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A two-priority single server retrial queue with additional items

  • * Corresponding author: A. N. Dudin

    * Corresponding author: A. N. Dudin 

The first author is supported by Kerala State Council for Science, Technology & Environment: 001-07/PDF/2016/KSCSTE in Department of Mathematics, CMS College, Kottayam-686001, India.
The second author is supported by "RUDN University Program 5-100".
The fourth author is supported by UGC No.F.6-6/2017-18/EMERITUS-2017-18-GEN-10822 (SA-Ⅱ) and DST project INT/RUS/RSF/P-15.

Abstract Full Text(HTML) Figure(8) / Table(7) Related Papers Cited by
  • In this paper, we study a priority queueing-inventory problem with two types of customers. Arrival of customers follows Marked Markovian arrival process and service times have phase-type distribution with parameters depending on the type of customer in service. For service of each type of customer, a certain number of additional items are needed. High priority customers do not have waiting space and so leave the system when on their arrival a priority 1 customer is in service or the number of available additional items is less than the required threshold. Preemptive priority is assumed. Type 2 customers, encountering a busy server or idle with the number of available additional items less than a threshold, go to an orbit of infinite capacity to retry for service. The customers in orbit are non-persistent: if on retrial the server is found to be busy/idle with the number of additional items less than the threshold, this customer abandons the system with certain probability. Such a system represents an accurate enough model of many real-world systems, including wireless sensor networks and system of cognitive radio with energy harvesting and healthcare systems. The probability distribution of the system states is computed, using which several of the characteristics are derived. A detailed numerical study of the system, including the analysis of the influence of the threshold, is performed.

    Mathematics Subject Classification: Primary: 60K25, 90B05; Secondary: 68M20, 90B22.

    Citation:

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  • Figure A.  Picture representation of the model

    Figure 1.  Dependence of average number of customers in the orbit $ N_{O} $ and average number of additional items in the stock $ N_{item} $ on $ N $

    Figure 2.  Dependence of probability that an arbitrary type 1 customer will be lost $ p_{1}^{loss} $ and probability that an arbitrary additional item will be lost $ p_{item}^{loss} $ on $ N $

    Figure 3.  Dependence of probability of an arbitrary arriving type 1 customer loss because the server is busy with type 1 customer $ p_{1}^{busy\ loss} $ and probability of an arbitrary arriving type 1 customer loss due to lack of additional items $ p_{1}^{lack\ loss} $ on $ N $

    Figure 4.  Dependence of average number of customers in the orbit $ N_{O} $ and average number of additional items in the stock $ N_{item} $ on $ q $

    Figure 5.  Dependence of probability that an arbitrary type 1 customer will be lost $ p_{1}^{loss} $ and probability that an arbitrary additional item will be lost $ p_{item}^{loss} $ on $ q $

    Figure 6.  Dependence of probability of an arbitrary arriving type 1 customer loss because the server is busy with type 1 customer $ p_{1}^{busy\ loss} $ and probability of an arbitrary arriving type 1 customer loss due to lack of additional items $ p_{1}^{lack\ loss} $ on $ q $

    Figure 7.  Dependence of $ N_O $ on $ \gamma $

    Table 1.  Dependence of $ N_O $ and Nitem on N for q = 0.2

    $ N $ $ MAP^{0}+ MMAP^{0}$ $ MAP^{0}+ MMAP^{0.4} $ $ MAP^{0.4}+ MMAP^{0.4} $ $ MAP^{0.4}+ MMAP^{0}$
    2 0.541687 1.367441 1.684218 1.271213
    4 0.549554 1.412635 1.703399 1.305689
    6 0.5646816 1.518695 1.755786 1.403333
    8 0.581552 1.588845 1.784279 1.466440
    10 0.60139 1.626319 1.805897 1.509774
    12 0.628908 1.656662 1.827878 1.551554
    14 0.669292 1.682243 1.849624 1.590796
    16 0.732318 1.709136 1.874669 1.631378
    18 0.831686 1.742311 1.904638 1.674188
    20 1.044837 1.802246 1.949755 1.728375
    (A) Dependence of NO
    $ N $ $ MAP^{0}+ MMAP^{0}$ $ MAP^{0}+ MMAP^{0.4} $ $ MAP^{0.4}+ MMAP^{0.4} $ $ MAP^{0.4}+ MMAP^{0}$
    2 15.265487 15.01827 12.685517 7.343982
    4 15.360819 15.39104 13.30253 8.478718
    6 15.489897 15.661518 13.504709 8.945458
    8 15.707897 16.001991 13.742329 9.415827
    10 16.008232 16.420983 13.98464 10.00213
    12 16.376142 16.857811 14.170774 10.51655
    14 16.821438 17.324321 14.475216 11.21957
    16 17.34789 17.797588 14.7440311 11.92004
    18 17.91796 18.255182 15.1044063 12.71894
    20 18.51922 18.665273 15.5122786 13.58646
    (B) Dependence of Nitem
     | Show Table
    DownLoad: CSV

    Table 2.  Dependence of $ p_1^{loss} $ and pitemloss on N for q = 0.2

    $ N $ $ MAP^{0}+ MMAP^{0}$ $ MAP^{0}+ MMAP^{0.4}$ $ MAP^{0.4}+ MMAP^{0.4}$ $ MAP^{0.4}+ MMAP^{0}$
    2 0.075438 0.594164 0.621250 0.584869
    4 0.069154 0.528047 0.612127 0.574752
    6 0.053754 0.343713 0.526146 0.420849
    8 0.044331 0.230872 0.488863 0.334311
    10 0.041304 0.184474 0.466582 0.291888
    12 0.039596 0.154385 0.442873 0.251298
    14 0.039017 0.138246 0.426295 0.224901
    16 0.038708 0.127744 0.407572 0.199159
    18 0.038589 0.12157 0.392962 0.179236
    20 0.038535 0.117919 0.37687 0.16082
    (A) Dependence of ploss of p1loss
    $ N $ $ MAP^{0}+ MMAP^{0}$ $ MAP^{0}+ MMAP^{0.4}$ $ MAP^{0.4}+ MMAP^{0.4}$ $ MAP^{0.4}+ MMAP^{0}$
    2 0.218984 0.596466 0.720285 0.570702
    4 0.22019 0.599877 0.725534 0.58051
    6 0.222009 0.60253 0.727949 0.584911
    8 0.225624 0.60626 0.731018 0.590137
    10 0.23144 0.611019 0.734444 0.596847
    12 0.23998 0.616327 0.737665 0.6033
    14 0.25284 0.62257 0.742163 0.61164
    16 0.272944 0.630337 0.747311 0.620533
    18 0.304933 0.641239 0.754921 0.631276
    20 0.373814 0.662226 0.767272 0.645899
    (B) Dependence of ploss of pitemloss
     | Show Table
    DownLoad: CSV

    Table 3.  Dependence of $ p_1^{busy\ loss} $and p1lack loss on N for q = 0.2

    $ N $ $ MAP^{0}+ MMAP^{0}$ $ MAP^{0}+ MMAP^{0.4}$ $ MAP^{0.4}+ MMAP^{0.4}$ $ MAP^{0.4}+ MMAP^{0}$
    2 0.036982 0.045292 0.042889 0.016605
    4 0.037234 0.054023 0.043846 0.01701
    6 0.03785 0.07846 0.054843 0.023166
    8 0.038227 0.093433 0.059606 0.026628
    10 0.038347 0.099601 0.062507 0.028324
    12 0.038416 0.103602 0.06565 0.029948
    14 0.038439 0.105748 0.06781 0.031004
    16 0.038452 0.107144 0.07029 0.032034
    18 0.038456 0.107965 0.07221 0.032831
    20 0.038459 0.10845 0.07433 0.033567
    (A) Dependence of p1busy loss
    $ N $ $ MAP^{0}+ MMAP^{0}$ $ MAP^{0}+ MMAP^{0.4}$ $ MAP^{0.4}+ MMAP^{0.4}$ $ MAP^{0.4}+ MMAP^{0}$
    2 0.038456 0.548872 0.578361 0.568264
    4 0.03192 0.474025 0.568281 0.557742
    6 0.015904 0.265254 0.471303 0.397683
    8 0.006105 0.13744 0.429257 0.307683
    10 0.002956 0.084873 0.404075 0.263563
    12 0.00118 0.050783 0.377223 0.22135
    14 5.77E-4 0.032498 0.358485 0.193897
    16 2.56E-4 0.020599 0.337281 0.167125
    18 1.33E-4 0.013605 0.320755 0.146405
    20 7.63E-5 0.009469 0.302539 0.127253
    (B) Dependence of p1lack loss
     | Show Table
    DownLoad: CSV

    Table 4.  Dependence of $ N_O $ and Nitem on q for N = 4

    $ q $ $ MAP^{0}+ MMAP^{0}$ $ MAP^{0}+MMAP^{0.4} $ $ MAP^{0.4}+MMAP^{0.4} $ $ MAP^{0.4}+ MMAP^{0} $
    0.1 0.758933 2.592361 3.261523 2.397430
    0.2 0.549554 1.412635 1.703399 1.305689
    0.3 0.441755 0.987633 1.162907 0.911481
    0.4 0.372968 0.765297 0.886588 0.705013
    0.5 0.324288 0.627350 0.718082 0.576937
    0.6 0.287627 0.532879 0.604271 0.489305
    0.7 0.258835 0.463871 0.522087 0.425367
    0.8 0.235527 0.411117 0.459869 0.376551
    0.9 0.216218 0.369402 0.411081 0.337998
    1 0.199928 0.335543 0.371769 0.306743
    (A) Dependence of NO
    $ q $ $ MAP^{0}+ MMAP^{0}$ $ MAP^{0}+MMAP^{0.4} $ $ MAP^{0.4}+MMAP^{0.4} $ $ MAP^{0.4}+ MMAP^{0} $
    0.1 14.631235 15.054343 13.047518 8.294399
    0.2 15.360819 15.391041 13.302530 8.478718
    0.3 15.760671 15.548513 13.408673 8.583818
    0.4 16.021155 15.649013 13.470799 8.656368
    0.5 16.206952 15.721749 13.512953 8.710798
    0.6 16.347252 15.778002 13.544014 8.753684
    0.7 16.457467 15.823323 13.568133 8.788605
    0.8 16.546609 15.860874 13.587549 8.817730
    0.9 16.620347 15.892632 13.603596 8.842468
    1 16.682445 15.919918 13.617128 8.863788
    (B) Dependence of Nitem
     | Show Table
    DownLoad: CSV

    Table 5.  Dependence of $ p_1^{loss} $ and pitemloss on q for N = 4

    $ q $ $ MAP^{0}+MMAP^{0} $ $ MAP^{0}+ MMAP^{0.4}$ $ MAP^{0.4}+ MMAP^{0.4}$ $ MAP^{0.4}+ MMAP^{0} $
    0.1 0.086069 0.557286 0.619555 0.588222
    0.2 0.069154 0.528047 0.612127 0.574752
    0.3 0.061659 0.506329 0.607792 0.566495
    0.4 0.057424 0.489551 0.604719 0.560590
    0.5 0.054707 0.476222 0.602361 0.556081
    0.6 0.052819 0.465381 0.600469 0.552495
    0.7 0.051434 0.456390 0.598905 0.549560
    0.8 0.050375 0.448809 0.597586 0.547105
    0.9 0.049540 0.442329 0.596454 0.545017
    1 0.048865 0.436726 0.595472 0.543215
    (A) Dependence of p1loss
    $ q $ $ MAP^{0}+MMAP^{0} $ $ MAP^{0}+ MMAP^{0.4}$ $ MAP^{0.4}+ MMAP^{0.4}$ $ MAP^{0.4}+ MMAP^{0} $
    0.1 0.187902 0.582932 0.711872 0.560124
    0.2 0.220190 0.599877 0.725534 0.580510
    0.3 0.241291 0.606780 0.731699 0.590660
    0.4 0.256630 0.610762 0.735528 0.597193
    0.5 0.268448 0.613446 0.738236 0.601898
    0.6 0.277902 0.615420 0.740290 0.605506
    0.7 0.285673 0.616952 0.741918 0.608389
    0.8 0.292192 0.618187 0.743249 0.610759
    0.9 0.297748 0.619209 0.744361 0.612748
    1 0.302546 0.620072 0.745308 0.614446
    (B) Dependence of pitemloss
     | Show Table
    DownLoad: CSV

    Table 6.  Dependence of $ p_1^{busy\ loss} $ and p1lack loss on q for N = 4

    $ q $ $ MAP^{0}+MMAP^{0} $ $ MAP^{0}+ MMAP^{0.4}$ $ MAP^{0.4}+ MMAP^{0.4}$ $ MAP^{0.4}+ MMAP^{0} $
    0.1 0.036557 0.050253 0.043100 0.016471
    0.2 0.037234 0.054023 0.043846 0.017010
    0.3 0.037534 0.056868 0.044325 0.017340
    0.4 0.037703 0.059078 0.044682 0.017576
    0.5 0.037812 0.060838 0.044963 0.017757
    0.6 0.037887 0.062272 0.045192 0.017900
    0.7 0.037943 0.063462 0.045384 0.018018
    0.8 0.037985 0.064466 0.045547 0.018116
    0.9 0.038018 0.065324 0.045688 0.018199
    1.0 0.038045 0.066067 0.045811 0.018271
    (A) Dependence of p1busy loss
    0.1 0.049511 0.507033 0.576455 0.571751
    0.2 0.031920 0.474025 0.568281 0.557742
    0.3 0.024125 0.449461 0.563466 0.549155
    0.4 0.019721 0.430472 0.560037 0.543014
    0.5 0.016895 0.415384 0.557398 0.538324
    0.6 0.014932 0.403110 0.555277 0.534595
    0.7 0.013491 0.392928 0.553521 0.531543
    0.8 0.012390 0.384343 0.552038 0.528990
    0.9 0.011521 0.377005 0.550766 0.526817
    1.0 0.010820 0.370659 0.549661 0.524944
    (B) Dependence of p1lack loss
     | Show Table
    DownLoad: CSV

    Table 7.  Dependence of $ N_O $ on $ \gamma $ and $ q $ for $ N = 4 $

    $ q $ $ \gamma $ $ MAP^{0}+ MMAP^{0}$ $ MAP^{0}+MMAP^{0.4} $ $ MAP^{0.4}+MMAP^{0.4} $ $ MAP^{0.4}+MMAP^{0} $
    0.1 1.5 0.959861 3.382764 4.300035 3.145578
    0.2 0.75 1.328719 3.583343 4.430688 3.384015
    0.3 0.5 1.628854 3.746986 4.534785 3.574155
    0.4 0.375 1.881123 3.887829 4.623184 3.731486
    0.5 0.3 2.098026 4.012896 4.700989 3.865063
    0.6 0.25 2.287694 4.126279 4.771088 3.980639
    0.7 0.2143 2.455731 4.230599 4.835292 4.082103
    0.8 0.1875 2.606171 4.327642 4.89482 4.172209
    0.9 0.1667 2.742017 4.418685 4.950534 4.252983
    1 0.1500 2.865570 4.504674 5.003066 4.325962
     | Show Table
    DownLoad: CSV
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