Transcript Poisson Distribution

Asian School of Business
PG Programme in Management (2005-06)
Course: Quantitative Methods in Management I
Instructor: Chandan Mukherjee
Session 8: Useful Theoretical Distributions (contd.)
Continuous Distribution
X is a continuous random variable
Density Function: f(x) ; a < x < b
x
Distribution Function:
 f ( y) dy
= F(x)
a
Failure Rate:
lim
x  0
1 F ( x  x)  F ( x)
f ( x)

x
1  F ( x)
1  F ( x)
Exponential Distribution
f ( x) 
1

exp(  x /  ) ; 0  x  
F ( x)  1  exp(  x /  )
E( X )  
V ( X )  2
Median ( X )  . ln( 2)
1
Failure Rate 

Poisson Process
X : Time interval between events
Example: Minutes between arrival of customers
X has an Exponential Distribution with Mean λ
Y: Number of customers arriving over a period T
Then, Y has a Poisson Distribution with Mean T/λ
T 
 
 T  
P(Y  k )  exp   
   k!
k
Poisson Process: Example
•Customers arrive at a shop according Exponential
Distribution
•On an average a customer arrives every two minutes
•i.e. λ = 2
•Number of arrivals in 30 minutes?
Queuing Process
Time between arrivals and time to serve both have
Exponential Distribution
Mean time between successive arrivals = λ
Mean time to serve = μ
If μ > λ then the queue will explode!
Let μ < λ
Utilisation Rate = μ / λ
= Probability that the service is busy = U (say)
Queuing Process (contd.)
N = Number of persons in the system including the
one being served
P(N=0) = (1- U)
P(N=1) = (1-U).U
P(N=2) = (1-U).U2
P(N=3) = (1-U).U3
P(N=k) = (1-U).Uk
Queuing Process (contd.)
E (N ) 


W = Time spent in the system

E (W ) 

2
Exercise 1: Poisson Distribution
•On an average there is 1 per cent of pieces lost in a
shipment;
•Order received for 592 pieces;
•The manufacturer shipped 600 pieces to cover for any
loss;
•What is the probability that the buyer will receive the
complete order?
Exercise 2: Poisson Distribution
•On an average 20 customers visit a shop in a day;
•If more than 30 customers visit in a day, then an
additional hand is hired from the next shop;
•What is the percentage of times the additional hand will
be required?
Exercise 3: Queuing Process
•Time between arrivals of successive patients at the OP and
time taken to attend to a patient, both have Exponential
distribution;
•There is only one Doctor at the OP in this small Nursing
Home;
•On an average a patient arrives every 35 minutes;
•Time taken to attend a patient is 22 minutes on an average;
•What is the probability that the Doctor is busy when a
patient arrives?
•What is the average (Expectation) number of patients at the
OP at any given time point?
Reference Sites For Theoretical Distributions
Specifications:
http://www.stat.vt.edu/~sundar/java/applets/Distributions.html
Probability Plotting:
http://www.chinarel.com/LifeDataWeb/probability_plotting.htm