sequencing and routing in multiclass queueing networks part ii workload relaxations part ii continues the development of policy synthesis techniques for multiclass queueing networks based upon a linear fluid model the following are shown a relaxation of the fluid model based on workload leads to an optimization problem of lower dimension an analogous workload relaxation is introduced for the stochastic model these relaxed control problems admit pointwise optimal solutions in many instances a translation to the original fluid model is almost optimal with vanishing relative error as the network load rho approaches one it is pointwise optimal after a short transient period provided a pointwise optimal solution exists for the relaxed control problem a translation of the optimal policy for the fluid model provides a policy for the stochastic network model that is almost optimal in heavy traffic over all solutions to the relaxed stochastic model again with vanishing relative error the regret is of order log 1 rho

propagation of reactions in inhomogeneous media consider reaction diffusion equation ut u f x u with x d and general inhomogeneous ignition reaction 0 vanishing at u 0 1 typical solutions 0 u 1 transition from 0 to 1 as time progresses and we study them in the region where this transition occurs under fairly general qualitative hypotheses on f we show that in dimensions d 3 the hausdorff distance of the superlevel sets u e and u 1 e remains uniformly bounded in time for each e 0 1 thus u remains uniformly in time close to the characteristic function of u in the sense of hausdorff distance of superlevel sets we also show that each u e expands with average speed over any long enough time interval between the two spreading speeds corresponding to any x independent lower and upper bounds on f on the other hand these results turn out to be false in dimensions d 4 at least without further quantitative hypotheses on f the proof for d 3 is based on showing that as the solution propagates small values of u cannot escape far ahead of values close to 1 the proof for d 4 is via construction of a counterexample for which this fails such results were before known for d 1 but are new for general non periodic media in dimensions d 2 some are also new for homogeneous and periodic media they extend in a somewhat weaker sense to monostable bistable and mixed reaction types as well as to transitions between general equilibria u u of the pde and to solutions not necessarily satisfying u u u 2016 wiley periodicals inc

some results on fuzzy supratopological spaces abstract in this note we study some results on the concepts of supracompactness and supracompactness in a fuzzy supratopological space we introduce the notions of hausdorffnessand lindelofness in a fuzzy supratopological space and characterize their properties interms of supracompactness we also introduce and study the concept of fuzzy supra regular space

sequencing and routing in multiclass queueing networks part i feedback regulation this paper establishes new criteria for stability and for instability of multiclass network models under a given stationary policy it also extends previous results on the approximation of the solution to the average cost optimality equations through an associated fluid model it is shown that an optimized network possesses a fluid limit model which is itself optimal with respect to a total cost criterion a general framework for constructing control algorithms for multiclass queueing networks is proposed based on these general results network sequencing and routing problems are considered as special cases the following aspects of the resulting feedback regulation policies are developed in the paper the policies are stabilizing and are in fact geometrically ergodic for a markovian model numerical examples are given in each case it is shown that the feedback regulation policy closely resembles the average cost optimal policy a method is proposed for reducing variance in simulation for a network controlled using a feedback regulation policy

speed up of combustion fronts in shear flows this paper is concerned with the analysis of speed up of reaction diffusion advection traveling fronts in infinite cylinders with periodic boundary conditions the advection is a shear flow with a large amplitude and the reaction is nonnegative with either positive or zero ignition temperature the unique or minimal speeds of the traveling fronts are proved to be asymptotically linear in the flow amplitude as the latter goes to infinity solving an open problem from berestycki nonlinear pdes in condensed matter and reactive flows kluwer doordrecht 2003 the asymptotic growth rate is characterized explicitly as the unique or minimal speed of traveling fronts for a limiting degenerate problem and the convergence of the regular traveling fronts to the degenerate ones is proved for positive ignition temperatures under an additional hormander type condition on the flow

generalized transition waves and their properties in this paper we generalize the usual notions of waves fronts and propagation speeds in a very general setting these new notions which cover all usual situations involve uniform limits with respect to the geodesic distance to a family of hypersurfaces that are parametrized by time we prove the existence of new such waves for some time dependent reaction diffusion equations as well as general intrinsic properties some monotonicity properties and some uniqueness results for almost planar fronts the classification results which are obtained under some appropriate assumptions show the robustness of our general definitions 2012 wiley periodicals inc

duality and linear programs for stability and performance analysis of queuing networks and scheduling policies we consider the problems of performance analysis and stability instability determination of queuing networks and scheduling policies we exhibit a strong duality relationship between the performance of a system and its stability analysis via mean drift we obtain a variety of linear programs lps to conduct such stability and performance analyses a certain lp called the performance lp bounds the performance of all stationary nonidling scheduling policies if it is bounded then its dual called the drift lp has a feasible solution which is a copositive matrix the quadratic form associated with this copositive matrix has a negative drift showing that all stationary nonidling scheduling policies result in a geometrically converging exponential moment these results carry over to fluid models allowing the study of networks with nonexponential distributions if a modification of the performance lp called the monotone lp is bounded then the system is stable finally there is a another modification of the performance lp called the finite time lp it provides transient bounds on the performance of the system from any initial condition

front blocking versus propagation in the presence of drift term varying in the direction of propagation abstract in this paper we derive quantitative conditions under which a compactly supported drift term depending on the direction of propagation blocks a traveling wave solution or lets it pass almost unchanged we give explicit conditions on the drift term for blocking as well as almost unchanged propagation in one spatial dimension

bistable transition fronts in rn abstract this paper is chiefly concerned with qualitative properties of some reaction diffusion fronts the recently defined notions of transition fronts generalize the standard notions of traveling fronts in this paper we show the existence and the uniqueness of the global mean speed of bistable transition fronts in r n this speed is proved to be independent of the shape of the level sets of the fronts the planar fronts are also characterized in the more general class of almost planar fronts with any number of transition layers these qualitative properties show the robustness of the notions of transition fronts but we also prove the existence of new types of transition fronts in r n that are not standard traveling fronts thus showing that the notions of transition fronts are broad enough to include other relevant propagating solutions

front blocking in the presence of gradient drift abstract this paper tries to understand how a drift influences the wave propagation of a bistable reaction diffusion equation in straight cylinders in dimension n 3 we derive a quantitative condition under which a compactly supported drift term blocks the propagation of a traveling wave we assume for this that the drift has a potential

fuzzy sets and type 2 under algebraic product and algebraic sum abstract the concept of fuzzy sets of type 2 has been proposed by l a zadeh as an extension of ordinary fuzzy sets a fuzzy set of type 2 can be defined by a fuzzy membership function the grade or fuzzy grade of which is taken to be a fuzzy set in the unit interval 0 1 rather than a point in 0 1 this paper investigates the algebraic properties of fuzzy grades that is fuzzy sets of type 2 under the operations of algebraic product and algebraic sum which can be defined by using the concept of the extension principle and shows that fuzzy grades under these operations do not form such algebraic structures as a lattice and a semiring moreover the properties of fuzzy grades are also discussed in the case where algebraic product and algebraic sum are combined with the well known operations of join and meet for fuzzy grades and it is shown that normal convex fuzzy grades form a lattice ordered semigroup under join meet and algebraic product

existence of planar flame fronts in convective diffusive periodic media we prove the existence of planar travelling wave solutions in a reaction diffusion convection equation with combustion nonlinearity and self adjoint linear part in rn n 1 the linear part involves diffusion convection terms and periodic coefficients these travelling waves have wrinkled flame fronts propagating with constant effective speeds in periodic inhomogeneous media we use the method of continuation spectral theory and the maximum principle uniqueness and monotonicity properties of solutions follow from a previous paper these properties are essential to overcoming the lack of compactness and the degeneracy in the problem

existence and convergence to a propagating terrace in one dimensional reaction diffusion equations we consider one dimensional reaction diffusion equations for a large class of spatially periodic nonlinearities including multistable ones and study the asymptotic behavior of solutions with heaviside type initial data our analysis reveals some new dynamics where the profile of the propagation is not characterized by a single front but by a layer of several fronts which we call a terrace existence and convergence to such a terrace is proven by using an intersection number argument without much relying on standard linear analysis hence on top of the peculiar phenomenon of propagation that our work highlights several corollaries will follow on the existence and convergence to pulsating traveling fronts even for highly degenerate nonlinearities that have not been treated before

the policy iteration algorithm for average reward markov decision processes with general state space the average cost optimal control problem is addressed for markov decision processes with unbounded cost it is found that the policy iteration algorithm generates a sequence of policies which are c regular where c is the cost function under consideration this result only requires the existence of an initial c regular policy and an irreducibility condition on the state space furthermore under these conditions the sequence of relative value functions generated by the algorithm is bounded from below and nearly decreasing from which it follows that the algorithm is always convergent under further conditions it is shown that the algorithm does compute a solution to the optimality equations and hence an optimal average cost policy these results provide elementary criteria for the existence of optimal policies for markov decision processes with unbounded cost and recover known results for the standard linear quadratic gaussian problem in particular in the control of multiclass queueing networks it is found that there is a close connection between optimization of the network and optimal control of a far simpler fluid network model

stability conditions for multiclass fluid queueing networks we introduce a new method to investigate stability of work conserving policies in multiclass queueing networks the method decomposes feasible trajectories and uses linear programming to test stability we show that this linear program is a necessary and sufficient condition for the stability of all work conserving policies for multiclass fluid queueing networks with two stations furthermore we find new sufficient conditions for the stability of multiclass queueing networks involving any number of stations and conjecture that these conditions are also necessary previous research had identified sufficient conditions through the use of a particular class of piecewise linear convex lyapunov functions using linear programming duality we show that for two station systems the lyapunov function approach is equivalent to ours and therefore characterizes stability exactly

existence and nonexistence of traveling waves and reaction diffusion front propagation in periodic media we review the existence results of traveling wave solutions to the reaction diffusion equations with periodic diffusion convection coefficients and combustion bistable nonlinearities we prove that whenever traveling waves exist the solutions of the initial value problem with either frontlike or pulselike data propagate with the constant effective speeds of traveling waves in all suitable directions in the case of bistable nonlinearity and one space dimension we give an example of nonexistence of traveling waves which causes quenching localization of wavefront propagation quenching localization only occurs when the variations of the media from their constant mean values are large enough our related numerical results also provide evidence for this phenomenon in the parameter regimes not covered by the analytical example finally we comment on the role of the effective wave speeds in determining the effective wavefront equation hamilton jacobi equation of the reactiondiffusion equations under the small diffusion fast reaction limit with a formal geometric optics expansion

generalized traveling waves in disordered media existence uniqueness and stability we prove existence uniqueness and stability of transition fronts generalized traveling waves for reaction diffusion equations in cylindrical domains with general inhomogeneous ignition reactions we also show uniform convergence of solutions with exponentially decaying initial data to time translations of the front in the case of stationary ergodic reactions the fronts are proved to propagate with a deterministic positive speed our results extend to reaction advection diffusion equations with periodic advection and diffusion

an invitation to topological structures of first order genuine sets the aim of this note is to construct several topological structures on first order genuine sets and it contains only the first steps to the topological properties on first order genuine sets

local minimisers and singular perturbations we construct local minimisers to certain variational problems the method is quite general and relies on the theory of convergence the approach is demonstrated through the model problem it is shown that in certain nonconvex domains n and for e small there exist nonconstant local minimisers u e satisfying u e 1 except in a thin transition layer the location of the layer is determined through the requirement that in the limit u e u 0 the hypersurface separating the states u 0 1 and u 0 1 locally minimises surface area generalisations are discussed with for example vector valued u and anisotropic perturbations replacing u 2

an analysis of temporal difference learning with function approximation we discuss the temporal difference learning algorithm as applied to approximating the cost to go function of an infinite horizon discounted markov chain the algorithm we analyze updates parameters of a linear function approximator online during a single endless trajectory of an irreducible aperiodic markov chain with a finite or infinite state space we present a proof of convergence with probability one a characterization of the limit of convergence and a bound on the resulting approximation error furthermore our analysis is based on a new line of reasoning that provides new intuition about the dynamics of temporal difference learning in addition to proving new and stronger positive results than those previously available we identify the significance of online updating and potential hazards associated with the use of nonlinear function approximators first we prove that divergence may occur when updates are not based on trajectories of the markov chain this fact reconciles positive and negative results that have been discussed in the literature regarding the soundness of temporal difference learning second we present an example illustrating the possibility of divergence when temporal difference learning is used in the presence of a nonlinear function approximator

workload models for stochastic networks value functions and performance evaluation this paper concerns control and performance evaluation for stochastic network models structural properties of value functions are developed for controlled brownian motion cbm and deterministic fluid workload models leading to the following conclusions outside of a null set of network parameters the following hold i the fluid value function is a smooth function of the initial state under further minor conditions the fluid value function satisfies the derivative boundary conditions that are required to ensure it is in the domain of the extended generator for the cbm model exponential ergodicity of the cbm model is demonstrated as one consequence ii the fluid value function provides a shadow function for use in simulation variance reduction for the stochastic model the resulting simulator satisfies an exact large deviation principle while a standard simulation algorithm does not satisfy any such bound iii the fluid value function provides upper and lower bounds on performance for the cbm model this follows from an extension of recent linear programming approaches to performance evaluation

stability and asymptotic optimality of generalized maxweight policies it is shown that stability of the celebrated maxweight or back pressure policies is a consequence of the following interpretation either policy is myopic with respect to a surrogate value function of a very special form in which the marginal disutility at a buffer vanishes for a vanishingly small buffer population this observation motivates the h maxweight policy defined for a wide class of functions h these policies share many of the attractive properties of the maxweight policy as follows i arrival rate data is not required in the policy ii under a variety of general conditions the policy is stabilizing when h is a perturbation of a monotone linear function a monotone quadratic or a monotone lyapunov function for the fluid model iii a perturbation of the relative value function for a workload relaxation gives rise to a myopic policy that is approximately average cost optimal in heavy traffic with logarithmic regret the first results are obtained for a general markovian network model asymptotic optimality is established for a general markovian scheduling model with a single bottleneck and with homogeneous servers

generalized transition fronts for one dimensional almost periodic fisher kpp equations this paper investigates the existence of generalized transition fronts for fisher kpp equations in one dimensional almost periodic media assuming that the linearized elliptic operator near the unstable steady state admits an almost periodic eigenfunction we show that such fronts exist if and only if their average speed is above an explicit threshold this hypothesis is satisfied in particular when the reaction term does not depend on x or in some cases is small enough moreover except for the threshold case the fronts we construct and their speeds are almost periodic in a sense when our hypothesis is no longer satisfied such generalized transition fronts still exist for an interval of average speeds with explicit bounds our proof relies on the construction of sub and super solutions based on an accurate analysis of the properties of the generalized principal eigenvalues

transition fronts of time periodic bistable reaction diffusion equations in rn abstract this paper is concerned with the existence and qualitative properties of transition fronts for time periodic bistable reaction diffusion equations in r n we first show that any almost planar transition front is actually planar regardless of the number of transition layers then we prove that all transition fronts admit a global mean speed and it holds c where c is the speed of the planar traveling front finally we establish the existence of a transition front in r n that is not a standard traveling front such a front behaves like three moving time periodic planar fronts as time goes to and like a time periodic v shaped traveling front as time goes to

bistable traveling waves around an obstacle we consider traveling waves for a nonlinear diffusion equation with a bistable or multistable nonlinearity the goal is to study how a planar traveling front interacts with a compact obstacle that is placed in the middle of the space n as a first step we prove the existence and uniqueness of an entire solution that behaves like a planar wave front approaching from infinity and eventually reaching the obstacle this causes disturbance on the shape of the front but we show that the solution will gradually recover its planar wave profile and continue to propagate in the same direction leaving the obstacle behind whether the recovery is uniform in space is shown to depend on the shape of the obstacle 2008 wiley periodicals inc

a heteroclinic orbit connecting traveling waves pertaining to different nonlinearities in a channel with decreasing cross section abstract in this paper we consider a semilinear parabolic equation in an infinite cylinder with shrinking cross section the portion where the domain is not a straight half cylinder is assumed to be compact the spatially varying nonlinearity is such that it connects two spatially independent bistable nonlinearities in a compact set in space we prove that given such a setting a traveling wave obeying the equation with the one bistable nonlinearity and starting at the respective side of the decreasing cylinder will converge to a traveling wave solution prescribed by the nonlinearity on the other side

regularity and stability of transition fronts in nonlocal equations with time heterogeneous ignition nonlinearity abstract the present paper is devoted to the investigation of various properties of transition fronts in one dimensional nonlocal equations in heterogeneous media of ignition type whose existence has been established by the authors of the present paper in a previous work it is first shown that transition fronts are continuously differentiable in space with uniformly bounded and uniformly lipschitz continuous space partial derivative this is the first time that space regularity of transition fronts in nonlocal equations is ever studied it is then shown that transition fronts are uniformly steep finally asymptotic stability in the sense of exponentially attracting front like initial data of transition fronts is studied

the linear programming approach to approximate dynamic programming the curse of dimensionality gives rise to prohibitive computational requirements that render infeasible the exact solution of large scale stochastic control problems we study an efficient method based on linear programming for approximating solutions to such problems the approach fits a linear combination of pre selected basis functions to the dynamic programming cost to go function we develop error bounds that offer performance guarantees and also guide the selection of both basis functions and state relevance weights that influence quality of the approximation experimental results in the domain of queueing network control provide empirical support for the methodology

entire solutions of the kpp equation this paper deals with the solutions defined for all time of the kpp equation ut uxx f u 0 0 f 0 1 0i n 0 1 and f 0 s f 0 0 in 0 1 this equation admits infinitely many traveling wave type solutions increasing or decreasing in x i t also admits solutions that depend only on t in this paper we build four other manifolds of solutions one is 5 dimensional one is 4 dimensional and two are 3 dimensional some of these new solutions are obtained by considering two traveling waves that come from both sides of the real axis and mix furthermore the traveling wave solutions are on the boundary of these four manifolds c 1999 john wiley sons inc

a heteroclinic orbit connecting traveling waves pertaining to different nonlinearities abstract in this paper we consider a semilinear parabolic equation in an infinite cylinder the spatially varying nonlinearity is such that it connects two spatially independent bistable nonlinearities in a compact set in space we prove that given such a setting a traveling wave obeying the equation with the one bistable nonlinearity and starting at the respective side of the cylinder will converge to a traveling wave solution prescribed by the nonlinearity on the other side

transition fronts in time heterogeneous and random media of ignition type abstract the current paper is devoted to the investigation of wave propagation phenomenon in reaction diffusion equations with ignition type nonlinearity in time heterogeneous and random media it is proven that such equations in time heterogeneous media admit transition fronts with time dependent profiles and that such equations in time random media admit transition fronts with random profiles important properties of transition fronts including the boundedness of propagation speeds and the uniform decaying estimates of the propagation fronts are also obtained