# Performance Analysis of Preemptive Priority Retrial Queueing System with Disaster under Working Breakdown Services

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Model Description

- The arrival process: Two sorts of customers were entered into the system: priority customers and regular customers. Priority customers have preemptive priorities over regular customers in relation to the service time of an occupied server. We supposed that both sorts of customers came as per three independent Poisson processes outgoing from the system with rates δ and λ, respectively.
- The retrial process: We supposed that there was no waiting capacity and consequently an arriving priority customer saw the server free and the customer started the service directly. By the time the priority customer arrived, the server presented service for a priority customer or working breakdowns. A priority customer that arrived later would leave the system instantly wanting service. However, since the orderly occupied server was working with a regular customer, the arriving priority customer would obstruct the service of the regular customer as the server would directly start providing service to the priority customer.
- We supposed that while an ordinary customer was preempted by a priority customer, the ordinary customer, who was solely being served before the service of the priority customer began, would wait in the service region for the residual service to finish. If an arriving ordinary customer discovered the server was occupied or was on working breakdown, the arrivals entered a pool of blocked customers, called an orbit, in accordance with FIFO discipline. This meant that only one customer at the head of the orbit queue was allowed to be served by the server. Then, measured from the instant the server became free, a potential priority customer, a regular customer and a retry customer competed to the entire server. Inter-retrial times are arbitrarily distributed $R(t)$ with the Laplace Stieltijes Transform (LST) ${R}^{\ast}(\vartheta ).$ The retrial ordinary customer wanted to try to abandon the service if the first external customer or the average customer arrived first. In this case, the retrial ordinary customer returned to its location in the retrial queue.
- The regular service process: Whenever a new primary (priority) customer or retry customer arrived at the server in idle state, then the server immediately started normal service for the arrivals. The service time of priority customers followed a general distribution denoted by the random variable S
_{p}with distribution function ${S}_{p}(t),$ having LST ${S}_{p}^{*}(\vartheta )$ and the first and second moments denoted as ${\beta}_{p}^{(1)}$ and ${\beta}_{p}^{(2)}.$ The service time of ordinary customers followed a general distribution denoted by the random variable S_{b}with distribution function ${S}_{b}(t)$ having LST ${S}_{b}^{*}(\vartheta ),$ and the first and second moments denoted as ${\beta}_{b}^{(1)}\text{}\mathrm{and}\text{}{\beta}_{b}^{(2)}.$ - The working breakdown process: The system had the potential to become deficient by disasters at any time when the regular occupied server was in task with exponentially appropriated with a rate of α. The event of disasters forced all present ordinary customers out of the system and caused the primary server to fail. At a failure moment, the major server was sent to the reform, and the reform period started immediately. The repair time followed an exponential distribution with the rate of γ. We supposed that the repaired server was as good as a new server. Yet, disaster occurred in the regular working server or when the orbit became vacuous, after which the server left for a working breakdown. Through the duration of the working breakdown, the replacement server worked at a minimal service rate for the arriving customers (μ
_{w}< μ). When the repair was completed, if there were customers in the orbit, the server changed to the normal working level. Alternatively, the server started another working breakdown period. Throughout the working breakdown periods (lower speed services), the service time pursued a general random variable S_{w}with distribution function ${S}_{w}(t)$ and LST ${S}_{w}^{*}(\vartheta )$.We supposed that inter-arrival times, retrial times, service times and working breakdown times were disjointed and independent of one another. - Throughout the rest of the paper, we denote by $\overline{F}(x)=1-F(x)$ the tail of distribution function F(x). We also denote ${F}^{*}(s)={\displaystyle \underset{0}{\overset{\infty}{\int}}{e}^{-sx}}dF(x),$ to be the LST of F(x) and $\stackrel{~}{F}(s)={\displaystyle \underset{0}{\overset{\infty}{\int}}{e}^{-sx}}F(x)dx,$ to be the Laplace transform of F(x). Ee assume the notation ${\overline{F}}^{*}(s)=\frac{1-{F}^{*}(s)}{s}.$

#### Practical Application of the Suggested Model

## 3. Steady State Analysis of the System

#### 3.1. Steady State Probabilities and Notations

- $N(t)\equiv $ the number of customers in the orbit at time t,
- $X(t)\equiv $ the state of the server at time t.

#### 3.2. Ergodicity Condition of the Model

_{n}; n = 1, 2, …} be the sequence of epochs at which either a normal service for priority/ordinary or a lower service completion occurs. The sequence of random vectors ${Z}_{n}=\left\{X\left({t}_{n}+\right),\text{}N\left({t}_{n}+\right)\right\}$ forms an embedded Markov chain in the retrial queueing system. It follows from Appendix A that $\left\{{Z}_{n};\text{}n\in N\right\}$ is ergodic if and only if $\rho <{A}_{1}^{*}(\lambda +\delta )$, where $\rho =\left({A}_{1}^{*}(\lambda +\delta )+\lambda {\overline{A}}_{1}^{*}(\lambda +\delta )\right)\left({B}^{\prime}{A}_{3}^{*}{}^{\prime}(\alpha )+{S}^{\prime}(1)\right)+\delta \lambda {\overline{A}}_{1}^{*}(\lambda +\delta ){\beta}_{p}^{(1)}.$

#### 3.3. Steady State Equations

#### 3.4. Computation of the Steady State Solution

**Theorem**

**1.**

**Proof.**

^{n}and Taking summation over n, (n = 0, 1, 2, …),

**Theorem**

**2.**

**Proof.**

_{0}). □

**Corollary**

**1.**

_{s}(z))is

_{o}(z))is:

## 4. Performance Measures

#### System State Probabilities and Mean System Size

- (i)
- Let P be the steady state probability that the server is idle during the retrial time:$${P}_{1}={P}_{1}(1)=\frac{(\lambda +\delta ){\overline{A}}_{1}^{*}(\lambda +\delta )}{\eta}\left\{\left(1-{A}_{4}^{*}(\gamma )\right)\left(\frac{\lambda}{\gamma}+\lambda {A}_{3}^{*}{}^{\prime}(\alpha )\left(1+\delta {\beta}_{p}^{(1)}\right)+{S}^{\prime}(1)\right)\right\}.$$
- (ii)
- The probability that the server is busy serving a priority customer without preempting an ordinary customer is given by:$${P}_{2}={P}_{2}(1)=\frac{\delta \left(\lambda +\delta \right){\overline{A}}_{1}^{*}(\lambda +\delta ){\beta}_{p}^{(1)}}{\eta}\left\{\left(1-{A}_{4}^{*}(\gamma )\right)\left(\frac{\lambda}{\gamma}+\lambda {A}_{3}^{*}{}^{\prime}(\alpha )\left(1+\delta {\beta}_{p}^{(1)}\right)+{S}^{\prime}(1)\right)\right\}.$$
- (iii)
- The probability that the server is busy serving a priority customer with preempting an ordinary customer is given by:$${P}_{3}={P}_{3}(1)=\frac{\delta (\lambda +\delta ){\beta}_{p}^{(1)}{\overline{A}}_{3}^{*}(\alpha )}{\eta}\left(1-{A}_{4}^{*}(\gamma )\right)\left\{{A}_{1}^{*}(\lambda +\delta )-\delta \lambda {\overline{A}}_{1}^{*}(\lambda +\delta ){\beta}_{p}^{(1)}+\frac{\lambda}{\gamma}\left(1-\delta {\overline{A}}_{1}^{*}(\lambda +\delta )\right)\right\}.$$
- (iv)
- The probability that the server is busy serving an ordinary customer is given by:$${P}_{4}={P}_{4}(1)=\frac{(\lambda +\delta ){\overline{A}}_{3}^{*}(\alpha )}{\eta}\left(1-{A}_{4}^{*}(\gamma )\right)\left\{{A}_{1}^{*}(\lambda +\delta )-\delta \lambda {\overline{A}}_{1}^{*}(\lambda +\delta ){\beta}_{p}^{(1)}+\frac{\lambda}{\gamma}\left(1-\delta {\overline{A}}_{1}^{*}(\lambda +\delta )\right)\right\}.$$
- (v)
- The probability that the server is on working vacation is given by:$${P}_{5}={P}_{5}(1)=\left\{\frac{(\lambda +\delta ){P}_{0}\left(1-{A}_{4}^{*}(\gamma )\right)}{\gamma}\right\}.$$
- (vi)
- The excepted number of customers in the orbit (L
_{q}) is obtained by differentiating (54) with respect to z and evaluating at z = 1:$${L}_{q}={{K}^{\prime}}_{o}(1)=\underset{z\to 1}{\mathrm{lim}}\frac{d}{dz}{K}_{o}(z).$$ - (vii)
- The excepted number of customers in the system (L
_{s}) is obtained by differentiating (53) with respect to z and evaluating at z = 1:$${L}_{s}\text{}={{K}^{\prime}}_{s}(1)\text{}=\text{}\underset{z\to 1}{\mathrm{lim}}\frac{d}{dz}{K}_{s}(z).$$ - (viii)
- The average time a customer spends in the system (W
_{s}) and queue (W_{q}) is expressed as:$${W}_{s}=\frac{{L}_{s}}{\lambda}\text{}\mathrm{and}\text{}{W}_{q}=\frac{{L}_{q}}{\lambda}.$$

## 5. Special Cases

## 6. Cost Optimization Analysis and Sensitivity Analysis

#### 6.1. Cost Optimization

- C
_{h}—the holding cost per unit time; - C
_{o}—the cost per unit time in operations; - C
_{s}—the setup cost per busy cycle; - C
_{a}—the startup cost per unit time.

_{b}= 5; μ

_{p}= 3; μ

_{w}= 2; γ = 3; θ = 2; C

_{h}= $5, C

_{o}= $60, C

_{s}= $500 and C

_{a}= $90 are chosen according to the satisfaction of the stability condition. Then the total expected cost per unit of time is TC = $310.3895.

_{h}, C

_{o}) and (C

_{s}, C

_{a}) on the expected cost function, respectively. The impact of the system parameters (λ, α) on the total expected cost per unit of time is shown in Figure 1 and Figure 2. From this, we infer that the expected cost function demonstrates a directly increasing trend with increasing cost parameters.

#### 6.2. Sensitivity Analysis

#### 6.3. Results and Discussion

_{b}= 5; μ

_{p}= 3; μ

_{w}= 2; γ = 3; θ = 2.

_{q}and P increase and the probability that the server is idle (P

_{0}) decreases. As can be seen in Table 4, there is a decreasing trend of P

_{0}as the priority rate (δ) increases. However, the other measures such as L

_{q}and Π

_{1}increase as the priority rate increases.

_{q}).

_{0}decreases with the increasing values of failure rate (α) and ordinary arrival rate (λ). As expected from Figure 5, the mean orbit size (L

_{q}) decreases with the increasing value of the lower service rate (μ

_{w}) and regular service rate (μ). Figure 6 shows that the idle probability increases with the increasing values of service rate (μ) and repair rate (η).

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

_{j}< ∞ for all j ≥ 0 and there exists j

_{0}∈ N such that ψ

_{j}≥ 0 for j ≥ j

_{0}. Notice that, in our case, Kaplan’s condition is satisfied because there is a k such that m

_{ij}= 0 for j < i − k and i > 0, where M = (m

_{ij}) is the one-step transition matrix of $\left\{{Z}_{n};\text{}n\in N\right\}.$ Then, $\rho \ge {R}^{*}(\lambda +\delta )$ implies the non-ergodicity of the Markov chain.

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**Table 1.**The impact of (C

_{h}, C

_{o}) on cost per unit of time (TC) with C

_{s}= $500 and C

_{a}= $90.

(C_{h}, C_{o}) | (5, 60) | (20, 60) | (15, 60) | (5, 75) | (5, 100) |
---|---|---|---|---|---|

TC | 310.3895 | 311.5580 | 313.1159 | 325.3895 | 350.3895 |

(C_{s}, C_{a}) | (500, 90) | (550, 90) | (600, 90) | (500, 100) | (500, 110) |
---|---|---|---|---|---|

TC | 310.3895 | 335.3985 | 360.3985 | 323.2147 | 326.3924 |

Retrial Rate (θ) | Exp | Erlang | Hyp-Exp | ||||||
---|---|---|---|---|---|---|---|---|---|

P_{0} | L_{q} | P | P_{0} | L_{q} | P | P_{0} | L_{q} | P | |

2.00 | 0.4673 | 4.3345 | 0.1732 | 0.3507 | 5.4842 | 0.3538 | 0.3986 | 4.7128 | 0.2318 |

3.00 | 0.5250 | 3.8563 | 0.1155 | 0.4773 | 4.0795 | 0.2271 | 0.4766 | 3.9451 | 0.1539 |

4.00 | 0.5539 | 3.6547 | 0.0866 | 0.5374 | 3.6448 | 0.1671 | 0.5153 | 3.6499 | 0.1151 |

5.00 | 0.5712 | 3.5434 | 0.0693 | 0.5724 | 3.4337 | 0.1321 | 0.5385 | 3.4937 | 0.0920 |

6.00 | 0.5828 | 3.4730 | 0.0577 | 0.5952 | 3.3090 | 0.1092 | 0.5539 | 3.3970 | 0.0766 |

Priority Arrival Rate (δ) | Exp | Erlang | Hyp-Exp | ||||||
---|---|---|---|---|---|---|---|---|---|

P_{0} | L_{q} | Π_{1} | P_{0} | L_{q} | Π_{1} | P_{0} | L_{q} | Π_{1} | |

1.00 | 0.7208 | 0.0831 | 0.0033 | 0.4249 | 0.2487 | 0.0250 | 0.6820 | 0.1625 | 0.0116 |

2.00 | 0.5804 | 0.1057 | 0.0042 | 0.2526 | 0.3450 | 0.0255 | 0.5361 | 0.2029 | 0.0281 |

3.00 | 0.4760 | 0.1263 | 0.0043 | 0.1614 | 0.4522 | 0.0223 | 0.4341 | 0.2211 | 0.0446 |

4.00 | 0.3965 | 0.1446 | 0.0042 | 0.1090 | 0.5675 | 0.0191 | 0.3595 | 0.2286 | 0.0598 |

5.00 | 0.3348 | 0.1608 | 0.0040 | 0.0769 | 0.6935 | 0.0165 | 0.3031 | 0.2310 | 0.0732 |

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**MDPI and ACS Style**

Ammar, S.I.; Rajadurai, P.
Performance Analysis of Preemptive Priority Retrial Queueing System with Disaster under Working Breakdown Services. *Symmetry* **2019**, *11*, 419.
https://doi.org/10.3390/sym11030419

**AMA Style**

Ammar SI, Rajadurai P.
Performance Analysis of Preemptive Priority Retrial Queueing System with Disaster under Working Breakdown Services. *Symmetry*. 2019; 11(3):419.
https://doi.org/10.3390/sym11030419

**Chicago/Turabian Style**

Ammar, Sherif I., and Pakkirisamy Rajadurai.
2019. "Performance Analysis of Preemptive Priority Retrial Queueing System with Disaster under Working Breakdown Services" *Symmetry* 11, no. 3: 419.
https://doi.org/10.3390/sym11030419