Computing Stationary Expectations in Level-Dependent QBD Processes

2013 ◽  
Vol 50 (01) ◽  
pp. 151-165 ◽  
Author(s):  
Hendrik Baumann ◽  
Werner Sandmann
Keyword(s):  
Performance Measures ◽  
Network Performance ◽  
Queueing Network ◽  
Birth And Death Processes ◽  
State Spaces ◽  
Additive Functionals ◽  
Long Run ◽  
Qbd Processes ◽  
Special Cases ◽  
Infinite State

Stationary expectations corresponding to long-run averages of additive functionals on level-dependent quasi-birth-and-death processes are considered. Special cases include long-run average costs or rewards, moments and cumulants of steady-state queueing network performance measures, and many others. We provide a matrix-analytic scheme for numerically computing such stationary expectations without explicitly computing the stationary distribution of the process, which yields an algorithm that is as quick as its counterparts for stationary distributions but requires far less computer storage. Specific problems arising in the case of infinite state spaces are discussed and the application of the algorithm is demonstrated by a queueing network example.

scholarly journals Computing Stationary Expectations in Level-Dependent QBD Processes

2013 ◽  
Vol 50 (1) ◽  
pp. 151-165 ◽  
Author(s):  
Hendrik Baumann ◽  
Werner Sandmann
Keyword(s):  
Performance Measures ◽  
Network Performance ◽  
Queueing Network ◽  
Birth And Death Processes ◽  
State Spaces ◽  
Additive Functionals ◽  
Long Run ◽  
Qbd Processes ◽  
Special Cases ◽  
Infinite State

Stationary expectations corresponding to long-run averages of additive functionals on level-dependent quasi-birth-and-death processes are considered. Special cases include long-run average costs or rewards, moments and cumulants of steady-state queueing network performance measures, and many others. We provide a matrix-analytic scheme for numerically computing such stationary expectations without explicitly computing the stationary distribution of the process, which yields an algorithm that is as quick as its counterparts for stationary distributions but requires far less computer storage. Specific problems arising in the case of infinite state spaces are discussed and the application of the algorithm is demonstrated by a queueing network example.

scholarly journals Bounded truncation error for long-run averages in infinite Markov chains

2015 ◽  
Vol 52 (3) ◽  
pp. 609-621 ◽  
Author(s):  
Hendrik Baumann ◽  
Werner Sandmann
Keyword(s):  
Markov Chains ◽  
Performance Measures ◽  
Network Performance ◽  
Truncation Error ◽  
Queueing Network ◽  
Discrete State ◽  
Additive Functionals ◽  
Long Run ◽  
Special Cases ◽  
Drift Conditions

We consider long-run averages of additive functionals on infinite discrete-state Markov chains, either continuous or discrete in time. Special cases include long-run average costs or rewards, stationary moments of the components of ergodic multi-dimensional Markov chains, queueing network performance measures, and many others. By exploiting Foster-Lyapunov-type criteria involving drift conditions for the finiteness of long-run averages we determine suitable finite subsets of the state space such that the truncation error is bounded. Illustrative examples demonstrate the application of this method.

scholarly journals Bounded truncation error for long-run averages in infinite Markov chains

2015 ◽  
Vol 52 (03) ◽  
pp. 609-621 ◽  
Author(s):  
Hendrik Baumann ◽  
Werner Sandmann
Keyword(s):  
Markov Chains ◽  
Performance Measures ◽  
Network Performance ◽  
Truncation Error ◽  
Queueing Network ◽  
Discrete State ◽  
Additive Functionals ◽  
Long Run ◽  
Special Cases ◽  
Drift Conditions

We consider long-run averages of additive functionals on infinite discrete-state Markov chains, either continuous or discrete in time. Special cases include long-run average costs or rewards, stationary moments of the components of ergodic multi-dimensional Markov chains, queueing network performance measures, and many others. By exploiting Foster-Lyapunov-type criteria involving drift conditions for the finiteness of long-run averages we determine suitable finite subsets of the state space such that the truncation error is bounded. Illustrative examples demonstrate the application of this method.

scholarly journals Kronecker-Based Infinite Level-Dependent QBD Processes

2012 ◽  
Vol 49 (4) ◽  
pp. 1166-1187 ◽  
Author(s):  
TuǦrul Dayar ◽  
M. Can Orhan
Keyword(s):  
Control Unit ◽  
Lyapunov Theory ◽  
Kronecker Products ◽  
Stochastic Chemical Kinetics ◽  
State Spaces ◽  
Qbd Processes ◽  
State Measure ◽  
Markovian Systems ◽  
Countably Infinite ◽  
Infinite State

Markovian systems with multiple interacting subsystems under the influence of a control unit are considered. The state spaces of the subsystems are countably infinite, whereas that of the control unit is finite. A recent infinite level-dependent quasi-birth-and-death model for such systems is extended by facilitating the automatic representation and generation of the nonzero blocks in its underlying infinitesimal generator matrix with sums of Kronecker products. Experiments are performed on systems of stochastic chemical kinetics having two or more countably infinite state space subsystems. Results indicate that, even though more memory is consumed, there are many cases where a matrix-analytic solution coupled with Lyapunov theory yields a faster and more accurate steady-state measure compared to that obtained with simulation.

scholarly journals Kronecker-Based Infinite Level-Dependent QBD Processes

2012 ◽  
Vol 49 (04) ◽  
pp. 1166-1187 ◽  
Author(s):  
TuǦrul Dayar ◽  
M. Can Orhan
Keyword(s):  
Control Unit ◽  
Lyapunov Theory ◽  
Kronecker Products ◽  
Stochastic Chemical Kinetics ◽  
State Spaces ◽  
Qbd Processes ◽  
State Measure ◽  
Markovian Systems ◽  
Countably Infinite ◽  
Infinite State

Markovian systems with multiple interacting subsystems under the influence of a control unit are considered. The state spaces of the subsystems are countably infinite, whereas that of the control unit is finite. A recent infinite level-dependent quasi-birth-and-death model for such systems is extended by facilitating the automatic representation and generation of the nonzero blocks in its underlying infinitesimal generator matrix with sums of Kronecker products. Experiments are performed on systems of stochastic chemical kinetics having two or more countably infinite state space subsystems. Results indicate that, even though more memory is consumed, there are many cases where a matrix-analytic solution coupled with Lyapunov theory yields a faster and more accurate steady-state measure compared to that obtained with simulation.

Multivariate Markov-switching score-driven models: an application to the global crude oil market

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Szabolcs Blazsek ◽  
Alvaro Escribano ◽  
Adrian Licht
Keyword(s):  
Crude Oil ◽  
Moving Average ◽  
Local Level ◽  
Markov Switching ◽  
Impulse Response Functions ◽  
Common Trends ◽  
Vector Autoregressive ◽  
Oil Market ◽  
Long Run ◽  
Special Cases

Abstract A new class of multivariate nonlinear quasi-vector autoregressive (QVAR) models is introduced. It is a Markov switching score-driven model with stochastic seasonality for the multivariate t-distribution (MS-Seasonal-t-QVAR). As an extension, we allow for the possibility of having common-trends and nonlinear co-integration. Score-driven nonlinear updates of local level and seasonality are used, which are robust to outliers within each regime. We show that VAR integrated moving average (VARIMA) type filters are special cases of QVAR filters. Using exclusion, sign, and elasticity identification restrictions in MS-Seasonal-t-QVAR with common-trends, we provide short-run and long-run impulse response functions for the global crude oil market.

scholarly journals Local model checking for infinite state spaces

1992 ◽  
Vol 96 (1) ◽  
pp. 157-174 ◽  
Author(s):  
Julian Bradfield ◽  
Colin Stirling
Keyword(s):  
Model Checking ◽  
Local Model ◽  
State Spaces ◽  
Infinite State ◽  
Local Model Checking

Optimal repair/replacement policy for a general repair model

2001 ◽  
Vol 33 (1) ◽  
pp. 206-222 ◽  
Author(s):  
Xiaoyue Jiang ◽  
Viliam Makis ◽  
Andrew K. S. Jardine
Keyword(s):  
Average Cost ◽  
Failure Time ◽  
Operating Time ◽  
Replacement Policy ◽  
Repair Cost ◽  
Virtual Age ◽  
Preventive Replacement ◽  
Long Run ◽  
Special Cases ◽  
Cost Limit

In this paper, we study a maintenance model with general repair and two types of replacement: failure and preventive replacement. When the system fails a decision is made whether to replace or repair it. The repair degree that affects the virtual age of the system is assumed to be a random function of the repair-cost and the virtual age at failure time. The system can be preventively replaced at any time before failure. The objective is to find the repair/replacement policy minimizing the long-run expected average cost per unit time. It is shown that a generalized repair-cost-limit policy is optimal and the preventive replacement time depends on the virtual age of the system and on the length of the operating time since the last repair. Computational procedures for finding the optimal repair-cost limit and the optimal average cost are developed. This model includes many well-known models as special cases and the approach provides a unified treatment of a wide class of maintenance models.

scholarly journals A Paradox for Admission Control of Multiclass Queueing Network with Differentiated Service

2007 ◽  
Vol 44 (02) ◽  
pp. 321-331
Author(s):  
Heng-Qing Ye
Keyword(s):  
Admission Control ◽  
Network Performance ◽  
Queueing Network ◽  
Differentiated Service ◽  
Service Station ◽  
Multiclass Queueing ◽  
Multiclass Queueing Network ◽  
Service Priority

In this paper we present counter-intuitive examples for the multiclass queueing network, where each station may serve more than one job class with differentiated service priority and each job may require service sequentially by more than one service station. In our examples, the network performance is improved even when more jobs are admitted for service.

scholarly journals A Paradox for Admission Control of Multiclass Queueing Network with Differentiated Service

2007 ◽  
Vol 44 (2) ◽  
pp. 321-331
Author(s):  
Heng-Qing Ye
Keyword(s):  
Admission Control ◽  
Network Performance ◽  
Queueing Network ◽  
Differentiated Service ◽  
Service Station ◽  
Multiclass Queueing ◽  
Multiclass Queueing Network ◽  
Service Priority

In this paper we present counter-intuitive examples for the multiclass queueing network, where each station may serve more than one job class with differentiated service priority and each job may require service sequentially by more than one service station. In our examples, the network performance is improved even when more jobs are admitted for service.

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