We first briefly introduce the following classic mean-field particle system given as a collection of Rd-valued stochastic processes {Xin(t) : t ∈ [0, T]}i=1n.Xin(t)=x0+1n∑j=1n∫0tbXin(s),Xjn(s) ds+W...
{"title":"Open Problem—Weakly Interacting Particle Systems on Dense Random Graphs","authors":"Ruoyu Wu","doi":"10.1287/stsy.2019.0049","DOIUrl":"https://doi.org/10.1287/stsy.2019.0049","url":null,"abstract":"We first briefly introduce the following classic mean-field particle system given as a collection of Rd-valued stochastic processes {Xin(t) : t ∈ [0, T]}i=1n.Xin(t)=x0+1n∑j=1n∫0tbXin(s),Xjn(s) ds+W...","PeriodicalId":36337,"journal":{"name":"Stochastic Systems","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1287/stsy.2019.0049","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42559160","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
For queueing systems, leveraging knowledge of job sizes to perform size-based scheduling leads to policies with attractive performance characteristics. Although there is a body of literature in thi...
{"title":"Open Problem—Size-Based Scheduling with Estimation Errors","authors":"D. Down","doi":"10.1287/stsy.2019.0041","DOIUrl":"https://doi.org/10.1287/stsy.2019.0041","url":null,"abstract":"For queueing systems, leveraging knowledge of job sizes to perform size-based scheduling leads to policies with attractive performance characteristics. Although there is a body of literature in thi...","PeriodicalId":36337,"journal":{"name":"Stochastic Systems","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1287/stsy.2019.0041","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46101807","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Central limit theorems represent among the most celebrated of limit theorems in probability theory (Lindeberg 1922, Feller 1945). It may be recalled that the sum of n independent and identically di...
{"title":"Open Problem—Iterative Schemes for Stochastic Optimization: Convergence Statements and Limit Theorems","authors":"A. Jalilzadeh, J. Lei, U. Shanbhag","doi":"10.1287/stsy.2019.0043","DOIUrl":"https://doi.org/10.1287/stsy.2019.0043","url":null,"abstract":"Central limit theorems represent among the most celebrated of limit theorems in probability theory (Lindeberg 1922, Feller 1945). It may be recalled that the sum of n independent and identically di...","PeriodicalId":36337,"journal":{"name":"Stochastic Systems","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1287/stsy.2019.0043","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41843310","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The problem of preemptively scheduling jobs in the M/G/1 queue to minimize mean response time is a classic problem in computer systems. Typically, one assumes that the time it takes to preempt a jo...
{"title":"Open Problem—M/G/1 Scheduling with Preemption Delays","authors":"Ziv Scully","doi":"10.1287/stsy.2019.0047","DOIUrl":"https://doi.org/10.1287/stsy.2019.0047","url":null,"abstract":"The problem of preemptively scheduling jobs in the M/G/1 queue to minimize mean response time is a classic problem in computer systems. Typically, one assumes that the time it takes to preempt a jo...","PeriodicalId":36337,"journal":{"name":"Stochastic Systems","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1287/stsy.2019.0047","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42602018","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the many-server queue shown in Figure 1(a). Customers from class j ∈ {1, 2} arrive according to renewal processes having rate λj > 0 and request processing. There are N ∈ {1, 2,…} servers,...
{"title":"Open Problem—Regarding Static Priority Scheduling for Many-Server Queues with Reneging","authors":"Amy R. Ward","doi":"10.1287/stsy.2019.0048","DOIUrl":"https://doi.org/10.1287/stsy.2019.0048","url":null,"abstract":"We study the many-server queue shown in Figure 1(a). Customers from class j ∈ {1, 2} arrive according to renewal processes having rate λj > 0 and request processing. There are N ∈ {1, 2,…} servers,...","PeriodicalId":36337,"journal":{"name":"Stochastic Systems","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1287/stsy.2019.0048","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45636690","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The relative value iteration scheme (RVI) for Markov decision processes (MDP) dates back to White (1963), a seminal work, which introduced an algorithm for solving the ergodic dynamic programming e...
{"title":"Open Problem—Convergence and Asymptotic Optimality of the Relative Value Iteration in Ergodic Control","authors":"A. Arapostathis","doi":"10.1287/stsy.2019.0040","DOIUrl":"https://doi.org/10.1287/stsy.2019.0040","url":null,"abstract":"The relative value iteration scheme (RVI) for Markov decision processes (MDP) dates back to White (1963), a seminal work, which introduced an algorithm for solving the ergodic dynamic programming e...","PeriodicalId":36337,"journal":{"name":"Stochastic Systems","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1287/stsy.2019.0040","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41500468","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider a single-server queue with unlimited waiting space, the first-come, first-served discipline, a periodic arrival-rate function, and independent and identically distributed service requir...
{"title":"Minimizing the Maximum Expected Waiting Time in a Periodic Single-Server Queue with a Service-Rate Control","authors":"Ni Ma, W. Whitt","doi":"10.1287/stsy.2018.0027","DOIUrl":"https://doi.org/10.1287/stsy.2018.0027","url":null,"abstract":"We consider a single-server queue with unlimited waiting space, the first-come, first-served discipline, a periodic arrival-rate function, and independent and identically distributed service requir...","PeriodicalId":36337,"journal":{"name":"Stochastic Systems","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1287/stsy.2018.0027","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45913483","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study rare event simulations of semimartingale reflecting Brownian motions (SRBMs) in an orthant. The rare event of interest is that a d-dimensional positive recurrent SRBM enters the set [Formula: see text] before hitting a small neighborhood of the origin [Formula: see text] as [Formula: see text] with a starting point outside the two sets and of order o(n). We show that, under two regularity conditions (the Dupuis–Williams stability condition of the SRBM and the Lipschitz continuity assumption of the associated Skorokhod problem), the probability of the rare event satisfies a large deviation principle. To study the variational problem (VP) for the rare event in two dimensions, we adapt its exact solution from developed by Avram, Dai, and Hasenbein in 2001. In three and higher dimensions, we construct a novel subsolution to the VP under a further assumption that the reflection matrix of the SRBM is a nonsingular [Formula: see text]-matrix. Based on the solution/subsolution, particle-based simulation algorithms are constructed to estimate the probability of the rare event. Our estimator is asymptotically optimal for the discretized problem in two dimensions and has exponentially superior performance over standard Monte Carlo in three and higher dimensions. In addition, we establish that the growth rate of the relative bias term arising from discretization is subexponential in all dimensions. Therefore, we can estimate the probability of interest with subexponential complexity growth in two dimensions. In three and higher dimensions, the computational complexity of our estimators has a strictly smaller exponential growth rate than the standard Monte Carlo estimators.
{"title":"Splitting Algorithms for Rare Events of Semimartingale Reflecting Brownian Motions","authors":"K. Leder, Xin Liu, Zicheng Wang","doi":"10.1287/stsy.2021.0076","DOIUrl":"https://doi.org/10.1287/stsy.2021.0076","url":null,"abstract":"We study rare event simulations of semimartingale reflecting Brownian motions (SRBMs) in an orthant. The rare event of interest is that a d-dimensional positive recurrent SRBM enters the set [Formula: see text] before hitting a small neighborhood of the origin [Formula: see text] as [Formula: see text] with a starting point outside the two sets and of order o(n). We show that, under two regularity conditions (the Dupuis–Williams stability condition of the SRBM and the Lipschitz continuity assumption of the associated Skorokhod problem), the probability of the rare event satisfies a large deviation principle. To study the variational problem (VP) for the rare event in two dimensions, we adapt its exact solution from developed by Avram, Dai, and Hasenbein in 2001. In three and higher dimensions, we construct a novel subsolution to the VP under a further assumption that the reflection matrix of the SRBM is a nonsingular [Formula: see text]-matrix. Based on the solution/subsolution, particle-based simulation algorithms are constructed to estimate the probability of the rare event. Our estimator is asymptotically optimal for the discretized problem in two dimensions and has exponentially superior performance over standard Monte Carlo in three and higher dimensions. In addition, we establish that the growth rate of the relative bias term arising from discretization is subexponential in all dimensions. Therefore, we can estimate the probability of interest with subexponential complexity growth in two dimensions. In three and higher dimensions, the computational complexity of our estimators has a strictly smaller exponential growth rate than the standard Monte Carlo estimators.","PeriodicalId":36337,"journal":{"name":"Stochastic Systems","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48925143","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Applied Probability Society Student Paper Competition: Abstracts of 2018 Finalists","authors":"","doi":"10.1287/stsy.2018.0025","DOIUrl":"https://doi.org/10.1287/stsy.2018.0025","url":null,"abstract":"","PeriodicalId":36337,"journal":{"name":"Stochastic Systems","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1287/stsy.2018.0025","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43880451","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Applied Probability Society Student Paper Competition: Abstracts of 2017 Finalists","authors":"","doi":"10.1287/stsy.2018.0024","DOIUrl":"https://doi.org/10.1287/stsy.2018.0024","url":null,"abstract":"","PeriodicalId":36337,"journal":{"name":"Stochastic Systems","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1287/stsy.2018.0024","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49386648","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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