Sheldon M. Ross's Stochastic Processes (2nd Edition) is widely regarded as a seminal text for its intuitive, non-measure theoretic approach. If you are reviewing a draft for its solutions manual, Core Content Overview
The transition rate $q_ij$ from state $i$ to $j$. The time spent in state $i$ before jumping is Exponential with rate $v_i = \sum_j \neq i q_ij$. --- Sheldon M Ross Stochastic Process 2nd Edition Solution
To illustrate what a high-quality looks like, consider a classic problem: Sheldon M
Independent contributors have compiled solution sets from various university courses (including Columbia and the University of Michigan) into central repositories like the Stochastic Process Ross 2nd edition GitHub Academic Course Sites: The time spent in state $i$ before jumping
Most students ignore this chapter. The problems here involve Borel-Cantelli lemmas and advanced expectation tricks that reappear in Chapter 8 (Brownian motion). A good solution set for Chapter 1 should show you how to handle "indicator variable" splitting—Ross’s favorite technique.
Find the probability that the 2nd arrival occurs before time $t$. Approach: Let $X_1, X_2$ be i.i.d. Exp($\lambda$). We want $P(X_1 + X_2 \le t)$. Since the sum of $n$ i.i.d. Exponential($\lambda$) variables is a Gamma($n, \lambda$) distribution: $$f_S_2(t) = \frac\lambda^2 t e^-\lambda t1! = \lambda^2 t e^-\lambda t$$ Integrate to find the CDF, or use the memoryless property arguments often used by Ross.