MA570 Spring 2021 Stochastic Methods in OR Krigman Final Exam Cheat Sheet
Please show all your work in the blue book provided and box off your answers.
Please be neat when writing down your answers. When it comes to your exam grade, neatness counts!!!
If you get stuck I recommend moving to the next problem and coming back to the one that gave you trouble later.
Some useful formulas are given next. Little’sLaw:W=L; Wq =Lq .
22 +2 Pollaczek- Khintchine formula: Lq = 2(1− ) .
Steady-state equations for birth-and-death process being described by M/M/1 Model: =s;
P =(1−)n; L= n
2 nP = n(1−)n = = ; L = (n−1)P =
q
n=0 n=0 1− − n=1 (−)
n
n
Steady-state probabilities for other (more general) types of birth-and-death systems with exponentially distributed service and arrival times that we had studied:
…
Exponential distribution pdf: f (t) = et ;t 0 The cumulative probability is: P(T t) = 1− e−t ; t 0 .
T 0 ; t 0
Expected value and variance are: E(T) = 1 and var(T) = 1
2 (t)ne−t
Poisson distribution function: P{X (t) = n} = n! ; n = 0,1,2… t 0 . Expected value is: E{X (t)} = t . Exponentialfunctionexpansion:𝑒𝑥 =1+𝑥 +𝑥2 +𝑥3 +⋯, −∞<𝑥<∞.
1! 2! 3!
Geometric series summation: 1 + x + 𝑥2 + 𝑥3 + ⋯ 𝑥𝑁 = 1−𝑥𝑁+1 𝑓𝑜𝑟 𝑎𝑛𝑦 𝑥 ≠ 1. 1−𝑥