Departure Process

Departure Process in Queueing and Discrete Event Simulation Models of Health Services

In a queuing model or discrete event simulation of a health service, the departure process records the moments at which patients finish with a stage of care and move on, whether that stage is a consultation room, a ward bed or the whole pathway. The analyst does not usually choose it. Departures follow from the arrival process, the service times and the capacity available, and in a stable system they leave at the same long-run rate as patients arrive. Their timing still matters to health economists, because the departures from one stage are the arrivals to the next, and a patient who cannot leave keeps a bed or slot that someone else is waiting for.

Departures as the output of arrivals, service and capacity

A departure process can be written as a running count of patients who have left a stage. Every patient who leaves must first have arrived, so the count is tied to arrivals and to the number still present.

$$D(t) = A(t) - N(t)$$

where $D(t)$ is the number of departures in the interval from 0 to $t$, $A(t)$ is the number of arrivals in the same interval, and $N(t)$ is the number of patients in the stage at time $t$, waiting or in service, for a stage that starts empty.

Whitt, describing the Queueing Network Analyzer, notes that over a long interval the number of departures is the number of arrivals minus the number in queue, and that the number in queue fluctuates around its steady-state distribution while arrivals keep growing. In a stable stage the long-run departure rate therefore equals the arrival rate, and that rate is the stage's throughput. Gallager's text on stochastic processes shows that the single-server queue with Poisson arrivals and exponential service needs a traffic intensity below 1 to reach a steady state. When demand exceeds capacity the queue grows instead and departures fall short of demand.

For a single server that sees patients in order of arrival, each departure time follows from three quantities.

$$D_n = \max(A_n, D_{n-1}) + S_n$$

where $D_n$ is the departure time of patient $n$, $A_n$ is the arrival time of patient $n$, $D_{n-1}$ is the departure time of the previous patient (taken as 0 for the first patient), and $S_n$ is the service time of patient $n$. Service starts when the patient has arrived and the server is free, whichever is later.

In an illustrative trace, three patients arrive at minutes 0, 5 and 7 and need 8, 4 and 6 minutes of service. The first leaves at 0 + 8 = 8 minutes, the second at max(5, 8) + 4 = 12 minutes and the third at max(7, 12) + 6 = 18 minutes. The gaps between arrivals are 5 and 2 minutes and those between departures are 4 and 6 minutes, so the patients leave in a different rhythm from the one in which they came.

How departures enter health economic models

The ISPOR-SMDM good practice report on discrete event simulation, by Karnon and colleagues, notes that individual entities can enter and leave a model between its start and end times, and lists Remove Entities as one of the usual parts of an implementation. Many models used in health technology assessment are what the report calls non-constrained, following the common assumption that every required resource is available as needed. In such an individual patient simulation a patient leaves at death or at the end of the time horizon, and, unless patients interact (for example through disease transmission), one patient's exit has no effect on anyone else.

In constrained-resource models patients compete for clinic appointments, beds or donor organs, so each departure releases a resource that a waiting patient can take. The report recommends that these models consider effects on health-related outcomes and not focus solely on resource utilisation and system capacity. In bed models the departure process sits behind familiar measures, since the average length of stay divides bed-days by discharges and the occupancy rate depends on how quickly beds are vacated as well as how often they are filled.

Burke's theorem for M/M/1 and M/M/c queues

An M/M/c queue has Poisson arrivals, exponential service times and c identical servers. Burke's 1956 paper showed that for Poisson input, a single waiting line without defections and independent exponential service times, the equilibrium distribution of the number of service completions in an arbitrary interval is the same as the input distribution, for any number of servers. Gallager states the result for the M/M/1 queue in steady state with arrival rate below service rate. The departure process is Poisson with the arrival rate, and the number in the system at any time is independent of earlier departures. He adds that both parts also hold for M/M/m queues.

$$P\{D(s + t) - D(s) = n\} = \frac{(\lambda t)^n e^{-\lambda t}}{n!}$$

where $D(s + t) - D(s)$ is the number of departures in a window of length $t$ starting at time $s$, $n$ is a whole number of departures, and $\lambda$ is the arrival rate, which in steady state is also the departure rate. The count follows a Poisson distribution and the gaps between departures follow an exponential distribution.

For pathways with several stages, Gallager shows that when the departures of one M/M/1 queue feed a second single-server queue with exponential service, the second queue is also M/M/1, the numbers in the two queues at any time are independent, and with first-come first-served service a patient's total time is the sum of two independent times, so each stage can be analysed on its own. The argument fails where a cycle of queues lets departures re-enter earlier queues. In a queue where some departures return for further service, the combined arrival stream is not Poisson, although the departures that leave the system are.

Departure variability when service times are not exponential

Once service times are not exponential, the departure stream from a stage where patients queue is in general no longer Poisson. Whitt notes one exception, the M/G/∞ queue, which has so many servers that no patient waits and whose departures stay Poisson. Whitt's Queueing Network Analyzer describes each stream by its rate and its squared coefficient of variation, which equals 1 for exponential gaps, 0 for constant gaps and more than 1 for mixtures of exponentials. For a single-server stage it approximates the variability of departures as follows.

$$c_d^2 = \rho^2 c_s^2 + (1 - \rho^2) c_a^2$$

where $c_d^2$ is the squared coefficient of variation of the time between departures, $c_s^2$ is that of the service time, $c_a^2$ is that of the time between arrivals, and $\rho$ is the traffic intensity, the arrival rate multiplied by the mean service time and divided by the number of servers.

Whitt describes the result as a weighted average of arrival variability and service-time variability. At high utilisation the weight moves towards the service times, so a busy stage passes its own variability downstream. With constant service times the departures are less variable than the arrivals, although Whitt cautions that the reduction seen in practice is often smaller than predicted. His extension for several servers gives a value of 1 for M/M/m queues, matching Burke's theorem.

The variability reaching the next stage then enters its waiting time. Whitt's waiting-time approximation for a single-server stage is as follows.

$$W_q = \frac{\tau \rho (c_a^2 + c_s^2)}{2(1 - \rho)}$$

where $W_q$ is the expected wait before service, $\tau$ is the mean service time, $\rho$ is the traffic intensity, and $c_a^2$ and $c_s^2$ are as above. Whitt uses this form when $c_a^2$ is at least 1 and a correction factor below 1 otherwise. With both coefficients equal to 1 it reduces to the exact M/M/1 result.

Worked example with a two-stage outpatient clinic

All figures in this example are illustrative. Patients arrive at a one-stop clinic as a Poisson stream at 4 per hour, so the squared coefficient of variation of the arrival gaps, called the coefficient below, is 1. Each sees an assessment nurse with a mean service time of 12 minutes and then a consultant whose service times are exponential with a mean of 13.5 minutes. The clinic is assumed to run long enough to reach a steady state. Two ways of running the assessment are compared, with the same mean service time.

1. Utilisation. At the nurse stage utilisation is 4 × 12 / 60 = 0.8. At the consultant stage it is 4 × 13.5 / 60 = 0.9.

2. Departure variability from the nurse stage. Under option A the assessment mixes brief and long cases, with a service coefficient of 2. Under option B a standard protocol gives exponential-like times with a coefficient of 1. For option A the departure coefficient is 0.64 × 2 + 0.36 × 1 = 1.64. For option B it is 0.64 × 1 + 0.36 × 1 = 1.00, the Poisson value given by Burke's theorem.

3. Wait for the nurse. The M/M/1 wait is 12 × 0.8 / 0.2 = 48 minutes. Option A multiplies this by (1 + 2) / 2 = 1.5, giving 72 minutes, and option B keeps it at 48 minutes.

4. Wait for the consultant. The M/M/1 wait is 13.5 × 0.9 / 0.1 = 121.5 minutes. Option A multiplies this by (1.64 + 1) / 2 = 1.32, giving 160.38 minutes, and option B keeps it at 121.5 minutes.

5. Totals. Expected waiting is 72 + 160.38 = 232.38 minutes per patient under option A and 48 + 121.5 = 169.5 minutes under option B, a difference of 62.88 minutes. Of that, 38.88 minutes (160.38 less 121.5) arises at the consultant stage, whose own service times are the same in both options. Over an 8-hour session of 4 × 8 = 32 patients, that downstream part comes to 32 × 38.88 = 1,244.16 patient-minutes, about 20.7 patient-hours.

6. Interpretation. Throughput is 4 patients an hour in both options, so a model tracking only departure rates would see no difference. On Whitt's approximation, a model treating the consultant's arrivals as Poisson under option A would understate the consultant wait by about 39 minutes per patient. Whitt notes that as the next stage nears full load its arrivals behave more like the original stream, so at a utilisation of 0.9 the true gap is likely to be smaller. Whether that waiting matters economically depends on whether it changes health, patient time costs or overtime. A real clinic that opens empty never reaches the assumed steady state, so a discrete event simulation of the session would give shorter waits early in the day.

Blocking, discharge timing and downstream capacity

In hospitals, departures often depend on what happens after a stage. Koizumi, Kuno and Smith modelled residential mental health services in Philadelphia as a queuing network with blocking, where patients turned away from the setting to which they are referred must stay in their present facility until space becomes available. They found that a shortage of one type of facility may have created upstream blocking, and that removing such bottlenecks may be the most efficient way to reduce congestion in the whole system. In a simulation, a patient ready to leave keeps holding a bed, so the ward's departures depend on capacity elsewhere.

Wong and colleagues built a system dynamics simulation of general internal medicine at an academic hospital in Toronto, noting that reduced weekend discharges are a reality for many hospitals. Spreading discharges evenly over the week cut the number of emergency department beds occupied by these patients by between about 27% and 57%, and emergency department length of stay by between 7 and 14 hours. For capacity planning, the pattern of discharges across the week therefore needs modelling alongside the pattern of admissions.

Specifying and checking departures in a model

In a discrete event simulation, departures are generated by the model logic and are not normally an input. A service completion sits on the event queue, and when it occurs the model frees the resource, records the patient's outcomes and routes the patient onwards or removes the patient. Observed discharge counts are then better used as validation targets than as inputs.

Arrivals should equal departures plus patients still in the system at the end of each run, and in a stable model the simulated departure rate should match the arrival rate within Monte Carlo error. A persistent gap signals an overloaded stage or patients lost in the model logic. Discharges recorded in a capacity-constrained service show what the service could release, which may be well below what it would release with more downstream places.

Where departure processes are misread

The commonest error is to assume Poisson departures by default. Burke's theorem needs Poisson arrivals, exponential service, no defections and a steady state, and lengths of stay that are more regular or more variable than exponential produce a different output stream unless beds are so plentiful that no patient waits. A related error is to read departure counts as demand, although throughput equals demand only in a stable stage and falls below it whenever the stage is overloaded or blocked.

Feedback paths are a second trap. Patients who return for review re-enter an earlier queue, so that queue's arrivals are not Poisson. Departures from the model should also be kept apart from departures from a stage. In a non-constrained cost-effectiveness model, death and the end of the time horizon are the only exits, whereas in a service model a discharge from one ward may be an arrival to another setting.

Sources

• Burke PJ. The output of a queuing system. Operations Research. 1956;4(6):699-704. https://doi.org/10.1287/opre.4.6.699
• Gallager RG. Discrete Stochastic Processes, course text for MIT 6.262, chapter 6 (Markov processes with countable state spaces). MIT OpenCourseWare; 2011. https://ocw.mit.edu/courses/6-262-discrete-stochastic-processes-spring-2011/
• Karnon J, Stahl J, Brennan A, Caro JJ, Mar J, Möller J. Modeling using discrete event simulation: a report of the ISPOR-SMDM Modeling Good Research Practices Task Force-4. Value in Health. 2012;15(6):821-827. https://doi.org/10.1016/j.jval.2012.04.013
• Koizumi N, Kuno E, Smith TE. Modeling patient flows using a queuing network with blocking. Health Care Management Science. 2005;8(1):49-60. https://doi.org/10.1007/s10729-005-5216-3
• Whitt W. The queueing network analyzer. Bell System Technical Journal. 1983;62(9):2779-2815. https://archive.org/details/bstj62-9-2779
• Wong HJ, Wu RC, Caesar M, Abrams H, Morra D. Smoothing inpatient discharges decreases emergency department congestion: a system dynamics simulation model. Emergency Medicine Journal. 2010;27(8):593-598. https://doi.org/10.1136/emj.2009.078543