# Analysis of MAP/PH/1 Queueing System with Degrading Service Rate and Phase Type Vacation

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## Abstract

**:**

## 1. Introduction

- All bold-faced letters will indicate either row or column vectors which will be clear from context.
**e**is the column vector with 1’s, of appropriate dimension. Where clarifications on the dimensions are needed, we will indicate accordingly.- ${\mathit{e}}_{i}\left(m\right)$ is a column vector of dimension m with all but one entries are zero and the nonzero entry in the ith position is taken to be 1. That is, ${\mathit{e}}_{i}\left(m\right)={\left(\begin{array}{ccccc}0& \cdots & 1& \cdots & 0\end{array}\right)}^{t},$ where the notation ‘t’ denotes the transpose notation.
- I is the identity matrix of appropriate order.

## 2. $\mathit{MAP}/\mathit{PH}/\mathbf{1}$ Model with Degradation Restored Instantaneously ($\mathit{Model}\phantom{\rule{3.33333pt}{0ex}}\mathbf{1}$)

#### 2.1. Generator of the Model

- ${N}_{1}\left(t\right)$ is the number of customers in the system,
- ${N}_{2}\left(t\right)$ is the phase of service, if any,
- ${N}_{3}\left(t\right)$ is the phase of arrival process.
- ${N}_{4}\left(t\right)$ is the type of service, if any.

#### 2.2. Stability Analysis

**Result**

**1.**

**Proof.**

**Remark**

**1.**

#### 2.3. Steady-State Probability Vector

#### 2.4. Performance Measures

- Mean number of the customers in the system ${L}_{s}={\sum}_{i=1}^{\infty}i\phantom{\rule{0.166667em}{0ex}}{\mathit{x}}_{i}\phantom{\rule{0.166667em}{0ex}}\mathit{e}={\mathit{x}}_{1}\phantom{\rule{0.166667em}{0ex}}{(I-R)}^{-2}\phantom{\rule{0.166667em}{0ex}}\mathit{e}.$
- Mean waiting time of the customer in the system ${W}_{s}=\frac{{L}_{s}}{\lambda}$.
- Probability of the server being in busy state ${P}_{busy}={\sum}_{i=1}^{\infty}{\mathit{x}}_{i}\phantom{\rule{0.166667em}{0ex}}\mathit{e}$.
- Probability of the server being in idle state ${P}_{idl\mathit{e}}={\mathit{x}}_{0}\phantom{\rule{0.166667em}{0ex}}\mathit{e}$.

## 3. $\mathit{MAP}/\mathit{PH}/\mathbf{1}$ Model with Degradation Restored after a Random Time (a.k.a. Vacation)—$\mathit{Model}\phantom{\rule{3.33333pt}{0ex}}\mathbf{2}$

#### 3.1. Generator of the Model

- ${N}_{1}\left(t\right)$ is the number of customers in the system,
- ${N}_{2}\left(t\right)$ is the phase of vacation or service depending on the server being idle or busy,
- ${N}_{3}\left(t\right)$ is the phase of arrival process,
- ${N}_{4}\left(t\right)$ is the type of service when the server is busy.

#### 3.2. Stability Analysis

**Result**

**2.**

**Proof.**

#### 3.3. Steady-State Probability Vector

#### 3.4. Performance Measures

- Mean number of customers in the system ${L}_{s}^{\left(v\right)}={\sum}_{i=1}^{\infty}i\phantom{\rule{0.166667em}{0ex}}{\mathit{x}}_{i}^{\left(v\right)}\phantom{\rule{0.166667em}{0ex}}\mathit{e}={\mathit{x}}_{1}^{\left(v\right)}\phantom{\rule{0.166667em}{0ex}}{(I-{R}_{v})}^{-2}\phantom{\rule{0.166667em}{0ex}}\mathit{e}.$
- Mean waiting time of the customer in the system ${W}_{s}^{\left(v\right)}=\frac{{L}_{s}^{\left(v\right)}}{\lambda}.$
- Probability of the server being in busy state ${P}_{busy}={\sum}_{i=1}^{\infty}{\mathit{x}}_{i}^{\left(v\right)}\phantom{\rule{0.166667em}{0ex}}\left(\begin{array}{c}\mathit{e}\\ \mathbf{0}\end{array}\right).$
- (a)
- Probability of the server being on vacation with no customer in the system ${P}_{NCvacation}={\mathit{x}}_{0}^{\left(v\right)}\phantom{\rule{0.166667em}{0ex}}\mathit{e}.$
- (b)
- Probability of the server being on vacation with at least one customer in the system ${P}_{OCvacation}={\sum}_{i=1}^{\infty}{\mathit{x}}_{i}^{\left(v\right)}\phantom{\rule{0.166667em}{0ex}}\left(\begin{array}{c}0\\ \mathit{e}\end{array}\right).$
- (c)
- Probability of the server being on vacation ${P}_{vacation}={\mathit{x}}_{0}^{\left(v\right)}\phantom{\rule{0.166667em}{0ex}}\mathit{e}+{\sum}_{i=1}^{\infty}{\mathit{x}}_{i}^{\left(v\right)}\phantom{\rule{0.166667em}{0ex}}\left(\begin{array}{c}0\\ \mathit{e}\end{array}\right).$

## 4. Numerical Examples

- (i)
- $ErA$—This is the Erlang distribution of order 5 for the arrival process. Here
- ${D}_{0}=\left(\begin{array}{ccccc}-5& 5& 0& 0& 0\\ 0& -5& 5& 0& 0\\ 0& 0& -5& 5& 0\\ 0& 0& 0& -5& 5\\ 0& 0& 0& 0& -5\end{array}\right)$ and ${D}_{1}=\left(\begin{array}{ccccc}0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 5& 0& 0& 0& 0\end{array}\right)$.
- The standard deviation for this process is 0.4472.

- (ii)
- $ExA$—This is the exponential distribution for the arrival process for which
- ${D}_{0}=-1$ and ${D}_{1}=1$.
- The standard deviation for this process is 1.

- (iii)
- $HeA$—This is the hyperexponential distribution for the arrival process for which
- ${D}_{0}=$$\left(\begin{array}{cccc}-68.5& 0& 0& 0\\ 0& -6.85& 0& 0\\ 0& 0& -0.685& 0\\ 0& 0& 0& -0.0685\end{array}\right)$ and ${D}_{1}=$$\left(\begin{array}{cccc}34.2500& 20.5500& 10.2750& 3.4250\\ 3.425& 2.0550& 1.0275& 0.3425\\ 0.3425& 0.2055& 0.1028& 0.0343\\ 0.0343& 0.0206& 0.0103& 0.0034\end{array}\right)$.
- The standard deviation for this process is 4.5787.

- (iv)
- $NeA$—It is the negatively correlated distribution for the arrival process, for which
- ${D}_{0}=\left(\begin{array}{ccc}-1.25& 1.25& 0\\ 0& -1.25& 0\\ 0& 0& -2.5\end{array}\right)$ and ${D}_{1}=\left(\begin{array}{ccc}0& 0& 0\\ 0.0125& 0& 1.2375\\ 2.4750& 0& 0.0250\end{array}\right)$.
- The standard deviation for this process is 1.0392 and since this arrival process is correlated, it can be verified that the 1-lag correlation is given by −0.3267.

- (v)
- $PoA$—It is the positively correlated distribution for the arrival process, for which
- ${D}_{0}=\left(\begin{array}{ccc}-1.25& 1.25& 0\\ 0& -1.25& 0\\ 0& 0& -2.5\end{array}\right)$ and ${D}_{1}=\left(\begin{array}{ccc}0& 0& 0\\ 1.2375& 0& 0.0125\\ 0.0250& 0& 2.4750\end{array}\right)$.
- The standard deviation for this process is 1.0392 and here the 1-lag correlation is given by 0.3267.

- (i)
- Erlang distribution (E). For this$\mathit{\beta}=\left(1\phantom{\rule{0.277778em}{0ex}}0\phantom{\rule{0.277778em}{0ex}}0\phantom{\rule{0.277778em}{0ex}}0\phantom{\rule{0.277778em}{0ex}}0\right)\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\mathrm{and}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}S=\left(\begin{array}{ccccc}-5\phantom{\rule{0.166667em}{0ex}}{a}_{1}& 5\phantom{\rule{0.166667em}{0ex}}{a}_{1}& 0& 0& 0\\ 0& -5\phantom{\rule{0.166667em}{0ex}}{a}_{1}& 5\phantom{\rule{0.166667em}{0ex}}{a}_{1}& 0& 0\\ 0& 0& -5\phantom{\rule{0.166667em}{0ex}}{a}_{1}& 5\phantom{\rule{0.166667em}{0ex}}{a}_{1}& 0\\ 0& 0& 0& -5\phantom{\rule{0.166667em}{0ex}}{a}_{1}& 5\phantom{\rule{0.166667em}{0ex}}{a}_{1}\\ 0& 0& 0& 0& -5\phantom{\rule{0.166667em}{0ex}}{a}_{1}\end{array}\right)$.
- (ii)
- Exponential distribution (X). For this$\mathit{\beta}=1$ and $S=-{a}_{1}$.
- (iii)
- Hyperexponential distribution (H). For this$\mathit{\beta}=\left(0.7\phantom{\rule{0.277778em}{0ex}}0.25\phantom{\rule{0.277778em}{0ex}}0.05\right)$ and $S=\left(\begin{array}{ccc}-8.2\phantom{\rule{0.166667em}{0ex}}{a}_{1}& 0& 0\\ 0& -0.82\phantom{\rule{0.166667em}{0ex}}{a}_{1}& 0\\ 0& 0& -0.082\phantom{\rule{0.166667em}{0ex}}{a}_{1}\end{array}\right)$.

**Example**

**1.**

- ${\mathit{\theta}}^{\left(\mathbf{1}\right)}={(1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1)}^{t}$,
- ${\mathit{\theta}}^{\left(\mathbf{2}\right)}=(1,0.99,0.98,0.97,0.96,0.95,0.94,0.92,0.90,0.88,0.86,0.84,0.82,0.80,0.78,0.76,$${0.74,0.72)}^{t},$
- ${\mathit{\theta}}^{\left(\mathbf{3}\right)}=(1,0.98,0.96,0.94,0.92,0.90,0.87,0.84,0.81,0.78,0.75,0.72,0.69,0.66,0.63,0.60,$${0.57,0.54)}^{t}$.

- As is to be expected, the mean number of customers in the system increases as L is increased. This is due to the fact that as L is increased, the rate of service is decreased and thus an increase in the mean number of customers in the system. However, the rate of increase (as a function of L) is higher as $\mu $ is decreased.
- ${L}_{s}$ increases as the arrival rate is increased. Furthermore, the rate of increase of ${L}_{s}$ (with respect to L) is higher as $\lambda $ is increased.
- First observe that ${\mathit{\theta}}^{\left(\mathbf{3}\right)}$ gives a higher degradation in service rates as compared to ${\mathit{\theta}}^{\left(\mathbf{1}\right)}$ and ${\mathit{\theta}}^{\left(\mathbf{2}\right)}$. We see from the third graph of Figure 1, Figure 2 and Figure 3 that more degradation in service rate gives more ${L}_{s}$ and more ${P}_{busy}$ but less ${P}_{idl\mathit{e}}$.
- The server will be more busy if there are more customers in the system and so is the measure ${P}_{busy}$. Also, an increase in L and in $\lambda $ will result in an increase in ${P}_{busy}$ and similarly an increase in $\mu $ will decrease the measure ${P}_{busy}$.
- Obviously, a higher degradation in the services provided will lead to the server being more busy and this is seen in the measure ${P}_{busy}$.

**Example**

**2.**

- Both the degradation and the corresponding classical vacation models have almost the same mean number of customers in the system, and also follow the same pattern as we change the service and vacation processes.
- Furthermore, the positively correlated arrival process gives the highest mean number of customers among all the considered arrival processes which reflects the role of correlation.
- The hyperexponential service gives more mean number of customers than the Erlang and exponential service.
- Erlang, exponential, and negatively correlated arrivals have the same pattern as the service and the vacation processes are varied. Similarly, the hyperexponential and positively correlated arrivals have the same pattern.

**Example**

**3.**

- For the negatively correlated arrivals, as the magnitude of the correlation coefficient decreases, the mean number of costumers in the system increases.
- For the positively correlated arrivals, as the magnitude of correlation coefficient increases, the mean number in the system increases.
- Furthermore, the effect of the positively correlated arrivals is seen more than the negatively correlated arrivals as the magnitude of the correlation is varied.

**Example**

**4.**

- For the mean number of customers in the system ${L}_{s}^{\left(v\right)}$:
- ${L}_{s}^{\left(v\right)}$ decreases if there is an increase in $\mu $ as well as in ${\mu}_{v}$.
- An increase in $\lambda $ results in an increase in ${L}_{s}^{\left(v\right)}$.
- More degradation in the service rate will result in a higher value for ${L}_{s}^{\left(v\right)}$.
- An increase in L will decrease the ${L}_{s}^{\left(v\right)}$ initially but after a certain point we notice an increase in ${L}_{s}^{\left(v\right)}$. The value of L for which the ${L}_{s}^{\left(v\right)}$ is smallest depends on all the system parameters.

- For the probability of the server being in busy state $\left({P}_{busy}\right)$:
- An increase in $\mu $ will decrease the probability, ${P}_{busy}$, but an increase in $\lambda $ increases ${P}_{busy}$.
- The more degradation is present in the services, the busier the server is.
- The parameter ${\mu}_{v}$ has a negligible effect on ${P}_{busy}$ as compared to other parameters. However, an increment in ${\mu}_{v}$ will decrease the ${P}_{busy}$.
- Increment in L will increase the ${P}_{busy}$.

- The probability of the server being in vacation state (${P}_{vacation}$) behaves opposite to the ${P}_{busy}$ as the system parameters are varied.

**Example**

**5.**

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

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**Figure 5.**Mean number of customer in the system for negatively correlated and positively correlated arrivals (Model 2).

Model | Processes | ${\mathit{L}}_{\mathit{s}}$ | ${\mathit{P}}_{\mathit{busy}}$ | ${\mathit{P}}_{\mathit{idle}}/{\mathit{P}}_{\mathit{vacation}}$ |
---|---|---|---|---|

Classical $MAP/PH/1$ model | NeA, ErS | 2.22305 | 0.790528 | 0.206349 |

NeA, HyS | 58.92956 | 0.82926 | 0.206349 | |

PoA, ErS | 24.16811 | 0.791647 | 0.206349 | |

PoA, HyS | 82.19010 | 0.848347 | 0.206349 | |

Model 1 | NeA, ErS | 2.07631 | 0.762403 | 0.237597 |

NeA, HyS | 60.92105 | 0.779311 | 0.220689 | |

PoA, ErS | 23.63030 | 0.773938 | 0.226062 | |

PoA, HyS | 84.05024 | 0.782954 | 0.217046 | |

Model 2 | NeA, ErS | 2.45311 | 0.768110 | 0.231890 |

NeA, HyS | 71.03092 | 0.781503 | 0.218497 | |

PoA, ErS | 26.56992 | 0.777106 | 0.222894 | |

PoA, HyS | 96.61144 | 0.784809 | 0.215191 |

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**MDPI and ACS Style**

Choudhary, A.; Chakravarthy, S.R.; Sharma, D.C.
Analysis of *MAP*/*PH*/1 Queueing System with Degrading Service Rate and Phase Type Vacation. *Mathematics* **2021**, *9*, 2387.
https://doi.org/10.3390/math9192387

**AMA Style**

Choudhary A, Chakravarthy SR, Sharma DC.
Analysis of *MAP*/*PH*/1 Queueing System with Degrading Service Rate and Phase Type Vacation. *Mathematics*. 2021; 9(19):2387.
https://doi.org/10.3390/math9192387

**Chicago/Turabian Style**

Choudhary, Alka, Srinivas R. Chakravarthy, and Dinesh C. Sharma.
2021. "Analysis of *MAP*/*PH*/1 Queueing System with Degrading Service Rate and Phase Type Vacation" *Mathematics* 9, no. 19: 2387.
https://doi.org/10.3390/math9192387