# How to Define Time and State Dependent Distributions¶

By defining custom distribution obejcts, both time-dependent and state-dependent distributions can be defined. They could be combined to create time-and-state-dependent distributions too.

This custom distribution object must inherit from the gerenic ciw.dists.Distribution object, and define a .sample method that returns a sampled time. This method needs to take in a time variable t, and also the concerned individual ind.

## Time Dependent Distributions¶

In Ciw we can get a time dependent distribution, that is a service time, inter-arrival time, or batching distribution that changes as the simulation time progresses. In order to do this a time dependent distribution object, that has a sample method to sample the time, must be defined.

For example, say we wish to have arrivals once every 30 minutes in the morning, every 15 minutes over lunch, every 45 minutes in the afternoon, and every 90 minutes throughout the night:

>>> import ciw
>>> class TimeDependentDist(ciw.dists.Distribution):
...     def sample(self, t, ind=None):
...         if t % 24 < 12.0:
...             return 0.5
...         if t % 24 < 14.0:
...             return 0.25
...         if t % 24 < 20.0:
...             return 0.75
...         return 1.5


This function returns inter-arrival times of 0.5 hrs between midnight (0) and 12, 0.25 hrs between 12 and 14, 0.75 hrs between 14 and 20, and 1.5 hrs between 20 and midnight (24). Then repeats. Testing this function we see:

>>> D = TimeDependentDist()
>>> D.sample(9.5)
0.5
>>> D.sample(11.0)
0.5
>>> D.sample(13.25)
0.25
>>> D.sample(17.0)
0.75
>>> D.sample(22.0)
1.5
>>> D.sample(33.2) # half 9 the next day
0.5


Let’s implement this into a one node infinite server queue:

>>> import ciw
>>> N = ciw.create_network(
...     arrival_distributions=[TimeDependentDist()],
...     service_distributions=[ciw.dists.Deterministic(value=0.0)],
...     number_of_servers=[float('Inf')]
... )


We’ll then simulate this for 1 day. We would expect 24 arrivals in the morning (12 hours, one every half an hour); 8 arrivals over lunch (2 hours, one every 15 minutes); 8 arrivals in the afternoon (6 hours, one every 45 mins); and 2 arrivals in the night (4 hours, one every hour and a half). Therefore a total of 42 customers passed through the system:

>>> Q = ciw.Simulation(N)
>>> Q.simulate_until_max_time(24.0)
>>> len(Q.nodes[-1].all_individuals)
42


## The Problem of Sampling Arrivals Across Thresholds¶

Problems can occur when using time dependent arrivals, in particular if the arrival rates suddenly change dramatically across time thresholds. This is due to the way in which arrivals are sampled, by sampling the _next_ arrival from the inter-arrival time of the previous sample.

For example, consider arrivals occurring once every 10 time units in the interval (0, 55) time units, and then once every 0.1 time units in the interval (55, 58) time units, and then back to once every 10 time units thereafter. Arrivals would occur at dates 10, 20, 30, 40, 50. At this point the simulation is still in the (0, 55) interval, and so only knows to sample every 10 time units. So the next sample will be at date 60, and the simulation has completely skipped the (55, 58) interval, which should have sampled around 30 arrivals. For example:

>>> class TimeDependentDist(ciw.dists.Distribution):
...     def sample(self, t, ind=None):
...         if t < 55:
...             return 10
...         if t < 58:
...             return 0.1
...         return 10

>>> N = ciw.create_network(
...     arrival_distributions=[TimeDependentDist()],
...     service_distributions=[ciw.dists.Deterministic(value=0.0)],
...     number_of_servers=[float('Inf')]
... )
>>> Q = ciw.Simulation(N)
>>> Q.simulate_until_max_time(101)
>>> recs = Q.get_all_records()
>>> [r.arrival_date for r in recs]
[10, 20, 30, 40, 50, 60, 70, 80, 90, 100]


To overcome this Ciw has the PoissonIntervals distribution, which allows different time intervals to sample number of arrivals from a Poisson distribution with different arrival rates. This does not use the same sampling logic and so can overcome this problem. It first sampled from a Poisson distribution to find the number of arrivals in each time interval, and then samples arrival dates within that time interval from a Uniform distribution. Therefore the whole schedule of arrivals is determined before the simulation has began.

For example, if we have a time interval (0, 3) with rate 1 customers per time unit, and an interval (3, 4) with 8 customers per time unit, which then repeats. We can use:

>>> ciw.seed(0)
>>> Pi = ciw.dists.PoissonIntervals(
...     rates=[1, 8],
...     endpoints=[3, 4],
...     max_sample_date=10
... )


Here they keyword argument max_sample_date is date where no samples will be sampled after this date. Here we can see Pi.dates gives a list of dates to sample

>>> [round(d, 3) for d in Pi.dates]
[0.0, 2.274, 2.533, 3.259, 3.303, 3.405, 3.421, 3.477, 3.511, 3.583, 3.784, 6.724, 7.251, 7.282, 7.505, 7.618, 7.756, 7.91]


Here in the interval (0, 3) 2 arrivals were sampled (expected value 3); in the interval (3, 4) 8 arrivals were sampled (expected value 8); in the interval (4, 7) 1 arrival was samples (expected value 3); and in the interval (7, 8) 6 arrivals were sampled (expected value 10); and in the interval (8, 10) 0 arrival was sampled (expeced value 2 ).

The distribution’s sample() method sampled the scheduled inter-arrival times for these dates:

>>> [round(Pi.sample(), 3) for _ in range(6)]
[2.274, 0.259, 0.726, 0.044, 0.102, 0.016]


## State Dependent Distributions¶

In addition to the time parameter t, the sample method takes in the concerned individual ind. Therefore individuals can use this individual’s attributes when sampling a service time (note it does not make sense to use this to sample inter-arrival times as that individual has not been created yet!). This individual has a ind.simulation attribute, which points to the Simulation object, meaning it has access to the whole state of the system.

Now we can take advantage of this to define state dependent distributions.

As an example, let’s define a distribution for a one node system that returns:
• 0.20 if there are 0 people at that node,

• 0.15 if there is 1 person at that node,

• 0.10 if there are 2 people at that node,

• 0.05 if there are 3 people at that node,

• 0.00 otherwise.

This corresponds the the function:

$$max(-0.05n + 0.2, 0)$$

where $$n$$ is the number of customers at that node. Write a distribution class to use:

>>> class StateDependentDist(ciw.dists.Distribution):
...     def sample(self, t=None, ind=None):
...         n = ind.simulation.statetracker.state
...         return max((-0.05*n) + 0.2, 0)


where we access the system’s state by considering the state tracker. Now to test if this is working, the average service time should be roughly equal to the above function applied to the average queue size:

>>> N = ciw.create_network(
...     arrival_distributions=[ciw.dists.Exponential(rate=4)],
...     service_distributions=[StateDependentDist()],
...     number_of_servers=
... )

>>> ciw.seed(0)
>>> Q = ciw.Simulation(N, tracker=ciw.trackers.SystemPopulation())
>>> Q.simulate_until_max_time(500)
>>> recs = Q.get_all_records()

>>> services = [r.service_time for r in recs if r.arrival_date > 100]
>>> sum(services) / len(services)
0.1549304...

>>> average_queue_size = sum(s*p for s, p in Q.statetracker.state_probabilities().items())
>>> (-0.05 * average_queue_size) + 0.2
0.1552347...


For arrival distributions - when creating the Simulation object, the distribution objects are given a .simulation attribute, so something similar can happen. For example, the following distribution will sample form an Exponential distribution until limit number of individuals has been sampled:

>>> class LimitedExponential(ciw.dists.Exponential):
...     def __init__(self, rate, limit):
...         super().__init__(rate)
...         self.limit = limit
...
...     def sample(self, t=None, ind=None):
...         if self.simulation.nodes.number_of_individuals < self.limit:
...             return super().sample()
...         else:
...             return float('Inf')


And to see it working, a limit of 44 individuals:

>>> N = ciw.create_network(
...     arrival_distributions=[LimitedExponential(rate=1, limit=44)],
...     service_distributions=[ciw.dists.Exponential(rate=3)],
...     number_of_servers=
... )

>>> ciw.seed(0)
>>> Q = ciw.Simulation(N)
>>> Q.simulate_until_max_time(3000)
>>> recs = Q.get_all_records()
>>> len(recs)
44