1,426 research outputs found
Entropy maximization and the busy period of some single-server vacation models
In this paper, information theoretic methodology for system modeling is applied to investigate the probability density function of the busy period in M/G/1 vacation models operating under the N-, T- and D-policies. The information about the density function is limited to a few mean value constraints (usually the first moments). By using the maximum entropy methodology one obtains the least biased probability density function satisfying the system's constraints. The analysis of the three controllable M/G/1 queueing models provides a parallel numerical study of the solution obtained via the maximum entropy approach versus “classical” solutions. The maximum entropy analysis of a continuous system descriptor (like the busy period) enriches the current body of literature which, in most cases, reduces to discrete queueing measures (such as the number of customers in the system)
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General queueing network models for computer system performance analysis. A maximum entropy method of analysis and aggregation of general queueing network models with application to computer systems.
In this study the maximum entropy formalism [JAYN 57] is suggested
as an alternative theory for general queueing systems of computer
performance analysis. The motivation is to overcome some of the
problems arising in this field and to extend the scope of the results
derived in the context of Markovian queueing theory.
For the M/G/l model a unique maximum entropy solution., satisfying
locALl balance is derived independent of any assumptions about the service
time distribution. However, it is shown that this solution is identical
to the steady state solution of the underlying Marko-v process when the
service time distribution is of the generalised exponential (CE) type.
(The GE-type distribution is a mixture of an exponential term and a unit
impulse function at the origin). For the G/M/1 the maximum entropy
solution is identical in form to that of the underlying Markov process,
but a GE-type distribution still produces the maximum overall similar
distributions.
For the GIG11 model there are three main achievements:
first, the spectral methods are extended to give exaft formulae for
the average number of customers in the system for any G/G/l with rational
Laplace transform. Previously, these results are obtainable only through
simulation and approximation methods.
(ii) secondly, a maximum entropy model is developed and used to obtain
unique solutions for some types of the G/G/l. It is also discussed how
these solutions can be related to the corresponding stochastic processes.
(iii) the importance of the G/GE/l and the GE/GE/l for the analysis of
general networks is discussed and some flow processes for these systems
are characterised.
For general queueing networks it is shown that the maximum entropy
solution is a product of the maximum entropy solutions of the individual
nodes. Accordingly, existing computational algorithms are extended to
cover general networks with FCFS disciplines. Some implementations are
suggested and a flow algorithm is derived. Finally, these results are
iised to improve existing aggregation methods.
In addition, the study includes a number of examples, comparisons,
surveys, useful comments and conclusions
Comparative Analysis of a Randomized N-policy Queue: An Improved Maximum Entropy Method
[[abstract]]We analyze a single removable and unreliable server in an M/G/1 queueing system operating under the 〈p, N〉-policy. As soon as the system size is greater than N, turn the server on with probability p and leave the server off with probability (1 − p). All arriving customers demand the first essential service, where only some of them demand the second optional service. He needs a startup time before providing first essential service until there are no customers in the system. The server is subject to break down according to a Poisson process and his repair time obeys a general distribution. In this queueing system, the steady-state probabilities cannot be derived explicitly. Thus, we employ an improved maximum entropy method with several well-known constraints to estimate the probability distributions of system size and the expected waiting time in the system. By a comparative analysis between the exact and approximate results, we may demonstrate that the improved maximum entropy method is accurate enough for practical purpose, and it is a useful method for solving complex queueing systems
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Entropy Maximisation and Queues With or Without Balking. An investigation into the impact of generalised maximum entropy solutions on the study of queues with or without arrival balking and their applications to congestion management in communication networks.
An investigation into the impact of generalised maximum entropy solutions on the study of queues with or without arrival balking and their applications to congestion management in communication networks
Keywords: Queues, Balking, Maximum Entropy (ME) Principle, Global Balance (GB), Queue Length Distribution (QLD), Generalised Geometric (GGeo), Generalised Exponential (GE), Generalised Discrete Half Normal (GdHN), Congestion Management, Packet Dropping Policy (PDP)
Generalisations to links between discrete least biased (i.e. maximum entropy (ME)) distribution inferences and Markov chains are conjectured towards the performance modelling, analysis and prediction of general, single server queues with or without arrival balking. New ME solutions, namely the generalised discrete Half Normal (GdHN) and truncated GdHN (GdHNT) distributions are characterised, subject to appropriate mean value constraints, for inferences of stationary discrete state probability distributions. Moreover, a closed form global balance (GB) solution is derived for the queue length distribution (QLD) of the M/GE/1/K queue subject to extended Morse balking, characterised by a Poisson prospective arrival process, i.i.d. generalised exponential (GE) service times and finite capacity, K. In this context, based on comprehensive numerical experimentation, the latter GB solution is conjectured to be a special case of the GdHNT ME distribution.
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Owing to the appropriate operational properties of the M/GE/1/K queue subject to extended Morse balking, this queueing system is applied as an ME performance model of Internet Protocol (IP)-based communication network nodes featuring static or dynamic packet dropping congestion management schemes. A performance evaluation study in terms of the model’s delay is carried out. Subsequently, the QLD’s of the GE/GE/1/K censored queue subject to extended Morse balking under three different composite batch balking and batch blocking policies are solved via the technique of GB. Following comprehensive numerical experimentation, the latter QLD’s are also conjectured to be special cases of the GdHNT. Limitations of this work and open problems which have arisen are included after the conclusion
Store-Forward and its implications for Proportional Scheduling
The Proportional Scheduler was recently proposed as a scheduling algorithm
for multi-hop switch networks. For these networks, the BackPressure scheduler
is the classical benchmark. For networks with fixed routing, the Proportional
Scheduler is maximum stable, myopic and, furthermore, will alleviate certain
scaling issued found in BackPressure for large networks. Nonetheless, the
equilibrium and delay properties of the Proportional Scheduler has not been
fully characterized.
In this article, we postulate on the equilibrium behaviour of the
Proportional Scheduler though the analysis of an analogous rule called the
Store-Forward allocation. It has been shown that Store-Forward has
asymptotically allocates according to the Proportional Scheduler. Further, for
Store-Forward networks, numerous equilibrium quantities are explicitly
calculable. For FIFO networks under Store-Forward, we calculate the policies
stationary distribution and end-to-end route delay. We discuss network
topologies when the stationary distribution is product-form, a phenomenon which
we call \emph{product form resource pooling}. We extend this product form
notion to independent set scheduling on perfect graphs, where we show that
non-neighbouring queues are statistically independent. Finally, we analyse the
large deviations behaviour of the equilibrium distribution of Store-Forward
networks in order to construct Lyapunov functions for FIFO switch networks
Nonlinear Markov Processes in Big Networks
Big networks express various large-scale networks in many practical areas
such as computer networks, internet of things, cloud computation, manufacturing
systems, transportation networks, and healthcare systems. This paper analyzes
such big networks, and applies the mean-field theory and the nonlinear Markov
processes to set up a broad class of nonlinear continuous-time block-structured
Markov processes, which can be applied to deal with many practical stochastic
systems. Firstly, a nonlinear Markov process is derived from a large number of
interacting big networks with symmetric interactions, each of which is
described as a continuous-time block-structured Markov process. Secondly, some
effective algorithms are given for computing the fixed points of the nonlinear
Markov process by means of the UL-type RG-factorization. Finally, the Birkhoff
center, the Lyapunov functions and the relative entropy are used to analyze
stability or metastability of the big network, and several interesting open
problems are proposed with detailed interpretation. We believe that the results
given in this paper can be useful and effective in the study of big networks.Comment: 28 pages in Special Matrices; 201
Effective temperature for finite systems
Under the Ansatz that the occupation times of a system with finitely many
states are given by the Gibbs distribution, an effective temperature is
uniquely determined (up to a choice of scale), and may be computed de novo,
without any reference to a Hamiltonian for empirically accessible systems. As
an example, the calculation of the effective temperature for a classical Bose
gas is outlined and applied to the analysis of computer network traffic.Comment: 9 pages, 10 figure
Closed queueing networks under congestion: non-bottleneck independence and bottleneck convergence
We analyze the behavior of closed product-form queueing networks when the
number of customers grows to infinity and remains proportionate on each route
(or class). First, we focus on the stationary behavior and prove the conjecture
that the stationary distribution at non-bottleneck queues converges weakly to
the stationary distribution of an ergodic, open product-form queueing network.
This open network is obtained by replacing bottleneck queues with per-route
Poissonian sources whose rates are determined by the solution of a strictly
concave optimization problem. Then, we focus on the transient behavior of the
network and use fluid limits to prove that the amount of fluid, or customers,
on each route eventually concentrates on the bottleneck queues only, and that
the long-term proportions of fluid in each route and in each queue solve the
dual of the concave optimization problem that determines the throughputs of the
previous open network.Comment: 22 page
A Statistical Mechanical Load Balancer for the Web
The maximum entropy principle from statistical mechanics states that a closed
system attains an equilibrium distribution that maximizes its entropy. We first
show that for graphs with fixed number of edges one can define a stochastic
edge dynamic that can serve as an effective thermalization scheme, and hence,
the underlying graphs are expected to attain their maximum-entropy states,
which turn out to be Erdos-Renyi (ER) random graphs. We next show that (i) a
rate-equation based analysis of node degree distribution does indeed confirm
the maximum-entropy principle, and (ii) the edge dynamic can be effectively
implemented using short random walks on the underlying graphs, leading to a
local algorithm for the generation of ER random graphs. The resulting
statistical mechanical system can be adapted to provide a distributed and local
(i.e., without any centralized monitoring) mechanism for load balancing, which
can have a significant impact in increasing the efficiency and utilization of
both the Internet (e.g., efficient web mirroring), and large-scale computing
infrastructure (e.g., cluster and grid computing).Comment: 11 Pages, 5 Postscript figures; added references, expanded on
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