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Article Contents

# A two-priority single server retrial queue with additional items

• * 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.

• 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:

• 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

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

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

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

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

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

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
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