Asymptotic sojourn time analysis of finite-source M/M/1 retrial queueing system with collisions and server subject to breakdowns and repairs

The aim of the present paper is to investigate the steady-state distribution of response and waiting time in a finite-source M / M / 1 retrial queuing system with collision of customers where the server is subjects to random breakdowns and repairs depending on whether it is idle or busy. An asymp...

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Veröffentlicht in:	Annals of operations research Jg. 288; H. 1; S. 417 - 434
Hauptverfasser:	Nazarov, Anatoly, Sztrik, János, Kvach, Anna, Tóth, Ádám
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	New York Springer US 01.05.2020 Springer Springer Nature B.V
Schlagworte:	Analysis Asymptotic methods Breakdown Business and Management Client/server architecture Combinatorics Control Customers Failure analysis Failure rates Markov analysis Methods Operations research Operations Research/Decision Theory Original Research Probability distribution functions Queues Queuing theory Repair Steady state Theory of Computation Time-domain analysis Russia Server breakdowns and repairs Finite-source queueing system Retrial queue Collision Limiting distribution Approximation of distributions Asymptotic analysis Accuracy and area of applicability of approximations
ISSN:	0254-5330, 1572-9338
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Abstract	The aim of the present paper is to investigate the steady-state distribution of response and waiting time in a finite-source M / M / 1 retrial queuing system with collision of customers where the server is subjects to random breakdowns and repairs depending on whether it is idle or busy. An asymptotic method is applied under the condition that the number of sources tends to infinity, the primary request generation rate, retrial rate tend to zero while service rate, failure rates, repair rate are fixed. As the result of the analysis it is shown that the steady-state probability distribution of the number of transitions/retrials of the customer into the orbit is geometric with a given parameter, and the normalized sojourn time of the customer in the system follows a generalized exponential distribution. It is also proved that the limiting distributions of the normalized sojourn time of the customer in the system and the normalized sojourn/waiting time of the customer in the orbit coincide. The novelty of this investigation is the introduction of failure and repair of the server. Approximations of prelimit distributions obtained with the help of stochastic simulation by asymptotic one are considered and several illustrative examples show the accuracy and range of applicability of the proposed asymptotic method.
AbstractList	The aim of the present paper is to investigate the steady-state distribution of response and waiting time in a finite-source M / M / 1 retrial queuing system with collision of customers where the server is subjects to random breakdowns and repairs depending on whether it is idle or busy. An asymptotic method is applied under the condition that the number of sources tends to infinity, the primary request generation rate, retrial rate tend to zero while service rate, failure rates, repair rate are fixed. As the result of the analysis it is shown that the steady-state probability distribution of the number of transitions/retrials of the customer into the orbit is geometric with a given parameter, and the normalized sojourn time of the customer in the system follows a generalized exponential distribution. It is also proved that the limiting distributions of the normalized sojourn time of the customer in the system and the normalized sojourn/waiting time of the customer in the orbit coincide. The novelty of this investigation is the introduction of failure and repair of the server. Approximations of prelimit distributions obtained with the help of stochastic simulation by asymptotic one are considered and several illustrative examples show the accuracy and range of applicability of the proposed asymptotic method. The aim of the present paper is to investigate the steady-state distribution of response and waiting time in a finite-source M / M / 1 retrial queuing system with collision of customers where the server is subjects to random breakdowns and repairs depending on whether it is idle or busy. An asymptotic method is applied under the condition that the number of sources tends to infinity, the primary request generation rate, retrial rate tend to zero while service rate, failure rates, repair rate are fixed. As the result of the analysis it is shown that the steady-state probability distribution of the number of transitions/retrials of the customer into the orbit is geometric with a given parameter, and the normalized sojourn time of the customer in the system follows a generalized exponential distribution. It is also proved that the limiting distributions of the normalized sojourn time of the customer in the system and the normalized sojourn/waiting time of the customer in the orbit coincide. The novelty of this investigation is the introduction of failure and repair of the server. Approximations of prelimit distributions obtained with the help of stochastic simulation by asymptotic one are considered and several illustrative examples show the accuracy and range of applicability of the proposed asymptotic method.
Audience	Academic
Author	Sztrik, János Nazarov, Anatoly Kvach, Anna Tóth, Ádám
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CitedBy_id	crossref_primary_10_1007_s10479_023_05648_0 crossref_primary_10_1016_j_ress_2021_107642 crossref_primary_10_3390_math9192415 crossref_primary_10_1007_s11009_021_09870_w crossref_primary_10_1016_j_ress_2022_108342 crossref_primary_10_1007_s12190_024_02074_8 crossref_primary_10_1016_j_matcom_2023_07_008
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Keywords	Server breakdowns and repairs Finite-source queueing system Retrial queue Collision Limiting distribution Approximation of distributions Asymptotic analysis Accuracy and area of applicability of approximations
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Snippet	The aim of the present paper is to investigate the steady-state distribution of response and waiting time in a finite-source M / M / 1 retrial queuing... The aim of the present paper is to investigate the steady-state distribution of response and waiting time in a finite-source M / M / 1 retrial queuing system...
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SubjectTerms	Analysis Asymptotic methods Breakdown Business and Management Client/server architecture Combinatorics Control Customers Failure analysis Failure rates Markov analysis Methods Operations research Operations Research/Decision Theory Original Research Probability distribution functions Queues Queuing theory Repair Steady state Theory of Computation Time-domain analysis
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Title	Asymptotic sojourn time analysis of finite-source M/M/1 retrial queueing system with collisions and server subject to breakdowns and repairs
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