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

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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


  • 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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