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Alexander L. Stolyar - One of the best experts on this subject based on the ideXlab platform.
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Large-scale join-idle-queue system with general service times
Journal of Applied Probability, 2017Co-Authors: Sergey Foss, Alexander L. StolyarAbstract:Abstract A parallel server system with n identical servers is considered. The service time distribution has a finite mean 1 / μ, but otherwise is arbitrary. Arriving Customers are routed to one of the servers immediately upon arrival. The join-idle-queue routeing algorithm is studied, under which an Arriving Customer is sent to an idle server, if such is available, and to a randomly uniformly chosen server, otherwise. We consider the asymptotic regime where n → ∞ and the Customer input flow rate is λn. Under the condition λ / μ < ½, we prove that, as n → ∞, the sequence of (appropriately scaled) stationary distributions concentrates at the natural equilibrium point, with the fraction of occupied servers being constant at λ / μ. In particular, this implies that the steady-state probability of an Arriving Customer waiting for service vanishes.
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Pull-based load distribution among heterogeneous parallel servers: the case of multiple routers
Queueing Systems, 2017Co-Authors: Alexander L. StolyarAbstract:The model is a service system, consisting of several large server pools. A server’s processing speed and buffer size (which may be finite or infinite) depend on the pool. The input flow of Customers is split equally among a fixed number of routers, which must assign Customers to the servers immediately upon arrival. We consider an asymptotic regime in which the total Customer arrival rate and pool sizes scale to infinity simultaneously, in proportion to a scaling parameter n , while the number of routers remains fixed. We define and study a multi-router generalization of the pull-based Customer assignment (routing) algorithm PULL, introduced in Stolyar (Queueing Syst 80(4): 341–361, 2015 ) for the single-router model. Under the PULL algorithm, when a server becomes idle it sends a “pull-message” to a randomly uniformly selected router; each router operates independently—it assigns an Arriving Customer to a server according to a randomly uniformly chosen available (at this router) pull-message, if there is any, or to a randomly uniformly selected server in the entire system otherwise. Under Markov assumptions (Poisson arrival process and independent exponentially distributed service requirements), and under subcritical system load, we prove asymptotic optimality of PULL: as $$n\rightarrow \infty $$ n → ∞ , the steady-state probability of an Arriving Customer experiencing blocking or waiting vanishes. Furthermore, PULL has an extremely low router–server message exchange rate of one message per Customer. These results generalize some of the single-router results in Stolyar ( 2015 ).
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pull based load distribution among heterogeneous parallel servers the case of multiple routers
arXiv: Probability, 2015Co-Authors: Alexander L. StolyarAbstract:The model is a service system, consisting of several large server pools. A server processing speed and buffer size (which may be finite or infinite) depend on the pool. The input flow of Customers is split equally among a fixed number of routers, which must assign Customers to the servers immediately upon arrival. We consider an asymptotic regime in which the Customer total arrival rate and pool sizes scale to infinity simultaneously, in proportion to a scaling parameter $n$, while the number of routers remains fixed. We define and study a multi-router generalization of the pull-based Customer assignment (routing) algorithm PULL, introduced in [11] for the single-router model. Under PULL algorithm, when a server becomes idle it send a "pull-message" to a randomly uniformly selected router; each router operates independently -- it assigns an Arriving Customer to a server according to a randomly uniformly chosen available (at this router) pull-message, if there is any, or to a randomly uniformly selected server in the entire system, otherwise. Under Markov assumptions (Poisson arrival process and independent exponentially distributed service requirements), and under sub-critical system load, we prove asymptotic optimality of PULL: as $n\to\infty$, the steady-state probability of an Arriving Customer experiencing blocking or waiting, vanishes. Furthermore, PULL has an extremely low router-server message exchange rate of one message per Customer. These results generalize some of the single-router results in [11].
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a service system with packing constraints greedy randomized algorithm achieving sublinear in scale optimality gap
arXiv: Probability, 2015Co-Authors: Alexander L. Stolyar, Yuan ZhongAbstract:A service system with multiple types of Arriving Customers is considered. There is an infinite number of homogeneous servers. Multiple Customers can be placed for simultaneous service into one server, subject to general packing constraints. Each new Arriving Customer is placed for service immediately, either into an occupied server, as long as packing constraints are not violated, or into an empty server. After service completion, each Customer leaves its server and the system. The basic objective is to minimize the number of occupied servers in steady state. We study a Greedy-Random (GRAND) placement (packing) algorithm, introduced in [23]. This is a simple online algorithm, which places each Arriving Customer uniformly at random into either one of the already occupied servers that can still fit the Customer, or one of the so-called zero-servers, which are empty servers designated to be available to new arrivals. In [23], a version of the algorithm, labeled GRAND($aZ$), was considered, where the number of zero servers is $aZ$, with $Z$ being the current total number of Customers in the system, and $a>0$ being an algorithm parameter. GRAND($aZ$) was shown in [23] to be asymptotically optimal in the following sense: (a) the steady-state optimality gap grows linearly in the system scale $r$ (the mean total number of Customers in service), i.e. as $c(a) r$ for some $c(a)> 0$; and (b) $c(a) \to 0$ as $a\to 0$. In this paper, we consider the GRAND($Z^p$) algorithm, in which the number of zero-servers is $Z^p$, where $p \in (1-1/(8\kappa),1)$ is an algorithm parameter, and $(\kappa-1)$ is the maximum possible number of Customers that a server can fit. We prove the asymptotic optimality of GRAND($Z^p$) in the sense that the steady-state optimality gap is $o(r)$, sublinear in the system scale. This is a stronger form of asymptotic optimality than that of GRAND($aZ$).
Xi Chen - One of the best experts on this subject based on the ideXlab platform.
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optimal policy for dynamic assortment planning under multinomial logit models
Mathematics of Operations Research, 2021Co-Authors: Xi Chen, Yining Wang, Yuan ZhouAbstract:We study the dynamic assortment planning problem, where for each Arriving Customer, the seller offers an assortment of substitutable products and the Customer makes the purchase among offered produ...
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An Optimal Policy for Dynamic Assortment Planning Under Uncapacitated Multinomial Logit Models
2019Co-Authors: Xi Chen, Wang Yining, Zhou YuanAbstract:We study the dynamic assortment planning problem, where for each Arriving Customer, the seller offers an assortment of substitutable products and Customer makes the purchase among offered products according to an uncapacitated multinomial logit (MNL) model. Since all the utility parameters of MNL are unknown, the seller needs to simultaneously learn Customers' choice behavior and make dynamic decisions on assortments based on the current knowledge. The goal of the seller is to maximize the expected revenue, or equivalently, to minimize the expected regret. Although dynamic assortment planning problem has received an increasing attention in revenue management, most existing policies require the estimation of mean utility for each product and the final regret usually involves the number of products $N$. The optimal regret of the dynamic assortment planning problem under the most basic and popular choice model---MNL model is still open. By carefully analyzing a revenue potential function, we develop a trisection based policy combined with adaptive confidence bound construction, which achieves an {item-independent} regret bound of $O(\sqrt{T})$, where $T$ is the length of selling horizon. We further establish the matching lower bound result to show the optimality of our policy. There are two major advantages of the proposed policy. First, the regret of all our policies has no dependence on $N$. Second, our policies are almost assumption free: there is no assumption on mean utility nor any "separability" condition on the expected revenues for different assortments. Our result also extends the unimodal bandit literature.Comment: 29 pages, 1 figure, 1 table. Removed an additional $O(\sqrt{\log\log T})$ term in the regret upper bound from the previous versio
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Dynamic Assortment Optimization with Changing Contextual Information
2019Co-Authors: Xi Chen, Wang Yining, Zhou YuanAbstract:In this paper, we study the dynamic assortment optimization problem under a finite selling season of length $T$. At each time period, the seller offers an Arriving Customer an assortment of substitutable products under a cardinality constraint, and the Customer makes the purchase among offered products according to a discrete choice model. Most existing work associates each product with a real-valued fixed mean utility and assumes a multinomial logit choice (MNL) model. In many practical applications, feature/contexutal information of products is readily available. In this paper, we incorporate the feature information by assuming a linear relationship between the mean utility and the feature. In addition, we allow the feature information of products to change over time so that the underlying choice model can also be non-stationary. To solve the dynamic assortment optimization under this changing contextual MNL model, we need to simultaneously learn the underlying unknown coefficient and makes the decision on the assortment. To this end, we develop an upper confidence bound (UCB) based policy and establish the regret bound on the order of $\widetilde O(d\sqrt{T})$, where $d$ is the dimension of the feature and $\widetilde O$ suppresses logarithmic dependence. We further established the lower bound $\Omega(d\sqrt{T}/K)$ where $K$ is the cardinality constraint of an offered assortment, which is usually small. When $K$ is a constant, our policy is optimal up to logarithmic factors. In the exploitation phase of the UCB algorithm, we need to solve a combinatorial optimization for assortment optimization based on the learned information. We further develop an approximation algorithm and an efficient greedy heuristic. The effectiveness of the proposed policy is further demonstrated by our numerical studies.Comment: 4 pages, 4 figures. Minor revision and polishing of presentatio
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dynamic assortment optimization with changing contextual information
Research Papers in Economics, 2018Co-Authors: Xi Chen, Yining Wang, Yuan ZhouAbstract:In this paper, we study the dynamic assortment optimization problem under a finite selling season of length $T$. At each time period, the seller offers an Arriving Customer an assortment of substitutable products under a cardinality constraint, and the Customer makes the purchase among offered products according to a discrete choice model. Most existing work associates each product with a real-valued fixed mean utility and assumes a multinomial logit choice (MNL) model. In many practical applications, feature/contexutal information of products is readily available. In this paper, we incorporate the feature information by assuming a linear relationship between the mean utility and the feature. In addition, we allow the feature information of products to change over time so that the underlying choice model can also be non-stationary. To solve the dynamic assortment optimization under this changing contextual MNL model, we need to simultaneously learn the underlying unknown coefficient and makes the decision on the assortment. To this end, we develop an upper confidence bound (UCB) based policy and establish the regret bound on the order of $\widetilde O(d\sqrt{T})$, where $d$ is the dimension of the feature and $\widetilde O$ suppresses logarithmic dependence. We further established the lower bound $\Omega(d\sqrt{T}/K)$ where $K$ is the cardinality constraint of an offered assortment, which is usually small. When $K$ is a constant, our policy is optimal up to logarithmic factors. In the exploitation phase of the UCB algorithm, we need to solve a combinatorial optimization for assortment optimization based on the learned information. We further develop an approximation algorithm and an efficient greedy heuristic. The effectiveness of the proposed policy is further demonstrated by our numerical studies.
Bocharov P. - One of the best experts on this subject based on the ideXlab platform.
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A Retrial Queueing System with a Finite Buffer, several Input Flows and a Customer Searching Server
SCS EUROPE, 2020Co-Authors: Bocharov P., D'apice C., Phong N., Rizelian G.Abstract:We consider a single-server retrial queueing system with K Poisson flows of Customers, which arrive to a buffer of finite capacity. If a Customer upon arrival finds the buffer full, he joins an orbit of limited capacity in order to return to the queue again after an exponentially distributed time interval. An Arriving Customer is lost if he finds the buffer and orbit fully occupied. The service time of an i-type Customer has an arbitrary distribution function Bi(x). Every service completion is followed by a search phase with exponentially distributed duration to seek for the next Customer for service. Customers are taken by the server from the queue according to the FCFS discipline. It is proved that the analysis of this queueing system is reduced to the analysis of a similar queueing system but with only one Poisson flow
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Stationary probabilities of the states of the MAP/G/1/r retrial system with priority servicing of primary Customers
Maik Nauka Publishing Springer SBM, 2020Co-Authors: Bocharov P., Pechinkin A.v., Phong N.h.Abstract:The queue with Markov input flow, retrying Customers, and limited queues both of the primary and retrying Customers was discussed. A new Arriving Customer is queued with the primary Customers, or is sent to the queue of retrying Customers if all places on the queue of primary Customers are occupied, or leaves the system if the Queue of retrying Customers also has no vacant places. A Customer from the queue of retrying Customers can be sent only to a free server. For this system, the stationary probabilities of states, as well as the stationary time distributions of the numbers of retrying and primary Customers in the system were determined for the embedded Markov chain which is generated by the instants of server release
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M/G/1/R retrial queueing systems with priority of primary Customers
2020Co-Authors: Bocharov P., Pavlova O.i., Puzikova D.a.Abstract:We analyze a single-server retrial queueing system with finite buffer, Poisson arrivals, and general distribution of service time. If an Arriving Customer finds the queue completely occupied he joins a retrial group (or orbit) in order to seek service again after an exponentially distributed amount of time. We obtain a stationary distribution of the primary queue size, a recurrent algorithm for the factorial moments of the number of retrial Customers and an expression for the expected number of Customers in the system
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Tandem queues with a Markov flow and blocking
SCS EUROPE, 2020Co-Authors: Bocharov P., Manzo R., Pechinkin A.Abstract:A tandem queueing system with two phases and a Markov flow entering into the first phase is studied. Both phases are characterized by one server with a buffer of finite capacity. The service times have an arbitrary distribution function and the service process in the second phase is of Markov-type. An Arriving Customer who finds the first buffer full is lost. A Customer served in the first phase blocks its operation if there is no free waiting place in the second phase at this moment. The stationary distribution of a Markov chain embedded at the instants of Customer transitions from the first phase to the second one is obtained. A computing algorithm was derived for PH-distribution of service time in the first server. Numerical examples are given
Yuan Zhou - One of the best experts on this subject based on the ideXlab platform.
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optimal policy for dynamic assortment planning under multinomial logit models
Mathematics of Operations Research, 2021Co-Authors: Xi Chen, Yining Wang, Yuan ZhouAbstract:We study the dynamic assortment planning problem, where for each Arriving Customer, the seller offers an assortment of substitutable products and the Customer makes the purchase among offered produ...
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dynamic assortment optimization with changing contextual information
Research Papers in Economics, 2018Co-Authors: Xi Chen, Yining Wang, Yuan ZhouAbstract:In this paper, we study the dynamic assortment optimization problem under a finite selling season of length $T$. At each time period, the seller offers an Arriving Customer an assortment of substitutable products under a cardinality constraint, and the Customer makes the purchase among offered products according to a discrete choice model. Most existing work associates each product with a real-valued fixed mean utility and assumes a multinomial logit choice (MNL) model. In many practical applications, feature/contexutal information of products is readily available. In this paper, we incorporate the feature information by assuming a linear relationship between the mean utility and the feature. In addition, we allow the feature information of products to change over time so that the underlying choice model can also be non-stationary. To solve the dynamic assortment optimization under this changing contextual MNL model, we need to simultaneously learn the underlying unknown coefficient and makes the decision on the assortment. To this end, we develop an upper confidence bound (UCB) based policy and establish the regret bound on the order of $\widetilde O(d\sqrt{T})$, where $d$ is the dimension of the feature and $\widetilde O$ suppresses logarithmic dependence. We further established the lower bound $\Omega(d\sqrt{T}/K)$ where $K$ is the cardinality constraint of an offered assortment, which is usually small. When $K$ is a constant, our policy is optimal up to logarithmic factors. In the exploitation phase of the UCB algorithm, we need to solve a combinatorial optimization for assortment optimization based on the learned information. We further develop an approximation algorithm and an efficient greedy heuristic. The effectiveness of the proposed policy is further demonstrated by our numerical studies.
Zhou Yuan - One of the best experts on this subject based on the ideXlab platform.
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An Optimal Policy for Dynamic Assortment Planning Under Uncapacitated Multinomial Logit Models
2019Co-Authors: Xi Chen, Wang Yining, Zhou YuanAbstract:We study the dynamic assortment planning problem, where for each Arriving Customer, the seller offers an assortment of substitutable products and Customer makes the purchase among offered products according to an uncapacitated multinomial logit (MNL) model. Since all the utility parameters of MNL are unknown, the seller needs to simultaneously learn Customers' choice behavior and make dynamic decisions on assortments based on the current knowledge. The goal of the seller is to maximize the expected revenue, or equivalently, to minimize the expected regret. Although dynamic assortment planning problem has received an increasing attention in revenue management, most existing policies require the estimation of mean utility for each product and the final regret usually involves the number of products $N$. The optimal regret of the dynamic assortment planning problem under the most basic and popular choice model---MNL model is still open. By carefully analyzing a revenue potential function, we develop a trisection based policy combined with adaptive confidence bound construction, which achieves an {item-independent} regret bound of $O(\sqrt{T})$, where $T$ is the length of selling horizon. We further establish the matching lower bound result to show the optimality of our policy. There are two major advantages of the proposed policy. First, the regret of all our policies has no dependence on $N$. Second, our policies are almost assumption free: there is no assumption on mean utility nor any "separability" condition on the expected revenues for different assortments. Our result also extends the unimodal bandit literature.Comment: 29 pages, 1 figure, 1 table. Removed an additional $O(\sqrt{\log\log T})$ term in the regret upper bound from the previous versio
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Dynamic Assortment Optimization with Changing Contextual Information
2019Co-Authors: Xi Chen, Wang Yining, Zhou YuanAbstract:In this paper, we study the dynamic assortment optimization problem under a finite selling season of length $T$. At each time period, the seller offers an Arriving Customer an assortment of substitutable products under a cardinality constraint, and the Customer makes the purchase among offered products according to a discrete choice model. Most existing work associates each product with a real-valued fixed mean utility and assumes a multinomial logit choice (MNL) model. In many practical applications, feature/contexutal information of products is readily available. In this paper, we incorporate the feature information by assuming a linear relationship between the mean utility and the feature. In addition, we allow the feature information of products to change over time so that the underlying choice model can also be non-stationary. To solve the dynamic assortment optimization under this changing contextual MNL model, we need to simultaneously learn the underlying unknown coefficient and makes the decision on the assortment. To this end, we develop an upper confidence bound (UCB) based policy and establish the regret bound on the order of $\widetilde O(d\sqrt{T})$, where $d$ is the dimension of the feature and $\widetilde O$ suppresses logarithmic dependence. We further established the lower bound $\Omega(d\sqrt{T}/K)$ where $K$ is the cardinality constraint of an offered assortment, which is usually small. When $K$ is a constant, our policy is optimal up to logarithmic factors. In the exploitation phase of the UCB algorithm, we need to solve a combinatorial optimization for assortment optimization based on the learned information. We further develop an approximation algorithm and an efficient greedy heuristic. The effectiveness of the proposed policy is further demonstrated by our numerical studies.Comment: 4 pages, 4 figures. Minor revision and polishing of presentatio
