Bounds and error bounds for queueing networks

Queueing networks are an important means to model and evaluate a variety of practical systems. Unfortunately, analytic results are often not available. Numerical computation may then have to be employed. Or, system modifications might be suggested to obtain simple bounds or computationally easy approximations. Formal analytic support for the accuaracy or nature of such modifications or approximations then becomes of interest. To this end, the Markov reward approach is surveyed and illustrated as a technique to conclude a priori error bounds as well as to formally prove bounds when comparing two related systems. More precisely, the technique can be applied to: perturbations, finite truncations, infinite approximations, system modifications, or system simplifications (bounds). A general comparison and error bound theorem are provided. The conditions and technical steps are illustrated in detail for a non-product form queueing network subject to breakdowns. This illustration highlights the technical difference with and extension of the stochastic comparison approach. In addition, some practical applications are given which illustrate the various types of applications. Copyright Kluwer Academic Publishers 1998

[1]  G. W. Stewart,et al.  Computable Error Bounds for Aggregated Markov Chains , 1983, JACM.

[2]  Nico M. van Dijk,et al.  Simple Bounds for Queueing Systems with Breakdowns , 1988, Perform. Evaluation.

[3]  Stanley B. Gershwin,et al.  An Efficient Decomposition Method for the Approximate Evaluation of Tandem Queues with Finite Storage Space and Blocking , 1987, Oper. Res..

[4]  Nico M. van Dijk.  A formal proof for the insensitivity of simple bounds for finite multi-server non-exponential tandem queues based on monotonicity results , 1987 .

[5]  Rajan Suri,et al.  A Concept of Monotonicity and Its Characterization for Closed Queueing Networks , 1985, Oper. Res..

[6]  Jan van der Wal,et al.  Simple bounds and monotonicity results for finite multi-server exponential tandem queues , 1989, Queueing Syst. Theory Appl..

[7]  D. Yao,et al.  The effect of increasing service rates in a closed queueing network , 1986 .

[8]  P. Schweitzer.  Perturbation theory and finite Markov chains , 1968 .

[9]  Jiri Kriz,et al.  Throughput bounds for closed queueing networks , 1984, Perform. Evaluation.

[10]  M. Haviv.  Aggregation/disaggregation methods for computing the stationary distribution of a Markov chain , 1987 .

[11]  Nico M. Van Dijk,et al.  The Importance of Bias Terms for Error Bounds and Comparison Results , 1989 .

[12]  Eugene Seneta.  The principle of truncations in applied probability , 1968 .

[13]  J. van der Wal.  Monotonicity of the throughput of a closed exponential queueing network in the number of jobs , 1989 .

[14]  A. Hordijk,et al.  Stochastic inequalities for an overflow model , 1987, Journal of Applied Probability.

[15]  Peter G. Taylor,et al.  Strong Stochastic Bounds for the Stationary Distribution of a Class of Multicomponent Performability Models , 1998, Oper. Res..

[16]  N. L. Lawrie,et al.  Comparison Methods for Queues and Other Stochastic Models , 1984 .

[17]  Boudewijn R. Haverkort,et al.  Product forms for availability models , 1992 .

[18]  Boudewijn R. Haverkort,et al.  Approximate Performability and Dependability Analysis Using Generalized Stochastic Petri Nets , 1993, Perform. Evaluation.

[19]  Eugene Seneta,et al.  Finite approximations to infinite non-negative matrices , 1967, Mathematical Proceedings of the Cambridge Philosophical Society.

[20]  Tayfur Altiok,et al.  Aggregate Modelling of Tandem Queues with Blocking , 1987, Computer Performance and Reliability.

[21]  Nico M. van Dijk.  Queueing networks and product forms - a systems approach , 1993, Wiley-Interscience series in systems and optimization.

[22]  William A. Massey,et al.  Stochastic ordering for Markov processes on partially ordered spaces with applications to queueing networks , 1991 .

[23]  Yung-Li Lily Jow,et al.  An approximation method for tandem queues with blocking , 1988 .

[24]  Nicolas D. Georganas,et al.  Decomposition and aggregation by class in closed queueing networks , 1986, IEEE Transactions on Software Engineering.

[25]  Valerie Isham,et al.  Non‐Negative Matrices and Markov Chains , 1983 .

[26]  J. Keilson,et al.  Monotone matrices and monotone Markov processes , 1977 .

[27]  J. Walrand,et al.  Monotonicity of Throughput in Non-Markovian Networks. , 1989 .

[28]  W. Whitt.  Comparing counting processes and queues , 1981, Advances in Applied Probability.

[29]  Harry G. Perros.  Queueing networks with blocking , 1994 .

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

[31]  Kishor S. Trivedi,et al.  Numerical transient analysis of markov models , 1988, Comput. Oper. Res..

[32]  Nico M. van Dijk,et al.  On product form approximations for communication networks with losses: Error bounds , 1992, Ann. Oper. Res..

[33]  Simonetta Balsamo,et al.  Bounded Aggregation in Markovian Networks , 1987, Computer Performance and Reliability.

[34]  Ward Whitt,et al.  Comparison methods for queues and other stochastic models , 1986 .

[35]  CARL D. MEYER,et al.  The Condition of a Finite Markov Chain and Perturbation Bounds for the Limiting Probabilities , 1980, SIAM J. Algebraic Discret. Methods.

[36]  Alexandre Brandwajn,et al.  An Approximation Method for Tandem Queues with Blocking , 1988, Oper. Res..

[37]  Pierre Semal,et al.  Computable Bounds for Conditional Steady-State Probabilities in Large Markov Chains and Queueing Models , 1986, IEEE J. Sel. Areas Commun..

[38]  Ward Whitt,et al.  Stochastic Comparisons for Non-Markov Processes , 1986, Math. Oper. Res..

[39]  M. Puterman,et al.  Perturbation theory for Markov reward processes with applications to queueing systems , 1988, Advances in Applied Probability.

[40]  Ijbf Ivo Adan,et al.  Monotonicity of the throughput in single server production and assembly networks with respect to the buffer sizes , 1987 .

[41]  J. Kemeny,et al.  Denumerable Markov chains , 1969 .

[42]  Simonetta Balsamo,et al.  Aggregation and Disaggregation in Queueing Networks: The Principle of Product-Form Synthesis , 1983, Computer Performance and Reliability.

[43]  L. V. D. Heyden,et al.  Perturbation bounds for the stationary probabilities of a finite Markov chain , 1984 .

[44]  Nico M. van Dijk,et al.  Simple Bounds and Monotnicity the Call Congestion of Finite Multiserver Delay Systems , 1988, Probability in the Engineering and Informational Sciences.