Exponential Distribution &
Poisson Process
Memorylessness & other exponential
distribution properties; Poisson process;
Nonhomogeneous & compound P.P.’s
Chapter 5
1
Exponential Distribution: Basic Facts
 e  x , x  0
,  0
• Density f  x   
0, x  0
1  e  x , x  0
• CDF F  x   
0, x  0

• MGF   t   E e  
 t
1
• Mean E  X  

1
• Variance Var  X   2
tX

Chapter 5
Coefficient of variation
EX 
Var  X 
1
2
Key Property: Memorylessness
P  X  s  t X  t  P  X  s for all s, t  0
• Reliability: Amount of time a component has been in
service has no effect on the amount of time until it fails
• Inter-event times: Amount of time since the last event
contains no information about the amount of time until the
next event
• Service times: Amount of remaining service time is
independent of the amount of service time elapsed so far
Chapter 5
3
Other Useful Properties
Sum of n independent exponential r.v.’s with common
parameter  has a gamma distribution w/parameters (n, )
Competing Exponentials:
If X1 and X2 are independent
exponential r.v.’s with
parameters 1 and 2, resp.,
then
P  X1  X 2 
1
1  2
(generalizes to any number
of competing r.v.’s)
Minimum of exponentials:
If X1, X2 , …, Xn are
independent exponential
r.v.’s where Xn has parameter
i, then
min(X1, X2 , …, Xn) is
exponential w/parameter
1 + 2 + … + n
Chapter 5
4
Counting Process
A stochastic process {N(t), t  0} is a counting process if N(t)
represents the total number of events that have occurred in
[0, t]
Then {N(t), t  0} must satisfy:
N(t)  0
N(t) is an integer for all t
If s < t, then N(s)  N(t)
For s < t, N(t) - N(s) is the number of events that occur in
the interval (s, t].
Chapter 5
5
Stationary & Independent Increments
• A counting process has independent increments if, for any
0  s  t  u  v, N  t   N  s  is independent of N  v   N u 
That is, the numbers of events that occur in nonoverlapping
intervals are independent random variables.
• A counting process has stationary increments if the
distribution if, for any s < t, the distribution of N(t) – N(s)
depends only on the length of the time interval, t – s.
Chapter 5
6
Poisson Process Definition 1
A counting process {N(t), t  0} is a Poisson process with
rate ,  > 0, if
N(0) = 0
The process has independent increments
The number of events in any interval of length t follows a
Poisson distribution with mean t (therefore, it has
stationary increments), i.e.,
n
 t
e  t 
P  N  t  s   N  s   n 
, n  0,1,...
n!
Chapter 5
7
Poisson Process Definition 2
f h
0
A function f is said to be o(h) (“Little oh of h”) if lim
h 0
h
A counting process {N(t), t  0} is a Poisson process with rate
,  > 0, if
N(0) = 0
The process has stationary and independent increments
P N  h   1  h  o  h 
P N  h   2  o  h 
Definitions 1 and 2 are equivalent!
Chapter 5
8
Interarrival and Waiting Times
N(t)
The times between arrivals
T1, T2, … are independent
exponential r.v.’s with mean
1/:
P T1  t  P N  t   0  et
P T2  t T1  s  e  t
S1 S2
T1 T2
S3 S4
T3
T4
t
The (total) waiting time until
the nth event has a gamma
dist’n: S  n T
n
Chapter 5

i 1 i
9
Other Poisson Process Properties
Poisson Splitting:
Suppose {N(t), t  0} is a P.P. with rate , and suppose that
each time an event occurs, it is classified as type I with
probability p and type II with probability 1-p,
independently of all other events. Let N1(t) and N2(t),
respectively, be the number of type I and type II events up
to time t.
Then {N1(t), t  0} and {N2(t), t  0} are independent
Poisson processes with respective rates p and (1-p).
Chapter 5
10
Other Poisson Process Properties
Competing Poisson Processes:
Suppose {N1(t), t  0} and {N2(t), t  0} are independent
Poisson processes with respective rates 1 and 2.
Let Sni be the time of the nth event of process i, i = 1,2.
P S  S
1
n
n  m 1
2
m
 
k n
k
 n  m  1  1   1 
 



k







 1 2   1 2 
Chapter 5
n  m 1 k
11
Other Poisson Process Properties
If Y1, Y2, …, Yn are random variables, then Y(1), Y(2), …, Y(n)
are their order statistics if Y(k) is the kth smallest value
among Y1, Y2, …, Yn, k = 1, …, n.
Conditional Distribution of Arrival Times:
Suppose {N(t), t  0} is a Poisson process with rate  and
for some time t we know that N(t) = n. Then the arrival
times S1, S2, …, Sn have the same conditional distribution
as the order statistics of n independent uniform random
variables on (0, t).
Chapter 5
12
Nonhomogeneous Poisson Process
A counting process {N(t), t  0} is a nonhomogeneous Poisson
process with intensity function t, t  0, if:
N(0) = 0
The process has independent increments (not stationary incr.)
P N  t  h   N  t   1   t  h  o  h 
P N  t  h   N t   2  o  h 
Let m  t     y dy
0
n
Then
  m s  t   m s    m  s  t   m  s  
P  N  t  s   N  s   n  e
, n  0,1,...
n!
t
Chapter 5
13
Compound Poisson Process
A counting process {X(t), t  0} is a compound Poisson process
if:
N t 
X  t    Yi , t  0
i 1
where {N(t), t  0} is a Poisson process and {Yi, i = 1, 2, …}
are independent, identically distributed r.v.’s that are
independent of {N(t), t  0}.
By conditioning on N(t), we can obtain:
E  X  t    tE Y1 
2
Var  X  t    tE Y1 
Chapter 5
14