1 Queue¶

Here, an example of an M/M/1 queue will be given, and results compared to to those obtained using standard queueing theory. This will walk through an example of an M/M/1 queue with Poisson arrivals of rate 3 and Exponential service times of rate 5.

Standard queueing theory gives the expected wait in an M/M/1 queue as \(\mathbb{E}[W] = \frac{\rho}{\mu(1-\rho)}\). With arrival rate \(\lambda = 3\) and service rate \(\mu = 5\), we get traffic intensity \(\rho = \frac{\lambda}{\mu} = 0.6\), and so \(\mathbb{E}[W] = 0.3\).

We set up the parameters in ASQ:

>>> params_dict = {'Queue_capacities': ['Inf'],
...                'Number_of_classes': 1,
...                'Arrival_rates': {'Class 0': [3.0]},
...                'Number_of_nodes': 1,
...                'Service_rates':{'Class 0': [['Exponential', 5.0]]},
...                'Simulation_time': 250,
...                'Transition_matrices': {'Class 0': [[0.0]]},
...                'detect_deadlock': False,
...                'Number_of_servers': [1]}

The following code repeats the experiment 100 times, only recording waits for those that arrived after a warm-up time of 50. It then returns the average wait in the system:

>>> def interation(warmup):
...     Q = asq.Simulation(params_dict)
...     Q.simulate_until_max_time()
...     inds = Q.get_all_individuals()
...     waits = [ind.data_records[1][0].wait for ind in inds if ind.data_records[1][0].arrival_date > warmup]
...     return sum(waits)/len(waits)

>>> seed(27)
>>> ws = []
>>> for i in range(100):
...     ws.append(interation(50))

>>> print sum(ws)/len(ws)
0.292014274888

We see that the results of the simulation are in agreement with those of standard queueing theory.