Question

Verify that if we use the hazard rate approach to simulate the event times of a nonhomogeneous Poisson process whose intensity function \(\lambda(t)\) is such that \(\lambda(t) \leqslant \lambda\), then we end up with the approach given in method 1 of Section \(11.5\).

Short answer

Using the hazard rate approach for simulating event times of a nonhomogeneous Poisson process with intensity function \(\lambda(t) \leqslant \lambda\), we generate exponentially distributed interarrival times with mean \(1/\lambda\), accept tentative interarrival times based on the computed probability \(p = \lambda(t) / \lambda\), and compare this method to the given approach in Section 11.5. Both methods follow the same core algorithm, generating tentative event times and accepting or rejecting them based on the intensity function conditions. Therefore, it has been verified that using the hazard rate approach for a nonhomogeneous Poisson process with intensity function \(\lambda(t) \leqslant \lambda\) results in the approach given in method 1 of Section 11.5.

Step-by-step solution

Introduce nonhomogeneous Poisson process and hazard rate approach

A nonhomogeneous Poisson process is a generalization of the Poisson process in which the arrival rate, in this case, the intensity function \(\lambda(t)\), is not constant and can be any non-negative function of time. The hazard rate approach is used to simulate the event times of a stochastic process, such as the arrival times of a nonhomogeneous Poisson process.

Simulate event times using the hazard rate approach

To simulate the event times using the hazard rate approach, we follow these steps: 1. Generate an exponentially distributed random variable with mean \(1/\lambda\), which represents an upper bound for the intensity function, i.e., \(\lambda(t) \leqslant \lambda\). 2. Define the tentative interarrival time, \(T_n\), as the sum of the exponential random variables generated in step 1. 3. Compute the probability of accepting the tentative interarrival time, which is given by \(p = \lambda(t)/\lambda\). 4. Generate a uniform random variable \(U\). 5. If \(U \le p\), accept the tentative interarrival time as the actual interarrival time; otherwise, reject it and go back to step 1. In this case, the nonhomogeneous Poisson process is such that its intensity function \(\lambda(t) \leqslant \lambda\).

Compare the hazard rate approach with the given approach in Section 11.5

Now let's verify that we end up with the approach given in method 1 of Section 11.5 while using hazard rate approach: Method 1 of Section 11.5 states that to simulate a nonhomogeneous Poisson process, first, we generate the time until the next event in a homogeneous Poisson process with rate \(\lambda\). Then, we accept the tentative event time if it is within the time constraints and the intensity function of the nonhomogeneous Poisson process is less than or equal to the intensity function of the homogeneous Poisson process. Comparing our hazard rate approach and this approach, we can see that both methods follow the same steps. The main difference is the terminology, but the core algorithm is exactly same. In both approaches, we first generate an exponentially distributed interarrival time. Then, we determine if the tentative event time meets the required intensity function conditions. If it does, we accept; otherwise, we reject the tentative event time and move to the next iteration. Since the hazard rate approach for simulating event times is consistent with the method in Section 11.5 as described above, we have verified that using the hazard rate approach for a nonhomogeneous Poisson process with intensity function \(\lambda(t) \leqslant \lambda\) indeed results in the approach given in method 1 of Section 11.5.