scholarly journals On a multilevel controlled bulk queueing system MX/Gr,R/1

1992 ◽  
Vol 5 (3) ◽  
pp. 237-260 ◽  
Author(s):  
Lev Abolnikov ◽  
Jewgeni H. Dshalalow
Keyword(s):  
Stationary Distribution ◽  
Queue Length ◽  
Transition Probability ◽  
Queueing Systems ◽  
Transition Probability Matrix ◽  
Queueing Process ◽  
Poisson Input ◽  
Service Delay ◽  
Associated Functions ◽  
Semi Markov Process

The authors introduce and study a class of bulk queueing systems with a compound Poisson input modulated by a semi-Markov process, multilevel control service time and a queue length dependent service delay discipline. According to this discipline, the server immediately starts the next service act if the queue length is not less than r; in this case all available units, or R (capacity of the server) of them, whichever is less, are taken for service. Otherwise, the server delays the service act until the number of units in the queue reaches or exceeds level r.The authors establish a necessary and sufficient criterion for the ergodicity of the embedded queueing process in terms of generating functions of the entries of the corresponding transition probability matrix and of the roots of a certain associated functions in the unit disc of the complex plane. The stationary distribution of this process is found by means of the results of a preliminary analysis of some auxiliary random processes which arise in the “first passage problem” of the queueing process over level r. The stationary distribution of the queueing process with continuous time parameter is obtained by using semi-regenerative techniques. The results enable the authors to introduce and analyze some functionals of the input and output processes via ergodic theorems. A number of different examples (including an optimization problem) illustrate the general methods developed in the article.

scholarly journals A bulk queueing system under N-policy with bilevel service delay discipline and start-up time

1993 ◽  
Vol 6 (4) ◽  
pp. 359-384 ◽  
Author(s):  
David C. R. Muh
Keyword(s):  
Queue Length ◽  
Queueing System ◽  
Probability Generating Function ◽  
Single Server ◽  
Numerical Examples ◽  
Queueing Process ◽  
Poisson Input ◽  
Service Delay ◽  
Start Up ◽  
Bulk Arrival

The author studies the queueing process in a single-server, bulk arrival and batch service queueing system with a compound Poisson input, bilevel service delay discipline, start-up time, and a fixed accumulation level with control operating policy. It is assumed that when the queue length falls below a predefined level r(≥1), the system, with server capacity R, immediately stops service until the queue length reaches or exceeds the second predefined accumulation level N(≥r). Two cases, with N≤R and N≥R, are studied.The author finds explicitly the probability generating function of the stationary distribution of the queueing process and gives numerical examples.

On Construction of Sparse Probabilistic Boolean Networks

2012 ◽  
Vol 2 (1) ◽  
pp. 1-18 ◽  
Author(s):  
Xi Chen ◽  
Hao Jiang ◽  
Wai-Ki Ching
Keyword(s):  
Objective Function ◽  
Stationary Distribution ◽  
Transition Probability ◽  
Additional Term ◽  
Control Policy ◽  
Transition Probability Matrix ◽  
Boolean Networks ◽  
Entropy Rate ◽  
Sparse Solution ◽  
Probabilistic Boolean Networks

AbstractIn this paper we envisage building Probabilistic Boolean Networks (PBNs) from a prescribed stationary distribution. This is an inverse problem of huge size that can be subdivided into two parts — viz. (i) construction of a transition probability matrix from a given stationary distribution (Problem ST), and (ii) construction of a PBN from a given transition probability matrix (Problem TP). A generalized entropy approach has been proposed for Problem ST and a maximum entropy rate approach for Problem TP respectively. Here we propose to improve both methods, by considering a new objective function based on the entropy rate with an additional term of La-norm that can help in getting a sparse solution. A sparse solution is useful in identifying the major component Boolean networks (BNs) from the constructed PBN. These major BNs can simplify the identification of the network structure and the design of control policy, and neglecting non-major BNs does not change the dynamics of the constructed PBN to a large extent. Numerical experiments indicate that our new objective function is effective in finding a better sparse solution.

scholarly journals Markov Chains with Hybrid Repeating Rows - Upper-Hessenberg, Quasi-Toeplitz Structure of the Block Transition Probability Matrix

2008 ◽  
Vol 45 (01) ◽  
pp. 211-225 ◽  
Author(s):  
Alexander Dudin ◽  
Chesoong Kim ◽  
Valentina Klimenok
Keyword(s):  
Markov Chains ◽  
Stationary Distribution ◽  
Discrete Time ◽  
Transition Probability ◽  
Sufficient Conditions ◽  
Transition Probability Matrix ◽  
Toeplitz Structure

In this paper we consider discrete-time multidimensional Markov chains having a block transition probability matrix which is the sum of a matrix with repeating block rows and a matrix of upper-Hessenberg, quasi-Toeplitz structure. We derive sufficient conditions for the existence of the stationary distribution, and outline two algorithms for calculating the stationary distribution.

scholarly journals Markov Chains with Hybrid Repeating Rows - Upper-Hessenberg, Quasi-Toeplitz Structure of the Block Transition Probability Matrix

2008 ◽  
Vol 45 (1) ◽  
pp. 211-225 ◽  
Author(s):  
Alexander Dudin ◽  
Chesoong Kim ◽  
Valentina Klimenok
Keyword(s):  
Markov Chains ◽  
Stationary Distribution ◽  
Discrete Time ◽  
Transition Probability ◽  
Sufficient Conditions ◽  
Transition Probability Matrix ◽  
Toeplitz Structure

In this paper we consider discrete-time multidimensional Markov chains having a block transition probability matrix which is the sum of a matrix with repeating block rows and a matrix of upper-Hessenberg, quasi-Toeplitz structure. We derive sufficient conditions for the existence of the stationary distribution, and outline two algorithms for calculating the stationary distribution.

On joint queue-length characteristics in infinite-server tandem queues with heavy traffic

1987 ◽  
Vol 19 (02) ◽  
pp. 474-486 ◽  
Author(s):  
Volker Schmidt
Keyword(s):  
Stationary Distribution ◽  
Simple Proof ◽  
Shot Noise ◽  
Queue Length ◽  
Heavy Traffic ◽  
Asymptotic Formulas ◽  
Tandem Queues ◽  
Infinite Server ◽  
Poisson Input ◽  
Infinite Server Queues

For m infinite-server queues with Poisson input which are connected in a series, a simple proof is given of a formula derived in [3] for the generating function of the joint customer-stationary distribution of the successive numbers of customers a randomly chosen customer finds at his arrival epochs at two queues of the system. In this connection, a shot-noise representation of the queue-length characteristics under consideration is used. Moreover, using this representation, corresonding asymptotic formulas are derived for infinite-server tandem queues with general high-density renewal input.

On joint queue-length characteristics in infinite-server tandem queues with heavy traffic

1987 ◽  
Vol 19 (2) ◽  
pp. 474-486 ◽  
Author(s):  
Volker Schmidt
Keyword(s):  
Stationary Distribution ◽  
Simple Proof ◽  
Shot Noise ◽  
Queue Length ◽  
Heavy Traffic ◽  
Asymptotic Formulas ◽  
Tandem Queues ◽  
Infinite Server ◽  
Poisson Input ◽  
Infinite Server Queues

For m infinite-server queues with Poisson input which are connected in a series, a simple proof is given of a formula derived in [3] for the generating function of the joint customer-stationary distribution of the successive numbers of customers a randomly chosen customer finds at his arrival epochs at two queues of the system. In this connection, a shot-noise representation of the queue-length characteristics under consideration is used. Moreover, using this representation, corresonding asymptotic formulas are derived for infinite-server tandem queues with general high-density renewal input.

On Generating Optimal Sparse Probabilistic Boolean Networks with Maximum Entropy from a Positive Stationary Distribution

2012 ◽  
Vol 2 (4) ◽  
pp. 353-372 ◽  
Author(s):  
Hao Jiang ◽  
Xi Chen ◽  
Yushan Qiu ◽  
Wai-Ki Ching
Keyword(s):  
Maximum Entropy ◽  
Stationary Distribution ◽  
Sparse Matrix ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Boolean Networks ◽  
Efficiency And Effectiveness ◽  
Probabilistic Boolean Networks ◽  
Transition Probability Matrices ◽  
Probability Matrices

Abstract.To understand a genetic regulatory network, two popular mathematical models, Boolean Networks (BNs) and its extension Probabilistic Boolean Networks (PBNs) have been proposed. Here we address the problem of constructing a sparse Probabilistic Boolean Network (PBN) from a prescribed positive stationary distribution. A sparse matrix is more preferable, as it is easier to study and identify the major components and extract the crucial information hidden in a biological network. The captured network construction problem is both ill-posed and computationally challenging. We present a novel method to construct a sparse transition probability matrix from a given stationary distribution. A series of sparse transition probability matrices can be determined once the stationary distribution is given. By controlling the number of nonzero entries in each column of the transition probability matrix, a desirable sparse transition probability matrix in the sense of maximum entropy can be uniquely constructed as a linear combination of the selected sparse transition probability matrices (a set of sparse irreducible matrices). Numerical examples are given to demonstrate both the efficiency and effectiveness of the proposed method.

scholarly journals Alternative approach based on roots for computing the stationary queue-length distributions in GIX/M(1, b)/1 single working vacation queue

2020 ◽  
Author(s):  
Miaomiao Yu
Keyword(s):  
Queue Length ◽  
Transition Probability ◽  
Length Distribution ◽  
Transition Probability Matrix ◽  
Batch Arrival ◽  
Embedded Markov Chain ◽  
Working Vacation ◽  
Queue Length Distribution ◽  
Stationary Queue Length ◽  
Number Of Customers

The purpose of this paper is to present an alternative algorithm for computing the stationary queue-length and system-length distributions of a single working vacation queue with renewal input batch arrival and exponential holding times. Here we assume that a group of customers arrives into the system, and they are served in batches not exceeding a specific number b. Because of batch arrival, the transition probability matrix of the corresponding embedded Markov chain for the working vacation queue has no skip-free-to-the-right property. Without considering whether the transition probability matrix has a special block structure, through the calculation of roots of the associated characteristic equation of the generating function of queue-length distribution immediately before batch arrival, we suggest a procedure to obtain the steady-state distributions of the number of customers in the queue at different epochs. Furthermore, we present the analytic results for the sojourn time of an arbitrary customer in a batch by utilizing the queue-length distribution at the pre-arrival epoch. Finally, various examples are provided to show the applicability of the numerical algorithm.

On calculating extinction probabilities for branching processes in random environments

1969 ◽  
Vol 6 (03) ◽  
pp. 478-492 ◽  
Author(s):  
William E. Wilkinson
Keyword(s):  
Generating Functions ◽  
Infinite Sequence ◽  
Branching Processes ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Random Environments ◽  
Real Numbers ◽  
Discrete Time Markov Chain ◽  
Extinction Probabilities ◽  
Image Position

Consider a discrete time Markov chain {Zn } whose state space is the non-negative integers and whose transition probability matrix ║Pij ║ possesses the representation where {Pr }, r = 1,2,…, is a finite or denumerably infinite sequence of non-negative real numbers satisfying , and , is a corresponding sequence of probability generating functions. It is assumed that Z 0 = k, a finite positive integer.

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