Steady State Analysis of Finite Fluid Flow Models Using Finite QBDs

The Markov modulated fluid model with finite buffer of size β is analyzed using a stochastic discretization yielding a sequence of finite waiting room queueing models with iid amounts of work distributed as exp (nλ). The n-th approximating queue’s system size is bounded at a value qn such that the corresponding expected amount of work qn/(nλ) → β as n → ∞. We demonstrate that as n → ∞, we obtain the exact performance results for the finite buffer fluid model from the processes of work in the system for these queues. The necessary (strong) limit theorems are proven for both transient and steady state results. Algorithms for steady state results are developed fully and illustrated with numerical examples.

[1]  D. Mitra,et al.  Stochastic theory of a data-handling system with multiple sources , 1982, The Bell System Technical Journal.

[2]  E. Coddington,et al.  Theory of Ordinary Differential Equations , 1955 .

[3]  Vaidyanathan Ramaswami,et al.  Fluid Flow Models and Queues—A Connection by Stochastic Coupling , 2003 .

[4]  Ward Whitt,et al.  An Introduction to Stochastic-Process Limits and their Application to Queues , 2002 .

[5]  Guy Latouche,et al.  Matrix-analytic methods for fluid queues with finite buffers , 2006, Perform. Evaluation.

[6]  Vaidyanathan Ramaswami,et al.  Introduction to Matrix Analytic Methods in Stochastic Modeling , 1999, ASA-SIAM Series on Statistics and Applied Mathematics.

[7]  Vaidyanathan Ramaswami,et al.  Matrix analytic methods for stochastic fluid flows , 1999 .

[8]  Marcel F. Neuts,et al.  Matrix-geometric solutions in stochastic models - an algorithmic approach , 1982 .

[9]  A. Skorokhod.  Limit Theorems for Stochastic Processes , 1956 .

[10]  Bo Friis Nielsen,et al.  A computational framework for a quasi birth and death process with a continuous phase variable , 1997 .

[11]  R. Tweedie.  Operator-geometric stationary distributions for markov chains, with application to queueing models , 1982, Advances in Applied Probability.

[12]  Erhan Çinlar,et al.  Introduction to stochastic processes , 1974 .

[13]  Vaidyanathan Ramaswami,et al.  Transient Analysis of Fluid Flow Models via Stochastic Coupling to a Queue , 2004 .

[14]  A. Shiryaev,et al.  Limit Theorems for Stochastic Processes , 1987 .

[15]  Ward Whitt,et al.  Heavy-Traffic Limits for Queues with Many Exponential Servers , 1981, Oper. Res..

[16]  Bruce Hajek,et al.  Birth-and-death processes on the integers with phases and general boundaries , 1982, Journal of Applied Probability.