Download - Discrete Event Simulation Modeling
Simulation and Modeling I Discrete Simulation 1
Overall Plan of Simulation and Modeling I
Chapters
Introduction to Simulation Discrete Simulation Analytical Modeling Modeling Paradigms Input Modeling Random Number Generation Output Analysis Continuous and Hybrid Simulation Simulation Software
Simulation and Modeling I Discrete Simulation 2
Discrete Simulation
Goals understand event list processing as the core of discrete simulation use of high-level simulation environments simulation at the programming language level
Contents Organization of Discrete-Event Simulation (DES) Simulation of a Single-Server Queue Hand Simulation of the Queue Simulation with AnyLogic Simulation with AutoMod Simulation with OPNET Simulation with OMNeT++ Simulation with Syntony Simulation in Java List Processing in Simulation
Simulation and Modeling I Discrete Simulation 3
Organization of Discrete-Event Simulation
Recall: in discrete-event simulation, events occur instantaneously at separate points in time and change the state variables
Two mechanisms to advance the simulated time: next-event time advance
– the simulation clock is always advanced to the time of the next event, the state variables are updated and future event times are determined, until termination – is used by all major simulation software and by most people coding their model in a general-purpose language – will be used throughout the lectures
fixed-increment time advance – the simulation clock is always advanced in increments of ∆t time units – useful for systems in which the time advances in fixed increments (e.g., economic systems with annual change, slotted communication systems)
Simulation and Modeling I Discrete Simulation 4
Organization of Discrete-Event Simulation (cont.)
Recall: the single-server queue (SSQ)
although this model seems simple compared with those usually of real interest, how it is simulated is actually quite representative
we will use it for illustrations throughout this and later lectures
arrival of customers
customers depart from system
waiting room (queue) service unit
Simulation and Modeling I Discrete Simulation 5
Organization of Discrete-Event Simulation (cont.)
Events and times in the single-server queue ti = time of arrival of the i'th customer (i=1,2,3,…) Ai = ti – ti-1 = inter-arrival time between the (i-1)st and the i'th arrivals of customers (t0 = 0) Si = time the server spends serving the i'th customer (without the customer´s waiting time) Wi = waiting time of the i'th customer Di = Wi + Si = time in system of the i'th customer (also often called total delay or response time of the system) ci = ti + Wi + Si = ti + Di = time i'th customer completes service and departs ei = time of occurrence of i'th event of any type
Simulation and Modeling I Discrete Simulation 6
Organization of Discrete-Event Simulation (cont.)
State in the single-server queue given by the number of customers in waiting room and service unit, i.e., number of customers in system at time t: N(t) in contrast, we denote by Q(t) the number of customers in the waiting room (queue length)
Simulation and Modeling I Discrete Simulation 7
Organization of Discrete-Event Simulation (cont.)
Next-event time advance the simulation clock jumps from ei to ei+1
e0 e1 e2 e3 e4 e5
0 t1 t2 c1 t3 c2
A1 A2 A3
S1 S2
Time
Simulation and Modeling I Discrete Simulation 8
Organization of Discrete-Event Simulation (cont.)
Fixed-increment time advance the simulation clock jumps ∆t time units after clock update, a check is made whether any events have occurred during the previous interval if one or more have occurred, they are considered to occur at the end of the interval and the state is updated accordingly if two or more events have occurred in the same interval, rules are required to decide in which order they are processed this complicates matters, therefore the approach is not common
e2 e3 e4 0 e1 ∆t Time 4∆t 2∆t 3∆t
Simulation and Modeling I Discrete Simulation 9
Organization of Discrete-Event Simulation (cont.)
Typical components in DES (with next-event time approach): system state: collection of state variables necessary to describe the system at a particular time simulation clock: variable giving the current value of simulated time event list: list containing the next time when each type of event will occur statistical counters: variables used for storing statistical information about system performance initialization routine: subprogram to initialize the simulation model at time 0 timing routine: subprogram that determines the next event from the event list and then advances the simulation clock to the time when that event is to occur
Simulation and Modeling I Discrete Simulation 10
Organization of Discrete-Event Simulation (cont.)
event routine: subprogram that updates the system state when a particular type of event occurs (there is one event routine for each event type) library routines: set of subprograms used to generate random observations from probability distributions that were determined as part of the simulation model (and other statistical features) report generator: subprogram that computes estimates (from the statistical counters) of the desired measures of performance and produces a report when the simulation ends main program: subprogram that invokes the timing routine to determine the next event and then transfers control to the corresponding event routine to update the system state appropriately. The main program may also check for termination and invoke the report generator when the simulation is over
Simulation and Modeling I Discrete Simulation 11
Organization of Discrete-Event Simulation (cont.)
Flow of control 0. Invoke the initialization routine 1. Invoke the timing routine 2. Invoke event routine i
Repeatedly
1. Set simulation clock = 0 2. Initialize system state and statistical counters 3. Initialize event list
1. Determine the next event type, say, i 2. Advance the simulation clock
1. Compute estimates of interest 2. Write report
Is simulation
over?
Stop
Initialization routine Main program Timing routine
1. Update system state 2. Update statistical counters 3. Generate future events and add to event list, eventually remove others from the event list
Event routine i
Library routines
Report generator Yes
Generate random variates
1
i
No
2
0
Start
Simulation and Modeling I Discrete Simulation 12
Discrete Simulation
Contents Organization of Discrete-Event Simulation (DES) Simulation of a Single-Server Queue Hand Simulation of the Queue Simulation with AnyLogic Simulation with AutoMod Simulation with OPNET Simulation with OMNeT++ Simulation with Syntony Simulation in Java List Processing in Simulation
Simulation and Modeling I Discrete Simulation 13
Simulation of a Single-Server Queue
Consider the single-server queue with
the random inter-arrival times A1, A2, ... the random service times S1, S2, ... upon completing service of a customer, the server chooses a customer from the queue (if any) in a FIFO manner at time 0 no customers are present and the server is idle termination of simulation when n'th customer enters service
Simulation and Modeling I Discrete Simulation 14
Simulation of a Single-Server Queue (cont.)
Define 3 measures of system performance for SSQ d(n) = expected mean time in system (delay) of the first n customers q(n) = expected mean queue length until n customers have been served u(n) = expected utilization until n customers have been served = expected average proportion of time that the server is busy until n customers have been served
Random characteristics of the measures on a given run of the simulation (or on a day of the real system) the observed means depend on the inter-arrival and service times on another run of the simulation (or on a different day of the real system), there would be probably different inter-arrival and service times, leading to different means thus, the means are random variables themselves we want to estimate the expected values of these random variables
Simulation and Modeling I Discrete Simulation 15
Simulation of a Single-Server Queue (cont.)
Estimation of the system time / delay / response time: from a single run of the simulation resulting in customer delays D1, D2, D3, ..., an obvious estimator of d(n) is
which is just the arithmetic mean of the n Di’s that were observed in the simulation as a common convention, ^ (circumflex) above a symbol denotes an estimator since the average (arithmetic mean) is taken over a finite number of values (a subcase of a countable set), this measure is an example of a discrete-time statistics the expected mean waiting time is similarly estimated from W1, W2,…
( ) ∑=
=n
1iiD
n1nd
Simulation and Modeling I Discrete Simulation 16
Simulation of a Single-Server Queue (cont.)
Estimation of the queue length: average is now taken over continuous time, rather than over customers thus we get a continuous-time statistics let Q(t) be the queue length at time t (customers in the waiting room), T(n) the termination time of the simulation (service completion of n'th customer), and t = 0 be the starting point, then formally
for computing it easily, let Ti be the total time during the simulation that the queue is of length i, then
the area can be accumulated as rectangles as the simulation progresses over time
( ) ( ) ( )( )
∫=nT
0dttQ
nT1nq
( ) ( )∑∞
==
0iiiT
nT1nq
Simulation and Modeling I Discrete Simulation 17
Simulation of a Single-Server Queue (cont.)
Q(t) for a possible realization with n = 5: Q(t)
3
1
2
1 9 8 7 6 5 4 3 2 0 t
e1=0.4
e5=3.1 e4=2.4
e3=2.1
e2=1.6
e9=4.9
e8=4.0
e6=3.3
e7=3.8
e13=8.6=T (5)
e12=7.2
e11=5.8
e10=5.6 Arrivals
Departures
Simulation and Modeling I Discrete Simulation 18
Simulation of a Single-Server Queue (cont.)
arrivals occur at times 0.4, 1.6, 2.1, 3.8, 4.0, 5.6, 5.8, 7.2 service completions occur at times 2.4, 3.1, 3.3, 4.9, 8.6 termination at T(6) = 8.6 T0 = (1.6-0.0) + (4.0-3.1) + (5.6-4.9) = 3.2 T1 = (2.1-1.6) + (3.1-2.4) + (4.9-4.0) + (5.8-5.6) = 2.3 T2 = (2.4-2.1) + (7.2-5.8) = 1.7 T3 = (8.6-7.2) = 1.4 Ti = 0 for i > 3
test:
numerator:
our estimate of the average number in queue gets:
)5(T6.84.17.13.22.3T3
0ii ==+++=∑
=
9.94.137.123.212.30iT3
0ii =×+×+×+×=∑
=
( ) 15.16.8/9.95q ==
Simulation and Modeling I Discrete Simulation 19
Simulation of a Single-Server Queue (cont.)
Estimation of the utilization define a “busy function” (an indicator function)
then the estimate can be expressed as a continuous-time statistics, as the proportion of time B(t) is equal to 1:
=t time at idle is server0
t time atbusy is server1)t(B
( ) ( )( )
∫=nT
0dt)t(B
nT1nu
Simulation and Modeling I Discrete Simulation 20
Simulation of a Single-Server Queue (cont.)
In the example realization we get
and ( ) 90.0
6.87.7
6.8)8.36.8()4.03.3(5u ==
−+−=
e1=0.4
e5=3.1 e4=2.4
e3=2.1
e2=1.6
e9=4.9
e8=4.0
e6=3.3 e7=3.8
e13=8.6=T (5)
e12=7.2 e11=5.8
e10=5.6 Arrivals
Departures
B(t)
1
0 1 3 5 4 2 1 9 8 7 6 t
Simulation and Modeling I Discrete Simulation 21
Discrete Simulation
Contents Organization of Discrete-Event Simulation (DES) Simulation of a Single-Server Queue Hand Simulation of the Queue Simulation with AnyLogic Simulation with AutoMod Simulation with OPNET Simulation with OMNeT++ Simulation with Syntony Simulation in Java List Processing in Simulation
Simulation and Modeling I Discrete Simulation 22
Hand Simulation of the Queue
Hand simulation we step through the first steps of a simulation to illustrate the changes and data structures involved in carrying out a DES the assumed inter-arrival times are: A1 = 0.4, A2 = 1.2, A3 = 0.5, A4 = 1.7, A5 = 0.2, A6 = 1.6, A7 = 0.2, A8 = 1.4, A9 = 1.9, ... the assumed service times are: S1 = 2.0, S2 = 0.7, S3 = 0.2, S4 = 1.1, S5 = 3.7, S6 = 0.6, ... inter-arrival times and service time are consistent with running example all time quantities are expressed in the same units, whatever one unit will be (minutes, hours, ...) each figure shows the situation after the occurrence of an event, the system and the computer representation are shown in the situation after all changes have been made the status of the system is on the left side the variables of the computer representation are on the right side
Simulation and Modeling I Discrete Simulation 23
Hand Simulation of the Queue (cont.)
System Computer representation
Initialization time = 0
idle
System state
Statistical counters Number in
queue
Server status
Time of last event
Event list Clock
Area under B(t)
Area under Q(t)
Total waiting time
Number served
0 0 0 0
0 0 0
0 0.4 ∞
Arrival Departure
poi-soning
server
Times of
arrival of
queued customers
Simulation and Modeling I Discrete Simulation 24
Hand Simulation of the Queue (cont.)
System Computer representation
Arrival time = 0.4
0.4
System state
Statistical counters Number in
queue
Server status
Times of
arrival of
queued customers
Time of last event
Event list Clock
Area under B(t)
Area under Q(t)
Total waiting time
Number served
0 0 0 0
1 0 0.4
0.4 1.6 2.4
Arrival Departure
arrival time
Simulation and Modeling I Discrete Simulation 25
Hand Simulation of the Queue (cont.)
System Computer representation
Arrival time = 1.6
0.4
System state
Statistical counters Number in
queue
Server status
Times of
arrival of
queued customers
Time of last event
Event list Clock
Area under B(t)
Area under Q(t)
Total waiting time
Number served
0 0 0 1.2
1 1 1.6
1.6 2.1 2.4
1.6
Arrival Departure
1.6
queued customer and
his arrival time
Simulation and Modeling I Discrete Simulation 26
Hand Simulation of the Queue (cont.)
System Computer representation
Arrival time = 2.1
0.4
System state
Statistical counters Number in
queue
Server status
Times of
arrival of
queued customers
Time of last event
Event list Clock
Area under B(t)
Area under Q(t)
Total waiting time
Number served
0 0 0.5 1.7
1 2 2.1
2.1 3.8 2.4
1.6 2.1
Arrival Departure
1.6
2.1
Simulation and Modeling I Discrete Simulation 27
Hand Simulation of the Queue (cont.)
System Computer representation
Departure time = 2.4
1.6
System state
Statistical counters Number in
queue
Server status
Times of
arrival of
queued customers
Time of last event
Event list Clock
Area under B(t)
Area under Q(t)
Total waiting time
Number served
1 0.8 1.1 2.0
1 1 2.4
2.4 3.8 3.1
2.1
Arrival Departure
2.1
Simulation and Modeling I Discrete Simulation 28
Hand Simulation of the Queue (cont.)
System Computer representation
Departure time = 3.1
2.1
System state
Statistical counters Number in
queue
Server status
Times of
arrival of
queued customers
Time of last event
Event list Clock
Area under B(t)
Area under Q(t)
Total waiting time
Number served
2 1.8 1.8 2.7
1 0 3.1
3.1 3.8 3.3
Arrival Departure
Simulation and Modeling I Discrete Simulation 29
Hand Simulation of the Queue (cont.)
System Computer representation
Departure time = 3.3
idle
System state
Statistical counters Number in
queue
Server status
Times of
arrival of
queued customers
Time of last event
Event list Clock
Area under B(t)
Area under Q(t)
Total waiting time
Number served
3 1.8 1.8 2.9
0 0 3.3
3.3 3.8 ∞
Arrival Departure
Simulation and Modeling I Discrete Simulation 30
Hand Simulation of the Queue (cont.)
System Computer representation
Arrival time = 3.8
3.8
System state
Statistical counters Number in
queue
Server status
Times of
arrival of
queued customers
Time of last event
Event list Clock
Area under B(t)
Area under Q(t)
Total waiting time
Number served
3 1.8 1.8 2.9
1 0 3.8
3.8 4.0 4.9
Arrival Departure
Simulation and Modeling I Discrete Simulation 31
Hand Simulation of the Queue (cont.)
System Computer representation
Arrival time = 4.0
3.8
System state
Statistical counters Number in
queue
Server status
Times of
arrival of
queued customers
Time of last event
Event list Clock
Area under B(t)
Area under Q(t)
Total waiting time
Number served
3 1.8 1.8 3.1
1 1 4.0
4.0 5.6 4.9
4.0
Arrival Departure
4.0
Simulation and Modeling I Discrete Simulation 32
Hand Simulation of the Queue (cont.)
System Computer representation
Departure time = 4.9
4.0
System state
Statistical counters Number in
queue
Server status
Times of
arrival of
queued customers
Time of last event
Event list Clock
Area under B(t)
Area under Q(t)
Total waiting time
Number served
4 2.7 2.7 4.0
1 0 4.9
4.9 5.6 8.6
Arrival Departure
Simulation and Modeling I Discrete Simulation 33
Hand Simulation of the Queue (cont.)
System Computer representation
Arrival time = 5.6
4.0
System state
Statistical counters Number in
queue
Server status
Times of
arrival of
queued customers
Time of last event
Event list Clock
Area under B(t)
Area under Q(t)
Total waiting time
Number served
4 2.7 2.7 4.7
1 1 5.6
5.6 5.8 8.6
5.6
Arrival Departure
5.6
Simulation and Modeling I Discrete Simulation 34
Hand Simulation of the Queue (cont.)
System Computer representation
Arrival time = 5.8
4.0
System state
Statistical counters Number in
queue
Server status
Times of
arrival of
queued customers
Time of last event
Event list Clock
Area under B(t)
Area under Q(t)
Total waiting time
Number served
4 2.7 2.9 4.9
1 2 5.8
5.8 7.2 8.6
5.6 5.8
Arrival Departure
5.6
5.8
Simulation and Modeling I Discrete Simulation 35
Hand Simulation of the Queue (cont.)
System Computer representation
Arrival time = 7.2
4.0
System state
Statistical counters Number in
queue
Server status
Times of
arrival of
queued customers
Time of last event
Event list Clock
Area under B(t)
Area under Q(t)
Total waiting time
Number served
4 2.7 5.7 6.3
1 3 7.2
7.2 9.1 8.6
7.2
5.6 5.8
Arrival Departure
5.6
5.8
7.2
Simulation and Modeling I Discrete Simulation 36
Hand Simulation of the Queue (cont.)
System Computer representation
Departure time = 8.6
5.6
System state
Statistical counters Number in
queue
Server status
Times of
arrival of
queued customers
Time of last event
Event list Clock
Area under B(t)
Area under Q(t)
Total waiting time
Number served
5 5.7 9.9 7.7
1 2 8.6
8.6 9.1 9.2
5.8 7.2
Arrival Departure
5.8
7.2
Simulation and Modeling I Discrete Simulation 37
Hand Simulation of the Queue (cont.)
Compute estimates of interest from statistical counters in report routine (simulation) clock = 8.6 (simulated time at service completion of 5th customer) estimate of utilization:
– û(5) = [ Area under B(t) ] / clock = 7.7/8.6 = 0.9 – compare with slide 20
estimate of average number in queue: – = [ Area under Q(t) ] / clock = 9.9/8.6 = 1.15 – compare with slide 18
estimate of expected mean waiting time: – ŵ(5) = [ Total waiting time ] / [ Number served ] = 5.7/5 = 1.14 – not computed before – estimate for expected mean system time is obtained similarly in hand simulation
)5(d
( )5q
Simulation and Modeling I Discrete Simulation 38
Hand Simulation of the Queue (cont.)
Remarks key element in the dynamics of a simulation is the interaction between the simulation clock and the event list while processing an event, no simulated time passes. Care must be taken to process updates of the state variables in the right order (e.g., first update the area calculators and only then the time of last event) it is sometimes easy to overlook certain updates of the state variables and counters (e.g., after a departure leaving the system empty, the server must be idled) it can happen that two (or more) entries in the event list are tied for smallest and a decision rule must be incorporated to break such time ties. The tie-breaking rule can sometimes significantly affect the result of the simulation next we turn to the use of computer software
Simulation and Modeling I Discrete Simulation 39
Discrete Simulation
Contents Organization of Discrete-Event Simulation (DES) Simulation of a Single-Server Queue Hand Simulation of the Queue Simulation with AnyLogic Simulation with AutoMod Simulation with OPNET Simulation with OMNeT++ Simulation with Syntony Simulation in Java List Processing in Simulation
Simulation and Modeling I Discrete Simulation 40
Simulation with AnyLogic
High-level simulation environment general purpose discrete, continuous, and hybrid modeling paradigm is a variant of the Unified Modeling Language for real-time systems (RT-UML)
Basic model elements active objects: model real-world objects statecharts: internal behavior of active objects ports: asynchronous message passing variables: shared variables all can be visualized
Java-based textual model parts in Java executable model + simulation engine mapped on Java other Java code can be linked
Simulation and Modeling I Discrete Simulation 41
Simulation with AnyLogic (cont.)
Visualization of active objects:
collection variable (may be used to implement port queue)
output variable (deprecated)
variable reference (deprecated)
input variable (deprecated)
dynamic event statechart
port reference
port (without queue)
port (with queue; deprecated)
encapsulated object
this object connector
text box
variable
parameter
event
Simulation and Modeling I Discrete Simulation 42
Simulation with AnyLogic (cont.)
A statechart:
Dispatching
do/initiate delivery
Checking
do/check item
Delivered Waiting
start /get first item
[All items checked && all items available]
transition
State
aktivity
self-transition
Item Received [some items not in stock]
[Not all items checked] /get next item
[All items checked && some items not in stock] Delivered
Simulation and Modeling I Discrete Simulation 43
Simulation with AnyLogic (cont.)
Interpretation of statecharts
states represent system states (states of object) transitions represent state changes transition label: event [guard] / action a transition may be taken
– if the trigger event occurs – and the specified guard condition is true
an action can be performed when the transition is taken all statecharts execute concurrently
more details later
Simulation and Modeling I Discrete Simulation 44
Simulation with AnyLogic (cont.)
Possible events timeout: after a fixed time has elapsed (timeout may be computed according to some distribution) rate: after an exponentially distributed time has elapsed (special case) condition: when an expression becomes true message: after arrival of a message at the statechart queue; occurs,
– when Java object arrives at port, which is connected to statechart (via port.map(statechart)) – when explicitly called from Java code (via statechart.fireEvent()or statechart.receiveMessage())
Underlying is discrete-event simulation simulation clock is always advanced to the time of the next event (anywhere in the model) and the event is then executed (transition is taken and the actions are performed) time ties are broken randomly
Simulation and Modeling I Discrete Simulation 45
Simulation with AnyLogic (cont.)
Single-server queue with RT-UML, conceptual:
2 active objects: client and server connected via ports, with a queue at server port 2 timed events: exponentially distributed inter-arrival and service times with means equal to 10 and 9
client server
generate
exp(1/10)/port.send()
port port
exp(1/9) (q.size()>0)/ q.removeFirst()
idle
busy
q
Simulation and Modeling I Discrete Simulation 46
Simulation with AnyLogic (cont.)
Implementation in AnyLogic:
Condition Trigger
CustomerQueue.size()>0
Action
CustomerQueue.removeFirst();
Rate Trigger
0.11111
Rate Trigger
0.1
Action
port.send(new Object()); On Receive Action
CustomerQueue.addLast(msg);
Simulation and Modeling I Discrete Simulation 47
Simulation with AnyLogic (cont.)
Runtime view windows to inspect model execution
Simulation and Modeling I Discrete Simulation 48
Simulation with AnyLogic (cont.)
Measures for the single-server queue mean waiting time = average time messages spend in port at server
mean queue length = average number of messages in port at server
utilization = average portion of time the server is in state busy
throughput = average number of transitions from busy to idle per time unit
Implementation in AnyLogic:
Simulation and Modeling I Discrete Simulation 49
Simulation with AnyLogic (cont.)
Messages get time stamps in Client : send a Double object with current time instead of object of class Object method time() delivers value of simulation clock
Rate Trigger
0.1
Action
port.send(new Double(time()));
Simulation and Modeling I Discrete Simulation 50
Simulation with AnyLogic (cont.) Statistical counters and state variables
plain variables are defined as orange circles with a `V` (e.g., Throughput in class Server)
collection variables (arrays or lists, three orange dots) may serve to implement queues for ports (see CustomerQueue)
variables may also be defined in Additional Class Code for each class
Simulation and Modeling I Discrete Simulation 51
Simulation with AnyLogic (cont.) Statistics
generated data for measure computation can be collected in Statistics obj. here: QueueLength.update() adds current number of customers in collection variable CustomerQueue (see Value field of QueueLength) functions for basic statistics (mean, variance, etc.) are built-in (see page 47)
select buttons for continuous-time statistics, like queue length or discrete-time statistics, like waiting time
Simulation and Modeling I Discrete Simulation 52
Simulation with AnyLogic (cont.) On Receive action
CustomerQueue.addLast(msg)
QueueLength.update(); Actions executed in Server: when message is received at port when service is started when service ends
Condition Trigger
CustomerQueue.size() > 0
Action
WaitingTime.add( time() –
CustomerQueue.removeFirst().doubleValue());
QueueLength.update();
BusyFunction.add(1, time());;
Rate Trigger
0.11111
Action
Throughput = (++CustomersServed)/time();
BusyFunction.add(0, time());
Simulation and Modeling I Discrete Simulation 53
Simulation with AnyLogic (cont.)
Graphical output (initial plots) and statistics (at t=10.000 sec) flexible visualization of data via various types of charts (time plots, histograms, bar, stack and pie charts, etc.) usual procedure:
– collect data in Data Set objects – connect data set to desired visualization object – data is displayed during simulation run
drawback: – each graphic below requires two data sets (actual value and mean) in addition to Statistics objects (WaitingTime, QueueLength, BusyFunction)
Simulation and Modeling I Discrete Simulation 54
Simulation with AnyLogic (cont.)
Summary: simulation with AnyLogic modeling
– identify main system objects and map them on active objects – identify states and actions of objects and map them on statechart states and transitions – model interaction between objects by message passing (and shared variables)
no care required for dealing with event handling identify measures of interest and collect relevant statistics in Statistics objects of appropriate type (basic properties like mean and variance are then available); sophisticated measures require additional Java coding graphical output and built-in statistical evaluation reveals system behavior
More information exercise class practice in various assignments detailed description in AnyLogic User‘s Manual distributed with the tool
Simulation and Modeling I Discrete Simulation 55
Discrete Simulation
Contents Organization of Discrete-Event Simulation (DES) Simulation of a Single-Server Queue Hand Simulation of the Queue Simulation with AnyLogic Simulation with AutoMod Simulation with OPNET Simulation with OMNeT++ Simulation with Syntony Simulation in Java List Processing in Simulation
Simulation and Modeling I Discrete Simulation 56
Simulation with AutoMod
High-level simulation environment a major commercial tool for manufacturing system simulation specialized model elements underlying discrete-event simulation
Basic model elements loads: dynamic entities which are moved around queues: buffers where loads can reside (i.e., wait or are being processed) resources: needed for processing loads processes: logical control of movement and resource usage of loads, executed by loads, causes events to happen
many more model elements for manufacturing systems conveyors, AGVs automated storage/retrieval systems (AS/RS) bridge cranes tanks & pipes (continuous) ...
Simulation and Modeling I Discrete Simulation 57
Simulation with AutoMod (cont.)
Single-Server queue with AutoMod queue Q_waitingroom models the waiting room, infinite capacity queue Q_service models the server room, capacity = 1 resource R_server models the server load L_customer models customers
– exponentially distributed inter-arrival times with mean 10 min. – first process: P_control
process P_control models the flow of customers:
begin P_control arriving
move into Q_waitingroom
move into Q_service
use R_server for exponential 9 min
send to die
end
Simulation and Modeling I Discrete Simulation 58
Simulation with AutoMod (cont.)
Graphical representation
Result display standard statistics about all modeling entities (queues, resources, ...): current value, mean, min, max, ... customized graphs (e.g., business graphics)
Simulation and Modeling I Discrete Simulation 59
Simulation with AutoMod (cont.)
Queue length of Q_waitingroom (1 day)
current value
averaged value
Simulation and Modeling I Discrete Simulation 60
Simulation with AutoMod (cont.)
Queue length of Q_waitingroom (1 month)
current value
averaged value, ≈ 5
Simulation and Modeling I Discrete Simulation 61
Simulation with AutoMod (cont.)
Queue length of Q_waitingroom (1 year)
current value
averaged value, ≈ 8.1
Simulation and Modeling I Discrete Simulation 62
Simulation with AutoMod (cont.)
Mean waiting time in Q_waitingroom (in sec)
≈ 79 min
Simulation and Modeling I Discrete Simulation 63
Simulation with AutoMod (cont.)
Utilization of Q_service:
≈ 0.9
Simulation and Modeling I Discrete Simulation 64
Simulation with AutoMod (cont.)
More information
exercise class practice in assignment tutorial: Getting Started with AutoMod with AutoMod installation, lecture Web page
Simulation and Modeling I Discrete Simulation 65
Discrete Simulation
Contents Organization of Discrete-Event Simulation (DES) Simulation of a Single-Server Queue Hand Simulation of the Queue Simulation with AnyLogic Simulation with AutoMod Simulation with OPNET Simulation with OMNeT++ Simulation with Syntony Simulation in Java List Processing in Simulation
Simulation and Modeling I Discrete Simulation 66
Simulation with OPNET High-level simulation environment
a major commercial simulation tool for communication systems underlying discrete-event simulation libraries for all relevant networking protocols and systems
Modeling similar to AnyLogic state-transition diagrams at the core, here called Finite-State Machines (FSMs) communication via packets (≈ messages) hierarchical structure (network, module and process level with dedicated editors: Project, Node and Process editor)
Main differences for modeling the M/M/1 queue packet streams are similar to connected ports, but the FSM itself must react to a reception (in AnyLogic port actions after a reception) initial states are needed in each FSM events have to be scheduled explicitly coding in C/C++ and Proto-C (OPNET-specific kernel procedures) even simple models require various kernel procedures
Simulation and Modeling I Discrete Simulation 67
Simulation with OPNET (cont.)
Single-server queue with OPNET, conceptual: client server
generate
op_dist_exponential(10)/ op_intrpt_schedule_self()
Packet
Stream
2 modules (here processes) with an extended FSM each connected via a packet stream with a queue the guards ARRIVAL and SCV_COMPLETION control which transition may be taken causing specific actions (C code) ARRIVAL: get packet from stream; start service if end of idle period SVC_COMPLETION: start new service unless end of busy period
init idle_and_busy init
ARRIVAL
SVC_COMPLETION
Simulation and Modeling I Discrete Simulation 68
Simulation with OPNET (cont.)
Implementation in OPNET:
Project/Network level Node/Module level
Attribute
(instance of SSQ_node)
Client.InterarrivalMean : 10
Server.ServiceMean : 9
Attribute
process model : SSQ_generator
InterarrivalMean : promoted
Attribute
process model : SSQ_service_unit
ServiceMean : promoted
Simulation and Modeling I Discrete Simulation 69
Simulation with OPNET (cont.)
Process level: Finite State Machines and their initialization procedures (for state init)
Simulation and Modeling I Discrete Simulation 70
Simulation with OPNET (cont.)
Packet Generation:
kernel procedures to generate
packet (op_pk_create()) and
send it to a packet stream
(op_pk_send())
generated packets implicitly get a time stamp!
kernel procedures to generate exponentially distributed interrarrival times (op_dist_exponential()) and to schedule an interrupt for the next arrival time (op_intrpt_schedule_self())
Simulation and Modeling I Discrete Simulation 71
Simulation with OPNET (cont.)
Packet Arrival and Service:
interrupt from
packet stream interrupt from
ended service
transition executives
Simulation and Modeling I Discrete Simulation 72
Discrete Simulation
Contents Organization of Discrete-Event Simulation (DES) Simulation of a Single-Server Queue Hand Simulation of the Queue Simulation with AnyLogic Simulation with AutoMod Simulation with OPNET Simulation with OMNeT++ Simulation with Syntony Simulation in Java List Processing in Simulation
QoS in Communication Systems Discrete Simulation 73
Simulation with OMNeT++ Simulation tool for communication systems
public-source, modular components embeddable discrete-event simulation kernel modeling frameworks for networking protocols mainly for scientific research
Modeling similar to OPNET Finite-State Machines (FSMs) as C++ code, no graphical representation communication via messages hierarchical structure (network, complex and simple modules) textual representation of model structure: ned files graphical model structure editor: GNED initial states are needed in each FSM events have to be scheduled explicitly coding in C/C++ using the OMNeT++ API INET framework for internet protocols, mobility, wireless channels (loss, fading) used in a joint simulation study of our group and Siemens Industry for sensor
networks in automation
QoS in Communication Systems Discrete Simulation 74
Simulation with OMNeT++
Single-server queue with OMNeT++, conceptual: client server
generate
exponential(10)/ scheduleAt()
connector
2 simple modules with an extended FSM for the server, no FSM for the client
connected gates queue explicitly represented in
model
busy
init
idle
handleMessage() timer
handleMessage()
queue
gate gate
QoS in Communication Systems Discrete Simulation 75
Simulation with OMNeT++ model structure in ned file:
simple modules simple client
parameters:
interarrivalMean: numeric;
gates:
out: out;
endsimple
simple server
parameters:
serviceMean: numeric;
gates:
in: in;
endsimple
simple module client : one parameter gate and direction
simple module server : one parameter gate and direction
QoS in Communication Systems Discrete Simulation 76
Simulation with OMNeT++ model structure in ned file:
module MM1
submodules:
Client: client;
Server: server;
connections:
Client.out --> Server.in;
endmodule
network mm1 : MM1
endnetwork
graphical representation of model structure:
complex module MM1 : submodules connection of gates
network structure mm1 : one complex module MM1
QoS in Communication Systems Discrete Simulation 77
Simulation with OMNeT++ C++ implementation
for the client:
kernel procedures to generate
packet (new cMessage()) and
send it to a port (send()) generated packets implicitly get a time stamp!
kernel procedures to generate exponentially distributed interrarrival times (exponential()) and to schedule an event for the next arrival time (scheduleAt())
QoS in Communication Systems Discrete Simulation 78
Simulation with OMNeT++ C++ implementation
for the server:
Declaration of the FSM, timers, queues and counters
Initial setup of server operation and statistics collection
Recording of statistics at the end of a simulation run
QoS in Communication Systems Discrete Simulation 79
Simulation with OMNeT++ C++ implementation for the server (cont.):
Entering and exiting the IDLE state
QoS in Communication Systems Discrete Simulation 80
Simulation with OMNeT++ C++ implementation for the server (cont.):
Entering and exiting the BUSY state
QoS in Communication Systems Discrete Simulation 81
Simulation with OMNeT++ graphical output during simulation:
QoS in Communication Systems Discrete Simulation 82
Simulation with OMNeT++ visualization of statistics:
Simulation and Modeling I Discrete Simulation 83
Discrete Simulation
Contents Organization of Discrete-Event Simulation (DES) Simulation of a Single-Server Queue Hand Simulation of the Queue Simulation with AnyLogic Simulation with AutoMod Simulation with OPNET Simulation with OMNeT++ Simulation with Syntony Simulation in Java List Processing in Simulation
QoS in Communication Systems Discrete Simulation 84
Simulation with Syntony
UML-based simulation environment for the network domain + general-purpose standard-compliant simulation approach allows integration into model driven design by our group in cooperation with Fraunhofer IIS
Modeling paradigm composite structure diagrams for hierarchical structuring state machine diagrams for behavioral specification detailed action specification with activity diagrams or action language Casual non-functional aspects (times, randomness,...) with MARTE profile (Modeling and
Analysis of Real-Time and Embedded systems) communication via signals sent over ports Syntony translates UML2 diagrams to C++ (for OMNeT++) or Java (for J-Sim) execution on simulation engines and backpropagation of measures tested for larger scenarios in inter-vehicle communication and for sensor
networks in logistics
QoS in Communication Systems Discrete Simulation 85
Simulation with Syntony
Single-server queue with Syntony, conceptual: Client Server
generate
exponential(10)/ create new client signal
Connector
2 composite structures with a state machine each connected via two ports and a connector the trigger client (arrival of a client signal) causes transition to busy state
where the service is performed completion of service causes transition back to idle; next client may be
served instantly
busy idle client
Completion
QoS in Communication Systems Discrete Simulation 86
Simulation with Syntony
UML model: Network
stereotype to record utilization and throughput of server cpu
stereotype to indicate which resources are present in the network
QoS in Communication Systems Discrete Simulation 87
Simulation with Syntony
UML model: Analysis
stereotype indicating which model (or part of the model) should be simulated
Definition of variables (and their values) that can be accessed in the model
QoS in Communication Systems Discrete Simulation 88
Simulation with Syntony
UML model: Client stereotype specifying the points in time when a new client is generated
Casual statement specifying the action to be taken on transition execution:
Create a new client signal (client.create()) and send it to the port called departures (sendTo(departures))
QoS in Communication Systems Discrete Simulation 89
Simulation with Syntony
UML model: Server
stereotypes to handle queueing of arriving client signals at the server
stereotype specifying service times demanded by clients on the server cpu
stereotype to record queue size statistics
stereotype to record waiting times for clients at the server
QoS in Communication Systems Discrete Simulation 90
Simulation with Syntony
User Interface: based on Eclipse
control translation process, simulation parameters and evaluation
analysis of model structure generated by the translator
select UML model for simulation
QoS in Communication Systems Discrete Simulation 91
Simulation with Syntony
User Interface: simulation animation (OMNeT++)
contents of the server module
event log
network view
QoS in Communication Systems Discrete Simulation 92
Simulation with Syntony
User Interface: evaluation of results
Histogram of the queue length at the server of a single-server queue
select simulation results to plot
set the options for the plot (name, labels, colors, ...)
QoS in Communication Systems Discrete Simulation 93
Simulation with Syntony
User Interface: evaluation of results
time series plot of the queue length at the server of a single-server queue
select simulation results to plot
set the options for the plot (name, labels, colors, ...)
Simulation and Modeling I Discrete Simulation 94
Discrete Simulation
Contents Organization of Discrete-Event Simulation (DES) Simulation of a Single-Server Queue Hand Simulation of the Queue Simulation with AnyLogic Simulation with AutoMod Simulation with OPNET Simulation with OMNeT++ Simulation in Java List Processing in Simulation
Simulation and Modeling I Discrete Simulation 95
Simulation in Java
Coding simulations in Java use of a general-purpose language we must pay attention to every detail and get a better understanding of how simulations operate sometimes necessary to code a simulation in a general-purpose language, since certain system aspects do not fit in the pre-programmed frameworks of simulation tools and languages still common to do the entire simulation this way
SSJ: Stochastic Simulation in Java Java library from Paul L‘Ecuyer, University of Montreal primarily for DES (event-/process-oriented), also continuous and hybrid classes for
– simulation clock and event list processing – generating random numbers – collecting statistics, writing output
Simulation and Modeling I Discrete Simulation 96
Simulation in Java (cont.)
Single-Server queue with SSJ inter-arrivals and services exponentially distributed 3 event classes
– Arrival
– Departure
– EndOfSim event list processing
– event instances are inserted into event list with time of occurrence and executed when simulation clock reaches this time – executing an event means invoking its actions method
statistical counters – Tally: discrete-time – Accumulate: continuous-time
Simulation and Modeling I Discrete Simulation 97
Simulation in Java (cont.)
Beginning of the code, basic definitions:
public class QueueEv { static final double meanArr = 10.0; static final double meanServ = 9.0; static final double timeHorizon = 1000.0; RandMrg genArr = new RandMrg (); RandMrg genServ = new RandMrg (); List waitList = new List ("Customers in queue"); List servList = new List ("Customers in service"); Tally queuingD = new Tally ("Queuing delay"); Accumulate queueL = new Accumulate ("Queue length"); class Customer { double arrivTime, servTime; } ...
2 random number streams 2 lists: queue + service unit 2 counters: mean queuing delay, mean queue length
customers with their arrival and service times
some constants
Simulation and Modeling I Discrete Simulation 98
Simulation in Java (cont.)
Construction of simulation:
... public static void main (String[] args) { new QueueEv(); } public QueueEv() { Sim.init(); new EndOfSim().schedule (timeHorizon); new Arrival().schedule (Rand1.expon (genArr, meanArr)); Sim.start(); } ...
constructor initialization schedule EndOfSim schedule Arrival start event processing
Simulation and Modeling I Discrete Simulation 99
Simulation in Java (cont.)
Flow of control when an Arrival event is executed:
Simulation and Modeling I Discrete Simulation 100
Simulation in Java (cont.)
Arrival event execution: ... class Arrival extends Event { public void actions() { new Arrival().schedule (Rand1.expon (genArr, meanArr)); Customer cust = new Customer(); cust.arrivTime = Sim.time(); cust.servTime = Rand1.expon (genServ, meanServ); if (servList.size() > 0) { waitList.insert (cust, List.LAST); queueL.update (waitList.size()); } else { servList.insert (cust, List.LAST); new Departure().schedule (cust.servTime); queuingD.update (0.0); } } } ...
schedule next arrival
create new customer, set arrival time, generate service time join queue, update counter enter service, schedule Departure, update counter
Simulation and Modeling I Discrete Simulation 101
Simulation in Java (cont.)
Flow of control when a Departure event is executed:
Simulation and Modeling I Discrete Simulation 102
Simulation in Java (cont.)
Departure event execution:
... class Departure extends Event { public void actions () { servList.remove (List.FIRST); if (waitList.size () > 0) { Customer cust = (Customer) waitList.remove (List.FIRST); servList.insert (cust, List.LAST); new Departure().schedule (cust.servTime); queuingD.update (Sim.time () - cust.arrivTime); queueL.update (waitList.size ()); } } } ...
remove customer
move customer from queue to service unit, schedule Departure, update counters
Simulation and Modeling I Discrete Simulation 103
Simulation in Java (cont.)
EndOfSim event execution:
... class EndOfSim extends Event { public void actions () { queuingD.report(); queueL.report(); Sim.stop(); } } }
write statistical report for the two counters stop event processing
Simulation and Modeling I Discrete Simulation 104
Simulation in Java (cont.)
Output of the program: Values differ from AutoMod results,
1000 minutes correspond to 16 hours only!
REPORT on Tally stat. collector ==> Queuing delay min max average standard dev nb. obs. 0 113.721 49.554 22.336 97 REPORT on Accumulate stat. collector ==> Queue length From time To time Min Max Average 0 1000 0 12 4.85
Simulation and Modeling I Discrete Simulation 105
Simulation in Java (cont.)
SSJ classes Sim: maintains simulation clock and event list EventList: event list, implemented as doubly linked list Event: abstract class for events, methods for scheduling and cancelling of events RandMrg: random number generators (uniform from 0 to 1) Rand1: random variate generators for various distributions Tally: discrete-time statistics Accumulate: continuous-time statistics List: lists of any type of object, implemented as doubly linked lists
Details in SSJ User‘s Guide
Simulation and Modeling I Discrete Simulation 106
Simulation in Java (cont.)
Main parts of the Sim class: public abstract class Sim implements Runnable { public static double currentTime = 0.0; public static EventList eventList = new DoublyLinked (); public static boolean stopped = false; public static double time () {return currentTime;} public static void init () {currentTime = 0.0; eventList.cleanup (); stopped = false;} public static void init (EventList evlist) {init(); eventList = evlist;} ...
initialization
initialization with event list
returns value of simulation clock
simulation clock event list
Simulation and Modeling I Discrete Simulation 107
Simulation in Java (cont.)
... public static void start () { if (eventList.isEmpty ()) error ("Sim.start with empty event list"); Event ev = eventList.removeFirst(); while (ev != null && !stopped) { currentTime = ev.eventTime; ev.actions(); ev = eventList.removeFirst(); } } public static void stop () {stopped = true;} }
start event list processing
set sim. clock, execute event, get next event
get first event
stop when start takes control
(dealing with processes is omitted)
Simulation and Modeling I Discrete Simulation 108
Simulation in Java (cont.)
Main parts of the Event class:
public abstract class Event implements Cloneable { protected double eventTime; public static String descriptor; public Event (double delay) { if (delay >= 0.0) { eventTime = Sim.time() + delay; Sim.eventList.insert (this); } else error ("Scheduling an event in the past."); } …
time of event occurrence
construct + schedule event
determine event time, insert in event list
event type identification
Simulation and Modeling I Discrete Simulation 109
Simulation in Java (cont.)
... public void schedule (double delay) { if (delay < 0.0) error ("Scheduling an event in the past."); eventTime = Sim.time() + delay; Sim.eventList.insert (this); } public final boolean cancel (String type) { Event ev = Sim.eventList.viewFirstOfClass (type); return ev.cancel(); } public final double time() {return eventTime;} public abstract void actions(); }
schedule event
remove event from event list
return event time
method actions, invoked when event is executed
Simulation and Modeling I Discrete Simulation 110
Simulation in Java (cont.)
Summary: programming in event-oriented style 1. identify
– system states – events – measures of interest
2. implement data structures for system states (typically lists) and statistical counters, in example:
– lists: waitList, servList – counters: here implicit as waitList.size(), servList.size()
3. write an event handling routine for each event – in SSJ a class with an actions method for each event – here: Arrival, Departure, EndOfSim
Higher level modeling paradigm and tool environment makes modeling less tedious
Simulation and Modeling I Discrete Simulation 111
Discrete Simulation
Contents Organization of Discrete-Event Simulation (DES) Simulation of a Single-Server Queue Hand Simulation of the Queue Simulation with AnyLogic Simulation with AutoMod Simulation with OPNET Simulation in Java List Processing in Simulation
Simulation and Modeling I Discrete Simulation 112
List Processing in Simulation
In the shown queue simulation event list (objects: events, variables: time of occurrence) list of customers waiting in the queue (objects: customers, variables: arrival time, service time)
General use of lists in simulations
most simulations require many lists which may contain many objects, consisting of several variables often necessary to process these lists other than FIFO lists are the dominating data structures in simulation programs usual implementation: pointers and dynamic memory allocation
Simulation and Modeling I Discrete Simulation 113
List Processing in Simulation (cont.)
Doubly linked lists can be implemented by using pointers (links) in C, Java (or arrays in languages without dynamic data structures such as FORTRAN) each object has a link to its successor and predecessor special links to the first and the last object of the list typical operations:
– insert an object in the list such that the list is sorted (increasing/decreasing) according to a certain variable – get an object at the i'th position of the list (get the first, get the last element) – remove a certain object according to its variable or position
object object object object first last
Simulation and Modeling I Discrete Simulation 114
List Processing in Simulation (cont.)
More sophisticated data structures for complex simulations involving a large number of events, much of the computer time required to perform the simulation can be expended on event-list processing implementing the event list as described leads to a linear search for events in it one way to improve the efficiency is to use other search techniques and the appropriate data structures for them
– binary search + search tree – a pointer to the middle of the lists – and other variants
Simulation and Modeling I Discrete Simulation 115
List Processing in Simulation (cont.)
Event list in SSJ
public interface EventList { public boolean isEmpty (); public void cleanup (); public void print (); public void insert (Event ev); public void insertFirst (Event ev); public void insertBefore (Event ev, Event other); public void insertAfter (Event ev, Event other); public Event viewFirst (); public Event viewFirstOfClass (String cl); public boolean remove (Event ev); public Event removeFirst (); }
Simulation and Modeling I Discrete Simulation 116
List Processing in Simulation (cont.) A part of its implementation:
public class DoublyLinked implements EventList {
private class Node {Event ev; Node prec, succ;} private Node first, last; private Node free = null;
public DoublyLinked () {first = last = null;}
public void insertFirst (Event ev) { Node newNode; if (free == null) newNode = new Node(); else {newNode = free; free = free.succ;} newNode.ev = ev; newNode.prec = null; if (first==null) {first=last=newNode; first.succ=null;} else {newNode.succ = first; first.prec = newNode; first = newNode;} } ...
insertion at first position
node with predecessor, successor
first and last pointers
pointer to stack of free nodes