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CPSC 601.43: Stochastic Processes
Instructor: Anirban Mahanti
Email: mahanti@cpsc.ucalgary.ca
Reference Book
“Computer Systems Performance Evaluation
and Prediction” by P. Fortier and H. Michel,
Digital Press, 2004.
Stochastic Processes
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Outline
 Definitions
 Discrete, continuous, independent, stationary
 Bernoulli Process
 Poisson Process
 Birth Death Process
 Markov Process (next time)
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Stochastic Processes
 Definition: A family of random variables,
denoted X(t), where one value of the
random variable X exists for each value
of t.
 Example
T = {heads, tails} <- the index set
x = {0, 1} <- the state space
X(heads) = 0; X(tails) = 1;
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Stochastic Processes (2)
Stochastic Processes
discrete
continuous
Examples
 Number of commands, N(t), received by a
time-sharing computer system during some
time interval (0, t) -> discrete state,
continuous index
 Number of heads returned, N(n), by
tossing a fair coin n times?
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Stochastic Processes (3)
t1 t2
t3
t4
s
t
s+h
t+h
Independent increments
Stationary Increments
x(t2) - x(t1) != x(t4) – x(t3)
x(t+h) – x(s+h) == x(t) – x(s)
E.g., number of phone calls,
N(t), handled by a call
center between noon and
3pm on a weekday.
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Bernoulli Process
Let Xi ( i>= 0) be an independent and
identically distributed Bernoulli random
variable, such that:
Xi = 1, with probability p, and,
Xi = 0, with probability (1-p)
Let Sn = X1 + X2 + … + Xn
(i.e., counting number of successes in n trials)
Sn is a Bernoulli process. Why?
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Poisson Process
A Poisson stochastic process has the following
characteri stics :
1. Events are independen t, and the interarriv al times
of events can be described using an exponentia l distributi on
N (t )  1  e t , where   rate of occurence of events.
2. Occurence of events in non - overlappin g intervals
of time are statistica lly independen t
3. For small increments of time, the probabilit y of
an event occuring is
t  o(t )
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Merging Poisson Streams
λ1
λ=λ1+λ2
λ2
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Dividing Poisson Streams
λ
λpa
pa
pb
λpb
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More on Poisson Process
 Number of occurrences in intervals of equal length
are identically distributed.
 Poisson process is “memory less”, i.e., past history
does not aid in predicting future events.
 Probability of k arrivals in an interval of length t (k is
an integer >= 0) follows the Poisson Density Function
P[ yt  k ]  e
 t
( t ) k
k!
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Birth Death (BD) Processes
A continuous parameter, discrete state space
stochastic process
{X(t), t >= 0}
E(n), n = 0, 1, 2, 3 … describe the state
X(t) = n means that X(t) is in state E(n) at
time t
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Properties of BD Processes
 State changes only in increments of +- 1
 En >= 0
 If the system is in state En at time t, the
probability of a transition to state En+1
during interval (t,t+h) is λnh + o(h), and to
state En-1 is µnh + o(h), where
λn = birth rate
µn = death rate
 Probability of more than one transition
during an interval of length h is o(h)
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Time Dependent Solutions for BD
Pn (t )  P[ X (t )  n]
Probabilit y that BD system is in state E n at time t.
Then, we can show the following :
dPn (t )
 ( n   n )  Pn (t )  n Pn 1 (t )   n 1 Pn 1 (t ), n  1
dt
dP0 (t )
  0 P0 (t )  1 P1 (t ), n  0
dt
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Equilibrium Solution for BD
λ0
1
0
µ1
 Rate entering = Rate leaving
λ0 p0= µn p1 and p0+ p1 = 1
So, now you can solve!
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