Transient analysis of a single server queueing system with infinite buffer

As a typical single server queueing system, computer integrated manufacturing system (CIMS) has been widely used in the field of intelligent manufacturing. However, how to derive its instantaneous index is still an important issue. This paper investigates the transient behavior of the CIMS with spectral method. By constructing an asymptotic system and analyzing the spectral distribution, we derive the explicit transient solution of the asymptotic system. Trotter–Kato theorem is used to prove that the transient solution of the CIMS is just the limitation of explicit transient solution of the asymptotic system. At the end of the paper, numerical examples are shown to illustrate the effectiveness of the proposed approximation.

[1]  Arka P. Ghosh,et al.  Optimal buffer size and dynamic rate control for a queueing system with impatient customers in heavy traffic , 2010 .

[2]  Gorjan Alagic,et al.  #p , 2019, Quantum information & computation.

[3]  P. Alam.  ‘A’ , 2021, Composites Engineering: An A–Z Guide.

[4]  Rakesh Kumar,et al.  Transient and steady-state analysis of a queuing system having customers' impatience with threshold , 2019, RAIRO Oper. Res..

[5]  B. Goyal.  Queuing Theory , 1999 .

[6]  C. Loan,et al.  Nineteen Dubious Ways to Compute the Exponential of a Matrix , 1978 .

[7]  Souvik Ghosh,et al.  A simple analysis of system characteristics in the batch service queue with infinite-buffer and Markovian service process using the roots method: GI/C-MSP(a,b)/1/∞ , 2016, RAIRO Oper. Res..

[8]  Kazufumi Ito,et al.  The Trotter-Kato theorem and approximation of PDEs , 1998, Math. Comput..

[9]  Wojciech M. Kempa,et al.  Transient solution for the queue-size distribution in a finite-buffer model with general independent input stream and single working vacation policy , 2018, Applied Mathematical Modelling.

[10]  Jens Braband,et al.  Waiting time distributions for closed M/M/N processor sharing queues , 1995, Queueing Syst. Theory Appl..

[11]  Weiwei Hu,et al.  Analysis and approximation of a reliable model , 2013 .

[12]  Gerardo Rubino,et al.  Transient analysis of the M/M/1 queue , 1993, Advances in Applied Probability.

[13]  Zhan-tao Li,et al.  Performance analysis of open general queuing networks with blocking and feedback , 2017 .

[14]  Ricardo Femat,et al.  On dynamical behaviour of two-dimensional biological reactors , 2012, Int. J. Syst. Sci..

[15]  Amnon Pazy,et al.  Semigroups of Linear Operators and Applications to Partial Differential Equations , 1992, Applied Mathematical Sciences.

[16]  D. Mitra.  Stochastic theory of a fluid model of producers and consumers coupled by a buffer , 1988, Advances in Applied Probability.

[18]  Reuven R. Levary.  Computer-integrated manufacturing: a complex information system , 1996 .

[19]  Prashant M. Ambad,et al.  Industry 4.0 – A Glimpse , 2018 .

[20]  P. Alam.  ‘E’ , 2021, Composites Engineering: An A–Z Guide.

[21]  Bruno Sericola,et al.  Transient Analysis of Stochastic Fluid Models , 1998, Perform. Evaluation.

[22]  Bruno Sericola,et al.  Exact transient solution of an M/M/1 driven fluid queue , 2005, Int. J. Comput. Math..

[23]  Benny Tjahjono,et al.  What does Industry 4.0 mean to Supply Chain , 2017 .

[24]  S. K. Samanta,et al.  Analyzing an infinite buffer batch arrival and batch service queue under batch-size-dependent service policy , 2016 .

[25]  F. Chatelin.  Spectral approximation of linear operators , 2011 .

[26]  J. W. Cohen.  Superimposed renewal processes and storage with gradual input , 1974 .

[27]  Ning Zhao,et al.  Analysis of dual tandem queues with a finite buffer capacity and non-overlapping service times and subject to breakdowns , 2015 .

[28]  Alan C. Krinik.  Taylor series solution of the M/M/1 queueing system , 1992 .

[30]  U. C. Gupta,et al.  Analysis of an infinite-buffer batch-size-dependent service queue with Markovian arrival process , 2019, Ann. Oper. Res..

[31]  Guido Lemos de Souza Filho,et al.  Multi-objective performance improvements of general finite single-server queueing networks , 2018, J. Heuristics.

[32]  Andrzej Chydzinski,et al.  The Single-Server Queue with the Dropping Function and Infinite Buffer , 2018, Mathematical Problems in Engineering.

[33]  N. Barbot,et al.  Stationary solution to the fluid queue fed by an M/M/1 queue , 2002, Journal of Applied Probability.

[35]  Bruno Sericola.  A Finite Buffer Fluid Queue Driven by a Markovian Queue , 2001, Queueing Syst. Theory Appl..

[36]  F. Huang.  Strong Asymptotic Stability of Linear Dynamical Systems in Banach Spaces , 1993 .

[37]  G. Reuter,et al.  Spectral theory for the differential equations of simple birth and death processes , 1954, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[38]  Philip J. Fleming,et al.  Interpolation Approximations of Sojourn Time Distributions , 1991, Oper. Res..