Math 311

Class 23

Conditional Distribution of Arrivals

Suppose \(\{N(t), t \ge 0\}\) is a Poisson process, and that \(N(t) = n\) for some time \(t\).

Then the arrival times of each of the \(n\) events that happened by the time \(t\) are independently uniformly distributed in the interval \((0,t]\).

Sampling a Poisson Process

Given a Poisson process \(\{N(t), t \ge 0\}\) with rate \(\lambda\). Suppose the events can be classified into one of \(k\) different types, with probabilities \(P_i(s)\), \(i = 1, 2, \ldots, k\), where \(s\) is the time the event occurred, independently of any other events.

Define \(N_i(t) = {}\) the number of events classified as type \(i\) that occurred by the time \(t\).

Then for each \(t\), the \(N_i(t)\)’s, \(i = 1, 2, \ldots, k\) are independent Poisson distributed random variables, with means (or rates)

\[\operatorname{E} \left(N_i(t)\right) = \lambda \int_0^t P_i(s)\;ds\]

Example (Replacing Machine Parts)

A machine has a large number of identical parts that fail according to a Poisson process with rate \(\lambda\).

  • Some failures can be immediately fixed.
  • In other cases, the part must be replaced.

Suppose the probability that the part can be fixed is given by the function \(P_1(s) = e^{-rs}\), where \(s\) is the time of the failure.

What is the expected number of parts that must be replaced by the time \(t = 20\)?

Example (Infinite Server Queue)

Suppose customers arriving to a service station can be modeled as a Poisson process with rate \(\lambda\).

  • When a customer arrives, they are immediately served by a server.
  • Service times are independent of each other and of the arrival time, distributed according a cumulative density function \(G\).

Define the following two random variables:

  • \(X(t) = {}\) the number of customers that completed the service by the time \(t\).
  • \(Y(t) = {}\) the number of customers that are still being served at the time \(t\).

How are those distributed?

Example (Number of Infections)

Consider a disease with a long incubation period and a steady infection rate (such as HIV or Tuberculosis).

  • Individuals contracting the disease can be modeled by a Poisson process with an unknown rate \(\lambda\).
  • Suppose the incubation period (time between the infection and the appearance of symptoms) can be modeled by a random variable having a known cumulative density function \(G\).

Define

  • \(N_1(t) = {}\) the number of people showed symptoms by the time \(t\)
  • \(N_2(t) = {}\) the number of people infected but not showing symptoms yet by the time \(t\)