1. 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.)

2. Continuous Distribution X is a continuous random variable Density Function: f(x) ; a < x < b Distribution Function: Failure Rate: = F(x)

3. Exponential Distribution

4. 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

5. 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?

6. 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)

7. 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).U 2 P(N=3) = (1-U).U 3 P(N=k) = (1-U).U k

8. Queuing Process (contd.) W = Time spent in the system

9. 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?

10. 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?

11. 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?

12. 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