3,936 research outputs found
Fixed points for multi-class queues
Burke's theorem can be seen as a fixed-point result for an exponential
single-server queue; when the arrival process is Poisson, the departure process
has the same distribution as the arrival process. We consider extensions of
this result to multi-type queues, in which different types of customer have
different levels of priority. We work with a model of a queueing server which
includes discrete-time and continuous-time M/M/1 queues as well as queues with
exponential or geometric service batches occurring in discrete time or at
points of a Poisson process. The fixed-point results are proved using
interchangeability properties for queues in tandem, which have previously been
established for one-type M/M/1 systems. Some of the fixed-point results have
previously been derived as a consequence of the construction of stationary
distributions for multi-type interacting particle systems, and we explain the
links between the two frameworks. The fixed points have interesting
"clustering" properties for lower-priority customers. An extreme case is an
example of a Brownian queue, in which lower-priority work only occurs at a set
of times of measure 0 (and corresponds to a local time process for the
queue-length process of higher priority work).Comment: 25 page
Waiting times in queueing networks with a single shared server
We study a queueing network with a single shared server that serves the
queues in a cyclic order. External customers arrive at the queues according to
independent Poisson processes. After completing service, a customer either
leaves the system or is routed to another queue. This model is very generic and
finds many applications in computer systems, communication networks,
manufacturing systems, and robotics. Special cases of the introduced network
include well-known polling models, tandem queues, systems with a waiting room,
multi-stage models with parallel queues, and many others. A complicating factor
of this model is that the internally rerouted customers do not arrive at the
various queues according to a Poisson process, causing standard techniques to
find waiting-time distributions to fail. In this paper we develop a new method
to obtain exact expressions for the Laplace-Stieltjes transforms of the
steady-state waiting-time distributions. This method can be applied to a wide
variety of models which lacked an analysis of the waiting-time distribution
until now
Metastability of Queuing Networks with Mobile Servers
We study symmetric queuing networks with moving servers and FIFO service
discipline. The mean-field limit dynamics demonstrates unexpected behavior
which we attribute to the meta-stability phenomenon. Large enough finite
symmetric networks on regular graphs are proved to be transient for arbitrarily
small inflow rates. However, the limiting non-linear Markov process possesses
at least two stationary solutions. The proof of transience is based on
martingale techniques
Analysis of a polling system modeling QoS differentiation in WLANs
This paper investigates a polling system with a random polling scheme, a 1-limited service discipline and deterministic service requirement modeling WLANs with QoS differentation capability. The system contains high and low priority queues that are distinguished via the probability of being served next. We propose a new iteration algorithm to approximate the waiting time of customers in the high and low priority queues. As shown by simulation results, our approximation is accurate for light to moderately loaded networks
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
- …