Queuing Model

Queuing Model: Concept Architecture

Orientation and learning roadmap

This page first identifies the demand, service capacity, and routing choices that shape a queue. It then shows what a simple model can calculate, how to interpret its assumptions, and how waiting and capacity enter a health economic decision.

System boundary and flow

Specify where an entity enters the system, which service stages it visits, and whether a reported delay includes only time waiting or the entire time in the system. A patient pathway may contain several queues, so a result for one consultation stage should not be labelled as an end-to-end patient wait.

The arrival process describes when patients request service; the service process describes how long each server takes; the queue discipline determines who is served next. Also state the number of servers, opening hours, queue capacity, priority rules, balking or abandonment, and any blocking between stages. Patients with different acuity or service requirements may need separate classes and explicit routing rather than one average patient.

Core measures and conservation relationship

Waiting time, queue length, service utilisation, throughput, and the probability of breaching a target answer different operational questions. Their definitions must specify the population, observation period, and whether they refer to a queue or the full system.

For a stable system with consistent boundaries, Little's law relates long-run average occupancy $L$, effective throughput $\lambda_{\mathrm{eff}}$, and mean time in the system $W$:

$$L=\lambda_{\mathrm{eff}}W.$$

Using queue-only quantities gives $L_q=\lambda_{\mathrm{eff}}W_q$, where $L_q$ is mean number waiting and $W_q$ is mean waiting time before service. The effective throughput is the rate of patients actually admitted to and completed by the defined system in steady state; it need not equal offered demand if patients are turned away or abandon the queue.

A transparent single-server example

An $M/M/1$ model assumes independent Poisson arrivals, exponentially distributed service times, one server, first-come-first-served service, and an effectively unlimited waiting space. Its stationary formulas apply only if the arrival rate $\lambda$ is below the server's service rate $\mu$.

With $\lambda=8$ patients per hour and $\mu=10$ patients per hour, utilisation is $\rho=\lambda/\mu=0.8$. The model gives:

$$W_q=\frac{\lambda}{\mu(\mu-\lambda)}=0.4\text{ hours}=24\text{ minutes},\qquad W=\frac{1}{\mu-\lambda}=0.5\text{ hours}=30\text{ minutes}.$$

Thus the model's mean service time is six minutes and its mean total time is 30 minutes; Little's law yields $L=8\times0.5=4$ patients on average in the system. If arrivals rose to nine per hour with service capacity unchanged, the same idealised model would predict $W=1$ hour, illustrating how delays can rise sharply as capacity approaches demand. These values are illustrative model outputs, not observed waiting times or a safe staffing recommendation.

Choosing and checking a model

The shorthand $A/S/c$ identifies an arrival distribution, service-time distribution, and number of servers, but additional assumptions still need to be stated. A many-server, priority, time-varying, finite-capacity, or network model may be necessary for a clinic, theatre pathway, or emergency department.

Compare predicted waits, throughput, and occupancy against observed data by time period and patient class, and examine extreme waits as well as means. Use sensitivity analysis for arrival patterns, service-time variability, staffing, and routing; simulation may be preferable when the pathway is too complex for a credible closed-form model.

Economic and decision use

Adding capacity can reduce waits but requires staff, space, and other resources that have alternative uses. A decision analysis should compare feasible staffing or scheduling options using operating costs, patient and carer time costs where the perspective calls for them, access and equity effects, and any defensible link from delay to health outcomes.

Utilisation alone is not an efficiency verdict: a near-fully occupied server can coexist with unacceptable waiting, while spare capacity may protect access when demand varies. Report whose waits improve, whether a bottleneck simply moves downstream, and whether forecast demand and cost assumptions match the decision horizon.

Interpretation and evidence base

A queuing model represents congestion arising from the interaction of arrivals, service, capacity, and service rules; it does not make patients' clinical outcomes or treatment effects identifiable by itself. Little's law is a broad steady-state accounting relationship, whereas the $M/M/1$ waiting-time formulas require much stronger distributional and service assumptions.

Little's original proof establishes the long-run flow relation. Queueing for healthcare discusses capacity and waiting in care delivery, while NHS England's demand and capacity models illustrate why demand variation matters in service planning.