Time dependent arrivals via thinning

This notebook provides an overview of how to use the time_dependent.NSPPThinning class.

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.

There are two random variables employed in sampling: an exponential distribution (used to sample IAT) and a uniform distibution (used to accept/reject samples).

All IATs are sampled from an Exponential distribution with the highest arrival rate (most frequent). These arrivals are then rejected (thinned) proportional to the ratio of the current arrival rate to the maximum arrival rate. The algorithm executes until a sample is accepted. The IAT returned is the sum of all the IATs that were sampled.

The thinning algorithm

A NSPP has arrival rate \(\lambda(t)\) where \(0 \leq t \leq T\)

Here \(i\) is the arrival number and \(\mathcal{T_i}\) is its arrival time.

1. Let \(\lambda^* = \max_{0 \leq t \leq T}\lambda(t)\) be the maximum of the arrival rate function and set \(t = 0\) and \(i=1\)

2. Generate \(e\) from the exponential distribution with rate \(\lambda^*\) and let \(t = t + e\) (this is the time of the next entity will arrive)

3. Generate \(u\) from the \(U(0,1)\) distribution. If \(u \leq \dfrac{\lambda(t)}{\lambda^*}\) then \(\mathcal{T_i} =t\) and \(i = i + 1\)

4. Go to Step 2.

sim-tools imports

from sim_tools.datasets import load_banks_et_al_nspp
from sim_tools.time_dependent import NSPPThinning

Example from Banks et al.

We will illustrate the use of NSPPThinning using an example from Banks et al.

The table below breaks an arrival process down into 60 minutes intervals.

t(min)	Mean time between arrivals (min)	Arrival Rate \(\lambda(t)\) (arrivals/min)
0	15	1/15
60	12	1/12
120	7	1/7
180	5	1/5
240	8	1/8
300	10	1/10
360	15	1/15
420	20	1/20
480	20	1/20

Interpretation: In the table above the fastest arrival rate is 1/5 customers per minute or 5 minutes between customer arrivals.

data = load_banks_et_al_nspp()
data

	t	mean_iat	arrival_rate
0	0	15	0.066667
1	60	12	0.083333
2	120	7	0.142857
3	180	5	0.200000
4	240	8	0.125000
5	300	10	0.100000
6	360	15	0.066667
7	420	20	0.050000
8	480	20	0.050000

# create arrivals and set random number seeds
SEED_1 = 42
SEED_2 = 101
arrivals = NSPPThinning(data, SEED_1, SEED_2)

# number of arrivals to simulate
n_arrivals = 15

# run simulation
simulation_time = 0.0
for _ in range(n_arrivals):
    iat = arrivals.sample(simulation_time)
    simulation_time += iat
    print(f'{simulation_time:.2f} Patient arrival (IAT={iat:.2f})')

37.46 Patient arrival (IAT=37.46)
73.02 Patient arrival (IAT=35.56)
89.41 Patient arrival (IAT=16.39)
96.79 Patient arrival (IAT=7.38)
98.37 Patient arrival (IAT=1.58)
104.94 Patient arrival (IAT=6.57)
112.30 Patient arrival (IAT=7.35)
121.49 Patient arrival (IAT=9.19)
127.62 Patient arrival (IAT=6.14)
135.25 Patient arrival (IAT=7.63)
135.64 Patient arrival (IAT=0.39)
141.91 Patient arrival (IAT=6.27)
148.23 Patient arrival (IAT=6.32)
155.26 Patient arrival (IAT=7.04)
158.47 Patient arrival (IAT=3.21)