Performance analysis of a discrete-time $ Geo/G/1$ retrial queue with non-preemptive priority, working vacations and vacation interruption

Shaojun Lan; Yinghui Tang

doi:10.3934/jimo.2018102

Article Contents

2019, Volume 15, Issue 3: 1421-1446. Doi: 10.3934/jimo.2018102

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Performance analysis of a discrete-time $ Geo/G/1$ retrial queue with non-preemptive priority, working vacations and vacation interruption

Shaojun Lan^1,2, , and
Yinghui Tang^3,

1.
School of Mathematics and Statistics, Sichuan University of Science and Engineering, Zigong 643000, Sichuan, China
2.
School of Mathematics and Software Science, Sichuan Normal University, Chengdu 610066, Sichuan, China
3.
School of Fundamental Education, Sichuan Normal University, Chengdu 610066, Sichuan, China

* Corresponding author: Shaojun Lan
* Corresponding author: Shaojun Lan

Received: September 30, 2017

Revised: February 28, 2018

Early access: June 2018

Published: April 2019

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Abstract

This paper is concerned with a discrete-time $ Geo/G/1$ retrial queueing system with non-preemptive priority, working vacations and vacation interruption where the service times and retrial times are arbitrarily distributed. If an arriving customer finds the server free, his service commences immediately. Otherwise, he either joins the priority queue with probability $ α$, or leaves the service area and enters the retrial group (orbit) with probability $ \mathit{\bar{\alpha }}\left( = 1-\alpha \right)$. Customers in the priority queue have non-preemptive priority over those in the orbit. Whenever the system becomes empty, the server takes working vacation during which the server can serve customers at a lower service rate. If there are customers in the system at the epoch of a service completion, the server resumes the normal working level whether the working vacation ends or not (i.e., working vacation interruption occurs). Otherwise, the server proceeds with the vacation. Employing supplementary variable method and generating function technique, we analyze the underlying Markov chain of the considered queueing model, and obtain the stationary distribution of the Markov chain, the generating functions for the number of customers in the priority queue, in the orbit and in the system, as well as some crucial performance measures in steady state. Furthermore, the relation between our discrete-time queue and its continuous-time counterpart is investigated. Finally, some numerical examples are provided to explore the effect of various system parameters on the queueing characteristics.

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Mathematics Subject Classification: Primary: 60K25, 68M20; Secondary: 90B22.

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Figure 1. Various time epochs in an early arrival system (EAS)

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Figure 2. The effect of $\alpha $ on $p(0,0,0)$ for different values of $r$, $\chi _b = 0.7$, $\chi _v = 0.4$, $\lambda = 0.2$, $\theta = 0.3$

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Figure 3. The effect of $\chi _v $ on $p(0,0,0)$ for different values of $\theta $, $\chi _b = 0.7$, $\alpha = 0.5$, $\lambda = 0.2$, $r = 0.35$

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Figure 4. The effect of $\alpha $ on $E\left[ {L_1 } \right]$ for different values of $\lambda $, $\chi _b = 0.7$, $\chi _v = 0.4$, $r = 0.35$, $\theta = 0.3$

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Figure 5. The effect of $\alpha $ on $E\left[ {L_2 } \right]$ for different values of $\lambda $, $\chi _b = 0.7$, $\chi _v = 0.4$, $r = 0.35$, $\theta = 0.3$

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Figure 6. The effect of $\chi _v $ on $E[L]$ for different values of $\theta $, $\chi _b = 0.7$, $\alpha = 0.5$, $\lambda = 0.2$, $r = 0.35$

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Figure 7. $E[L]$ versus $\chi _v $ and $\lambda $, $\chi _b = 0.7$, $\alpha = 0.5$, $\theta = 0.3$, $r = 0.35$

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Figure 8. $E[L]$ versus $r$ and $\lambda $, $\chi _b = 0.7$, $\chi _v = 0.4$, $\alpha = 0.5$, $\theta = 0.3$

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Figure 9. The effect of $\alpha $ on the expected operating cost per unit time

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Table 1. The parabolic method in searching for the optimum solution

	No. of iterations
	0	1	2	3	4	5
$\alpha^{(l)} $	0.55000	0.60000	0.61667	0.61712	0.61734	0.61735
$\alpha^{(m)} $	0.60000	0.61667	0.61712	0.61734	0.61735	0.61735
$\alpha^{(r)} $	0.65000	0.65000	0.65000	0.65000	0.65000	0.65000
$TC\left( {\alpha^{(l)} } \right)$	443.08917	443.06412	443.06226	443.06226	443.06226	443.06226
$TC\left( {\alpha^{(m)} } \right)$	443.06412	443.06226	443.06226	443.06226	443.06226	443.06226
$TC\left( {\alpha^{(r)} } \right)$	443.06912	443.06912	443.06912	443.06912	443.06912	443.06912
$\alpha^\ast $	0.61667	0.61712	0.61734	0.61735	0.61735	0.61735
$TC\left( {\alpha^\ast } \right)$	443.06226	443.06226	443.06226	443.06226	443.06226	443.06226
Tolerance	1.66744×10$^{-2}$	4.40785×10$^{-4}$	2.27516×10$^{-4}$	8.98158×10$^{-6}$	3.20017×10$^{-6}$	1.62729×10$^{-7}$

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