Question

Let \(\left\\{N_{1}(t), t \geqslant 0\right\\}\) and \(\left[N_{2}(t), t \geqslant 0\right\\}\) be independent renewal processes. Let \(N(t)=\) \(N_{1}(t)+N_{2}(t)\) (a) Are the interarrival times of \(\\{N(t), t \geqslant 0\\}\) independent? (b) Are they identically distributed? (c) Is \(\\{N(t), t \geqslant 0\\}\) a renewal process?

Short answer

In conclusion, for the process \(N(t) = N_{1}(t) + N_{2}(t)\), the interarrival times of \(N(t)\) are not independent and not identically distributed, making \(N(t)\) not a renewal process.

Step-by-step solution

A renewal process is a stochastic process defined by a sequence of non-negative, independent, and identically distributed random variables \(\{X_i\}\), where \(X_i\) represents the time between the \((i-1)\)-th event and the \(i\)-th event, called the interarrival time. The process counts the number of events that have occurred by time \(t\), denoted as \(N(t) = \sum_{i=1}^{\infty} I(X_i \leq t)\). #Step 2: Analyze the new process N(t) and its interarrival times#

Let \(Y_i\) denote the \(i\)-th interarrival time of the process \(N(t)\). Since \(N(t) = N_{1}(t) + N_{2}(t)\), the events in process \(N(t)\) are formed by merging the events from both processes \(N_1(t)\) and \(N_2(t)\). Therefore, the interarrival times \(Y_i\) of process N(t) are formed by merging the interarrival times of both sub-processes \(N_1(t)\) and \(N_2(t)\). #Step 3: Determine if the interarrival times of N(t) are independent#

The interarrival times of \(N_1(t)\) and \(N_2(t)\) are independent by definition. However, when merging the two processes to form \(N(t)\), the interarrival times of \(N(t)\) are directly influenced by the interarrival times from both sub-processes. Therefore, the interarrival times of \(N(t)\) are not independent. #Step 4: Determine if the interarrival times of N(t) are identically distributed#

To determine whether the interarrival times of \(N(t)\) are identically distributed, we will look for any similarities in their distributions. Since the interarrival times of \(N_1(t)\) and \(N_2(t)\) are independent and identically distributed, their sum, the interarrival times of \(N(t)\), will have a distinct distribution. Thus, the interarrival times of \(N(t)\) are not identically distributed. #Step 5: Determine if N(t) is a renewal process#

As established in steps 3 and 4, the interarrival times of \(N(t)\) are not independent and not identically distributed. Since the interarrival times do not satisfy the definition of a renewal process, \(N(t)\) is not a renewal process. #Summary#

In conclusion, for the process \(N(t) = N_{1}(t) + N_{2}(t)\), the interarrival times of \(N(t)\) are not independent, not identically distributed, and \(N(t)\) is not a renewal process.

Key concepts

Stochastic Process

Imagine playing a game where the outcome is uncertain and varies every time you play. A stochastic process is somewhat like that game but in a mathematical sense. It is a collection of random variables representing a process that evolves over time, subject to some randomness. Each random variable in the collection corresponds to the state of the process at a particular time.

For example, the queue at your favorite coffee shop can be modeled as a stochastic process, where the number of people in the line changes randomly over time. In the context of renewal processes, we are particularly interested in counting processes, which tally the number of certain events occurring by a specific time. The unpredictability of these events is what makes such a process stochastic.

Within stochastic processes, renewal processes have a unique feature. They reset upon the occurrence of each event, much like a stopwatch that's reset after every lap. This 'renewal' property means that past behavior doesn't affect future evolution directly, which the independence of the interarrival times ideally signifies.

Interarrival Times

When studying any process where events occur sporadically over time, the timing between consecutive events holds great significance. These durations are known as interarrival times. To visualize, picture the arrival of buses at a station; the time gap between one bus leaving and the next one arriving is the interarrival time.

In a renewal process, interarrival times are crucial for understanding the dynamics of event occurrences. A critical assumption for such processes is that these times should be independent and identically distributed (i.i.d.). This would mean that no matter when you start timing (after any given bus has departed), the statistical properties of the wait time until the next bus (like the average wait time, the variance, etc.) remains constant, and the wait time for the next bus doesn't depend on how long the previous wait times were.

However, in the exercise provided, merging two independent renewal processes complicates things. Each process separately may have i.i.d. interarrival times, but when combined, the resulting process does not inherit this property, as the next event could come from either of the two original processes with no predictable pattern.

Independence of Random Variables

The concept of independence in probability theory is analogous to two events not influencing each other. For random variables, independence implies that knowing the outcome of one does not provide any useful information about the outcome of another. This concept is fundamental in the definition of a renewal process.

Coin Tosses as an Illustration

Think about tossing a fair coin. The result of one toss does not affect the result of the next toss; they are independent events. This same principle applies to the interarrival times in renewal processes—they should each be like a new coin toss, unaffected by the results of previous 'tosses' or interarrival times.

In the exercise, while the original processes had independent interarrival times, merging them breaks this independence. As events begin to intertwine from both processes, the outcome (the time until the next event) from the merged process becomes a function of the times from both original processes, and thus, they no longer exhibit independence.