Ciw is a Python package for simulating queueing networks.
The two G’s in G/G/1 do not have to be the same distribution, and respectively can be any distribution with non-negative support.
We will use a Hyperexponential distribution for the arrival distribution. A hyperexponential distribution is exactly a mixture distribution of exponential distributions. This has an interpretation of there being an implicit set of arrival processes that each have their own distinct but independent exponential arrival times. In this case we will choose a mixture of four such arrival processes with distinct arrival rates:
\[\begin{align} U_1 \sim & \text{Exponential}\left( \frac{1}{9} \right) \\ U_2 \sim & \text{Exponential}\left( \frac{1}{5} \right) \\ U_3 \sim & \text{Exponential}\left( \frac{1}{6} \right) \\ U_4 \sim & \text{Exponential}\left( 1 \right) \end{align}\]The following mixture distribution for the arrival times will be used:
\[T_{\text{arrivals}} \sim \frac{1}{5} f_{U_1} + \frac{1}{10} f_{U_2} + \frac{3}{5} f_{U_3} + \frac{1}{10} f_{U_4}\]We will use a gamma distribution for the sake of example. A gamma distribution is the result of a sum of independent exponentially-distributed random variable, and thus for this example we have an interpretation that the servicing is implicitly a multi-step process where each step has an exponentially-distributed completion time.
A G/G/1 queue as described can be implemented and simulated using Ciw in the following way.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
import ciw
ciw.seed(2018)
arrival_dist = ciw.dists.HyperExponential(rates=[9, 5, 6, 1], probs=[0.2, 0.1, 0.6, 0.1])
service_dist = ciw.dists.Gamma(shape=0.6, scale=1.2)
HORIZON = 365
network = ciw.create_network(
arrival_distributions = [arrival_dist],
service_distributions = [service_dist],
number_of_servers = [1]
)
simulation = ciw.Simulation(network)
simulation.simulate_until_max_time(HORIZON)
records = simulation.get_all_records()