Interarrival Times
LECTURE 15
• Yk time of kth arrival
!
• •Erlang
distribution:
Defining
characteristics
0
Review
Poisson process — II
– Time homogeneity:
λk y k−1e−λyP (k, τ )
f (y) =
,
y≥0
Yk
– Independence
(k − 1)!
– Small interval probabilities (small δ):
fY (y)

k


1 − λδ, if k = 0,
P (k, δ) ≈ λδ,
if k = 1,
k=1



0,
if k > 1.
k=2
• Readings: Finish Section 6.2.
• Review of Poisson process
• Merging and splitting
• Examples
• Nτ is a Poisson r.v., with parameter λτ :
k=3
(λτ )k e−λτ
P (k, τ ) =
,
k = 0, 1, . . .
y
k!
• Random incidence
[Nτ ] = var(N
τ ) = λτ
• E
First-order
interarrival
times (k = 1):
exponential
• Interarrival times (k = 1): exponential:
y≥0
fY1 (y) = λe−λy ,
E[T1] = 1/λ
fT1 (t) = λe−λt, t ≥ 0,
– Memoryless property: The time to the
next arrival is independent of the past
• Time Yk to kth arrival: Erlang(k):
fYk (y) =
λk y k−1e−λy
,
(k − 1)!
y≥0
Adding Poisson Processes
• Sum of independent Poisson random variables is Poisson
Poisson fishing
Merging Poisson Processes (again)
• •Sum
of independent
Poisson
processes
Merging
of independent
Poisson
processes
is is
Poisson
Poisson
• Assume: Poisson, λ = 0.6/hour.
– Fish for two hours.
Red bulb flashes
(Poisson)
– if no catch, continue until first catch.
"1
a) P(fish for more than two hours)=
All flashes
(Poisson)
"2
b) P(fish for more than two and less than
five hours)=
Green bulb flashes
(Poisson)
– –What
next
Whatis isthe
theprobability
probability that
that the
the next
arrival
comes
from
the
first
process?
arrival comes from the first process?
c) P(catch at least two fish)=
d) E[number of fish]=
e) E[future fishing time | fished for four hours]=
f) E[total fishing time]=
1
Light bulb example
Splitting of
of Poisson
Poisson processes
Splitting
processes
• Each light bulb has independent,
exponential(λ) lifetime
Assume
that email
traffic through
a server
• • Each
message
is routed
along the
first
is
a
Poisson
process.
stream with probability p
Destinations of different messages are
• Install three light bulbs.
Find expected time until last light bulb
dies out.
– independent.
Routings of different messages are independent
USA
Email Traffic
leaving MIT
!
MIT
Server
p!
(1 - p) !
Foreign
• Each output stream is Poisson
• Each output stream is Poisson.
Renewal processes
Random incidence for Poisson
Random incidence in
processes”
• Series of“renewal
successive
arrivals
• Poisson process that has been running
forever
–
i.i.d. interarrival
times
• Series
of successive
arrivals
(but not necessarily exponential)
– i.i.d. interarrival times
(but not necessarily exponential)
• Show up at some “random time”
(really means “arbitrary time”)
x
x
x
x
• Example:
interarrival times are equally likely to
• Bus
Example:
be
5
10 minutes
Bus or
interarrival
times are equally likely to
x
be 5 or 10 minutes
Time
• If you arrive at a “random time”:
Chosen
time instant
• If you arrive at a “random time”:
– what is the probability that you selected
– awhat
is the probability
you selected
5 minute
interarrivalthat
interval?
a 5 minute interarrival interval?
– what is the expected time to next ar– rival?
what is the expected time
• What is the distribution of the length of
the chosen interarrival interval?
to next arrival?
2
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6.041SC Probabilistic Systems Analysis and Applied Probability
Fall 2013
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