Optimal Hysteresis Control via a Queuing System with Two Heterogeneous Energy-Consuming Servers
Abstract
:1. Introduction
1.1. Short Literature Overview
1.1.1. Single-Server Queues with Variable Service Rates
1.1.2. Multi-Server Queues with a Variable Number of Homogeneous Active Service Devices
1.1.3. Multi-Server Queues with a Variable Number of Heterogeneous Service Devices
1.1.4. Queuing Systems with Energy Consumption and Harvesting
1.1.5. Queuing Systems with Demand Impatience
1.2. Contributions of the Paper
1.3. Possible Applications of the Model
1.4. Structure of the Text
2. Mathematical Model
3. The Process Describing the Dynamics of the System and Its Generator
- is the number of demands in the buffer,
- is the number of e.u. in the stock,
- is the state of the underlying process of the of demands,
- is the state of the underlying process of the of e.u.,
- vector process defines the states of the underlying processes of service in the devices. We will distinguish four macro-states of the process : the first macro-state means that both service devices are idle; the second macro-state contains the states when the first service device is busy while the second service device is idle; the third macro-state contains the states when the second service device is busy while the first one is idle; and the fourth macro-state contains the states when both service devices are busy. Here, denotes the current state of the underlying process of service in the rth service device,
- is the elements’ number of the state space of the components of the
- ⊗ and ⊕ mean the operations of the Kronecker product and the sum for the matrices, see, e.g., [47];
- is the diagonal matrix with the diagonal entries given in the brackets;
- is equal to 1 if and equal to 0 if ;
- ;
- ;
- sub-generator of the form
- matrix of the form
- matrix of the form
- matrix of the form
- matrix of the form
- matrices defining the transitions of the vector process when it transits from the group of the states consisting of r macro-states to the group of the states consisting of macro-states:
- matrices defining the transitions of the vector process when service by the second service device starts and the process transits from the group of the states consisting of r macro-states to the group of the states consisting of macro-states:
- matrices defining the transitions of the vector process when the first service device completes service and starts the new one and the process transits from the group of the states consisting of r macro-states to the group of the states consisting of macro-states:
- matrices defining the transitions of the vector process when service by the second service device finishes service and starts the new one and the process transits from the group of the states consisting of r macro-states to the group of the states consisting of macro-states:
- matrix defining the transitions of the vector process when service by the second service device finishes service and the new service does not start, so the process continues to stay in the group of the states consisting of two macro-states:
- square matrix of size has:the diagonal blocks
- matrix has only diagonal blocks.If this matrix is defined byIf the formula is changed to
- Matrices are square matrices of size if and size ifMatrices are the block-tridiagonal ones. Their diagonal blocks are defined as follows.If then for allIf then for allIf then for , we haveThe sub-diagonal blocks for are given byFor these blocks have the formThe up-diagonal blocks for have the following form:The blocks for are given by
- Matrix ’s non-zero blocks are already defined by the formulas above (when ) and it has the complementary non-zero blocks
- Matrices have the entries caused by the demands’ departure from the queue due to impatience and also non-zero blocksIn the case of the matrix is the sum of both the matrix and the matrix having the non-zero blocksFor there exist the blocksFor , blocks for are equal to zero.In the case of the matrix is the sum of both the matrix and the matrix having the non-zero blocksFormulas for have different forms depending on the relation between the numbers , andIn the case of , the formula is:In the case of , we have the formula:Formulas for have the following form:As it was already stated above, the matrix is not square. Its size for is It is defined by the formula:Formulas for have different forms depending on the relation between the numbers andIn the case of , the formula is:In the case of , we have the formula:Formulas for have the following form:
- Matrix has different forms for various values of the threshold k: and Its form is not obtained from the above formula for via formal setting because level 0 of the chain has another cardinality than the other levels.In the case of the size of this matrix is and the matrix is equal to the sum of the matrix where the matrix has the blocks defined by the formulaBlock is defined byBlocks are defined by the formulaIf the matrix of size is equal to the sum of the matrix whereBlocks are defined by the formula
- Matrix has different sizes and forms in the three cases: , and The size is equal to for for , and forIn the case of is the block diagonal matrix of size with the diagonal blocks defined byIn the case of it is the matrix of size , with the blocks defined byIn the case of matrix is a square matrix of size , with the diagonal blocks defined by
4. The Case of Absolutely Patient Demands—The Ergodicity Condition of the System and Its Stationary Distribution
5. The Case of Impatient Demands—The Ergodicity Condition of the System and Its Stationary Distribution
6. Performance Measures
7. Numerical Examples and Optimization Problem
7.1. Dependence of Performance Measures on Demands’ Arrival Rate for Different Values of e.u. Arrival Rate
7.2. Illustration of the Impact of Correlation in Arrival Processes of Demands and e.u.
7.3. Illustration of the Effect of the First Service Device’s Coefficient of Variation in Service Time
7.4. Solution of Optimization Problem
8. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Appendix A. Proof of Theorem 1
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D’Apice, C.; D’Arienzo, M.P.; Dudin, A.; Manzo, R. Optimal Hysteresis Control via a Queuing System with Two Heterogeneous Energy-Consuming Servers. Mathematics 2023, 11, 4515. https://doi.org/10.3390/math11214515
D’Apice C, D’Arienzo MP, Dudin A, Manzo R. Optimal Hysteresis Control via a Queuing System with Two Heterogeneous Energy-Consuming Servers. Mathematics. 2023; 11(21):4515. https://doi.org/10.3390/math11214515
Chicago/Turabian StyleD’Apice, Ciro, Maria Pia D’Arienzo, Alexander Dudin, and Rosanna Manzo. 2023. "Optimal Hysteresis Control via a Queuing System with Two Heterogeneous Energy-Consuming Servers" Mathematics 11, no. 21: 4515. https://doi.org/10.3390/math11214515
APA StyleD’Apice, C., D’Arienzo, M. P., Dudin, A., & Manzo, R. (2023). Optimal Hysteresis Control via a Queuing System with Two Heterogeneous Energy-Consuming Servers. Mathematics, 11(21), 4515. https://doi.org/10.3390/math11214515