# Optimization of Queueing Model with Server Heating and Cooling

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## Abstract

**:**

## 1. Introduction

## 2. Description of the Model

## 3. Process of System States and Its Analysis

- ${n}_{t}$ is the number of customers in the buffer, ${n}_{t}\ge 0.$
- ${r}_{t},\phantom{\rule{0.166667em}{0ex}}{r}_{t}=\overline{0,2},$ is the server state: ${r}_{t}=0$ if the server is idle, ${r}_{t}=1$ if the server is busy, and ${r}_{t}=2$ if the server is blocked.
- ${k}_{t}$ is the temperature of the server, ${k}_{t}=\overline{0,K}.$
- ${\nu}_{t}$ is the state of the underlying process of the $MAP$, ${\nu}_{t}=\overline{0,W}.$
- ${m}_{t}$ is the state of the underlying process of the $PH$ service process, ${m}_{t}=\overline{1,M}.$

**Theorem**

**1.**

- I is the identity matrix, and O is a zero matrix of an appropriate dimension.
- $\overline{W}=W+1;$
- ⊗ and ⊕ are the symbols of the Kronecker product and the sum of matrices, respectively.
- ${E}_{l}^{-}$ is a square matrix of size l with all zero entries except the entries ${\left({E}_{l}^{-}\right)}_{k,k-1}={\gamma}_{k},\phantom{\rule{0.166667em}{0ex}}k=\overline{1,l-1}$.
- ${C}_{l}$ is a square matrix of size l with all zero entries except the entries ${\left({C}_{l}\right)}_{k,k}={\gamma}_{k},\phantom{\rule{0.166667em}{0ex}}k=\overline{1,l-1}$.
- ${I}_{{K}_{2},K}$ is a matrix of size ${K}_{2}\times K$ with all zero entries except the entries ${\left({I}_{{K}_{2},K}\right)}_{n,n},\phantom{\rule{0.166667em}{0ex}}n=\overline{0,{K}_{2}-1},$ which are equal to 1.
- ${I}_{K,{K}_{2}}$ is a matrix of size $K\times {K}_{2}$ with all zero entries except the entries ${\left({I}_{K,{K}_{2}}\right)}_{n,n},\phantom{\rule{0.166667em}{0ex}}n=\overline{0,{K}_{2}-1},$ which are equal to 1.
- ${E}^{+}$ is a square matrix of size K with all zero entries except the entries ${\left({E}^{+}\right)}_{k,k+1},\phantom{\rule{0.166667em}{0ex}}k=\overline{0,K-2},$ which are equal to 1.
- ${\overline{I}}_{K,K-{K}_{1}}$ is a matrix of size $K\times (K-{K}_{1})$ with all zero entries except the entries ${\left({\overline{I}}_{K,K-{K}_{1}}\right)}_{n,n-{K}_{1}-1},\phantom{\rule{0.166667em}{0ex}}n=\overline{{K}_{2},K-1},$ which are equal to 1.
- $\widehat{I}$ is a matrix of size $K\times (K-{K}_{1})$ with all zero entries except the entry ${(\widehat{I})}_{K-1,K-{K}_{1}-1},$ which is equal to 1.
- ${\tilde{I}}_{K-{K}_{1},{K}_{2}}$ is a matrix of size $(K-{K}_{1})\times {K}_{2}$ with all zero entries except the entry ${({\tilde{I}}_{K-{K}_{1},{K}_{2}})}_{0,{K}_{1}},$ which is equal to 1.
- $\tilde{E}$ is a square matrix of size $K-{K}_{1}$ with all zero entries except the entries ${(\tilde{E})}_{k,k-1}={\gamma}_{{K}_{1}+k+1},\phantom{\rule{0.166667em}{0ex}}k=\overline{1,K-{K}_{1}-1}$.
- $\tilde{C}$ is a square matrix of size $K-{K}_{1}$ with all zero entries except the entries ${(\tilde{C})}_{k,k}={\gamma}_{{K}_{1}+k+1},\phantom{\rule{0.166667em}{0ex}}k=\overline{0,K-{K}_{1}-1}$.
- B is a square matrix of size K with all zero entries except the entries ${(B)}_{k,k}=1,\phantom{\rule{0.166667em}{0ex}}k=\overline{0,{K}_{2}-1}$.
- ${\tilde{I}}_{K-{K}_{1},K}$ is a matrix of size $(K-{K}_{1})\times K$ with all zero entries except the entry ${({\tilde{I}}_{K-{K}_{1},K})}_{0,{K}_{1}}=1$.

**Theorem**

**2.**

## 4. Performance Indicators

## 5. Numerical Example

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Dependence of the average number N of customers in the buffer on the values of ${K}_{1}$ and ${K}_{2}$.

**Figure 3.**Dependence of the average average intensity ${\lambda}_{out}$ of the flow of served customers on the values of ${K}_{1}$ and ${K}_{2}$.

**Figure 4.**Dependence of the average temperature T of the server on the values of ${K}_{1}$ and ${K}_{2}$.

**Figure 5.**Dependence of the probability ${P}_{idle}$ that the server is idle on the values of ${K}_{1}$ and ${K}_{2}$.

**Figure 6.**Dependence of the probability ${P}_{busy}$ that the server is busy on the values of ${K}_{1}$ and ${K}_{2}$.

**Figure 7.**Dependence of the probability ${P}_{block}$ that the server is blocked on the values of ${K}_{1}$ and ${K}_{2}$.

**Figure 8.**Dependence of the probability ${P}_{imp}$ that an arbitrary customer is lost due to impatience on the values of ${K}_{1}$ and ${K}_{2}$.

**Figure 9.**Dependence of the probability ${P}_{overheating}$ that an arbitrary customer is lost due to server overheating on the values of ${K}_{1}$ and ${K}_{2}$.

**Figure 10.**Dependence of the probability ${P}_{loss}$ of an arbitrary customer loss on the values of ${K}_{1}$ and ${K}_{2}$.

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**MDPI and ACS Style**

Dudina, O.; Dudin, A. Optimization of Queueing Model with Server Heating and Cooling. *Mathematics* **2019**, *7*, 768.
https://doi.org/10.3390/math7090768

**AMA Style**

Dudina O, Dudin A. Optimization of Queueing Model with Server Heating and Cooling. *Mathematics*. 2019; 7(9):768.
https://doi.org/10.3390/math7090768

**Chicago/Turabian Style**

Dudina, Olga, and Alexander Dudin. 2019. "Optimization of Queueing Model with Server Heating and Cooling" *Mathematics* 7, no. 9: 768.
https://doi.org/10.3390/math7090768