Chapter Summary

Chapter Summary

Key Points

• 1.

  The Poisson process is the fundamental continuous-time arrival model. It is defined by independent increments, stationary increments, and orderliness ($\mathbb{P}(N(h)=1) = \lambda h + o(h)$). The count in any interval of length $t$ is Poisson($\lambda t$), and inter-arrival times are i.i.d. Exp($\lambda$).

• 2.

  Superposition and thinning preserve the Poisson property. Merging $m$ independent Poisson processes yields a Poisson process with the sum of rates. Independently retaining each arrival with probability $p$ produces a Poisson process with rate $p\lambda$, independent of the deleted process.

• 3.

  Compound and non-homogeneous extensions handle real traffic. The compound Poisson process $X(t) = \sum_{k=1}^{N(t)} Y_k$ models random-sized arrivals. The non-homogeneous Poisson process with $\lambda(t)$ models time-varying intensity (e.g., diurnal traffic).

• 4.

  The generator matrix $\mathbf{G}$ completely characterizes a CTMC. Off-diagonal entries $g_{ij} \geq 0$ are transition rates; rows sum to zero. The transition matrix is $\mathbf{P}(t) = e^{\mathbf{G} t}$, solving both Kolmogorov equations $\mathbf{P}'(t) = \mathbf{P}(t)\mathbf{G}$ and $\mathbf{P}'(t) = \mathbf{G}\mathbf{P}(t)$.

• 5.

  The stationary distribution of a CTMC solves $\boldsymbol{\pi}\mathbf{G} = \mathbf{0}$. This is the continuous-time analog of $\boldsymbol{\pi}\mathbf{P} = \boldsymbol{\pi}$ for DTMCs. For birth-death processes, detailed balance gives $\pi_n = \pi_0 \prod_{k=0}^{n-1}\lambda_k/\mu_{k+1}$.

• 6.

  The M/M/1 queue has a geometric stationary distribution. $\pi_n = (1-\rho)\rho^n$ with $\rho = \lambda/\mu < 1$. Mean number in system: $L = \rho/(1-\rho)$. Mean delay: $W = 1/(\mu - \lambda)$.

• 7.

  Little's law $L = \lambda W$ is universal. It holds for any stable queue, regardless of arrival process, service distribution, number of servers, or scheduling discipline. It links mean occupancy, arrival rate, and mean sojourn time.

• 8.

  Erlang-B and Erlang-C are the dimensioning formulas for multi-server systems. Erlang-B gives blocking probability (loss system, no buffer). Erlang-C gives waiting probability (delay system, infinite buffer). Both are parameterized by offered load $A = \lambda/\mu$ and number of servers $c$.