Stochastically Dominating

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Stochastically Dominating

• upper processes

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The Idea
We want to simulate one process on a unbounded
state space. (Say 0, ).)
Suppose there is another process that is

• always above the process we care about

• time-reversible

• has a stationary distribution that we know and
  can draw from

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Then starting at time zero, we draw a value from
where is the stationary distribution of
the upper process.
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We run the upper process backwards one step to
time 1.
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We then use this point to be the highest point
for which the process of interest can start at
time 1.
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If coupling for the process of interest is
achieved after one time step, we are done.
Otherwise, we run the upper process back another
time step, etc
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A Trivial Example of a Reversible Process
(and the non-trivial problem of reversing it in
a way that would dominate a forward moving
process)

• Any two-state Markov chain is reversible

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A Trivial Example Reversible Process?

• Suppose that the process of interest is given by

• Note that this process is monotone.

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A Trivial Example Reversible Process?

• An upper bounding process is given by Pu

• Note that

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A Trivial Example Reversible Process?

• We draw from at time zero.

ie we draw Uunif(0,1) at let

• Draw U1unif(0,1) to send the upper process back
  one step.

ie if then
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A Trivial Example Reversible Process?
On the other hand, if then

• Find and store U-1. U-1 is related to, but not
  equal to U1.

• Use U-1 to run a path from forward to time
  zero using P.

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Stochastically Dominating Upper Processes
Example The Infinite Storage Model with
Exponential Release

• input arriving according to a Poisson process
  with rate

• input in exponential amounts with mean

• release rate r(u)u

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Stochastically Dominating Upper Processes
Example (continued)

• We dominate this process with the storage model
  with identical input parameters and simple unit
  linear release rate r(u)1.

• When run with the same jumps sizes at the same
  Poisson arrival times, this linear process will
  stay above the process of interest as long as
  both sample paths are above 1.

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Stochastically Dominating Upper Processes
Example (continued)

• So, we set the bottom of the boundng process to
  be 1.

(ie whenever the sample path of the upper
process hits 1, it will stay flat until the next
input time)
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Stochastically Dominating Upper Processes
Example (continued)
The stationary distribution for this upper
process is known
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Stochastically Dominating Upper Processes
Example (continued)
This upper process is reversible.
We will check the tricky case (at the bottom of
the space). Checking for the rest of the space is
left as an exercise!
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Stochastically Dominating Upper Processes
Example (continued)
For xgt1,
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Stochastically Dominating Upper Processes
Example (continued) on the other hand
For xgt1,
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Stochastically Dominating Upper Processes
So,
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Stochastically Dominating Upper Processes
and

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Stochastically Dominating Upper Processes
This reversibility is not going to help us in
practice though

• we could write down the transition probabilities
  for the linear upper process

• we may even be able to draw directly from these
  transition probabilities

• we cant, however, draw directly from the
  transition probabilities for the exponential
  release process of interest

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Stochastically Dominating Upper Processes
. hmmm

• we need to draw from the upper bounding process
  and the process of interest in the same way in
  order to preserve monotonicity

• we would like to make the transitions based on
  drawing some exponential inter-arrival time and
  exponential replenishment amounts

• this procedure is not reversible

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Stochastically Dominating Upper Processes
The Workaround

• the linear upper process can be interpreted as
  the workload process for the M/M/1 queue (shifted
  up by one unit)

• just because we gave it a different name does not
  suddenly make it reversible!

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Stochastically Dominating Upper Processes
However

• with this interpretation, we can shift to a
  consideration of the process Nn, of the number of
  customers in the queue

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Stochastically Dominating Upper Processes
Continued

• Note here are some imaginary transitions.
  (When the chain is at zero, we can not generate a
  departure, we just remain at zero with
  probability p.)

• This chain has transition law

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Stochastically Dominating Upper Processes
Still going

• It is easy to check that this chain is reversible.

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Stochastically Dominating Upper Processes
and more

• The backward construction on Nn

• Now generate the Poisson process backwards by
  moving up or down w.p. q or p, respectively
  (staying at zero w.p. p).

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Stochastically Dominating Upper Processes
The Workload Process

• Given this sample path of the customer process,
  we generate a sample path of the workload process
  conditional on the upper process.

• This requires a little thought At the times of
  arrivals, the input is precisely the exponential
  variable that gives the next forward time of
  departure.

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The Workload Process
workload W-2 W-3-A-2W-2
1
0
0