Probability, Stochastic Processes, and Queueing Theory
This domain tests probability modeling and stochastic system performance. Expect conditional probability, Bayes rule, random variables, Poisson processes, Markov chains, Little's law, and queueing measures such as utilization, waiting time, and system length.
Core rules
- State the sample space and conditioning event before applying Bayes rule.
- For a Poisson process, counts in disjoint intervals are independent and interarrival times are exponential.
- A queue is stable only when arrival rate is below service capacity; utilization ρ=λ/μ must be less than 1 for a stable single-server queue.
- Little's law L=λW applies to stable systems; W includes service time, while Wq is waiting before service.
- M/M/1 formulas assume Poisson arrivals, exponential service, one server, unlimited capacity, and FIFO discipline.
Worked example
An M/M/1 queue has λ=6 arrivals per hour and μ=10 services per hour. Utilization is ρ=0.6. Average number in system is L=ρ/(1-ρ)=0.6/0.4=1.5. By Little's law, W=L/λ=1.5/6=0.25 hour, or 15 minutes. Average number waiting is Lq=ρ²/(1-ρ)=0.36/0.4=0.9, so Wq=Lq/λ=0.9/6=0.15 hour, or 9 minutes. The 6-minute difference is average service time.
Traps
Applying M/M/1 when service is not exponential, using λ≥μ and expecting a finite steady state, mixing minutes and hours, and reporting W when the question asks for Wq.
Key points
- Bayes rule needs explicit conditioning.
- Poisson arrivals imply exponential interarrivals.
- Stability requires λ<μ for one server.
- Little's law links L, λ, and W.
- W includes service; Wq does not.