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# Open Queueing Network and MVA - PowerPoint PPT Presentation

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

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### Open Queueing Network and MVA

Cheng-Fu Chou

• 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

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 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.

• 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