Discuss the congruential method as used in generating random numbers.
(a) Differentiate between the following: (i) Physical and mathematical model (2 Marks) (ii) Static and Dynamic model (2 Marks) (iii) Reliability and validity (2 Marks) (b) Highlight the steps involved in Monte-Carlo simulation. (5 Marks) (c) A bakery shop keeps stock of a popular brand of cake. Daily demand based on the past experience is given below: Daily demand 0 15 5 35 45 50 frequency 1 15 20 50 12 2
Simulate the demand for the next ten days using the following random numbers 48, 78, 09, 87, 99, 77, 15, 14, 68 and 89. Find out the stock situation if the owner of the bakery decides to make 35 cakes every day. Unmet demand on any day is lost. (6 Marks) (d) Customers arrive at a watch repair shop according to a Poisson process at a rate of one per every 10 minutes and the service time is an exponential random variable with mean 8 minutes.
(i) Compute the following measures of performance L, Lq, W, Wq. (5 Marks) (ii) Suppose the arrival rate of the customer increases 10 percent. Find the corresponding changes in L, Lq, W, Wq. (5 Marks) (iii) Is the system stable? Determine the idle time of the server. (3 Marks)
QUESTION TWO (20 MARKS)
(a) Discuss the congruential method as used in generating random numbers. (8 Marks) (b) With the aid of a diagram, highlight the key steps involved in simulation. (5 Marks) (c) Explain how random numbers can be generated from the following functions using the inverse transform method.
(i) b xa ab xF , 1 )( (4 Marks) (ii) 0 1 x exf x (3 Marks)