Research Article
A Product-Form Solution for a Two-Class\(Geo^{Geo}/D/1\)Queue with Random Routing and Randomly Alternating Service
@INPROCEEDINGS{10.1007/978-3-031-31234-2_6, author={Arnaud Devos and Michiel De Muynck and Herwig Bruneel and Joris Walraevens}, title={A Product-Form Solution for a Two-Class\textbackslash(Geo\^{}\{Geo\}/D/1\textbackslash)Queue with Random Routing and Randomly Alternating Service}, proceedings={Performance Evaluation Methodologies and Tools. 15th EAI International Conference, VALUETOOLS 2022, Virtual Event, November 2022, Proceedings}, proceedings_a={VALUETOOLS}, year={2023}, month={5}, keywords={Two-class queueing model Non-work-conserving Product-form solution}, doi={10.1007/978-3-031-31234-2_6} }- Arnaud Devos
Michiel De Muynck
Herwig Bruneel
Joris Walraevens
Year: 2023
A Product-Form Solution for a Two-Class\(Geo^{Geo}/D/1\)Queue with Random Routing and Randomly Alternating Service
VALUETOOLS
Springer
DOI: 10.1007/978-3-031-31234-2_6
Abstract
We analyze a discrete-time queueing system, consisting of two queues and a single server. The server randomly distributes its time between the two queues. Service times of any customer of either queue are deterministically equal to 1 time slot. In general, the joint analysis of such a two-queue system turns out to be very hard. In this paper, we assume that the total number of arrivals into the system constitutes a series of i.i.d. random variables with common geometric distribution. Each arriving customer is routed probabilistically to a queue. By means of a state-of-the-art approach, we obtain a closed-form expression of the steady-state joint PGF of the number of customers present (“system contents”) in both queues, at the beginning of a random slot. We find that the joint PGF is of product form, which proves that the system contents in both queues are independent. We provide an additional intuitive stochastic explanation for this remarkable result. We discuss several model extensions using the stochastic analysis.
