Pub Date : 2022-05-26DOI: 10.1017/s0269964822000171
K. Preethi, A. Shophia Lawrence, B. Sivakumar
In this article, we provide a comprehensive analyses of two continuous review lost sales inventory system based on different replenishment policies, namely $(s,S)$ and $(s,Q)$ . We assume that the arrival times of demands form a Poisson process and that the demand sizes have i.i.d. exponential distribution. We assume that the items in stock may obsolete after an exponential time. The lead time for replenishment is exponential. We also assume that the excess demands and the demands that occurred during stock out periods are lost. Using the system point method of level crossing and integral equation method, we derive the steady-state probability distribution of inventory level explicitly. After deriving some system performance measures, we computed the total expected cost rate. We also provide numerical examples of sensitivity analyses involving different parameters and costs. As a result of our numerical analysis, we provide several insights on the optimal $(s,S)$ and $(s,Q)$ policies for inventory systems of obsolescence items with positive lead times. The better policy for maintaining inventory can be quantified numerically.
{"title":"Lost sales obsolescence inventory systems with positive lead time: a system-point level-crossing approach","authors":"K. Preethi, A. Shophia Lawrence, B. Sivakumar","doi":"10.1017/s0269964822000171","DOIUrl":"https://doi.org/10.1017/s0269964822000171","url":null,"abstract":"\u0000 In this article, we provide a comprehensive analyses of two continuous review lost sales inventory system based on different replenishment policies, namely \u0000 \u0000 \u0000 $(s,S)$\u0000 \u0000 and \u0000 \u0000 \u0000 $(s,Q)$\u0000 \u0000 . We assume that the arrival times of demands form a Poisson process and that the demand sizes have i.i.d. exponential distribution. We assume that the items in stock may obsolete after an exponential time. The lead time for replenishment is exponential. We also assume that the excess demands and the demands that occurred during stock out periods are lost. Using the system point method of level crossing and integral equation method, we derive the steady-state probability distribution of inventory level explicitly. After deriving some system performance measures, we computed the total expected cost rate. We also provide numerical examples of sensitivity analyses involving different parameters and costs. As a result of our numerical analysis, we provide several insights on the optimal \u0000 \u0000 \u0000 $(s,S)$\u0000 \u0000 and \u0000 \u0000 \u0000 $(s,Q)$\u0000 \u0000 policies for inventory systems of obsolescence items with positive lead times. The better policy for maintaining inventory can be quantified numerically.","PeriodicalId":54582,"journal":{"name":"Probability in the Engineering and Informational Sciences","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2022-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78425221","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On approximation of the analytic fixed finite time large t probability distributions in an extreme renewal process with no-mean inter-renewals","authors":"P. H. Brill, M. Huang","doi":"10.1017/s0269964822000122","DOIUrl":"https://doi.org/10.1017/s0269964822000122","url":null,"abstract":"We consider an extreme renewal process with no-mean heavy-tailed Pareto(II) inter-renewals and shape parameter \u0000 \u0000 \u0000 $alpha$\u0000 \u0000 where \u0000 \u0000 \u0000 $0<alpha leq 1$\u0000 \u0000 . Two steps are required to derive integral expressions for the analytic probability density functions (pdfs) of the fixed finite time \u0000 \u0000 \u0000 $t$\u0000 \u0000 excess, age, and total life, and require extensive computations. Step 1 creates and solves a Volterra integral equation of the second kind for the limiting pdf of a basic underlying regenerative process defined in the text, which is used for all three fixed finite time \u0000 \u0000 \u0000 $t$\u0000 \u0000 pdfs. Step 2 builds the aforementioned integral expressions based on the limiting pdf in the basic underlying regenerative process. The limiting pdfs of the fixed finite time \u0000 \u0000 \u0000 $t$\u0000 \u0000 pdfs as \u0000 \u0000 \u0000 $trightarrow infty$\u0000 \u0000 do not exist. To reasonably observe the large \u0000 \u0000 \u0000 $t$\u0000 \u0000 pdfs in the extreme renewal process, we approximate them using the limiting pdfs having simple well-known formulas, in a companion renewal process where inter-renewals are right-truncated Pareto(II) variates with finite mean; this does not involve any computations. The distance between the approximating limiting pdfs and the analytic fixed finite time large \u0000 \u0000 \u0000 $t$\u0000 \u0000 pdfs is given by an \u0000 \u0000 \u0000 $L_{1}$\u0000 \u0000 metric taking values in \u0000 \u0000 \u0000 $(0,1)$\u0000 \u0000 , where “near \u0000 \u0000 \u0000 $0$\u0000 \u0000 ” means “close” and “near \u0000 \u0000 \u0000 $1$\u0000 \u0000 ” means “far”.","PeriodicalId":54582,"journal":{"name":"Probability in the Engineering and Informational Sciences","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2022-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84485473","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-05-20DOI: 10.1017/s0269964822000134
Soumaya Idriss
In this paper, we treat a nonlinear and unbalanced $2$ -color urn scheme, subjected to two different nonlinear drawing rules, depending on the color withdrawn. We prove a central limit theorem as well as a law of large numbers for the urn composition. We also give an estimate of the mean and variance of both types of balls.
{"title":"Nonlinear and unbalanced urn models with two types of strategies: a stochastic approximation point of view","authors":"Soumaya Idriss","doi":"10.1017/s0269964822000134","DOIUrl":"https://doi.org/10.1017/s0269964822000134","url":null,"abstract":"\u0000 In this paper, we treat a nonlinear and unbalanced \u0000 \u0000 \u0000 $2$\u0000 \u0000 -color urn scheme, subjected to two different nonlinear drawing rules, depending on the color withdrawn. We prove a central limit theorem as well as a law of large numbers for the urn composition. We also give an estimate of the mean and variance of both types of balls.","PeriodicalId":54582,"journal":{"name":"Probability in the Engineering and Informational Sciences","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2022-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82088535","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-05-02DOI: 10.1017/s0269964822000523
K. R. Bhutani, Ravi Kalpathy, H. Mahmoud
We introduce a broad class of multi-hooking networks, wherein multiple copies of a seed are hooked at each step at random locations, and the number of copies follows a predetermined building sequence of numbers. We analyze the degree profile in random multi-hooking networks by tracking two kinds of node degrees—the local average degree of a specific node over time and the global overall average degree in the graph. The former experiences phases and the latter is invariant with respect to the type of building sequence and is somewhat similar to the average degree in the initial seed. We also discuss the expected number of nodes of the smallest degree. Additionally, we study distances in the network through the lens of the average total path length, the average depth of a node, the eccentricity of a node, and the diameter of the graph.
{"title":"Random multi-hooking networks","authors":"K. R. Bhutani, Ravi Kalpathy, H. Mahmoud","doi":"10.1017/s0269964822000523","DOIUrl":"https://doi.org/10.1017/s0269964822000523","url":null,"abstract":"\u0000 We introduce a broad class of multi-hooking networks, wherein multiple copies of a seed are hooked at each step at random locations, and the number of copies follows a predetermined building sequence of numbers. We analyze the degree profile in random multi-hooking networks by tracking two kinds of node degrees—the local average degree of a specific node over time and the global overall average degree in the graph. The former experiences phases and the latter is invariant with respect to the type of building sequence and is somewhat similar to the average degree in the initial seed. We also discuss the expected number of nodes of the smallest degree. Additionally, we study distances in the network through the lens of the average total path length, the average depth of a node, the eccentricity of a node, and the diameter of the graph.","PeriodicalId":54582,"journal":{"name":"Probability in the Engineering and Informational Sciences","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2022-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88698274","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-04-01DOI: 10.1017/s0269964822000110
{"title":"PES volume 36 issue 2 Cover and Back matter","authors":"","doi":"10.1017/s0269964822000110","DOIUrl":"https://doi.org/10.1017/s0269964822000110","url":null,"abstract":"","PeriodicalId":54582,"journal":{"name":"Probability in the Engineering and Informational Sciences","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2022-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78206532","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-04-01DOI: 10.1017/s0269964822000109
{"title":"PES volume 36 issue 2 Cover and Front matter","authors":"","doi":"10.1017/s0269964822000109","DOIUrl":"https://doi.org/10.1017/s0269964822000109","url":null,"abstract":"","PeriodicalId":54582,"journal":{"name":"Probability in the Engineering and Informational Sciences","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2022-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74886606","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-03-31DOI: 10.1017/S0269964822000080
Eric C. K. Cheung, Haibo Liu
This paper studies a generalization of the Gerber-Shiu expected discounted penalty function [Gerber and Shiu (1998). On the time value of ruin. North American Actuarial Journal 2(1): 48–72] in the context of the perturbed compound Poisson insurance risk model, where the moments of the total discounted claims and the discounted small fluctuations (arising from the Brownian motion) until ruin are also included. In particular, the latter quantity is represented by a stochastic integral and has never been analyzed in the literature to the best of our knowledge. Recursive integro-differential equations satisfied by our generalized Gerber-Shiu function are derived, and these are transformed to defective renewal equations where the components are identified. Explicit solutions are given when the individual claim amounts are distributed as a combination of exponentials. Numerical illustrations are provided, including the computation of the covariance between discounted claims and discounted perturbation until ruin.
本文研究了Gerber-Shiu期望折现惩罚函数[Gerber and Shiu(1998)]的推广。论毁灭的时间价值。北美精算杂志2(1):48-72]在扰动复合泊松保险风险模型的背景下，其中也包括了总贴现索赔和贴现小波动(由布朗运动引起)的矩，直到破产。特别是，后一个量由随机积分表示，据我们所知，在文献中从未对其进行过分析。导出了由广义Gerber-Shiu函数满足的递推积分-微分方程，并将其转化为有缺陷的更新方程。当单个索赔金额作为指数组合分布时，给出显式解决方案。给出了数值实例，包括贴现索赔与贴现扰动之间的协方差计算，直至破产。
{"title":"Joint moments of discounted claims and discounted perturbation until ruin in the compound Poisson risk model with diffusion","authors":"Eric C. K. Cheung, Haibo Liu","doi":"10.1017/S0269964822000080","DOIUrl":"https://doi.org/10.1017/S0269964822000080","url":null,"abstract":"This paper studies a generalization of the Gerber-Shiu expected discounted penalty function [Gerber and Shiu (1998). On the time value of ruin. North American Actuarial Journal 2(1): 48–72] in the context of the perturbed compound Poisson insurance risk model, where the moments of the total discounted claims and the discounted small fluctuations (arising from the Brownian motion) until ruin are also included. In particular, the latter quantity is represented by a stochastic integral and has never been analyzed in the literature to the best of our knowledge. Recursive integro-differential equations satisfied by our generalized Gerber-Shiu function are derived, and these are transformed to defective renewal equations where the components are identified. Explicit solutions are given when the individual claim amounts are distributed as a combination of exponentials. Numerical illustrations are provided, including the computation of the covariance between discounted claims and discounted perturbation until ruin.","PeriodicalId":54582,"journal":{"name":"Probability in the Engineering and Informational Sciences","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2022-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87250436","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-03-15DOI: 10.1017/S0269964821000504
G. S. Fishman, S. Stidham
A company with $n$ geographically widely dispersed sites seeks insurance that pays off if $m$ out of the $n$ sites experience rarely occurring catastrophes (e.g., earthquakes) during a year. This study describes an adaptive dynamic strategy that enables an insurance company to offer the policy with smaller loss probability than more conventional static policies induce, but at a comparable or lower premium. The strategy accomplishes this by periodically purchasing reinsurance on individual sites. Exploiting rarity, the policy induces zero loss with probability one if no more than one quake occurs during any review interval. The policy also may induce a profit if $m$ or more quakes occur in an interval if no quakes have occurred in previous intervals. The study also examines the benefit of more than one reinsurance policy per active site. The study relies on expected utility to determine indifference premiums and derives an upper bound on loss probability independent of premium.
{"title":"An adaptive strategy for offering m-out-of-n insurance policies","authors":"G. S. Fishman, S. Stidham","doi":"10.1017/S0269964821000504","DOIUrl":"https://doi.org/10.1017/S0269964821000504","url":null,"abstract":"A company with $n$ geographically widely dispersed sites seeks insurance that pays off if $m$ out of the $n$ sites experience rarely occurring catastrophes (e.g., earthquakes) during a year. This study describes an adaptive dynamic strategy that enables an insurance company to offer the policy with smaller loss probability than more conventional static policies induce, but at a comparable or lower premium. The strategy accomplishes this by periodically purchasing reinsurance on individual sites. Exploiting rarity, the policy induces zero loss with probability one if no more than one quake occurs during any review interval. The policy also may induce a profit if $m$ or more quakes occur in an interval if no quakes have occurred in previous intervals. The study also examines the benefit of more than one reinsurance policy per active site. The study relies on expected utility to determine indifference premiums and derives an upper bound on loss probability independent of premium.","PeriodicalId":54582,"journal":{"name":"Probability in the Engineering and Informational Sciences","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2022-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86470075","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-03-11DOI: 10.1017/s0269964822000043
M'hamed Gaïgi, Idris Kharroubi, T. Lim
In this paper, we consider the problem of sustainable harvesting. We explain how the manager maximizes his/her profit according to the quantity of natural resource available in a harvesting area and under the constraint of penalties and fines when the quota is exceeded. We characterize the optimal values and some optimal strategies using a verification result. We then show by numerical examples that this optimal strategy is better than naive ones. Moreover, we define a level of fines which insures the double objective of the sustainable harvesting: a remaining quantity of available natural resource to insure its sustainability and an acceptable income for the manager.
{"title":"Optimal management and valuation of a natural resource: the case of optimal harvesting","authors":"M'hamed Gaïgi, Idris Kharroubi, T. Lim","doi":"10.1017/s0269964822000043","DOIUrl":"https://doi.org/10.1017/s0269964822000043","url":null,"abstract":"\u0000 In this paper, we consider the problem of sustainable harvesting. We explain how the manager maximizes his/her profit according to the quantity of natural resource available in a harvesting area and under the constraint of penalties and fines when the quota is exceeded. We characterize the optimal values and some optimal strategies using a verification result. We then show by numerical examples that this optimal strategy is better than naive ones. Moreover, we define a level of fines which insures the double objective of the sustainable harvesting: a remaining quantity of available natural resource to insure its sustainability and an acceptable income for the manager.","PeriodicalId":54582,"journal":{"name":"Probability in the Engineering and Informational Sciences","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2022-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86695968","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-02-22DOI: 10.1017/S0269964822000018
Yu Chen, Yu Xing
Abstract In this paper, we study the credit default swap (CDS) pricing with counterparty risk in a reduced form model. The default jump intensities of the reference firm and counterparty are both assumed to follow the mean-reverting CIR processes with independent jumps respectively and a common jump. The approximate closed-form solutions of the joint survival probability density and the probability density of the first default can be obtained by using the PDE method. Then with the expressions of the probability densities, we can get the formula for the CDS price with counterparty risk in a reduced form model with a common jump. In the numerical analysis part, we find that the default of the reference asset has a greater impact on the CDS price than that of the default of counterparty after introducing the common jump process.
{"title":"Credit default swap pricing with counterparty risk in a reduced form model with a common jump process","authors":"Yu Chen, Yu Xing","doi":"10.1017/S0269964822000018","DOIUrl":"https://doi.org/10.1017/S0269964822000018","url":null,"abstract":"Abstract In this paper, we study the credit default swap (CDS) pricing with counterparty risk in a reduced form model. The default jump intensities of the reference firm and counterparty are both assumed to follow the mean-reverting CIR processes with independent jumps respectively and a common jump. The approximate closed-form solutions of the joint survival probability density and the probability density of the first default can be obtained by using the PDE method. Then with the expressions of the probability densities, we can get the formula for the CDS price with counterparty risk in a reduced form model with a common jump. In the numerical analysis part, we find that the default of the reference asset has a greater impact on the CDS price than that of the default of counterparty after introducing the common jump process.","PeriodicalId":54582,"journal":{"name":"Probability in the Engineering and Informational Sciences","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2022-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91298464","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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