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Article Contents

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

• * Corresponding author: Shaojun Lan
• 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.

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

 Citation:

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

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

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

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

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

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

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

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

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

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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