This paper studies a service system in which arriving customers are provided with information about the delay they will experience. Based on this information, they decide to wait for service or leave the system. Specifically, every customer has a patience threshold, and they balk if the observed delay is above the threshold. The main objective is to estimate the parameters of the customers’ patience-level distribution and the corresponding potential arrival rate, using knowledge of the actual queue-length process only. The main complication and distinguishing feature of our setup lies in the fact that customers who decide not to join are not observed, and remarkably, we manage to devise a procedure to estimate the underlying patience and arrival rate parameters. The model is a multiserver queue with a Poisson stream of customers, enabling evaluation of the corresponding likelihood function of the state-dependent effective arrival process. We establish strong consistency of the MLE and derive the asymptotic distribution of the estimation error. Several applications and extensions of the method are discussed. The performance is further assessed through a series of numerical experiments. By fitting parameters of hyperexponential and generalized hyperexponential distributions, our method provides a robust estimation framework for any continuous patience-level distribution. Funding: The research of Yoshiaki Inoue is supported in part by JSPS KAKENHI [Grant JP18K18007]. The research of Liron Ravner and Michael Mandjes is partly funded by NWO Gravitation Project Networks [Grant 024.002.003]. Supplemental Material: The e-companion is available at https://doi.org/10.1287/stsy.2022.0101 .
{"title":"Estimating Customer Impatience in a Service System With Unobserved Balking","authors":"Yoshiaki Inoue, L. Ravner, M. Mandjes","doi":"10.1287/stsy.2022.0101","DOIUrl":"https://doi.org/10.1287/stsy.2022.0101","url":null,"abstract":"This paper studies a service system in which arriving customers are provided with information about the delay they will experience. Based on this information, they decide to wait for service or leave the system. Specifically, every customer has a patience threshold, and they balk if the observed delay is above the threshold. The main objective is to estimate the parameters of the customers’ patience-level distribution and the corresponding potential arrival rate, using knowledge of the actual queue-length process only. The main complication and distinguishing feature of our setup lies in the fact that customers who decide not to join are not observed, and remarkably, we manage to devise a procedure to estimate the underlying patience and arrival rate parameters. The model is a multiserver queue with a Poisson stream of customers, enabling evaluation of the corresponding likelihood function of the state-dependent effective arrival process. We establish strong consistency of the MLE and derive the asymptotic distribution of the estimation error. Several applications and extensions of the method are discussed. The performance is further assessed through a series of numerical experiments. By fitting parameters of hyperexponential and generalized hyperexponential distributions, our method provides a robust estimation framework for any continuous patience-level distribution. Funding: The research of Yoshiaki Inoue is supported in part by JSPS KAKENHI [Grant JP18K18007]. The research of Liron Ravner and Michael Mandjes is partly funded by NWO Gravitation Project Networks [Grant 024.002.003]. Supplemental Material: The e-companion is available at https://doi.org/10.1287/stsy.2022.0101 .","PeriodicalId":36337,"journal":{"name":"Stochastic Systems","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47612031","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 many-server queueing model with server vacations, where the population size dynamics of servers and customers are coupled: a server may leave for vacation only when no customers await, and the capacity available to customers is directly affected by the number of servers on vacation. We focus on scaling regimes in which server dynamics and queue dynamics fluctuate at matching time scales so that their limiting dynamics are coupled. Specifically, we argue that interesting coupled dynamics occur in (a) the Halfin–Whitt regime, (b) the nondegenerate slowdown regime, and (c) the intermediate near Halfin–Whitt regime, whereas the dynamics asymptotically decouple in the other heavy-traffic regimes. We characterize the limiting dynamics, which are different for each scaling regime. We consider relevant respective performance measures for regimes (a) and (b)—namely, the probability of wait and the slowdown. Although closed-form formulas for these performance measures have been derived for models that do not accommodate server vacations, it is difficult to obtain closed-form formulas for these performance measures in the setting with server vacations. Instead, we propose formulas that approximate these performance measures and depend on the steady-state mean number of available servers and previously derived formulas for models without server vacations. We test the accuracy of these formulas numerically.
{"title":"Customer-Server Population Dynamics in Heavy Traffic","authors":"R. Atar, Prasenjit Karmakar, David Lipshutz","doi":"10.1287/stsy.2021.0079","DOIUrl":"https://doi.org/10.1287/stsy.2021.0079","url":null,"abstract":"We study a many-server queueing model with server vacations, where the population size dynamics of servers and customers are coupled: a server may leave for vacation only when no customers await, and the capacity available to customers is directly affected by the number of servers on vacation. We focus on scaling regimes in which server dynamics and queue dynamics fluctuate at matching time scales so that their limiting dynamics are coupled. Specifically, we argue that interesting coupled dynamics occur in (a) the Halfin–Whitt regime, (b) the nondegenerate slowdown regime, and (c) the intermediate near Halfin–Whitt regime, whereas the dynamics asymptotically decouple in the other heavy-traffic regimes. We characterize the limiting dynamics, which are different for each scaling regime. We consider relevant respective performance measures for regimes (a) and (b)—namely, the probability of wait and the slowdown. Although closed-form formulas for these performance measures have been derived for models that do not accommodate server vacations, it is difficult to obtain closed-form formulas for these performance measures in the setting with server vacations. Instead, we propose formulas that approximate these performance measures and depend on the steady-state mean number of available servers and previously derived formulas for models without server vacations. We test the accuracy of these formulas numerically.","PeriodicalId":36337,"journal":{"name":"Stochastic Systems","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-03-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49249865","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 this paper we develop to our best knowledge the first perfect sampling algorithm for queues with Hawkes input (i.e., single-server queues with Hawkes arrivals and independent and identically distributed service times of general distribution). In addition to the stability condition, we also assume the excitation function of the Hawkes process has a light tail and the service time has finite moment-generating function in the neighborhood of the origin. In this procedure, we also propose a new perfect sampling algorithm for Hawkes processes with improved computational efficiency compared with the existing algorithm. Theoretical analysis and numerical tests on the algorithms’ correctness and efficiency are also included.
{"title":"Perfect Sampling of Hawkes Processes and Queues with Hawkes Arrivals","authors":"Xinyun Chen","doi":"10.1287/stsy.2021.0070","DOIUrl":"https://doi.org/10.1287/stsy.2021.0070","url":null,"abstract":"In this paper we develop to our best knowledge the first perfect sampling algorithm for queues with Hawkes input (i.e., single-server queues with Hawkes arrivals and independent and identically distributed service times of general distribution). In addition to the stability condition, we also assume the excitation function of the Hawkes process has a light tail and the service time has finite moment-generating function in the neighborhood of the origin. In this procedure, we also propose a new perfect sampling algorithm for Hawkes processes with improved computational efficiency compared with the existing algorithm. Theoretical analysis and numerical tests on the algorithms’ correctness and efficiency are also included.","PeriodicalId":36337,"journal":{"name":"Stochastic Systems","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46690365","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 generic class of chance-constrained optimization problems with heavy-tailed (i.e., power-law type) risk factors. As the most popular generic method for solving chance constrained optimization, the scenario approach generates sampled optimization problem as a precise approximation with provable reliability, but the computational complexity becomes intractable when the risk tolerance parameter is small. To reduce the complexity, we sample the risk factors from a conditional distribution given that the risk factors are in an analytically tractable event that encompasses all the plausible events of constraints violation. Our approximation is proven to have optimal value within a constant factor to the optimal value of the original chance constraint problem with high probability, uniformly in the risk tolerance parameter. To the best of our knowledge, our result is the first uniform performance guarantee of this type. We additionally demonstrate the efficiency of our algorithm in the context of solvency in portfolio optimization and insurance networks. Funding: The research of B. Zwart is supported by the NWO (Dutch Research Council) [Grant 639.033.413]. The research of J. Blanchet is supported by the Air Force Office of Scientific Research [Award FA9550-20-1-0397], the National Science Foundation [Grants 1820942, 1838576, 1915967, and 2118199], Defense Advanced Research Projects Agency [Award N660011824028], and China Merchants Bank.
{"title":"Efficient Scenario Generation for Heavy-Tailed Chance Constrained Optimization","authors":"J. Blanchet, Fan Zhang, B. Zwart","doi":"10.1287/stsy.2021.0021","DOIUrl":"https://doi.org/10.1287/stsy.2021.0021","url":null,"abstract":"We consider a generic class of chance-constrained optimization problems with heavy-tailed (i.e., power-law type) risk factors. As the most popular generic method for solving chance constrained optimization, the scenario approach generates sampled optimization problem as a precise approximation with provable reliability, but the computational complexity becomes intractable when the risk tolerance parameter is small. To reduce the complexity, we sample the risk factors from a conditional distribution given that the risk factors are in an analytically tractable event that encompasses all the plausible events of constraints violation. Our approximation is proven to have optimal value within a constant factor to the optimal value of the original chance constraint problem with high probability, uniformly in the risk tolerance parameter. To the best of our knowledge, our result is the first uniform performance guarantee of this type. We additionally demonstrate the efficiency of our algorithm in the context of solvency in portfolio optimization and insurance networks. Funding: The research of B. Zwart is supported by the NWO (Dutch Research Council) [Grant 639.033.413]. The research of J. Blanchet is supported by the Air Force Office of Scientific Research [Award FA9550-20-1-0397], the National Science Foundation [Grants 1820942, 1838576, 1915967, and 2118199], Defense Advanced Research Projects Agency [Award N660011824028], and China Merchants Bank.","PeriodicalId":36337,"journal":{"name":"Stochastic Systems","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45968877","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 propose and analyze a recipient-anonymous stochastic routing model to study a fundamental trade-off between anonymity and routing delay. An agent wants to quickly reach a goal vertex in a network through a sequence of routing actions, whereas an overseeing adversary observes the agent’s entire trajectory and tries to identify the agent’s goal among those vertices traversed. We are interested in understanding the probability that the adversary can correctly identify the agent’s goal (anonymity) as a function of the time it takes the agent to reach it (delay). A key feature of our model is the presence of intrinsic uncertainty in the environment, so that each of the agent’s intended steps is subject to random perturbation and thus may not materialize as planned. Using large-network asymptotics, our main results provide near-optimal characterization of the anonymity–delay trade-off under a number of network topologies. Our main technical contributions are centered on a new class of “noise-harnessing” routing strategies that adaptively combine intrinsic uncertainty from the environment with additional artificial randomization to achieve provably efficient obfuscation.
{"title":"Anonymous Stochastic Routing","authors":"Mine Su Erturk, Kuang Xu","doi":"10.1287/stsy.2021.0074","DOIUrl":"https://doi.org/10.1287/stsy.2021.0074","url":null,"abstract":"We propose and analyze a recipient-anonymous stochastic routing model to study a fundamental trade-off between anonymity and routing delay. An agent wants to quickly reach a goal vertex in a network through a sequence of routing actions, whereas an overseeing adversary observes the agent’s entire trajectory and tries to identify the agent’s goal among those vertices traversed. We are interested in understanding the probability that the adversary can correctly identify the agent’s goal (anonymity) as a function of the time it takes the agent to reach it (delay). A key feature of our model is the presence of intrinsic uncertainty in the environment, so that each of the agent’s intended steps is subject to random perturbation and thus may not materialize as planned. Using large-network asymptotics, our main results provide near-optimal characterization of the anonymity–delay trade-off under a number of network topologies. Our main technical contributions are centered on a new class of “noise-harnessing” routing strategies that adaptively combine intrinsic uncertainty from the environment with additional artificial randomization to achieve provably efficient obfuscation.","PeriodicalId":36337,"journal":{"name":"Stochastic Systems","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47317600","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}
This Special Section of Stochastic Systems includes a number of open problems that were presented at a session organized by the Applied Probability Society during the INFORMS Annual Meeting held in...
随机系统的这一特别部分包括一些开放性问题,这些问题是在应用概率学会于。。。
{"title":"Introduction to the Applied Probability Society’s “Open Problems in Applied Probability” Session at the INFORMS Annual Meeting, Phoenix, Arizona, November 4–7, 2018","authors":"R. Atar, Harsha Honappa","doi":"10.1287/stsy.2019.0039","DOIUrl":"https://doi.org/10.1287/stsy.2019.0039","url":null,"abstract":"This Special Section of Stochastic Systems includes a number of open problems that were presented at a session organized by the Applied Probability Society during the INFORMS Annual Meeting held in...","PeriodicalId":36337,"journal":{"name":"Stochastic Systems","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1287/stsy.2019.0039","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41987041","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}
Suppose f:ℝd→ℝ is a smooth function that is bounded from below. The classic stochastic approximation (SA) recursion used to identify a stationary point of f is given byXk+1=Xk–ηkGk+1(Xk), k≥0, (1)...
{"title":"Open Problem—Adaptive Constant-Step Stochastic Approximation","authors":"R. Pasupathy, Harsha Honnappa, S. R. Hunter","doi":"10.1287/stsy.2019.0046","DOIUrl":"https://doi.org/10.1287/stsy.2019.0046","url":null,"abstract":"Suppose f:ℝd→ℝ is a smooth function that is bounded from below. The classic stochastic approximation (SA) recursion used to identify a stationary point of f is given byXk+1=Xk–ηkGk+1(Xk), k≥0, (1)...","PeriodicalId":36337,"journal":{"name":"Stochastic Systems","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1287/stsy.2019.0046","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45085259","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}
Load-balancing algorithms are important for efficiently routing jobs in systems of parallel queues; however, there has been relatively little attention devoted to developing algorithms in the prese...
负载均衡算法对于并行队列系统中作业的高效路由至关重要。然而,目前很少有人关注开发算法。
{"title":"Open Problem—Load Balancing Using Delayed Information","authors":"David Lipshutz","doi":"10.1287/stsy.2019.0045","DOIUrl":"https://doi.org/10.1287/stsy.2019.0045","url":null,"abstract":"Load-balancing algorithms are important for efficiently routing jobs in systems of parallel queues; however, there has been relatively little attention devoted to developing algorithms in the prese...","PeriodicalId":36337,"journal":{"name":"Stochastic Systems","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1287/stsy.2019.0045","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45781999","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 shortest remaining processing time (SRPT) scheduling policy has been deployed in many computer systems, such as web servers (Harchol-Balter et al. 2003), networks (Montazeri et al. 2018), datab...
{"title":"Open Problem—M/G/k/SRPT Under Medium Load","authors":"Isaac Grosof","doi":"10.1287/stsy.2019.0042","DOIUrl":"https://doi.org/10.1287/stsy.2019.0042","url":null,"abstract":"The shortest remaining processing time (SRPT) scheduling policy has been deployed in many computer systems, such as web servers (Harchol-Balter et al. 2003), networks (Montazeri et al. 2018), datab...","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.2019.0042","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44248068","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}