813 research outputs found
A Fluid Limit for an Overloaded X Model Via a Stochastic Averaging Principle
We prove a many-server heavy-traffic fluid limit for an overloaded Markovian
queueing system having two customer classes and two service pools, known in the
call-center literature as the X model. The system uses the
fixed-queue-ratio-with-thresholds (FQR-T) control, which we proposed in a
recent paper as a way for one service system to help another in face of an
unexpected overload. Under FQR-T, customers are served by their own service
pool until a threshold is exceeded. Then, one-way sharing is activated with
customers from one class allowed to be served in both pools. After the control
is activated, it aims to keep the two queues at a pre-specified fixed ratio.
For large systems that fixed ratio is achieved approximately. For the fluid
limit, or FWLLN, we consider a sequence of properly scaled X models in overload
operating under FQR-T. Our proof of the FWLLN follows the compactness approach,
i.e., we show that the sequence of scaled processes is tight, and then show
that all converging subsequences have the specified limit. The characterization
step is complicated because the queue-difference processes, which determine the
customer-server assignments, remain stochastically bounded, and need to be
considered without spatial scaling. Asymptotically, these queue-difference
processes operate in a faster time scale than the fluid-scaled processes. In
the limit, due to a separation of time scales, the driving processes converge
to a time-dependent steady state (or local average) of a time-varying
fast-time-scale process (FTSP). This averaging principle (AP) allows us to
replace the driving processes with the long-run average behavior of the FTSP.Comment: There are 55 pages, 46 references and 0 figure
A Switching Fluid Limit of a Stochastic Network Under a State-Space-Collapse Inducing Control with Chattering
Routing mechanisms for stochastic networks are often designed to produce
state space collapse (SSC) in a heavy-traffic limit, i.e., to confine the
limiting process to a lower-dimensional subset of its full state space. In a
fluid limit, a control producing asymptotic SSC corresponds to an ideal sliding
mode control that forces the fluid trajectories to a lower-dimensional sliding
manifold. Within deterministic dynamical systems theory, it is well known that
sliding-mode controls can cause the system to chatter back and forth along the
sliding manifold due to delays in activation of the control. For the prelimit
stochastic system, chattering implies fluid-scaled fluctuations that are larger
than typical stochastic fluctuations. In this paper we show that chattering can
occur in the fluid limit of a controlled stochastic network when inappropriate
control parameters are used. The model has two large service pools operating
under the fixed-queue-ratio with activation and release thresholds (FQR-ART)
overload control which we proposed in a recent paper. We now show that, if the
control parameters are not chosen properly, then delays in activating and
releasing the control can cause chattering with large oscillations in the fluid
limit. In turn, these fluid-scaled fluctuations lead to severe congestion, even
when the arrival rates are smaller than the potential total service rate in the
system, a phenomenon referred to as congestion collapse. We show that the fluid
limit can be a bi-stable switching system possessing a unique nontrivial
periodic equilibrium, in addition to a unique stationary point
Analysis of Large Unreliable Stochastic Networks
In this paper a stochastic model of a large distributed system where users'
files are duplicated on unreliable data servers is investigated. Due to a
server breakdown, a copy of a file can be lost, it can be retrieved if another
copy of the same file is stored on other servers. In the case where no other
copy of a given file is present in the network, it is definitively lost. In
order to have multiple copies of a given file, it is assumed that each server
can devote a fraction of its processing capacity to duplicate files on other
servers to enhance the durability of the system.
A simplified stochastic model of this network is analyzed. It is assumed that
a copy of a given file is lost at some fixed rate and that the initial state is
optimal: each file has the maximum number of copies located on the servers
of the network. Due to random losses, the state of the network is transient and
all files will be eventually lost. As a consequence, a transient
-dimensional Markov process with a unique absorbing state describes
the evolution this network. By taking a scaling parameter related to the
number of nodes of the network. a scaling analysis of this process is
developed. The asymptotic behavior of is analyzed on time scales of
the type for . The paper derives asymptotic
results on the decay of the network: Under a stability assumption, the main
results state that the critical time scale for the decay of the system is given
by . When the stability condition is not satisfied, it is
shown that the state of the network converges to an interesting local
equilibrium which is investigated. As a consequence it sheds some light on the
role of the key parameters , the duplication rate and , the maximal
number of copies, in the design of these systems
Stochastic Systems ACHIEVING RAPID RECOVERY IN AN OVERLOAD CONTROL FOR LARGE-SCALE SERVICE SYSTEMS
We consider an automatic overload control for two large service systems modeled as multi-server queues, such as call centers. We assume that the two systems are designed to operate independently, but want to help each other respond to unexpected overloads. The proposed overload control automatically activates sharing (sending some customers from one system to the other) once a ratio of the queue lengths in the two systems crosses an activation threshold (with ratio and activation threshold parameters for each direction). To prevent harmful sharing, sharing is allowed in only one direction at any time. In this paper, we are primarily concerned with ensuring that the system recovers rapidly after the overload is over, either (i) because the two systems return to normal loading or (ii) because the direction of the overload suddenly shifts in the opposite direction. To achieve rapid recovery, we introduce lower thresholds for the queue ratios, below which one-way sharing is released. As a basis for studyin
Random Fluid Limit of an Overloaded Polling Model
In the present paper, we study the evolution of an overloaded cyclic polling
model that starts empty. Exploiting a connection with multitype branching
processes, we derive fluid asymptotics for the joint queue length process.
Under passage to the fluid dynamics, the server switches between the queues
infinitely many times in any finite time interval causing frequent oscillatory
behavior of the fluid limit in the neighborhood of zero. Moreover, the fluid
limit is random. Additionally, we suggest a method that establishes finiteness
of moments of the busy period in an M/G/1 queue.Comment: 36 pages, 2 picture
Moment Closure - A Brief Review
Moment closure methods appear in myriad scientific disciplines in the
modelling of complex systems. The goal is to achieve a closed form of a large,
usually even infinite, set of coupled differential (or difference) equations.
Each equation describes the evolution of one "moment", a suitable
coarse-grained quantity computable from the full state space. If the system is
too large for analytical and/or numerical methods, then one aims to reduce it
by finding a moment closure relation expressing "higher-order moments" in terms
of "lower-order moments". In this brief review, we focus on highlighting how
moment closure methods occur in different contexts. We also conjecture via a
geometric explanation why it has been difficult to rigorously justify many
moment closure approximations although they work very well in practice.Comment: short survey paper (max 20 pages) for a broad audience in
mathematics, physics, chemistry and quantitative biolog
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