introduction to matrix analytic methods in stochastic modeling preface part i quasi birth and death processes 1 examples part ii the method of phases 2 ph distributions 3 markovian point processes part iii the matrix geometric distribution 4 birth and death processes 5 processes under a taboo 6 homogeneous qbds 7 stability condition part iv algorithms 8 algorithms for the rate matrix 9 spectral analysis 10 finite qbds 11 first passage times part v beyond simple qbds 12 nonhomogeneous qbds 13 processes skip free in one direction 14 tree processes 15 product form networks 16 nondenumerable states bibliography index

generalized hyperelliptic surfaces this article presents some results on the surfaces of general type whose albanese morphism is a holomorphic fibre bundle

on a discrete risk model with two sided jumps in this paper we consider a discrete renewal risk model with phase type interarrival times and two sided jumps in this model downward jumps represent claim loss while upward jumps are also allowed to represent random gains assume that the downward jumps have an arbitrary probability function and the upward jumps have a rational probability generating function we study the gerber shiu discounted penalty function the generating function the recursive formula as well as an explicit expression for the discounted penalty function are obtained

nonlinear matrix equations and structured linear algebra different kinds of nonlinear matrix equations are analyzed and their interplay with certain structured matrices is pointed out by using the point of view and the tools of structured linear algebra both theoretical results and numerical methods are derived for solving different classes of nonlinear matrix equations

recursive markov chains stochastic grammars and monotone systems of nonlinear equations we define recursive markov chains rmcs a class of finitely presented denumerable markov chains and we study algorithms for their analysis informally an rmc consists of a collection of finite state markov chains with the ability to invoke each other in a potentially recursive manner rmcs offer a natural abstract model for probabilistic programs with procedures they generalize in a precise sense a number of well studied stochastic models including stochastic context free grammars scfg and multi type branching processes mt bp we focus on algorithms for reachability and termination analysis for rmcs what is the probability that an rmc started from a given state reaches another target state or that it terminatesq these probabilities are in general irrational and they arise as least fixed point solutions to certain monotone systems of nonlinear equations associated with rmcs we address both the qualitative problem of determining whether the probabilities are 0 1 or in between and the quantitative problems of comparing the probabilities with a given bound or approximating them to desired precision we show that all these problems can be solved in pspace using a decision procedure for the existential theory of reals we provide a more practical algorithm based on a decomposed version of multi variate newton s method and prove that it always converges monotonically to the desired probabilities we show this method applies more generally to any monotone polynomial system we obtain polynomial time algorithms for various special subclasses of rmcs among these for scfgs and mt bps equivalently for 1 exit rmcs the qualitative problem can be solved in p time for linearly recursive rmcs the probabilities are rational and can be computed exactly in p time we show that our pspace upper bounds cannot be substantially improved without a breakthrough on long standing open problems the square root sum problem and an arithmetic circuit decision problem that captures p time on the unit cost rational arithmetic ram model we show that these problems reduce to the qualitative problem and to the approximation problem to within any nontrivial error for termination probabilities of general rmcs and to the quantitative decision problem for termination extinction of scfgs mt bps

a discrete time inventory system with postponed demands in this article we consider a discrete time inventory model in which demands arrive according to a discrete markovian arrival process the inventory is replenished according to an s s policy and the lead time is assumed to follow a discrete phase type distribution the demands that occur during stock out periods either enter a pool which has a finite capacity n or leave the system with a predefined probability any demand that arrives when the pool is full and the inventory level is zero is assumed to be lost the demands in the pool are selected one by one if the on hand inventory level is above s 1 and the interval time between any two successive selections is assumed to have discrete phase type distribution the joint probability distribution of the number of customers in the pool and the inventory level is obtained in the steady state case the measures of system performance in the steady state are derived and the total expected cost rate is also calculated the results are illustrated numerically

a fluid limit for an overloaded x model via a stochastic averaging principle we prove a many server heavy traffic fluid limit for an overloaded markovian queueing system having two customer classes and two service pools known in the call center literature as the x model the system uses the fixed queue ratio with thresholds fqr t control which we proposed in a recent paper as a way for one service system to help another in face of an unexpected overload under fqr t customers are served by their own service pool until a threshold is exceeded then one way sharing is activated with customers from one class allowed to be served in both pools after the control is activated it aims to keep the two queues at a prespecified fixed ratio for large systems that fixed ratio is achieved approximately for the fluid limit or fwlln functional weak law of large numbers we consider a sequence of properly scaled x models in overload operating under fqr t our proof of the fwlln follows the compactness approach i e we show that the sequence of scaled processes is tight and then show that all converging subsequences have the specified limit the characterization step is complicated because the queue difference processes which determine the customer server assignments need to be considered without spatial scaling asymptotically these queue difference processes operate on a faster time scale than the fluid scaled processes in the limit because of a separation of time scales the driving processes converge to a time dependent steady state or local average of a time varying fast time scale process ftsp this averaging principle allows us to replace the driving processes with the long run average behavior of the ftsp

a ph ph n c c state dependent queuing model for metro station corridor width design abstract metro station corridor width design considering demand fluctuation as well as the randomness and state dependence of service time is an urgent concern and a complicated random planning issue this paper confirms the accuracy of phase type distribution ph fitting for passenger arrival intervals and service times with randomness and state dependence in metro station corridors a ph ph n c c state dependent queuing model is thus established by a finite level dependent quasi birth death qbd process the existing m g n c c m g 1 c and d d 1 c models are proved to be special cases of the ph ph n c c model through theoretical derivation and the precision of the proposed model is analyzed through simulation tests the quantitative relationship between the level of service los and the corridor width is established based on the proposed model a total of 81 experiments are designed to compare the calculations between the proposed model and the m g n c c m g 1 c and d d 1 c models comparison results demonstrate that 1 the value of effective width of the ph ph n c c queuing model is higher than those of the m g n c c m g 1 c and d d 1 c models 2 the real area occupied per person in the corridor of the ph ph n c c queuing model is mostly proximate to the designed los whereas those of the m g n c c m g 1 c and d d 1 c models fail to meet the designed los and 3 the performance measures of the ph ph n c c queuing model enjoy high performance width elasticity and are significantly improved compared with those of the m g n c c m g 1 c and d d 1 c models

quasi birth and death processes level geometric distributions an aggregation disaggregation approach a special class of homogeneous continuous time quasi birth and death qbd markov chains mcs which possess level geometric lg stationary distribution are considered a functional analytic approach is applied which provides not only a clear analytic interpretation of the concepts introduced elsewhere but also represents a suitable basis for computations an iterative aggregation disaggregation method is proposed as tool for numerical computations

random matrix ensembles with column row constraints ii we analyze statistical properties of a complex system subjected to conditions which manifests through specific constraints on the column row sum of the matrix elements of its hermitian operators the presence of additional constraints besides real symmetric nature leads to new correlations among their eigenfunctions hinders a complete delocalization of dynamics and affects the eigenvalues too the statistical analysis of the latter indicates the presence of a new universality class analogous to that of a special type of brownian ensemble appearing between poisson and gaussian orthogonal ensemble

on the exponential of semi infinite quasi toeplitz matrices let a z sum i in mathbb z a iz i be a complex valued function defined for z 1 such that sum i in mathbb z a i infty define t a t i j i j in mathbb z t i j a j i for i j in mathbb z the semi infinite toeplitz matrix associated with the symbol a z let e e i j i j in mathbb z be a compact operator in ell p with 1 le p le infty a semi infinite matrix of the kind a t a e is said quasi toeplitz qt the problem of the computation of exp a or exp a v with a quasi toeplitz and v a vector arises in many applications we prove that the exponential of a qt matrix a is qt that is exp a t exp a f where f is a compact operator in ell p this property allows the design of an algorithm for computing exp a and exp a v up to any precision the case of families of n times n matrices obtained by truncating infinite qt matrices to finite size is also considered numerical experiments show the effectiveness of this approach

recursive markov decision processes and recursive stochastic games we introduce recursive markov decision processes rmdps and recursive simple stochastic games rssgs which are classes of finitely presented countable state mdps and zero sum turn based perfect information stochastic games they extend standard finite state mdps and stochastic games with a recursion feature we study the decidability and computational complexity of these games under termination objectives for the two players one player s goal is to maximize the probability of termination at a given exit while the other player s goal is to minimize this probability in the quantitative termination problems given an rmdp or rssg and probability p we wish to decide whether the value of such a termination game is at least p or at most p in the qualitative termination problem we wish to decide whether the value is 1 the important 1 exit subclasses of these models 1 rmdps and 1 rssgs correspond in a precise sense to controlled and game versions of classic stochastic models including multitype branching processes and stochastic context free grammars where the objective of the players is to maximize or minimize the probability of termination extinction we provide a number of upper and lower bounds for qualitative and quantitative termination problems for rmdps and rssgs we show both problems are undecidable for multi exit rmdps but are decidable for 1 rmdps and 1 rssgs specifically the quantitative termination problem is decidable in pspace for both 1 rmdps and 1 rssgs and is at least as hard as the square root sum problem a well known open problem in numerical computation we show that the qualitative termination problem for 1 rmdps i e a controlled version of branching processes can be solved in polynomial time both for maximizing and minimizing 1 rmdps the qualitative problem for 1 rssgs is in np conp and is at least as hard as the quantitative termination problem for condon s finite state simple stochastic games whose complexity remains a well known open problem finally we show that even for 1 rmdps more general qualitative and quantitative model checking problems with respect to linear time temporal properties are undecidable even for a fixed property

stochastic darboux transformations for quasi birth and death processes and urn models abstract we consider stochastic ul and lu block factorizations of the one step transition probability matrix for a discrete time quasi birth and death process namely a stochastic block tridiagonal matrix the simpler case of random walks with only nearest neighbors transitions gives a unique lu factorization and a one parameter family of factorizations in the ul case the block structure considered here yields many more possible factorizations resulting in a much enlarged class of potential applications by reversing the order of the factors also known as a darboux transformation we get new families of quasi birth and death processes where it is possible to identify the matrix valued spectral measures in terms of a geronimus ul or a christoffel lu transformation of the original one we apply our results to one example going with matrix valued jacobi polynomials arising in group representation theory we also give urn models for some particular cases

relaxation and diffusion models with non singular kernels abstract anomalous relaxation and diffusion processes have been widely quantified by fractional derivative models where the definition of the fractional order derivative remains a historical debate due to its limitation in describing different kinds of non exponential decays e g stretched exponential decay meanwhile many efforts by mathematicians and engineers have been made to overcome the singularity of power function kernel in its definition this study first explores physical properties of relaxation and diffusion models where the temporal derivative was defined recently using an exponential kernel analytical analysis shows that the caputo type derivative model with an exponential kernel cannot characterize non exponential dynamics well documented in anomalous relaxation and diffusion a legitimate extension of the previous derivative is then proposed by replacing the exponential kernel with a stretched exponential kernel numerical tests show that the caputo type derivative model with the stretched exponential kernel can describe a much wider range of anomalous diffusion than the exponential kernel implying the potential applicability of the new derivative in quantifying real world anomalous relaxation and diffusion processes

consensus over random graph processes network borel cantelli lemmas for almost sure convergence distributed consensus computation over random graph processes is considered the random graph process is defined as a sequence of random variables which take values from the set of all possible dig

moments of random variables a systems theoretic interpretation moments of continuous random variables admitting a probability density function are studied we show that under certain assumptions the moments of a random variable can be characterized in terms of a sylvester equation and of the steady state output response of a specific interconnected system this allows to interpret well known notions and results of probability theory and statistics in the language of systems theory including the sum of independent random variables the notion of mixture distribution and results from renewal theory the theory developed is based on tools from center manifold theory the theory of the steady state response of nonlinear systems and the theory of output regulation our formalism is illustrated by means of several examples and can be easily adapted to the case of discrete and multivariate random variables

the intercept term of the asymptotic variance curve for some queueing output processes we consider the output processes of some elementary queueing models such as the m m 1 k queue and the m g 1 queue an important performance measure for these counting processes is their variance curve v t which gives the variance of the number of customers in the time interval 0 t recent work has revealed some non trivial properties dealing with the asymptotic rate at which the variance curve grows in this paper we add to these results by finding explicit expressions for the intercept term of the linear asymptote for m m 1 k queues our results are based on the deviation matrix of the generator it turns out that by viewing output processes as markovian point processes and considering the deviation matrix one can obtain explicit expressions for the intercept term together with some further insight regarding the bravo balancing reduces asymptotic variance of outputs effect for m g 1 queues our results are based on a classic transform of d j daley in this case we represent the intercept term of the variance curve in terms of the first three moments of the service time distribution in addition we shed light on a conjecture of daley dealing with characterization of stationary m m 1 queues within the class of stationary m g 1 queues based on the variance curve

gerbershiu analysis with two sided acceptable levels in this paper insurers surplus process moved within upper and lower levels is analyzed to this end a truncated type of gerbershiu function is proposed by further incorporating the minimum and the maximum surplus before ruin into the existing ones e g gerber and shiu 1998 cheung et al 2010a a key component in our analysis of this proposed gerbershiu function is the so called transition kernel explicit expressions of the transition function under two different risk models are obtained these two models are both generalizations of the classical poisson risk model i the first model provides flexibility in the net premium rate which is dependent on the surplus such as linear or step function and ii the second model assumes that claims arrive according to a markovian arrival process map finally we discuss some applications of the truncated gerbershiu function with numerical examples under various scenarios

convergence of a second order markov chain in this paper we consider convergence properties of a second order markov chain similar to a column stochastic matrix being associated to a markov chain a transition probability tensor p of order 3 and dimension n is associated to a second order markov chain with n states for this p define f p as f p x px 2 on the n 1 dimensional standard simplex d n if 1 is not an eigenvalue of f p on d n and p is irreducible then there exists a unique fixed point of f p on d n in particular if every entry of p is greater than 12n then 1 is not an eigenvalue of f p on d n under the latter condition we further show that the second order power method for finding the unique fixed point of f p on d n is globally linearly convergent and the corresponding second order markov process is globally r linearly convergent

real eigenvalues of certain tridiagonal matrix polynomials with queueing applications many queueing problems lead to tridiagonal lambda matrices containing polynomials that have except for the diagonal non negative coefficients this paper deals with the question addressed in the literature only for special cases whether the eigenvalues corresponding to such lambda matrices are real in most cases they are as the theorems of this paper show but sometimes complex eigenvalues occur our results are derived by using sturm sequences in addition to simplifying the proofs of our theorems sturm sequences are also valuable to verify whether or not a given interval contains eigenvalues

survival in a quasi death process we consider a markov chain in continuous time with an absorbing coffin state and a finite set s of transient states when s is irreducible the limiting distribution of the chain as t to infty conditional on survival up to time t is known to equal the unique quasi stationary distribution of the chain we address the problem of generalizing this result to a setting in which s may be reducible and obtain a complete solution if the eigenvalue with maximal real part of the generator of the sub markov chain on s has multiplicity one the result is applied to pure death processes and more generally to quasi death processes

rate tilting for fast simulation of level phase processes abstract this paper study the efficient simulation methods for estimating some metrics that depend heavily on a certain rarely occurring event the process considered is the so called level phase process a markov process in which the level and the phase are two state variables furthermore changes of level and phase are induced by events which have rates that are independent of the level except at a boundary if a system typically stays at lower levels then reaching a high level n is a rare event thus direct simulation for the related metrics is very inefficient we change the events rates in a level phase process to accelerate simulation and find from simulation the so called hitting probability the probability of entering a rare event set this method is called rate tilting and in our approach a proper construct of rate tilting relates to a generalized eigenvalue problem involving the infinitesimal generator matrix of the process being considered we can show that the relative estimation error of the hitting probability resulting from the proposed simulation remains bounded as the level increases provided that the boundary set of the state space satisfies certain conditions if these conditions are met rate tilting will be advantageous