Multiclass queueing networks (McQNs) extend the classical concept of the Jackson network by allowing jobs of different classes to visit the same server. Although such a generalization seems rather ...
{"title":"Stochastic Monotonicity of Markovian Multiclass Queueing Networks","authors":"Leahu Haralambie, M. Mandjes","doi":"10.1287/STSY.2018.0022","DOIUrl":"https://doi.org/10.1287/STSY.2018.0022","url":null,"abstract":"Multiclass queueing networks (McQNs) extend the classical concept of the Jackson network by allowing jobs of different classes to visit the same server. Although such a generalization seems rather ...","PeriodicalId":36337,"journal":{"name":"Stochastic Systems","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2016-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1287/STSY.2018.0022","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66531183","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 service system with an infinite number of exponential servers sharing a finite service capacity. The servers are ordered according to their speed, and arriving customers join the fastest idle server. A capacity allocation is an infinite sequence of service rates. We study the probabilistic properties of this system by considering overflows from sub-systems with a finite number of servers. Several stability measures are suggested and analysed. The tail of the series of service rates that minimizes the average expected delay (service time) is shown to be approximately geometrically decreasing. We use this property in order to approximate the optimal allocation of service rates by constructing an appropriate dynamic program.
{"title":"Delay-Minimizing Capacity Allocation in an Infinite Server-Queueing System","authors":"Refael Hassin, L. Ravner","doi":"10.1287/STSY.2018.0020","DOIUrl":"https://doi.org/10.1287/STSY.2018.0020","url":null,"abstract":"We consider a service system with an infinite number of exponential servers sharing a finite service capacity. The servers are ordered according to their speed, and arriving customers join the fastest idle server. A capacity allocation is an infinite sequence of service rates. We study the probabilistic properties of this system by considering overflows from sub-systems with a finite number of servers. Several stability measures are suggested and analysed. The tail of the series of service rates that minimizes the average expected delay (service time) is shown to be approximately geometrically decreasing. We use this property in order to approximate the optimal allocation of service rates by constructing an appropriate dynamic program.","PeriodicalId":36337,"journal":{"name":"Stochastic Systems","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2016-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1287/STSY.2018.0020","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66531466","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 paper considers optimization over multiple renewal systems coupled by time average constraints. These systems act asynchronously over variable length frames. For each system, at the beginning of each renewal frame, it chooses an action which affects the duration of its own frame, the penalty, and the resource expenditure throughout the frame. The goal is to minimize the overall time average penalty subject to several overall time average resource constraints which couple these systems. This problem has applications to task processing networks, coupled Markov decision processes(MDPs) and so on. We propose a distributed algorithm so that each system can make its own decision after observing a global multiplier which is updated slot-wise. We show that this algorithm satisfies the desired constraints and achieves $mathcal{O}(varepsilon)$ near optimality with $mathcal{O}(1/varepsilon^2)$ convergence time.
{"title":"Asynchronous Optimization over Weakly Coupled Renewal Systems","authors":"Xiaohan Wei, M. Neely","doi":"10.1287/STSY.2018.0013","DOIUrl":"https://doi.org/10.1287/STSY.2018.0013","url":null,"abstract":"This paper considers optimization over multiple renewal systems coupled by time average constraints. These systems act asynchronously over variable length frames. For each system, at the beginning of each renewal frame, it chooses an action which affects the duration of its own frame, the penalty, and the resource expenditure throughout the frame. The goal is to minimize the overall time average penalty subject to several overall time average resource constraints which couple these systems. This problem has applications to task processing networks, coupled Markov decision processes(MDPs) and so on. We propose a distributed algorithm so that each system can make its own decision after observing a global multiplier which is updated slot-wise. We show that this algorithm satisfies the desired constraints and achieves $mathcal{O}(varepsilon)$ near optimality with $mathcal{O}(1/varepsilon^2)$ convergence time.","PeriodicalId":36337,"journal":{"name":"Stochastic Systems","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2016-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1287/STSY.2018.0013","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66531241","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}
Given an ODE and its perturbation, the Alekseev formula expresses the solutions of the latter in terms related to the former. By exploiting this formula and a new concentration inequality for martingale-differences, we develop a novel approach for analyzing nonlinear Stochastic Approximation (SA). This approach is useful for studying a SA's behaviour close to a Locally Asymptotically Stable Equilibrium (LASE) of its limiting ODE; this LASE need not be the limiting ODE's only attractor. As an application, we obtain a new concentration bound for nonlinear SA. That is, given $epsilon >0$ and that the current iterate is in a neighbourhood of a LASE, we provide an estimate for i.) the time required to hit the $epsilon-$ball of this LASE, and ii.) the probability that after this time the iterates are indeed within this $epsilon-$ball and stay there thereafter. The latter estimate can also be viewed as the `lock-in' probability. Compared to related results, our concentration bound is tighter and holds under significantly weaker assumptions. In particular, our bound applies even when the stepsizes are not square-summable. Despite the weaker hypothesis, we show that the celebrated Kushner-Clark lemma continues to hold. %
{"title":"A Concentration Bound for Stochastic Approximation via Alekseev’s Formula","authors":"Gugan Thoppe, V. Borkar","doi":"10.1287/STSY.2018.0019","DOIUrl":"https://doi.org/10.1287/STSY.2018.0019","url":null,"abstract":"Given an ODE and its perturbation, the Alekseev formula expresses the solutions of the latter in terms related to the former. By exploiting this formula and a new concentration inequality for martingale-differences, we develop a novel approach for analyzing nonlinear Stochastic Approximation (SA). This approach is useful for studying a SA's behaviour close to a Locally Asymptotically Stable Equilibrium (LASE) of its limiting ODE; this LASE need not be the limiting ODE's only attractor. As an application, we obtain a new concentration bound for nonlinear SA. That is, given $epsilon >0$ and that the current iterate is in a neighbourhood of a LASE, we provide an estimate for i.) the time required to hit the $epsilon-$ball of this LASE, and ii.) the probability that after this time the iterates are indeed within this $epsilon-$ball and stay there thereafter. The latter estimate can also be viewed as the `lock-in' probability. Compared to related results, our concentration bound is tighter and holds under significantly weaker assumptions. In particular, our bound applies even when the stepsizes are not square-summable. Despite the weaker hypothesis, we show that the celebrated Kushner-Clark lemma continues to hold. %","PeriodicalId":36337,"journal":{"name":"Stochastic Systems","volume":"18 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2015-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1287/STSY.2018.0019","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66531431","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}
Probability distributions over rankings are crucial for the modeling and design of a wide range of practical systems. In this work, we pursue a nonparametric approach that seeks to learn a distribution over rankings (aka the ranking model) that is consistent with the observed data and has the sparsest possible support (i.e., the smallest number of rankings with nonzero probability mass). We focus on first-order marginal data, which comprise information on the probability that item i is ranked at position j, for all possible item and position pairs. The observed data may be noisy. Finding the sparsest approximation requires brute force search in the worst case. To address this issue, we restrict our search to, what we dub, the signature family, and show that the sparsest model within the signature family can be found computationally efficiently compared with the brute force approach. We then establish that the signature family provides good approximations to popular ranking model classes, such as the multinomial logit and the exponential family classes, with support size that is small relative to the dimension of the observed data. We test our methods on two data sets: the ranked election data set from the American Psychological Association and the preference ordering data on 10 different sushi varieties.
{"title":"Inferring Sparse Preference Lists from Partial Information","authors":"V. Farias, Srikanth Jagabathula, D. Shah","doi":"10.1287/stsy.2019.0060","DOIUrl":"https://doi.org/10.1287/stsy.2019.0060","url":null,"abstract":"Probability distributions over rankings are crucial for the modeling and design of a wide range of practical systems. In this work, we pursue a nonparametric approach that seeks to learn a distribution over rankings (aka the ranking model) that is consistent with the observed data and has the sparsest possible support (i.e., the smallest number of rankings with nonzero probability mass). We focus on first-order marginal data, which comprise information on the probability that item i is ranked at position j, for all possible item and position pairs. The observed data may be noisy. Finding the sparsest approximation requires brute force search in the worst case. To address this issue, we restrict our search to, what we dub, the signature family, and show that the sparsest model within the signature family can be found computationally efficiently compared with the brute force approach. We then establish that the signature family provides good approximations to popular ranking model classes, such as the multinomial logit and the exponential family classes, with support size that is small relative to the dimension of the observed data. We test our methods on two data sets: the ranked election data set from the American Psychological Association and the preference ordering data on 10 different sushi varieties.","PeriodicalId":36337,"journal":{"name":"Stochastic Systems","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2010-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66531202","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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