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Customers' joining behavior in an unobservable GI/Geo/m queue

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  • This paper studies the equilibrium balking strategies of impatient customers in a discrete-time multi-server renewal input queue with identical servers. Arriving customers are unaware of the number of customers in the queue before making a decision whether to join or balk the queue. We model the decision-making process as a non-cooperative symmetric game and derive the Nash equilibrium mixed strategy and optimal social strategies. The stationary system-length distributions at different observation epochs under the equilibrium structure are obtained using the roots method. Finally, some numerical examples are presented to show the effect of the information level together with system parameters on the equilibrium and social behavior of impatient customers.

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


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  • Figure 1.  A schematic representation of an ATM switch

    Figure 2.  Various time epochs in early-arrival system (EAS)

    Figure 3.  λ vs mixed strategies with m = 2, µ = 0.2, R = 6, C = 1

    Figure 4.  R vs mixed strategies with m = 2, λ = 0.4, µ = 0.2, C = 1

    Figure 5.  C vs mixed strategies with m = 2, λ = 0.3, µ = 0.2, R = 40

    Figure 6.  λ vs benefit with m = 2, µ = 0.2, R = 6, C = 1

    Figure 7.  C vs benefit with m = 2, λ = 0.3, µ = 0.2, R = 40

    Figure 8.  µ vs expected waiting time with λ = 0.3

    Figure 9.  R vs PoA with m = 2, λ = 0.5, µ = 0.4, C = 1

    Figure 10.  λ vs PoA with m = 2, µ = 0.4, R = 6, C = 1

    Table 1.  Survey on queueing models related to game-theoretic analysis

    Reference Model Buffer size Findings
    [29] M/M/1 Finite Individual and social
    optimal behavior
    [6] M/M/1 Infinite Individual and social
    optimal behavior
    [8] M/M/1 Finite Price of Anarchy
    [11] Geo/Geo/1 Finite & infinite Individual & social optimal
    behavior and PoA
    [34] GI/M/s finite & infinite Self & social optimization
    [19] M/M/s Infinite Individual & social optimization
    behavior of customers under a
    non-linear holding cost
    [23] M/M/s Infinite Individual & social optimization
    behavior of customers under a
    linear holding cost
    [1] M/G/s Infinite Individual & social optimization
    with holding cost
    [14] GI/M/c Infinite Equilibrium balking strategy
    with reneging
    Present study GI/Geo/m Infinite Individual & social optimal
    behavior and PoA
     | Show Table
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    Table 2.  Notations and model parameters

    Operational parameters
    $ 1/\lambda $ mean arrival times
    $ 1/\mu $ mean service time of each server
    $ m $ number of independent homogeneous servers
    $ d \in [0,1] $ probability of joining in unobservable case
    Economic parameters
    $ R $ customers gets a reward after completion of service
    $ C $ waiting cost per time unit in the system
    Performance measures
    $ W_s $ average sojourn time in the system
    $ L_s $ mean system-length
    $ \Delta_e(d) $ net benefit of the tagged customer
    $ \Delta_s(d) $ social benefit per time unit
    $ PoA $ price of anarchy
    $ d_e $ equilibrium joining probability
    $ d^* $ socially optimal joining probability
     | Show Table
    DownLoad: CSV
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