of one large pool of O(r) identical servers, where r?? is the scaling parameter. The arriving customers belong to one of several classes which determines the service times in the Distributional Sense. The system is heavily loaded in the Halfin---Whitt Sense, namely the nominal utilization is $1-a/\sqrt{r}$ where a>0 is the spare capacity parameter. Our goal is to obtain bounds on the steady state performance metrics such as the number of customers waiting in the queue Q r (?). While there is a rich literature on deriving process level (transient) scaling limits for such systems, the results for steady state are primarily limited to the single class case. This paper is the first one to address the case of heterogeneity in the steady state regime. Moreover, our results hold for any service policy which does not admit server idling when there are customers waiting in the queue. We assume that the interarrival and service times have exponential distribution, and that customers of each class may abandon while waiting in the queue at a certain rate (which may be zero). We obtain upper bounds of the form $O(\sqrt{r})$ on both Q r (?) and the number of idle servers. The bounds are uniform w.r.t. parameter r and the service policy. In particular, we show that $\limsup_{r} \mathbb {E}\exp(\theta r^{-{1\over2}}Q^{r}(\infty)) 0.
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multiclass multiserver queueing system in the halfin whitt heavy traffic regime asymptotics of the stationary distribution
arXiv: Probability, 2011Co-Authors: David Gamarnik, Alexander L. StolyarAbstract:We consider a heterogeneous queueing system consisting of one large pool of $O(r)$ identical servers, where $r\to\infty$ is the scaling parameter. The arriving customers belong to one of several classes which determines the service times in the Distributional Sense. The system is heavily loaded in the Halfin-Whitt Sense, namely the nominal utilization is $1-a/\sqrt{r}$ where $a>0$ is the spare capacity parameter. Our goal is to obtain bounds on the steady state performance metrics such as the number of customers waiting in the queue $Q^r(\infty)$. While there is a rich literature on deriving process level (transient) scaling limits for such systems, the results for steady state are primarily limited to the single class case. This paper is the first one to address the case of heterogeneity in the steady state regime. Moreover, our results hold for any service policy which does not admit server idling when there are customers waiting in the queue. We assume that the interarrival and service times have exponential distribution, and that customers of each class may abandon while waiting in the queue at a certain rate (which may be zero). We obtain upper bounds of the form $O(\sqrt{r})$ on both $Q^r(\infty)$ and the number of idle servers. The bounds are uniform w.r.t. parameter $r$ and the service policy. In particular, we show that $\limsup_r E \exp(\theta r^{-1/2}Q^r(\infty)) 0$.
David Gamarnik - One of the best experts on this subject based on the ideXlab platform.
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multiclass multiserver queueing system in the halfin whitt heavy traffic regime asymptotics of the stationary distribution
Queueing Systems, 2012Co-Authors: David Gamarnik, Alexander L. StolyarAbstract:We consider a heterogeneous queueing system consisting of one large pool of O(r) identical servers, where r?? is the scaling parameter. The arriving customers belong to one of several classes which determines the service times in the Distributional Sense. The system is heavily loaded in the Halfin---Whitt Sense, namely the nominal utilization is $1-a/\sqrt{r}$ where a>0 is the spare capacity parameter. Our goal is to obtain bounds on the steady state performance metrics such as the number of customers waiting in the queue Q r (?). While there is a rich literature on deriving process level (transient) scaling limits for such systems, the results for steady state are primarily limited to the single class case. This paper is the first one to address the case of heterogeneity in the steady state regime. Moreover, our results hold for any service policy which does not admit server idling when there are customers waiting in the queue. We assume that the interarrival and service times have exponential distribution, and that customers of each class may abandon while waiting in the queue at a certain rate (which may be zero). We obtain upper bounds of the form $O(\sqrt{r})$ on both Q r (?) and the number of idle servers. The bounds are uniform w.r.t. parameter r and the service policy. In particular, we show that $\limsup_{r} \mathbb {E}\exp(\theta r^{-{1\over2}}Q^{r}(\infty)) 0.
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multiclass multiserver queueing system in the halfin whitt heavy traffic regime asymptotics of the stationary distribution
arXiv: Probability, 2011Co-Authors: David Gamarnik, Alexander L. StolyarAbstract:We consider a heterogeneous queueing system consisting of one large pool of $O(r)$ identical servers, where $r\to\infty$ is the scaling parameter. The arriving customers belong to one of several classes which determines the service times in the Distributional Sense. The system is heavily loaded in the Halfin-Whitt Sense, namely the nominal utilization is $1-a/\sqrt{r}$ where $a>0$ is the spare capacity parameter. Our goal is to obtain bounds on the steady state performance metrics such as the number of customers waiting in the queue $Q^r(\infty)$. While there is a rich literature on deriving process level (transient) scaling limits for such systems, the results for steady state are primarily limited to the single class case. This paper is the first one to address the case of heterogeneity in the steady state regime. Moreover, our results hold for any service policy which does not admit server idling when there are customers waiting in the queue. We assume that the interarrival and service times have exponential distribution, and that customers of each class may abandon while waiting in the queue at a certain rate (which may be zero). We obtain upper bounds of the form $O(\sqrt{r})$ on both $Q^r(\infty)$ and the number of idle servers. The bounds are uniform w.r.t. parameter $r$ and the service policy. In particular, we show that $\limsup_r E \exp(\theta r^{-1/2}Q^r(\infty)) 0$.
Sorin Mardare - One of the best experts on this subject based on the ideXlab platform.
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The fundamental theorem of surface theory for surfaces with little regularity
2020Co-Authors: Sorin MardareAbstract:Abstract. Consider a symmetric, positive definite matrix field of order two and a symmetric matrix field of order two that satisfy together the Gauss and Codazzi-Mainardi equations in a connected and simply connected open subset of R 2 . If the matrix fields are respectively of class C 2 and C 1 , the fundamental theorem of surface theory asserts that there exists a surface immersed in the threedimensional Euclidean space with these fields as its first and second fundamental forms. The purpose of this paper is to prove that this theorem still holds under the weaker regularity assumptions that the matrix fields are respectively of class W 1,∞ loc and L ∞ loc , the Gauss and Codazzi-Mainardi equations being then understood in a Distributional Sense. Mathematics Subject Classification
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on pfaff systems with lp coefficients and their applications in differential geometry
Journal de Mathématiques Pures et Appliquées, 2005Co-Authors: Sorin MardareAbstract:Abstract We prove that a Pfaff system with coefficients in L loc p , p > 2 , in a simply-connected open subset Ω of R 2 has at least a nontrivial solution of class W loc 1 , p ( Ω ) provided that its coefficients satisfies a compatibility condition in the Distributional Sense. If in addition the set Ω is connected, the Cauchy problem associated with the Pfaff system has a unique solution. An application of this result is that the fundamental theorem of surface theory holds under the assumption that the first and second fundamental forms are respectively of class W loc 1 , p and L loc p , with p > 2 , and satisfy together the Gauss and Codazzi–Mainardi equations in the Distributional Sense.
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the fundamental theorem of surface theory for surfaces with little regularity
Journal of Elasticity, 2003Co-Authors: Sorin MardareAbstract:Consider a symmetric, positive definite matrix field of order two and a symmetric matrix field of order two that satisfy together the Gauss and Codazzi-Mainardi equations in a connected and simply connected open subset of R2. If the matrix fields are respectively of class C2 and C1, the fundamental theorem of surface theory asserts that there exists a surface immersed in the three-dimensional Euclidean space with these fields as its first and second fundamental forms. The purpose of this paper is to prove that this theorem still holds under the weaker regularity assumptions that the matrix fields are respectively of class W1,∞loc and L∞loc, the Gauss and Codazzi-Mainardi equations being then understood in a Distributional Sense.
Sakajo Takashi - One of the best experts on this subject based on the ideXlab platform.
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Rotating equilibria of vortex sheets
'Elsevier BV', 2020Co-Authors: Protas Bartosz, Sakajo TakashiAbstract:We consider relative equilibrium solutions of the two-dimensional Euler equations in which the vorticity is concentrated on a union of finite-length vortex sheets. Using methods of complex analysis, more specifically the theory of the Riemann–Hilbert problem, a general approach is proposed to find such equilibria which consists of two steps: first, one finds a geometric configuration of vortex sheets ensuring that the corresponding circulation density is real-valued and also vanishes at all sheet endpoints such that the induced velocity field is well-defined; then, the circulation density is determined by evaluating a certain integral formula. As an illustration of this approach, we construct a family of rotating equilibria involving different numbers of straight vortex sheets rotating about a common center of rotation and with endpoints at the vertices of a regular polygon. This equilibrium generalizes the well-known solution involving single rotating vortex sheet. With the geometry of the configuration specified analytically, the corresponding circulation densities are obtained in terms of a integral expression which in some cases lends itself to an explicit evaluation. It is argued that as the number of sheets in the equilibrium configuration increases to infinity, the equilibrium converges in a certain Distributional Sense to a hollow vortex bounded by a constant-intensity vortex sheet, which is also a known equilibrium solution of the two-dimensional Euler equations
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Rotating Equilibria of Vortex Sheets
'Elsevier BV', 2019Co-Authors: Protas Bartosz, Sakajo TakashiAbstract:We consider relative equilibrium solutions of the two-dimensional Euler equations in which the vorticity is concentrated on a union of finite-length vortex sheets. Using methods of complex analysis, more specifically the theory of the Riemann-Hilbert problem, a general approach is proposed to find such equilibria which consists of two steps: first, one finds a geometric configuration of vortex sheets ensuring that the corresponding circulation density is real-valued and also vanishes at all sheet endpoints such that the induced velocity field is well-defined; then, the circulation density is determined by evaluating a certain integral formula. As an illustration of this approach, we construct a family of rotating equilibria involving different numbers of straight vortex sheets rotating about a common center of rotation and with endpoints at the vertices of a regular polygon. This equilibrium generalizes the well-known solution involving single rotating vortex sheet. With the geometry of the configuration specified analytically, the corresponding circulation densities are obtained in terms of a integral expression which in some cases lends itself to an explicit evaluation. It is argued that as the number of sheets in the equilibrium configuration increases to infinity, the equilibrium converges in a certain Distributional Sense to a hollow vortex bounded by a constant-intensity vortex sheet, which is also a known equilibrium solution of the two-dimensional Euler equations.Comment: 22 pages, 5 figures; accepted for publication in Physica
Alberto Lanconelli - One of the best experts on this subject based on the ideXlab platform.
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absolute continuity and fokker planck equation for the law of wong zakai approximations of ito s stochastic differential equations
Journal of Mathematical Analysis and Applications, 2020Co-Authors: Alberto LanconelliAbstract:Abstract We investigate the regularity of the law of Wong-Zakai-type approximations for Ito stochastic differential equations. These approximations solve random differential equations where the diffusion coefficient is Wick-multiplied by the smoothed white noise. Using criteria based on the Malliavin calculus we establish absolute continuity and a Fokker-Planck-type equation solved in the Distributional Sense by the density. The parabolic smoothing effect typical of the solutions of Ito equations is lacking in this approximated framework; therefore, in order to prove absolute continuity, the initial condition of the random differential equation needs to possess a density itself.
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absolute continuity and fokker planck equation for the law of wong zakai approximations of it o s stochastic differential equations
arXiv: Probability, 2019Co-Authors: Alberto LanconelliAbstract:We investigate the regularity of the law of Wong-Zakai-type approximations for It\^o stochastic differential equations. These approximations solve random differential equations where the diffusion coefficient is Wick-multiplied by the smoothed white noise. Using a criteria based on the Malliavin calculus we establish absolute continuity and a Fokker-Planck-type equation solved in the Distributional Sense by the density. The parabolic smoothing effect typical of the solutions of It\^o equations is lacking in this approximated framework; therefore, in order to prove absolute continuity, the initial condition of the random differential equation needs to possess a density itself.