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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

• 2 Advanced Queueing Theory Today (lecture 1): queue length based on Burke Nelson, sec 10.1-10.3.3, Kelly, sec 1.1-2.2 Continuous time Markov chain Birth-death process Example: M/M/1 queue Birth-death process: equilibrium distribution Reversibility, stationarity Time reversed process Truncation of reversible processes Example: two M/M/1 queues Output M/M/1 queue Tandem network of M/M/1 queues Sojourn time in a tandem network of M/M/1 queues

• 3 Continuous time Markov chain

stochastic process X=(X(t), t0) evolution random variable

countable or finite state space S={0,1,2,}

Markov property

Markov chain: Stochastic process satisfying Markov property

Transition probability time homogeneous

Chapman-Kolmogorov equations

• 4 Continuous time Markov chain

Chapman-Kolmogorov equations

transition rates or jump rates

Kolmogorov forward equations: (REGULAR)

• 5 Continuous time Markov chain

Assume ergodic and regular

global balance equations (equilibrium equations)

is invariant measure (stationary measure)

solution that can be normalised is stationary distribution

if stationary distribution exists, then it is unique and is limiting distribution

• 6

Advanced Queueing Theory Today (lecture 1): Continuous time Markov chain Birth-death process Example: M/M/1 queue Birth-death process: equilibrium distribution Reversibility, stationarity Time reversed process Truncation of reversible processes Example: two M/M/1 queues Output M/M/1 queue Tandem network of M/M/1 queues Sojourn time in a tandem network of M/M/1 queues

• 7

Birth-death process

State space Markov chain, transition rates

Kolmogorov forward equations

Global balance equations

• 8

Advanced Queueing Theory Today (lecture 1): Continuous time Markov chain Birth-death process Example: M/M/1 queue Birth-death process: equilibrium distribution Reversibility, stationarity Time reversed process Truncation of reversible processes Example: two M/M/1 queues Output M/M/1 queue Tandem network of M/M/1 queues Sojourn time in a tandem network of M/M/1 queues

• 9 M/M/1 queue

Poisson arrival process rate , single server, exponential service times, mean 1/

State space S={0,1,2,} Markov chain? Assume initially empty: P(X(0)=0)=1,

Transition rates :

• 10 M/M/1 queue

Kolmogorov forward equations, j>0

Global balance equations, j>0

• 11

M/M/1 queue

j j+1

Equilibrium distribution: 0, then the resulting Markov process is reversible in equilibrium and has equilibrium distribution B is the normalizing constant.

If c=0 then the reversible Markov process is truncated to A and the resulting Markov process is reversible with equilibrium distribution

Truncation of reversible processes

A

S\A

• 25

Advanced Queueing Theory Today (lecture 1): Continuous time Markov chain Birth-death process Example: M/M/1 queue Birth-death process: equilibrium distribution Reversibility, stationarity Time reversed process Truncation of reversible processes Example: two M/M/1 queues Output M/M/1 queue Tandem network of M/M/1 queues Sojourn time in a tandem network of M/M/1 queues

• 26 Example: two M/M/1 queues

Consider two M/M/1 queues, queue i with Poisson arrival process rate i, service rate i

Independence:

Now introduce a common capacity restriction

Queues no longer independent, but

( j1, j2 ) =i=1

2

(1 i)i ji , , j1, j2 0 ,i = /i

• 27

Advanced Queueing Theory Today (lecture 1): Continuous time Markov chain Birth-death process Example: M/M/1 queue Birth-death process: equilibrium distribution Reversibility, stationarity Time reversed process Truncation of reversible processes Example: two M/M/1 queues Output M/M/1 queue Tandem network of M/M/1 queues Sojourn time in a tandem network of M/M/1 queues

• 28 Output M/M/1 queue: Burkes theorem

M/M/1 queue, Poisson( ) arrivals, exponential( ) service

X(t) number of customers in M/M/1 queue:

in equilibrium reversible Markov process.

Forward process: upward jumps Poisson ( )

Reversed process X(-t): upward jumps Poisson ( )

= downward jump of forward process

Downward jump process of X(t) Poisson ( ) process

• 29 Output M/M/1 queue (2)

Let t0 fixed. Arrival process Poisson, thus arrival process after t0 independent of number in queue at t0.

For reversed process X(-t): arrival process after t0 independent of number in queue at t0

Reversibility: joint distribution departure process up to t0 and number in queue at t0 for X(t) have same distribution as arrival process to X(-t) up to t0 and number in queue at t0.

Burkes theorem: In equilibrium the departure process from an M/M/1 queue is a Poisson process, and the number in the queue at time t0 is independent of the departure process prior to t0

Holds for each reversible Markov process with Poisson arrivals as long as an arrival causes the process to change state

• 30

Advanced Queueing Theory Today (lecture 1): Continuous time Markov chain Birth-death process Example: M/M/1 queue Birth-death process: equilibrium distribution Reversibility, stationarity Time reversed process Truncation of reversible processes Example: two M/M/1 queues Output M/M/1 queue Tandem network of M/M/1 queues Sojourn time in a tandem network of M/M/1 queues

• 31 Tandem network of M/M/1 queues

M/M/1 queue, Poisson( ) arrivals, exponential( ) service Equilibrium distribution Tandem of J M/M/1 queues, exp( i) service queue i Xi(t) number in queue i at time t Queue 1 in isolation: M/M/1 queue. Departure process queue 1 Poisson,

thus queue 2 in isolation: M/M/1 queue State X1(t0) independent departure process prior to t0,

but this determines (X2(t0),, XJ(t0)), hence X1(t0) independent (X2(t0),, XJ(t0)). Similar Xj(t0) independent (Xj+1(t0),, XJ(t0)). Thus X1(t0), X2(t0),, XJ(t0) mutually independent, and

( j1,..., jJ ) =i=1

J

(1 i)i ji , ,i = /i

• 32

Example: feed forward network of M/M/1 queues

( j1,..., j5 ) =i=1

5

(1 i)i ji , ,i = i /i

• 33

Advanced Queueing Theory Today (lecture 1): Continuous time Markov chain Birth-death process Example: M/M/1 queue Birth-death process: equilibrium distribution Reversibility, stationarity Time reversed process Truncation of reversible processes Example: two M/M/1 queues Output M/M/1 queue Tandem network of M/M/1 queues Sojourn time in a tandem network of M/M/1 queues

• 34 Waiting time M/M/1 queue (1)

Consider simple queue FCFS discipline W : waiting time typical customer in M/M/1

(excludes service time) N customers present upon arrival Sr (residual) service time of customers present PASTA

Voor j=0,1,2,

• 35 Waiting time M/M/1 queue (2)

Thus

is exponential ( )

• 36

Recall: In equilibrium the departure process from an M/M/1 queue is a Poisson process, and the number in the queue at time t0 is independent of the departure process prior to t0

Theorem: If service discipline at each queue in tandem of J M/M/1 queues is FCFS, then in equilibrium the waiting times of a customer at each of the J queues are independent

Proof: Kelly p. 38

Tandem M/M/s queues: overtaking

Sojourn time tandem M/M/1 queues

• 37

Summary / next:

Detailed balance or reversibility and their consequences Birth-death process M/M/1 queue Truncation Kolmogorovs criteria Tandem network Sojourn time in tandem network

Next on AQT.. Quasi-reversibility and partial balance and their consequences Network of queues Blocking protocols Customer types Queue disciplines

( j1,..., jJ ) =i=1

J

(1 i)i ji , ,i = i /i

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