Stability of a queue with discriminatory random order service discipline and heterogeneous servers

Jeongsim Kim; Bara Kim

doi:10.3934/jimo.2016070

Article Contents

2017, Volume 13, Issue 3: 1237-1254. Doi: 10.3934/jimo.2016070

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Stability of a queue with discriminatory random order service discipline and heterogeneous servers

Jeongsim Kim^1, and
Bara Kim^2, ,

1.
Department of Mathematics Education, Chungbuk National University, 1 Chungdae-ro, Seowon-gu, Cheongju, Chungbuk, 28644, Korea
2.
Department of Mathematics, Korea University, 145 Anam-ro, Seongbuk-gu, Seoul, 02841, Korea

* Corresponding author: Bara Kim
* Corresponding author: Bara Kim

Received: September 2015

Published: September 2017

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Abstract

We consider a queueing system with two classes of customers, two heterogeneous servers, and discriminatory random order service (DROS) discipline. The two servers may have either the same or different DROS weights for each class. Customers of each class arrive according to a Poisson process and the service times of each class of customers are assumed to be exponentially distributed with service rate depending on both the customer's class and the servers. We provide stability and instability conditions for this two-class two-server queue with DROS discipline.

Keywords:

Stability,
discriminatory random order service discipline,
Markov process,
test function.

Mathematics Subject Classification: Primary: 60K25; Secondary: 60J27.

Citation:

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Figure 1. The curve dividing the two regions

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Figure 2. The stable and unstable regions of Example 1 (when $\mu_{11}=0.9$, $\mu_{12}=0.1$, and $\mu_{21}=\mu_{22}=0.5)$

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Figure 3. Simulation results for $N_1(t)$ and $N_2(t)$ in Example 1 (when $\mu_{11}=0.9$, $\mu_{12}=0.1$, $\mu_{21}=\mu_{22}=0.5$, and $\lambda_1=\lambda_2=0.5$)

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Figure 4. The curve dividing the stable and unstable regions

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Figure 5. The stable and unstable regions of Example 2 (when $\mu_{11}=0.9$ and $\mu_{12}=0.1$)

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