scholarly journals Internet-Scale Storage Systems under Churn -- A Study of the Steady-State using Markov Models

2006 ◽  
Author(s):  
A. Datta ◽  
K. Aberer
Keyword(s):  
Steady State ◽  
Markov Models ◽  
Storage Systems

scholarly journals A Duality Approach to Queues with Service Restrictions and Storage Systems with State-Dependent Rates

2013 ◽  
Vol 50 (3) ◽  
pp. 612-631 ◽  
Author(s):  
D. Perry ◽  
W. Stadje ◽  
S. Zacks
Keyword(s):  
Steady State ◽  
Storage Systems ◽  
Storage System ◽  
Queueing Systems ◽  
Local Maxima ◽  
State Dependent ◽  
State Densities ◽  
Duality Approach ◽  
And Storage ◽  
Level Process

Based on pathwise duality constructions, several new results on truncated queues and storage systems of the G/M/1 type are derived by transforming the workload (content) processes into certain ‘dual’ M/G/1-type processes. We consider queueing systems in which (a) any service requirement that would increase the total workload beyond the capacity is truncated so as to keep the associated sojourn time below a certain constant, or (b) new arrivals do not enter the system if they have to wait more than one time unit in line. For these systems, we derive the steady-state distributions of the workload and the numbers of customers present in the systems as well as the distributions of the lengths of busy and idle periods. Moreover, we use the duality approach to study finite capacity storage systems with general state-dependent outflow rates. Here our duality leads to a Markovian finite storage system with state-dependent jump sizes whose content level process can be analyzed using level crossing techniques. We also derive a connection between the steady-state densities of the non-Markovian continuous-time content level process of the G/M/1 finite storage system with state-dependent outflow rule and the corresponding embedded sequence of peak points (local maxima).

Equivalence of aggregated Markov models of ion-channel gating

1989 ◽  
Vol 236 (1284) ◽  
pp. 269-309 ◽  
Keyword(s):  
Steady State ◽  
Ion Channel ◽  
Markov Models ◽  
Similarity Transformation ◽  
Variable Parameter ◽  
Transition Matrices ◽  
Steady State Kinetic ◽  
Ion Channel Kinetics ◽  
Generator Matrices ◽  
State Kinetic

One cannot always distinguish different Markov models of ion-channel kinetics solely on the basis of steady-state kinetic data. If two generator (or transition) matrices are related by a similarity transformation that does not combine states with different conductances, then the models described by these generator matrices have the same observable steady-state statistics. This result suggests a procedure for expressing the model in a unique form, and sometimes reducing the number of parameters in a model. I apply the similarity transformation procedure to a number of simple models. When a model specifies the dependence of the rates of transition on an experimentally variable parameter such as the concentration of a ligand or the membrane potential, the class of equivalent models may be further restricted, but a model is not always uniquely determined even under these conditions. Voltage-step experiments produce non-stationary data that can also be used to distinguish models.

Loss tandem networks with blocking - a semi-Markov approach

2010 ◽  
Vol 58 (4) ◽  
pp. 673-681
Author(s):  
W. Oniszczuk
Keyword(s):  
Steady State ◽  
Markov Process ◽  
Markov Model ◽  
Markov Models ◽  
State Graph ◽  
State Equations ◽  
Semi Markov Process ◽  
Tandem Networks ◽  
Measures Of Effectiveness

Loss tandem networks with blocking - a semi-Markov approachBased on the semi-Markov process theory, this paper describes an analytical study of a loss multiple-server two-station network model with blocking. Tasks arrive to the tandem in a Poisson fashion at a rate λ, and the service times at the first and second stations are non-exponentially distributed with means sAand sB, respectively. Between these two stations there is a buffer with finite capacity. In this type of network, if the buffer is full, the accumulation of new tasks (jobs) by the second station is temporarily suspended (blocking factor) and tasks must wait on the first station until the transmission process is resumed. Any new task that finds all service lines at the first station occupied is turned away and is lost (loss factor). Initially, in this document, a Markov model of the loss tandem with blocking is investigated. Here, a two-dimensional state graph is constructed and a set of steady-state equations is created. These equations allow the calculation of state probabilities for each graph state. A special algorithm for transforming the Markov model into a semi-Markov process is presented. This approach allows calculating steady-state probabilities in the semi-Markov model. In the next part of the paper, the algorithms for calculation of the main measures of effectiveness in the semi-Markov model are presented. Finally, the numerical part of this paper contains an investigation of some special semi-Markov models, where the results are presented of the calculation of the quality of service (QoS) parameters and the main measures of effectiveness.

The solution of certain two-dimensional Markov models

1982 ◽  
Vol 14 (02) ◽  
pp. 295-308 ◽  
Author(s):  
G. Fayolle ◽  
P. J. B. King ◽  
I. Mitrani
Keyword(s):  
Steady State ◽  
Solution Method ◽  
Markov Models ◽  
Transition Rates ◽  
Two Dimensional ◽  
Steady State Distribution ◽  
State Distribution ◽  
Birth And Death Processes ◽  
State Dependent ◽  
Modelling Problems

A class of two-dimensional birth-and-death processes, with applications in many modelling problems, is defined and analysed in the steady state. These are processes whose instantaneous transition rates are state-dependent in a restricted way. Generating functions for the steady-state distribution are obtained by solving a functional equation in two variables. That solution method lends itself readily to numerical implementation. Some aspects of the numerical solution are discussed, using a particular model as an example.

The solution of certain two-dimensional Markov models

1982 ◽  
Vol 14 (2) ◽  
pp. 295-308 ◽  
Author(s):  
G. Fayolle ◽  
P. J. B. King ◽  
I. Mitrani
Keyword(s):  
Steady State ◽  
Solution Method ◽  
Markov Models ◽  
Transition Rates ◽  
Two Dimensional ◽  
Steady State Distribution ◽  
State Distribution ◽  
Birth And Death Processes ◽  
State Dependent ◽  
Modelling Problems

A class of two-dimensional birth-and-death processes, with applications in many modelling problems, is defined and analysed in the steady state. These are processes whose instantaneous transition rates are state-dependent in a restricted way. Generating functions for the steady-state distribution are obtained by solving a functional equation in two variables. That solution method lends itself readily to numerical implementation. Some aspects of the numerical solution are discussed, using a particular model as an example.

scholarly journals Tail Asymptotics for a Random Sign Lindley Recursion

2010 ◽  
Vol 47 (1) ◽  
pp. 72-83 ◽  
Author(s):  
Maria Vlasiou ◽  
Zbigniew Palmowski
Keyword(s):  
Steady State ◽  
Storage Systems ◽  
Queueing Systems ◽  
Service Systems ◽  
Tail Asymptotics ◽  
Steady State Distribution ◽  
State Distribution ◽  
Random Sign ◽  
Special Cases ◽  
Tail Behaviour

We investigate the tail behaviour of the steady-state distribution of a stochastic recursion that generalises Lindley's recursion. This recursion arises in queueing systems with dependent interarrival and service times, and includes alternating service systems and carousel storage systems as special cases. We obtain precise tail asymptotics in three qualitatively different cases, and compare these with existing results for Lindley's recursion and for alternating service systems.

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