ISE-230 Fall 2015 Homework 3 Q1. Consider the birth-and-death process with = 5 for n=1,2,3,... and 0 = 4, 1 = 3, 2 = 2, and = 1 for = 3,4,5, ... 1. Draw the rate diagram (observe the slides or the textbook for conventions on how to draw rate diagrams with an infinite number of possible states). 2. Calculate the steady state probabilities 0 , 1 , 2 , 3 , and for a general > 3. 3. Calculate the average measures of performance: , , , . Note: You have to make an assumption on the number of servers, which was not specified in the question. Assume the most likely, given that = for all . Q2. We consider a small watch repair shop with one server. Customers arrive at a rate of 3 per hour according to a Poisson input process. The chance they leave immediately without waiting is /3 where is the number of customer in the shop, including the customer being served. The server tends to rush when he sees people waiting: his service time follows an exponential distribution with rate = 2 + . 1. Draw the rate diagram. 2. Write the balance equations 3. Solve the balance equations (you need as many linearly independent equations as you have unknowns so you will need to add an equation). 4. Calculate