290 research outputs found
Validity of heavy traffic steady-state approximations in generalized Jackson Networks
We consider a single class open queueing network, also known as a generalized
Jackson network (GJN). A classical result in heavy-traffic theory asserts that
the sequence of normalized queue length processes of the GJN converge weakly to
a reflected Brownian motion (RBM) in the orthant, as the traffic intensity
approaches unity. However, barring simple instances, it is still not known
whether the stationary distribution of RBM provides a valid approximation for
the steady-state of the original network. In this paper we resolve this open
problem by proving that the re-scaled stationary distribution of the GJN
converges to the stationary distribution of the RBM, thus validating a
so-called ``interchange-of-limits'' for this class of networks. Our method of
proof involves a combination of Lyapunov function techniques, strong
approximations and tail probability bounds that yield tightness of the sequence
of stationary distributions of the GJN.Comment: Published at http://dx.doi.org/10.1214/105051605000000638 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Correction. Brownian models of open processing networks: canonical representation of workload
Due to a printing error the above mentioned article [Annals of Applied
Probability 10 (2000) 75--103, doi:10.1214/aoap/1019737665] had numerous
equations appearing incorrectly in the print version of this paper. The entire
article follows as it should have appeared. IMS apologizes to the author and
the readers for this error. A recent paper by Harrison and Van Mieghem
explained in general mathematical terms how one forms an ``equivalent workload
formulation'' of a Brownian network model. Denoting by the state vector
of the original Brownian network, one has a lower dimensional state descriptor
in the equivalent workload formulation, where can be chosen as
any basis matrix for a particular linear space. This paper considers Brownian
models for a very general class of open processing networks, and in that
context develops a more extensive interpretation of the equivalent workload
formulation, thus extending earlier work by Laws on alternate routing problems.
A linear program called the static planning problem is introduced to articulate
the notion of ``heavy traffic'' for a general open network, and the dual of
that linear program is used to define a canonical choice of the basis matrix
. To be specific, rows of the canonical are alternative basic optimal
solutions of the dual linear program. If the network data satisfy a natural
monotonicity condition, the canonical matrix is shown to be nonnegative,
and another natural condition is identified which ensures that admits a
factorization related to the notion of resource pooling.Comment: Published at http://dx.doi.org/10.1214/105051606000000583 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Fluid and Diffusion Limits for Bike Sharing Systems
Bike sharing systems have rapidly developed around the world, and they are
served as a promising strategy to improve urban traffic congestion and to
decrease polluting gas emissions. So far performance analysis of bike sharing
systems always exists many difficulties and challenges under some more general
factors. In this paper, a more general large-scale bike sharing system is
discussed by means of heavy traffic approximation of multiclass closed queueing
networks with non-exponential factors. Based on this, the fluid scaled
equations and the diffusion scaled equations are established by means of the
numbers of bikes both at the stations and on the roads, respectively.
Furthermore, the scaling processes for the numbers of bikes both at the
stations and on the roads are proved to converge in distribution to a
semimartingale reflecting Brownian motion (SRBM) in a -dimensional box,
and also the fluid and diffusion limit theorems are obtained. Furthermore,
performance analysis of the bike sharing system is provided. Thus the results
and methodology of this paper provide new highlight in the study of more
general large-scale bike sharing systems.Comment: 34 pages, 1 figure
On the Convergence of Multiclass Queueing Networks in Heavy Traffic
The subject of this paper is the heavy traffic behavior of a general class of queueing networks with first-in-first-out (FIFO) service discipline. For special cases which require various assumptions on the network structure, several authors have proved heavy traffic limit theorems to justify the approximation of queueing networks by reflected Brownian motions (RBM's). Based on these theorems, some have conjectured that the Brownian approximation may in fact be valid for a more general class of queueing networks. In this paper, we prove that the Brownian approximation does not hold for such a general class of networks. Our findings suggest that studying Brownian models of non-FIFO queueing networks may perhaps be more fruitful
Product-form solutions for integrated services packet networks and cloud computing systems
We iteratively derive the product-form solutions of stationary distributions
of priority multiclass queueing networks with multi-sever stations. The
networks are Markovian with exponential interarrival and service time
distributions. These solutions can be used to conduct performance analysis or
as comparison criteria for approximation and simulation studies of large scale
networks with multi-processor shared-memory switches and cloud computing
systems with parallel-server stations. Numerical comparisons with existing
Brownian approximating model are provided to indicate the effectiveness of our
algorithm.Comment: 26 pages, 3 figures, short conference version is reported at MICAI
200
Sample path large deviations for multiclass feedforward queueing networks in critical loading
We consider multiclass feedforward queueing networks with first in first out
and priority service disciplines at the nodes, and class dependent
deterministic routing between nodes. The random behavior of the network is
constructed from cumulative arrival and service time processes which are
assumed to satisfy an appropriate sample path large deviation principle. We
establish logarithmic asymptotics of large deviations for waiting time, idle
time, queue length, departure and sojourn-time processes in critical loading.
This transfers similar results from Puhalskii about single class queueing
networks with feedback to multiclass feedforward queueing networks, and
complements diffusion approximation results from Peterson. An example with
renewal inter arrival and service time processes yields the rate function of a
reflected Brownian motion. The model directly captures stationary situations.Comment: Published at http://dx.doi.org/10.1214/105051606000000439 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Modeling a healthcare system as a queueing network:The case of a Belgian hospital.
The performance of health care systems in terms of patient flow times and utilization of critical resources can be assessed through queueing and simulation models. We model the orthopaedic department of the Middelheim hospital (Antwerpen, Belgium) focusing on the impact of outages (preemptive and nonpreemptive outages) on the effective utilization of resources and on the flowtime of patients. Several queueing network solution procedures are developed such as the decomposition and Brownian motion approaches. Simulation is used as a validation tool. We present new approaches to model outages. The model offers a valuable tool to study the trade-off between the capacity structure, sources of variability and patient flow times.Belgium; Brownian motion; Capacity management; Decomposition; Health care; Healthcare; Impact; Model; Models; Performance; Performance measurement; Queueing; Queueing theory; Simulation; Stochastic processes; Structure; Studies; Systems; Time; Tool; Validation; Variability;
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