4.2 Arrival times and interarrival times

Arrival times and interarrival times

Arrival times and interarrival times describe when events occur and how much time passes between them. These two perspectives are mathematically equivalent: if you know one, you can recover the other. For Poisson processes specifically, the interarrival times turn out to be exponentially distributed, which gives the process its characteristic memoryless behavior.

This topic connects counting processes (how many events by time $t$?) to the timing of individual events. That connection is central to queueing theory, reliability analysis, and simulation.

Arrival Times

The arrival time $T_n$ is the exact moment the $n$-th event occurs. Think of it as a timestamp: the 3rd customer walks in at $T_3 = 7.2$ minutes, the 4th at $T_4 = 9.5$ minutes, and so on.

Definition and Notation

• The $n$-th arrival time is denoted $T_n$, with the convention $T_0 = 0$.
• The full sequence $\{T_n, n \geq 0\}$ is an increasing sequence of random variables: $T_0 \leq T_1 \leq T_2 \leq \cdots$
• Each $T_n$ can be written as a cumulative sum of interarrival times:

$T_n = \sum_{i=1}^{n} X_i$

This sum is the key link between arrival times and interarrival times.

Connection to the Counting Process

The counting process $N(t)$ and the arrival times carry the same information, just viewed differently:

• $N(t) = \max\{n : T_n \leq t\}$ counts how many events have occurred by time $t$
• $T_n = \inf\{t : N(t) \geq n\}$ gives the time of the $n$-th event

These two expressions are inverses of each other. The event $\{N(t) \geq n\}$ happens exactly when $\{T_n \leq t\}$.

Examples

• In a bank, $T_n$ is the clock time when the $n$-th customer arrives.
• In reliability engineering, $T_n$ is when the $n$-th component failure occurs.
• In a network, $T_n$ is when the $n$-th data packet reaches a router.

Interarrival Times

The interarrival time $X_n$ is the gap between consecutive events:

$X_n = T_n - T_{n-1}, \quad n \geq 1$

If the 5th customer arrives at $T_5 = 12.0$ min and the 6th at $T_6 = 14.3$ min, then $X_6 = 2.3$ min.

Distribution and Notation

• The sequence of interarrival times is $\{X_n, n \geq 1\}$.
• Each $X_n$ has a CDF $F_n(x) = P(X_n \leq x)$ and, when it exists, a PDF $f_n(x)$.
• In many models (including the Poisson process), the $X_n$ are assumed to be independent and identically distributed (i.i.d.), so a single distribution $F(x)$ describes all of them.

Recovering Arrival Times from Interarrival Times

Since $T_n = X_1 + X_2 + \cdots + X_n$, the distribution of $T_n$ is the $n$-fold convolution of the interarrival distribution. For i.i.d. interarrivals, you can also use moment generating functions or Laplace transforms to compute this.

Examples

• In a call center, $X_n$ is the time between the $(n-1)$-th and $n$-th incoming calls.
• On a production line, $X_n$ is the time between completing consecutive units.
• In traffic analysis, $X_n$ is the time gap between successive vehicles passing a sensor.

Counting Processes

A counting process $\{N(t), t \geq 0\}$ tracks the total number of events up to time $t$. It must satisfy three properties:

1. $N(t) \geq 0$ and takes integer values for all $t \geq 0$
2. $N(t)$ is non-decreasing: if $t_1 \leq t_2$, then $N(t_1) \leq N(t_2)$
3. $N(t)$ is right-continuous: $\lim_{s \downarrow t} N(s) = N(t)$

The increment $N(t_2) - N(t_1)$ counts events in the interval $(t_1, t_2]$. The distribution of $N(t)$ can be derived from the interarrival time distribution.

Poisson Process: Arrival and Interarrival Structure

A Poisson process with rate $\lambda > 0$ is a counting process where:

• $N(0) = 0$
• The process has independent increments: event counts in non-overlapping intervals are independent
• The process has stationary increments: the distribution of $N(t+s) - N(s)$ depends only on the length $t$, not on $s$
• $N(t) \sim \text{Poisson}(\lambda t)$, so $P(N(t) = k) = \frac{(\lambda t)^k}{k!} e^{-\lambda t}$

The expected number of events in an interval of length $t$ is $E[N(t)] = \lambda t$.

Exponential Interarrival Times

The defining feature for this topic: in a Poisson process, the interarrival times $X_1, X_2, \ldots$ are i.i.d. Exponential($\lambda$) random variables.

• PDF: $f(x) = \lambda e^{-\lambda x}$ for $x \geq 0$
• CDF: $F(x) = 1 - e^{-\lambda x}$ for $x \geq 0$
• Mean: $E[X] = 1/\lambda$
• Variance: $\text{Var}(X) = 1/\lambda^2$

So if events arrive at rate $\lambda = 3$ per hour, the average wait between events is $1/3$ hour (20 minutes).

Gamma-Distributed Arrival Times

Since $T_n = X_1 + \cdots + X_n$ is a sum of $n$ i.i.d. Exponential($\lambda$) variables, $T_n$ follows a Gamma($n, \lambda$) distribution:

$f_{T_n}(t) = \frac{\lambda^n t^{n-1} e^{-\lambda t}}{(n-1)!}, \quad t \geq 0$

This is also called the Erlang-$n$ distribution (since $n$ is a positive integer). Its mean is $n/\lambda$ and variance is $n/\lambda^2$.

The Memoryless Property

The exponential distribution is the only continuous distribution with the memoryless property:

$P(X > s + t \mid X > s) = P(X > t) \quad \text{for all } s, t \geq 0$

This says that if you've already waited $s$ time units without an event, the remaining wait has the same distribution as if you'd just started. No "aging" occurs. This property is what makes the Poisson process so tractable, and it's the reason the process has independent and stationary increments.

Relationship Between Poisson Process and Poisson Distribution

The Poisson distribution and the Poisson process are connected but distinct:

• The Poisson distribution is a discrete probability distribution: $P(X = k) = \frac{\mu^k}{k!} e^{-\mu}$ for $k = 0, 1, 2, \ldots$
• The Poisson process is a stochastic process whose counts over any interval of length $t$ follow a Poisson distribution with parameter $\mu = \lambda t$.

The process generates Poisson-distributed counts; the distribution describes those counts.

Renewal Processes

A renewal process generalizes the Poisson process by allowing the i.i.d. interarrival times to follow any distribution $F(x)$, not just exponential.

Definition

• $N(0) = 0$
• The interarrival times $\{X_n, n \geq 1\}$ are i.i.d. with common CDF $F(x)$ and mean $\mu = E[X]$
• $N(t) = \max\{n : T_n \leq t\}$

The renewal function $m(t) = E[N(t)]$ gives the expected number of renewals by time $t$ and satisfies the renewal equation:

$m(t) = F(t) + \int_0^t m(t - x)\, dF(x)$

Key Results

• Elementary renewal theorem: $\lim_{t \to \infty} \frac{N(t)}{t} = \frac{1}{\mu}$ almost surely, and $\lim_{t \to \infty} \frac{m(t)}{t} = \frac{1}{\mu}$
• Excess life $Y(t)$: the time from $t$ until the next renewal. Its limiting distribution is $\lim_{t \to \infty} P(Y(t) \leq x) = \frac{1}{\mu} \int_0^x (1 - F(y))\, dy$

How Renewal Processes Differ from Poisson Processes

The Poisson process is the special case of a renewal process where $F(x) = 1 - e^{-\lambda x}$.

Applications

Queueing Theory

Arrival and interarrival times determine the input side of any queueing model. The standard Kendall notation (e.g., M/M/1) specifies the interarrival distribution as its first letter: "M" for memoryless (exponential), "G" for general, "D" for deterministic. Changing the interarrival distribution directly affects waiting times, queue lengths, and server utilization.

Reliability Theory

Interarrival times between failures model component lifetimes. Exponential interarrivals imply a constant failure rate; Weibull or lognormal interarrivals can capture increasing or decreasing failure rates. These models drive maintenance scheduling and warranty analysis.

Inventory Theory

Customer demand arrivals determine reorder policies. A Poisson demand process leads to tractable (s, Q) or (s, S) inventory models. When demand is bursty or correlated, renewal or compound Poisson models provide better fits.

Simulation Modeling

Discrete-event simulations generate synthetic interarrival times to drive the model forward. You sample from the fitted interarrival distribution (exponential, uniform, triangular, etc.) to schedule each next event. Getting this distribution right is critical for the simulation's validity.