9,380 research outputs found
A product form for the general stochastic matching model
We consider a stochastic matching model with a general compatibility graph,
as introduced in \cite{MaiMoy16}. We show that the natural necessary condition
of stability of the system is also sufficient for the natural matching policy
'First Come, First Matched' (FCFM). For doing so, we derive the stationary
distribution under a remarkable product form, by using an original dynamic
reversibility property related to that of \cite{ABMW17} for the bipartite
matching model
Stability of the stochastic matching model
We introduce and study a new model that we call the {\em 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 {\em matched}. There is
a finite set of classes \maV for the items, and the allowed matchings depend
on the classes, according to a {\em matching graph} on \maV. Upon arrival, an
item may find several possible matches in the buffer. This indeterminacy is
resolved by a {\em matching policy}. When the sequence of classes of the
arriving items is i.i.d., the sequence of buffer-contents 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 non-bipartite
Reversibility and further properties of FCFS infinite bipartite matching
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 and an independent
sequence of items that are chosen i.i.d. from ,
and a bipartite compatibility graph between and
. 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 of matches between compatible types
and 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.Comment: 33 pages, 12 figure
Locally Stable Marriage with Strict Preferences
We study stable matching problems with locality of information and control.
In our model, each agent is a node in a fixed network and strives to be matched
to another agent. An agent has a complete preference list over all other agents
it can be matched with. Agents can match arbitrarily, and they learn about
possible partners dynamically based on their current neighborhood. We consider
convergence of dynamics to locally stable matchings -- states that are stable
with respect to their imposed information structure in the network. In the
two-sided case of stable marriage in which existence is guaranteed, we show
that the existence of a path to stability becomes NP-hard to decide. This holds
even when the network exists only among one partition of agents. In contrast,
if one partition has no network and agents remember a previous match every
round, a path to stability is guaranteed and random dynamics converge with
probability 1. We characterize this positive result in various ways. For
instance, it holds for random memory and for cache memory with the most recent
partner, but not for cache memory with the best partner. Also, it is crucial
which partition of the agents has memory. Finally, we present results for
centralized computation of locally stable matchings, i.e., computing maximum
locally stable matchings in the two-sided case and deciding existence in the
roommates case.Comment: Conference version in ICALP 2013; to appear in SIAM J. Disc Mat
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