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

• Autorzy: Chakravarthy S. R.

Identyfikatory

DOI

10.14708/ma.v41i2.389

Warianty tytułu

PL

Analiza systemu kolejkowego z wieloma stanowiskami obsługi dostępnymi na żądanie i fakultatywnym spektrum czynności

• Języki publikacji: EN

Abstrakty

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.

Słowa kluczowe

EN

Markovian arrival process; queueing; waiting time; vacation; secondary batch services; multi-service; phase type distribution; algorithmic probability

PL

Markowski proces zgłoszeń; kolejka; czas oczekiwania; wyłączenia; drugoplanowe zestawy usług; obsługa wielostanowiskowa; mieszanki rozkładów wykładniczego typu; prawdopodobieństwo algorytmiczne

• Wydawca: Polskie Towarzystwo Matematyczne
• Czasopismo: Mathematica Applicanda
• Rocznik: 2013
• Tom: Vol. 41, No. 2
• Strony: 145--169
• Opis fizyczny: Bibliogr. 60 poz., rys.

Twórcy

autor

Chakravarthy S. R.

• Kettering University Department of Industrial and Manufacturing Engineering Flint, MI-48504, USA

Bibliografia

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