程序代写代做代考 Lecture 19 (April 2, 2019)¶

Lecture 19 (April 2, 2019)¶
In this lecture, we discuss how to model a discrete event system
In [0]:
import numpy as np
import matplotlib.pyplot as plt
import math
import scipy.special as spsp

import scipy.stats as spst

Introduction to Basics of Discrete Event System¶
A discrete event system is a system that operates as a discrete sequence of events in time.
For a discrete event system, the system status will only change when a specific event happens at a specific time. Because of this, the key points are the following:
(1) Determining the time when the next event happens
(2) When an event happens, how to update the system states
For a discrete event system simulation, there are three key components.
• Clock
A variable that keeps track of the current simulation time.
• Event
The events are those activities that will change the status of a system. For example, for a queueing system, the events can be the arrival of a customer, the completion of the service for a customer, etc. To simulate a system, we need to know the pending time of different events.
• States
A list of variables that capture the status of the system. For example, for a queueing system, the system states could be how many customers are currently waiting in the system.
• History
Variables that records the history of the system simulation. We use these variables to produce our system output, determine the termination of the system, etc.

General structure of discrete-event simulation¶
The following structure can be used for the most of discrete-event simulation.
Initialization
Ending Condition to FALSE
Clock t=0
State_variable
History
Initialize the time for the events to happen for the first time
….
While loop (Ending Condition is FALSE)
update clock t
update history
update state_variable
update the pending time for events
Ending system
Return the system statistics

Simulating a simple system¶
A system will have 20 customers arriving a building following a homogeneous Poisson process with $\lambda=5$. When the customer arrives, they will swipe their ID. They will either enter the building (p=0.4) or fail to enter the building (p=0.6). Assume that the time to swipe their time and enter the building is negligible.
• Use a function to simulate the process. This function should return (1) how many customers entered the venue (2) the time of the last person who successfully entered the venue.
• Run the function 10000 times. Visualize the distributions of these two system outputs.

Analyze the system.¶
• Events:
Customer arriving
• States:
In this simple system, we do not need any state variables to keep the simulation going
• History:
The number of people who have arrived (n_arrived)
The number of people who entered the building (n_entered)
The arrival time of the last person who entered the building (t_entered)

When an event happens, what should we update?¶
Updating the history
The number of customers arrived already. Add 1 to the value
The number of customers entered the venue already simulate whether the customer entered. if so, add 1 to the value
The time of the latest customer entering the venue
Updating the pending event
Generating the arrival time for the next customer

In [0]:
def system1():
###initialization

#clock:
t=0

#history:
##number of customers arrived
n_arrived=0

##number of customers successfully entered
n_entered=0
##time of the latest successful arrival
t_entered=np.inf

##event:
##next arrival time
lmbda=5
t_arriving=t-1/lmbda*np.log(np.random.rand())

#while stop condition is false
while(n_arrived<20): #updating clock t=t_arriving #updating history n_arrived=n_arrived+1 if (np.random.rand()<0.4): n_entered=n_entered+1 t_entered=t #generating the next arrival time t_arriving=t-1/lmbda*np.log(np.random.rand()) return n_entered, t_entered system1() Out[0]: (6, 2.637936573473268) In [0]: outputs=[system1() for i in range(10000)] print(outputs) #collect output1 and output2 #number of entries nentry=[output[0] for output in outputs] #time of the last entry lastentry=[output[1] for output in outputs] print(nentry) plt.hist(np.array(lastentry)) plt.show() [(7, 3.576936164664284), (9, 4.178725074739943), (8, 5.7984501915218125), (10, 3.334509447469686), (7, 4.497507180907048), (6, 2.4560702392204448), (7, 4.54675672243451), (6, 4.949007884496898), (10, 2.987070771422634), (8, 3.4630920593358496), (7, 3.0077937251943205), (13, 5.382509082003923), (11, 3.4058003405049653), (8, 2.6760172302928997), (8, 2.8708878119104275), (9, 3.6208882848099577), (6, 4.184925277557326), (8, 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9, 7, 4, 7, 9, 7, 7, 6, 8, 5, 6, 12, 6, 5, 10, 8, 6, 11, 9, 10, 9, 5, 7, 8, 8, 8, 9, 7, 8, 8, 7, 4, 5, 6, 4, 10, 6, 7, 6, 4, 3, 6, 10, 11, 9, 8, 7, 8, 9, 10, 10, 6, 6, 8, 6, 11, 9, 7, 7, 9, 7, 10, 11, 7, 7, 10, 6, 8, 8, 8, 5, 9, 7, 11, 10, 11, 8, 12, 5, 8, 6, 8, 5, 8, 11, 10, 6, 8, 9, 10, 8, 10, 11, 9, 8, 8, 10, 10, 10, 6, 6, 8, 7, 10, 11, 5, 10, 5, 11, 5, 9, 9, 8, 8, 4, 6, 6, 6, 11, 9, 5, 9, 5, 10, 9, 5, 9, 10, 8, 5, 8, 7, 12, 11, 9, 7, 12, 12, 7, 6, 8, 10, 6, 10, 11, 7, 5, 7, 4, 9, 6, 9, 6, 7, 7, 10, 7, 9, 12, 9, 8, 8, 8, 4, 6, 8, 9, 4, 11, 7, 5, 10, 8, 13, 6, 10, 14, 6, 6, 9, 7, 12, 9, 2, 5, 10, 8, 8, 3, 9, 14, 7, 12, 6, 8, 11, 10, 8, 10, 6, 7, 13, 7, 2, 5, 6, 13, 10, 13, 6, 12, 8, 8, 7, 8, 13, 6, 6, 10, 10, 8, 5, 8, 7, 10, 5, 8, 5, 6, 6, 11, 10, 13, 9, 10, 11, 2, 3, 9, 10, 12, 6, 10, 8, 8, 7, 7, 10, 9, 10, 8, 9, 6, 9, 7, 15, 9, 8, 5, 10, 4, 6, 11, 6, 8, 12, 8, 10, 8, 6, 9, 8, 9, 5, 7, 7, 12, 8, 10, 5, 7, 8, 7, 11, 9, 9, 7, 8, 11, 10, 4, 6, 6, 5, 9, 5, 9, 12, 10, 10, 10, 9, 10, 7, 8, 15, 11, 6, 6, 7, 8, 8, 5, 8, 10, 7, 8, 6, 9, 6, 9, 10, 6, 4, 14, 8, 8, 9, 8, 10, 6, 11, 4, 10, 7, 6, 7, 10, 7, 4, 8, 8, 5, 8, 8, 11, 6, 10, 9, 7, 6, 6, 7, 9, 7, 9, 10, 8, 5, 11, 9, 6, 7, 7, 8, 12, 6, 7, 8, 10, 7, 9, 6, 8, 10, 8, 7, 10, 6, 3, 6, 8, 10, 6, 8, 10, 9, 9, 5, 12, 9, 11, 11, 8, 6, 8, 5, 9, 9, 5, 8, 8, 7, 11, 11, 8, 5, 12, 10, 4, 10, 6, 4, 9, 8, 10, 8, 4, 10, 6, 5, 6, 4, 7, 7, 5, 12, 5, 5, 6, 8, 4, 7, 8, 9, 8, 11, 6, 4, 4, 5, 11, 8, 10, 6, 6, 6, 8, 9, 9, 12, 11, 10, 9, 7, 10, 6, 6, 7, 7, 7, 9, 5, 6, 10, 7, 10, 9, 6, 7, 8, 7, 5, 7, 10, 6, 7, 10, 11, 10, 6, 7, 7, 8, 9, 8, 8, 11, 3, 10, 6, 5, 8, 4, 8, 7, 10, 11, 5, 8, 4, 9, 9, 4, 12, 10, 12, 7, 5, 8, 8, 11, 6, 10, 7, 12, 7, 9, 10, 8, 5, 10, 9, 6, 7, 5, 8, 8, 6, 4, 9, 9, 6, 11, 7, 4, 11, 10, 10, 8, 7, 9, 8, 11, 11, 7, 7, 11, 8, 6, 4, 6, 7, 8, 7, 8, 5, 8, 11, 8, 12, 9, 7, 5, 8, 10, 8, 12, 9, 9, 9, 10, 10, 9, 10, 8, 5, 10, 8, 9, 6, 9, 9, 6, 7, 11, 7, 9, 6, 13, 7, 8, 7, 9, 7, 10, 6, 12, 8, 5, 8, 9, 12, 8, 11, 6, 8, 8, 12, 8, 6, 7, 8, 10, 6, 8, 8, 7, 4, 10, 8, 5, 11, 9, 7, 8, 7, 7, 5, 5, 8, 7, 7, 8, 11, 9, 9, 7, 6, 8, 4, 7, 5, 7, 7, 10, 9, 10, 10, 4, 6, 10, 6, 7, 7, 10, 6, 7, 4, 8, 7, 10, 9, 8, 6, 9, 8, 4, 12, 12, 13, 8, 7, 9, 8, 7, 5, 9, 9, 6, 9, 13, 6, 8, 11, 5, 7, 9, 10, 5, 7, 5, 9, 8, 8, 6, 12, 15, 8, 10, 5, 7, 11, 8, 11, 10, 9, 8, 10, 8, 9, 5, 7, 9, 9, 10, 6, 6, 10, 12, 7, 10, 11, 14, 7, 7, 8, 9, 9, 6, 11, 7, 8, 6, 9, 11, 9, 5, 9, 9, 5, 9, 7, 11, 5, 10, 11, 5, 7, 10, 7, 6, 11, 9, 8, 8, 8, 9, 8, 8, 6, 6, 10, 8, 11, 7, 8, 5, 5, 10, 12, 10, 7, 6, 5, 11, 9, 8, 8, 10, 7, 9, 10, 13, 6, 10, 9, 13, 7, 7, 9, 7, 7, 6, 11, 6, 8, 7, 6, 9, 10, 7, 8, 12, 5, 6, 9, 6, 8, 8, 7, 12, 8, 10, 7, 9, 7, 7, 11, 7, 4, 5, 13, 9, 8, 5, 10, 10, 5, 4, 7, 10, 7, 7, 11, 10, 7, 6, 9, 6, 7, 7, 7, 9, 9, 8, 8, 10, 5, 8, 9, 7, 3, 7, 9, 7, 12, 8, 8, 8, 3, 7, 12, 10, 11, 7, 7, 7, 7, 10, 6, 5, 8, 7, 6, 8, 7, 7, 8, 8, 9, 7, 12, 8, 6, 8, 9, 8, 8, 5, 13, 6, 9, 11, 10, 8, 8, 8, 6, 8, 7, 10, 4, 5, 7, 3, 10, 8, 7, 5, 11, 8, 9, 5, 4, 8, 9, 6, 5, 10, 6, 7, 8, 8, 7, 9, 10, 7, 4, 7, 5, 11, 8, 6, 10, 5, 6, 10, 10, 8, 6, 9, 12, 8, 6, 6, 8, 9, 8, 7, 6, 6, 8, 12, 7, 5, 4, 10, 7, 7, 9, 9, 7, 9, 8, 8, 7, 9, 6, 11, 9, 11, 9, 7, 6, 11, 9, 5, 8, 9, 8, 4, 7, 9, 7, 10, 7, 9, 9, 7, 11, 6, 10, 7, 8, 11, 15, 10, 10, 6, 7, 8, 11, 11, 7, 7, 12, 9, 8, 5, 8, 9, 9, 6, 7, 5, 9, 10, 12, 7, 6, 5, 8, 7, 6, 13, 5, 9, 8, 8, 6, 7, 11, 9, 5, 7, 10, 7, 8, 6, 11, 9, 10, 6, 6, 9, 10, 9, 8, 8, 9, 9, 12, 8, 6, 8, 9, 9, 7, 9, 9, 10, 7, 8, 8, 8, 5, 10, 9, 7, 5, 13, 7, 10, 8, 12, 2, 8, 7, 11, 8, 11, 11, 9, 11, 10, 11, 13, 8, 7, 9, 9, 11, 12, 11, 5, 8, 11, 8, 4, 5, 7, 10, 6, 8, 7, 10, 8, 7, 7, 7, 8, 4, 7, 6, 8, 6, 10, 8, 3, 7, 11, 7, 8, 5, 11, 5, 10, 11, 7, 9, 12, 10, 6, 7, 7, 9, 9, 8, 8, 10, 7, 7, 5, 10, 8, 4, 6, 8, 10, 5, 8, 6, 7, 9, 6, 9, 11, 13, 2, 9, 8, 10, 6, 9, 10, 6, 6, 10, 10, 4, 11, 11, 11, 9, 10, 9, 11, 3, 9, 6, 7, 11, 7, 7, 8, 8, 7, 6, 9, 6, 7, 4, 12, 8, 9, 7, 6, 4, 6, 10, 11, 9, 6, 7, 8, 9, 11, 7, 5, 5, 13, 7, 3, 10, 6, 10, 5, 4, 6, 7, 7, 5, 9, 4, 10, 11, 9, 11, 7, 14, 6, 7, 7, 5, 10, 9, 8, 11, 8, 6, 6, 4, 8, 6, 6, 10, 7, 8, 5, 8, 9, 7, 10, 6, 7, 10, 10, 7, 7, 2, 7, 10, 10, 9, 8, 6, 6, 9, 5, 11, 11, 10, 7, 7, 8, 10, 10, 10, 5, 7, 10, 9, 11, 6, 7, 7, 5, 8, 9, 9, 6, 8, 8, 11, 8, 11, 5, 5, 9, 12, 8, 8, 8, 8, 9, 10, 7, 9, 9, 7, 9, 6, 8, 9, 7, 9, 7, 8, 9, 4, 6, 9, 6, 4, 9, 9, 10, 8, 10, 5, 6, 8, 7, 7, 12, 9, 10, 7, 8, 5, 6, 7, 11, 7, 6, 8, 4, 6, 10, 6, 4, 5, 5, 9, 4, 8, 7, 5, 6, 8, 10, 12, 11, 9, 9, 7, 2, 5, 4, 6, 4, 10, 8, 9, 11, 12, 8, 7, 9, 8, 8, 8, 8, 9, 9, 6, 8, 9, 4, 8, 8, 7, 7, 11, 7, 7, 9, 8, 12, 10, 5, 5, 7, 7, 9, 9, 10, 7, 7, 8, 8, 7, 12, 3, 9, 11, 6, 6, 9, 5, 10, 9, 4, 7, 10, 4, 6, 7, 9, 7, 4, 9, 4, 8, 8, 9, 11, 6, 9, 1, 9, 7, 7, 8, 9, 6, 7, 11, 10, 7, 9, 7, 9, 11, 8, 3, 9, 11, 10, 8, 6, 10, 10, 9, 11, 8, 8, 8, 7, 5, 9, 10, 7, 5, 7, 7, 8, 7, 8, 7, 7, 10, 7, 6, 7, 8, 8, 3, 2, 12, 12, 7, 3, 12, 7, 10, 9, 7, 10, 9, 11, 8, 6, 7, 4, 9, 7, 6, 8, 8, 7, 9, 10, 10, 7, 7, 7, 6, 8, 3, 9, 9, 11, 11, 13, 12, 8, 12, 11, 9, 8, 8, 12, 4, 6, 8, 6, 6, 7, 10, 7, 3, 4, 10, 13, 9, 10, 7, 5, 8, 6, 7, 6, 10, 10, 7, 6, 6, 10, 9, 7, 4, 10, 6, 9, 8, 7, 7, 6, 10, 10, 9, 10, 8, 5, 8, 6, 6, 9, 7, 5, 6, 6, 10, 11, 6, 10, 7, 9, 7, 6, 8, 9, 5, 8, 3, 10, 6, 7, 10, 10, 8, 7, 4, 6, 12, 10, 10, 10, 8, 10, 10, 12, 8, 6, 8, 12, 11, 7, 7, 7, 8, 8, 11, 7, 8, 8, 10, 4, 6, 5, 10, 10, 11, 11, 13, 9, 11, 11, 10, 9, 6, 10, 7, 12, 8, 7, 8, 7, 6, 10, 6, 6, 9, 10, 8, 11, 6, 10, 8, 3, 9, 6, 9, 5, 4, 10, 8, 7, 6, 8, 4, 6, 10, 7, 10, 10, 9, 6, 9, 9, 5, 9, 6, 9, 11, 8, 7, 8, 8, 7, 8, 7, 8, 6, 2, 9, 9, 11, 7, 6, 6, 11, 10, 7, 9, 8, 5, 4, 8, 8, 6, 6, 5, 5, 9, 7, 10, 8, 11, 9, 8, 10, 6, 5, 9, 8, 8, 9, 9, 9, 9, 7, 7, 7, 8, 7, 13, 10, 7, 6, 8, 5, 6, 9, 12, 4, 8, 8, 4, 13, 6, 11, 9, 9, 11, 8, 5, 8, 8, 3, 8, 11, 6, 5, 9, 8, 7, 7, 9, 6, 7, 5, 14, 8, 4, 8, 10, 8, 11, 9, 8, 8, 9, 6, 6, 7, 7, 10, 5, 7, 5, 7, 5, 10, 6, 7, 9, 4, 9, 8, 7, 9, 5, 4, 7, 7, 7, 10, 8, 6, 10, 7, 6, 5, 7, 7, 5, 6, 7, 5, 9, 9, 3, 13, 9, 12, 10, 9, 6, 8, 9, 6, 8, 8, 11, 13, 9, 8, 7, 6, 6, 7, 10, 7, 4, 6, 6, 8, 13, 4, 8, 4, 9, 5, 9, 6, 9, 5, 11, 9, 7, 6, 9, 9, 3, 5, 11, 8, 9, 7, 8, 6, 7, 7, 7, 6, 10, 11, 8, 11, 9, 9, 8, 10, 5, 10, 9, 5, 5, 9, 6, 7, 8, 9, 7, 10, 5, 9, 7, 10, 11, 8, 6, 10, 10, 8, 7, 8, 10, 7, 9, 7, 3, 9, 12, 6, 6, 9, 7, 8, 9, 8, 4, 9, 5, 10, 7, 6, 12, 10, 5, 9, 8, 9, 6, 5, 10, 10, 8, 9, 5, 10, 4, 10, 8, 11, 5, 9, 8, 10, 9, 5, 9, 11, 10, 8, 6, 6, 6, 12, 14, 9, 11, 8, 8, 8, 5, 8, 8, 10, 8, 7, 4, 7, 9, 13, 8, 7, 8, 8, 4, 7, 6, 8, 11, 7, 8, 6, 8, 6, 5, 7, 9, 9, 7, 10, 5, 8, 8, 7, 4, 11, 7, 5, 9, 6, 7, 7, 11, 6, 9, 9, 10, 4, 8, 7, 8, 11, 7, 8, 9, 6, 4, 5, 7, 3, 8, 7, 8, 8, 10, 7, 7, 10, 11, 7, 7, 6, 7, 7, 8, 11, 4, 6, 10, 8, 10, 10, 9, 11, 7, 8, 9, 10, 3, 7, 6, 5, 7, 12, 7, 8, 13, 11, 10, 8, 7, 9, 4, 7, 7, 5, 8, 9, 9, 13, 8, 6, 6, 5, 6, 5, 7, 7, 8, 10, 2, 11, 10, 9, 9, 9, 5, 8, 7, 7, 6, 6, 7, 8, 6, 7, 7, 7, 8, 11, 7, 6, 5, 7, 6, 7, 7, 5, 7, 10, 4, 13, 8, 4, 10, 6, 7, 7, 7, 8, 8, 12, 7, 7, 6, 7, 9, 11, 8, 8, 8, 9, 7, 5, 6, 4, 12, 11, 9, 7, 8, 9, 9, 4, 6, 11, 3, 6, 6, 9, 10, 8, 9, 9, 7, 8, 4, 7, 5, 6, 7, 8, 8, 12, 10, 7, 9, 7, 9, 11, 4, 7, 5, 7, 10, 12, 8, 7, 6, 4, 9, 7, 10, 9, 6, 6, 7, 6, 7, 8, 11, 7, 5, 8, 6, 8, 8, 9, 6, 10, 7, 6, 7, 6, 8, 9, 12, 5, 9, 8, 6, 9, 7, 9, 9, 12, 9, 6, 7, 10, 7, 10, 5, 7, 9, 6, 8, 5, 7, 6, 8, 10, 9, 7, 7, 9, 7, 8, 7, 2, 6, 10, 6, 5, 5, 7, 8, 7, 7, 9, 11, 10, 9, 7, 8, 5, 9, 10, 6, 4, 8, 3, 6, 7, 7, 4, 10, 9, 8, 8, 9, 8, 5, 8, 8, 5, 9, 8, 6, 10, 8, 9, 6, 5, 11, 5, 14, 9, 12, 10, 7, 7, 7, 9, 5, 5, 11, 12, 7, 6, 9, 10, 6, 6, 7, 8, 7, 10, 11, 11, 9, 8, 12, 8, 7, 8, 5, 5, 6, 12, 10, 7, 8, 7, 10, 10, 6, 6, 10, 7, 5, 10, 11, 11, 8, 8, 8, 7, 7, 5, 9, 5, 6, 11, 11, 6, 4, 7, 5, 8, 9, 8, 4, 5, 8, 8, 9, 6, 5, 8, 5, 5, 11, 11, 12, 12, 9, 10, 8, 9, 9, 7, 10, 12, 10, 8, 6, 11, 8, 9, 7, 10, 8, 4, 10, 6, 7, 6, 8, 8, 3, 8, 6, 11, 6, 10, 9, 5, 7, 9, 9, 7, 7, 8, 8, 12, 9, 5, 8, 10, 9, 9, 5, 8, 7, 6, 6, 6, 5, 11, 6, 10, 5, 13, 8, 8, 8, 7, 10, 8, 3, 4, 8, 7, 12, 7, 8, 7, 7, 4, 5, 5, 13, 10, 7, 7, 9, 5, 11, 9, 8, 13, 8, 7, 7, 6, 9, 8, 5, 15, 8, 8, 10, 4]  In [0]: plt.hist(np.array(nentry)) plt.show()  Single-server Queueing system¶ In a queueing system, customers arrive following an arrival process. When they arrive, they will even be immediately served when there is nobody in the front. Otherwise, they will need to wait until people arrived earlier are served. The service time follows a specific distribution. We compute the average time in the system for customers who received their service. Analyzing the system¶ Events: Arriving of a customer (t_arrival) Departure of a customer (t_depart) States How many customers are currently in the system (n_current) History The arrival time of each customer (t_arrivalA) The departure time of each customer (t_departA) How to update clock/states/pending time of events?¶ **Case 1: arrival event** clock t=t_arrival States n_current=n_current+1 History Store the arrival time in the t_arrivalA array pending events Generate the next arrival time If no customers in the front (n_current==1), update the next departure time ** Case 2: departure event** clock t=depart States n_current=n_current-1 History Store the depart time in the t_departA array pending events If no customers are in the front, update the next departure time to np.inf. Otherwise, generate the next departure time. Stopping of the distribution¶ There are many ways we would stop the simulation. (1) Simulating a single-server Queueing system from 0 to T. (2) The arrival could only happen before T. However, we will finish serving all the consumers (3) We will only allow the first n customers to enter the system ... In [2]: ## Arrival: poisson process with λ=100 for this hour. ## Service: Customers will be served in a counter based on first-come first serve with the service time following exponential distribution with λ=110 ## Simulation stops when t>T
## Return the average waiting time for customers who finished receiving the service

def single_server():

#initialize clock
t=0

#initialize the states
n_c=0

#generate the pending event times
#arrival rate
lmbda_a=100
t_arrival=t-1/lmbda_a*np.log(np.random.rand())
t_depart=np.inf
#service rate
lmbda_s=110
N_arrivals=0

#other initialization

##Collecting system output (history)

##array to store Arrival times
arrival_ts=np.array([])
##array to store Departure times
depart_ts=np.array([])

T=1

while (t<=T): #while ( condition to continue the simulation): #conditional statement that identifies arrival event happens: if t_arrival==min(t_arrival,t_depart): #update clock t=t_arrival #store arrival time (history) arrival_ts=np.append(arrival_ts,t) #update system states n_c=n_c+1 N_arrivals=N_arrivals+1 #update pending event times t_arrival=t-1/lmbda_a*np.log(np.random.rand()) if n_c==1: t_depart=t-1/lmbda_s*np.log(np.random.rand()) #conditional statement that identifies departure event happens: else: #update clock t=t_depart #store departure time (history) depart_ts=np.append(depart_ts,t) #update system states n_c=n_c-1 #update pending event times if n_c==0: t_depart=np.inf else: t_depart=t-1/lmbda_s*np.log(np.random.rand()) arrival_ts_cleaned=arrival_ts[arrival_ts<=1] depart_ts_cleaned=depart_ts[depart_ts<=1] avg_time=np.mean(depart_ts_cleaned-arrival_ts_cleaned[0:len(depart_ts_cleaned)]) ##when simulation stops, we compute the average waiting time return avg_time,N_arrivals single_server() Out[2]: (0.0322620710061288, 96) In [0]: In [0]: