A recursive algorithm for generating the transition matrices of multistation multiserver exponential reliable queueing networks

Abstract This paper is concerned with reliable multistation series queueing networks. Items arrive at the first station according to a Poisson distribution and an operation is performed on each item by a server at each station. Every station is allowed to have more than one server with the same characteristics. The processing times at each station are exponentially distributed. Buffers of nonidentical finite capacities are allowed between successive stations. The structure of the transition matrices of these specific type of queueing networks is examined and a recursive algorithm is developed for generating them. The transition matrices are block-structured and very sparse. By applying the proposed algorithm the transition matrix of a K-station network can be created for any K. This process allows one to obtain the exact solution of the large sparse linear system by the use of the Gauss–Seidel method. From the solution of the linear system the throughput and other performance measures can be calculated. Scope and purpose The exact analysis of queueing networks with multiple servers at each workstation and finite capacities of the intermediate queues is extremely difficult as for even the case of exponential operation (service or processing) times the Markovian chain that models the system consists of a huge number of states which grows exponentially with the number of stations, the number of servers at each station and the queue capacity of each intermediate queue of the resulting system. The scope and purpose of the present paper is to analyze and provide a recursive algorithm for generating the transition matrices of multistation multiserver exponential reliable queueing networks. By applying the proposed algorithm one may create the transition matrix of a K-station queueing network for any K. This process allows one to obtain the exact solution of the resulting large sparse linear system by the use of the Gauss–Seidel method. From the solution of the linear system the throughput and other performance measures of the system can be obtained.

[1]  Nico M. van Dijk,et al.  A Simple Throughput Bound for Large Closed Queueing Networks with Finite Capacities , 1989, Perform. Evaluation.

[2]  Harry G. Perros,et al.  Approximate analysis of arbitrary configurations of open queueing networks with blocking , 1987 .

[3]  Yves Dallery,et al.  Manufacturing flow line systems: a review of models and analytical results , 1992, Queueing Syst. Theory Appl..

[4]  Yves Dallery,et al.  On Decomposition Methods for Tandem Queueing Networks with Blocking , 1993, Oper. Res..

[5]  Yves Dallery,et al.  An Analytical Method for Performance Evaluation of Kanban Controlled Production Systems , 1996, Oper. Res..

[6]  Harry G. Perros.  Queueing Networks with Blocking: Exact and Approximate Solutions , 1994 .

[7]  Sushant Jain,et al.  Open finite queueing networks with M/M/C/K parallel servers , 1994, Comput. Oper. Res..

[8]  Stanley B. Gershwin,et al.  Throughput estimation in cyclic queueing networks with blocking , 1998, Ann. Oper. Res..

[9]  Li Zhuang,et al.  Approximate mean value performance analysis of cyclic queueing networks with production blocking , 1994, Queueing Syst. Theory Appl..

[10]  Demetres D. Kouvatsos,et al.  Arbitrary open queueing networks with server vacation periods and blocking , 1998, Ann. Oper. Res..

[11]  Victor L. Wallace,et al.  Algebraic Techniques for Numerical Solution of Queueing Networks , 1974 .

[12]  Donald F. Towsley,et al.  Properties of fork/join queueing networks with blocking under various operating mechanisms , 1997, IEEE Trans. Robotics Autom..

[13]  Yves Dallery,et al.  Approximate Analysis of General Open Queuing Networks with Restricted Capacity , 1990, Perform. Evaluation.

[14]  Attahiru Sule Alfa,et al.  A Simple And Quick Approximation Algorithm For Tandem, Split And Merge Queueing Networks With Blocking , 1994 .

[15]  Hirotaka Sakasegawa,et al.  On the equivalence of three types of blocking in non-Markovian tandem queues , 1994, Queueing Syst. Theory Appl..

[16]  John A. Buzacott,et al.  Stochastic models of manufacturing systems , 1993 .

[17]  Chii-Lian Lin,et al.  An efficient two-phase approximation method for exponential tandem queueing systems with blocking , 1995, Comput. Oper. Res..

[18]  H. T. Papadopoulos,et al.  Throughput rate of multistation reliable production lines with inter station buffers. (I) Exponential case , 1989 .

[19]  Frederick S. Hillier,et al.  On the optimal design of tandem queueing systems with finite buffers , 1995, Queueing Syst. Theory Appl..

[20]  Alexandre Brandwajn,et al.  A Note on Approximate Iterative Solution of Open Tandem Networks with Blocking , 1989, Perform. Evaluation.

[21]  H. T. Papadopoulos,et al.  A recursive algorithm for generating the transition matrices of multistation series production lines , 1989 .

[22]  J. MacGregor Smith,et al.  An algorithm for sub-optimal routeing in series-parallel queueing networks , 1997 .

[23]  Tayfur Altiok,et al.  Performance analysis of manufacturing systems , 1996 .

[24]  Simonetta Balsamo,et al.  A survey of product form queueing networks with blocking and their equivalences , 1994, Ann. Oper. Res..

[25]  Stefan Helber,et al.  Decomposition of unreliable assembly/disassembly networks with limited buffer capacity and random processing times , 1998, Eur. J. Oper. Res..

[26]  Stanley B. Gershwin,et al.  Manufacturing Systems Engineering , 1993 .

[27]  Surendra M. Gupta,et al.  Tandem manufacturing systems with machine vacations , 1998 .

[28]  Ben Atkinson,et al.  Queueing theory in manufacturing systems analysis and design , 1993 .

[29]  Raif O. Onvural,et al.  Survey of closed queueing networks with blocking , 1990, CSUR.

[30]  Jürgen Becker,et al.  Analysis of queueing networks with blocking using a new aggregation technique , 1998, Ann. Oper. Res..

[31]  Surendra M. Gupta,et al.  Manufacturing systems with machine vacations, arbitrary topology and finite buffers , 1999 .

[32]  Harry G. Perros,et al.  Approximate Analysis of Product-Form Type Queueing Networks with Blocking and Deadlock , 1988, Perform. Evaluation.

[33]  E. Muth.  The Reversibility Property of Production Lines , 1979 .

[34]  Richard V. Evans.  Geometric Distribution in Some Two-Dimensional Queuing Systems , 1967, Oper. Res..

[35]  Mostafa H. Ammar,et al.  Equivalence Relations in Queueing Models of Fork/Join Networks with Blocking , 1989, Perform. Evaluation.

[36]  Surendra M. Gupta,et al.  A methodology for analyzing finite buffer tandem manufacturing systems with N -policy , 1998 .

[37]  Irfan-Ullah Awan,et al.  MEM for arbitrary closed queueing networks with RS-blocking and multiple job classes , 1998, Ann. Oper. Res..

[38]  Frederick S. Hillier,et al.  Finite Queues in Series with Exponential or Erlang Service Times - A Numerical Approach , 1966, Oper. Res..

[39]  Gabriel R. Bitran,et al.  A review of open queueing network models of manufacturing systems , 1992, Queueing Syst. Theory Appl..

[40]  Yves Dallery,et al.  Performance evaluation of open queueing networks with arbitrary configuration and finite buffers , 1998, Ann. Oper. Res..

[41]  H. T. Papadopoulos,et al.  The throughput rate of multistation reliable production lines , 1993 .

[42]  Boudewijn R. Haverkort,et al.  Approximate analysis of networks of PH|PH|1|K queues with customer losses: Test results , 1998, Ann. Oper. Res..

[43]  Tayfur Altiok,et al.  Approximate Analysis of Queues in Series with Phase-Type Service Times and Blocking , 1989, Oper. Res..

[44]  Rajeev Agrawal,et al.  Cyclic networks with general blocking and starvation , 1994, Queueing Syst. Theory Appl..