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

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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

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0 =

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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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Phase and level

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Quasi-birth-death process (QBD)

Qi blocks of size M x M

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πi blocks of size M

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Theorem: equilibrium distribution

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Stability

Behaviour in phase direction

x stat distrib over phases

downward drift

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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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Proof, ctd

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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

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Generalisations: different first row

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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