Advanced Queueing Theory

Transcription

1 Advanced Queueing Theory 1 Networks of queues (reversibility, output theorem, tandem networks, partial balance, product-form distribution, blocking, insensitivity, BCMP networks, mean-value analysis, Norton's theorem, sojourn times) Analytical-numerical techniques (matrix-analytical methods, compensation method, error bound method, approximate decomposition method) Polling systems (cycle times, queue lengths, waiting times, conservation laws, service policies, visit orders) Richard J. Boucherie department of Applied Mathematics University of Twente Queueing Theory/AQT.html

2 Advanced Queueing Theory Today (lecture 6): Compensation approach 2 IJBF Adan, J. Wessels, and WHM Zijm. Analysis of the symmetric shortest queue problem. Stochastic Models, 6: , OJ Boxma, GJ van Houtum. The compensation approach applied to a 2x2 switch, Prob. Inf Sci, 7: , Model Global balance equations as difference equations The boundary Shortest queue problem

3 Model 3 Consider Markov chain with transition rates: Global balance equations

4 Advanced Queueing Theory Today (lecture 6): Compensation approach 4 IJBF Adan, J. Wessels, and WHM Zijm. Analysis of the symmetric shortest queue problem. Stochastic Models, 6: , OJ Boxma, GJ van Houtum. The compensation approach applied to a 2x2 switch, Prob. Inf Sci, 7: , Model Global balance equations as difference equations The boundary Shortest queue problem

5 Global balance as difference equations 5 Consider Markov chain with transition rates: Global balance equations Difference equations Solution

6 Global balance as difference equations 6 Consider Markov chain with transition rates: Global balance equations Difference equations Solution

7 Global balance as difference equations 7 Quadratic form: on curve through (1,1), shape depending on drift

8 Global balance as difference equations 8 Consider curve C with all solutions of α m β n {q 10 + q 11 + q 01 + q 11 + q 10 + q q q 1 1 } = α m 1 β n q 10 + α m 1 β n 1 q 11 + α m β n 1 q 01 + α m +1 β n 1 q 11 +α m +1 β n q 10 + α m +1 β n +1 q α m β n +1 q α m 1 β n +1 q 1 1 Theorem: Let p(m,n) be an invariant measure. Then there exists a unique Borel probability measure ζ on (0, ) 2 such that p(m,n) = α m β n dζ(α,β) C

9 Advanced Queueing Theory Today (lecture 6): Compensation approach 9 IJBF Adan, J. Wessels, and WHM Zijm. Analysis of the symmetric shortest queue problem. Stochastic Models, 6: , OJ Boxma, GJ van Houtum. The compensation approach applied to a 2x2 switch, Prob. Inf Sci, 7: , Model Global balance equations as difference equations The boundary Shortest queue problem

10 Global balance as difference equations 10 Consider Markov chain with transition rates: Global balance equations Difference equations Solution

11 Influence of the boundary 11 Jackson network Horizontal boundary: linear equation Vertical boundary: linear equation Unique solution that satisfies both boundary equations

12 Influence of the boundary 12 General setting: For each stable geometric solution of interior rates, there exist boundary rates such that geometric solution is equilibrium distribution. Boundary rates not unique (determined up to linear equation at each boundary separately) Given selection for boundary rates: product form determined Rates for sum of product forms? When rates are prescribed?

13 Advanced Queueing Theory Today (lecture 6): Compensation approach 13 IJBF Adan, J. Wessels, and WHM Zijm. Analysis of the symmetric shortest queue problem. Stochastic Models, 6: , OJ Boxma, GJ van Houtum. The compensation approach applied to a 2x2 switch, Prob. Inf Sci, 7: , Model Global balance equations as difference equations The boundary Shortest queue problem

14 Exercise on compensation approach and boundary modification Consider the Markov chain with transition rates as depicted. A) Let h(-1,1)=q(-1,1), h(0,1)=q(0,1), h(1,1)=q(1,1), v(1,1)=q(1,1), v(1,0)=q(1,0), v(1,-1)=q(1,-1), r(1,1)=q(1,1).show that for each stable geometric solution of the global balance equations in the interior, boundary rates can be found such that this geometric solution is the unique equilibrium distribution. B) Investigate whether or not boundary rates can be established such that the equilibrium distribution is a linear combination of stable geometric solutions of the interior global balance equations. 2. Consider the shortest queue problem. Obtain an expression for the coefficient of correlation of the two queue lengths.