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1 Advanced Queueing Theory • 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 http://wwwhome.math.utwente.nl/~boucherierj/onderwijs/Advanced Queueing Theory/AQT.html
• Doe na de m/m/1 eerst even de M/E_r/1 expliciet uit notes
• Laat dan expliciet zien dat generator een blok structuur heeft
• Ga dan pas naar QBD
2
3 Advanced Queueing Theory Today (lecture 7): Matrix analytical techniques
• G. Latouche, V Ramaswami. Introduction to Matrix Analytic Methods in Stochastic Modeling, SIAM, Philadelphia, 1999
• Tutorial on Matrix analytic methods: http://www-net.cs.umass.edu/pe2002/papers/nelson.pdf
• M/M/1 queue • Quasi birth death process • Generalisations
4 M/M/1 queue
• Poisson arrival process rate λ, single server, exponential service times, mean 1/μ
• State space S={0,1,2,…} • transition rates :
• Global balance
• Detailed balance
• Equilibrium distribution
7 Advanced Queueing Theory Today (lecture 7): Matrix analytical techniques
• G. Latouche, V Ramaswami. Introduction to Matrix Analytic Methods in Stochastic Modeling, SIAM, Philadelphia, 1999
• Tutorial on Matrix analytic methods: http://www-net.cs.umass.edu/pe2002/papers/nelson.pdf
• M/M/1 queue • Quasi birth death process • Generalisations
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Vector state process: example M/E_k/1 • Let service requirement in single server queue be Erlang (k,) • Augment state description with phase of Erlang distribution • State (n,j): n= # customers, j = #remaining phases • Transitions
(n,j)(n+1,j) arrival (rate ) (n,j)(n,j-1) completion of phase (j>1) (rate ) (n,j)(n-1,k) completion in last phase, dept (n>1,j=1) (rate ) (n,j)(0) completion for n=1, (j=1) (rate ) (0)(1,k) arrival to empty system (rate )
• Picture • Generator in block structure • M/Ph/1
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QBD: Proof of equilibrium distribution
For the discrete time case, R(i,j) is the expected number of visits to phase j in level 1 before absorption in level 0 for the process that starts at level 0 in phase i
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Computing R • For computation of R, rearrange
• Note that Q1 is indeed invertible, since it is a transient generator
• Fixed point equation solved by successive substitution
• It can be shown that
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Example: Ek/M/1 queue
• Let service requirement in single server queue be Exp() • Let interarrival time be Erlang (k, ) • Augment state description with phase of Erlang distribution • State (n,j): n=# customers, j =#remaining phases • Transitions
(n,1)(n+1,k) arrival (rate ) (n,j)(n,j-1) completion of phase (j>1) (rate ) (n,j)(n-1,j) service completion (n>1) (rate )
• Picture • Generator in block structure
21 Advanced Queueing Theory Today (lecture 7): Matrix analytical techniques
• G. Latouche, V Ramaswami. Introduction to Matrix Analytic Methods in Stochastic Modeling, SIAM, Philadelphia, 1999
• Tutorial on Matrix analytic methods: http://www-net.cs.umass.edu/pe2002/papers/nelson.pdf
• M/M/1 queue • Quasi birth death process • Generalisations
24
Generalisations: GI/M/1-type Markov chains
• Consider GI/M/1 at arrival epochs • Interarrival time has general distribution FA with mean 1/ Service time exponential with rate
• Probability exactly n customers served during intarr time
• Probability more than n served during intarr time
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Generalisations: GI/M/1-type Markov chains
• Prob n cust served during intarr time
• Prob more than n served during intarr time
• Transition probability matrix
• Equilibrium probabilities
• Where σ is unique root in (0,1) of where A has distribution FA
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Generalisations: GI/M/1-type Markov chains
• Markov chain with transition matrix (suitably ordered states) of the form
is called Markov chain of the GI/M/1 type
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Generalisations: GI/M/1-type Markov chains
• Equilibrium distribution
• Where R is minimal non-negative solution of
• Computation: truncate
• And use successive approximation
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Generalisations: M/G/1 type Markov chains
• Embedding of M/Q/1 at departure epochs gives upper triangular structure for transition matrix
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Generalisations: Level dependent rates
• For Markov chain of the GI/M/1 type, we may generalise to allow for level dependent matrices, i.e. Ai(n) at level n, i=0,1,2,…, n=0,1,2,…
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References and Exercise
• http://www.ms.unimelb.edu.au/~pgt/Stochworkshop2004.pdf • http://www.ms.unimelb.edu.au/~pgt/Stochworkshop2004-2.pdf
• Exercise: Consider the Ph/Ph/1 queue. Formulate as Matrix Analytic queue (i.e. specify the transition matrix, and the blocks in that matrix). For the E2/E2/1 queue, obtain explicit expression for R, and give the equilibrium distribution
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