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A queueing system with heterogeneous impatient customers and consumable additional items

Baek J., Dudina O., Kim C.

International Journal of Applied Mathematics and Computer Science

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2017

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Vol. 27, no. 2

367--384

EN

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

Dudin A., Lee M. H., Dudin S.

International Journal of Applied Mathematics and Computer Science

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2016

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Vol. 26, no. 2

367--378

EN

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.

3

Analysis of a multi-server queueing model with vacations and optional secondary services

Chakravarthy S. R.

Mathematica Applicanda

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2013

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Vol. 41, No. 2

145--169

EN

In this paper we study a multi-server queueing model in which the customer arrive according to a Markovian arrival process. The customers may require, with a certain probability, an optional secondary service upon completion of a primary service. The secondary services are offered (in batches of varying size) when any of the following conditions holds good: (a) upon completion of a service a free server finds no primary customer waiting in the queue and there is at least one secondary customer (including possibly the primary customer becoming a secondary customer) waiting for service; (b) upon completion of a primary service, the customer requires a secondary service and at that time the number of customers needing a secondary service hits a pre-determined threshold value; (c) a server returning from a vacation finds no primary customer but at least one secondary customer waiting. The servers take vacation when there are no customers (either primary or secondary) waiting to receive service. The model is studied as a QBD-process using matrix-analytic methods and some illustrative examples arediscussed.

PL

Ten artykuł poświęcony jest modelom kolejkowym dla systemów z wieloma serwerami z Markowskim strumieniem zgłoszeń. Klienci żądają, aby obsługa świadczyła również pewne opcjonalne usługi po zakończeniu podstawowego procesu. Te usługi dodatkowe (o różnym zakresie) mają być dostępne i oferowane z pewnym prawdopodobieństwem, gdy którykolwiek z następujących warunków jest spełniony: (a) po zakończeniu obsługi na darmowy, podstawowy, serwis nie czeka klient w kolejce i jest co najmniej jeden chętny klient na serwis wtórny (tym chętnym prawdopodobnie jest klientem, który właśnie otrzymał podstawową usługę), (b) po zakończeniu podstawowego serwisu, klient wymaga dodatkowego serwisu i w tym czasie liczba klientów, którzy reflektują na tę dodatkową usługę przekroczy wcześniej ustaloną wartość progową; (c) serwer który wznawia obsługę po przerwie nie ma klientów na podstawową usługę, ale przynajmniej jeden klient czeka na dodatkowy serwis. Serwery mogą zostać wyłączone na pewien czas, gdy nie ma klientów (podstawowych lub chętnych na serwis dodatkowy) czekających na obsługę. Model jest badane jako uogólniony proces urodzin i śmierci (quasi-birth-death-matrix-process) analizowany analitycznie. Podane są przykłady ilustrujące zastosowane podejście.