19 research outputs found
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
Local stability in a transient Markov chain
We prove two propositions with conditions that a system, which is described
by a transient Markov chain, will display local stability. Examples of such
systems include partly overloaded Jackson networks, partly overloaded polling
systems, and overloaded multi-server queues with skill based service, under
first come first served policy.Comment: 6 page
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
Many Server Scaling of the N-System Under FCFS-ALIS
The N-System with independent Poisson arrivals and exponential
server-dependent service times under first come first served and assign to
longest idle server policy has explicit steady state distribution. We scale the
arrival and the number of servers simultaneously, and obtain the fluid and
central limit approximation for the steady state. This is the first step
towards exploring the many server scaling limit behavior of general parallel
service systems
FCFS Parallel Service Systems and Matching Models
We consider three parallel service models in which customers of several types
are served by several types of servers subject to a bipartite compatibility
graph, and the service policy is first come first served. Two of the models
have a fixed set of servers. The first is a queueing model in which arriving
customers are assigned to the longest idling compatible server if available, or
else queue up in a single queue, and servers that become available pick the
longest waiting compatible customer, as studied by Adan and Weiss, 2014. The
second is a redundancy service model where arriving customers split into copies
that queue up at all the compatible servers, and are served in each queue on
FCFS basis, and leave the system when the first copy completes service, as
studied by Gardner et al., 2016. The third model is a matching queueing model
with a random stream of arriving servers. Arriving customers queue in a single
queue and arriving servers match with the first compatible customer and leave
immediately with the customer, or they leave without a customer. The last model
is relevant to organ transplants, to housing assignments, to adoptions and many
other situations.
We study the relations between these models, and show that they are closely
related to the FCFS infinite bipartite matching model, in which two infinite
sequences of customers and servers of several types are matched FCFS according
to a bipartite compatibility graph, as studied by Adan et al., 2017. We also
introduce a directed bipartite matching model in which we embed the queueing
systems. This leads to a generalization of Burke's theorem to parallel service
systems