Controlling distribution inventory systems with shipment consolidation and compound Poisson demand

We consider a one-warehouse-multiple-retailer inventory system where the retailers face stochastic customer demand, modelled as compound Poisson processes. Deliveries from the central warehouse to groups of retailers are consolidated using a time based shipment consolidation policy. This means that replenishment orders have to wait until a vehicle departures, which increases the lead time for the retailers and therefore also the safety stock. Thus, a trade-off exists between expected shipment costs and holding costs. Our aim is to determine the shipment intervals and the required amount of safety stock for each retailer and the warehouse to minimize total cost, both for backorder costs and fill rate constraints. Previous work has focused on exact solutions which are computationally demanding and not applicable for larger real world problems. The focus of our present work is on the development of computationally attractive heuristics that can be applied in practice. A numerical study shows that the proposed heuristics perform well compared to the exact cost minimizing solutions. We also illustrate that the approaches are appropriate for solving real world problems using data from a large European company.     

动力系统的Poisson结构和可积变形

利用动力系统的守恒积分构造Poisson结构,将动力系统表示为广义Hamilton系统的形式,并以一个三维动力系统为例,通过添加任意可微函数推广守恒积分,构造系统的可积变形,并给出变形后系统的Poisson结构,由此得到了新的刘维尔可积系统.     

Hamiltonian dynamical systems with even and odd Poisson brackets

Physicotechnical Institute, Ukrainian SSR Academy of Sciences, Kharkov. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 79, No. 1, pp. 117–126, April, 1989.     

Poisson stable motions of monotone nonautonomous dynamical systems

In this paper, we study the Poisson stability(in particular, stationarity, periodicity, quasiperiodicity, Bohr almost periodicity, almost automorphy, recurrence in the sense of Birkhoff, Levitan almost periodicity, pseudo periodicity, almost recurrence in the sense of Bebutov, pseudo recurrence, Poisson stability) of motions for monotone nonautonomous dynamical systems and of solutions for some classes of monotone nonautonomous evolution equations(ODEs, FDEs and parabolic PDEs). As a byproduct, some of our results indicate that all the trajectories of monotone systems converge to the above mentioned Poisson stable trajectories under some suitable conditions, which is interesting in its own right for monotone dynamics.     

First exit times for compound Poisson dams with a general release rule

An infinite dam with compound Poisson inputs and a state-dependent release rate is considered. For this dam, we solve Kolmogorov’s backward differential equation to obtain the Laplace transforms of the first exit times in terms of a certain positive kernel. This allows us to provide an explicit expression for the Laplace transform of the wet period for a finite dam.     

Sojourn times in laminar states of deterministic dynamical systems with intermittency

We consider intermittent dynamical systems on [0, 1] with two laminar states A and B. The dynamical systems considered in this article are the iterates T^k of mappings T from [0, 1] onto itself. The trajectory {T^k(x)} _k=0 ⁿ belongs to A and B for an extremely long time. We study the limit of the ratio of the sojourn time of the trajectory {T^k(x)} _k=0 ⁿ in A to that in B. The dynamical systems mainly studied in this paper have ergodic infinite invariant measures. Proceedings of the Seminar on Stability Problems for Stochastic Models, Hajdúszoboszló, Hungary, 1997, Part II.     

Entropy of open quantum systems and the Poisson distribution

The entropies of a harmonic oscillator and a quantum Klein-Gordon-Fock field with a static source are found in the coherent state. In both cases, the obtained expressions coincide up to a numerical factor with the Bekenstein-Hawking entropy for a black hole or, more precisely, for the physical vacuum around the hole. Such a coincidence and the property of a gravitational field to make a quantum system placed in this field decoherent lead to the assumption that the vacuum in the vicinity of a black hole is in a coherent state. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 123, No. 1, pp. 107–115, April, 2000.     

Poisson and compound Poisson approximations in conventional and nonconventional setups

It was shown in Kifer (Israel J Math, 2013) that for any subshift of finite type considered with a Gibbs invariant measure the numbers of multiple recurrencies to shrinking cylindrical neighborhoods of almost all points are asymptotically Poisson distributed. Here we not only extend this result to all \(\psi \) -mixing shifts with countable alphabet but actually show that for all points the distributions of these numbers are asymptotically close either to Poisson or to compound Poisson distributions. It turns out that for all nonperiodic points a limiting distribution is always Poisson while at the same time for periodic points there may be no limiting distribution at all unless the shift invariant measure is Bernoulli in which case the limiting distribution always exists. Thus we describe, essentially completely, limiting distributions of multiple recurrence numbers in this setup. As a corollary we obtain also that the first occurence time of the multiple recurrence event is asymptotically exponentially distributed. Most of the results are new also for the widely studied single recurrencies case (see, for instance, Haydn and Vaienti Discret Contin Dyn Syst A 10:589–616, 2004; Probab Theory Relat Fields 144:517–542, 2009; Abadi and Saussol Stoch Process Appl 121:314–323, 2011; Abadi and Vergne Nonlinearity 21:2871–2885, 2008), as well.     

Transient and stationary waiting times in (max,+)-linear systems with Poisson input

We consider a certain class of vectorial evolution equations, which are linear in the (max,+) semi-field. They can be used to model several Types of discrete event systems, in particular queueing networks where we assume that the arrival process of customers (tokens, jobs, etc.) is Poisson. Under natural Cramér Type conditions on certain variables, we show that the expected waiting time which the nth customer has to spend in a given subarea of such a system can be expanded analytically in an infinite power series with respect to the arrival intensity λ. Furthermore, we state an algorithm for computing all coefficients of this series expansion and derive an explicit finite representation formula for the remainder term. We also give an explicit finite expansion for expected stationary waiting times in (max,+)-linear systems with deterministic queueing services. This revised version was published online in June 2006 with corrections to the Cover Date.     

Equilibrium of stellar dynamical systems in the context of the Vlasov–Poisson model

This short review is devoted to the problem of the equilibrium of stellar dynamical systems in the context of the Vlasov–Poisson model. In a first part we will review some classical problems posed by the application of the Vlasov–Poisson model to the astrophysical systems like globular clusters or galaxies. In a second part we will recall some recent numerical results which may give us some quantitative hints about the equilibrium state associated to those systems.     

Customer waiting times in an (R,S) inventory system with compound Poisson demand

Besides service level and mean physical stock, customer waiting time is an important performance characteristic for an inventory system. In this paper we discuss the calculation of this waiting time in case a periodic review control policy with order-up-to-levelS is used and customers arrive according to a Poisson process. For the case of Gamma distributed demand per customer, we obtain (approximate) expressions for the waiting time characteristics. The approach clearly differs from the traditional approaches. It can also be used to obtain other performance characteristics such as the mean physical stock and the service level.     

Periodic review lost-sales inventory models with compound Poisson demand and constant lead times of any length

In almost all literature on inventory models with lost sales and periodic reviews the lead time is assumed to be either an integer multiple of or less than the review period. In a lot of practical settings such restrictions are not satisfied. We develop new models allowing constant lead times of any length when demand is compound Poisson. Besides an optimal policy, we consider pure and restricted base-stock policies under new lead time and cost circumstances. Based on our numerical results we conclude that the latter policy, which imposes a restriction on the maximum order size, performs almost as well as the optimal policy. We also propose an approximation procedure to determine the base-stock levels for both policies with closed-form expressions.     

Asymptotic efficiency of joint estimation of parameters of a compound Poisson distribution

Translated from Chislennye Metody Resheniya Obratnykh Zadach Matematicheskoi Fiziki, pp. 46–57, 1988.     

Asymptotic efficiency of the moment method for estimating the compound Poisson distribution

Point estimation of distribution parameters is considered for a three-parameter compound Poisson process. Formulas are derived for moment method estimation, including estimate biases and the covariance matrix. Asymptotic efficiency of the parameter estimates with a series informant is examined. The efficiency of moment method estimation is computed and analyzed for typical parameter values. __________ Translated from Prikladnaya Matematika i Informatika, No. 24, pp. 44–53, 2006.     

The compound Poisson risk model with multiple thresholds

In this paper we consider a multi-threshold compound Poisson risk model. A piecewise integro-differential equation is derived for the Gerber-Shiu discounted penalty function. We then provide a recursive approach to obtain general solutions to the integro-differential equation and its generalizations. Finally, we use the probability of ruin to illustrate the applicability of the approach.     

Quantum Poisson processes and dilations of dynamical semigroups

Summary The notion of a quantum Poisson process over a quantum measure space is introduced. This process is used to construct new quantum Markov processes on the matrix algebraM _n with stationary faithful state . If (, ) is the quantum measure space in question ( a von Neumann algebra and a faithful normal weight), then the semigroupe ^tL of transition operators on (M _n , ) has generator whereu is an arbitrary unitary element of the centraliser of (M _n ,).Supported by the Netherlands Organization for Scientific Research NWO