Concept Architecture
Concept
Theoretically, Arrival Process is a stochastic process describing the occurrence of entities entering a system over time. It forms a fundamental component of queueing theory, stochastic modelling and discrete-event simulation, representing the probabilistic mechanism governing the timing and frequency of arrivals. In health economics, arrival processes are used to model patient arrivals to healthcare facilities, incident disease cases, referrals and demand for healthcare services.
Mathematically, an Arrival Process is commonly represented as a counting process N(t), where N(t) denotes the cumulative number of arrivals by time t. The most widely recognised model is the homogeneous Poisson process, characterised by a constant arrival rate ? and exponentially distributed inter-arrival times. More general formulations include non-homogeneous Poisson processes and renewal processes when arrival rates vary over time or inter-arrival times follow alternative probability distributions.
In practice, arrival processes are estimated using observational data, routinely collected healthcare activity data or epidemiological surveillance systems. Parameters are estimated using maximum likelihood estimation or related statistical methods and incorporated into queueing models, discrete-event simulation and capacity planning models to evaluate healthcare demand, waiting times and resource utilisation.
Purpose
Used to model the timing and frequency of patient, disease or service arrivals within healthcare systems to support queueing analysis, simulation modelling, capacity planning and health economic evaluation.
Mathematical Formulae
Primary Formula
N(t) ~ Poisson(?t)
Supporting Formulae
Probability of n arrivals:
P(N(t) = n) = e???(?t)� / n!
Expected number of arrivals:
E[N(t)] = ?t
Variance:
Var(N(t)) = ?t
Inter-arrival time:
T ~ Exp(?)
E(T) = 1/?
Related Mathematical Methods
- Poisson Process
- Renewal Process
- Queueing Theory
- Discrete-Event Simulation
- Stochastic Processes
- Maximum Likelihood Estimation
Example
An emergency department receives patients at an average rate of ? = 12 patients per hour. Assuming a Poisson arrival process, the expected number of arrivals over a four-hour period is:
E[N(4)] = 12 ? 4 = 48 patients.
The probability of observing exactly 50 arrivals during this period is calculated using the Poisson distribution with mean ?t = 48.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| POISSON.DIST | =POISSON.DIST(A2,$B$1*4,FALSE) | Calculate the probability of a specified number of arrivals over a time interval |
| EXP | =EXP(-$B$1*A2) | Calculate exponential inter-arrival probabilities |
| RAND | =RAND() | Generate random arrivals in simulation models |
| -LN | =-LN(RAND())/$B$1 | Simulate exponentially distributed inter-arrival times |
VBA (Optional)
Automate generation of stochastic patient arrival streams for queueing and discrete-event simulation models.
Sources
- Ross SM. Introduction to Probability Models.
- Law AM. Simulation Modeling and Analysis.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation.
- Banks J, Carson JS, Nelson BL, Nicol DM. Discrete-Event System Simulation.
- ISPOR-SMDM Modeling Good Research Practices Task Force Reports.
Related Concepts (2)
Library
Publications
1
Modeling Using Discrete Event Simulation: A Report of the ISPOR-SMDM Modeling Good Research Practices Task Force-4 — Karnon, Stahl, Brennan, Caro, Mar & Moller, Task Force Report 4 ed., 2012 (Value in Health / Medical Decision Making)
Best-practice guidance on discrete event simulation (DES) for health economic evaluation — when DES is preferable to cohort approaches, and how to structure, populate and validate such models.
Journal ArticleView source →
Frequently Asked Questions (6)
What is an arrival process?
The pattern by which entities, such as patients, enter a discrete event simulation over time, governed by a probability distribution of intervals.
Source: Law & Kelton 2000
How does the arrival process affect a simulated service?
The rate and pattern with which entities arrive drive how heavily a simulated service is loaded and how much queuing forms. Arrivals that come steadily are easier to serve than the same number bunched together, since a sudden cluster can overwhelm capacity and create waits even when average demand is modest. Because arrivals often occur at random rather than evenly, the process is usually described by a probability distribution of the gaps between them. Getting it right matters for the waits the model predicts. Pidd (2004) describes this influence.
Source: Pidd 2004
How is an arrival process represented?
An arrival process is represented by a probability distribution of the times between successive arrivals, or equivalently of the number of arrivals in a period. A common choice is the Poisson process, in which arrivals occur randomly and independently at a constant average rate, giving exponentially distributed inter-arrival times. Other distributions represent more regular or more clustered arrivals, or arrival rates that vary over time, such as by hour or day. The distribution is chosen to match the observed pattern of arrivals.
Source: Law & Kelton 2000
Why does the arrival process matter in a simulation?
The arrival process matters because it drives the demand placed on the system, determining how many entities need service and when, which in turn affects queues, waiting times, and resource use. A simulation's results, such as congestion and delays, depend heavily on the arrival pattern, since bursts of arrivals or high average rates strain resources. Representing the arrival process accurately is therefore important, as errors in it would misestimate the load on the system and hence the performance measures the simulation produces.
Source: Law & Kelton 2000
How is an arrival process estimated?
An arrival process is estimated from data on when entities actually arrive, such as records of patient arrivals, by examining the pattern of inter-arrival times and fitting a suitable distribution. Whether arrivals are random, regular, or clustered, and whether the rate varies over time, is assessed and represented. Where the rate changes, for instance by time of day, a time-varying arrival process is used. The estimated process should reproduce the observed arrival pattern, so the simulation's demand reflects reality.
Source: Law & Kelton 2000
How does the arrival process relate to queuing?
The arrival process relates directly to queuing because it determines the rate and pattern at which entities demand service, which, together with the service process and the number of resources, governs whether queues form and how long they grow. If arrivals exceed the capacity to serve them, queues build up and waiting times rise. The arrival process is thus one of the key inputs, alongside service times and resources, that determine congestion in a simulation or queuing model.
Source: Law & Kelton 2000
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 6 Oct 2025
Content version: 1.0.0
Canonical Identity
- Persistent URI
- https://healtheconomics.wiki/concept/arrival-process
- Term code
- HE-EM-DES-001
Stable URI · Machine-readable · Resolvable · CC BY 4.0