Random Walk Processes in a Bilevel (M-N)-Policy Queue with Multiple Vacations

Abstract

In this article we introduce a bulk queue with a multiple vacations policy. When the queue is exhausted, the server goes on multiple vacations until the queue replenishes to M or more customers. On his return, if the queue crosses an N as well, the server resumes his service. Otherwise, he waits until the queue to reach or exceed N. We use fluctuation analysis and game-theoretic approach to obtain a closed form functional of the queue length at the beginning of a busy period followed by a Kendall-like formula for the queueing process upon departures. The formation of the queueing process during vacations and waiting periods can be modeled as a multistage game of two players, with the use of past results from stochastic games to arrive at explicit functionals of the queueing process. Then we continue investigating the queueing process, now with continuous time parameter, using time sensitive analysis and semi-regenerative techniques. This enables us to work on potentials of related processes and obtain various performance measures, including the mean buffer load, switchovers rate, and mean stationary service cycle. Special cases demonstrate analytical tractability of the results.

Key Words:

• Multiple vacations
• random walk analysis
• time-sensitive analysis

Additional information

Notes on contributors

Jewgeni H. Dshalalow

Jewgeni H. Dshalalow is a Professor of Mathematical Sciences at Florida Institute of Technology. His areas of interests include stochastic processes, queueing theory, real analysis, stochastic finance, and stochastic games. He is the author of three books and the author and co-author of over 100 research articles. He has been a recipient of numerous awards, including the National Science Foundation (twice), NSA, and the US Army. From 1986 to 2009 he was the chief editor of the Journal of Applied Mathematics and Stochastic Analysis. He currently serves as a regional and associate editor in seven academic journals in mathematics and engineering. Dr. Dshalalow was the major advisor to 16 doctoral students in mathematics and operations research, all of whom graduated from 1993 to 2010.

Ramy Motir

Ramy Motir received his bachelor and master degrees in Electrical Engineering and Computer Engineering, respectively, from King Abdulaziz University. In 2010 he received his doctoral degree in Operations Research from Florida Institute of Technology. Dr. Motir is currently employed by the Saudi Arabian Airline at the Operation Support Department. His area of interest is queuing theory.