Pub Date : 2022-04-04DOI: 10.1080/15326349.2022.2049822
Ann M. Noblesse, Nikki Sonenberg, R. Boute, M. Lambrecht, B. Van Houdt
Abstract In this paper we analyze a continuous review finite capacity production-inventory system with two products in inventory. With stochastic order quantities and time between orders, the model reflects a supply chain that operates in an environment with high levels of volatility. The inventory is replenished using an independent order-up-to (s, S) policy or a can-order (s, c, S) joint replenishment policy in which the endogenously determined lead times drive the parameters of the replenishment policy. The production facility is modeled as a multi-type MMAP[K]/PH[K]/1 queue in which there are K possible inventory positions when the order is placed and the age process of the busy queue has matrix-exponential distribution. We characterize the system and determine the steady state distribution using matrix analytic methods. Using numerical methods we obtain the inventory parameters that minimize the total ordering and inventory related costs. We present numerical comparisons of independent and joint replenishment policies with varying lead times, order quantities, and cost reductions. We further demonstrate the interplay between the two products in terms of lead times, order quantities and costs.
{"title":"A joint replenishment production-inventory model as an MMAP[K]/PH[K]/1 queue","authors":"Ann M. Noblesse, Nikki Sonenberg, R. Boute, M. Lambrecht, B. Van Houdt","doi":"10.1080/15326349.2022.2049822","DOIUrl":"https://doi.org/10.1080/15326349.2022.2049822","url":null,"abstract":"Abstract In this paper we analyze a continuous review finite capacity production-inventory system with two products in inventory. With stochastic order quantities and time between orders, the model reflects a supply chain that operates in an environment with high levels of volatility. The inventory is replenished using an independent order-up-to (s, S) policy or a can-order (s, c, S) joint replenishment policy in which the endogenously determined lead times drive the parameters of the replenishment policy. The production facility is modeled as a multi-type MMAP[K]/PH[K]/1 queue in which there are K possible inventory positions when the order is placed and the age process of the busy queue has matrix-exponential distribution. We characterize the system and determine the steady state distribution using matrix analytic methods. Using numerical methods we obtain the inventory parameters that minimize the total ordering and inventory related costs. We present numerical comparisons of independent and joint replenishment policies with varying lead times, order quantities, and cost reductions. We further demonstrate the interplay between the two products in terms of lead times, order quantities and costs.","PeriodicalId":21970,"journal":{"name":"Stochastic Models","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46919122","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-03-25DOI: 10.1080/15326349.2022.2041037
Kaloyan N. Vitanov, M. Slavtchova-Bojkova
Abstract Biological populations under stress often face certain extinction unless they adapt toward unfavorable circumstances. In some scenarios such adaptation can assume the form of mutations within the genome of the population (e.g., cancer cells resisting chemotherapy, viruses developing resistance toward a vaccine), while in other scenarios adaptation can be in the form of movement toward some physical location (e.g., spreading of cancer cells to parts of the organism unaffected by treatment, animal populations fleeing polluted areas or areas struck by disaster). Regardless of the particular situation, it is often the case that cells/individuals with different levels of adaptation (which we may group into types) emerge among the cells/individuals of a stressed population. We propose a decomposable multi-type Sevastyanov branching process (possibly with multiple supercritical types) for modeling relevant aspects of the dynamics of such populations. The branching process developed within this paper is a generalization of the decomposable multi-type age-dependent branching process with a single supercritical type considered in Slavtchova-Bojkova and Vitanov. With respect to Slavtchova-Bojkova and Vitanov, we introduce additional, possibly supercritical, types into the interaction scheme between types, further, we incorporate possible dependence of the reproductive capabilities of cells/individuals from their age. We obtain a system of integral equations for the probability generating function of the new process and accordingly expand previous results from Slavtchova-Bojkova and Vitanov concerning probabilities of extinction, number of occurred mutations, waiting time to escape mutant, and immediate risk of escaping extinction. We also provide a general numerical scheme for calculating obtained systems of integral equations.
{"title":"Modeling escape from extinction with decomposable multi-type Sevastyanov branching processes","authors":"Kaloyan N. Vitanov, M. Slavtchova-Bojkova","doi":"10.1080/15326349.2022.2041037","DOIUrl":"https://doi.org/10.1080/15326349.2022.2041037","url":null,"abstract":"Abstract Biological populations under stress often face certain extinction unless they adapt toward unfavorable circumstances. In some scenarios such adaptation can assume the form of mutations within the genome of the population (e.g., cancer cells resisting chemotherapy, viruses developing resistance toward a vaccine), while in other scenarios adaptation can be in the form of movement toward some physical location (e.g., spreading of cancer cells to parts of the organism unaffected by treatment, animal populations fleeing polluted areas or areas struck by disaster). Regardless of the particular situation, it is often the case that cells/individuals with different levels of adaptation (which we may group into types) emerge among the cells/individuals of a stressed population. We propose a decomposable multi-type Sevastyanov branching process (possibly with multiple supercritical types) for modeling relevant aspects of the dynamics of such populations. The branching process developed within this paper is a generalization of the decomposable multi-type age-dependent branching process with a single supercritical type considered in Slavtchova-Bojkova and Vitanov. With respect to Slavtchova-Bojkova and Vitanov, we introduce additional, possibly supercritical, types into the interaction scheme between types, further, we incorporate possible dependence of the reproductive capabilities of cells/individuals from their age. We obtain a system of integral equations for the probability generating function of the new process and accordingly expand previous results from Slavtchova-Bojkova and Vitanov concerning probabilities of extinction, number of occurred mutations, waiting time to escape mutant, and immediate risk of escaping extinction. We also provide a general numerical scheme for calculating obtained systems of integral equations.","PeriodicalId":21970,"journal":{"name":"Stochastic Models","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49309950","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-03-16DOI: 10.1080/15326349.2022.2045205
Nicolás Piña, T. Caraballo, E. Porcu
Abstract The paper provides conditions for the fractional Laplacian and its spectral representation on stationary Gaussian random fields to be well-defined. In addition, we study existence and uniqueness of the weak solution for a stochastic fractional elliptic equation driven by an additive colored noise over an open bounded set. Both spectral and variational approaches are used to provide a solution. Further, the functional covariance associated with the solution is derived.
{"title":"A stochastic fractional Laplace equation driven by colored noise on bounded domain, and its covariance functional","authors":"Nicolás Piña, T. Caraballo, E. Porcu","doi":"10.1080/15326349.2022.2045205","DOIUrl":"https://doi.org/10.1080/15326349.2022.2045205","url":null,"abstract":"Abstract The paper provides conditions for the fractional Laplacian and its spectral representation on stationary Gaussian random fields to be well-defined. In addition, we study existence and uniqueness of the weak solution for a stochastic fractional elliptic equation driven by an additive colored noise over an open bounded set. Both spectral and variational approaches are used to provide a solution. Further, the functional covariance associated with the solution is derived.","PeriodicalId":21970,"journal":{"name":"Stochastic Models","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-03-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47785015","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-03-11DOI: 10.1080/15326349.2022.2032755
C. Gutiérrez, C. Minuesa
Abstract Two density-dependent branching processes are considered to model predator–prey populations. For both models, preys are considered to be the main food supply of predators. Moreover, in each generation the number of individuals of each species is distributed according to a binomial distribution with size given by the species population size and probability of success depending on the density of preys per predator at the current generation. The difference between the two proposed processes lies in the food supply of preys. In the first one, we consider that preys have all the food they need at their disposal while in the second one, we assume that the natural resources of the environment are limited and therefore there exists a competition among preys for food supplies. Results on the fixation and extinction of both species as well as conditions for the coexistence are provided for the first model. On the event of coexistence of both populations and on the prey fixation event, the limiting growth rates are obtained. For the second model, we prove that the extinction of the entire system occurs almost surely. Finally, the evolution of both models over the generations and our analytical findings are illustrated by simulated examples.
{"title":"Predator–prey density-dependent branching processes","authors":"C. Gutiérrez, C. Minuesa","doi":"10.1080/15326349.2022.2032755","DOIUrl":"https://doi.org/10.1080/15326349.2022.2032755","url":null,"abstract":"Abstract Two density-dependent branching processes are considered to model predator–prey populations. For both models, preys are considered to be the main food supply of predators. Moreover, in each generation the number of individuals of each species is distributed according to a binomial distribution with size given by the species population size and probability of success depending on the density of preys per predator at the current generation. The difference between the two proposed processes lies in the food supply of preys. In the first one, we consider that preys have all the food they need at their disposal while in the second one, we assume that the natural resources of the environment are limited and therefore there exists a competition among preys for food supplies. Results on the fixation and extinction of both species as well as conditions for the coexistence are provided for the first model. On the event of coexistence of both populations and on the prey fixation event, the limiting growth rates are obtained. For the second model, we prove that the extinction of the entire system occurs almost surely. Finally, the evolution of both models over the generations and our analytical findings are illustrated by simulated examples.","PeriodicalId":21970,"journal":{"name":"Stochastic Models","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42500876","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-02-28DOI: 10.1080/15326349.2022.2036192
Lei Dang, Xamxinur Abdurahman, Z. Teng
Abstract Brucellosis is one of the major infective and contagious bacterial diseases among animals in pastoral areas of some countries. In this paper, we introduce the effect of environment white noise in the spatial propagation process of brucellosis, and consider a stochastic two-patch brucellosis model. On one hand, we get existence and uniqueness of the global positive solution to the stochastic systems. On the other hand, by using the stochastic Lyapunov function theory we obtain a series of stochastic threshold dynamics results, incorporating extinction of the disease, existence of a unique ergodic stationary distribution of the positive solutions to systems in both patch 1 and patch 2. Furthermore, we find that stochastic perturbation is contribute to extinction of the disease to some extent by numerical simulations.
{"title":"The threshold dynamics of a stochastic two-patch brucellosis model","authors":"Lei Dang, Xamxinur Abdurahman, Z. Teng","doi":"10.1080/15326349.2022.2036192","DOIUrl":"https://doi.org/10.1080/15326349.2022.2036192","url":null,"abstract":"Abstract Brucellosis is one of the major infective and contagious bacterial diseases among animals in pastoral areas of some countries. In this paper, we introduce the effect of environment white noise in the spatial propagation process of brucellosis, and consider a stochastic two-patch brucellosis model. On one hand, we get existence and uniqueness of the global positive solution to the stochastic systems. On the other hand, by using the stochastic Lyapunov function theory we obtain a series of stochastic threshold dynamics results, incorporating extinction of the disease, existence of a unique ergodic stationary distribution of the positive solutions to systems in both patch 1 and patch 2. Furthermore, we find that stochastic perturbation is contribute to extinction of the disease to some extent by numerical simulations.","PeriodicalId":21970,"journal":{"name":"Stochastic Models","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42813706","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-02-23DOI: 10.1080/15326349.2022.2040365
Chunmao Huang, Chen Wang, Xiaoqiang Wang
Abstract We consider a supercritical discrete-time branching process with immigration Y in a stationary and ergodic environment ξ. Let mn be the mean of the reproduction distribution at time n conditioned on the environment ξ and be the natural submartingale of the model. We show sufficient conditions for the boundedness of the moments and for and discover the exponential Lp decay rates of as well as the rates of Then, as an application of the moment results, we show the exponential decay rates of and the convergence rates of the average of ratios
{"title":"Moments and asymptotic properties for supercritical branching processes with immigration in random environments","authors":"Chunmao Huang, Chen Wang, Xiaoqiang Wang","doi":"10.1080/15326349.2022.2040365","DOIUrl":"https://doi.org/10.1080/15326349.2022.2040365","url":null,"abstract":"Abstract We consider a supercritical discrete-time branching process with immigration Y in a stationary and ergodic environment ξ. Let mn be the mean of the reproduction distribution at time n conditioned on the environment ξ and be the natural submartingale of the model. We show sufficient conditions for the boundedness of the moments and for and discover the exponential Lp decay rates of as well as the rates of Then, as an application of the moment results, we show the exponential decay rates of and the convergence rates of the average of ratios","PeriodicalId":21970,"journal":{"name":"Stochastic Models","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-02-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43893984","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-02-21DOI: 10.1080/15326349.2021.2022496
H. Jiang, N. Gibson, Y. Chen
Abstract This paper considers the revenue maximization problem for a hydropower company. The company can generate excess electricity by releasing water from a reservoir and then sell it to the energy market. On the other hand, the company has an obligation to keep the reservoir level above a pre-determined level, which may require the company to purchase electricity in order to fulfill the customers’ power demand. The electricity price and reservoir level are both represented by diffusion processes. We refer to a one-factor diffusion model for electricity price, which is known to fit the data well. After applying Bellman dynamic programming principle, we derive the associated state-constrained Hamilton-Jacobi-Bellman (HJB) equation to characterize the value function. Then we prove that the value function is the viscosity solution of the state-constrained HJB equation and it is unique in this constrained optimization problem.
{"title":"A stochastic model for the optimal allocation of hydropower flexibility in renewable energy markets","authors":"H. Jiang, N. Gibson, Y. Chen","doi":"10.1080/15326349.2021.2022496","DOIUrl":"https://doi.org/10.1080/15326349.2021.2022496","url":null,"abstract":"Abstract This paper considers the revenue maximization problem for a hydropower company. The company can generate excess electricity by releasing water from a reservoir and then sell it to the energy market. On the other hand, the company has an obligation to keep the reservoir level above a pre-determined level, which may require the company to purchase electricity in order to fulfill the customers’ power demand. The electricity price and reservoir level are both represented by diffusion processes. We refer to a one-factor diffusion model for electricity price, which is known to fit the data well. After applying Bellman dynamic programming principle, we derive the associated state-constrained Hamilton-Jacobi-Bellman (HJB) equation to characterize the value function. Then we prove that the value function is the viscosity solution of the state-constrained HJB equation and it is unique in this constrained optimization problem.","PeriodicalId":21970,"journal":{"name":"Stochastic Models","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45242335","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-02-21DOI: 10.1080/15326349.2022.2033628
A. Imomov, E. E. Tukhtaev
Abstract We consider the Galton-Watson branching process allowing immigration. We are dealing with the critical case, in which the immigration law has infinite mean and the offspring law have an infinite variance. An explicit-integral form of the generating function of a stationary measure for the process without immigration is found. We study the asymptotic properties of transition probabilities and their convergence to stationary measures in the case of processes with immigration, when the process is ergodic. And also we define a rate of speed of this convergence.
{"title":"On asymptotic structure of critical Galton-Watson branching processes allowing immigration with infinite variance","authors":"A. Imomov, E. E. Tukhtaev","doi":"10.1080/15326349.2022.2033628","DOIUrl":"https://doi.org/10.1080/15326349.2022.2033628","url":null,"abstract":"Abstract We consider the Galton-Watson branching process allowing immigration. We are dealing with the critical case, in which the immigration law has infinite mean and the offspring law have an infinite variance. An explicit-integral form of the generating function of a stationary measure for the process without immigration is found. We study the asymptotic properties of transition probabilities and their convergence to stationary measures in the case of processes with immigration, when the process is ergodic. And also we define a rate of speed of this convergence.","PeriodicalId":21970,"journal":{"name":"Stochastic Models","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42335913","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-02-07DOI: 10.1080/15326349.2022.2026790
S. Ankirchner, Christophette Blanchet-Scalliet, Diana Dorobantu, Laura Gay
Abstract We consider an Ornstein–Uhlenbeck process with different drift rates below and above zero. We derive an analytic expression for the density of the first time, where the process hits a given level. The passage time density is linked to the joint law of the process and its running supremum, and we also provide an analytic formula of the joint density/distribution function. Results from a numerical experiment reveal that our formulas allow to numerically evaluate the joint law and the density of the first passage time faster than a simulation-based method.
{"title":"First passage time density of an Ornstein–Uhlenbeck process with broken drift","authors":"S. Ankirchner, Christophette Blanchet-Scalliet, Diana Dorobantu, Laura Gay","doi":"10.1080/15326349.2022.2026790","DOIUrl":"https://doi.org/10.1080/15326349.2022.2026790","url":null,"abstract":"Abstract We consider an Ornstein–Uhlenbeck process with different drift rates below and above zero. We derive an analytic expression for the density of the first time, where the process hits a given level. The passage time density is linked to the joint law of the process and its running supremum, and we also provide an analytic formula of the joint density/distribution function. Results from a numerical experiment reveal that our formulas allow to numerically evaluate the joint law and the density of the first passage time faster than a simulation-based method.","PeriodicalId":21970,"journal":{"name":"Stochastic Models","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47347357","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-21DOI: 10.1080/15326349.2021.2016446
M. Slavtchova-Bojkova, O. Hyrien, N. Yanev
Abstract We consider multitype Markov branching processes with immigration occurring at time points generated by Poisson random measures. These models find applications to study evolution of multitype cell populations in which new cells join the population according to a time-varying immigration mechanism. The focus of this paper is the supercritical case. We investigate the limiting behavior of the process for different rates of the Poisson random measures. In particular, we prove a result analogous to a strong LLN and establish limiting normal distributions.
{"title":"Poisson random measures and supercritical multitype Markov branching processes","authors":"M. Slavtchova-Bojkova, O. Hyrien, N. Yanev","doi":"10.1080/15326349.2021.2016446","DOIUrl":"https://doi.org/10.1080/15326349.2021.2016446","url":null,"abstract":"Abstract We consider multitype Markov branching processes with immigration occurring at time points generated by Poisson random measures. These models find applications to study evolution of multitype cell populations in which new cells join the population according to a time-varying immigration mechanism. The focus of this paper is the supercritical case. We investigate the limiting behavior of the process for different rates of the Poisson random measures. In particular, we prove a result analogous to a strong LLN and establish limiting normal distributions.","PeriodicalId":21970,"journal":{"name":"Stochastic Models","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49099878","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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