Concept Architecture
Concept
Theoretically, Departure Process is a stochastic process describing the timing and sequence of entities leaving a system following service, treatment or completion of an activity. It forms a fundamental component of queueing theory, stochastic processes and discrete-event simulation by characterising system outflow. The concept exists because healthcare systems require mathematical representation of patient discharge, service completion and resource release to evaluate throughput, waiting times and system performance.
Mathematically, a Departure Process is represented as a counting process D(t), where D(t) denotes the cumulative number of departures by time t. The mathematical characteristics of the departure process depend upon both the arrival process and the service-time distribution. In many queueing models, such as the M/M/1 queue operating in steady state, the departure process follows a Poisson process with rate ?, consistent with Burke's Theorem.
In practice, departure processes are estimated from hospital discharge records, clinic activity data, electronic health records or simulation outputs. They are incorporated into queueing models, capacity planning models and discrete-event simulations to estimate patient throughput, bed occupancy, waiting times and healthcare resource utilisation.
Purpose
Used to model patient discharge, treatment completion and service throughput within healthcare systems, supporting queueing analysis, simulation modelling, operational planning and health economic evaluation.
Mathematical Formulae
Primary Formula
D(t) ~ Poisson(?t)
(For an M/M/1 queue in steady state under Burke's Theorem.)
Supporting Formulae
Expected departures:
E[D(t)] = ?t
Variance:
Var(D(t)) = ?t
Queue stability condition:
? = ? / ? < 1
Departure rate in steady state:
? = ??
Related Mathematical Methods
- Queueing Theory
- Poisson Process
- Burke's Theorem
- Markov Processes
- Discrete-Event Simulation
- Stochastic Processes
Example
An outpatient clinic operates with an average arrival rate of 18 patients per hour and a service rate of 24 patients per hour. Because utilisation is:
? = 18 � 24 = 0.75
the clinic operates under steady-state conditions. Under an M/M/1 queue, the expected departure rate equals the arrival rate of 18 patients per hour. Over an eight-hour clinic session:
E[D(8)] = 18 ? 8 = 144 departures.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| POISSON.DIST | =POISSON.DIST(A2,$B$1*8,FALSE) | Calculate the probability of a specified number of departures |
| SUM | =SUM(B2:B25) | Calculate cumulative patient departures |
| IF | =IF(B2/C2<1,"Stable","Unstable") | Assess queue stability |
| RAND | =RAND() | Generate stochastic service completion events within simulation models |
VBA (Optional)
Automate simulation of patient departures and healthcare throughput within queueing and discrete-event simulation models.
Sources
- Gross D, Shortle JF, Thompson JM, Harris CM. Fundamentals of Queueing Theory.
- 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.
- Burke PJ. The Output of a Queueing System. Operations Research. 1956.
Related Concepts (2)
Frequently Asked Questions (6)
What is a departure process?
The pattern by which entities, such as patients, leave a discrete event simulation after completing the services or processes being modelled.
Source: Law & Kelton 2000
Why does the departure process affect a system's capacity?
How quickly entities leave a service governs how soon the resources they held become free for others. If departures are slow, resources stay occupied and new arrivals must wait even when few are coming in, so the effective capacity of the system depends on the departure rate as much as the arrival rate. Delays in discharge from a hospital bed, for instance, can block admissions upstream. The pattern of departures therefore shapes throughput and queuing. Pidd (2004) explains this effect.
Source: Pidd 2004
How does the departure process work?
The departure process works as the outflow of entities from the simulation once they have completed their required services or reached an exit point. An entity departs after passing through the modelled activities and any queues and resources, at the time its final service concludes. The pattern of departures depends on the arrival process, the service times, and the resources available, since these determine how quickly entities are processed and leave. The departure process thus emerges from the system's dynamics rather than being specified directly.
Source: Law & Kelton 2000
Why does the departure process matter?
The departure process matters because it determines the throughput of the system, how many entities complete their processing over time, and, together with arrivals, the number present at any moment. It reflects how effectively the system serves its entities, so a departure rate that lags arrivals indicates congestion and growing queues. Analysing the departure process helps assess the system's capacity and performance, and in connected systems the departures from one part form the arrivals to the next, linking the flow through the whole.
Source: Law & Kelton 2000
How does the departure process relate to the arrival process?
The departure process relates to the arrival process as the outflow relates to the inflow of a system. If entities depart at least as fast as they arrive, the system keeps pace and queues stay bounded, whereas if arrivals outpace departures, entities accumulate and queues grow. In networks of connected systems, the departure process from one stage becomes the arrival process to the next, so the two processes link the stages. Together, arrivals and departures govern the number of entities in the system over time.
Source: Law & Kelton 2000
How is the departure process used in analysis?
The departure process is used in analysis to measure throughput and to assess whether a system can serve its arrivals without excessive congestion, since the rate and pattern of departures reflect the system's capacity and performance. Comparing departures with arrivals reveals whether queues are stable or growing. In modelling connected systems, the departure process from one stage feeds the next as arrivals, so it links the analysis of the parts. It thus informs judgements about capacity, waiting, and the flow of entities through the system.
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/departure-process
- Term code
- HE-EM-DES-004
Stable URI · Machine-readable · Resolvable · CC BY 4.0