A two-priority single server retrial queue with additional items

Dhanya Shajin; A. N. Dudin; Olga Dudina; A. Krishnamoorthy

doi:10.3934/jimo.2019085

Article Contents

2020, Volume 16, Issue 6: 2891-2912. Doi: 10.3934/jimo.2019085

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A two-priority single server retrial queue with additional items

1.
Department of Mathematics, S. N. College, Chempazhanthy, Trivandrum, Kerala-695587, India
2.
Belarusian State University, 4, Nezavisimosti Ave., Minsk, 220030, Belarus
3.
Peoples Friendship University of Russia, 6 Miklukho-Maklaya St, Moscow, 117198, Russia
4.
Department of Mathematics, CMS College, Kottayam-686001, India

^* Corresponding author: A. N. Dudin
^* Corresponding author: A. N. Dudin

Received: September 30, 2018

Revised: February 28, 2019

Early access: July 2019

Published: October 2020

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.

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Abstract

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.

Keywords:

Markovian arrival process (MAP),
marked Markovian arrival process (MMAP),
phase-type distribution,
additional items,
retrial queue.

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

Citation:

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Figure A. Picture representation of the model

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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 $

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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 $

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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 $

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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 $

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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 $

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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 $

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

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Table 1. Dependence of $ N_O $ and N_item 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 N_O
$ 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 N_item

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Table 2. Dependence of $ p_1^{loss} $ and p_item^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	0.075438	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 p₁^loss
$ 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 p_item^loss

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Table 3. Dependence of $ p_1^{busy\ loss} $and p₁^{lack 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 p₁^{busy 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 p₁^{lack loss}

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Table 4. Dependence of $ N_O $ and N_item 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 N_O
$ 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 N_item

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Table 5. Dependence of $ p_1^{loss} $ and p_item^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	0.086069	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 p₁^loss
$ 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 p_item^loss

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Table 6. Dependence of $ p_1^{busy\ loss} $ and p₁^{lack 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 p₁^{busy 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 p₁^{lack loss}

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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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