Arrival Rate¶

Definition¶

Let $\{N_t\colon t\ge 0\}$ be a Poisson process. The arrival rate is the parameter $\lambda>0$ such that the expected number of arrivals in a time interval of length $t$ equals $\lambda t$ .

Interpretation¶

The arrival rate tells how quickly arrivals occur on average: $\lambda$ is the expected number of arrivals per unit of time. A larger rate means more frequent arrivals and shorter waits between them.

One realization of a counting process N t as a staircase rising by one at each arrival, together with the straight line of its expected value lambda t with slope 0.8.

The blue staircase is one realization of $N_t$ : it rises by one at each arrival, and the arrival times are marked on the time axis. The red line is $\mathbb{E}[N_t]=\lambda t$ with slope $\lambda=0.8$ $0.8$ . The staircase wanders around the line: by time $8$ it has counted six arrivals, while $\mathbb{E}[N_8]=0.8\cdot8=6.4$ $6.4$ .

The value of $\lambda$ depends on the time unit: $12$ per hour is $0.2$ per minute. It is a rate, not a probability, and can be larger than $1$ . Since $N_t\sim\operatorname{Pois}(\lambda t)$ $Pois (λ t)$ , both $\mathbb{E}[N_t]$ and $\operatorname{Var}(N_t)$ equal $\lambda t$ , and the probability of no arrival in $[0,t]$ is $\mathbb{P}(N_t=0)=\mathrm{e}^{-\lambda t}$ $e^{- λ t}$ . That is also the probability that the first arrival comes after time $t$ , which is why the first waiting time has the exponential distribution with mean $1/\lambda$ .

Rates of independent Poisson arrival streams add: customers arriving through one door at rate $\lambda_1$ and through another at rate $\lambda_2$ arrive at rate $\lambda_1+\lambda_2$ in total. The same addition of rates appears in the minimum of independent exponentials, which gives the waiting time until the next arrival in either stream.

Key Results¶

In a Poisson process with rate $\lambda$ , $\mathbb{E}[N_t]=\lambda t$ and $\operatorname{Var}(N_t)=\lambda t$ $λ t$ .
The first waiting time in a Poisson process with rate $\lambda$ has an exponential distribution with mean $1/\lambda$ .
If independent Poisson processes have rates $\lambda_1$ and $\lambda_2$ , their superposition has rate $\lambda_1+\lambda_2$ $λ_{2}$ .

Examples¶

Customer arrivals at a help desk can be modelled with rate $12$ per hour, giving an expected $6$ arrivals in half an hour.
If failures occur at rate $0.02$ per machine-hour, then $100$ machine-hours give an expected $2$ failures.
Radioactive decays with larger arrival rate have shorter expected waiting times between successive decays.

Prerequisites¶

Poisson Process

Sources¶

Probability Theory for EOR · Week 6

Aliases: intensity, rate parameter, arrival intensity, mean arrival rate Keywords: lambda rate Symbols: lambda

Last updated: 20 October 2026 03:02