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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

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## 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**• Consider the open network in Fig. 3.6 with single-server queues A, B, and C with branching prob. pA, pB, pC, respectively.**Let pdone = 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**• 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**• 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**• 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**• The Iterative Solution Method

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