Prerequisites & Notation

Before You Begin

This chapter moves from the discrete-time Markov chains of Chapter 17 to continuous-time stochastic models. You need strong familiarity with the exponential distribution (memoryless property), discrete-time Markov chain theory (stationary distributions, detailed balance), and matrix exponentials. If any item below feels uncertain, review it before proceeding.

  • Discrete-time Markov chains: transition matrix P\mathbf{P}, stationary distribution π\boldsymbol{\pi}(Review fsp-ch17)

    Self-check: Can you write πP=π\boldsymbol{\pi} \mathbf{P} = \boldsymbol{\pi} and explain what it means?

  • Exponential distribution: XsimtextExp(lambda)X \\sim \\text{Exp}(\\lambda), memoryless property(Review fsp-ch06)

    Self-check: Can you prove that P(X>s+t∣X>s)=P(X>t)\mathbb{P}(X > s + t \mid X > s) = \mathbb{P}(X > t) for X∼Exp(λ)X \sim \text{Exp}(\lambda)?

  • Matrix exponential: eAt=∑k=0∞(At)k/k!e^{\mathbf{A}t} = \sum_{k=0}^{\infty} (\mathbf{A}t)^k / k!

    Self-check: Can you compute eAte^{\mathbf{A}t} for a diagonal 2×22 \times 2 matrix?

  • Conditional probability, total probability, Bayes' theorem(Review fsp-ch02)

    Self-check: Can you partition a sample space and apply the law of total probability?

  • Geometric series and convergence of power series

    Self-check: Do you know ∑n=0∞ρn=1/(1−ρ)\sum_{n=0}^{\infty} \rho^n = 1/(1-\rho) for ∣ρ∣<1|\rho| < 1?

Notation for This Chapter

Symbols introduced in this chapter. We follow Caire's conventions throughout. The generator matrix is denoted G\mathbf{G} (some texts use Q\mathbf{Q}); the stationary distribution is π\boldsymbol{\pi} as in Chapter 17.

SymbolMeaningIntroduced
{N(t):t≥0}\{N(t) : t \geq 0\}Poisson counting processs01
λ\lambdaArrival rate (Poisson intensity); also birth rate λn\lambda_n in state nns01
μ\muService/death rate; μn\mu_n in state nns04
G\mathbf{G}Generator (rate) matrix of a CTMC: G=[gij]\mathbf{G} = [g_{ij}]s03
P(t)\mathbf{P}(t)Transition probability matrix of CTMC: Pij(t)=P(X(t)=j∣X(0)=i)P_{ij}(t) = \mathbb{P}(X(t)=j \mid X(0)=i)s03
π\boldsymbol{\pi}Stationary distribution of the CTMC: πG=0\boldsymbol{\pi} \mathbf{G} = \mathbf{0}s03
ρ\rhoTraffic intensity: ρ=λ/μ\rho = \lambda / \mu (M/M/1) or ρ=λ/(cμ)\rho = \lambda / (c\mu) (M/M/c)s04
LLMean number of customers in systems04
WWMean sojourn time (waiting + service)s04
B(A,c)B(A, c)Erlang-B blocking probability with offered load AA and cc serverss05
C(A,c)C(A, c)Erlang-C waiting probability with offered load AA and cc serverss05
← Ch 17The Poisson Process