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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Zusammenfassung: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.
Bibliographie:ObjectType-Article-1
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ISSN:0254-5330
1572-9338
DOI:10.1007/s10479-019-03463-0