Leistungsanalyse

1
Leistungsanalyse

• Kap 11
• Networks of Queues
• open networks

2
Classes of Queuing Networks

• Tandem Queues
• Open queuing networks (Jacksons network)
• Closed queuing networks (Gordon-Newell network)
• Multi class queuing networks (BCMP queues)
• Non-product-form queuing networks.
• In the last segment we will discuss the
  computation of response time distribution (or
  percentiles) in networks

3
Networks of Queues

• Two types of networks Open and Closed
• An open queuing network is characterized by one
  or more sources of job arrivals and
  correspondingly one or more sinks that absorb
  jobs departing from the network.
• In a closed queuing network, jobs neither enter
  nor depart from the network.
• The behavior of jobs within the network is
  characterized by
• the distribution of job service times at each
  center
• the probabilities of transitions between service
  centers
• For each center the number of servers, the
  scheduling discipline, and the size of the queue
  must be specified.
• For an open network, a characterization of
  job-arrival processes is needed.
• For a closed network, the number of jobs in the
  network must be specified.

4
Open Queuing Networks

• M/M/1, etc. single node queuing network
• Two M/M/1 queues in tandem
• Exponentially dist. service times at s0 and s1.
• Underlying stochastic process is an HCTMC
• State (k0,k1),
• ki number of jobs at node i, i0,1
• The changes of state occur upon a completion of
  service at one of the two servers or upon an
  external arrival.
• Since all inter-event times are exponentially
  distributed (by our assumptions), the underlying
  stochastic process is a homogeneous CTMC with the
  state diagram shown on the next page.

Poisson stream
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Tandem queue-state diagram

• CTMC state diagram

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Tandem queue CTMC solution

• Following steady state solution can be shown to
  satisfy the CTMC balance equations
• Solution has a product form, i.e., product of the
  solution of two independent M/M/1 queues.
• For an M/M/1 queue Burke showed that the output
  process is also Poisson with rate ?
• Therefore, for the two queue tandem network, the
  second queue is also an independent M/M/1 queue.
  Hence the product form of the result holds.

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Tandem queue product form

• Generalization to an n-node tandem network
• The solution can be shown to satisfy the balance
  equations of the underlying CTMC
• The solution can also be derived by repeatedly
  invoking Burkes result.

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Tandem Queues-example

• Repair facility three sequential repair stations

cumulative failures per hr
Littles formula gives the mean (repairwaiting)
time at each station,
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General Feed Forward Networks

• Burkes result together with the following two
  properties of Poisson process can be utilized to
  derive product form solution for any feed forward
  queuing network
• Probabilistically splitting a Poisson stream
  gives rise to two or more Poisson streams
• Joining two or more Poisson streams produces a
  single Poisson stream

10
Open Queuing Networks with Feedback

• Open queuing network is one in which jobs may
  arrive from the outside world and on completion,
  jobs may leave the network.
• Jacksons result Product form solution is
  applicable to open queuing networks with any
  arbitrary feed-forward/feedback connections.
  (assuming each node is an M/M/1 or M/M/m queue).
  Such networks are called open PFQNs.
• Assumptions
• Poisson arrival process(es)
• Exponentially distributed service times
• Each node follows FCFS queuing discipline
• Infinite storage space at each node.

11
M/M/1 queue with Bernoulli feedback

• Burkes second result the queue above does not
  have Poisson input process (I) even though
  processes at points A and D are both Poisson.
• Hence, the queues within a network with feedback
  will not be M/M/m queues in general (as their
  arrival process may not be Poisson), but they
  behave as if they are independent M/M/m queues.
• This is the beauty of Jacksons remarkable result
  of product form of networks with feedback.

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Open PFQN with Feedback

• Open queuing network with 2 nodes
• CTMC state diagram with state (k0,k1)

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Example (contd.)

• The following product form solution can be shown
  to satisfy the balance equations of the
  underlying CTMC
• Where li is the average arrival rate at the ith
  node
• In the steady state, the departure rate from the
  ith node is also li.
• Equations relating lis are called Traffic
  equations which are dealt with next.

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Traffic Equations for the Example

• In this example with two nodes, and l0 and l1 are
  the arrival rates at the CPU and I/O node,
  respectively.
• Arrivals to CPU are either from outside at a rate
  l or from the I/O node at rate l1, therefore,
• l0 l l1
• Also a job after completion of the CPU burst
  would go to the I/O node with probability p1,
  therefore,
• l1 l0p1 l0(1-p0).
• Solving these two traffic equations, we get,

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Unfolding the Open PFQN with Feedback

• Hence the product form solution for the 2-node
  network with feedback
• Above solution suggests an equivalence with the
  following network without feedback

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Meaning of Equivalence

• The previous equivalence established between an
  open network with feedback and an open network
  without feedback is restricted to only
• Steady state behavior and
• Mean response times, queue length distribution
• !!! This equivalence does not apply to other
  analysis, e.g.,
• Response time distributions
• Transient behavior, etc.

17
Open PFQN General CSM

• Consider the central server model example
• This is open queuing network of m1 nodes
• Single CPU node and m I/O nodes

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General Open PFQN Solution

• Generating and solving the underlying CTMC for
  such a queuing network is neither feasible nor
  necessary due to Jacksons result
• Consider a single tagged program executing on the
  system (without any queuing or interference from
  other programs), moving from one node to another
• Observe the system only at times when a job
  completes service at a node.
• The underlying stochastic process forms a DTMC
  which is characterized by its transition
  probability matrix.

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DTMC model for the Open PFQN

• The DTMC transition probability matrix

Routing matrix X
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DTMC Solution

• Using the technique of visits
• Vj Av. no. of visits made by a job to node j
  before leaving the system
• ? jobs/unit time enter the system (from outside),
  arrival rate ?j of jobs at node j is,

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DTMC Solution(contd.)

• Jackson has shown that the steady state joint
  probability is given by,
• Where the probability of finding kj jobs at node
  j is given by the M/M/1 formula
• So the solution for the this open queuing network
  has product form

The validity can be shown using direct approach
as for open tandem networks
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General Open PFQN Measures

• Note Despite the product form solution, the
  arrival process at a node may not be Poisson and
  still the Jacksons result holds!
• Average Queue length at node j
• Average response time at node j

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Open PFQN Measures (contd.)

• Av. no. of jobs in the system
• Av. system response time (Littles formula)
• An equivalent unfolded tandem network can thus
  be derived

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Open PFQN Equivalence

• The total service requirement at each node j is
  given by
• The service rate at node j in the equivalent
  unfolded network is given by .
• Hence the unfolded open network is as below
• Note that only the total service requirement at
  each node is needed to find the rates in the
  equivalent unfolded network