16  Modelling Variable Arrival Rates

Authors

Sammi Rosser

Dan Chalk

It is often the case that arrivals to a system do not occur completely regularly throughout the day.

For example, in an emergency department, we may find that the number of arrivals climb in the afternoon and early evening before dropping off again overnight.

One way to implement this is to return to the sim-tools package by Tom Monks, which we used in (Chapter 12): Choosing Distributions and Chapter 13: Reproducibility. We will use a class that creates a non-stationary poisson process via thinning.

16.1 The principle used

The sim-tools documentation states the following about the approach we will be using:

Thinning is an acceptance-rejection approach to sampling inter-arrival times (IAT) from a time dependent distribution where each time period follows its own exponential distribution.

The NSPP thinning class takes in a dataframe with two columns:

  • The first, called ‘t’, is a list of time points at which the arrival rate changes.
  • The second, called ‘arrival_rate’, is the arrival rate in the form of the average inter-arrival time in time units.

Let’s look at an example of the sort of dataframe we would create and pass in.

t mean_iat
0 0 15
1 60 15
2 120 20
3 180 23
4 240 25
5 300 28
6 360 25
7 420 22
8 480 18
9 540 16
10 600 15
11 660 13
12 720 10
13 780 8
14 840 10
15 900 11
16 960 8
17 1020 8
18 1080 7
19 1140 6
20 1200 9
21 1260 11
22 1320 13
23 1380 14

Let’s add a few more columns so we can better understand what’s going on.

t time_minutes mean_iat arrival_rate arrivals_per_hour
0 0 00:00:00 15 1/15 4.0
1 60 01:00:00 15 1/15 4.0
2 120 02:00:00 20 1/20 3.0
3 180 03:00:00 23 1/23 2.6
4 240 04:00:00 25 1/25 2.4
5 300 05:00:00 28 1/28 2.1
6 360 06:00:00 25 1/25 2.4
7 420 07:00:00 22 1/22 2.7
8 480 08:00:00 18 1/18 3.3
9 540 09:00:00 16 1/16 3.8
10 600 10:00:00 15 1/15 4.0
11 660 11:00:00 13 1/13 4.6
12 720 12:00:00 10 1/10 6.0
13 780 13:00:00 8 1/8 7.5
14 840 14:00:00 10 1/10 6.0
15 900 15:00:00 11 1/11 5.5
16 960 16:00:00 8 1/8 7.5
17 1020 17:00:00 8 1/8 7.5
18 1080 18:00:00 7 1/7 8.6
19 1140 19:00:00 6 1/6 10.0
20 1200 20:00:00 9 1/9 6.7
21 1260 21:00:00 11 1/11 5.5
22 1320 22:00:00 13 1/13 4.6
23 1380 23:00:00 14 1/14 4.3

Let’s visualise this in a graph.