We consider a long-term average profit–maximizing admission control problem in an M/M/1 queuing system with unknown service and arrival rates. With a fixed reward collected upon service completion and a cost per unit of time enforced on customers waiting in the queue, a dispatcher decides upon arrivals whether to admit the arriving customer or not based on the full history of observations of the queue length of the system. Naor [Naor P (1969) The regulation of queue size by levying tolls. Econometrica 37(1):15–24] shows that, if all the parameters of the model are known, then it is optimal to use a static threshold policy: admit if the queue length is less than a predetermined threshold and otherwise not. We propose a learning-based dispatching algorithm and characterize its regret with respect to optimal dispatch policies for the full-information model of Naor [Naor P (1969) The regulation of queue size by levying tolls. Econometrica 37(1):15–24]. We show that the algorithm achieves an O(1) regret when all optimal thresholds with full information are nonzero and achieves an [Formula: see text] regret for any specified [Formula: see text] in the case that an optimal threshold with full information is 0 (i.e., an optimal policy is to reject all arrivals), where N is the number of arrivals.Funding: A. Cohen is partially supported by the National Science Foundation [Grant DMS-2006305]. V. Subramanian is supported in part by the NSF [Grants CCF-2008130, ECCS-2038416, CNS-1955777, and CMMI-2240981].
我们考虑的是一个具有未知服务率和到达率的 M/M/1 排队系统中的长期平均利润最大化接纳控制问题。调度员在服务完成后收取固定奖励，并对排队等候的顾客强制执行单位时间成本，调度员根据对系统排队长度的完整历史观察，在到达时决定是否接纳到达的顾客。Naor [Naor P (1969) The regulation of queue size by levying tolls.Econometrica 37(1):15-24] 表明，如果模型的所有参数都是已知的，那么使用静态阈值策略是最优的：如果队列长度小于预定阈值，则接纳，否则不接纳。我们提出了一种基于学习的调度算法，并描述了其与 Naor [Naor P (1969) The regulation of queue size by levying tolls.经济计量学》37（1）：15-24]。我们证明，当所有全信息最优阈值都不为零时，该算法的遗憾值为 O(1)，而在全信息最优阈值为 0（即最优策略是拒绝所有到达者）的情况下，对于任何指定的[公式：见正文]遗憾值，其中 N 是到达者的数量，该算法的遗憾值为[公式：见正文]：A. Cohen 由美国国家科学基金会 [Grant DMS-2006305] 提供部分资助。V. Subramanian 部分获得了美国国家科学基金会 [CCF-2008130, ECCS-2038416, CNS-1955777 和 CMMI-2240981] 的资助。
{"title":"Learning-Based Optimal Admission Control in a Single-Server Queuing System","authors":"Asaf Cohen, Vijay Subramanian, Yili Zhang","doi":"10.1287/stsy.2022.0042","DOIUrl":"https://doi.org/10.1287/stsy.2022.0042","url":null,"abstract":"We consider a long-term average profit–maximizing admission control problem in an M/M/1 queuing system with unknown service and arrival rates. With a fixed reward collected upon service completion and a cost per unit of time enforced on customers waiting in the queue, a dispatcher decides upon arrivals whether to admit the arriving customer or not based on the full history of observations of the queue length of the system. Naor [Naor P (1969) The regulation of queue size by levying tolls. Econometrica 37(1):15–24] shows that, if all the parameters of the model are known, then it is optimal to use a static threshold policy: admit if the queue length is less than a predetermined threshold and otherwise not. We propose a learning-based dispatching algorithm and characterize its regret with respect to optimal dispatch policies for the full-information model of Naor [Naor P (1969) The regulation of queue size by levying tolls. Econometrica 37(1):15–24]. We show that the algorithm achieves an O(1) regret when all optimal thresholds with full information are nonzero and achieves an [Formula: see text] regret for any specified [Formula: see text] in the case that an optimal threshold with full information is 0 (i.e., an optimal policy is to reject all arrivals), where N is the number of arrivals.Funding: A. Cohen is partially supported by the National Science Foundation [Grant DMS-2006305]. V. Subramanian is supported in part by the NSF [Grants CCF-2008130, ECCS-2038416, CNS-1955777, and CMMI-2240981].","PeriodicalId":36337,"journal":{"name":"Stochastic Systems","volume":"52 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139374276","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 solution of Poisson’s equation plays a key role in constructing the martingale through which sums of Markov correlated random variables can be analyzed. In this paper, we study three different representations for the solution for countable state space irreducible Markov chains, two based on entry time expectations, and the other based on a potential kernel. Our consideration of null recurrent chains allows us to extend our theory to positive recurrent nonexplosive Markov jump processes. We also develop the martingale structure induced by these solutions to Poisson’s equation, under minimal conditions, and establish verifiable Lyapunov conditions to support our theory. Finally, we provide a central limit theorem for Markov dependent sums, under conditions weaker than have previously appeared in the literature.
{"title":"Solution Representations for Poisson’s Equation, Martingale Structure, and the Markov Chain Central Limit Theorem","authors":"Peter W. Glynn, Alex Infanger","doi":"10.1287/stsy.2022.0001","DOIUrl":"https://doi.org/10.1287/stsy.2022.0001","url":null,"abstract":"The solution of Poisson’s equation plays a key role in constructing the martingale through which sums of Markov correlated random variables can be analyzed. In this paper, we study three different representations for the solution for countable state space irreducible Markov chains, two based on entry time expectations, and the other based on a potential kernel. Our consideration of null recurrent chains allows us to extend our theory to positive recurrent nonexplosive Markov jump processes. We also develop the martingale structure induced by these solutions to Poisson’s equation, under minimal conditions, and establish verifiable Lyapunov conditions to support our theory. Finally, we provide a central limit theorem for Markov dependent sums, under conditions weaker than have previously appeared in the literature.","PeriodicalId":36337,"journal":{"name":"Stochastic Systems","volume":"8 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139065896","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 system consisting of n particles, moving forward in jumps on the real line. System state is the empirical distribution of particle locations. Each particle “jumps forward” at some time points, with the instantaneous rate of jumps given by a decreasing function of the particle’s location quantile within the current state (empirical distribution). Previous work on this model established, under certain conditions, the convergence, as [Formula: see text], of the system random dynamics to that of a deterministic mean-field model (MFM), which is a solution to an integro-differential equation. Another line of previous work established the existence of MFMs that are traveling waves, as well as the attraction of MFM trajectories to traveling waves. The main results of this paper are: (a) We prove that, as [Formula: see text], the stationary distributions of (recentered) states concentrate on a (recentered) traveling wave; (b) we obtain a uniform across n moment bound on the stationary distributions of (recentered) states; and (c) we prove a convergence-to-MFM result, which is substantially more general than that in previous work. Results (b) and (c) serve as “ingredients” of the proof of (a), but also are of independent interest.
{"title":"A Particle System with Mean-Field Interaction: Large-Scale Limit of Stationary Distributions","authors":"Alexander L. Stolyar","doi":"10.1287/stsy.2023.0108","DOIUrl":"https://doi.org/10.1287/stsy.2023.0108","url":null,"abstract":"We consider a system consisting of n particles, moving forward in jumps on the real line. System state is the empirical distribution of particle locations. Each particle “jumps forward” at some time points, with the instantaneous rate of jumps given by a decreasing function of the particle’s location quantile within the current state (empirical distribution). Previous work on this model established, under certain conditions, the convergence, as [Formula: see text], of the system random dynamics to that of a deterministic mean-field model (MFM), which is a solution to an integro-differential equation. Another line of previous work established the existence of MFMs that are traveling waves, as well as the attraction of MFM trajectories to traveling waves. The main results of this paper are: (a) We prove that, as [Formula: see text], the stationary distributions of (recentered) states concentrate on a (recentered) traveling wave; (b) we obtain a uniform across n moment bound on the stationary distributions of (recentered) states; and (c) we prove a convergence-to-MFM result, which is substantially more general than that in previous work. Results (b) and (c) serve as “ingredients” of the proof of (a), but also are of independent interest.","PeriodicalId":36337,"journal":{"name":"Stochastic Systems","volume":"39 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135944526","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}
Fernanda Bravo, C. Rudin, Yaron Shaposhnik, Yuting Yuan
We study the problem of predicting congestion risk in intensive care units (ICUs). Congestion is associated with poor service experience, high costs, and poor health outcomes. By predicting future congestion, decision makers can initiate preventive measures, such as rescheduling activities or increasing short-term capacity, to mitigate the effects of congestion. To this end, we consider well-established queueing models of ICUs and define “high-risk states” as system states that are likely to lead to congestion in the near future. We strive to formulate rules for determining whether a given system state is high risk. We design the rules to be interpretable (informally, easy to understand) for their practical appeal to stakeholders. We show that for simple Markovian queueing systems, such as the [Formula: see text] queue with multiple patient classes, our rules take the form of linear and quadratic functions on the state space. For more general queueing systems, we employ methods from queueing theory, simulation, and machine learning (ML) to devise interpretable prediction rules, and we demonstrate their effectiveness through an extensive computational study, which includes a large-scale ICU model validated using data. Our study shows that congestion risk can be effectively and transparently predicted using linear ML models and interpretable features engineered from the queueing model representation of the system. History: This paper has been accepted for the Service Science/Stochastic Systems Joint Special Issue. Supplemental Material: The online appendix is available at https://doi.org/10.1287/stsy.2022.0018 .
{"title":"Interpretable Prediction Rules for Congestion Risk in Intensive Care Units","authors":"Fernanda Bravo, C. Rudin, Yaron Shaposhnik, Yuting Yuan","doi":"10.1287/stsy.2022.0018","DOIUrl":"https://doi.org/10.1287/stsy.2022.0018","url":null,"abstract":"We study the problem of predicting congestion risk in intensive care units (ICUs). Congestion is associated with poor service experience, high costs, and poor health outcomes. By predicting future congestion, decision makers can initiate preventive measures, such as rescheduling activities or increasing short-term capacity, to mitigate the effects of congestion. To this end, we consider well-established queueing models of ICUs and define “high-risk states” as system states that are likely to lead to congestion in the near future. We strive to formulate rules for determining whether a given system state is high risk. We design the rules to be interpretable (informally, easy to understand) for their practical appeal to stakeholders. We show that for simple Markovian queueing systems, such as the [Formula: see text] queue with multiple patient classes, our rules take the form of linear and quadratic functions on the state space. For more general queueing systems, we employ methods from queueing theory, simulation, and machine learning (ML) to devise interpretable prediction rules, and we demonstrate their effectiveness through an extensive computational study, which includes a large-scale ICU model validated using data. Our study shows that congestion risk can be effectively and transparently predicted using linear ML models and interpretable features engineered from the queueing model representation of the system. History: This paper has been accepted for the Service Science/Stochastic Systems Joint Special Issue. Supplemental Material: The online appendix is available at https://doi.org/10.1287/stsy.2022.0018 .","PeriodicalId":36337,"journal":{"name":"Stochastic Systems","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47409658","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}
Antoine Lesage-Landry, Félix Pellerin, Duncan S. Callaway, Joshua A. Taylor
In an effort to reduce power system-caused wildfires, utilities carry out public safety power shutoffs (PSPSs), in which portions of the grid are deenergized to mitigate the risk of ignition. The decision to call a PSPS must balance reducing ignition risks and the negative impact of service interruptions. In this work, we consider three PSPS scheduling scenarios, which we model as dynamic programs. In the first two scenarios, we assume that N PSPSs are budgeted as part of the investment strategy. In the first scenario, a penalty is incurred for each PSPS declared past the Nth event. In the second, we assume that some costs can be recovered if the number of PSPSs is below N while still being subject to a penalty if above N. In the third, the system operator wants to minimize the number of PSPSs such that the total expected cost is below a threshold. We provide optimal or asymptotically optimal policies for each case, the first two of which have closed-form expressions. Lastly, we establish the applicability of the first PSPS model’s policy to critical peak pricing and obtain an optimal scheduling policy to reduce the peak demand based on weather observations. Funding: This work was funded in part by the Natural Sciences and Engineering Research Council of Canada, the Institute for Data Valorization, the National Science Foundation [Award 1351900], the Advanced Research Projects Agency-Energy [Award DE-AR0001061], and the University of California Office of the President Laboratory Fees Program [Grant LFR-20-652467].
{"title":"Optimally Scheduling Public Safety Power Shutoffs","authors":"Antoine Lesage-Landry, Félix Pellerin, Duncan S. Callaway, Joshua A. Taylor","doi":"10.1287/stsy.2022.004","DOIUrl":"https://doi.org/10.1287/stsy.2022.004","url":null,"abstract":"In an effort to reduce power system-caused wildfires, utilities carry out public safety power shutoffs (PSPSs), in which portions of the grid are deenergized to mitigate the risk of ignition. The decision to call a PSPS must balance reducing ignition risks and the negative impact of service interruptions. In this work, we consider three PSPS scheduling scenarios, which we model as dynamic programs. In the first two scenarios, we assume that N PSPSs are budgeted as part of the investment strategy. In the first scenario, a penalty is incurred for each PSPS declared past the Nth event. In the second, we assume that some costs can be recovered if the number of PSPSs is below N while still being subject to a penalty if above N. In the third, the system operator wants to minimize the number of PSPSs such that the total expected cost is below a threshold. We provide optimal or asymptotically optimal policies for each case, the first two of which have closed-form expressions. Lastly, we establish the applicability of the first PSPS model’s policy to critical peak pricing and obtain an optimal scheduling policy to reduce the peak demand based on weather observations. Funding: This work was funded in part by the Natural Sciences and Engineering Research Council of Canada, the Institute for Data Valorization, the National Science Foundation [Award 1351900], the Advanced Research Projects Agency-Energy [Award DE-AR0001061], and the University of California Office of the President Laboratory Fees Program [Grant LFR-20-652467].","PeriodicalId":36337,"journal":{"name":"Stochastic Systems","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135409508","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}
Antoine Lesage-Landry, Félix Pellerin, Duncan S. Callaway, Joshua A. Taylor
In an effort to reduce power system-caused wildfires, utilities carry out public safety power shutoffs (PSPSs), in which portions of the grid are deenergized to mitigate the risk of ignition. The decision to call a PSPS must balance reducing ignition risks and the negative impact of service interruptions. In this work, we consider three PSPS scheduling scenarios, which we model as dynamic programs. In the first two scenarios, we assume that N PSPSs are budgeted as part of the investment strategy. In the first scenario, a penalty is incurred for each PSPS declared past the Nth event. In the second, we assume that some costs can be recovered if the number of PSPSs is below N while still being subject to a penalty if above N. In the third, the system operator wants to minimize the number of PSPSs such that the total expected cost is below a threshold. We provide optimal or asymptotically optimal policies for each case, the first two of which have closed-form expressions. Lastly, we establish the applicability of the first PSPS model’s policy to critical peak pricing and obtain an optimal scheduling policy to reduce the peak demand based on weather observations. Funding: This work was funded in part by the Natural Sciences and Engineering Research Council of Canada, the Institute for Data Valorization, the National Science Foundation [Award 1351900], the Advanced Research Projects Agency-Energy [Award DE-AR0001061], and the University of California Office of the President Laboratory Fees Program [Grant LFR-20-652467].
{"title":"Optimally Scheduling Public Safety Power Shutoffs","authors":"Antoine Lesage-Landry, Félix Pellerin, Duncan S. Callaway, Joshua A. Taylor","doi":"10.1287/stsy.2022.0004","DOIUrl":"https://doi.org/10.1287/stsy.2022.0004","url":null,"abstract":"In an effort to reduce power system-caused wildfires, utilities carry out public safety power shutoffs (PSPSs), in which portions of the grid are deenergized to mitigate the risk of ignition. The decision to call a PSPS must balance reducing ignition risks and the negative impact of service interruptions. In this work, we consider three PSPS scheduling scenarios, which we model as dynamic programs. In the first two scenarios, we assume that N PSPSs are budgeted as part of the investment strategy. In the first scenario, a penalty is incurred for each PSPS declared past the Nth event. In the second, we assume that some costs can be recovered if the number of PSPSs is below N while still being subject to a penalty if above N. In the third, the system operator wants to minimize the number of PSPSs such that the total expected cost is below a threshold. We provide optimal or asymptotically optimal policies for each case, the first two of which have closed-form expressions. Lastly, we establish the applicability of the first PSPS model’s policy to critical peak pricing and obtain an optimal scheduling policy to reduce the peak demand based on weather observations. Funding: This work was funded in part by the Natural Sciences and Engineering Research Council of Canada, the Institute for Data Valorization, the National Science Foundation [Award 1351900], the Advanced Research Projects Agency-Energy [Award DE-AR0001061], and the University of California Office of the President Laboratory Fees Program [Grant LFR-20-652467].","PeriodicalId":36337,"journal":{"name":"Stochastic Systems","volume":"25 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135449343","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}
Distributed ledger technologies provide a mechanism to achieve ordering among transactions that are scattered on multiple participants with no prerequisite trust relations. This mechanism is essentially based on the idea of new transactions referencing older ones in a chain structure. Recently, directed acyclic graph (DAG)-type distributed ledgers that are based on DAGs were proposed to increase the system scalability through sacrificing the total order of transactions. In this paper, we develop a mathematical model to study the process that governs the addition of new transactions to the DAG-type distributed ledger. We propose a simple model for DAG-type distributed ledgers that are obtained from a recursive young-age preferential attachment scheme (i.e., new connections are made preferably to transactions that have not been in the system for very long). We determine the asymptotic degree structure of the resulting graph and show that a forward component of linear size arises if the edge density is chosen sufficiently large in relation to the “young-age preference” that tunes how quickly old transactions become unattractive. Funding: The research of C. Mönch is supported by the Deutsche Forschungsgemeinschaft [Grant 443916008].
{"title":"Directed Acyclic Graph-Type Distributed Ledgers via Young-Age Preferential Attachment","authors":"Christian Mönch, Amr Rizk","doi":"10.1287/stsy.2022.0005","DOIUrl":"https://doi.org/10.1287/stsy.2022.0005","url":null,"abstract":"Distributed ledger technologies provide a mechanism to achieve ordering among transactions that are scattered on multiple participants with no prerequisite trust relations. This mechanism is essentially based on the idea of new transactions referencing older ones in a chain structure. Recently, directed acyclic graph (DAG)-type distributed ledgers that are based on DAGs were proposed to increase the system scalability through sacrificing the total order of transactions. In this paper, we develop a mathematical model to study the process that governs the addition of new transactions to the DAG-type distributed ledger. We propose a simple model for DAG-type distributed ledgers that are obtained from a recursive young-age preferential attachment scheme (i.e., new connections are made preferably to transactions that have not been in the system for very long). We determine the asymptotic degree structure of the resulting graph and show that a forward component of linear size arises if the edge density is chosen sufficiently large in relation to the “young-age preference” that tunes how quickly old transactions become unattractive. Funding: The research of C. Mönch is supported by the Deutsche Forschungsgemeinschaft [Grant 443916008].","PeriodicalId":36337,"journal":{"name":"Stochastic Systems","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48445773","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}
Distributed ledger technologies provide a mechanism to achieve ordering among transactions that are scattered on multiple participants with no prerequisite trust relations. This mechanism is essentially based on the idea of new transactions referencing older ones in a chain structure. Recently, directed acyclic graph (DAG)-type distributed ledgers that are based on DAGs were proposed to increase the system scalability through sacrificing the total order of transactions. In this paper, we develop a mathematical model to study the process that governs the addition of new transactions to the DAG-type distributed ledger. We propose a simple model for DAG-type distributed ledgers that are obtained from a recursive young-age preferential attachment scheme (i.e., new connections are made preferably to transactions that have not been in the system for very long). We determine the asymptotic degree structure of the resulting graph and show that a forward component of linear size arises if the edge density is chosen sufficiently large in relation to the “young-age preference” that tunes how quickly old transactions become unattractive. Funding: The research of C. Mönch is supported by the Deutsche Forschungsgemeinschaft [Grant 443916008].
{"title":"Directed Acyclic Graph-Type Distributed Ledgers via Young-Age Preferential Attachment","authors":"Christian Mönch, Amr Rizk","doi":"10.1287/stsy.2022.005","DOIUrl":"https://doi.org/10.1287/stsy.2022.005","url":null,"abstract":"Distributed ledger technologies provide a mechanism to achieve ordering among transactions that are scattered on multiple participants with no prerequisite trust relations. This mechanism is essentially based on the idea of new transactions referencing older ones in a chain structure. Recently, directed acyclic graph (DAG)-type distributed ledgers that are based on DAGs were proposed to increase the system scalability through sacrificing the total order of transactions. In this paper, we develop a mathematical model to study the process that governs the addition of new transactions to the DAG-type distributed ledger. We propose a simple model for DAG-type distributed ledgers that are obtained from a recursive young-age preferential attachment scheme (i.e., new connections are made preferably to transactions that have not been in the system for very long). We determine the asymptotic degree structure of the resulting graph and show that a forward component of linear size arises if the edge density is chosen sufficiently large in relation to the “young-age preference” that tunes how quickly old transactions become unattractive. Funding: The research of C. Mönch is supported by the Deutsche Forschungsgemeinschaft [Grant 443916008].","PeriodicalId":36337,"journal":{"name":"Stochastic Systems","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135703140","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 a Markovian system of two stations in tandem and two flexible servers who are capable of working at both stations. The novelty of our work is that jobs waiting in line to be processed by the second station may leave the system (without being processed at the second station) after an exponential amount of time. We show that the dynamic server assignment policy that maximizes the long-run average throughput assigns one server to each station unless station 2 is starved (i.e., has no work) or the number of jobs in the buffer reaches a certain threshold (both servers work together at station 1 when station 2 is starved and at station 2 when the threshold is reached). We specify the criterion for the assignment of the servers to the stations and characterize the threshold. Finally, we investigate how the threshold depends on the service and abandonment rates.
{"title":"Optimal Server Assignment in Queues with Flexible Servers and Abandonments","authors":"S. Andradóttir, H. Ayhan","doi":"10.1287/stsy.2023.0109","DOIUrl":"https://doi.org/10.1287/stsy.2023.0109","url":null,"abstract":"We study a Markovian system of two stations in tandem and two flexible servers who are capable of working at both stations. The novelty of our work is that jobs waiting in line to be processed by the second station may leave the system (without being processed at the second station) after an exponential amount of time. We show that the dynamic server assignment policy that maximizes the long-run average throughput assigns one server to each station unless station 2 is starved (i.e., has no work) or the number of jobs in the buffer reaches a certain threshold (both servers work together at station 1 when station 2 is starved and at station 2 when the threshold is reached). We specify the criterion for the assignment of the servers to the stations and characterize the threshold. Finally, we investigate how the threshold depends on the service and abandonment rates.","PeriodicalId":36337,"journal":{"name":"Stochastic Systems","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-04-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44369158","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 assemble-to-order (ATO) inventory systems with a general bill of materials and general deterministic lead times. Unsatisfied demands are always backlogged. We apply a four-step asymptotic framework to develop inventory policies for minimizing the long-run average expected total inventory cost. Our approach features a multistage stochastic program (SP) to establish a lower bound on the inventory cost and determine parameter values for inventory control. Our replenishment policy deviates from the conventional constant base stock policies to accommodate nonidentical lead times. Our component allocation policy differentiates demands based on backlog costs, bill of materials, and component availabilities. We prove that our policy is asymptotically optimal on the diffusion scale, that is, as the longest lead time grows, the percentage difference between the average cost under our policy and its lower bound converges to zero. In developing these results, we formulate a broad stochastic tracking model and prove general convergence results from which the asymptotic optimality of our policy follows as specialized corollaries. Funding: This study is based on work supported by the National Science Foundation [Grant CMMI-1363314].
{"title":"Asymptotically Optimal Inventory Control for Assemble-to-Order Systems","authors":"Martin I. Reiman, Haohua Wan, Qiong Wang","doi":"10.1287/stsy.2022.0099","DOIUrl":"https://doi.org/10.1287/stsy.2022.0099","url":null,"abstract":"We consider assemble-to-order (ATO) inventory systems with a general bill of materials and general deterministic lead times. Unsatisfied demands are always backlogged. We apply a four-step asymptotic framework to develop inventory policies for minimizing the long-run average expected total inventory cost. Our approach features a multistage stochastic program (SP) to establish a lower bound on the inventory cost and determine parameter values for inventory control. Our replenishment policy deviates from the conventional constant base stock policies to accommodate nonidentical lead times. Our component allocation policy differentiates demands based on backlog costs, bill of materials, and component availabilities. We prove that our policy is asymptotically optimal on the diffusion scale, that is, as the longest lead time grows, the percentage difference between the average cost under our policy and its lower bound converges to zero. In developing these results, we formulate a broad stochastic tracking model and prove general convergence results from which the asymptotic optimality of our policy follows as specialized corollaries. Funding: This study is based on work supported by the National Science Foundation [Grant CMMI-1363314].","PeriodicalId":36337,"journal":{"name":"Stochastic Systems","volume":"173 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136132076","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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