Analysis of an MMAP/PH1, PH2/N/∞ queueing system operating in a random environment

Kim, Che Soong; Dudin, Alexander; Dudin, Sergey; Dudina, Olga

doi:10.2478/amcs-2014-0036

Cited by 18 publications

(9 citation statements)

References 21 publications

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“…These numerical evidences show one of the directions of further research: explanation of these effects from an analytical point of view. Another direction is the generalization of the proposed method for non-Markovian systems and other types of inhomogeneous queueing systems (e.g., Almasi et al, 2005;Gudkova et al, 2016). One of the appealing candidates are queueing systems with balking, in which arrival intensities decrease with the growth of the total number of customers in the system.…”

Section: Resultsmentioning

confidence: 99%

Bounds on the rate of convergence for one class of inhomogeneous Markovian queueing models with possible batch arrivals and services

Zeifman

Razumchik

Satin

et al. 2018

International Journal of Applied Mathematics and Computer Science

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In this paper we present a method for the computation of convergence bounds for four classes of multiserver queueing systems, described by inhomogeneous Markov chains. Specifically, we consider an inhomogeneous M/M/S queueing system with possible state-dependent arrival and service intensities, and additionally possible batch arrivals and batch service. A unified approach based on a logarithmic norm of linear operators for obtaining sharp upper and lower bounds on the rate of convergence and corresponding sharp perturbation bounds is described. As a side effect, we show, by virtue of numerical examples, that the approach based on a logarithmic norm can also be used to approximate limiting characteristics (the idle probability and the mean number of customers in the system) of the systems considered with a given approximation error.

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Section: Resultsmentioning

confidence: 99%

Bounds on the rate of convergence for one class of inhomogeneous Markovian queueing models with possible batch arrivals and services

Zeifman

Razumchik

Satin

et al. 2018

International Journal of Applied Mathematics and Computer Science

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show abstract

“…How large should the capacity of the stock and the speed of additional items delivering be? The results are planned to be extended to the case of the discrete time system similar to the one by Atencia (2014), the batch arrivals and more complicated strategies of control as in the work Gaidamaka et al (2014), the system operating in the random environment (Kim et al, 2014).…”

Section: Resultsmentioning

confidence: 99%

A Queueing System with Heterogeneous Impatient Customers and Consumable Additional Items

Baek

Dudina

Kim

2017

International Journal of Applied Mathematics and Computer Science

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A single-server queueing system with a marked Markovian arrival process of heterogeneous customers is considered. Type-1 customers have limited preemptive priority over type-2 customers. There is an infinite buffer for type-2 customers and no buffer for type-1 customers. There is also a finite buffer (stock) for consumable additional items (semi-products, half-stocks, etc.) which arrive according to the Markovian arrival process. Service of a customer requires a fixed number of consumable additional items depending on the type of the customer. The service time has a phase-type distribution depending on the type of the customer. Customers in the buffer are impatient and may leave the system without service after an exponentially distributed amount of waiting time. Aiming to minimize the loss probability of type-1 customers and maximize throughput of the system, a threshold strategy of admission to service of type-2 customers is offered. Service of type-2 customer can start only if the server is idle and the number of consumable additional items in the stock exceeds the fixed threshold. Stationary distributions of the system states and the waiting time are computed. In the numerical example, we show some interesting effects and illustrate a possibility of application of the presented results for solution of optimization problems.

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“…The results can be extended to the case of cross-correlation of arrivals of customers and energy units via consideration of the Marked Markovian arrival process (see, e.g., Kim et al, 2014). An extension to the case of a discrete time system with energy harvesting is possible by means of the technique described by Atencia (2014).…”

Section: Resultsmentioning

confidence: 99%

Optimization of the service strategy in a queueing system with energy harvesting and customers’ impatience

Dudin

Lee

Dudin

2016

International Journal of Applied Mathematics and Computer Science

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A single-server queueing system with an infinite buffer is considered. The service of a customer is possible only in the presence of at least one unit of energy, and during the service the number of available units decreases by one. New units of energy arrive in the system at random instants of time if the finite buffer for maintenance of energy is not full. Customers are impatient and leave the system without service after a random amount of waiting time. Such a queueing system describes, e.g., the operation of a sensor node which harvests energy necessary for information transmission from the environment. Aiming to minimize the loss of customers due to their impatience (and maximize the throughput of the system), a new strategy of control by providing service is proposed. This strategy suggests that service temporarily stops if the number of customers or units of energy in the system becomes zero. The server is switched off (is in sleep mode) for some time. This time finishes (the server wakes up) if both the number of customers in the buffer and the number of energy units reach some fixed threshold values or when the number of energy units reaches some threshold value and there are customers in the buffer. Arrival flows of customers and energy units are assumed to be described by an independent Markovian arrival process. The service time has a phase-type distribution. The system behavior is described by a multi-dimensional Markov chain. The generator of this Markov chain is derived. The ergodicity condition is presented. Expressions for key performance measures are given. Numerical results illustrating the dependence of a customer's loss probability on the thresholds defining the discipline of waking up the server are provided. The importance of the account of correlation in arrival processes is numerically illustrated.

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Analysis of an MMAP/PH₁, PH₂/N/∞ queueing system operating in a random environment

Cited by 18 publications

References 21 publications

Bounds on the rate of convergence for one class of inhomogeneous Markovian queueing models with possible batch arrivals and services

Bounds on the rate of convergence for one class of inhomogeneous Markovian queueing models with possible batch arrivals and services

A Queueing System with Heterogeneous Impatient Customers and Consumable Additional Items

Optimization of the service strategy in a queueing system with energy harvesting and customers’ impatience

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