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Network of Quasi-reversible queues

Networks of queues

- 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

Richard J. Boucherie

Stochastic Operations Research

department of Applied Mathematics

University of Twente

Networks of Queues: lecture 5

Nelson, sec 10.3.5-10.6.6

- Last time on NoQ …
- Kelly-Whittle network
- Partial balance
- Quasi reversibility

- PASTA / MUSTA
- Norton’s theorem

Queue disciplines

- Operation of the queue j:(i) Each job requires exponential(1) amount of service.(ii) Total service effort supplied at rate φj(nj)(iii) Proportion Υj(k,nj) of this effort directed to job in position k, k=1,…, nj ; when this job leaves, his service is completed, jobs in positions k+1,…, nj move to positions k,…, nj -1.(iv) When a job arrives at queue j he moves into position k with probability δj(k,nj + 1), k=1,…, nj +1; jobs previously in positions k,…, nj move to positions k+1,…, nj +1.
- FCFS, LCFS, PS, infinite server queue
- BCMP network

Network of queues

- Multiclass queueing network, type or class i=1,..,I
- J queues
- Customer type identifies route
- Poisson arrival rate per type ν(i), i=1,…,I
- Route r(i,1),r(i,2),…,r(i,S(i))
- Fixed number of visits; cannot use Markov routing
- 1, 2, or 3 visits to queue: use 3 types
- Type i at stage s in queue r(i,s)
- tj(l): type of customer in position l in queue j
- sj(l): stage of this customer along his route
- cj(l)=(tj(l),sj(l)): class of this customer
- cj=(cj(1),…, cj(nj)): state of queue j
- C=(c1,…, cJ): Markov process representing states of the system

- Arrival rate class i to network ν(i), so also for each stage, say
- Transition rates
- Equilibrium distribution queue j
- Theorem: equilibrium distribution for open network (closed):(and departure process from the network is Poisson)

Network of Quasi-reversible queues

- Rates
- Theorem : For an open network of QR queues(i) the states of individual queues are independent at fixed time(ii) an arriving customer sees the equilibrium distribution(ii’) the equibrium distribution for a queue is as it would be in isolation with arrivals forming a Poisson process.(iii) time-reversal: another open network of QR queues(iv) system is QR, so departures form Poisson process

Network of Quasi-reversible queues

- Rates
- Theorem : For an open network of QR queues(i) the states of individual queues are independent at fixed time(ii) an arriving customer sees the equilibrium distribution(ii’) the equibrium distribution for a queue is as it would be in isolation with arrivals forming a Poisson process.(iii) time-reversal: another open network of QR queues(iv) system is QR, so departures form Poisson process

Network of Quasi-reversible queues

- Rates
- Theorem : For an open network of QR queues(i) the states of individual queues are independent at fixed time(ii) an arriving customer sees the equilibrium distribution(ii’) the equibrium distribution for a queue is as it would be in isolation with arrivals forming a Poisson process.(iii) time-reversal: another open network of QR queues(iv) system is QR, so departures form Poisson process

- Rates

Networks of Queues: lecture 6

Nelson, sec 10.3.5-10.6.6

- Last time on NoQ …
- Kelly-Whittle network
- Partial balance
- Quasi reversibility

- PASTA / MUSTA
- Norton’s theorem

fraction of time system in staten

probability outside observer seesncustomers at timet

probability that arriving customer sees n customers at time t (just before arrival at time t there are n customers in the system)

in general

PASTA: Poisson Arrivals See Time AveragesLet C(t,t+h) event customer arrives in (t,t+h)

For Poisson arrivals q(n,n+1)=λ so that

Alternative explanation; PASTA holds in general!

PASTA: Poisson Arrivals See Time AveragesPalm probabilities: Each type of transition nn’ for Markov chain associated with subset H of SxS \diag(SxS)

Example:transition in which customer queue i queue j

Transition in which customer leaves queue i

Transition in which customer enters queue j

MUSTA: Moving Units See Time AveragesNH process counting the H-transitions

Palm probability PH (C) of event C given that H occurs:

Probability customer queue i queue j sees state m

Probability customer arriving to queue j sees state m

MUSTA: Moving Units See Time AveragesLast time on NoQ :Kelly Whittle network

Theorem: The equilibrium distribution for the Kelly Whittle network is where and πsatisfies partial balance

MUSTA :Kelly Whittle network

Theorem: The distribution seen by a customer

moving from queue i tot queue j is

Entering queue j is

where

Networks of Queues: lecture 5

Nelson, sec 10.3.5-10.6.6

- Last time on NoQ …
- Kelly-Whittle network
- Partial balance
- Quasi reversibility

- PASTA / MUSTA
- Norton’s theorem

Insensitivity and partial balance, Norton’s theorem

- Theorem: InsensitivitySuppose subsets Ai of S are such that there is no single transition of positive intensity in which more than one Ai is vacated or more than one Ai entered. Then the equilibrium distribution π(x) is insensitive to the nominal sojourn times in the Ai if and only if the Markov process shows partial balance in all the Ai.
- Norton’s theorem: state aggregation, flow equivalent servers, Nelson, sec 10.6.14-15
- Consider network of subnetworks, each subnetwork represented by auxiliary process. Then we may lump subnet into single node if and only if partial balance over the subnets

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