Arrival & Service Times for Assignment 3 Byung-Hyun Ha

What We’ll Do Generate input data for your own Part Number Arrival Time Inter-arrival Time Service Time

Overview Use your student ID as a seed, i.e. Z 0 For i th inter-arrival time (A i ) and service time (S i ) –Generate random integers (Z 2i–1, Z 2i ) –Get random numbers (U 2i–1, U 2i ) from integers –Generate A i and S i from random numbers

Generate Random Integer Linear congruential generator (LCG) –Consult 12.1 of our textbook –Z i = (aZ i-1 + c) mod m –For us  a = 13821, c = 0, m = 2 15 = Knuth - and Borosh and Niederreiter LCGs –

Generate Random Integer Example (Zi =  Z i-1 mod 32768) –Z 0 =  seed: my employee id –Z 1 =  mod = –Z 2 =  mod = –Z 3 = … You can use a calculator or an excel sheet 

Get Random Number U i ~ distributed uniformly in [0,1] –U i = Z i / m = Z i / Example –U 1 = Z 1 / = /  0.98 –U 2 = Z 2 /  0.54 –U 3 = Z 3 /  0.92 –U 4 = Z 4 /  0.59 –…

Generate A i and S i Generating random variates –Consult 12.2 of our textbook In case of exponential dist. with  = –PDF: f(x) = (1/ )e -x/ –CDF: F(x) = 1 - e -x/ –with U ~ distributed uniformly in [0,1] U = F(X) = 1 - e -X/  X = -  ln(1 – U)

Generate A i and S i Pictorial illustration

Generate A i and S i Assumption –A i ~ distributed exponential with  = 5 –S i ~ distributed exponential with  = 4 Example –A 1 = -5  ln(1-U 1 ) = -5  ln(1-0.98)  –S 1 = -4  ln(1-U 2 ) = -4  ln(1-0.54)  3.12 –A 2 = -5  ln(1-U 3 ) = -5  ln(1-0.92)  –S 2 = -4  ln(1-U 4 ) = -4  ln(1-0.59)  3.61

What We Have Done Generate input data for my own Part Number Arrival Time Inter-arrival Time Service Time

Further Readings Chapter 12 of the textbook Linear congruential generator from Wikipedia – al_generatorhttp://en.wikipedia.org/wiki/Linear_congruenti al_generator Knuth - and Borosh and Niederreiter LCGs – de3.htmlhttp://random.mat.sbg.ac.at/~charly/server/no de3.html