The COVID-19 pandemic has profoundly boosted the use of hybrid healthcare settings, which orchestrate face-to-face services together with virtual ones. The advantages of virtual healthcare services are clear: they are less costly and less disruptive for patients who can receive the service in the comfort of their home and reduce patients’ exposure to illnesses prevalent in healthcare facilities. Nevertheless, there is evidence that patients are likely to require a supplementary in-person service upon completion of their virtual service. Motivated by such settings, we study a multiservice queueing system with face-to-face, virtual, and supplementary service channels. The service operator needs to allocate service capacity among the three classes and decide how to prioritize the patients when a service provider becomes available. The strong dependency between virtual and supplementary visits makes the problem challenging. Based on a fluid relaxation, we develop an index-based policy, the [Formula: see text] rule (or the [Formula: see text] rule in short), which, in addition to the holding cost, service time, abandonment rate, and service reward, also carefully balances the return probability and associated penalty. The theoretical results along with numerical experiments demonstrate the effectiveness of the proposed policy and the importance of capacity coordination when managing hybrid service settings. Our work provides insights on the trade-off between convenience and the value of care when offering virtual healthcare services. Funding: The author was supported in part by an Israel Science Foundation [Grant 277/21] and the Israel National Institute for Health Policy Research [Grant 2021/160/R].
{"title":"Managing Queues with Reentrant Customers in Support of Hybrid Healthcare","authors":"Noa Zychlinski","doi":"10.1287/stsy.2022.0105","DOIUrl":"https://doi.org/10.1287/stsy.2022.0105","url":null,"abstract":"The COVID-19 pandemic has profoundly boosted the use of hybrid healthcare settings, which orchestrate face-to-face services together with virtual ones. The advantages of virtual healthcare services are clear: they are less costly and less disruptive for patients who can receive the service in the comfort of their home and reduce patients’ exposure to illnesses prevalent in healthcare facilities. Nevertheless, there is evidence that patients are likely to require a supplementary in-person service upon completion of their virtual service. Motivated by such settings, we study a multiservice queueing system with face-to-face, virtual, and supplementary service channels. The service operator needs to allocate service capacity among the three classes and decide how to prioritize the patients when a service provider becomes available. The strong dependency between virtual and supplementary visits makes the problem challenging. Based on a fluid relaxation, we develop an index-based policy, the [Formula: see text] rule (or the [Formula: see text] rule in short), which, in addition to the holding cost, service time, abandonment rate, and service reward, also carefully balances the return probability and associated penalty. The theoretical results along with numerical experiments demonstrate the effectiveness of the proposed policy and the importance of capacity coordination when managing hybrid service settings. Our work provides insights on the trade-off between convenience and the value of care when offering virtual healthcare services. Funding: The author was supported in part by an Israel Science Foundation [Grant 277/21] and the Israel National Institute for Health Policy Research [Grant 2021/160/R].","PeriodicalId":36337,"journal":{"name":"Stochastic Systems","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41584737","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 global optimization problem where the objective function can be observed exactly at individual design points with no derivative information. We suppose that the design points are determined sequentially using an epsilon-greedy algorithm, that is, by sampling uniformly on the design space with a certain probability and otherwise sampling in a local neighborhood of the current estimate of the best solution. We study the rate at which the estimate converges to the global optimum and derive two types of bounds: an asymptotic pathwise rate and a concentration inequality measuring the likelihood that the asymptotic rate has not yet gone into effect. The order of the rate becomes faster when the width of the local search neighborhood is made to shrink over time at a suitably chosen speed.
{"title":"Convergence Rates of Epsilon-Greedy Global Optimization Under Radial Basis Function Interpolation","authors":"Jialin Li, I. Ryzhov","doi":"10.1287/stsy.2022.0096","DOIUrl":"https://doi.org/10.1287/stsy.2022.0096","url":null,"abstract":"We study a global optimization problem where the objective function can be observed exactly at individual design points with no derivative information. We suppose that the design points are determined sequentially using an epsilon-greedy algorithm, that is, by sampling uniformly on the design space with a certain probability and otherwise sampling in a local neighborhood of the current estimate of the best solution. We study the rate at which the estimate converges to the global optimum and derive two types of bounds: an asymptotic pathwise rate and a concentration inequality measuring the likelihood that the asymptotic rate has not yet gone into effect. The order of the rate becomes faster when the width of the local search neighborhood is made to shrink over time at a suitably chosen speed.","PeriodicalId":36337,"journal":{"name":"Stochastic Systems","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46286003","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}
Externalities are the costs that a user of a common resource imposes on others. In the context of an FCFS M/G/1 queue, where a customer with service demand [Formula: see text] arrives when the workload level is [Formula: see text], the externality [Formula: see text] is the total waiting time that could be saved if this customer gave up on their service demand. In this work, we analyze the externalities process [Formula: see text]. It is shown that this process can be represented by an integral of a (shifted in time by v) compound Poisson process with a positive discrete jump distribution, so that [Formula: see text] is convex. Furthermore, we compute the Laplace-Stieltjes transform of the finite-dimensional distributions of [Formula: see text] and its mean and auto-covariance functions. We also identify conditions under which a sequence of normalized externalities processes admits a weak convergence on [Formula: see text] equipped with the uniform metric to an integral of a (shifted in time by v) standard Wiener process. Finally, we also consider the extended framework when v is a general nonnegative random variable which is independent from the arrival process and the service demands. Our analysis leads to substantial generalizations of the results presented in the existing literature. Funding: This research was supported by the European Union’s Horizon 2020 research and innovation programme [Marie Skłodowska-Curie Grant Agreement 945045] and the NWO Gravitation project NETWORKS [Grant 024.002.003].
{"title":"Externalities in Queues as Stochastic Processes: The Case of FCFS M/G/1","authors":"R. Jacobovic, M. Mandjes","doi":"10.1287/stsy.2022.0021","DOIUrl":"https://doi.org/10.1287/stsy.2022.0021","url":null,"abstract":"Externalities are the costs that a user of a common resource imposes on others. In the context of an FCFS M/G/1 queue, where a customer with service demand [Formula: see text] arrives when the workload level is [Formula: see text], the externality [Formula: see text] is the total waiting time that could be saved if this customer gave up on their service demand. In this work, we analyze the externalities process [Formula: see text]. It is shown that this process can be represented by an integral of a (shifted in time by v) compound Poisson process with a positive discrete jump distribution, so that [Formula: see text] is convex. Furthermore, we compute the Laplace-Stieltjes transform of the finite-dimensional distributions of [Formula: see text] and its mean and auto-covariance functions. We also identify conditions under which a sequence of normalized externalities processes admits a weak convergence on [Formula: see text] equipped with the uniform metric to an integral of a (shifted in time by v) standard Wiener process. Finally, we also consider the extended framework when v is a general nonnegative random variable which is independent from the arrival process and the service demands. Our analysis leads to substantial generalizations of the results presented in the existing literature. Funding: This research was supported by the European Union’s Horizon 2020 research and innovation programme [Marie Skłodowska-Curie Grant Agreement 945045] and the NWO Gravitation project NETWORKS [Grant 024.002.003].","PeriodicalId":36337,"journal":{"name":"Stochastic Systems","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-07-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41326505","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 make a detailed analysis in a special case of the Boston algorithm, which is widely used around the world to assign students to schools. We compute the limiting distribution in large random markets of both the utilitarian welfare and the order bias, a recently introduced average-case fairness measure. Our results show that the differences in utilitarian welfare between the Boston algorithms and the serial dictatorship (SD) algorithm are small and positive, whereas the differences in terms of order bias are large and positive. The naive implementation of the Boston algorithm beats its adaptive implementation on both utilitarian welfare and order bias, and both apparently beat SD on both criteria. In order to establish our results, we derive several basic results on the time evolution of the assignments made by the algorithms, which we expect to be useful for other applications. For example, we compute limiting distributions as a function of [Formula: see text] of the exit time and preference rank obtained for an arbitrary agent whose initial relative position in the tiebreak order is θ.
{"title":"Asymptotic Welfare Performance of Boston Assignment Algorithms","authors":"G. Pritchard, Mark C. Wilson","doi":"10.1287/stsy.2022.0104","DOIUrl":"https://doi.org/10.1287/stsy.2022.0104","url":null,"abstract":"We make a detailed analysis in a special case of the Boston algorithm, which is widely used around the world to assign students to schools. We compute the limiting distribution in large random markets of both the utilitarian welfare and the order bias, a recently introduced average-case fairness measure. Our results show that the differences in utilitarian welfare between the Boston algorithms and the serial dictatorship (SD) algorithm are small and positive, whereas the differences in terms of order bias are large and positive. The naive implementation of the Boston algorithm beats its adaptive implementation on both utilitarian welfare and order bias, and both apparently beat SD on both criteria. In order to establish our results, we derive several basic results on the time evolution of the assignments made by the algorithms, which we expect to be useful for other applications. For example, we compute limiting distributions as a function of [Formula: see text] of the exit time and preference rank obtained for an arbitrary agent whose initial relative position in the tiebreak order is θ.","PeriodicalId":36337,"journal":{"name":"Stochastic Systems","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47814883","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 problem of Bayesian learning in a dynamical system involving strategic agents with asymmetric information. In a series of seminal papers in the literature, this problem has been investigated under a simplifying model where selfish players appear sequentially and act once in the game. It has been shown that there exist information cascades where users discard their private information and mimic the action of their predecessor. In this paper, we provide a framework for studying Bayesian learning dynamics in a more general setting than the one just described. In particular, our model incorporates cases where players can act repeatedly and there is strategic interaction in that each agent’s payoff may also depend on other players’ actions. The proposed framework hinges on a sequential decomposition methodology for finding structured perfect Bayesian equilibria of a general class of dynamic games with asymmetric information. Using this methodology, we study a specific dynamic learning model where players make decisions about public investment based on their estimates of everyone’s states. We characterize a set of informational cascades for this problem where learning stops for the team as a whole. Moreover, we show that such cascades occur almost surely.
{"title":"A Framework for Studying Decentralized Bayesian Learning with Strategic Agents","authors":"Deepanshu Vasal, A. Anastasopoulos","doi":"10.1287/stsy.2021.0092","DOIUrl":"https://doi.org/10.1287/stsy.2021.0092","url":null,"abstract":"We study the problem of Bayesian learning in a dynamical system involving strategic agents with asymmetric information. In a series of seminal papers in the literature, this problem has been investigated under a simplifying model where selfish players appear sequentially and act once in the game. It has been shown that there exist information cascades where users discard their private information and mimic the action of their predecessor. In this paper, we provide a framework for studying Bayesian learning dynamics in a more general setting than the one just described. In particular, our model incorporates cases where players can act repeatedly and there is strategic interaction in that each agent’s payoff may also depend on other players’ actions. The proposed framework hinges on a sequential decomposition methodology for finding structured perfect Bayesian equilibria of a general class of dynamic games with asymmetric information. Using this methodology, we study a specific dynamic learning model where players make decisions about public investment based on their estimates of everyone’s states. We characterize a set of informational cascades for this problem where learning stops for the team as a whole. Moreover, we show that such cascades occur almost surely.","PeriodicalId":36337,"journal":{"name":"Stochastic Systems","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45033647","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}
In Hurtado-Lange and Maguluri (2020) [Transform methods for heavy-traffic analysis. Stochastic Systems 10(4):275–309], the statement of Claim 4 has a typo; it should say that the expression therein is bounded by a constant. The statement of Lemma 11, and Lemma 14 is incorrect. The lemmas should show the existence of the moment generating function in an interval around the origin with the length being independent of the heavy-traffic parameter. The correct statements and proofs are provided.
{"title":"Erratum: “Transform Methods for Heavy-Traffic Analysis”","authors":"Daniela Hurtado-Lange, S. T. Maguluri","doi":"10.1287/stsy.2021.0089","DOIUrl":"https://doi.org/10.1287/stsy.2021.0089","url":null,"abstract":"In Hurtado-Lange and Maguluri (2020) [Transform methods for heavy-traffic analysis. Stochastic Systems 10(4):275–309], the statement of Claim 4 has a typo; it should say that the expression therein is bounded by a constant. The statement of Lemma 11, and Lemma 14 is incorrect. The lemmas should show the existence of the moment generating function in an interval around the origin with the length being independent of the heavy-traffic parameter. The correct statements and proofs are provided.","PeriodicalId":36337,"journal":{"name":"Stochastic Systems","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47738083","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 introduce a new class of policies called “delay-join the shortest queue (delay-JSQ)” for use in parallel processing networks with removable servers. When jobs arrive to the system while all servers are on, jobs should be routed to the shortest queue. However, when servers are off, they take a random time to turn back on, which we allow to occur only when the number of jobs in each of the nonempty queues exceeds a fixed threshold. This new class of policies balances the load among all servers that are currently on and balances the capacity by keeping servers off until they are needed. A detailed numerical study shows that at moderate loads (where server farms and increasingly manufacturing facilities operate), delay-JSQ outperforms JSQ by up to 80%. In addition, it does so without precise knowledge of the input parameters and even when the input process is nonstationary.
{"title":"Delay-Join the Shortest Queue Routing for a Parallel Queueing System with Removable Servers","authors":"Pamela Badian-Pessot, D. Down, M. Lewis","doi":"10.1287/stsy.2021.0090","DOIUrl":"https://doi.org/10.1287/stsy.2021.0090","url":null,"abstract":"We introduce a new class of policies called “delay-join the shortest queue (delay-JSQ)” for use in parallel processing networks with removable servers. When jobs arrive to the system while all servers are on, jobs should be routed to the shortest queue. However, when servers are off, they take a random time to turn back on, which we allow to occur only when the number of jobs in each of the nonempty queues exceeds a fixed threshold. This new class of policies balances the load among all servers that are currently on and balances the capacity by keeping servers off until they are needed. A detailed numerical study shows that at moderate loads (where server farms and increasingly manufacturing facilities operate), delay-JSQ outperforms JSQ by up to 80%. In addition, it does so without precise knowledge of the input parameters and even when the input process is nonstationary.","PeriodicalId":36337,"journal":{"name":"Stochastic Systems","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42621850","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 bound the rate at which the steady-state distribution of the join-the-shortest-queue (JSQ) system converges, in the Halfin-Whitt regime, to its diffusion limit. Our proof uses Stein’s method and, specifically, the recently proposed prelimit generator comparison approach. The JSQ system is nontrivial and high-dimensional and has a state-space collapse component; our analysis may serve as a helpful example to readers wishing to apply the approach to their own setting.
{"title":"The Join-the-Shortest-Queue System in the Halfin-Whitt Regime: Rates of Convergence to the Diffusion Limit","authors":"Anton Braverman","doi":"10.1287/stsy.2022.0102","DOIUrl":"https://doi.org/10.1287/stsy.2022.0102","url":null,"abstract":"We bound the rate at which the steady-state distribution of the join-the-shortest-queue (JSQ) system converges, in the Halfin-Whitt regime, to its diffusion limit. Our proof uses Stein’s method and, specifically, the recently proposed prelimit generator comparison approach. The JSQ system is nontrivial and high-dimensional and has a state-space collapse component; our analysis may serve as a helpful example to readers wishing to apply the approach to their own setting.","PeriodicalId":36337,"journal":{"name":"Stochastic Systems","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47484486","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}
Lucas van Kreveld, M. Mandjes, Jan-Pieter L. Dorsman
A common assumption in the vast literature on the extremes of spectrally one-sided Markov additive processes (MAPs) is that the continuous-time Markov chain that serves as the background process is irreducible. In the present paper, we consider, motivated by, for example, applications in credit risk, the case in which the irreducibility condition has been lifted, thus allowing the presence of one or more transient classes. More specifically, we consider the distribution of the maximum when the MAP under study has only positive jumps (the spectrally positive case) or negative jumps (the spectrally negative case). The methodology used relies on two crucial previous results: (i) the Wiener–Hopf decomposition for Lévy processes and, in particular, its explicit form in spectrally one-sided cases and (ii) a result on the number of singularities of the matrix exponent of a spectrally one-sided MAP. In both the spectrally positive and negative cases, we derive a system of linear equations of which the solution characterizes the distribution of the maximum of the process. As a by-product of our results, we develop a procedure for calculating the maximum of a spectrally one-sided Lévy process over a phase-type distributed time interval.
{"title":"Extreme Value Analysis for a Markov Additive Process Driven by a Nonirreducible Background Chain","authors":"Lucas van Kreveld, M. Mandjes, Jan-Pieter L. Dorsman","doi":"10.1287/stsy.2021.0086","DOIUrl":"https://doi.org/10.1287/stsy.2021.0086","url":null,"abstract":"A common assumption in the vast literature on the extremes of spectrally one-sided Markov additive processes (MAPs) is that the continuous-time Markov chain that serves as the background process is irreducible. In the present paper, we consider, motivated by, for example, applications in credit risk, the case in which the irreducibility condition has been lifted, thus allowing the presence of one or more transient classes. More specifically, we consider the distribution of the maximum when the MAP under study has only positive jumps (the spectrally positive case) or negative jumps (the spectrally negative case). The methodology used relies on two crucial previous results: (i) the Wiener–Hopf decomposition for Lévy processes and, in particular, its explicit form in spectrally one-sided cases and (ii) a result on the number of singularities of the matrix exponent of a spectrally one-sided MAP. In both the spectrally positive and negative cases, we derive a system of linear equations of which the solution characterizes the distribution of the maximum of the process. As a by-product of our results, we develop a procedure for calculating the maximum of a spectrally one-sided Lévy process over a phase-type distributed time interval.","PeriodicalId":36337,"journal":{"name":"Stochastic Systems","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-01-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44393894","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 journal is pleased to publish the abstracts of the winner and finalists of the 2019 Applied Probability Society’s student paper competition. The 2019 student paper prize committee was chaired by Amy Ward. The 2019 committee members are (in alphabetical order by last name): Reza Aghajani, Pelin Canbolat, Jing Dong, Johan van Leeuwaarden, Ilya Ryzhov, Assaf Zeevi, Jiheng Zhang, and Serhan Ziya.
{"title":"Applied Probability Society Student Paper Competition: Abstracts of 2019 Finalists","authors":"","doi":"10.1287/stsy.2021.0073","DOIUrl":"https://doi.org/10.1287/stsy.2021.0073","url":null,"abstract":"The journal is pleased to publish the abstracts of the winner and finalists of the 2019 Applied Probability Society’s student paper competition. The 2019 student paper prize committee was chaired by Amy Ward. The 2019 committee members are (in alphabetical order by last name): Reza Aghajani, Pelin Canbolat, Jing Dong, Johan van Leeuwaarden, Ilya Ryzhov, Assaf Zeevi, Jiheng Zhang, and Serhan Ziya.","PeriodicalId":36337,"journal":{"name":"Stochastic Systems","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45558910","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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