Stability of the stochastic matching model
01 Dec 2016-Journal of Applied Probability (Applied Probability Trust)-Vol. 53, Iss: 4, pp 1064-1077
TL;DR: The matching model is introduced and it is proved that the model may be stable if and only if the matching graph is nonbipartite.
Abstract: We introduce and study a new model that we call the matching model. Items arrive one by one in a buffer and depart from it as soon as possible but by pairs. The items of a departing pair are said to be matched. There is a finite set of classes đ± for the items, and the allowed matchings depend on the classes, according to a matching graph on đ±. Upon arrival, an item may find several possible matches in the buffer. This indeterminacy is resolved by a matching policy. When the sequence of classes of the arriving items is independent and identically distributed, the sequence of buffer-content is a Markov chain, whose stability is investigated. In particular, we prove that the model may be stable if and only if the matching graph is nonbipartite.
Citations
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TL;DR: Three parallel service models in which customers of several types are served by several types of servers subject to a bipartite compatibility graph are considered, and Burke's theorem is generalized to parallel service systems.
53Â citations
TL;DR: In this article, it was shown that there always exists a matching policy that is strictly smaller than the set of arrival intensities satisfying NCOND, which is not the case in general.
Abstract: A matching queue is described via a graph, an arrival process and a matching policy. Specifically, to each node in the graph there is a corresponding arrival process of items, which can either be queued or matched with queued items in neighboring nodes. The matching policy specifies how items are matched whenever more than one matching is possible. Given the matching graph and the matching policy, the stability region of the system is the set of intensities of the arrival processes rendering the underlying Markov process positive recurrent. In a recent paper, a condition on the arrival intensities, which was named NCOND, was shown to be necessary for the stability of a matching queue. That condition can be thought of as an analogue to the usual traffic condition for traditional queueing networks, and it is thus natural to study whether it is also sufficient. In this paper, we show that this is not the case in general. Specifically, we prove that, except for a particular class of graphs, there always exists a matching policy rendering the stability region strictly smaller than the set of arrival intensities satisfying NCOND. Our proof combines graph- and queueing-theoretic techniques: After showing explicitly, via fluid-limit arguments that the stability regions of two basic models is strictly included in NCOND, we generalize this result to any graph inducing either one of those two basic graphs.
41Â citations
TL;DR: This work proposes approximation methods based on fluid and diffusion limits using different scalings and shows that some performance measures are insensitive to the matching probability, agreeing with the existing results.
Abstract: This paper focuses on probabilistic matching systems where two classes of users arrive at the system to match with users from the other class. The users are selective and the matchings occur probabilistically. Recently, Markov chain models were proposed to analyze these systems; however, an exact analysis of these models to completely characterize the performance is not possible due to the probabilistic matching structure. In this work, we propose approximation methods based on fluid and diffusion limits using different scalings. We analyze the basic properties of these approximations and show that some performance measures are insensitive to the matching probability, agreeing with the existing results. We also perform numerical experiments with our approximations to gain insight into probabilistic matching systems.
32Â citations
Cites background from "Stability of the stochastic matchin..."
...Mairesse and Moyal [15] generalize the bipartite matching model and develop necessary conditions for matching networks with general topology....
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Posted Contentâą
TL;DR: In this article, a pathwise Loynes' type construction is proposed to prove the existence of a unique matching for the infinite bipartite matching model defined over all the integers.
Abstract: The model of FCFS infinite bipartite matching was introduced in caldentey-kaplan-weiss 2009. In this model there is a sequence of items that are chosen i.i.d. from $\mathcal{C}=\{c_1,\ldots,c_I\}$ and an independent sequence of items that are chosen i.i.d. from $\mathcal{S}=\{s_1,\ldots,s_J\}$, and a bipartite compatibility graph $G$ between $\mathcal{C}$ and $\mathcal{S}$. Items of the two sequences are matched according to the compatibility graph, and the matching is FCFS, each item in the one sequence is matched to the earliest compatible unmatched item in the other sequence. In adan-weiss 2011 a Markov chain associated with the matching was analyzed, a condition for stability was verified, a product form stationary distribution was derived and the rates $r_{c_i,s_j}$ of matches between compatible types $c_i$ and $s_j$ were calculated.
In the current paper, we present several new results that unveil the fundamental structure of the model. First, we provide a pathwise Loynes' type construction which enables to prove the existence of a unique matching for the model defined over all the integers. Second, we prove that the model is dynamically reversible: we define an exchange transformation in which we interchange the positions of each matched pair, and show that the items in the resulting permuted sequences are again independent and i.i.d., and the matching between them is FCFS in reversed time. Third, we obtain product form stationary distributions of several new Markov chains associated with the model. As a by product, we compute useful performance measures, for instance the link lengths between matched items.
32Â citations
TL;DR: In this article, the authors considered a matching system with random arrivals of items of different types and proposed an optimal matching scheme that asymptotically maximizes the long-term average matching reward, while keeping the queues stable.
Abstract: We consider a matching system with random arrivals of items of different types. The items wait in queuesâone per item typeâuntil they are âmatched.â Each matching requires certain quantities of items of different types; after a matching is activated, the associated items leave the system. There exists a finite set of possible matchings, each producing a certain amount of âreward.â This model has a broad range of important applications, including assemble-to-order systems, Internet advertising, and matching web portals. We propose an optimal matching scheme in the sense that it asymptotically maximizes the long-term average matching reward, while keeping the queues stable. The scheme makes matching decisions in a specially constructed virtual system, which in turn controls decisions in the physical system. The key feature of the virtual system is that, unlike the physical one, it allows the queues to become negative. The matchings in the virtual system are controlled by an extended version of the greedy primalâdual (GPD) algorithm, which we prove to be asymptotically optimalâthis in turn implies the asymptotic optimality of the entire scheme. The scheme is real time; at any time, it uses simple rules based on the current state of the virtual and physical queues. It is very robust in that it does not require any knowledge of the item arrival rates and automatically adapts to changing rates. The extended GPD algorithm and its asymptotic optimality apply to a quite general queueing network framework, not limited to matching problems, and therefore are of independent interest.
22Â citations
References
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TL;DR: The stability of a queueing network with interdependent servers is considered and a policy is obtained which is optimal in the sense that its Stability Region is a superset of the stability region of every other scheduling policy, and this stability region is characterized.
Abstract: The stability of a queueing network with interdependent servers is considered. The dependency among the servers is described by the definition of their subsets that can be activated simultaneously. Multihop radio networks provide a motivation for the consideration of this system. The problem of scheduling the server activation under the constraints imposed by the dependency among servers is studied. The performance criterion of a scheduling policy is its throughput that is characterized by its stability region, that is, the set of vectors of arrival and service rates for which the system is stable. A policy is obtained which is optimal in the sense that its stability region is a superset of the stability region of every other scheduling policy, and this stability region is characterized. The behavior of the network is studied for arrival rates that lie outside the stability region. Implications of the results in certain types of concurrent database and parallel processing systems are discussed. >
3,018Â citations
"Stability of the stochastic matchin..." refers background in this paper
...In spirit, it is related to the general models of âconstrained queueing networksâ[12], âinput-queued crossbar switchesâ [11], or âcall centers with skills-based routingâ [8, Section 5]....
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...The result (i) on the optimality of âmatch the longestâ, has connections with a result of [12], which stated that in their âconstrained queueing networkâ, the âmax-weightâ policy has a maximal stability region....
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Bookâą
01 Jan 1999TL;DR: This book describes the development of Markov models for discrete-time Carlo simulation and some of the models used in this study had problems with regard to consistency and Ergodicity.
Abstract: Preface * 1 Probability Review * 2 Discrete Time Markov Models * 3 Recurrence and Ergodicity * 4 Long Run Behavior * 5 Lyapunov Functions and Martingales * 6 Eigenvalues and Nonhomogeneous Markov Chains * 7 Gibbs Fields and Monte Carlo Simulation * 8 Continuous-Time Markov Models 9 Poisson Calculus and Queues * Appendix * Bibliography * Author Index * Subject Index
1,584Â citations
TL;DR: This work begins with a tutorial on how call centers function and proceed to survey academic research devoted to the management of their operations, which identifies important problems that have not been addressed and identifies promising directions for future research.
Abstract: Telephone call centers are an integral part of many businesses, and their economic role is significant and growing. They are also fascinating sociotechnical systems in which the behavior of customers and employees is closely intertwined with physical performance measures. In these environments traditional operational models are of great value--and at the same time fundamentally limited--in their ability to characterize system performance.We review the state of research on telephone call centers. We begin with a tutorial on how call centers function and proceed to survey academic research devoted to the management of their operations. We then outline important problems that have not been addressed and identify promising directions for future research.
1,415Â citations
TL;DR: This paper introduces two maximum weight matching algorithms: longest queue first (LQF) and oldest cell first (OCF), which achieve 100% throughput for all independent arrival processes.
Abstract: It is well known that head-of-line blocking limits the throughput of an input-queued switch with first-in-first-out (FIFO) queues. Under certain conditions, the throughput can be shown to be limited to approximately 58.6%. It is also known that if non-FIFO queueing policies are used, the throughput can be increased. However, it has not been previously shown that if a suitable queueing policy and scheduling algorithm are used, then it is possible to achieve 100% throughput for all independent arrival processes. In this paper we prove this to be the case using a simple linear programming argument and quadratic Lyapunov function. In particular, we assume that each input maintains a separate FIFO queue for each output and that the switch is scheduled using a maximum weight bipartite matching algorithm. We introduce two maximum weight matching algorithms: longest queue first (LQF) and oldest cell first (OCF). Both algorithms achieve 100% throughput for all independent arrival processes. LQF favors queues with larger occupancy, ensuring that larger queues will eventually be served. However, we find that LQF can lead to the permanent starvation of short queues. OCF overcomes this limitation by favoring cells with large waiting times.
851Â citations
24 Mar 1996
TL;DR: This paper proves that if a suitable queueing policy and scheduling algorithm are used then it is possible to achieve 100% throughput for all independent arrival processes.
Abstract: It is well known that head-of-line (HOL) blocking limits the throughput of an input-queued switch with FIFO queues. Under certain conditions, the throughput can be shown to be limited to approximately 58%. It is also known that if non-FIFO queueing policies are used, the throughput can be increased. However it has not been previously shown that if a suitable queueing policy and scheduling algorithm are used then it is possible to achieve 100% throughput for all independent arrival processes. In this paper we prove this to be the case using a simple linear programming argument and quadratic Lyapunov function. In particular we assume that each input maintains a separate FIFO queue for each output and that the switch is scheduled using a maximum weight bipartite matching algorithm.
829Â citations
"Stability of the stochastic matchin..." refers background in this paper
...In spirit, it is related to the general models of âconstrained queueing networksâ[12], âinput-queued crossbar switchesâ [11], or âcall centers with skills-based routingâ [8, Section 5]....
[...]