scholarly journals Decay property of stopped Markovian bulk-arriving queues

2008 ◽  
Vol 40 (1) ◽  
pp. 95-121 ◽  
Author(s):  
Junping Li ◽  
Anyue Chen
Keyword(s):  
Generating Functions ◽  
Busy Period ◽  
Invariant Measures ◽  
Geometric Interpretation ◽  
Decay Property ◽  
Decay Parameter ◽  
Decay Properties ◽  
Invariant Vectors ◽  
Quasistationary Distributions

We consider decay properties including the decay parameter, invariant measures, invariant vectors, and quasistationary distributions of a Markovian bulk-arriving queue that stops immediately after hitting the zero state. Investigating such behavior is crucial in realizing the busy period and some other related properties of Markovian bulk-arriving queues. The exact value of the decay parameter λC is obtained and expressed explicitly. The invariant measures, invariant vectors, and quasistationary distributions are then presented. We show that there exists a family of invariant measures indexed by λ ∈ [0, λC]. We then show that, under some conditions, there exists a family of quasistationary distributions, also indexed by λ ∈ [0, λC]. The generating functions of these invariant measures and quasistationary distributions are presented. We further show that a stopped Markovian bulk-arriving queue is always λC-transient and some deep properties are revealed. The clear geometric interpretation of the decay parameter is explained. A few examples are then provided to illustrate the results obtained in this paper.

scholarly journals Decay property of stopped Markovian bulk-arriving queues

2008 ◽  
Vol 40 (01) ◽  
pp. 95-121 ◽  
Author(s):  
Junping Li ◽  
Anyue Chen
Keyword(s):  
Generating Functions ◽  
Busy Period ◽  
Invariant Measures ◽  
Geometric Interpretation ◽  
Decay Property ◽  
Decay Parameter ◽  
Decay Properties ◽  
Invariant Vectors ◽  
Quasistationary Distributions

We consider decay properties including the decay parameter, invariant measures, invariant vectors, and quasistationary distributions of a Markovian bulk-arriving queue that stops immediately after hitting the zero state. Investigating such behavior is crucial in realizing the busy period and some other related properties of Markovian bulk-arriving queues. The exact value of the decay parameter λCis obtained and expressed explicitly. The invariant measures, invariant vectors, and quasistationary distributions are then presented. We show that there exists a family of invariant measures indexed by λ ∈ [0, λC]. We then show that, under some conditions, there exists a family of quasistationary distributions, also indexed by λ ∈ [0, λC]. The generating functions of these invariant measures and quasistationary distributions are presented. We further show that a stopped Markovian bulk-arriving queue is always λC-transient and some deep properties are revealed. The clear geometric interpretation of the decay parameter is explained. A few examples are then provided to illustrate the results obtained in this paper.

Uniqueness and Decay Properties of Markov Branching Processes with Disasters

2014 ◽  
Vol 51 (03) ◽  
pp. 613-624 ◽  
Author(s):  
Anyue Chen ◽  
Kai Wang Ng ◽  
Hanjun Zhang
Keyword(s):  
Branching Process ◽  
Invariant Measures ◽  
Branching Processes ◽  
Decay Parameter ◽  
Markov Branching Process ◽  
Decay Properties ◽  
Invariant Vectors ◽  
Quasistationary Distributions

In this paper we discuss the decay properties of Markov branching processes with disasters, including the decay parameter, invariant measures, and quasistationary distributions. After showing that the corresponding q-matrix Q is always regular and, thus, that the Feller minimal Q-process is honest, we obtain the exact value of the decay parameter λ C . We show that the decay parameter can be easily expressed explicitly. We further show that the Markov branching process with disaster is always λ C -positive. The invariant vectors, the invariant measures, and the quasidistributions are given explicitly.

scholarly journals Uniqueness and Decay Properties of Markov Branching Processes with Disasters

2014 ◽  
Vol 51 (3) ◽  
pp. 613-624 ◽  
Author(s):  
Anyue Chen ◽  
Kai Wang Ng ◽  
Hanjun Zhang
Keyword(s):  
Branching Process ◽  
Invariant Measures ◽  
Branching Processes ◽  
Decay Parameter ◽  
Markov Branching Process ◽  
Decay Properties ◽  
Invariant Vectors ◽  
Quasistationary Distributions

In this paper we discuss the decay properties of Markov branching processes with disasters, including the decay parameter, invariant measures, and quasistationary distributions. After showing that the corresponding q-matrix Q is always regular and, thus, that the Feller minimal Q-process is honest, we obtain the exact value of the decay parameter λC. We show that the decay parameter can be easily expressed explicitly. We further show that the Markov branching process with disaster is always λC-positive. The invariant vectors, the invariant measures, and the quasidistributions are given explicitly.

scholarly journals Analysis of a Single-Sever Queue with Disasters and Repairs Under Bernoulli Vacation Schedule

2016 ◽  
Vol 4 (6) ◽  
pp. 547-559
Author(s):  
Jingjing Ye ◽  
Liwei Liu ◽  
Tao Jiang
Keyword(s):  
Performance Measures ◽  
Generating Functions ◽  
Sojourn Time ◽  
Time Distribution ◽  
Busy Period ◽  
Balance Equations ◽  
Sojourn Time Distribution ◽  
The Mean ◽  
System Length ◽  
Number Of Customers

AbstractThis paper studies a single-sever queue with disasters and repairs, in which after each service completion the server may take a vacation with probabilityq(0≤q≤1), or begin to serve the next customer, if any, with probabilityp(= 1− q). The disaster only affects the system when the server is in operation, and once it occurs, all customers present are eliminated from the system. We obtain the stationary probability generating functions (PGFs) of the number of customers in the system by solving the balance equations of the system. Some performance measures such as the mean system length, the probability that the server is in different states, the rate at which disasters occur and the rate of initiations of busy period are determined. We also derive the sojourn time distribution and the mean sojourn time. In addition, some numerical examples are presented to show the effect of the parameters on the mean system length.

INFINITE-SERVER QUEUES WITH SYSTEM'S ADDITIONAL TASKS AND IMPATIENT CUSTOMERS

2008 ◽  
Vol 22 (4) ◽  
pp. 477-493 ◽  
Author(s):  
Eitan Altman ◽  
Uri Yechiali
Keyword(s):  
Quality Of Service ◽  
Generating Functions ◽  
Busy Period ◽  
Impatient Customers ◽  
Partial Probability ◽  
The Mean ◽  
Infinite Server Queues ◽  
Mean Cycle ◽  
Number Of Customers

A system is operating as an M/M/∞ queue. However, when it becomes empty, it is assigned to perform another task, the duration U of which is random. Customers arriving while the system is unavailable for service (i.e., occupied with a U-task) become impatient: Each individual activates an “impatience timer” having random duration T such that if the system does not become available by the time the timer expires, the customer leaves the system never to return. When the system completes a U-task and there are waiting customers, each one is taken immediately into service. We analyze both multiple and single U-task scenarios and consider both exponentially and generally distributed task and impatience times. We derive the (partial) probability generating functions of the number of customers present when the system is occupied with a U-task as well as when it acts as an M/M/∞ queue and we obtain explicit expressions for the corresponding mean queue sizes. We further calculate the mean length of a busy period, the mean cycle time, and the quality of service measure: proportion of customers being served.

scholarly journals On Estimation of the Convergence Rate to Invariant Measures in Markov Branching Processes with Possibly Infinite Variance and Allowing Immigration

2021 ◽  
Vol 14 (5) ◽  
pp. 573-583
Author(s):  
Azam A. Imomov ◽  
Keyword(s):  
Generating Functions ◽  
Continuous Time ◽  
Branching Process ◽  
Critical Case ◽  
Invariant Measures ◽  
Branching Processes ◽  
Immigration Law ◽  
Infinite Variance ◽  
Transition Functions ◽  
First Moment

The paper discusses the continuous-time Markov Branching Process allowing Immigration. We are considering a critical case for which the second moment of offspring law and the first moment of immigration law are possibly infinite. Assuming that the nonlinear parts of the appropriate generating functions are regularly varying in the sense of Karamata, we prove theorems on convergence of transition functions of the process to invariant measures. We deduce the speed rate of these convergence providing that slowly varying factors are with remainder

A Discrete-Time Queueing Model with Service Upgrade

2014 ◽  
Vol 513-517 ◽  
pp. 806-811
Author(s):  
Ivan Atencia ◽  
Inmaculada Fortes ◽  
Sixto Sánchez
Keyword(s):  
Discrete Time ◽  
Generating Functions ◽  
Busy Period ◽  
Queueing System ◽  
Queueing Model ◽  
Numerical Examples ◽  
Extensive Analysis ◽  
Number Of Customers ◽  
Incoming Message

In this paper we analyze a discrete-time queueing system where the server decides whento upgrade the service depending on the information carried by the incoming message. We carry outan extensive analysis of the system developing recursive formulae and generating functions for thestationary distribution of the number of customers in the queue, the system, the busy period and thesojourntimeas well as some numerical examples.

Family of measures on a space of curves that are quasi-invariant with respect to some action of diffeomorphisms group

2016 ◽  
Vol 19 (03) ◽  
pp. 1650019
Author(s):  
Evgenii Dmitrievich Romanov
Keyword(s):  
Invariant Measures ◽  
Functional Space ◽  
Geometric Interpretation ◽  
Unitary Representations ◽  
Multidimensional Case ◽  
Dimensional Euclidean Space ◽  
Future Research ◽  
Finite Dimensional ◽  
Irreducible Unitary Representations ◽  
Nikodym Derivative

A family of quasi-invariant measures on the special functional space of curves in a finite-dimensional Euclidean space with respect to the action of diffeomorphisms is constructed. The main result is an explicit expression for the Radon–Nikodym derivative of the transformed measure relative to the original one. The stochastic Ito integral allows to express the result in an invariant form for a wider class of diffeomorphisms. These measures can be used to obtain irreducible unitary representations of the diffeomorphisms group which will be studied in future research. A geometric interpretation of the action considered together with a generalization to the multidimensional case makes such representations applicable to problems of quantum mechanics.

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