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 2 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

3 Advanced Queueing Theory Today (lecture 7): Matrix analytical techniques 3 G. Latouche, V Ramaswami. Introduction to Matrix Analytic Methods in Stochastic Modeling, SIAM, Philadelphia, 1999 Tutorial on Matrix analytic methods: M/M/1 queue Quasi birth death process Generalisations

4 M/M/1 queue 4 Poisson arrival process rate, single server, exponential service times, mean 1/ State space S={0,1,2, } transition rates : Global balance Detailed balance Equilibrium distribution

5 0 = 5

6 6

7 Advanced Queueing Theory Today (lecture 7): Matrix analytical techniques 7 G. Latouche, V Ramaswami. Introduction to Matrix Analytic Methods in Stochastic Modeling, SIAM, Philadelphia, 1999 Tutorial on Matrix analytic methods: M/M/1 queue Quasi birth death process Generalisations

8 8 Vector state process: example M/E_k/1 Let service requirement in single server queue be Erlang (k,apple) Augment state description with phase of Erlang distribution State (n,j): n= # customers, j = #remaining phases Transitions (n,j) (n+1,j) arrival (rate apple) (n,j) (n,j-1) completion of phase (j>1) (rate apple) (n,j) (n-1,k) completion in last phase, dept (n>1,j=1) (rate apple) (n,j) (0) completion for n=1, (j=1) (rate apple) (0) (1,k) arrival to empty system (rate apple) Picture Generator in block structure M/Ph/1

9 Phase and level 9

10 10 Quasi-birth-death process (QBD) Q i blocks of size M x M

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12 π i blocks of size M 12

13 Theorem: equilibrium distribution 13

14 14 Stability Behaviour in phase direction x stat distrib over phases downward drift

15 15 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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18 Proof, ctd 18

19 19 Computing R For computation of R, rearrange Note that Q 1 is indeed invertible, since it is a transient generator Fixed point equation solved by successive substitution It can be shown that

20 20 Example: E k /M/1 queue Let service requirement in single server queue be Exp(apple) Let interarrival time be Erlang (k, apple) Augment state description with phase of Erlang distribution State (n,j): n=# customers, j =#remaining phases Transitions (n,1) (n+1,k) arrival (rate apple) (n,j) (n,j-1) completion of phase (j>1) (rate apple) (n,j) (n-1,j) service completion (n>1) (rate apple) Picture Generator in block structure

21 Advanced Queueing Theory Today (lecture 7): Matrix analytical techniques 21 G. Latouche, V Ramaswami. Introduction to Matrix Analytic Methods in Stochastic Modeling, SIAM, Philadelphia, 1999 Tutorial on Matrix analytic methods: M/M/1 queue Quasi birth death process Generalisations

22 Generalisations: different first row 22

23 23

24 24 Generalisations: GI/M/1-type Markov chains Consider GI/M/1 at arrival epochs Interarrival time has general distribution F A with mean 1/ apple Service time exponential with rate apple Probability exactly n customers served during intarr time Probability more than n served during intarr time

25 Generalisations: GI/M/1-type Markov chains 25 Prob n cust served during intarr time Prob more than n served during intarr time Transition probability matrix Equilibrium probabilities Where is unique root in (0,1) of where A has distribution F A

26 26 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

27 27 Generalisations: GI/M/1-type Markov chains Equilibrium distribution Where R is minimal non-negative solution of Computation: truncate And use successive approximation

28 28 Generalisations: M/G/1 type Markov chains Embedding of M/Q/1 at departure epochs gives upper triangular structure for transition matrix

29 29 Generalisations: Level dependent rates For Markov chain of the GI/M/1 type, we may generalise to allow for level dependent matrices, i.e. A i (n) at level n, i=0,1,2,, n=0,1,2,

30 30 References and Exercise 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 E 2 /E 2 /1 queue, obtain explicit expression for R, and give the equilibrium distribution