CPE 345: Modeling and simulation

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CPE 345 Modeling and simulation

• Lecture 12

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Todays topics

• Sample final
• interactive problem solving

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Problem 1

• We would like to generate a sequence of 200
  uniformly distributed random numbers, using a
  linear congruential random number generator with
  parameters a 11, and c0.
• We want to implement a maximum period
  generator.
• (a) What is the minimum value for m that we can
  use such that the 200 numbers generated should
  not repeat?
• (b) What additional condition you need such that
  the period of the generator would be greater than
  200?

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What do we know ?

• N 200 numbers. We want P gt N 200 (maximal
  length)
• a 11, c0, m ?
• Linear congruential generator
• Properties - Maximum period generator conditions
• For m 2b, and c?0, the longest possible period
  is P m 2b, achieved if c is relatively prime
  to m (the greatest common factor of c and m is
  1), and a 14k, k integer.
• For m 2b, and c 0, the longest possible
  period is P m/4 2b-2, achieved if the seed X0
  is odd, and a 38k or a 58k, k0,1,
• For m a prime number and c 0, the longest
  possible period is P m-1, achieved if a has the
  property that the smallest integer k s.t. ak-1 is
  divisible by m is k m-1.

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Problem 1 solution

• Answer
• (a)
• (b) Additional condition?
• For m 2b, and c 0, the longest possible
  period is P m/4 2b-2, achieved if the seed X0
  is odd, and a 38k or a 58k, k0,1,
• The seed X0 should be selected to be an odd
  number
• e.g. X0 5

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Problem 2

• At a Supermarket express line, self-checkout is
  employed there are two scanning machines, that
  each customer can use, and then the customers get
  a receipt and need to pay to a physical person,
  which serves both queues. Assume that the
  customers arrive with a Poisson distribution of
  ? 0.1 per minute, and the time required for
  each self-scanning order (one customer service)
  is exponential distributed with mean 1/?10
  minutes, also the time of service for the
  physical cashier person is also exponentially
  distributed with mean 1/?1 minute.
• (a) Draw the queueing system
• (b) Is this system stable? Motivate your
  answer using formulas.
• (c) What is the average delay experienced by
  the customers?
• (d) How could you improve the system (reduce
  the delay)?
• (e) Suppose you want to estimate the
  customers delay by means of simulations. After
  the arrival of 10000 customers, you may compute
  an estimator for the average delay experienced by
  the customers. How accurate you may expect your
  estimator to be approximately (e.g. in the order
  of 0.1 (1 decimal accuracy), 0.01 (two decimals),
  0.001 (3 decimals), etc.)?

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What do we know?

• Self checkout pay cashier
• Self-checkout infinite number of servers ?
• No! We have only 2 scanning machines!
• Poisson arrivals ? 0.1 per minute
• Exponential service times
• Self scanning 1/?10 minutes ? ?s 0.1
• Cashier 1/?1 minute ? ?C 1

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Stability

• (b) Stable if both queues are stable
• First queue M/M/2 ? ?1 ?/c?s ?/2?s
  0.1/0.2 0.5 ? stable
• Second queue M/M/1 ? ?2 ?/?C 0.1/1 0.1 ?
  stable

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Average delay
w1

• w w1w2
• First queue M/M/2

w2
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Average delay continued

• Second queue M/M/1
• Delay given mostly by the self-checkout queue
• Buying more scanning machines may reduce the
  queueing time for queue 1
• Not a significant reduction though the
  bottleneck is the speed of the customers
  operating the self-scanning machines

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Simulation performance

• (e) Suppose you want to estimate the customers
  delay by means of simulations. After the arrival
  of 10000 customers, you may compute an estimator
  for the average delay experienced by the
  customers. How accurate you may expect your
  estimator to be approximately (e.g. in the order
  of 0.1 (1 decimal accuracy), 0.01 (two decimals),
  0.001 (3 decimals), etc.)?
• Standard error
• Your accuracy is in the order of 1/sqrt(10000)
  0.01

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Problem 3

• We have analyzed a computer network performance
  based on the assumption that the traffic
  generated at the output of each server is
  exponentially distributed. To validate that, we
  experimentally build a histogram as in the figure

Estimated mean
5-6
7-8
9-10
8-9
(a) Can you estimate the mean of the
distribution? (b) Run a chi-square test to
determine if the distribution is indeed
exponential. The desired significance level is ?
0.05.
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Chi-square test

• Exponential expected mean 1/? 6.26, Ei npi
• n 50 samples,

(ai, bi) Oi Ei (Oi-Ei)2/Ei
5-6 27 3.32 169.55
6-7 13 2.83 36.74
7-8 6 2.43 5.296
8-9 3 2.07 0.42
9-10 1 1.76 0.33
Total 50 12.37 212.34
For ? 0.05 ? critical value is 7.815 ? the
exponential distrib. hypothesis is rejected
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Chi-square test-cont.

• The critical value (7.815) was obtained from the
  table chi_square_critical_value.doc, for ?
  0.05, and the number of degrees of freedom,
• k number of cells
• s number of parameters estimated (we have
  estimated the mean)
• If (Oi-Ei)2/Ei gt critical value, the hypothesis
  is rejected
• 212.34 gt 7.815 ? distribution is not exponential

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Problem 4

• By analyzing a very complex computer network, you
  find that the average delay experienced by each
  packet in queue i can be determined as
• where n is the total number of queues in the
  network, each having arrival rates ?i and service
  rates ?i.
• Is this formula valid on its face?
• What other validity tests can you use? Is this
  formula valid according to these new tests?

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Face validity

• Units test rate2/(raterate) ? units test
  passed
• Does model behavior change in expected ways with
  modification of parameters?
• if arrival rate in queue i , then wi ?
• if service rate for queue i , then wi
  ?
• if total arrival in the other queues is 0, wi ?
  ? ?
• The formula is not valid on its face
• Other test some form of Turing test observe
  delay produce by the real system, compare with
  the computed expected value - we expect the
  validity test to fail

(b)
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Problem 5

• We are designing a simulation in which we have
  implemented an exponential random number
  generator. To validate this generator, we collect
  its outputs and generate a histogram. If the
  generated exponential r.v. has mean 5, and we
  select 10 cells for the histogram of width 1, and
  the range starts at 3, what is the probability
  that the random number generated may fall outside
  the range?
• Range 3, 13

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Problem 6

• Suppose you are simulating the delay experienced
  by the customers of a bank with one single
  teller. If customers arrive with a rate of 1
  every 15 minutes (and the arrivals are Poisson
  distributed), and the tellers service is
  exponentially distributed with mean 10 minutes,
• (a) Determine the 95 confidence interval for
  delay estimation, if the simulation has collected
  100 samples in 4 different runs, and the obtained
  values for the delay are
• w1 31.2 w2 29.42 w3 33.1 w4 28.7.

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What do we know?

• N 100 samples, 4 runs ? 25 samples/run
• Arrivals Poisson, ? 1/15
• Departures exponential ? 0.1
• Average delays for the 4 runs w1 31.2 w2
  29.42 w3 33.1 w4 28.7.
• Need to determine 95 confidence intervals
• Lots of redundant information!

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Confidence intervals

• A 100(1-?) confidence interval is given by
• With f R-1 3
• Need to determine
• ? ? 100(1-?) 95 ? ? 0.05
• 3.18 (Table A.5. appendix in your
  book)

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Confidence intervals
run Delay
1 31.2
2 29.42
3 33.1
4 28.7
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Confidence intervals cont.

• The 95 confidence interval
• For extra credit, you may compute the theoretical
  delay and compare the results M/M/1

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Next class

• Projects are due
• Final in class open books, open notes
• Good luck to everybody!