Court backlog arithmetic: Little's law on a docket

For cases with filing times arrivals and disposition times dispositions inside a window \([0, T]\), the time-average number of pending cases equals the filing rate times the mean time to disposition, with no probability model at all: the integral of the pending count is the sum of the case durations (Research.P16.occupancy_integral, little). A court that publishes a mean disposition time and a filing rate has therefore published its average backlog (little_backlog), and a backlog target fixes the mean disposition time that achieves it (little_target). Cases still pending at \(T\) make the mean of disposed cases a truncated estimate; the function reports how many are censored so the reader sees the dark-figure problem that follows.

Usage

morie_court_backlog(
  arrivals,
  dispositions,
  horizon = NULL,
  target_backlog = NULL
)

Arguments

arrivals

Filing times (non-negative).

dispositions

Disposition times, at least the filing times; NA for cases still pending at horizon.

horizon

Window end \(T\); by default the latest disposition.

target_backlog

Optional backlog target \(L^*\) for which the required mean disposition time is reported.

Value

A list with n, n_censored, horizon, filing_rate, mean_disposition_time (over disposed cases), average_backlog (\(\lambda \bar W\)), occupancy_integral, pending_at_horizon, required_mean_time (when a target is given) and theorems.

Examples

a <- c(0, 1, 2, 4, 5, 7); d <- c(3, 2.5, 6, 5, 9, 10)
morie_court_backlog(a, d, horizon = 10, target_backlog = 1)
#> $n
#> [1] 6
#> 
#> $n_censored
#> [1] 0
#> 
#> $horizon
#> [1] 10
#> 
#> $filing_rate
#> [1] 0.6
#> 
#> $mean_disposition_time
#> [1] 2.75
#> 
#> $average_backlog
#> [1] 1.65
#> 
#> $occupancy_integral
#> [1] 16.5
#> 
#> $pending_at_horizon
#> [1] 0
#> 
#> $little_identity_check
#> [1] 0
#> 
#> $required_mean_time
#> [1] 1.666667
#> 
#> $theorems
#> [1] "Research.P16.occupancy_integral" "Research.P16.little"            
#> [3] "Research.P16.little_backlog"     "Research.P16.little_target"     
#>