Title: Waiting List Metrics Using Queuing Theory
Version: 0.1.2
Maintainer: Chris Mainey <c.mainey1@nhs.net>
Description: Waiting list management using queuing theory to analyse, predict and manage queues, based on the approach described in Fong et al. (2022) <doi:10.1101/2022.08.23.22279117>. Aimed at UK National Health Service (NHS) applications, waiting list summary statistics, target-value calculations, waiting list simulation, and scheduling functions are included.
License: MIT + file LICENSE
Encoding: UTF-8
RoxygenNote: 7.3.2
VignetteBuilder: knitr
URL: https://nhs-r-community.github.io/NHSRwaitinglist/, https://github.com/nhs-r-community/NHSRwaitinglist
BugReports: https://github.com/nhs-r-community/NHSRwaitinglist/issues
Config/testthat/edition: 3
Imports: cli, dplyr, rlang, utils, stats, randomNames
Suggests: ggplot2, knitr, rmarkdown, testthat (≥ 3.0.0)
Depends: R (≥ 4.1.0)
LazyData: true
Language: en-GB
NeedsCompilation: no
Packaged: 2025-07-15 13:28:29 UTC; ChrisMainey(Birmingh
Author: Neil Walton ORCID iD [aut], Jacqueline Grout [ctb], Zoë Turner ORCID iD [ctb], Matt Dray [aut], Paul Fenton [ctb], Peter Shakeshaft [ctb], David Foord [ctb], Tom Smith [aut], Chris Mainey ORCID iD [cre, aut], Mohammed Mohammed [ctb], NHS-R community [cph]
Repository: CRAN
Date/Publication: 2025-07-15 15:50:02 UTC

Calculate Queue Load

Description

Calculates the queue load. The queue load is the number of arrivals that occur for every patient leaving the queue (given that the waiting list did not empty). It could also be described as the rate of service at the queue. The queue load is calculated by dividing the demand by the capacity: queue_load = demand / capacity.

Usage

calc_queue_load(demand, capacity)

Arguments

demand

Numeric value of rate of demand in same units as target wait - e.g. if target wait is weeks, then demand in units of patients/week.

capacity

Numeric value of the number of patients that can be served (removals) from the waiting list each week.

Value

Numeric value of load which is the ratio between demand and capacity.

Examples

# If 30 patients are added to the waiting list each week (demand) and 27
# removed (capacity) this results in a queue load of 1.11 (30/27).
calc_queue_load(30, 27)

Relief Capacity

Description

Calculates required relief capacity to achieve target queue size in a given period of time as a function of demand, queue size, target queue size and time period. Relief Capacity is required if Queue Size > 2 * Target Queue Size.

Relief Capacity = Current Demand + (Queue Size - Target Queue Size)/Time Steps

WARNING!: make sure units match. I.e. if demand is measured per week then time_to_target should be weeks or if demand is per day then time_to_target is per day

Usage

calc_relief_capacity(
  demand,
  queue_size,
  target_queue_size,
  time_to_target = 26,
  num_referrals = 0,
  cv_demand = 0
)

Arguments

demand

Numeric value of rate of demand in same units as target wait e.g. if target wait is weeks, then demand in units of patients/week.

queue_size

Numeric value of current number of patients in queue.

target_queue_size

Numeric value of desired number of patients in queue.

time_to_target

Numeric value of desired number of time-steps to reach the target queue size by.

num_referrals

Numeric value of the number of referrals per time step.

cv_demand

To be completed

Value

A numeric value of the required rate of capacity to achieve a target queue size in a given period of time.

Examples

# If demand is 30 patients per week, the current queue size is 1200 and the
# target is to achieve a queue size of 390 in 26 weeks, then

# Relief Capacity = 30 + (1200 - 390)/26 = 61.15 patients per week.

calc_relief_capacity(30, 1200, 390, 26)

Target Capacity

Description

Applies Kingman/Marchal's Formula : 

 capacity = demand + (cvd**2 + cvc**2) / waiting_time

where cvd = coefficient of variation of time between arrivals cvd = coefficient of variation of service times waiting_time = target_wait / factor

Usage

calc_target_capacity(
  demand,
  target_wait,
  factor = 4,
  cv_demand = 1,
  cv_capacity = 1
)

Arguments

demand

Numeric value of rate of demand in same units as target wait e.g. if target wait is weeks, then demand in units of patients/week.

target_wait

Numeric value of number of weeks that has been set as the target within which the patient should be seen.

factor

the amount we divide the target by in the waiting list e.g. if target is 52 weeks the mean wait should be 13 for a factor of 4

cv_demand

coefficient of variation of time between arrivals

cv_capacity

coefficient of variation between removals due to operations completed

Value

numeric. The capacity required to achieve a target waiting time.

Examples


demand <- 4 # weeks
target_wait <- 52 # weeks

# number of operations per week to have mean wait of 52/4
calc_target_capacity(demand, target_wait)

Average Waiting Time

Description

This calculates the target mean wait given the two inputs of target_wait and a numerical value for factor. The average wait is actually the target mean wait and is calculated as follows: target_wait / factor. If we want to have a chance between 1.8%-0.2% of making a waiting time target, then the average patient should have a waiting time between a quarter and a sixth of the target. Therefore: The mean wait should sit somewhere between target_wait/factor=6 < Average Waiting Time < target_wait/factor=4.

Usage

calc_target_mean_wait(target_wait, factor = 4)

Arguments

target_wait

Numeric value of the number of weeks that has been set as the target within which the patient should be seen.

factor

Numeric factor used in average wait calculation - to get a quarter of the target use factor=4 and one sixth of the target use factor = 6 etc. Defaults to 4.

Value

Numeric value of target mean waiting time to achieve a given target wait.

Examples

# If the target wait is 52 weeks then the target mean wait with a factor of 4
# would be 13 weeks and with a factor of 6 it would be 8.67 weeks.
calc_target_mean_wait(52, 4)

Target Queue Size

Description

Uses Little's Law to calculate the target queue size to achieve a target waiting time as a function of observed demand, target wait and a variability factor used in the target mean waiting time calculation.

Target Queue Size = Demand * Target Wait / 4.

The average wait should sit somewhere between target_wait/factor=6 < Average Waiting Time < target_wait/factor=4 The factor defaults to 4.

Only applicable when Capacity > Demand.

Usage

calc_target_queue_size(demand, target_wait, factor = 4)

Arguments

demand

Numeric value of rate of demand in same units as target wait e.g. if target wait is weeks, then demand in units of patients/week.

target_wait

Numeric value of number of weeks that has been set as the target within which the patient should be seen.

factor

Numeric factor used in average wait calculation

  • to get a quarter of the target use factor=4

  • to get one sixth of the target use factor = 6 etc. Defaults to 4.

Value

Numeric target queue length.

Examples

# If demand is 30 patients per week and the target wait is 52 weeks, then the
# Target queue size = 30 * 52/4 = 390 patients.

calc_target_queue_size(30, 52, 4)

Calculate Waiting List Pressure

Description

For a waiting list with target waiting time, the pressure on the waiting list is twice the mean delay divided by the waiting list target. The pressure of any given waiting list should be less than 1. If the pressure is greater than 1 then the waiting list is most likely going to miss its target. The waiting list pressure is calculated as follows: pressure = 2 * mean_wait / target_wait.

Usage

calc_waiting_list_pressure(mean_wait, target_wait)

Arguments

mean_wait

Numeric value of target mean waiting time to achieve a given target wait.

target_wait

Numeric value of the number of weeks that has been set as the target within which the patient should be seen.

Value

Numeric value of wait_pressure which is the waiting list pressure.

Examples

calc_waiting_list_pressure(63, 52)

Create Waiting List

Description

Creates a waiting list using the parameters specified

Usage

create_waiting_list(
  n,
  mean_arrival_rate,
  mean_wait,
  start_date = Sys.Date(),
  limit_removals = TRUE,
  sd = 0,
  rott = 0,
  ...
)

Arguments

n

Numeric value of rate of demand in same units as target wait

  • e.g. if target wait is weeks, then demand in units of patients/week.

mean_arrival_rate

Numeric value of mean daily arrival rate.

mean_wait

Numeric value of mean wait time for treatment/on waiting list.

start_date

Character value of date from which to start generated waiting list.

limit_removals

Defaults to TRUE

sd

Numeric value, standard deviation. Defaults to 0.

rott

Numeric value, proportion of referrals to be randomly flagged as ROTT. Defaults to 0.

...

Container for the list

Value

A tibble with randomly generated patient records and the following columns:

pat_id

Integer. Unique identifier for the patient.

addition_date

Date. The date the patient was added to the waiting list.

removal_date

Date. The date the patient was removed from the waiting list.

wait_length

Numeric. Number of days between the addition and removal dates.

rott

Logical. Whether the removal was for reasons other than treatment (ROTT).

Additional columns may be included if supplied via ..., where named vectors (e.g., patient-level variables) of compatible length are merged into the output tibble.

Examples

create_waiting_list(366, 50, 21, "2024-01-01", 10, 0.1)

demographic data

Description

demographic data

Usage

data(demographic_data)

Format

Data frame with 9 columns

hospital_site

ODS hospital site code

...

Others to do with file is updated

Examples


data(demographic_data)

OPCS4 data

Description

OPCS4 data

Usage

data(opcs4)

Format

Data frame with 9 columns

code_1digit

The first digit of the OPCS4 code, or 'chapter'

name_1digit

The name/group of 'chapter' of the OPCS4 code

...

Others to do with file is updated

Source

https://biobank.ndph.ox.ac.uk/ukb/coding.cgi?id=240

Examples


data(opcs4)

Generator of NHS patients

Description

Generates simulated NHS patients

Usage

sim_patients(n_rows = 10, start_date = NULL)

Arguments

n_rows

Number of rows/patients to generate

start_date

Start date (needed to generate patient ages)

Value

A data.frame representing an empty waiting list with the following columns:

Referral

Date. Referral date; all values are NA.

Removal

Date. Removal date; all values are NA.

Withdrawal

Date. Patient withdrawal date; all values are NA

Priority

Numeric. Waiting list priority level, from 1 (most urgent) to 4 (least urgent).

Target_wait

Numeric. Target number of days the patient should wait at the assigned priority level (e.g., 28 days for priority 2)

Name

Character. Patient name in the format "Last, First".

Birth_date

Date. Date of birth.

NHS_number

Integer. Patient identifier, up to 100,000,000.

Specialty_code

Character. One-letter code representing the specialty of the procedure.

Specialty

Character. Full name of the specialty associated with the procedure.

OPCS

Character. OPCS-4 code of the selected procedure.

Procedure

Character. Name of the selected procedure.

Consultant

Character. Consultant name in the format "Last, First".

Examples

sim_patients()

Generator a list of dates to schedule

Description

Generates a list if dates in a given range

Usage

sim_schedule(n_rows = 10, start_date = NULL, daily_capacity = 1)

Arguments

n_rows

Number of rows/patients to generate

start_date

Start date (needed to generate patient ages)

daily_capacity

Number of patients per day

Value

A vector of Date values representing scheduled procedure dates. The length of the vector is equal to n_rows, and the dates are spaced according to the specified daily_capacity.

Insert new referrals into the waiting list

Description

Adds new referrals, with other columns set as NA.

Usage

wl_insert(waiting_list, additions, referral_index = 1)

Arguments

waiting_list

data.frame. A df of referral dates and removals

additions

Date or character vector (in format 'YYYY-MM-DD'). A list of referral dates to add to the waiting list

referral_index

The index of the column in waiting_list which contains the referral dates. Defaults to the first column.

Value

A data.frame representing the updated waiting list, with additional referrals dates in the column specified by referral_index. Other columns are filled with NA in the new rows. The result is sorted by the referral column.

Examples

referrals <- c.Date("2024-01-01", "2024-01-04", "2024-01-10", "2024-01-16")
removals <- c.Date("2024-01-08", NA, NA, NA)
waiting_list <- data.frame("referral" = referrals, "removal" = removals)
additions <- c.Date("2024-01-03", "2024-01-05", "2024-01-18")
longer_waiting_list <- wl_insert(waiting_list, additions)

Join two waiting list

Description

Take two waiting list and sorting in date order

Usage

wl_join(wl_1, wl_2, referral_index = 1)

Arguments

wl_1

a waiting list: dataframe consisting addition and removal dates

wl_2

a waiting list: dataframe consisting addition and removal dates

referral_index

the column index where referrals are listed

Value

A data.frame representing the combined waiting list, created by joining wl_1 and wl_2. The result is sorted by the referral date column specified by referral_index. The column structure is preserved from the input data frames.

Examples


referrals <- c.Date("2024-01-01","2024-01-04","2024-01-10","2024-01-16")
removals <- c.Date("2024-01-08",NA,NA,NA)
wl_1 <- data.frame("referral" = referrals ,"removal" = removals )

referrals <- c.Date("2024-01-04","2024-01-05","2024-01-16","2024-01-25")
removals <- c.Date("2024-01-09",NA,"2024-01-19",NA)
wl_2 <- data.frame("referral" = referrals ,"removal" = removals )
wl_join(wl_1,wl_2)

Queue size calculator

Description

Calculates queue sizes from a waiting list

Usage

wl_queue_size(
  waiting_list,
  start_date = NULL,
  end_date = NULL,
  referral_index = 1,
  removal_index = 2
)

Arguments

waiting_list

data.frame consisting addition and removal dates

start_date

Date or character (in format 'YYYY-MM-DD'); start of calculation period

end_date

Date or character (in format 'YYYY-MM-DD'); end of calculation period

referral_index

the index of referrals in waiting_list

removal_index

the index of removals in waiting_list

Value

A data.frame containing the size of the waiting list for each day in the specified date range. If start_date and/or end_date are NULL, the function uses the earliest and latest referral dates in the input data.frame. The returned data.frame has the following columns:

dates

Date. Each date within the computed range, starting from the first referral.

queue_size

Numeric. Number of patients on the waiting list on that date.

Examples

referrals <- c.Date("2024-01-01", "2024-01-04", "2024-01-10", "2024-01-16")
removals <- c.Date("2024-01-08", NA, NA, NA)
waiting_list <- data.frame("referral" = referrals, "removal" = removals)
wl_queue_size(waiting_list)

Calculate some stats about referrals

Description

Calculate some stats about referrals

Usage

wl_referral_stats(
  waiting_list,
  start_date = NULL,
  end_date = NULL,
  referral_index = 1
)

Arguments

waiting_list

data.frame. A df of referral dates and removals

start_date

Date or character (in format 'YYYY-MM-DD'); The start date to calculate from

end_date

Date or character (in format 'YYYY-MM-DD'); The end date to calculate to

referral_index

the column index of referrals

Value

A data.frame with the following summary statistics on referrals/demand:

demand_weekly

Numeric. Mean number of additions to the waiting list per week.

demand_daily

Numeric. Mean number of additions to the waiting list per day.

demand_cov

Numeric. Coefficient of variation in the time between additions to the waiting list.

demand_count

Numeric. Total demand over the full time period.

Examples

referrals <- c.Date("2024-01-01", "2024-01-04", "2024-01-10", "2024-01-16")
removals <- c.Date("2024-01-08", NA, NA, NA)
waiting_list <- data.frame("referral" = referrals, "removal" = removals)
referral_stats <- wl_referral_stats(waiting_list)

Calculate some stats about removals

Description

Calculate some stats about removals

Usage

wl_removal_stats(
  waiting_list,
  start_date = NULL,
  end_date = NULL,
  referral_index = 1,
  removal_index = 2
)

Arguments

waiting_list

data.frame. A df of referral dates and removals

start_date

Date or character (in format 'YYYY-MM-DD'); The start date to calculate from.

end_date

Date or character (in format 'YYYY-MM-DD'); The end date to calculate to.

referral_index

Index of the referral column in waiting_list.

removal_index

Index of the removal column in waiting_list.

Value

A data.frame with the following summary statistics on removals/capacity:

capacity_weekly

Numeric. Mean number of removals from the waiting list per week.

capacity_daily

Numeric. Mean number of removals from the waiting list per day.

capacity_cov

Numeric. Coefficient of variation in the time between removals from the waiting list.

removal_count

Numeric. Total number of removals from the waiting list over the full time period.

Examples

referrals <- c.Date("2024-01-01", "2024-01-04", "2024-01-10", "2024-01-16")
removals <- c.Date("2024-01-08", NA, NA, NA)
waiting_list <- data.frame("referral" = referrals, "removal" = removals)
removal_stats <- wl_removal_stats(waiting_list)

A simple operation scheduler

Description

Takes a list of dates and schedules them to a waiting list, by adding a removal date to the data.frame. This is done in referral date order, I.e. earlier referrals are scheduled first (FIFO).

Usage

wl_schedule(
  waiting_list,
  schedule,
  referral_index = 1,
  removal_index = 2,
  unscheduled = FALSE
)

Arguments

waiting_list

data.frame. A df of referral dates and removals

schedule

Date or character vector. Should be formatted as year-month-date, e.g. "2024-04-01". The dates to schedule open referrals into (i.e. dates of unbooked future capacity)

referral_index

The column index in the waiting_list which contains the referral dates

removal_index

The column index in the waiting_list which contains the removal dates

unscheduled

logical. If TRUE, returns a list of scheduled and unscheduled procedures If FALSE, only returns the updated waiting list

Value

The updated waiting list with removal dates assigned based on the given schedule, either as a single data.frame (default) or as part of a list (if unscheduled = TRUE).

If unscheduled = TRUE, returns a list with two data frames:

  1. A data.frame. The updated waiting list with scheduled removals.

  2. A data.frame showing which slots were used, with columns:

    schedule

    Date. The available dates from the input schedule.

    scheduled

    Numeric. 1 if the slot was used to schedule a patient, 0 if not.

Examples

referrals <- c.Date("2024-01-01", "2024-01-04", "2024-01-10", "2024-01-16")
removals <- c.Date("2024-01-08", NA, NA, NA)
waiting_list <- data.frame("referral" = referrals, "removal" = removals)
schedule <- c.Date("2024-01-03", "2024-01-05", "2024-01-18")
updated_waiting_list <- wl_schedule(waiting_list, schedule)

Simple simulator to create a waiting list

Description

Creates a simulated waiting list comprising referral dates, and removal dates

Usage

wl_simulator(
  start_date = NULL,
  end_date = NULL,
  demand = 10,
  capacity = 11,
  waiting_list = NULL,
  withdrawal_prob = NA_real_,
  detailed_sim = FALSE
)

Arguments

start_date

Date or character (in format 'YYYY-MM-DD'); The start date to calculate from

end_date

Date or character (in format 'YYYY-MM-DD'); The end date to calculate to

demand

numeric. Weekly demand (i.e., typical referrals per week).

capacity

numeric. Weekly capacity (i.e., typical removals per week).

waiting_list

data.frame. Waiting list where each row is a pathway/patient with date columns 'Referral' and 'Removal'.

withdrawal_prob

numeric. Probability of a patient withdrawing.

detailed_sim

logical. If TRUE, simulation provides detailed output.

Value

A data.frame simulating a waiting list, with columns:

Referral

Date. The date each patient was added to the waiting list.

Removal

Date. The date each patient was removed from the waiting list (may be NA if unscheduled).


If detailed_sim = TRUE, returns a more detailed data.frame with the following additional fields:

Withdrawal

Date. The date the patient withdrew from the waiting list.

Priority

Numeric. Waiting list priority level, from 1 (most urgent) to 4 (least urgent).

Target_wait

Numeric. Target number of days the patient should wait at the assigned priority level (e.g., 28 days for priority 2)

Name

Character. Patient name in the format "Last, First".

Birth_date

Date. Date of birth.

NHS_number

Integer. Patient identifier, up to 100,000,000.

Specialty_code

Character. One-letter code representing the specialty of the procedure.

Specialty

Character. Full name of the specialty associated with the procedure.

OPCS

Character. OPCS-4 code of the selected procedure.

Procedure

Character. Name of the selected procedure.

Consultant

Character. Consultant name in the format "Last, First".

Examples


over_capacity_simulation <-
  wl_simulator("2024-01-01", "2024-03-31", 100, 110)
under_capacity_simulation <-
  wl_simulator("2024-01-01", "2024-03-31", 100, 90)

Calculate some stats about the waiting list

Description

A summary of all the key stats associated with a waiting list

Usage

wl_stats(waiting_list, target_wait = 4, start_date = NULL, end_date = NULL)

Arguments

waiting_list

data.frame. A df of referral dates and removals

target_wait

numeric. The required waiting time

start_date

Date or character (in format 'YYYY-MM-DD'); The start date to calculate from

end_date

Date or character (in format 'YYYY-MM-DD'); The end date to calculate to

Value

A data.frame of key waiting list summary statistics based on queueing theory:

mean_demand

Numeric. Mean number of additions to the waiting list per week.

mean_capacity

Numeric. Mean number of removals from the waiting list per week.

load

Numeric. Ratio between demand and capacity.

load_too_big

Logical. Whether the load is greater than or equal to 1, indicating whether the waiting list is unstable and expected to grow.

count_demand

Numeric. Total demand (i.e., number of referrals) over the full time period.

queue_size

Numeric. Number of patients on the waiting list at the end of the time period.

target_queue_size

Numeric. The recommended size of the waiting list to achieve approximately 98.2% of patients being treated within their target wait time. This is based on Little’s Law, assuming the system is in equilibrium, with the average waiting time set to one-quarter of the target_wait.

queue_too_big

Logical. Whether queue_size is more than twice the target_queue_size. A value of TRUE indicates the queue is at risk of missing its targets.

mean_wait

Numeric. Mean waiting time in weeks.

cv_arrival

Numeric. Coefficient of variation in the time between additions to the waiting list.

cv_removal

Numeric. Coefficient of variation in the time between removals from the waiting list.

target_capacity

Numeric. The weekly treatment capacity required to maintain the waiting list at its target equilibrium, assuming the target queue size has been reached.

relief_capacity

Numeric. The temporary weekly capacity required to reduce the waiting list to its target_queue_size within 26 weeks, assuming current demand remains steady. Calculated only if queue_too_big is TRUE; otherwise returns NA.

pressure

Numeric. A measure of pressure on the system, defined as 2 × mean_wait / target_wait. Values greater than 1 suggest the system is unlikely to meet its waiting time targets.

Examples


referrals <- c.Date("2024-01-01", "2024-01-04", "2024-01-10", "2024-01-16")
removals <- c.Date("2024-01-08", NA, NA, NA)
waiting_list <- data.frame("referral" = referrals, "removal" = removals)
waiting_list_stats <- wl_stats(waiting_list)