Analysis of the M/G/1 queue with exponentially working vacations—a matrix analytic approach

In this paper, an M/G/1 queue with exponentially working vacations is analyzed. This queueing system is modeled as a two-dimensional embedded Markov chain which has an M/G/1-type transition probability matrix. Using the matrix analytic method, we obtain the distribution for the stationary queue length at departure epochs. Then, based on the classical vacation decomposition in the M/G/1 queue, we derive a conditional stochastic decomposition result. The joint distribution for the stationary queue length and service status at the arbitrary epoch is also obtained by analyzing the semi-Markov process. Furthermore, we provide the stationary waiting time and busy period analysis. Finally, several special cases and numerical examples are presented.

[1]  Marcel F. Neuts,et al.  Matrix-Geometric Solutions in Stochastic Models , 1981 .

[2]  Hideaki Takagi,et al.  Queueing analysis: a foundation of performance evaluation , 1993 .

[3]  Yutaka Baba,et al.  Analysis of a GI/M/1 queue with multiple working vacations , 2005, Oper. Res. Lett..

[4]  Robert B. Cooper,et al.  Stochastic Decompositions in the M/G/1 Queue with Generalized Vacations , 1985, Oper. Res..

[5]  Carl M. Harris,et al.  Fundamentals of queueing theory (2nd ed.). , 1985 .

[6]  Hideaki Takagi,et al.  Stochastic Analysis of Computer and Communication Systems , 1990 .

[7]  Naishuo Tian,et al.  The discrete-time GI/Geo/1 queue with working vacations and vacation interruption , 2007, Appl. Math. Comput..

[8]  Naishuo Tian,et al.  Vacation Queueing Models Theory and Applications , 2006 .

[9]  Vincent Hodgson,et al.  The Single Server Queue. , 1972 .

[10]  B. T. Doshi,et al.  Queueing systems with vacations — A survey , 1986, Queueing Syst. Theory Appl..

[11]  Naishuo Tian,et al.  Discrete-time GI/Geo/1 queue with multiple working vacations , 2007, Queueing Syst. Theory Appl..

[12]  Marcel F. Neuts,et al.  Structured Stochastic Matrices of M/G/1 Type and Their Applications , 1989 .

[13]  Leslie D. Servi,et al.  M/M/1 queues with working vacations (M/M/1/WV) , 2002, Perform. Evaluation.

[14]  Hideaki Takagi,et al.  M/G/1 queue with multiple working vacations , 2006, Perform. Evaluation.

[15]  Yutaka Takahashi,et al.  Queueing analysis: A foundation of performance evaluation, volume 1: Vacation and priority systems, Part 1: by H. Takagi. Elsevier Science Publishers, Amsterdam, The Netherlands, April 1991. ISBN: 0-444-88910-8 , 1993 .

[16]  Naishuo Tian,et al.  Vacation Queueing Models: Theory and Applications (International Series in Operations Research & Management Science) , 2006 .

[17]  J. George Shanthikumar,et al.  On Stochastic Decomposition in M/G/1 Type Queues with Generalized Server Vacations , 1988, Oper. Res..

[18]  U. C. Gupta,et al.  On the GI/M/1/N queue with multiple working vacations—analytic analysis and computation , 2007 .

[19]  Carl M. Harris,et al.  Fundamentals of queueing theory , 1975 .

[20]  Naishuo Tian,et al.  Stochastic decompositions in the M/M/1 queue with working vacations , 2007, Oper. Res. Lett..