Stationary Waiting Times in Simple Fork-and-Join Queues with Finite Buffers and Communication Blocking

In this study, we consider stationary waiting times in a simple fork-and-join type queue which consists of three single-server machines, Machine 1, Machine 2, and Assembly Machine. We assume that the queue has a renewal arrival process and that independent service times at each node are either deterministic or non-overlapping. We also assume that the Machines 1 and 2 have an infinite buffer capacity whereas the Assembly Machine has two finite buffers, one for each machine. Services at each machine are given by FIFO service discipline and a communication blocking policy. We derive the explicit expressions for stationary waiting times at all nodes as a function of finite buffer capacities by using (max,+)-algebra. Various characteristics of stationary waiting times such as mean, higher moments, and tail probability can be computed from these expressions.

[1]  Hayriye Ayhan,et al.  Laplace Transform and Moments of Waiting Times in Poisson Driven (max,+) Linear Systems , 2001, Queueing Syst. Theory Appl..

[2]  Ronald W. Wolff,et al.  Bounds for Different Arrangements of Tandem Queues with Nonoverlapping Service Times , 1993 .

[3]  L. Flatto,et al.  Two parallel queues created by arrivals with two demands. II , 1984 .

[4]  Armand M. Makowski,et al.  Simple computable bounds for the fork-join queue , 1985 .

[5]  B. Heidergott Max-plus linear stochastic systems and perturbation analysis , 2006 .

[6]  Geert Jan Olsder,et al.  Synchronization and Linearity: An Algebra for Discrete Event Systems , 1994 .

[7]  François Baccelli,et al.  Expansions for steady-state characteristics of (max, +)-linear systems , 1998 .

[8]  N. R. Srinivasa Raghavan,et al.  Generalized queueing network analysis of integrated supply chains , 2001 .

[9]  F. Baccelli,et al.  The fork-join queue and related systems with synchronization constraints: stochastic ordering and computable bounds , 1989, Advances in Applied Probability.

[10]  Simonetta Balsamo,et al.  Bound Performance Models of Heterogeneous Parallel Processing Systems , 1998, IEEE Trans. Parallel Distributed Syst..

[11]  Seo Dong-Won Application of (Max, +)-algebra to the Waiting Times in Deterministic 3-node Tandem Queues with Blocking , 2005 .

[12]  Asser N. Tantawi,et al.  Approximate Analysis of Fork/Join Synchronization in Parallel Queues , 1988, IEEE Trans. Computers.

[13]  Ward Whitt,et al.  The Best Order for Queues in Series , 1985 .

[14]  Hayriye Ayhan,et al.  Tail probability of transient and stationary waiting times in (max, +)-linear systems , 2002, IEEE Trans. Autom. Control..

[15]  Armand M. Makowski,et al.  Interpolation Approximations for Symmetric Fork-Join Queues , 1994, Perform. Evaluation.

[16]  L. Flatto,et al.  Erratum: Two Parallel Queues Created by Arrivals with Two Demands I , 1985 .

[17]  Volker Schmidt,et al.  Transient and stationary waiting times in (max,+)-linear systems with Poisson input , 1997, Queueing Syst. Theory Appl..

[18]  F. Baccelli,et al.  Taylor series expansions for Poisson-driven $(\max,+$)-linear systems , 1996 .