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Performance Evaluation of Computer Systems

In the Name of the Most High. Performance Evaluation of Computer Systems. Introduction to Probabilities: Discrete Random Variables. By Behzad Akbari Tarbiat Modares University Spring 2009. These slides are based on the slides of Prof. K.S. Trivedi (Duke University). Random Variables.

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Performance Evaluation of Computer Systems

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  1. In the Name of the Most High Performance Evaluation of Computer Systems Introduction to Probabilities: Discrete Random Variables By Behzad Akbari Tarbiat Modares University Spring 2009 These slides are based on the slides of Prof. K.S. Trivedi (Duke University)

  2. Random Variables • Sample space is often too large to deal with directly • Recall that in the sequence of Bernoulli trials, if we don’t need the detailed information about the actual pattern of 0’s and 1’s but only the number of 0’s and 1’s, we are able to reduce the sample space from size 2n to size (n+1). • Such abstractions lead to the notion of a random variable

  3. Discrete Random Variables • A random variable(rv) X is a mapping (function) from the sample space S to the set of real numbers • If image(X ) finite or countable infinite, X is a discrete rv • Inverse image of a real number x is the set of all sample points that are mapped by X into x: • It is easy to see that

  4. Probability Mass Function (pmf) • Ax : set of all sample points such that, • pmf

  5. pmf Properties • Since a discrete rv X takes a finite or a countable infinite set values, the last property above can be restated as,

  6. Distribution Function • pmf: defined for a specific rv value, i.e., • Probability of a set • Cumulative Distribution Function (CDF)

  7. Distribution Function properties

  8. Discrete Random Variables • Equivalence: • Probability mass function • Discrete density function (consider integer valued random variable) • cdf: • pmf:

  9. Commondiscrete random variables • Constant • Uniform • Bernoulli • Binomial • Geometric • Poisson

  10. Constant Random Variable • pmf • CDF 1.0 c 1.0 c

  11. Discrete Uniform Distribution • Discrete rv X that assumes n discrete value with equal probability 1/n • Discrete uniform pmf • Discrete uniform distribution function:

  12. Bernoulli Random Variable • RV generated by a single Bernoulli trial that has a binary valued outcome {0,1} • Such a binary valued Random variable X is called the indicator or Bernoulli random variable so that • Probability mass function:

  13. Bernoulli Distribution • CDF p+q=1 q x 0.0 1.0

  14. Binomial Random Variable • Binomial rv  a fixed no. n of Bernoulli trials (BTs) • RV Yn: no. of successes in n BTs • Binomial pmf b(k;n,p) • Binomial CDF

  15. In fact, the number of successes in n Bernoulli trials can be seen as the sum of the number of successes in each trial: where Xi’s are independent identically distributed Bernoulli random variables. Binomial Random Variable

  16. Binomial Random Variable: pmf pk

  17. Binomial Random Variable: CDF

  18. Applications of the binomial • Reliability of a k out of n system • Series system: • Parallel system:

  19. Applications of the binomial • Transmitting an LLC frame using MAC blocks • p is the prob. of correctly transmitting one block • Let pK(k) be the pmf of the rv K that is the number of LLC transmissions required to transmit n MAC blocks correctly; then

  20. Geometric Distribution • Number of trials upto and including the 1st success. • In general, S may have countably infinite size • Z has image {1,2,3,….}. Because of independence,

  21. Geometric Distribution (contd.) • Geometric distribution is the only discrete distribution that exhibits MEMORYLESS property. • Future outcomes are independent of the past events. • n trials completed with all failures. Y additional trials are performed before success, i.e. Z = n+Y or Y=Z-n

  22. Geometric Distribution (contd.) • Z rv: total no. of trials upto and including the 1st success. • Modified geometric pmf: does not include the successful trial, i.e. Z=X+1. Then X is a modified geometric random variable.

  23. Applications of the geometric • The number of times the following statement is executed: repeat S until B is geometrically distributed assuming …. • The number of times the following statement is executed: while B do S is modified geometrically distributed assuming ….

  24. Negative Binomial Distribution • RV Tr: no. of trials until rthsuccess. • Image of Tr = {r, r+1, r+2, …}. Define events: • A: Tr = n • B: Exactly r-1 successes in n-1 trials. • C: The nthtrial is a success. • Clearly, since B and C are mutually independent,

  25. PoissonRandom Variable • RV such as “no. of arrivals in an interval (0,t)” • In a small interval, Δt, prob. of new arrival= λΔt. • pmf b(k;n, λt/n); CDF B(k;n, λt/n)= • What happens when

  26. Poisson Random Variable (contd.) • Poisson Random Variable often occurs in situations, such as, “no. of packets (or calls) arriving in t sec.” or “no. of components failing in t hours” etc.

  27. Let N(t) be the number of (failure) events that occur in the time interval (0,t). Then a (homogeneous) Poisson model for N(t) assumes: 1. The probability mass function (pmf) of N(t) is: Where l> 0 is the expected number of event occurrences per unit time 2. The number of events in two non-overlapping intervals are mutually independent Poisson Failure Model

  28. Note: • For a fixed t, N(t) is a random variable (in this case a discrete random variable known as the Poisson random variable). • The family {N(t), t 0} is a stochastic process, in this case, the homogeneous Poisson process.

  29. Poisson Failure Model (cont.) • The successive interevent times X1, X2, … in a homogenous Poisson model, are mutually independent, and have a common exponential distribution given by: • To show this: • Thus, the discrete random variable, N(t), with the Poisson distribution, is related to the continuous random variable X1, which has an exponential distribution • The mean interevent time is 1/l, which in this case is the mean time to failure

  30. Poisson Distribution • Probability mass function (pmf) (or discrete density function): • Distribution function (CDF):

  31. Poisson pmf pk t=1.0

  32. CDF 1 t=1.0 0.5 0.1 1 2 3 4 5 6 7 8 9 t 10 Poisson CDF

  33. Poisson pmf pk t=4.0 t=4.0

  34. CDF 1 t=4.0 0.5 0.1 1 2 3 4 5 6 7 8 9 t 10 Poisson CDF

  35. Probability Generating Function (PGF) • Helps in dealing with operations (e.g. sum) on rv’s • Letting, P(X=k)=pk , PGF of X is defined by, • One-to-one mapping: pmf (or CDF) PGF • See page 98 for PGF of some common pmfs

  36. Discrete Random Vectors • Examples: • Z=X+Y, (X and Y are random execution times) • Z = min(X, Y) or Z = max(X1, X2,…,Xk) • X:(X1, X2,…,Xk) is a k-dimensional rv defined on S • For each sample point s in S,

  37. Discrete Random Vectors (properties)

  38. IndependentDiscrete RVs • X and Y are independent iff the joint pmf satisfies: • Mutual independence also implies: • Pair wise independence vs. set-wide independence

  39. Discrete Convolution • Let Z=X+Y . Then, if X and Y are independent, • In general, then,

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