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# Strategic behavior and optimal strategies in an M/G/1 queue with Bernoulli vacations

• * Corresponding author:Jinting Wang
This work is supported in part by the National Natural Science Foundation of China (Grant nos. 71571014,71390334.)
• This paper considers an unobservable M/G/1 queue with Bernoulli vacations in which the server begins a vacation when the system is empty or upon completing a service. In the latter case, the server takes a vacation with p or serves the next customer, if any, with 1-p. We first give the steady-state equations and some performance measures, and then study the customer strategic behavior and obtain customers' Nash equilibrium strategies. From the viewpoint of the social planner, we derive the socially optimal joining probability, the socially optimal vacation probability and the socially optimal vacation rate. The socially optimal joining probability is found not greater than the equilibrium probability. In addition, if the vacation scheme does not incur any cost, the socially optimal decision is that the server does not take either a Bernoulli vacation or the normal vacation. On the other hand, if the server incurs the costs due to the underlying loss and the technology upgrade, proper vacations are beneficial to the social welfare maximization. Finally, sensitivity analysis is also performed to explore the effect of different parameters, and some managerial insights are provided for the social planner.

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

 Citation: • • Figure 1.  Social welfare $SW$ vs. joining probability $q$ for $R = 4, C = 1, \lambda = 0.5, \beta_{1} = 1.2, \beta_{2} = 1, p = 0.1$

Figure 2.  Equilibrium joining probability $q^{e}$ and socially optimal joining probability $q^{*}$ vs. potential arrival rate $\lambda$ for $R = 2, C = 1, \beta_{1} = 0.56, \beta_{2} = 0.25, \theta = 0.83, p = 0.1$

Figure 3.  Equilibrium joining probability $q^{e}$ and socially optimal joining probability $q^{*}$ vs. the first moment of server time $\beta_{1}$ for $R = 2, C = 1, \lambda = 0.5, \beta_{2} = 0.25, \theta = 0.83, p = 0.1$

Figure 4.  Equilibrium joining probability $q^{e}$ and socially optimal joining probability $q^{*}$ vs. $p$ for $R = 2, C = 1, \lambda = 0.5, \beta_{1} = 0.56, \beta_{2} = 0.25, \theta = 0.83$

Figure 5.  Equilibrium joining probability $q^{e}$ and socially optimal joining probability $q^{*}$ vs. the second moment of server time $\beta_{2}$ for $R = 2, C = 1, \lambda = 0.5, \beta_{1} = 0.56, \theta = 0.83, p = 0.1$

Figure 6.  Equilibrium joining probability $q^{e}$ and socially optimal joining probability $q^{*}$ vs. vacation rate $\theta$ for $R = 2, C = 1, \beta_{1} = 0.56, \beta_{2} = 0.25, \lambda = 0.5, p = 0.1$

Figure 7.  Equilibrium joining probability $q^{e}$ and socially optimal joining probability $q^{*}$ vs. waiting cost per time $C$ for $R = 2, \theta = 0.83, \beta_{1} = 0.56, \beta_{2} = 0.25, \lambda = 0.5, p = 0.1$

Figure 8.  Equilibrium joining probability $q^{e}$ and socially optimal joining probability $q^{*}$ vs. $R$ for $\theta = 0.83, \beta_{1} = 0.56, \beta_{2} = 0.25, \lambda = 0.5, p = 0.1$

Figure 9.  Social welfare $SW$ vs. vacation probability $p$ for $R = 10, C = 1, \lambda = 0.8, \beta_{1} = 0.6, \beta_{2} = 1, q = 0.85, \theta = 1, C_{s} = 0.1$

Figure 10.  Social welfare $SW$ vs. vacation rate $\theta$ for $R = 10, C = 1, \lambda = 0.8, \beta_{1} = 0.6, \beta_{2} = 1, q = 0.85, p = 0.3, C_{s} = 0.5$

Figure 11.  Socially optimal vacation probability $p^{*}$ vs. $C_{b}$ for $R = 10, C = 1, \theta = 1, \lambda = 0.8, \beta_{1} = 0.6, \beta_{2} = 1$

Figure 12.  Socially optimal vacation probability $p^{*}$ vs. joining probability $q$ for $R = 10, C = 1, \theta = 1, \lambda = 0.8, \beta_{1} = 0.6, \beta_{2} = 1$

Figure 13.  Socially optimal vacation probability $p^{*}$ vs. vacation rate $\theta$ for $R = 10, C = 1, \theta = 1, \lambda = 0.8, \beta_{1} = 0.6, \beta_{2} = 1, C_{b} = 1$

Figure 14.  Socially optimal vacation rate $\theta^{*}$ vs. $C_{s}$ for $R = 10, C = 1, \lambda = 0.8, \beta_{1} = 0.6, \beta_{2} = 1, p = 0.32$

Figure 15.  Socially optimal vacation rate $\theta^{*}$ vs. joining probability $q$ for $R = 10, C = 1, \lambda = 0.8, \beta_{1} = 0.6, \beta_{2} = 1, p = 0.32$

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