• Many useful applications, especially in queueing systems, inventory management, and reliability analysis.
• A connection between discrete time Markov chains and continuous time Markov chains.

A continuous random variable X is said to have an exponential distribution
  
0, if its probability density function is given by




 e
  x x

0
0, x

0

 
 x  
1
 e
  x x

0, x

0
0

( )

 

 
0

   x x e dx
Integrating by parts leads to
( )
  xe
  x
0

 
0

  x e dx

1



 t

E e tX
( ) [ ]
 
0
 tx
   x e e dx



0

 e

(
 


 t dx
, for t

[ ]
 d

( ) dt t

0

1

 
E X
2
[ ]
 d
2

( ) dt
2 t

0

(

2

 t )
3
Var X )

E X
2
]

E X ]
2 

1
2 t

0


2
2

A random variable is said to be memoryless if
(
 
|
 t )

(
 s ) for all ,

0


(
 
(

, t )
 t )

(
(
 
 t ) t )
(
  t )

(

) (
 s )

(
 s )

If X has the exponential distrbution, then
( s t )
 e
 
(

)    s e e
  t 
(

) (
 s )
 Exponentially distributed random variables are memoryless.

It can be shown that the exponential distribution is the only distribution that has the memoryless property.

Example 1: The amount of time one spends in a bank is exponentially distributed with mean ten minutes (  = 1/10).
What is the probability that a customer spends more than 15 minutes? What is the probability that the customer spends more than 15 minutes given that she is in the bank after 10 minutes?

Example 1: The amount of time one spends in a bank is exponentially distributed with mean ten minutes (  = 1/10).
What is the probability that a customer spends more than 15 minutes? What is the probability that the customer spends more than 15 minutes given that she is in the bank after 10 minutes?
(

15)
 e

15
  e

1.5

0.220
(

15 | X

10)

(

5)
 e

5
  e

0.5

0.604

Example 2: Consider a branch of a bank with two agents serving customers. The time an agent takes with a customer is exponentially distributed with mean 1/  . A customer enters and finds the two agents busy serving two other customers. What is the probability that the customer that just entered would be last to leave?

The minimum of n exponentially distributed random variables
Suppose that X
1
, X random variables, with X
What is P(min(X
1
2
, ..., X
, X
2 n i
, ..., X are independent exponential n having rate
)>x)?
m i
, i=1, ..., n.

The minimum of n exponentially distributed random variables
Suppose that X
1
, X random variables, with X
What is P(min(X
1
2
, ..., X
, X
2 n i
, ..., X are independent exponential n having rate
)>x)?
m i
, i=1, ..., n.
P (min( X
1
, X
2
, ..., X n
)
 x )

P {( X
1
 x ), ( X
2
 x ), ..., ( X n
 x )}

P {( X
1

)} {( X
2
 x )} ... {( n
 x )}
  
1 x e e
 
2 x
...
e
 
N x

 e

(
 
...

N
) x
P X
1
, X
2
, ..., X n
) is exponentially distributed with mean
1
 
1
  
2
...
 n
.

Comparing two exponentially distributed random variables
Suppose that X
1 and X
2 are independent exponentially distributed random variables with rates
What is P(X
1
< X
2
)?

1 and 
2
.

Comparing two exponentially distributed random variables
Suppose that X
1 and X
2 are independent exponentially distributed random variables with rates
What is P(X
1
< X
2
)?

1 and 
2
.
(
1

X
2
)



0


0

(
1

X
2
| X
1

)
X
1
( )
(
1

X
2
| X
1
 x )

1 e
 
1 x dx




0
 e
 
2 x

1
 
1 x e dx

1
 
1

2
.
 
0


1 e

(
 
2
) x dx

0

(

X
2
)

1 e
 
1 x dx

The sum of 2 identical exponentially distributed random variables
Suppose that X
1 and X
2 are independent exponential random variables with rates distribution of f
X1+X2
(x)?)?

1
= 
2
=  . What is the

The sum of 2 identical exponentially distributed random variables
Suppose that X
1 and X
2 are independent exponential random variables with rates distribution of f
X1+X2
(x)?

1
= 
2
=  . What is the f
X
1

X
2
 
0 x f
X
1
( )
X
2
( x

)
 
0 x  e
  s
 e
 
(

) ds
 
2 e
  x

0 x ds
 
2 xe
  x

The sum of n identical exponentially distributed random variables
Suppose that X
1
,..., X
N variables with rates distribution of f
 i
=
X1+...+Xn are independent exponential random
 for i =1, ..., n. What is the
(x)?
f
X
1

X
2
...
X n
( )
  e
  x
(
 x ) n

1
( n

1)!
 X
1 and 
,..., X
N
. has the gamma distribution with parameters n

The sum of 2 exponentially distributed random variables
Suppose that X
1 and X
2 are independent exponential random variables with rates of f
X1+X2
(x)?

1
≠ 
2
. What is the distribution

The sum of 2 exponentially distributed random variables
Suppose that X
1 and X
2 are independent exponential random variables with rates of f
X1+X2
(x)?

1
≠ 
2
. What is the distribution f
X
1

X
2
 
0 x f
X
1 s f
X
2
( x
 s ds
 
0 x 
1 e
 
1 s

2 e
 
2
(

) ds


 
1 2 e
 
2 x

1
 
1

2

0 x e

(
 
2
) s ds

2 e
 
2 x 

2
 
1

2

1 e
 
1 x

A stochastic process {N(t), t ≥ 0} is said to be a counting process if N(t) represents the total number of “events” that occur by time t (i.e., in the time interval [0, t]).

A stochastic process {N(t), t ≥ 0} is said to be a counting process if N(t) represents the total number of “events” that occur by time t (i.e., in the time interval [0, t]).
Example 1: N(t) is the number of customers that enter a store at or prior to time t. An event corresponds to a person entering the store.

A stochastic process {N(t), t ≥ 0} is said to be a counting process if N(t) represents the total number of “events” that occur by time t (i.e., in the time interval [0, t]).
Example 1: N(t) is the number of customers that enter a store at or prior to time t. An event corresponds to a person entering the store.
Example 2: N(t) is the number of individuals born at or prior to time t. An event occurs whenever a child is born.

A stochastic process {N(t), t ≥ 0} is said to be a counting process if N(t) represents the total number of “events” that occur by time t (i.e., in the time interval [0, t]).
Example 1: N(t) is the number of customers that enter a store at or prior to time t. An event corresponds to a person entering the store.
Example 2: N(t) is the number of individuals born at or prior to time t. An event occurs whenever a child is born.
Example 3: N(t) is the number of calls made to a technical help line at or prior to time t. An event occurs whenever a call is placed.

A counting process satisfies the following properties.
(i) N(t) ≥ 0.
(ii) N(t) is integer valued.
(iii) If s < t, then N(s) ≤ N(t).
(iv) For s < t, N(t) – N(s) equals the number of events that occurs in the time interval (s, t].

A counting process is said to possess independent increments if the number of events that occur in disjoint intervals are independent.

A counting process is said to possess independent increments if the number of events that occur in disjoint intervals are independent.
Example 1: The number of customers N(10) that enter the store in the interval [0, 10] is independent from the number of customers N(15) – N(10) that enter the store in the interval
(10, 15].

A counting process is said to possess independent increments if the number of events that occur in disjoint intervals are independent.
Example 1: The number of customers N(10) that enter the store in the interval [0, 10] is independent from the number of customers N(15) – N(10) that enter the store in the interval
(10, 15].
Example 2: The number of individuals N(10) born in the interval [1996, 2000] is not independent from the number of individuals N(2004) – N(2000) that enter the store in the interval (2000, 2004].

A counting process is said to possess stationary increments if the distribution of the number of events that occur in an interval depend only on the length of the interval and not the starting time of the interval.

A counting process is said to possess stationary increments if the distribution of the number of events that occur in an interval depend only on the length of the interval and not the starting time of the interval.
Example 1: The number of customers N(t) – N(s) that enter the store in the interval (s, t] does not depend on s (this is true if there is not a particular time of day where more customers enter the store).

The counting process {N(t) t ≥ 0} is said to be a Poisson process having rate  ,  > 0, if
(i) N(0) = 0.
(ii) The process has independent increments.
(iii) The number of events in any interval of length t is Poisson distributed with mean  t. That is for all s, t ≥ 0
{ (
 
( )}
 e
  t
(
 t ) n !
n
, for n

0,1,...

• Let T n describe the time that elapses between (n-1)th event and the nth event for n > 1 and let T
1 be the time of the first event.
• The sequence {T n interarrival times.
, n = 1, 2, ...} is called the sequence of

• Let T n describe the time that elapses between (n-1)th event and the nth event for n > 1 and let T
1 be the time of the first event.
• The sequence {T n interarrival times.
, n = 1, 2, ...} is called the sequence of
Example: if T
1
= 5 and T
2
= 10  the first event arrives at time t
= 5 and event 2 occurs at time t = 15.

• P(T
1
> t) = P(N(t) = 0) = e distribution.
 t  T
1 has the exponential

• P(T
1
> t) = P(N(t) = 0) = e distribution.
 t  T
1 has the exponential
• P(T
2
> t) = E[P(T
2
> t|T
1
) ]
Since P(T
2
Then, P(T
2
> t|T
1
=s) = P(0 events in (s, s+t]|T
1
=s)
= P(0 events in (s, s+t])
= e  t
> t) = E[P(T
2
> t|T
1
) ] = e  t  T
2 has the exponential

• P(T
1
> t) = P(N(t) = 0) = e distribution.
 t  T
1 has the exponential
• P(T
2
> t) = E[P(T
2
> t|T
1
) ]
Since P(T
2
Then, P(T
2
> t|T
1
=s) = P(0 events in (s, s+t]|T
1
=s)
= P(0 events in (s, s+t])
= e  t
> t) = E[P(T
2
> t|T
1
) ] = e  t  T
2 has the exponential
•The same applies to other values of n  T n distribution. has the exponential

Let T n denote the inter-arrival time between the (n-1)th event and the nth event of a Poisson process, then the T n
(n=1, 2, ...) are independent, identically distributed exponential random variables having mean 1/  .