626 research outputs found
Exact Sampling of Stationary and Time-Reversed Queues
We provide the first algorithm that under minimal assumptions allows to
simulate the stationary waiting-time sequence of a single-server queue
backwards in time, jointly with the input processes of the queue (inter-arrival
and service times). The single-server queue is useful in applications of DCFTP
(Dominated Coupling From The Past), which is a well known protocol for
simulation without bias from steady-state distributions. Our algorithm
terminates in finite time assuming only finite mean of the inter-arrival and
service times. In order to simulate the single-server queue in stationarity
until the first idle period in finite expected termination time we require the
existence of finite variance. This requirement is also necessary for such idle
time (which is a natural coalescence time in DCFTP applications) to have finite
mean. Thus, in this sense, our algorithm is applicable under minimal
assumptions.Comment: 30 pages, 3 figures, Journa
Perfect Simulation of Queues
In this paper we describe a perfect simulation algorithm for the stable
queue. Sigman (2011: Exact Simulation of the Stationary Distribution of
the FIFO M/G/c Queue. Journal of Applied Probability, 48A, 209--213) showed how
to build a dominated CFTP algorithm for perfect simulation of the super-stable
queue operating under First Come First Served discipline, with
dominating process provided by the corresponding queue (using Wolff's
sample path monotonicity, which applies when service durations are coupled in
order of initiation of service), and exploiting the fact that the workload
process for the queue remains the same under different queueing
disciplines, in particular under the Processor Sharing discipline, for which a
dynamic reversibility property holds. We generalize Sigman's construction to
the stable case by comparing the queue to a copy run under Random
Assignment. This allows us to produce a naive perfect simulation algorithm
based on running the dominating process back to the time it first empties. We
also construct a more efficient algorithm that uses sandwiching by lower and
upper processes constructed as coupled queues started respectively from
the empty state and the state of the queue under Random Assignment. A
careful analysis shows that appropriate ordering relationships can still be
maintained, so long as service durations continue to be coupled in order of
initiation of service. We summarize statistical checks of simulation output,
and demonstrate that the mean run-time is finite so long as the second moment
of the service duration distribution is finite.Comment: 28 pages, 5 figure
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
Two extensions of Kingman's GI/G/1 bound
A simple bound in GI/G/1 queues was obtained by Kingman using a discrete martingale transform. We extend this technique to 1) multiclass queues and 2) Markov Additive Processes (MAPs) whose background processes can be time-inhomogeneous or have an uncountable state-space. Both extensions are facilitated by a necessary and sufficient ordinary differential equation (ODE) condition for MAPs to admit continuous martingale transforms. Simulations show that the bounds on waiting time distributions are almost exact in heavy-traffic, including the cases of 1) heterogeneous input, e.g., mixing Weibull and Erlang-k classes and 2) Generalized Markovian Arrival Processes, a new class extending the Batch Markovian Arrival Processes to continuous batch sizes
- …