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Synchronized abandonment in discrete-time renewal input queues with vacations

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  • We consider a discrete-time infinite buffer renewal input queue with multiple vacations and synchronized abandonment. Waiting customers get impatient during the server's vacation and decide whether to take service or abandon simultaneously at the vacation completion instants. Using the supplementary variable technique and difference operator method, we obtain an explicit expression to find the steady-state system-length distributions at pre-arrival, random, and outside observer's observation epochs. We provide the stochastic decomposition structure for the number of customers and discuss the various performance measures. With the help of numerical experiments, we show that the method formulated in this work is analytically elegant and computationally tractable. The results are appropriate for light-tailed inter-arrival distributions and can also be leveraged to find heavy-tailed inter-arrival distributions.

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

    Citation:

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  • Figure 1.  Various time epochs in a late arrival system with delayed access (LAS-DA)

    Figure 2.  Effect of $ \mu $ on customer abandonment in different queueing systems with $ \lambda = 0.2, \phi = 0.3, q = 0.6 $

    Figure 3.  Effect of $ \mu $ on the mean sojourn time in different queueing systems with $ \lambda = 0.2, \phi = 0.3, q = 0.6 $

    Figure 4.  Effect of $ \phi $ on customer abandonment in different queueing systems with $ \lambda = 0.2, \mu = 0.6, q = 0.6 $

    Figure 5.  Effect of $ \phi $ on the mean sojourn time in different queueing systems with $ \lambda = 0.2, \mu = 0.6, q = 0.6 $

    Figure 6.  Effect of $ q $ on customer abandonment in different queueing systems with $ \lambda = 0.2, \mu = 0.6, \phi = 0.3 $

    Figure 7.  Effect of $ q $ on the mean sojourn time in different queueing systems with $ \lambda = 0.2, \mu = 0.6, \phi = 0.3 $

    Figure 8.  $ \lambda $ vs $ L_s $ for different arrival streams with $ \mu = 0.6, \phi = 0.3, q = 0.5 $

    Figure 9.  $ \lambda $ vs abandonment rate for systems with $ \mu = 0.6, \phi = 0.3, q = 0.5 $

    Table 1.  System-length distributions at pre-arrival and arbitrary epochs for geometric and deterministic arrival distributions

    Geo/Geo/1 D/Geo/1
    n $ P_{n, 0}^- $ $ P_{n, 1}^- $ $ P_{n, 0} $ $ P_{n, 1} $ $ P_{n, 0}^- $ $ P_{n, 1}^- $ $ P_{n, 0} $ $ P_{n, 1} $
    0 0.313492 0.313492 0.288979 0.138613
    1 0.201531 0.064165 0.201531 0.064165 0.170639 0.152011 0.213012 0.143164
    2 0.129555 0.030556 0.129555 0.030556 0.100761 0.080583 0.125781 0.108959
    3 0.083286 0.014474 0.083286 0.014474 0.059498 0.035638 0.074273 0.051092
    4 0.053541 0.006856 0.053541 0.006856 0.035133 0.015139 0.043857 0.022010
    5 0.034419 0.003248 0.034419 0.003248 0.020746 0.006365 0.025897 0.009288
    6 0.022127 0.001538 0.022127 0.001538 0.012250 0.002669 0.015292 0.003899
    7 0.014224 0.000729 0.014224 0.000729 0.007234 0.001119 0.009030 0.001634
    8 0.009144 0.000345 0.009144 0.000345 0.004271 0.000469 0.005332 0.000685
    9 0.005878 0.000164 0.005878 0.000164 0.002522 0.000196 0.003149 0.000287
    10 0.003779 0.000077 0.003779 0.000077 0.001489 0.000082 0.001859 0.000120
    15 0.000415 0.000002 0.000415 0.000002 0.000107 0.000001 0.000133 0.000002
    20 0.000046 0 0.000046 0 0.000008 0 0.000010 0
    $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$
    $ \Sigma $ 0.877778 0.122222 0.877778 0.122222 0.705671 0.294329 0.658776 0.341224
    $ L_s $=1.812532, $ W_s $= 9.062658 $ L_s $=1.964151, $ W_s $= 9.820755
     | Show Table
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    Table 2.  System-length distributions at pre-arrival and arbitrary epochs for negative binomial and discrete phase-type arrival distributions

    NB/Geo/1 PH/Geo/1
    $ n $ $ P_{n, 0}^- $ $ P_{n, 1}^- $ $ P_{n, 0} $ $ P_{n, 1} $ $ P_{n, 0}^- $ $ P_{n, 1}^- $ $ P_{n, 0} $ $ P_{n, 1} $
    0 0.288979 0.138613 0.231585 0.228347
    1 0.170639 0.152011 0.213012 0.143164 0.148479 0.119171 0.149592 0.117569
    2 0.100761 0.080583 0.125781 0.108959 0.095196 0.099585 0.095909 0.099895
    3 0.059498 0.035638 0.074273 0.051092 0.061034 0.062761 0.061491 0.063295
    4 0.035133 0.015139 0.043857 0.022010 0.039131 0.035351 0.039425 0.035744
    5 0.020746 0.006365 0.025897 0.009288 0.025089 0.018767 0.025277 0.019004
    6 0.012250 0.002669 0.015292 0.003899 0.016085 0.009615 0.016206 0.009745
    7 0.007234 0.001119 0.009030 0.001634 0.010313 0.004813 0.010390 0.004881
    8 0.004271 0.000469 0.005332 0.000685 0.006612 0.002371 0.006662 0.002406
    9 0.002522 0.000196 0.003149 0.000287 0.004239 0.001155 0.004271 0.001173
    10 0.001489 0.000082 0.001859 0.000120 0.002718 0.000558 0.002738 0.000567
    15 0.000107 0.000001 0.000133 0.000002 0.000294 0.000014 0.000297 0.000014
    20 0.000008 0 0.000010 0 0.000032 0 0.000032 0
    $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$
    $ \Sigma $ 0.705671 0.294329 0.658776 0.341224 0.645338 0.354662 0.645201 0.354799
    $ L_s $=1.964151, $ W_s $=9.820755 $ L_s $=2.041164, $ W_s $=10.205818
     | Show Table
    DownLoad: CSV

    Table 3.  System-length distributions at pre-arrival and arbitrary epochs for discrete standard log normal (SLN) and discrete Weibull (Wb) arrival distributions

    SLN/Geo/1 Wb/Geo/1
    $ n $ $ P_{n, 0}^- $ $ P_{n, 1}^- $ $ P_{n, 0} $ $ P_{n, 1} $ $ P_{n, 0}^- $ $ P_{n, 1}^- $ $ P_{n, 0} $ $ P_{n, 1} $
    0 0.181276 0.248765 0.103221 0.132529
    1 0.123225 0.108283 0.112313 0.104447 0.085123 0.113309 0.082833 0.109539
    2 0.083763 0.108336 0.076346 0.099997 0.070198 0.088184 0.068309 0.084544
    3 0.056939 0.081316 0.051897 0.073931 0.057890 0.062539 0.056332 0.059909
    4 0.038705 0.054268 0.035277 0.048965 0.047740 0.043931 0.046455 0.042080
    5 0.026310 0.033964 0.023980 0.030504 0.039369 0.030828 0.038310 0.029529
    6 0.017885 0.020412 0.016301 0.018276 0.032467 0.021630 0.031593 0.020719
    7 0.012157 0.011930 0.011081 0.010658 0.026774 0.015177 0.026054 0.014537
    8 0.008264 0.006832 0.007532 0.006094 0.022080 0.010649 0.021486 0.010200
    9 0.005618 0.003853 0.005120 0.003432 0.018208 0.007472 0.017718 0.007157
    10 0.003819 0.002146 0.003480 0.001910 0.015016 0.005242 0.014612 0.005021
    15 0.000554 0.000102 0.000505 0.000091 0.005727 0.000891 0.005573 0.000854
    20 0.000080 0.000004 0.000073 0.000004 0.002184 0.000152 0.002126 0.000145
    30 0.000002 0 0.000002 0 0.000318 0.000004 0.000309 0.000004
    40 0 0 0 0 0.000046 0 0.000045 0
    60 0 0 0 0 0.000001 0 0.000001 0
    $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$
    $ \Sigma $ 0.566067 0.433933 0.599480 0.400520 0.588712 0.411288 0.604956 0.395044
    $ L_s $=2.280859, $ W_s $=10.610305 $ L_s $=4.048721, $ W_s $=7.961503
     | Show Table
    DownLoad: CSV
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