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# Equilibrium and optimal balking strategies for low-priority customers in the M/G/1 queue with two classes of customers and preemptive priority

• * Corresponding author: Xiuli Xu
• This paper investigates the low-priority customers' strategic behavior in the single-server queueing system with general service time and two customer types. The priority system is preemptive resume, which means that if a high-priority customer enters the system that are serving a low-priority customer, the arriving customer preempts the service facility and the preempted customer returns to the head of the queue for his own class. The customer who is preempted resumes service at the point of interruption upon reentering the system. The low-priority customer's dilemma is whether to join or balk based on a linear reward-cost structure. Two cases are distinguished based on the different levels of information that the low-priority customers acquire before joining the system. The equilibrium threshold strategy in the observable case and the equilibrium balking strategy as well as the socially optimal balking strategy in the unobservable case for the low-priority customers are derived finally.

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

 Citation: • • Figure 1.  The service process of an arbitrary arriving class-1 customer

Figure 2.  The service process of an arbitrary arriving class-2 customer

Figure 3.  The equilibrium thresholds of the class-1 customers vs.$K_1$ for ${\lambda _2} = 0.5, E\left[ {{G_1}} \right] = E\left[ {{G_2}} \right] = 1, {C_1} = 8$

Figure 4.  The equilibrium thresholds of the class-1 customers vs.$C_1$ for ${\lambda _2} = 0.5, E\left[ {{G_1}} \right] = E\left[ {{G_2}} \right] = 1, {K_1} = 100$

Figure 5.  The equilibrium thresholds of the class-1 customers vs.$E\left[ {{G_1}} \right]$ for ${\lambda _2} = 0.5, E\left[ {{G_2}} \right] = 1, {K_1} = 100, {C_1} = 5$

Figure 6.  The expected net social benefit vs.$q$ for ${\lambda _1} = {\lambda _2} = 0.5, E\left[ {{G_1}} \right] = E\left[ {{G_2}} \right] = 1, E\left[ {{G_1^2}} \right] = E\left[ {{G_2^2}} \right] = 1.2, {C_1} = 10, {K_2} = 100, {C_2} = 50$

Figure 7.  The expected net social benefit vs.$q$ for ${\lambda _1} = {\lambda _2} = 0.5, E\left[ {{G_1}} \right] = E\left[ {{G_2}} \right] = 1, E\left[ {{G_1^2}} \right] = E\left[ {{G_2^2}} \right] = 1.2, {K_1} = 10, {K_2} = 100, {C_2} = 50$

Figure 8.  Equilibrium and socially optimal joining probabilities of the class-1 customers vs. ${K_1}$ for ${\lambda _1} = {\lambda _2} = 0.5, E\left[ {{G_1}} \right] = E\left[ {{G_2}} \right] = 1, E\left[ {{G_1^2}} \right] = E\left[ {{G_2^2}} \right] = 1.2, {C_1} = 10$

Figure 9.  Equilibrium and socially optimal joining probabilities of the class-1 customers vs. ${C_1}$ for ${\lambda _1} = {\lambda _2} = 0.5, E\left[ {{G_1}} \right] = E\left[ {{G_2}} \right] = 1, E\left[ {{G_1^2}} \right] = E\left[ {{G_2^2}} \right] = 1.2, {K_1} = 50$

Figure 10.  Equilibrium and socially optimal joining probabilities of the class-1 customers vs. ${\lambda _1}$ for ${\lambda _2} = 0.2, E\left[ {{G_1}} \right] = E\left[ {{G_2}} \right] = 1, E\left[ {{G_1^2}} \right] = E\left[ {{G_2^2}} \right] = 1.2, {K_1} = 15, {C_1} = 10$

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