2,176 research outputs found
Spare parts provisioning for multiple k-out-of-n:G systems
In this paper, we consider a repair shop that fixes failed components from different k-out-of-n:G systems. We assume that each system consists of the same type of component; to increase availability, a certain number of components are stocked as spare parts. We permit a shared inventory serving all systems and/or reserved inventories for each system; we call this a hybrid model. Additionally, we consider two alternative dispatching rules for the repaired component. The destination for a repaired component can be chosen either on a first-come-first-served basis or by following a static priority rule. Our analysis gives the steady-state system size distribution of the two alternative
models at the repair shop. We conduct numerical examples minimizing the spare parts held while subjecting the availability of each system to exceed a targeted value. Our findings show that unless the availabilities of systems are close, the HP policy is better than the HF policy
Exact Solutions for M/M/c/Setup Queues
Recently multiserver queues with setup times have been extensively studied
because they have applications in power-saving data centers. The most
challenging model is the M/M//Setup queue where a server is turned off when
it is idle and is turned on if there are some waiting jobs. Recently, Gandhi et
al.~(SIGMETRICS 2013, QUESTA 2014) present the recursive renewal reward
approach as a new mathematical tool to analyze the model. In this paper, we
derive exact solutions for the same model using two alternative methodologies:
generating function approach and matrix analytic method. The former yields
several theoretical insights into the systems while the latter provides an
exact recursive algorithm to calculate the joint stationary distribution and
then some performance measures so as to give new application insights.Comment: Submitted for revie
Analysis of the finite-source multiclass priority queue with an unreliable server and setup time
In this article, we study a queueing system serving multiple classes of customers. Each class has a finite-calling population. The customers are served according to the preemptive-resume priority policy. We assume general distributions for the service times. For each priority class, we derive the steady-state system size distributions at departure/arrival and arbitrary time epochs. We introduce the residual augmented process completion times conditioned on the number of customers in the system to obtain the system time distribution. We then extend the model by assuming that the server is subject to operation-independent failures upon which a repair process with random duration starts immediately. We also demonstrate how setup times, which may be required before resuming interrupted service or picking up a new customer, can be incorporated in the model
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
Many-Sources Large Deviations for Max-Weight Scheduling
In this paper, a many-sources large deviations principle (LDP) for the
transient workload of a multi-queue single-server system is established where
the service rates are chosen from a compact, convex and coordinate-convex rate
region and where the service discipline is the max-weight policy. Under the
assumption that the arrival processes satisfy a many-sources LDP, this is
accomplished by employing Garcia's extended contraction principle that is
applicable to quasi-continuous mappings.
For the simplex rate-region, an LDP for the stationary workload is also
established under the additional requirements that the scheduling policy be
work-conserving and that the arrival processes satisfy certain mixing
conditions.
The LDP results can be used to calculate asymptotic buffer overflow
probabilities accounting for the multiplexing gain, when the arrival process is
an average of \emph{i.i.d.} processes. The rate function for the stationary
workload is expressed in term of the rate functions of the finite-horizon
workloads when the arrival processes have \emph{i.i.d.} increments.Comment: 44 page
Propagation of epistemic uncertainty in queueing models with unreliable server using chaos expansions
In this paper, we develop a numerical approach based on Chaos expansions to
analyze the sensitivity and the propagation of epistemic uncertainty through a
queueing systems with breakdowns. Here, the quantity of interest is the
stationary distribution of the model, which is a function of uncertain
parameters. Polynomial chaos provide an efficient alternative to more
traditional Monte Carlo simulations for modelling the propagation of
uncertainty arising from those parameters. Furthermore, Polynomial chaos
expansion affords a natural framework for computing Sobol' indices. Such
indices give reliable information on the relative importance of each uncertain
entry parameters. Numerical results show the benefit of using Polynomial Chaos
over standard Monte-Carlo simulations, when considering statistical moments and
Sobol' indices as output quantities
Models and algorithms for transient queueing congestion at a hub airport
Includes bibliographical references (p. 35-37).Supported by a grant from Draper Laboratory and a National Science Foundation Graduate Fellowship.Dimitris Bertsimas, Michael D. Peterson and Amedeo R. Odoni
A Queueing Characterization of Information Transmission over Block Fading Rayleigh Channels in the Low SNR
Unlike the AWGN (additive white gaussian noise) channel, fading channels
suffer from random channel gains besides the additive Gaussian noise. As a
result, the instantaneous channel capacity varies randomly along time, which
makes it insufficient to characterize the transmission capability of a fading
channel using data rate only. In this paper, the transmission capability of a
buffer-aided block Rayleigh fading channel is examined by a constant rate input
data stream, and reflected by several parameters such as the average queue
length, stationary queue length distribution, packet delay and overflow
probability. Both infinite-buffer model and finite-buffer model are considered.
Taking advantage of the memoryless property of the service provided by the
channel in each block in the the low SNR (signal-to-noise ratio) regime, the
information transmission over the channel is formulated as a \textit{discrete
time discrete state} queueing problem. The obtained results show that
block fading channels are unable to support a data rate close to their ergodic
capacity, no matter how long the buffer is, even seen from the application
layer. For the finite-buffer model, the overflow probability is derived with
explicit expression, and is shown to decrease exponentially when buffer size is
increased, even when the buffer size is very small.Comment: 29 pages, 11 figures. More details on the proof of Theorem 1 and
proposition 1 can be found in "Queueing analysis for block fading Rayleigh
channels in the low SNR regime ", IEEE WCSP 2013.It has been published by
IEEE Trans. on Veh. Technol. in Feb. 201
Stability Analysis of GI/G/c/K Retrial Queue with Constant Retrial Rate
We consider a GI/G/c/K-type retrial queueing system with constant retrial
rate. The system consists of a primary queue and an orbit queue. The primary
queue has identical servers and can accommodate the maximal number of
jobs. If a newly arriving job finds the full primary queue, it joins the orbit.
The original primary jobs arrive to the system according to a renewal process.
The jobs have general i.i.d. service times. A job in front of the orbit queue
retries to enter the primary queue after an exponentially distributed time
independent of the orbit queue length. Telephone exchange systems, Medium
Access Protocols and short TCP transfers are just some applications of the
proposed queueing system. For this system we establish minimal sufficient
stability conditions. Our model is very general. In addition, to the known
particular cases (e.g., M/G/1/1 or M/M/c/c systems), the proposed model covers
as particular cases the deterministic service model and the Erlang model with
constant retrial rate. The latter particular cases have not been considered in
the past. The obtained stability conditions have clear probabilistic
interpretation
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