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Open Queueing Network and MVAPowerPoint Presentation

Open Queueing Network and MVA

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### Open Queueing Network and MVA

Cheng-Fu Chou

Jackson Networks

- Assume each queue has one or more servers with expo. distributed service time, and Poisson arrival of jobs to the network
- If the network has Q queues and if ni is the number of jobs at queue I, then a Jackson network in steady state has the surprising property that
- A product of marginal prob., such a network is called product-form or separable

- Each queue behaves like an independent M/M/K queue or M/G/1 queue
- We can use a Markov chain derive an expression for Prob(ni=ki) separately and substitute it into the above eqn.
- Note that the arrival process at some queue may not be Poisson

Ex queue

- Consider the open network in Fig. 3.6 with single-server queues A, B, and C with branching prob. pA, pB, pC, respectively.

- Let p queuedone = 1 – (pA +pB +pC), then, the number of visits to A has distribution Geometric(pdone).
- Let VA, VB, and VC be the expected number of visits to A, B, and C, respetively.
- VA =1/ pdone , VB =pB VA = pB / pdone , and VC = pC / pdone
- Let lA, lB, abd lC be the job arrival rates at A, B, and C respectively.
- lA = l VA , lB = l VB , lC = l VC
- After we can get prob(nA=i), prob(nB=j), and prob(nC=h) , we get prob(nA=i, nB=j, nC=h)

Closed Queueing Network queue

- Closed systems are used for the interactive systems
- In a closed network, we can model a set of users submitting requests to a system, waiting for results, then submitting more requests
- human users interacting with a system,
- threads contending for a lock,
- processes blocking for I/O,
- networked servers waiting for a response message.

Product Forum Queueing Networks queue

- A PFQN consists of a collection of queueing and delay centers. It satisfies the following conditions.
- All queueing centers : FCFS, PS, or LCFSPR
- Any delay centers
- FCFS with exponential distribution
- If a FCFS center has multiple service classes, they must all have the same average service time
- External arrivals, if any, are Poisson
- Routing is state-independent

Arrival Theorem queue

- For a separated closed network with N jobs, an arrival at a queue sees a network state that is (distribution-wise) the same as that seen by an outside observer of the same network with N-1 jobs.

Mean Value Analysis queue

- The Iterative Solution Method