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# Performance evaluation and analysis of a discrete queue system with multiple working vacations and non-preemptive priority

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• In this paper, we introduce a discrete time Geo/Geo/1 queue system with non-preemptive priority and multiple working vacations. We assume that there are two types of customers in this queue system named "Customers of type-Ⅰ" and "Customers of type-Ⅱ". Customer of type-Ⅱ has a higher priority with non-preemption than Customer of type-Ⅰ. By building a discrete time four-dimensional Markov Chain which includes the numbers of customers with different priorities in the system, the state of the server and the service state, we obtain the state transition probability matrix. Using a birth-and-death chain and matrix-geometric method, we deduce the average queue length, the average waiting time of the two types of customers, and the average busy period of the system. Then, we provide some numerical results to evaluate the effect of the parameters on the system performance. Finally, we develop some benefit functions to analyse both the personal and social benefit, and obtain some optimization results within a certain range.

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

 Citation: • • Figure 1.  Schematic illustration for the service process of the non-preemptive priority queue

Figure 2.  Schematic diagram for the model description

Figure 3.  Relation of $E(L_1)$ with $\lambda$ and $\alpha$

Figure 4.  Relation of $E(L_2)$ with $\lambda$ and $\alpha$

Figure 5.  Relation of $E(L)$ with $\theta$ and $\mu_1$

Figure 6.  Relation of $E(B)$ with $\mu _1$ and $\mu_2$

Figure 7.  Relation of $B_1$ with $\lambda$ and $\alpha$

Figure 8.  Relation of $B_2$ with $\lambda$ and $\alpha$

Figure 9.  Relation of $D$ with $\mu _1$ and $\alpha$

Figure 10.  Relation of $D$ with $\lambda$ and $\alpha$

Table 1.  Relation of $E(W_1)$ with $\mu _1$ and $\mu _2$

 $\mu _2$ $\mu _1 =0.30$ $\mu _1 =0.32$ $\mu _1 =0.34$ $\mu _1 =0.36$ $\mu _1 =0.38$ $\mu _1 =0.40$ 0.45 4.7081 2.9925 2.1313 1.6271 1.3023 1.0787 0.50 2.5403 1.7894 1.3538 1.0748 0.8836 0.7459 0.55 1.7483 1.2914 1.0067 0.8156 0.6800 0.5797

Table 2.  Relation of $E(W_2)$ with $\mu _1$ and $\mu _2$

 $\mu _2$ $\mu _1 =0.30$ $\mu _1 =0.32$ $\mu _1 =0.34$ $\mu _1 =0.36$ $\mu _1 =0.38$ $\mu _1 =0.40$ 0.45 3.6164 2.9022 2.3961 2.0311 1.7626 1.5610 0.50 2.4961 1.9948 1.6473 1.4001 1.2199 1.0853 0.55 1.8492 1.4802 1.2263 1.0464 0.9154 0.8175
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Tables(2)

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